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Wy lee 1G) matiel, ite Wethe le toAdl det ie ete H 4 ead eh ae led Soy Aloha if i A Geiysel chang seta Ae SS STE at: ae Ww Lia Mi sett \ a aun a\'5 ite ee = i i y rr ate po iM ip i a i ret + iw act ak 1 ne hy neg 'y) ener te Gal het “i aaa lett \ PE DEER taal oi Neti (a 14 tel att at AG A utet eka eas tet CGE NWT BSN cot ah ety eh ‘ te } a { ‘ Puree een teed rik ed TA aa Vii nails PY ep) td) (ometede nites ome “I He dhe - . ’ 1 " i (ays | SS apete tele aly al tole itd a i abs 8 hg as ‘ . iat aan! eon tab eget a : oa Tie a ‘ re puri oe alg? bal Me b Sigil Mates \tuie 118 aut imaay onnte ante ites Aas tte tet TE wey Ny ae nee i COs Lab ele Mbaulse aia Haan Caaeiean te vat wii ie ety Ma Atal y k it i} GUN UU Nawal ay erent pa fo rena anu Li : sana ratte as ieeeuyae Lah Radner itt ate LEI Jette dard He * ay fh EDN ih hat i ae 4 ) tj pet dy’ iqanie if Lah pha leet F Hy 3 J we ee ae 1347 ai 48 Vives Ht i We | Reap ats myey eld a) aN i ‘i au 44 vant hy ai ‘a eae ata , ah : i My i lis ae 7 inet iia ta} AMaha ; MF 1 ' PP 4 # y patie Vie fairies Per ag hae Me mit Misi nh te | yuh ruleta te ae sen Care ha ih ia ashi nites ’ dy ances tt ‘ +h Liga yreaeee Nehetee: Neu yy tha a EY nee i tig ah eagles UB bk ios teat) q Laat is aa hl ELEM hee 1 aid nh angen! a mints Lah inate ‘odes fe ike a oe Hi al my ’ Ea pated vias vitae ae ee te iheredy int aa Hy EP , eens Wyllie ai have , Yalan at aad Atel) Pie uy ’ (he ait Saat tse Ry earn Cai Sa rtgyy sean waht’ 4 sahaw\er lated Fe be 1 PPUPUr ue. Al H enated gt! 7}: re J Me oe » Wala iF tidy { sing! ag aL ah iat ' Me 0 " breatin sige hey Pasty) Pgh ak! My u toed bratie ae oe Pn Pn ea , arr lidelors yak.) yee et Mam ge eth) Ms Pre Pry OLS As RAD AL ba pean, ad Pee Pn PRT P TO LeaeL SL pb tails Dy RPE NYE MRE REL! yah i hind Deeeiae Banas ati | aeke dey eee iyi erie apt nyt Navarates (as th Ses anata ai: vd ; oe aantnoa ata weet te rl Seetee Wee Tiny hae ste take ae ind te ait 1% a hiaettt Uap hy | ALA Ay . phates He WT Chahidt ihe ‘haaanet aay! wzhe TY. th Suk ‘eh, % vas Sates seas Nath ‘ ' detetans Hates sige ‘e heya heey habits t ay haeer At Lt Mb aif + bak idait tate nual ete * J tah any a \ Ady ated NEM Ve phe ee teetepe OP! PATE pial Ata. ®. O48. 8708 1 jaar y are nbdae MALHEMATICS The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN ee ae ee eee NOV 28 iF7S oct 31% MAR 3 0 (9 . MAR 02 fect AUG 2 3 1988 SEP 4 4 REC’ MAY 24 1999 _ | MAY 0.2 RECD 2.0 1989 .- « l- MA \f 9 vy rent i iv & { BBUB Neg L161— O-1096 F oy ais 7 ’ Far i ii er yh ae nae ‘Digitized by the Internet Archive in 2022 with funding rom a ae of Illinois Ne pChameaign mY 7) a a ae ’ Me ie Teg S lt Z| archive, mea elanenaticigg calpaibar \ | Boi oi a SE ta mae) Sik ——_— A . NEW MATHEMATICAL AND PHILOSOPHICAL DICTIONARY; path AN EXPLANATION OF THE TERMS AND PRINCIPLES OF PURE AND MIXED MATHEMATICS, AND SUCH BRANCHES OF Datural Philagaphy as are susceptible of Wathematical GYnbvestiyation. WITH HISTORICAL SKETCHES OF THE RISE, PROGRESS, AND PRESENT STATE OF THE SEVERAL ' DEPARTMENTS OF THESE SCIENCES. AND AN ACCOUNT OF THE DISCOVERIES AND WRITINGS OF THE MOST CELEBRATED AUTHORS, BOTH ANCIENT AND MODERN. BY PETER BARLOW, ~ THE ROYAL MILITARY ACADEMY, WOOLWICH; AUTHOR OF AN ELEMENTARY INVESTIGATION OF THE THEORY OF NUMBERS, Kc. &c. &c. iss. LONDON: PRINTED BY WHITTINGHAM AND ROWLAND, Goswell Street ; FOR G. AND S. ROBINSON, NO. 25, PATERNOSTER ROW; SCATCHERD AND LETTERMAN, AVE MARIA LANE; T. EGERTON, CHARING CROSs ; CRADOCK AND JOY, PATERNOSTER ROW; C. WHITTINGHAM, GOSWELL STREET ; AND JAMES CARPENTER AND SON, OLD BOND STREET, Ee 1314, ‘ ' cao aw bee , we fis pal? 4 yaaa. “ah SOT ee ea i had i = t \ Mea sR Sa is gti dy 8A KORE an ae i ert , 4g a ’ < , : A Cd aati ay" tales We areata oa « ie co a a « (nahn. f he, é Michoned es Toi a ae i spunea Boma : > Lee | nb eh ere A if Saad * Eh ee Gaisrice Hi Eu) Pairs 48 sid ae eM dpe pe iaytan A | fe Waa een at Acoma. mee SURV EL. . if >>) 4 “¥ - ke SZ Any me Ly Ou y) MATHEMATICS LIGA by: PREFACE. Pa: progress of the Mathematical Sciences, through a long ; succession of ages, during which they have been cultivated by ) nearly all the civilized nations of the earth, has given rise to a f multiplicity of peculiar terms and expressions, only to be found in the works of those authors who have written expressly on these subjects. Whence it must necessarily happen that the oemathematical student will frequently meet with words and > phrases, to which he is unable to affix any precise ideas, and © for an explanation of which it will be in vain to consult common > < dictionaries, into which they are either not introduced, or too ® slightl y defined to afford him the requisite degree of information. i ee And besides this inconvenience, which more particularly af- ~ fects the student, there are others experienced even by proficients, arising Out of the numerous rules, principles, and results, so frequently called for in practice; but the ready recollection of _which bids defiance to the most retentive memory, and renders frequent references, if not absolutely necessary, at least extremely convenient. ‘Be 16 98194 297372 1V PREFACE. These and other impediments which might be enumerated, and which have been more or less experienced at all times, have given rise to several distinct works, under the titles of Dic- tionaries, Lexicons, &c. the object of which has been the ex- clusive illustration of the terms and principles of these sciences : together with such other information as accorded with the views of their respective authors. Of these we have several in our own language, but the only two, which are here entitled to notice, are Stone’s ‘“ Ma- thematical Dictionary,” in one small octavo volume, and Dr. Hutton’s “ Mathematical and Philosophical Dictionary,” in two volumes quarto. But the first of these works is evidently too small to convey much essential information on such an extensive subject, and of too remote a date to detail any of the numerous and important discoveries of more modern times. And the latter, though an excellent and comprehensive performance, cannot of course contain many of the more recent improvements, besides its being necessarily limited to a comparatively small number of purchasers, its size and price being such as not to meet the views or to suit the conveniences of a number of persons, who would, notwithstanding, wish to possess the requisite informa- tion on these subjects, provided it could be comprised in a moderate compass, and purchased at a reasonable expense. The present work is therefore intended to form a mean between these two extremes; and while equally remote from brevity on the one hand, and from digression on the other, will, it is pre- sumed, be found to contain much important information relating to Mathematical and Physico-Mathematical subjects. In order the more effectually to accomplish this part of the design, a PREFACE. Vv number of terms usually introduced into works of this descrip- tion, have been omitted. Such are all those relating to the exploded science of Astrology, and those which are merely tech- nical in Architecture, Fortification, Music, and Military Affairs; which, though proper articles for an Universal Dictionary, do not necessarily, nor properly, form a part of a work. professedly mathematical. In all cases, however, where these subjects are either practically or theoretically connected with the Mathematical Sciences they have been retained, and their relations and depend- ences fully illustrated. This exclusion of a number of irrelevant terms, usually in- troduced into works of this kind, has afforded the author an opportunity of dwelling more at length on subjects of greater interest, and of introducing many articles not to be found in other works of a similar description; as also of combining with the explanatory matter such historical facts, connected with these sciences, as seemed most conducive to the reader’s information and instruction. With this view, besides the historical sketches, found under the heads of the several distinct branches, there is also given an alphabetical arrangement of the names of the most eminent authors, from the earliest period of authentic history to. the present time; with an account of their several discoveries and improvements, the dates and titles of their respective publica- tions, the several editions through which they have passed, &e. &e ; a part which, it is presumed, may be advantageously consulted, not only by the student, but also by those who may have already made some progress in these sciences. vi PREFACE. ‘To accomplish this in such a manner as should be most advan- tageous to the reader, it has been necessary to have recourse to a great number of original works, as well as to many heavy volumes of Histories, Dictionaries, Transactions, &c.; which it would be useless to enumerate in this place, as frequent references are made to them in the body of the work. It is, however, but justice to particularize Montucla’s ‘“ His- 99 toire des Mathematiques,” the mathematical volumes of the « Encyclopedié Methodique,” Dr. Hutton’s “‘ Mathematical and Philosophical Dictionary” above mentioned, and the abridg- ment of the Philosophical Transactions by Drs. Hutton, Shaw, and Pearson; from each of which much important information has been obtained. New principles and new subjects of investigation will not be expected ina work, the professed object of which is to detail the discoveries and improvements of preceding writers. At the same time, a different method of illustrating certain articles, as also in the arrangement and classification of them, may be con- sidered as the necessary consequences of the peculiar opinions which different writers form on those subjects. If, therefore, in these respects, any deviation be observed, from the excellent models which the author had before him, he trusts the reader will do him the justice to attribute it to the above cause, and not to any desire of introducing novelty at the expense of simplicity and truth. In the composition of the scientific articles, perspicuity and conciseness have always been kept in view, and in the historical — parts every possible means have been used to obtain the most PREFACE. Vil accurate information; where any contrariety has appeared in dif- ferent accounts, it has been duly considered, and the conclusion drawn on the side of the strongest evidence ; or where this could not be determined with safety, the reader is put in pos- session of the statements on both sides of the question, and thence enabled to draw his own conclusions. Having thus explained the motive which led to the present undertaking, and in some measure the means which have been employed in its execution: the author has only further to add his acknowledgements to those who have been pleased to express their approbation of the performance during the publication of the parts ; from which, and the success it has already experienced, he has reason to flatter himself that the work now completed will be favourably received by the public, to whom it is most respect- fully submitted. ROYAL MILITARY ACADEMY, January 24th, 1814. Pri, jie obuer te yt soa. sie | patiiareyto OK ite | fay sdbesop ait ‘yo aa bie alte 0. ‘waorgosete “gud WO in| PS Ties aa ere Ho. ait. wna | of holeaces abetgal 3A aD oe ge eta ee, Tica ae et | He: hey’ it 63 iret nai sive bonintgne: aude geil : Malad wiriat doi: mee ba8) qld otieigies aio: a ‘baw’ vane proba M aii 0: piled lao ash vo Nard: rr tor: Lae aif ‘tis bon onal a ae signa, Oo} Liaadisly ieaaeh avutt culys oegds ov cine ausghsin scent * yy) howe’) dius Sis at y eonetaeenechen ori er ‘doitadonqiger : A ies Mov toile eal, t tiie fatale: = — NV 2 4 ahh ey bey oere stl? fiser laid we io) mame 5) gO siti aad ago Monin dovatid ape eed i q aia pipe donut eS saison wb dildag, alt wa honing its ‘ eS a pate ay RS) oy ut is at . pe roa %, Tape Pee 1 es al ons . 4 ikl ' Mi > ¥ iy sei at Jag - ui ; PRN i i . rae ' j F At. t w al ® ‘ te K — / a, os ie. Oh oe bi ® ‘wt 32 a aka : (Bs anise ana gi eh i en wt iq re tA e, 4 pie © ae in crs 4 PAs .. thy! ya ie ry (i : nee ’ oH cig nie iy! oni” en we amremncnnnpanl us: : 4 HG ty Ly ak - 4 ie wh iis a : € ee \ hat ns at vit pit ort wii im Se ey, A wes ; ’ St hs afin a>! Si BY Ae nih hee ae as "ger shethno i aie vah.8 : 7 LWP i” ny nt Wr ent ts pi he nh oa mae) ee me vn dae pe we hk yh: oval vie Pua or ts : Hug ( Pah si ¥ Li. cat fe ‘ i ha < aioe: : ie as MATHEMATICAL AND PHILOSOPHICAL DICTIONARY. a ee ABA ABACUS, an instrument used by the an- cients, for facilitating arithmetical calculations. “This seems to have been, at first, nothing more than a small table or board, covered thinly with dust, on which their figures and schemes were drawn, and their calculations performed ; answering the purpose of our slates; and hence it appears to have derived its name, viz. from the Phoenician pbx dust ; though others derive it, with less probability, from the Greek aCaé. This instrument seems to be almost as an- cient as arithmetic itself: if it be later than methods of computing by the fingers, and by dapilli, or stones, (which obtained amongst the #Heyptians) it is, at least, much prior to the numerical letters or figures wrought with the en. its forms have been very various amongst different nations, under one or other of which at was employed by the Greeks, Romans, Chi- nese, Germans, French, &c.; and, in some eases, it excels the common method in point of facility and neatness of operation, as work- ing without any strokes or blots of the pen, or waste of paper; and it is even said by some to be more expeditious ; but this is very doubt- ful under the present form of arithmetic, though it might be true with respect to the ‘method of notation employed by the ancients. There are, however, instances in modern times, of instruments of a similar kind, that were found very useful in performing long multiplications; such were Napier’s, Bone’s, Saunderson's palpable Arithmetic, and some others. | The ancient abacus was variously contrived: that which was principally used in European 2ountries, was made by drawing any number: of parallel lines at pleasure, at a convenient jistance from each other; that is, equal to bout twice the diameter of the calculus or ‘ounters employed. Then, a cqunter placed on he first or lowermost line, signifies 1; on the econd, 10; on the third, 100; and so on.. In he Spaces between the lines, the counters ignified half of what they did in the next Iperior line; viz. in the space between the ABE first and second line, 5; between the second and third, 50; between the third and fourth, 500, &c. Thus in the following position of the counters on the abacus, the number 37392 is represented.— The abacus is also di- vided crosswise in- to areolae, by means whereof — subtrac- tions are performed. Wolf. Lex. Math. p. 171, et seq. Apacus Logisticus is a right-angled tri- angle, the sides of which forming the right angle, contain the numbers from 1 to 60; and its area, the product of each two of the oppo- site numbers: this is also called a canon of sexagesimals, and is nothing more than a mul- tiplication table carried to 60 both ways. Asacus Pythagoricus, a table of numbers contrived for facilitating arithmetical com- putations ; which was probably nothing more than our multiplication table, carried to a little further extent. ABBREVIATION of Fractions, in Arith- metic and Algebra, is the reduction of them to lower terms. See REDUCTION. ABENEZRA, a name sometimes given te the star Aldebaran. ABERRATION, in Astronomy, an apparent motion of the celestial bodies, occasioned by the progressive motion of light, and the earth’s annual motion in its orbit. The word is com- pounded of ab from, and erro to wander from their situations. ‘This apparent motion is so minute, that it could never have been disco- vered by observations, unless they had been made with extreme care and accuracy; and although it naturally arises from the combina- tions of the two causes above mentioned, yet as it was not even suggested by theorists, until it was discovered by observation, it furnishes us with one of the strongest proofs of the truth of the Copernican system. ‘The discovery was first made by that ingenious and indefatigable observer Dr. Bradley, Astronomer Royal, whe was led to it accidentally by the result of some careful observations, which he made with a yiew of determining the annual parallex of the B ABE fixed stars; the history of which discovery is related by the doctor’ himself in No. 406 of the Philosophical Transactions. The theory of aberration may be illustrated in the following manner :—-Iflight be supposed to have a progressive motion, the position of the telescope, through which any celestial ob- ject is viewed, must be different from that which it must have been, if light had been instantaneous; and therefore the place mea- sured in the heavens will be different from the true place. Thus if ; y S’ be a fixed star, ‘ ; V F the direetion of the earth’s mo- tion, S’ F the direc- tion of a particle of light, entering the axis ac of the tele- scope at a, and moving through a F while the earth moves from e¢ to F, and if the telescope = be kept parallel to itself, the light will descend in the axis. For, Jet the axis nm, Fw continue parallel to ae, and if each motion be considered as uniform, that of the spectator, occasioned by the earth’s rotation, being disregarded, because it is so small as to preduce no sensible effect, the spaces described in the same time will pre- serve the same proportion; eF and aE being described in the same time, and as we have eF : aF:: en: ao, en and ao will be de- scribed in the same time; and therefore when the telescope is in the situation nm, the par- ticle of light will be at o in the telescope; and the case being the same at every instant of its descent, the place measured by the tele- scope at F is.s’, and the angle S’ F's’ is the aberration, or the difference between the true place of the star, and the place measured by the instrument. Hence it appears, that if we take SF: Ft :: the velocity of light : the velocity of the earth, join St and complete the parallelograms Ft Ss, the aberration will be equal to SF¢, S will be the true place of the star, and s the place measured by the in- strument: and this latter is the same as the apparent place of the object, as it would be seen by the naked eye. In order to prove this, let it be considered, that if a particle of light fall upon the eye in motion, its relative motion with regard to the eye will be the same as if equal motions in the same direc- tion were impressed upon each at the moment of contact ; for it is a well-known principle in Mechanics, that equal motions, in the same direction impressed upon two bodies, will not affect their relative motions, and consequently the effect of one upon the other will not be altered. Let VF then be a tangent to the earth’s orbit at F, and represent the direction of the earth’s motion at F, and $8’ a star; join S’F and produce it to G, and take FG: Fn +: the velocity of light : the velocity of the earth in its orbit; complete parallelogram -motion with respect to the eye itself. ABE I'G Tin, and draw the diagonal FH. Since i'G and 2 F represent the motions of light and of the earth, if we conceive a motion F 2 equal and“opposite «I to be impressed upon the eye at F, and upon the particle of light, then the eye will be at rest, and the particle of light, by two motions FG and Fn, will describe the diagonal I’ H, which is its relative Hence it follows, that the objeet appears in the di- rection H F, and consequently that its appa- rent place differs from its true place by the angle GFHo—FSt. | But by trigonometry, sine FSé: sine FzS 7: Ft: FS:: the velocity of the earth : the velocity of light, and therefore the sine of vel. of earth— a vel. of hight ’ m4 if these velocities be considered as constant, the sine of aberration or the aberration itself, which never exceeds 20”, varies as the sine’ FéS, and is therefore greatest when F¢S is aright angle. Let then s express the sine of Ft, and we shall have radius (1): s 1; 20": 20 x s the aberration, and since St = 20”, when F'¢5 = 90°, we shall have the velocity of the earth : the velocity of light :: sine 20” ; 1::1: 10314. It appears that the aberration S's’ lies from the true place of a star, in a direction parallel to that of the earth’s motion, and towards the same part. ; Maupertius, in his Elements of Geography, familiarly illustrates the aberration, by the di- rection in which a gun must be pointed in- order to shoot a bird in its flight. Instead of” pointing straight to the bird, the sportsman” directs the gun a little before it in the path of its flight; and so much the more as the flight of the bird is the more rapid with respect to that of the shot. Po Clairaut likewise, in the Memoir of the Aca-_ demy of Sciences for 1746, explains the aber-_ ration, by supposing drops of rain to fall rapidly after each other from a cloud, under which a person moves with a very narrow tube; in which case it is evident that the tube must have a certain inclination, in order to admit a drop which enters at the top, to fall freely through the axis of the tube, without touching” the sides of it; and this inclination must be greater or less, according to the velocity of, the drops in respect to that of the tube. In this case the angle made by the direction of, the tube, and that of the falling drops, is the aberration, arising from the combination of aberration — sine F4S x these two motions. | To jind the Aberration of a Star in Latitude and Longitude. : ‘This problem has been investigated by D Bradley, and various other authors, the results, of which are as follow; viz. a 1. The greatest aberration in latitude equal to 20’ multiplied by the sine of star’s latitude. 2. The aberration in latitude for any time 1 equal to 20’ multiplied by the sine of th star’s latitude, and the sine of elongation found for the same time. ne ABE Note. The aberration is subtractive before #pposition, and additive after it. 3. The greatest aberration in longitude is equal to 20” divided by the cosin@ of the star’s latitude; and the aberration for any time is equal to the above quotient multiplied by the cosine of the elongation of the star. Note. This aberration is subtractive in the first and last quadrants of the argument, or of the difference between the longitudes of the sun and stars, and additive in the second and fourth quadrants. EXAMPLES. 1. ‘To find the greatest aberration of y Urse Minoris, whose latitude is 75° 13’. Here the sine 75° 13’ = -9669; consequently 20” x 9669 = 19.34", the greatest aberration in latitude. Also cosine 75° 13’ =*2551; and au? 2 therefore =-— = 784”, the greatest aberra- "2551 tion mn longitude. 2. 'To tind the aberration of the same star in latitude and longitude, when the earth is 30° from syzygies. Here sine of 36° =°5; and therefore 19.34” x ‘56 = 9°67", the aberration in latitude. If the earth be 36° beyond conjunction, or be- fore opposition, the latitude is diminished ; but if it be 80° before conjunction, or after opposition, the latitude is increased. . Again, cosine 30° = ‘866; consequently 78°4" «x ‘866 = 67°89", the aberration in longitude. If the earth be 30° from conjunction, the longitude is diminished ; but ifit be 30° from opposition, it is increased. To find the Aberration of a Star in Declination and Right Ascension. Dr. Bradley has annexed to his Theory the rules or formule for this purpose; and these rules have been variously demonstrated, and reduced to other practical forms, by M. Clai- raut, in the Memoirs of the Academy of Sci- ences for 1737; by Simpson, in his Essays, in 1740; by Euler, in the Memoirs of Berlin, tom. il. p. 14, &e.; and by several other per- sons, the results of which are as follow :— 1. The greatest aberration in declination is 20", muitiplied by the sine of the angle of position A at the star, and divided by the sine of B, the difference of longitude between the sun and star, when the aberration in declina- tion is nothing. 2. The aberration in declination at any other time, will be equal to the greatest aberration multiplied by the sine of the difference, be- tween the sun’s place at the given time, and its place when the aberration is nothing. 3. The sine of the latitude of a star : ra- ' dius :: the tangent of A : the tangent of B. 4. The greatest aberration in right ascension is equal to 20” multiplied by the cosine of A, ' the angle of position, and divided by the sine of C, the difference in longitude between-the sun and star, when the aberration in right ascension is nothing. 4. The aberration in right ascension at any other time, is equal to the greatest aberration ABE multiplied by the sine of the difference be- tween the sun’seplace at the given time, and his place when the aberration is nothing. Also the sine of the latitude of the star : the radius :: the co-tangent of A: the tangent of C. ABERRATION Of the Planets is their geo- centric motion, or the space through which they appear to move, as seen from the earth, during the time of light’s passing from the pla- net to the earth.— Let $8 be the sun, 'T the earth, P the cor- responding place of the planet; and let the earth be sup- posed to move in the direction 'l, pa- rallel to whichdraw PQ, and let it be equal to the space through which the earth has moved, whilst the light s passes from P to'T, and Q will be the apparent place of the planct. If Pp represent the motion of the planet in the same time, Q being the apparent, and p the corresponding true place, the angle QT p will be the aberration arising from the pro- gressive motion of light and the motion of the planet. Since P Q and Pp represent the mo- tions of the earth and planet, Qp will repre- sent their relative motions; and hence the motion of the planet about the earth in the time which the light takes to pass from the planet to the earth is the aberration. With respect to the sun, the aberration in longitude is constantly 20”, that being the space moved through by the sun, or by the earth, in 87’, which is the time in which light passes from the sun to the earth. In like manner, if we know the distance of any other planet from the earth, we shall obtain its aberration; for let ST =1, PT—d, m=the angle described by the planet about the earth, or its geocentric motion, in latitude, longitude, right ascension or declination, in 24 hours; then 1: d:: 8'7''2 : 872d, the time in which light moves from P to T; consequently 24h. : 8' 7’ d::m: the it kc = 00564 dm. There- 24h. fore, taking the geocentric motion from the Nautical Almanac, and estimating the dis- tance (in doing which no great accuracy is required), we shall find the aberration of a planet in latitude, longitude, right ascension, or declination. When m =o, or when the pla- net is stationary, the aberration is evidently equal to nothing. It is evident that the aberration will be greatest in the longitude, and very small in latitude, because the planets deviate very little from the plane of the ecliptic, so that this aber- ration is almost imensible and disregarded ; the greatest in Mercury being only about 4” 4, and much less in the other planets. As to the aberration in right ascension and declina- tion, they must depend upon the place of the B 2 ‘aberration = ABE planet in the zodiac. The aberration in longi- tude being equal to the geocentric motion, will be greater or less according to this mo- tion: it will be greatest in the superior pla- nets, Mars, Jupiter, Saturn, and Uranus, when they are in opposition to the sun; but in the inferior planets, Mercury and Venus, the aber- ration is greatest at the time of their superior conjunction. These maxima of aberration for the several planets, when their distance from the sun is the least, are as follow; viz. OWT sus claws oe SRS 27"0 MEDIEET g Cntlacn ce coach te 29” ¢ PE Meee so 5 SS oo RR 37" ¢ RVG, chs s cbt xe Mee 43" 4 TRLGPCUEY ys < oo'5 22 so ote laad 59"0 Moon o”2 eeervrvvreeetewr ee 8 @ 3 And between these numbers and nothing the aberration of the planets, in longitude, varies according to their situation. That of the sun, however, is invariable, being constantly 20"; and this may alter his declination, by a quan- tity which varies from 0 to 8”, being the greatest or 8” at the equinoxes, and vanishing in the solstices. For the methods of computing these, see a paper by Clairaut in the Memoirs of the Aca- demy of Sciences for 1746, and another by Euler in the Memoirs of Berlin, vol. ii. 1746. See also Vince’s Astronomy, vol. i. p. 332. 338; and a paper by Dr. Price in the Philoso- phical Transactions, vol. lx. art. 47. ARERRATION, in Optics; that error, or de- viation of the rays of light when inflected by a lens or speculum, whereby they are pre- vented from meeting or uniting in the same point, called the geometrical focus; it is either lateral or longitudinal. ‘The lateral aberration is measured by a perpendicular to the axis of the speculum, produced from the focus, to meet the refracted ray. The longitudinal aberration is the distance of the focus from the point in which the same ray intersects the axis. If the focal distance of any lenses be given, their aperture be small, and the inci- dent ray homogeneous and parallel, the longi- tudinal aberrations will be as the squares, and the lateral aberrations as the cubes, of the linear apertures, There are two species of aberration, distin- guished according to their different causes; the one arises from the figures of the speculum, or lens, producing a geometrical dispersion of the rays, when these are perfectly equal in all respects: the other arising from the unequal refrangibility of the rays of light themselves ; a discovery that was made by Sir I. Newton, for which reason it is frequently called the Newtonian aberration. As to the former spe- cies of aberration, or that arising from the figure of the lens, it is well known that if rays issue from a point at a given distance, they must be reflected into the focus of an ellipse, having the given luminous point for the other focus, or directly from the focus of an hyper- bole; and will be variously dispersed by all other figures. But if the luminous point be ABE indefinitely distant, or, which is the same, if the incident rays be parallel, then they will be reflected by a parabola into its focus, and variously dispersed by all other figures. ‘Those figures are, however, very diflicult to make, and therefore curved specula are generally made spherical, the figure of which is gene- rated by the rotation of a cireular arc, which produces an aberration of all the rays, whether they are parallel or not, and therefore it has no accurate geometrical focus which is com- mon to all the rays. Let B V represent a concave spherical-spe- culum, whose centre is C; and let AB, EF be incident rays parallel to the axis CV. Be- cause the angle of in- cidence is equal to the angle of reflection in all cases; therefore if the radii CB, BF be drawn to the points of incidence, and thence B D making the angle C BD equal to the angle CBA; and FG making the angle CI’'G equal to the angle CFE; then BD, FG will be the reflected rays, and D, G, the points in which they meet the axis. Hence it appears that the point of coincidence with the axis, is equally distant from the point of incidence and the centre ; for because the angle CBD is equal to the angle C BA, which is equal to the alternate angle BCD, therefore their opposite sides CD and DB are equal; and in like manner, in any other, G F is equal to GC. And hence it is evident, that when B is indefinitely near the vertex V, then D is the middle of the radius C V; and the nearer the incident ray is to the vertex C V, the nearer will the re- flected ray come to the middle point D; and the contrary. So that the aberration DG of any ray EF G, is always more and more, as the incident ray is further from the axis, or the incident point F from the vertex V; till when the distance VD is 60°, then the re- flected ray falls in the vertex V, making the aberration equal to the whole length DV. And this shows the reason why specula are made of a very small segment of a sphere, namely, that all their reflected rays may arrive very near the middle point or focus D, in order to produce an image the most distinct, by the least aberration of the rays. And in like man- ner for rays refracted through lenses. Huy- gens has demonstrated, that the aberration from the figure in different spherical lenses is as follows: 1. In all plano convex lenses, having their plane surfaces exposed to parallel rays, the Ex longitudinal aberration of the extreme ray, or — that most remote from the axis, is equal to 4% times the thickness of the lens. ) 2. In all plano convex lenses, having their convex surfaces exposed to the parallel rays, the longitudinal aberration of the extreme ray — { is equal to 11 of the thickness of the lens. , . 3. In all double convex lenses of equal A | | ABE spheres, the aberration of the extreme ray is equal to 14 of the thickness of the lens. 4. In a double convex lens, the radius of whose spheres are as 6 to 1, if the more con- vex surface be exposed to the parallel rays, the aberration from the figure is less than that of any other spherical lens, being no more than 35 of its thickness. Huygens has also shown, that the same aberration is produced by concave lenses as by similar convex ones. It has been asserted, principally on the authority of Sir I. Newton, that this aberration, arising from the figure of the glass, is very inconsiderable when com- pared with that arising from the unequal re- frangibility of the rays of light; and Smith, in his Optics, book ii. chap. 6, makes the latter to the former as 5449 to 1. Admitting the truth of this, it was thought strange that ob- jects should appear through refracting tele- scopes so distinctly as they are found to do; and indeed many persons despaired of success in the use and construction of Jenses. Buta little attentive consideration of the subject will probably convince us, that the above pro- portion is over-rated. In consequence of the aberration, a geometrical point after refraction is not represented by another geometrical point, but by a very small circle, which is sometimes called the circle of diffusion. And as in the performance of optical instruments, it is necessary that this extended representa- tion of any point P be so small that it may not sensibly interfere with the representation of the points adjacent to P, and thus cause indistinct vision, a limit is set to the extent of the refracting surface (that is, to the aper- ture) which must be employed to produce the representation. But this evidently diminishes the quantity of light, and renders the vision less vivid, though distinct. The nature of these aberrations may be finely illustrated, by receiving on a piece of white paper the light of the sun refracted through a globe or cylin- der of glass filled with water. If the paper be held parallel to the axis of the cylinder, and close to it, the illuminated part will be bounded by two very bright parallel lines, where it is cut by a diacaustic curve; and these lines will gradually approach each other, as the paper is withdrawn from the vessel, till they coalesce into one very bright line. If the paper be held with its end touching the vessel, and its plane nearly perpendicular to the axis, the whole progress at the curve will be distinctly seen. We know that the doc- trine of aberrations has been considered in a manner independent of caustic curves. But | whoever considers the progress of rays in the eye-piece of optical instruments, will see that the knowledge of diacaustic curves determines directly, and almost accurately, the foci and images that are formed there. It is of great importance to attend to the manner in which the light is distributed over the surface of the circle of smallest diffusion; for this is the representation of one point of the infinitely distant radiant object. Each point of a plane, ABR for instance, is represented by this little cir cle; and as the circles representing the difé ferent adjacent points’ must thus interfere with each other, an indistinctness must arise similar to what is observed when we view an object through a pair of spectacles, which do not suit the eye; and this indistinctness must be in proportion to the number of points whose circles of diffusion interfere; that is, to the area of those circles, provided that the light is uniformly diffused over them: but if it be very rare at the circumference, the impressions made by the circles belonging to the adjacent points must be less sensible. Accordingly Sir I. Newton, supposing it incomparably rarer at the circumference than towards the centre, affirms, that the indistinctness of telescopes arising from the spherical figure of the object- glass was incomparably less than that arising from the unequal refrangibility of light as be- fore stated. It is much to be regretted that this truly eminent philosopher should have committed such an oversight, as the authority of his name prevented others from examining the matter, trusting to his assertion, that the light was so rare at the borders of this circle, contrary to the very nature of a caustic, in which the light is infinitely dense at these borders. ‘The first who discovered this error was the celebrated Abbe Boscovieh, who, in a dissertation, published at Vienna in 1767, showed, by a very beautiful analysis, that the distribution was very different from what Newton had supposed, and that the superior indistinctness, arising from unequal refrangi- bility, was considerably less than had been stated. Many ingenious philosophers, in different parts of Europe, have paid great attention to the subject of optical aberration, particularly with a view of improving the construction of refracting telescopes, and among those no one has more distinguished himself than Mr. Dol- land, who, in 1758, made a considerable im- provement in this instrument, by introducing: two object-glasses of crown glass, and one of flint.—For various papers on this subject, see Phil. Trans. vols. xxxv. xlviii. 1. li. lil. Iv. Ivi. Hist. Ac. Par. for 1737, 46, 52, 55, 56, 57, 62, 64, 65, 67, 70. Swed. Mem. vol. xvi. Ac. Berlin, 1746, 62, 66. Com. Nov. Petrop. 1762. Euler’s Dioptrics, and M. d’Alembert’s Opus. Mathem. See also the articles Aberration, &c. Rees’s Cyclopedia; to which latter work we have been much indebted for many of our preceding remarks. ABRACHALEUS, given to the star Pollus. ABRIDGING, in Algebra, is the reduction of a compound quantity, or equation, to a more simple expression: thus to abridge the equation, a name sometimes ay ab xz? — 5 Cat ucla —abe=0. c be Instead of the known compound co-eflicient (a +b +c), some single letter, as p, is sub-~ stituted; and instead of ab + a¢ + be, another ABS single letter, as q; and for —abe, is written another, as 7; Whereby the original equation is expressed mé@re simply, as follows: v—px’+qr—r—0. This abridging of an algebraic equation, or expression, ts done principally to save room, or the trouble of writing a number of symbols, or to simplify the expression for relieving the attention and memory, or in order to render the formula more easy and general. ABSCISS, Anscis®, Asseissa, (from Ad- scindere, to cut off) is any part of the diameter or axis of a curve, comprised between any fixed point, where all the abseisses begin, and another line called the ordinate, which is terminated in the curve. Commonly the ab- scisses are considered as commencing at the vertex of the curve; but this is not necessary, as they may have their origin m any other point; but generally, when no condition Is specified, thay are understood as commencing at the vertex. Thus if AQ be any curve, and A, or A’, or O A’, certain fixed points, of which, in the annexed figures A and A’, are at thevertices, but A” any point whatever in the axis; also P Qany line perpendicu- lar, or otherwise, to the axis A, A, and terminating in the curve; then < is PQ an ordi- A PA A nate, and A P, A’P, A’ P, adscisses. The absciss and corresponding ordinate, consi- dered together, are called co-ordmates, and by means of these the equation of the curve is defined. See Curve. ABSIDES. See ApsIDEs. ABSOLUTE Equation, in Astronomy, is the sum of the optic and eccentric equations. The apparent inequality of a planet’s mo- tion, arising from its not being equally distant from the earth at all times, is called its optic equation; and this would subsist if the pla- net’s real motion were uniform. The eccen- tric inequality is caused by_the planct’s motion not being uniform. For the illustration of this, conceive the sun to move, or appear to move, in the circumference of a circle, in the centre of which the earth is placed. Then it is ma- nifest, that if the sun move uniformly in this circle, he must appear to move uniformly to a spectator on the earth; and in this case, there would be no optic or eccentric equation. But suppose the earth to be placed out of the centre of the circle; and then, though the sun’s motion should be really uniform, it would not appear to be so when seen from the earth; and in this case there would be an optic equa- tion, but not an eccentric one. Again, let us imagine the sun’s orbit not to be circular, but elliptical, and the earth to be in its focus, then it is evident th at the sun cannot appear e bo ACA to have a uniform motion in such ellipse, and therefore his notion will be subject te two equations, viz. the Optic and Uecentrie Liquations, the sum of which is the Absolute Equation. ApsoLutTe Term or Number, in Algebra, is that which is completely known, and to which all the other part of the equation is made equal; thus in the equation 2? + ax 6, the latter quantity 6, is called the absolute term, which in every equation is equal to. the con- tinued product of all its roots. This term Victa calls the homogenewm comparationis ; but modern algebraists class it with the co-effici- ents, considering it as the co-efficient of x°. ABSOLUTE Gravity, Motion, Space, Time, &c. see the respective substantives.’ ABSTRACT Mathematics, or Pure iathe- maties, is that which treats of the properties of magnitude, figure, or quantity, absolutely and generally considered, without restriction to any species in partieular, such as Arithmetic aid Geometry. [tis thus distinguished from Mixed Matkematics, 1 whieh simple and abstract quantities, primitively considered in Pure Ma- thematics, are applied to sensible sbjects, as in Astronomy, Mechanics, Opties, &e. Axstract Numbers, are assemblages of units, considered independently of any thing or things, that they might otherwise be sup- posed to represent. Lor example, 5 is an ab- stract number, while it remains independent ; but if we say 5 feet, or 5 miles, it is no longer an abstract, but a concrete rinmber. ABSURD, or ApsurpuM, a term commonly employed in demonstrating converse proposi- tions, in which the proposition is not proved in a direct manner, by principles before laid down; but it proves that the contrary is ab- surd, or impossible; and thus indirectly de- monstrates the truth of the proposition itself. The 4th prop. of the Ist book of Euclid is demonstrated by this method, by. showing, that if the extremities of two right lines coin- cide, the lines themselves will coincide in all their parts, otherwise they would enclose a space, which is absurd, being contrary to the 10th axiom. Converse propositions are com- monly proved in this way, which mode of de- monstration is called reductio ad absurdum ;: but it is objected to by some mathematicians, as inconclusive and unsatisfactory: it would, however, be very difficult, perhaps impossible, to devise any system of geometry in which such a method should be entirely excluded. ABUNDANT Number, in Arithmetic, is a number the sum of whose aliquot parts is greater than the number itself: thus 12 is an abundant number; for the sum of its aliquot parts 14+243+4+46=16, whichis greater than 12. It is thus distinguished from a per- fect number, which is equal to the sum of all its aliquot parts, and from a deficient number, which is less than the sum of all its aliquot parts. ACADEMY, Acapemia in Antigqnity, a fine villa or pleasure-house in one of the suburbs of Athens, about a mile from the city, where _ ACC Plato and his followers. held assemblies, for disputes and philosophical discussions. ‘The house took its name from its original pro- prietor Academus, or Ecademus, who lived there in the time of Theseus. Amongst the moderns this term is applied to a society of learned men, instituted under the protection of some prince or other public authority, for the cultivation and improvement of the arts and sciences.—The reader will find. an interesting history of the several Eu- ropean institutions of this kind, under the article Academy, in Dr. Hutton’s Math. and Phil. Dict. AcADEMY is also used amongst us fora kind of collegiate school or seminary, where youth are instructed in the liberal arts and sciences in a private manner. ACAMPTE, a word used by Leibnitz to denote a figure, which being epaque and po- lished, and consequently possessing properties necessary for reflecting light, yet does not re- flect it. See Op. Leib. tome iii. p. 203. ACCELERATED Motion (from accelero, to make quick,) is that which continually re- ceives fresh accessions of velocity, and is either equably or unequably accelerated. If the accessions of velocity be always equal in equal times, the motion is said to be equably or uniformly accelerated ; but if the accessious in equal times either increase or decrease, the motion is unequably or variably accelerated. Acceleration stands directly opposite to Re- tardation, which denotes a diminution of ve- locity. ACCELERATED Motion, of Pendulums, of Pre- jectiles, of Compressed Bodies. See PENDULUM, PROJECTILES, DILATION, COMPRESSION, EXLAS- Ticity, &e. ACCELERATING Force, in Mechanics. See Force. ACCELERATION is principally used in Physics, in respect of falling bodies, viz. of heavy bodies falling, or tending towards the centre of the earth, by the force of GRAvITy. That natural bodies are accelerated in their descent, is evident from various considera- tions, both @ priort and posteriort. ‘Thus we actually find, that the greater height a body descends from, the greater impression it makes, and the more intense will be the blow with which it strikes the plane or other ob- stacle on which it impinges. Various have been the systems and opinions which philosophers have produced to account for this acceleration. Some have attributed it to the pressure of the air; others to an in- herent principle in matter, by which all bodies tend to the centre of the earth as their proper seat or element, where they would be at rest ; and hence, say they, the nearer bodies ap- proach to it, the more is their motion acce- jerated. Another class held, that the carth emitted a sort of attractive effluvia, innumer- able threads of which continually ascend and descend, proceeding, like radii, from a com- mon centre, and diverging the more the fur- ACC ther they go; so that the nearer a heavy body is to the centre, the more of these magnetic threads it receives; and hence the more its motion is accelerated. The Cartesians accounted for acceleration, from the repeated pulses of a subtile ethereal matter, which they supposed to be continually acting on the falling body, and impelling it downwards. But leaving all such visionary theories, and only admitting the existence of such a force as gravity, so evidently inherent in all bodies, without regard to what may be the cause of it, the whole mystery of acceleration will be cleared up, and the theory of it established on the most obvious principles. Suppose a body let fall from any height, the primary cause of its begimning to descend is doubtless the power of gravity; but when once the descent is commenced, that state becomes in some measure natural to the body; so that if left to itself, it would persevere in it for ever, even though the action of the first cause should cease; as we see ina stone cast from the hand, which continues to move after it is left by the cause that first gave it motion; and which motion would continue for ever, was it not destroyed by resistance and gra- vity, which cause it to fall to the earth. But besides this tendeney to descend, impressed by the first cause, and which of itself is suffi- cient to continue the same degree of motion once begun, i jinitum, there is a constant accession of subsequent efforts of the same principle, gravity, which continues to act on the body already in motion, in the same man- ner as if it were at rest. Here, then, being two causes of motion, and both acting in the same direction, the motion they jointly pre- duce must necessarily be greater than that of one of them; and the same cause of increase acting still on the body, the descent of it must of course be continually accelerated. For supposing gravity, whatever it be, to act uniformly on all bodies, at equal distances from the earth’s centre; and that the time in which a heavy body falls to the earth be di- vided into equal parts, indefinitely small: let this gravity incline the body towards the earth’s centre, while it moves in the first in- definitely small space of time of its descent; if after this the action of gravity be supposed to cease, the bedy would proceed uniformly towards the centre of the earth, with a velo- city equal to that which results from the force of the first impression. But now, ‘since the action of gravity is here supposed still to continue, in the second mo- ment of time the body will receive a new im- pulse downwards, equal to what it received at first; and thus its velocity will be double of what it was in the first moment: in the third moment, it will be treble; in the fourth, quadruple ; and so on continually: for the im- pression made in one moment, is not at all altered by what is made in another; but they are, as it were, united into one sum. Ae Hence, since the portions of time are sup- posed indefinitely small, and all equal to one another, the velocity acquired by the falling body will be every where proportional to the times from the beginning of the descent ; that is, the velocity will be proportional to the time in which it is acquired. Thus if a body, by means of this constant force, acquire a velocity v, in one second of time, it will in two seconds acquire a velocity 2v; in three seconds, it will acquire a 37; and so on: and all bodies, whatever be their quantity of matter, will acquire, by the force of gravity, the same velocity in the same time ; that is, supposing no resistance from the atmosphere,-or that the bodies are falling in a perfect vacuum. For every particle of matter being endued with an equal impelling force, viz. its gravity or weight, the sum of all the forces in any compound mass of mat- ter, will be proportional to the sum of all the weights or quantities of matter to be moved ; and consequently, the forces and masses moved, being thus constantly increased in the same proportion, the velocities generated will be the same in all bodies, great or small; 7. e. a double force will move a double mass of matter, with the same velocity as the single force moves the single mass, &c. as the whole compound mass falls together with the same velocity, and in the same time, as if its par- ticles were not united. Hence, if t be made to represent the time a body has been falling, and v the velocity generated in one second, then willtv repre- sent the velocity at the end of ¢ seconds. And if now we represent by ¢? the velocity of a body when it first begins to fall, or its velocity acquired in an indefinitely small space of time, or first instant of its descent, then will the terms of the series %?, 29, 39, 49, &ce. t4, represent the successive velocities at each successive instant; and since the velocity v into the fime in uniform motions is equal to the space, and since we may consider the motions as uniform during any indefinite small instant of time, therefore the above may also be taken to represent the spaces deseribed at each successive instant, and hence the sum of them will be the whole space described in the time ¢t. Now the number of terms in this series being t, and ¢ @ being the final ve- locity, it may be represented by v, and thus the sum of the series will be expressed by 4¢(@+v). But since ¢? represents the first velocity of the descending body, this is inde- finitely small, and may be considered as no- thing with regard to v, and may therefore be cancelled out of the expression, and conse- quently the whole space will be represented by £¢v, or taking s to represent the space, we shall have s = itv, that is, the space de- scribed by a body uniformly accelerated in any time, is equal to half the space that would be described by the same body with a uniform yelocity equal to that last acquired. ACC And since it has been found by experimen?’ that a body falling freely by the force of gra- vity, in the latitude of London, passes throug!’ a space equal to 163, feet in the first second: © of time, we have, by representing this space by g, g=itv, or g=4v; because t=—1, whence v = 2g, is the velocity acquired at the end of the first second of time, and therefore, from what has been above demonstrated, v —=2 gt, will represent the acquired velocity at the end of any time ¢. Also we have seen, that s = #tv; substi- — tuting therefore for v, as above, we have s = it (2g¢t), ors—tg; and since g is constant, it follows that the spaces described by falling bodies are to each other as the squares of the time of descent, the spaces themselves being _ accurately expressed by the formula s =? gy — where g represents 16;, feet, the space a body falls through in the first second of time. Hence then, from the preceding principles andreasoning, we deduce the following general laws of uniformly accelerated motions ; viz. 1. The velocities accquired are constantly proportional to the times. 2: That the spaces are proportional to the squares of the time; so that if a body describe any given space ina given time, it will de- scribe four times that space in a double time, nine times that space in a treble time, and so on. And universally, if the times be in arithmetical proportion, as_ 1, 2, 3, °4, &e. t, the spaces described will be as 1, 4, 9, 16, &c. ¢*. Thus a body, which falls by gravity through 164, feet in the first second, will fall through 644 feet in two seconds, and so on. And since the velocities acquired in falling are as the times, the spaces will be as the square of the velocities ; and both the times and veloci- ties will be as the square roots of the spaces. 3. The spaces described by a falling body in a series of equal moments, or intervals of time, will be as the odd numbers 1, 3, 5, 7, 9, &c. which are the differences of the squares or whole spaces; that is, a body which falls through 16,4 feet in the first second, will fall through 3 x 163, in the second, 5 x 164 in the third, and so on. These properties may be otherwise repre- sented, thus:—Let S, V be put for the space and velocity of a falling body in any time T; and s,2, the same for the time t; then we shall have S o.3 T2 t? S gi ST Fe si Vv D4 se as J/Ss a ie fis ay Js S ge sey: tv Fr * e 8s s {i oe $ ee V v 4 Weg 424 he ae oy, Vs When the accelerating forces are different, but constant, the spaces will be as the pro- ducts of the forces into the square of the times; and the times will be as the square — roots of the spaces directly, and of the forces. ACC mversely. For when the force is given or constant, the velocity (V) is as the time (T); and when the forces are different, but con- stant, and the time is given, the velocity (v) will be as the force (Ff). But when neither the force nor the time is given, the velocity ‘v) will be as the product of the force into the time that is as (fF x T). Hence, Wi: CPT si xt conseq. EXT’: fxenVxTivxt::S:s T? . t?:: Ss . bd f S « it tings From the properties above demonstrated, ve obtain the following practical theorems : Let g denote the space passed over in the first iecond of time, by a body urged by any uni- orm force, denoted by 1; and let ¢ denote the ime or number of seconds in which the body asses Over any other space s, and v the velo- ity acquired at the end of that time: then ve shall have v = 2gt, and s=¢@?; and from hese two equations we obtain the following ‘eneral formule : therefore, or, . ’ . I passe v 2s Ss eee aas ity a , z S 2 ie Sok Stewed AS dear : v? tv ee ee od She teesnt ‘= 2 = Pere 4 Be rhaly oth “ait a nh 9 . eeteee QI B — Qt “45 _Letus now illustrate ‘Ww examples. Exam. 1. How far will a body fall in 10”, ithe latitude of London; and what will be i's last velocity? Here g = 161,, and t=10; therefore s — gt 716, x 100 = 16085 feet, the space required. Also, v= 2gt=2x 162, x 10 = 3213, feet jie last acquired velocity. Exam. 2. How long will a body be in fall- § through a space of 1608 feet? — ,/ 1608 eee Here s = 1608, and t — es 210 seconds nearly. Exam. 3. How far must a body fall to ac- lire a velocity of 321 feet per second? + Let MR 9 flere v =321, and t= Qg = 301 these formule by a s & = 10 se- nds nearly. The same law of acceleration obtains equally the descents of bodies down inclined planes, cept that the force of gravity will in that case ry as the sines of the angles of inclination the planes, that is, the force down the in- ned plane, is to the whole force of gravity, the sine of the angle of inclination of the me, to radius. If, therefore, the angle of slination of the plane be a, the force down » plane will be sin. a x §; and by using S instead of g, the above formule will be aally applicable to the descents of heavy lies down inclined planes. See IncLINED ANES. he theory that we have been investigating, ACC was first discovered by Galileo, and many in genious contrivances which he made use of in order to deduce this law from experiments, may be seen in his work, entitled Dialoghi delle Scienze nuove, &c. printed first at Leyden in 1637. ; Of variably accelerated motion. H aving above illustrated the laws of accelerated motion, when the accelerating forces are constant, and deduced the formula for expressing them in finale determinate quantities, we shall now subjoin those that belong to the cases of variably accelerated motions. Here the formule will be fluxionary ex- pressions, the fluents of which, adapted to par- ticular cases, will give the relation of time, space, velocity, &c. Let ¢ denote the time of motion; v the velocity generated by any force; s the space passed over; and 2g the variable force at any part of the motion, or the velocity that the force would generate in one second of time, if it should continue invariably like the force of gravity during that one second, and the value of this velocity, 2g, will be in proportion to 32: feet, as that variable force is to the force of gravity. Then, because the force may be supposed constant during the indefinitely small space of time t, and spaces and velocities in uniform’ motions being pro- portional to the times, we shall have these two fundamental proportions, viz. ORL" 53 or s=iv 2 ae a eg or v= 2g Whence are deduced the following formule, in which the value of each quantity is ex- pressed in terms of the rest: F s v 1. POOH reo tee eeererseey = ap = pS ra ° v 2f : . Was 2; Pee eeesertseseee ee pee V=2gt = Vv We oe Seore SOC e ran etene . § 5 vt Rae sake 2g Vv DU Ab GNM Sadi kG PF os 7 ———— $ These formule, liké those in the preceding part of this article, are equally applicable to the destruction of motion and velocity, by means of retarding forces, as to the generation of them by means of accelerating forces. The motion of a body ascending, or im- pelled upwards, is diminished or retarded from the same principle of gravity acting in a con- trary direction, in the same manner as a fall- ing body is accelerated. See RETARDATION. A body, thus projected upwards, rises till it has lost all its motion; which it does in the same time that a falling body would have ac- quired a velocity equal to that with which the body was thrown up. Hence the same body projected up, will rise to the same height, from which, if it fell, it would have acquired the velocity with which it was projected up- wards. And hence the heights to which bo- dies, thrown up with different velocities, as- cend, are to each other as the square of those velocities, ACCELERATION, in Mechanics, the increase of velocity in a moving body, ACC ACCELERATION, in Astronomy, is a term applied to the fixed stars. Thus, the diur- nal acceleration is the time by which the stars, in one diurnal revolution, anticipate the mean diurnal revolution of the sun, which is 3) 55,2," of mean time, or nearly 3 56”; viz. a star rises or sets, or passes the meridian, 3’ 56" sooner each day. This apparent acceleration of the stars is owing to the real retardation of the sun, which depends upon his apparent motion towards the east, which is at the rate of about 59’ 82” of a degree every day. In consequence of this the star which passed the meridian at the same moment with the sun yesterday, is about 59’ 81” beyond the meri- dian to the west, when the sun arrives at it; and this distance will require about 3’ 56" of time for the sun to pass over, and therefore the star will anticipate the motion of the sun at this rate every day. AccELERATION of a Planet. A planet is said to be accelerated in its motion, when its real diurnal motion exceeds its mean diurnal motion. And on the other hand, the planet is said to be vetarded in its motion, when the mean exceeds the real diurnal motion. ‘This inequality arises from the change in the dis- tance of the planet from the sun, which is continually varying; the planet moving always quicker in its orbit when nearest the sun, and slower when furthest off. ACCELERATION of the Moon, is a term used to express the increase of the moon’s mean motion from the sun, compared with the diurnal motion of the earth, by which it appears that, from a certain cause, it Is now a little quicker than it was formerly. Dr. Hal- ley was led to the discovery, or suspicion of this acceleration, by comparing the ancient eclipses observed at Babylon, &e. and those observed by Albategnius in the ninth cen- tury, with some of his own time (Phil. Trans. No. 218). He was not able to ascertain the quantity of this acceleration, because the lon- gitudes of Bagdat, Alexandria, and Aleppo, where the observations were made, had not then been accurately determined. But since his time, the longitude of Alexandria has been ascertained by Chazeller; and Babylon, ac- cording to Ptolemy’s account, lies 50’ east from Alexandria. From which data Mr. Dun- thorn (Phil. Tras. No. 492. abs. vol. x. p. 84.) compared several ancient and modern eclipses With the calculations of them by his own tables, and thus verified Dr. Halley’s opinion ; for he found that the same tables represent the moon’s place in the ancient eclipses behind her true place, and before it in latter eclipses ; and hence, justly inferred that her mofion in ancient times was slower, but in latter times quicker than the tables give it; and therefore that it must have been accelerated. But he did not content himself with barely ascertaining the fact. He proceeded to determine the quan- tity of the acceleration; and by means of the most ancient eclipses, of which any authentic record remains, observed in Babylon 721 years before Christ, he concluded that the observed ACC beginning of this eclipse was not above an hour and three quarters before the beginning by the tables; and therefore the moon’s true place could precede her place by computa tion but a little more than 50’ of a degree a that time. Admitting the acceleration to be uniform, and the aggregate of it as the squar of the time, it will be at the rate of abow 10” in 100 years: Lalande makes it 9.886 In Mayer’s Tables it is 9”, beginning fron 1700. La Place (Mem. del’Acad. Roy. des Science: for 1786) has shown this acceleration of thy moon’s motion to arise from the action of thy sun upon the moon, combined with the varia tion of the eccentricity of the earth’s orbit. B: the present diminution of the eccentricity, th moon’s mean motion is accelerated; but whei the eccentricity is arrived at the minimum, th acceleration will cease; after this the eccen tricity will increase, and the moon’s mea motion will be retarded. | M. de Lambre found, by comparing moder observations at about the distance of a cer tury, that the secular mean motion of th moon, in the last tables of Mayer, was to great by 25"; and that the place of the moo calculated by these tables, ought to be corree ed by the quantity — 25” (nm + 2”), 1357? +0 04398 n3, » being the number of centuries froy 1760. Vince’s Astron, vol. i. p. 206. ACCIDENS. Per aceidens is frequent! used by philosophers to denote what does ny follow from the nature of a thing, but froi some accidental quality of it; in which sens it stands opposed to per se, which denotes tl nature and essence of a thing. ACCIDENTAL Colows, so called by Bu fon, are those which depends upon the affe tion of the eyes, in contradistinction to suc as belong to light itself. The impressioi made upon the eye by looking steadfastly « objects of a particular colour are vario according to the single colour, or assemblas of colours, in the object; and they contim for some time after the eye is withdrawn, at give a false colouring to other objects that a. viewed during their continuance. See sever ingenious papers on this subject in the Ac Par. 1743 and 1765 ; Nov. Com. Petrop. vol.3 see also Priestley’s History, &c. of Discoy ries relating to Vision, &c. p. 631. AccIDENTAL Point, in Perspective, is a poi in the horizontal line, in which a right lp drawn from the eye, parallel to another rig line, intersect the picture or perspective plar This is also the accidental point of all oth lines parallel to the original line, since t same line drawn from the eye is parallel them all; and the representations of all the parallels, when produced, concur in the ae dental point. See PERSPECTIVE. | ACCLIVITY, the slope or steepness ol line or plane inclined to the horizon, ta upwards, in contradistinction to declivi which is taken downwards. So the ascent a hill is an acelivity, and the descent of: same a declivity, — , ACR . ‘ACCOUNT, or Accompet, in Arithmetic, ». a caleulation or computation. ‘Account, in Chronology, is nearly synony- nus with style ; thus, we say the old or new ‘le, or old or new account. ACHERNES, or ACHARNES, in Astronomy, star of the first magnitude, in the southern ‘tremity of the consteliation Sridanus ; urked « by Bayer. ‘ACHILLES, in Philosophy, a name that 's schools gave to the principal argument, ‘eged by each sect of philosophers in behalf ‘their system. In this sense, we say this is 1; Achilles ; that is his master-proof, alluding ) the strength and importance of Achilles hongst the Greeks. Zeno’s argument against tion is particularly termed Achilles. ‘That lilosopher made a comparison between the ‘iftness of Achilles and the slowness of a toise ; maintaining that if the tortoise were ‘e mile before Achilles, and his motion 100 j1es swifter than that of the tortoise, yet he 'uld never overtake it; and thence he con- fided that there was no such thing as motion. {ut this it is evident was only amere sophism; ‘d is easily solved by expressing the whole ‘ative distance run by the tortoise before hilles overtook him, by the following series : 1 ‘7 * Too00 * 1000000 ich is equal to 3, of a mile; and the dis- fice run by Achilles is just one mile more; ‘that when the latter had run 13, miles, he vuld have overtaken the tortoise. ‘ACHLUSCHEMALL, an Arabian term for + constellation Corona Borealis. ACHROMATIC, (from « privative and vue colour, without colour) in Optics, a term it used by Lalande, in his astronomy, to de- te telescopes of a new invention; contrived remedy aberrations and colowrs. See ABER- TION and ‘TELESCOPE. ACLASTE, a term used by Leibnitz to de- ‘te figures which possess the necessary pro- ‘ties for refracting the rays of light; through lich, notwithstanding, they pass without ‘raction. See Op. Leib. tome iii. p. 203. “ACOUSTICS, (from axew, to hear, in Philo- hy) signifies the doctrine of hearing, and : art of assisting that sense by means of vaking trumpets, hearing trumpets, whis- (ing galleries, &c. See Ecuo and Sounp. /ACRE (from the German acker, field) de- tes a quantity of land containing 4 square ‘ds, or 160 square rods or perches. By the *stoms of countries, the perch differs in quan- 7; but by the ordnance for measuring land, ide in the 33d and 34th of Edward I. the } tute length of the pereh is fixed at 51 yards, 1 16} feet; and therefore the statute acre . 4840 square yards, or 43,560 square +, &c.; the sum of Che French acre, arpent, is equal to 11 Eng- 1 acres. ‘he Strasburg contains about half an Eng- t acre. The Welsh acre is equal to nearly 2 Eing- i acres, . ACT The Jrish acre contains la. 27.194 p, Eng lish. ‘The number of square acres contained in England and Wales is supposed to be about 47,000,000, in Scotland 26,000,000, and in Treland 13,578,000; but there is a very con- siderable. difference in the estimates of diffe- rent writers on this subject. See Phil. Trans, No. 320, and Sir Will. Petty’s Political Arith- metic. . ACROAMATICI, in Philosophy, a deno- mination given to the disciples of Aristotle, &c. who were admitted into the secrets of the inner, 6r acroamatic plilosophy. ; ACRONYCAL, ACHRONYCAL, or ACHRO- NICAL, in Astronomy, is said of a star, or planet, when it is opposite to the sun. It is derived from the Greek, axpovvyo;, the point or ex- tremity of night. The acronycal is one of the three poetic risings and settings of the stars, and stands distinguished from Cosmical and Heliacal. And by means of which, for want of accurate in- struments, they might regulate the length of the year. ACTION, in Mechanics, denotes either the. effort or power which one body exerts against another, or the effect resulting from such effort ; or, more accurately, the motion which a hody really produces, or tends to produce, on another. ‘The action of a body becomes apparent only by its motion; and we cannot affix any precise idea to the term action, be- sides that either of actual motion, or a simple tendency to motion. Leibnitz and his dis- ciples, for want of duly attending to the pro- per and discriminating idea of the word action, have perplexed themselves and others with unprofitable and indecisive disputes concern- ing vis viva and vis mortua. See Force. | ‘The Cartesians resolve all physical action into metaphysical. According to them, bodies act not upon one another, but the action pre< ceeds immediately from the Deity: the mo- tions of bodies, which seem to be the cause, being only the consequence of it. Action is either instantaneous or continued; that is, either by percussion or pressure. ‘These two forces, or sorts of action, have been generally considered as incongruous, no more admitting of comparison than a line and a surface, or a surface and a solid; but this does not appear to be well established, as we have shown un- der the articles PRESSURE and PERCUSSION. * It is one of the first laws of nature, that action and reaction are equal and contrary to each other. If a body be urged by equal and contrary actions, it will remain at rest; but if one of the actions be greater, then its opposite mo- tion will ensue towards the part least urged. It is to be observed, that the actions of bodies on each other, ina space that is car- ried uniformly forward, are the same as if the spaces were at rest; and any power or forces that act upon all bodies, so as to produce equal velocities in them, in the same or in parallel right lines, have no effect on the ACU mutual actions or relative motions. Thus the motion of bodies on board a ship, which is car- ried steadily and uniformly forward, are per- formed in the same manner as if the ship were at rest. The motion of the earth round its axis has no effect on the actions of bodies and agents at its surface, except so far as it is not uniform and rectilineal; and, in gene- ral, the actions of bodies upon each other de- pend not upon their absolute but relative mo- tion. See Force, Power, Motion, &c. AcTION, quantity of, in Mechanics, an ex- pression used by Maupertuis in the Mem. of the Acad. of Sciences of Paris for 1744, and in those of Berlin for 1746, to denote the con- tinual product of the mass of a body, by the space through which it runs, and by its cele- rity. He lays it down as a general principle, that ‘whenever any changes happen in nature, the quantity of action necessary to produce this change is always the least possible.’ And this, he says, is a law indicating the highest wisdom. ‘This principle he applies to the ix- vestigation of the laws of refraction, the laws of collision of hard and elastic bodies, and even the laws of rest, as he calls them; that is, of the equilibrium of, or equipollency of pressures: and thus investigating the laws of motion, referring these and the laws of equi- librium to the same principle, and connecting the metaphysical consideration of final causes with the fundamental doctrine of‘ mechanics, he deduces what he conceives to be a stronger proof of the existence of a Deity, or of a first intelligent cause, than the other arguments commonly alleged, and derived from the order of nature. It may be observed, however, that the quan- tity of action, according to the definition of Maupertuis, is in reality the same with the product of the mass into the square of the celerity, when the space passed over is equal to that by which the celerity is measured ; and therefore the force or quantity of motion will be proportional to the mass into the square of the velocity ; since the space is measured by the velocity continued for a cer- tain time. In the same year that Maupertuis commu- nicated his principle, Euler also demonstrated, in the supplement to his Treatise, entitled “‘ Methodus inveniendi Lineas curvas,” &c. ; that in the trajectories, described by bodies urged by central forces, the velocity multi- plied by what the foreign mathematicians call the element of the curve, is always a minimum ; which Maupertuis considered as an applica- tion of his principle to the motion of the planets. ACTIVE, something that communicates motion or action to another. In this sense the word stands opposed to passive. Thus we say, an active cause, active principles, &c. ACUBENE, in Astronomy, a name some- times given to a star of the fourth mag- nitude, in the claw of Cancer; marked « by Bayer. ACUTE, or SHARP, a term opposed to ob- ADD tuse; thus we say, acute-angled triangle, acui angled cone, &c. AcutE Angle. GLE, &e. AcutE Angular Section of a cone, an t lipse, a term applied by the ancient geomet to the plane cutting both sides of an acui angled cone; but they did not consider, bef it was pointed out by Apollonius, that such section might be obtained in any cone whi ever. See Wallis’s Oper. vol. i. ADAMANT, a term sometimes given the magnet, or loadstone. ADAR, in the Hebrew Chronology, is t 6th month of their civil year, but the 12th their ecclesiastical year. It contains only! days, and it answers to our February; b sometimes it enters into the month of Mare according to the course of the moon. ADDITION, the joining or uniting of ty or more things together; or the finding of 0) quantity equal to the sum of any number other things taken together. ADDITION, in Arithmetic, is the first of tl four fundamental rules of that science; ai consists in finding a number equal to sever others taken together; which number is ge rally called the sum. Addition is either sit ple or compound; the first relates to qua tities which are all of the same kind, or den mination; and the latter, to quantities of d ferent denominations. Simple ADDITION, is the method of colle¢ ing several numbers of the same denomin tion into one sum. . Rule 1. Place the numbers under eay other, so that units may stand under uni tens under tens, hundreds under hundre¢ and so on; drawing a line below them. Add up the figures in the row of units, al find how many tens are contained in tht sum. 3. Set down what remains above t tens, or if nothing remains, a cipher; al carry as many units, as there are tens, to t next row. 4. Add up the second row, tog ther with the number carried, in the sar manner as the first; and proceed thus till f whole is finished. ) See ANGLE, Cone, TRIA EXAMPLES. 7468 2468 3417 1357 5432 9135 6847 2468 4132 7523 1489 6814 28785 Sums 29765 ae ASCE Proof of Addition, Cast up the seve; lines a second time, omitting one of the and, finally, add the line thus omitted to t sum of the other quantities; and if in tl manner the same result is obtained as in t first operation, it is highly probable that work is right. Other methods of proof m be seen in Bonyeastle’s Arithmetic; and m«¢ authors on this subject. 4 Compound Appi7Tion is the method of © ADD ding several quantities of different denomi- tions into one sum. Rule 1. Place the quantities so that those the same denominations stand directly un- r each other; and draw a line below them. Add up the figures in the lowest denomi- tion, and find how many units of the next ‘her denomination are contained in their n. 3. Write down the remainder, and ‘ry these units to the next higher denomi- tion; which add up in the same manner as fore: and thus proceed through all the dif- ent denominations to the highest; and this n, together with the several remainders, il give the answer required. EXAMPLES. a ae dad. OSs Wks 1D. 8413 4 ty eal Eek § | 64 9 115 16 2 9 36 14 9! 13.1 725 1715 72 14 0 63 1913 64 20 0 0 223 7 3 Sums 82 0 104 Che method of proof is the same as in the mer rule. Appition of Fractions is the adding into » sum several fractions of the same, or of ferent denominations. Rule 1. If the fractions are of different de- ninations, they must be first reduced to same denomination. 2. Reduce all mix- ‘numbers to improper fractions; and all tions, having different denominators, to same denominator. 3. Add all the nume- ors into one sum; which, placed over the nmon denominator, will be the answer uired. . EXAMPLES. . Add together the fraction 4, 4, and 3. First, $= 2; 1-10 pT eae ear 35 14. 10__59 then, 75 t as 707 70° . Required the sum of 3 of a shilling, 2 of uinea, and Z of a pound. 3 of a guinea = of a shilling, # of a pound = ~ of a shilling ; 63 , 35 189 , 210 Pat ao tees 8 pete |S pital Cherefore, 3 + 7 + 5 =8+ is + 3 ‘or the method of the reducing the frac- is, see REDUCTION. DDITION Of Decimals, is finding the sum several quantities, consisting partly of in- ers and partly of decimals, or of decimals y. (tule 1. Arrange all the quantities so that several decimal points may fall in a line setly under each other. 2. Add up the Feral lines as in simple addition, and place ‘decimal point in the result exactly under ADD those in the numbers proposed, and it will be the sum required. EXAMPLES, 1, Find the sum of 34.17; 19.143; 167.18; and 143.5. 2. Find the sum of .1176; .1344; .746; and 1468. ist Exam, 2d. Exam, 34.17 -1176 19.143 1344 167.13 :746 143.5 1468 © 363.943 Sums 1.1448 Appition of Circulating Decimals, is the finding the sum of any number of circulating decimals. Rule 1. Reduce all the decimals to their equivalent fractions; and the sum of these will be the answer required. Rule 2. Carry on the repetends till they be- come conterminous; that is, till they all begin their periods of circulation in the same linc, and let also the circulation of each be carried two figures beyond this place; then add them up as in the former rule, observing not to set down any thing in the first two places; only carry the proper number from them to the conterminous period; and the result will give the true period of circulation in the sum required, EXAMPLE. Add3.6; 78.3476; 42.84; and 15.5 together, 3.6 = 3.6666666 | 66 78.3476 = 78.3476476 | 47 42.84 428484848 | 48 15.5 = 15.5000000 | 00 The sum...... 140.362799)} In this example, the periods of circulation do not commence together till after the seventh place of decimals; they are then carried on two places further, in order to ascertain what ought to be carried to the conterminous period; which in the present case is 1, as will appear from the above operation. Note. ‘There may arise cases in which it will be necessary to carry the circulation on to three or more places beyond the conter- minous period, but in general two or three places are sufficient. ADDITION, in Algebra, is finding the sum of several algebraical quantities, and connecting those quantities together with their proper signs. And this is generally divided into the following cases. | Case 1. When all the indeterminate letters are the same, and affected with the same sign. Rule. Add the co-efficients of the several quantities together, and prefix before the sum the proper sign, whether it be plus or minus. EXAMPLES. + 7a+ 36 + 5x—13y + 5a+ 4b + 4x—lly + 6a+11b +142r— 7y +18a¢+186 Sums +232—3ly ADD Case 2. When the quantities are ihe same, but affected with different signs. Rule. Add all the like quantities together that have also the same sign, and thus two separate sums will be obtained; then subtract the less of these from the greater, and prefix before the remained the same sign as belong to the greatest sum. EXAMPLES. +7ab— 8c? —4ry+ 323 —6ab+ 7e* +6ry— 1123 +4ab—13c? + 4ry7— 1423 +5ab— 9c* Sums +62y—2223 Note 1. When the leading quantity of any algebraical expression has no sign, it is sup- posed to be affected with the sign +. Note 2. Unlike quantities can only be add- ed by means of the sign + placed between them. Appition of Algebraic Fractions, is find- ing the sum of any number of fractions ex- pressed by means of indeterminate letters. Rule. Reduce all the given fractions to a common denominator, and then add the nu- merators together for the -sum required. EXAMPLES. ba», oa 5e 1. Add together AG + 0 + , 5e _ 5e 66> 1203’ 4b — 1263’ 1265 — 1203 10ab , Ga?b* | 5e _ 10ab4+ 9a*h? +5e 1263 1263 1253 1263 t 2. 'Thus also we find, Va Av Ble 44x Goxr Thy "By B8y " BBy— 3387 7x+4y | d5x—3y__TVexr+4ey Tt yd nBabareadidabe da unRlbeon 15x—9y _ Tex + 15x 4+ 4ey—9y _ 3abe 3abe Hi (7e + 15) a + (4e—9)y 3abe : ADDITION of Surds, is the finding the sum of any number of surd expressions, which are inexpressible in rational numbers. Rule. Reduce all the given quantities to their most simple form; then add together the co-ellicients of the equal radicals for the sum required, EXAMPLES. Thus, / 8+ /18=2V243V72=—5y2 / 12+ /27—2/3 +373 =—by3 AY 108a* +4) 32a Bay/ 4a +247 4a = (3a + 2)/4a. vote. When the quantities are reduced to their lowest terms, or simplest form, and have different radicals, they can only be added to- gether by means of the sign + placed between them. Thus, 418+ /108=38 /2+6,//38, can- not be reduced to a simpler form than that above; and the same with various others. ADH AppiITion of Latios, is the same as e¢ position of ratios ; thus, ifa:b::¢:d; then by addition, or composition, a+b:art:e+d:c or,a+b:6::¢+d:d. See Ratio ADDITIVE, denotes something to bea ed to another, in contradistinction to soi thing to be taken away, or subtracted. TI astronomers speak of additive equations ; ; gceometricians, of additive ratios, &c. Ke. ADERAIMIN, or ALDERAIMIN, a n given to « Cepheus. ADYECTED, or Arrectep Equation, Algebra, is that in which the unknown q tity is found in two or more different grees or powers; thus, 77—pa*+4+qaxma an adfected equation, having three diffe powers of the unknown quantity x enter int¢ composition. ‘The term affected is also so. times used to denote a certain condition an indeterminate quantity; such as in the pressions 7x, =k 38a, &c.; we say that ; affected with the co-efficient 7; and in: latter, that the quantity, 3a, is affected y the ambiguous sign +, &c. This term, adj ed, is said to have been introduced into alge by Vieta. ‘ ADHESION, in Philosophy, (adhesio, fi adherio, to stick to), is a species of attrae which takes place between the surface; bodies, whether similar or dissimilar, which, in a certain degree, connects tl together; differing in this respect from hesion, which, uniting particle to particle, tains together the component. parts of same mass.. It has been proved that power of adhesion is proportional to then ber of touching points, which depends u the figures of the particles that form bodies ; and in solid bodies, upon the dey in which their surfaces are polished and ¢ pressed. The effects of this power are tremely curious, and in many instances tonishing. Musschenbroek relates that | cylinders of glass, whose diameters were quite two inches, being heated to the s; degree as boiling water, and joined toge by means of melted tallow lightly put betw them, adhered with a force equal to pounds: lead, of the same diameter an similar circumstances, adhered with a f of 275 pounds; and soft iron with one of pounds. (Musschenbroek’s Philosophy, Colson.) And Martin, in his Philoso Britannica, vol. i. says, That with two lea balls, not weighing above a pound each, touching upon more than one-thirtieth square inch surface, he has lifted more 1 150 pounds weight. The balls were planed very finely with the edge of a sl penknife, and then equally pressed toge with a considerable force and a gentle | of the hand. The force of adhesion bet) two brass planes, each four and a qué inches diameter, and smeared with gre fat, was so great, that Mr. Martin as that he never could meet with two men sti —= ADH nough to separate them by pulling against tach other. ‘These instances are suflicient to ive an idea of the nature of this power. ‘hose who wish for more experiments may nd them detailed in the two works above sferred to, and in Rees’s Cyclopedia, under 1e article Adhesion. ‘To measure the force f adhesion between different substances, and 1 different temperatures and circumstances, arious methods have been contrived; but the estis that which was suggested by Dr. Brook ‘aylor, whose experiments led him to conclude 1at the force of adhesion might be determined y the weight necessary to produce a separa- son, and which has since been pursued and xtended by M. de Morveau (now M. Guyton), ith considerable success. He constructed ylinders of different metals, perfectly round, ninch in diameter, and the same in thickness, aving a small ring in their upper surface, by hich they might be hung exactly in equili- rium. ‘These cylinders were suspended, one ‘ter another, to the beam of a balance; and hen they counterpoised exactly, were ap- lied to mereury, placed about one-sixth of n inch below them. After sliding them long the surface, to prevent any air from dging between them and the mercury, he iarked exactly the weight necessary to over- ome their adhesion; taking care to change 16 mercury after every experiment. ‘The ‘sults were as follow : | ; Grains, Gold adheres to mercury, with a force DE i shes Siig’ tyes Vedi sisSel sp -pebseieesuey 448 DALVET ......+ teidsh babins eaxds ine oes és ike ies 420 SAT an 8S vs chirys pein VES oc cbsesce scosdch ALS SY, BP eae eee ree site «ubivude ene g, MASISINUEN o..ccccceccece Stattads = abaeeeder mavens O82 MPALING *......ci000 wehsbblach Wb iuhsawesdatesse 202 eons dala chin stale ewhias yds ash «pudailcsane OE BE 2051. ate hietin beniros ns de series ohfeeoey), CAPD MR oh Dix ets vi Fale ipase ovsians ds sates 126 This method which, when it can be applied, | the most direct and accurate of any that 1s yet been devised, has been pursued to Ul greater lengths and degrees of nicety, by . Achard, and others; whose experiments iw limits will not allow us to detail. From hat has been done altogether, we may de- ice the following conclusions :—That there ‘ist a tendency to adhesion between many, jd probably between all substances in na- | re, absolutely independent of atmospherical any other external pressure; that the forec ) this adhesion between solids, is in the order their chemical affinities ; and between solids id fluids, in an inverse ratio to the thermo- rical temperature, and a direct ratio to e squares of the surfaces; that every solid (heres with a peculiar force to each fluid; at this force is expressed by the weight cessary to break the adhesion, in all cases aere the solid comes out clear from the id; but that whenever any particles of the JE.O.L fluid adhere to the solid, the weight of the - counterpoise is then expressive of the mixed forces of the adhesion between the surfaces of the solid and the fluid, and of the cohesion between the component parts of the fluid. ADHIL, in Astronomy, a star of the sixth magnitude, in the garment of Andromeda, under the last star in her foot. ADJACENT Angle, in Geometry. ANGLE. ADJUTAGE. See Asurace. ADSCRIPTS, an ancient mathematical Sce term for tangents of arcs; called also, by. Vieta, prosines. JEGOC EROS, or ASGoceRUvUS, a name some- times given to the constellation Capricorn. JEOLIPILE, A@oLiriLa, an instrument consisting of a hollow metalline ball, with a slender neck or pipe arising from it. This being filled with water, and thus exposed to the fire, produces a vehement blast of wind. This instrument, Des Cartes and others have made use of to account for the natural cause and generation of wind; and hence, its name Alolipila, that is, pila Aoli, Avolos’s bali, feolus being reputed the god of the winds. The best way of fitting up this instrument, is to have the neck screw off and on, for the convenience of introducing the water into the inside ; for by unscrewing the pipe or neck, and immersing the ball in water, it readily fills, the hole being pretty large; and then the pipe is screwed on. But if the pipe do not screw off, its orifice is too small to force its way in against the included.air. The water being thus introduced, the ball is set upon - the fire, which gradually heats the water, and converts it into elastic steam, which rushes out of the pipe with a great violence and noise; and thus continues till all the water is discharged ; though not with a constant uni- form blast, but by sudden gusts: and the stronger the fire is, the more elastic will be the steam and the force of the blast. Care should be taken that the ballis not set upon a violent fire with very little water in it, and that the small pipe be not stopped with any thing; for in such cases the included elastic steam will burst the ball with a very dangerous ex- plosion. ‘This instrument was known to the ancients; being mentioned by Vitruvius, lib. i. cap. 6. It is also treated of, or mentioned, by several modern authors, as Des Cartes, Father Mersennus, Varenius, &e. In Italy, it is said the /®olipile is often used to prevent chimnies smoking ; for being hung over the fire, the blast arising from it carries up the loitering smoke along with it. And some have imagined that the Aolipile might be employed as bellows to blow the fire ; but experience would soon convince them of their mistake ; for it would rather blow the fire out than up, as it is not air, but rarified water, that is thus violently blown Y/ through the pipe. JLOLUS, in Mechanies, a small porta machine, invented by Mr. Tidd, for refresh4 and changing the air inrooms. This ma fi RO ‘ys adapted in its dimensions to supply the place of a square of glass in a sash window ; and is executed in so small a compass, as to project but a little way from the sash ; and in so neat a manner, says the inventor, as to be ‘an elegant ornament to the place where it is fixed. It works without the least noise, re- quires no attendance, and occasions neither trouble nor expense to keep it in order. It throws in only such a quantity of air as is agreeable; and leaves off working of itself whenever the door or window is opened. ~ AOLUS’S Harp, an instrument so named from its producing agreeable harmony, merely ~by the action of the wind. It is thus con- structed: Let a box be made of as thin deal ‘as possible, of the exact length answering to the width of the window in which it is in- tended to be placed, five or six inches deep, and seven or eight inches wide; Jet there be glued upon it two pieces of wainscot, about half an inch high and a quarter of an inch thick, to serve as bridges for the strings; and, within side, at each end, glue two pieces of beech, about an inch square, of length equal to the width of the box, which are to sustain the pegs; into these fix as many pins, such as are used in a harpsichord, as there are to be strings in the instrument, half at one end and half at the other, at equal distances: it now remains to string it with small catgut, or blue first fiddle strings, fixing one end to a small ‘brass pin, and twisting the other round the opposite pin. When these strings are tuned in unison, and the instrument placed with the strings outward in the window to which it is fitted, it will, provided the air blows on that window, give a sound like a distant choir, increasing or decreasing according to the strength of the wind. JERA, in Chronology, is the same as Epoch, and means a fixed point of time, from which to begin a computation of the succeeding years. /ERA also means the way or mode of ac- counting time. Thus, we say, such a year of the Christian @ra, Ke. Christian Ansa. It is generally allowed by chronologers, that the computation of time from the birth of Christ, was/only introduced in the sixth century, in the reign of Justinian; and is generally ascribed to Dionysius Exiguus. See Epocn. JERIAL, Perspective, is that which repre- sents bodies diminished and weakened in pro- portion ‘to their distance from the eye ; but it relates principally to the colours of objects, which are less distinct the greater the dis- tance at which they are viewed. JEROGRAPHY, (from anp, air, and ypagu, I describe) a description of the air or atmos- phere, its limits, dimensions, properties, &c. JEROLITHS, (from ang, the air, and Arbos, a stone) air stones, a name lately given to those solid semi-metallic substances which hil from the atmosphere. The descent of Stch bodies have been long reported ; but the AC was not considered as authentic till with- AL RO in a few years, during which it has been | well established, that “there is now no lon any doubt on the subject, several cases havi been attested by eye-witnesses of undeniab veracity. The larger sort of these stones ha. been seen as luminous bodies, to move wi great velocities, descending in oblique dire tions, and frequently witha ‘loud hissing nois’ resembling that of a mor ar-shell when pr jected from a piece of ordnance; they a sometimes surrounded with a blaze or flan tapering off to a narrow stream at the hind part, are heard to explode, and seen to | in pieces. ‘The velocity with which th strike the earth is very great, frequently p netrating to a considerable depth, and whe taken up have been, in some cases, found be still hot, and bearing evident marks recent fusion. Sometimes such falls hay happened during a storm of thunder ar lightning, at others, when the sky has bee clear and serene; whence one may infer th these phenomena are unconnected with tl state of the atmosphere. One of the most remarkable and disth guishing characteristics of these stones is, th they ‘perfectly resemble each other; at th same time, that they are totally different fro! any known terrestrial substance; and pri sent, in all cases, the same appearance of sem metallic matter, coated on the outside with thin black incrustation. It*‘is not, however, appearance only that their similarity is to I discovered; they have been submitted to th most accurate chemical analysis, in all cas have yielded the same substances, and in ve nearly the same proportions. The — stor which fell at l’Aigle in France, in 1803, we found to contain 54 parts of silica — BE in ccecene oxyde of iron D+ seinassiss . magnesia 3 .seeeee OXYAE of nickel Q eis sesestia sulphur bes esisssties lime And all that have been analized have give nearly the same results. The specific gravity of these bodies is als found to be nearly the same, much exceedin that of common stone, and approaching th of metallic ores, being about 3.400, or nearl 3% times that of common water. These general and constant characte) strongly indicate a common origin, and vi rious hypotheses have been advanced to at count for these singular phenomena. Som have attributed them to terrestrial, and othe to lunar, volcanoes ; they have again been su posed to be concretions actually formed the regions of our atmosphere; while othe have considered them as small planets ¢ culating about the sun or earth, which comin in contact with our atmosphere, take’ fir from the resistance and friction they expe rience in passing through it. Our limits will not admit of a very minut examination of these several hypotheses, - few remarks however will not be uninteres sl ‘ JE RO ing. With regard to the first supposition, viz. that these stones proceed from terrestrial vol- canoes, it will be sufficient to observe, that no remarkable eruption has been known to have happened at or near the time of their fall, and that-such bodies have been found at the dis- tance of some thousand miles from any known voleano; beside the immense force that would be necessary to j/*oject bodies, some of them of many hundred weight, to so great a dis- tance, far exceeds any force that we can con- ceive to arise from volcanic eruptions. As to the lunar theory, that is, that they proceed from yolcanoes in the moon, it has certainly a much greater degree of probability. The same force that would project a body from the moon to the earth, would not, if it were exerted at the earth’s surface, send the same body to the distance of ten miles; in conse- quence of the superior gravity of our planet, and the density of the atmosphere. It is readily computed, that a body projected from 2 favourable spot on the moon’s surface, that 's, from the centre of her disc opposite the sarth, with a velocity about four times what $s commonly given to a cannon ball, or, wubout 8220 fect per second, would carry it seyond the centre of attraction, and conse- uently into the sphere of the earth’s activity; whence it must necessarily either fall to our surface, or circulate about us as a satellite. Che time that a body, so projected, would be nits passage from the moon to the earth, is uso found to be three days; which is not so ong but that it might retain its heat, particu- arly as it is doubtful whether in passing through a vacuum, or very attenuated medium, t would be possible for the caloric to escape, 10t to say that it might acquire afresh accumu- ation of heat, by passing through the denser arts of our atmosphere. Add to this, that sruptions resembling those of our volcanoes ave been frequently observed in the moon, __ these phenomena. Substances, Places where they fell. shower of stones ........ Shower of stones..... .. at Rome....ccccosesees Shower of iron. ........6. 1 Lacania...cccccceees : -‘Thrace..... 1G Large stone of 260]b. ethan ERO and that her atmosphere is extremely rare, and consequently presenting but little resistance to projected bodies; and it will then be seen that this hypothesis, though at first sight appa- rently extravagant, is notwithstanding much more probable than the one we before exa- mined, It must be acknowledged, however, that the explosions of which we have spoken, are difficult to account for upon this sup- position. With regard to these bodies being concre- tions formed in the air, there is one principal objection, viz. that the velocity with which they strike the earth, estimated by the depth to which they have been known to penetrate, is so great as to indicate their having fallen from heights far exceeding the limits of the terres- trial atmosphere. It only remains now to make afew remarks concerning the supposition of their being little planets, which in the course of their revolu~ tions get entangled in our atmosphere; and by the resistance they experience, have their projectile velocity destroyed, which brings them to the earth. On this hypothesis it is very difficult to account for the strong ap- pearance of recent fusion that these bodies possess, which can hardly be attributed to the friction experienced in passing through the medium of our air. Beside, it would ex- hibit the first mark, which any where appears, of want of order, uniformity, and durability, in the works of the Deity: to have this grand machine thus falling to pieces and to decay, from the mere operation of external causes, but ill accords with that wisdom, power, and omniscience which are so strongly indicated in all the works of the creation. » It will be seen from the preceding remarks, that we incline towards the lunar theory; the difficulties, however, are fairly stated, and the reader of course will draw his own cons clusions. . The following is an abstract of a Table, drawn up by M. Izarn, a philosopher, who has paid great | attention to this subject; it exhibits a collection of the best authenticated modern instances of Period of their fall. Testimony, AL ROME ...4.0.scsccsieders underTullius Hostilius, Livy. Consuls, C. Martius and M. Torquatus \ J. Obsequens. year before the at Pliny, feat of Crassus..... Shower of mercury...... in Italy ..........c0sceeee oe TBR DOWD ini 10a: e050 .. Dion. near the river Negos, : second year78Olym- 2 Pliny 7 d e OC eessteoseresesesere AER AER Substances. Places wheré they fefl. Period of thoir fall. Testimony, reilly serie em in the Atlantic.......0.. April 6, 1709.,....000-. Pére la Feuillée. Shower of stones...... ' Paces near Royy y WY, A teisiasta rue ninast ... Darcet, jun. Lomet,&¢, Shower of stones......... at Plann, Bohemia..... July 3, 1753 ...........008 B. de Born. ¢ A stony MASS.........0000 Niort, in Normandy.... in 1750 ..........s0000+088. De Lalande. pa. oF200nnd 2. is cay Verona «.ansiset dest 1762 ......00 sean hibee se Acad. de Bourd. A stone of 72 Ib............ at Luce, in Le Maine.. September 13, 1768 .... Bachelay~ PORONO Wide wide isd. .. at Aix, in Artois......... ity ie | rurson de Boyavail. PPBTONG . cnsesensn cesses ..- In LapContentin......... in 1766 .2iids,..s-....-.....: Morand. . Great shower of stones.. Environs of Agen....... July 24, 1790..........085 St.Amand, Bandin, ke, Twelve stones ........000 Sienna, ‘Tuscany........ ULV, 1794... cceseeeeceee .. Earl of Bristol. x A stone of 56]b....... .... Wold Cottage, Yorks. December 13, 1795..... Captain Topham. A stone of 10Ib.......... in Portugal............«. February 19, 1796...... Southey. ‘ v Salé, Department of t Rie ae age om : A stone of 20Ib........ § the Rhone.........,,¢ areh 17, 1798.......... De Drée. Shower of stones......... Benares, East Indies.. December 19, 1798..... Lloyd Williams, Esqd d Salés, “near. Ville? pi ek. rs A stone of es anise $ |. Frariches) aeaeucat March 12, 1798.......... De Drée. | Mass of iron, 70 cub. ft.. Ameriea........ te EL . April 5, 1800.............. Philosoph. Magazine, Several stones, from) near VAigle, Nor-) 4. . j 10 to 17ib...... swe ; mandy spi oe Sms ie Apnil 26, 1803........-.. Foureroy. | AEROLOGY, (from anp and Aoyos, dis- course) the doctrine or science of the air, and its phenomena, properties, good‘and bad qua- lities, &c, AEROMETRY, (from np, wir, and werpov, measure) the art of measuring the: air, its powers and properties ; including the Jaws of motion, gravitation, pressure, elasticity, re- fraction, condensation, &c. of the atmosphe- rical fluid. See PNEUMATICS. AERONAUT, a person who sails or navi- gates through the air. AERONAUTICA, the pretended art of sail- ing through the air, or atmosphere, in a ves- sel, sustained as a ship at sea. AEROSTATICA, the same as AEROMETRY. AEROSTATION, (from enp, air, and carixn, of isnus, L weigh, the science of weights), in its primitive and proper sense, denotes the science of weights suspended in the air;-but in the modern application of the term, it sig- nifies the art of navigating through the air, both in the principles and practice of it. Hence also the machines, which are employed for this purpose, are called @rostats, or eros- tatic machines; and on account of their round figure, air-balloons. And the person who navigates them is called an @ronaut. The Principles of Aerostation—The fan- damental principles of this art have been long known; although the application of them to practice scems to be altogether a modern dis- covery. These chiefly relate to the weight, pressure, and elasticity of the air, its. specific gravity, and that of the other bodies to be raised or floated in it; the particulars of which will be found under the articles Air and Ar- MOSPHERE: it will therefore be sufficient, in this place, to observe, that any body which is specifically, or bulk for bulk, lighter than the atmospheric air encompassing the earth, will be buoyed up by it, and ascend; the same as wood, or cork, ascends in water; but as the 4 density of the atmosphere decreases, on @ count of the diminished pressure of the super incumbent air, and the elastic property whiel it possesses at different elevations above thi ‘earth, this body can rise only to a height ii which the surreunding air will be of the sam specific gravity with itself: in this situation ]| will either float, or be driven in the directioi} of the wind or current of air to which it i} exposed. An air balloon is a body of thi kind, the whole mass of which, including th -covering, contents, and appendages, is of les specific gravity than that of the air in whid “it rises. Heat is well known to rarify and expan and consequently to lessen the specific gravit of the air to which it is applied; and the di minution of its weight is proportional to th heat. According to the scale of Fahrenheit thermometer, heat seems to expand the ai about one four-hundredth part; and 400, ¢ rather 435, degrees of heat, will just doubl the bulk of a quantity of air. If, therefor the air enclosed in any kind of covering b heated, and consequently dilated, to such degree as that the excess of the weight of a} equal bulk of common air above the weigl of the heated air, is greater than the weigl] of the covering and its appendages, this whol mass will ascend ‘in the atmosphere ; till b the cooling and condensation of the include air, or the diminished density of the surrount ing fluid, it becomes of the same specific gr -vity with the air in which it floats, and witl out renewed heat it will then gradually di scend. If instead of heating common ai inclosed in any covering, and thus diminisl ing its gravity, the covering be filled with a elastic fluid lighter than atmospheric air, § that the excess of the weight of an equal b of the latter, above that of the inclosed elast fluid, be greater than the weight of the cove ing and its appendages, the whole mass Wi ¥ ey | td a AE ROSTATIO "Plate i | o sxssstncentsnsareitta ace a tte i tit NN Ae CN ALE ALLL LLL ILO AL LL AL ae See ee ee “s _——, a ce ty a a ap ew renege nn ene a A ON —_~ 89 eE—eEe———e————————— ace a eee See es HH] | | | i, | | | ! ig | i | ' | iD 2 Ne / | TV ee ~ “wa Ahh i, H = a cg — 3) 4 —— ee ee -_-— —— — —- Engraved by Sam# Lacey - 31,2813, bv Xs Robinson, Faternoster Row, kth rest of the Propreers. ‘ + mak #) a ? oO NOEL F LS a ws ascend in the atmosphere, as in the former ease, and continue to rise in it till it attains an altitude at which the surrounding air is of the same specific gravity with itself—For the knowledge of many of the properties of in- flammable air, which is a fluid of this kind, we are much indebted to modern philosophers, particularly to Priestley and Cavendish, the latter of Whom made many important disco- veries relating to this subject, which the limits of this work will not allow us to enter upon ; we must therefore refer the reader to the 66th yol. of the Phil. Trans. in which he will find an interesting paper on this subject by the gentleman above mentioned. See also Ca- vallo’s History and Progress of Atrostation, 8vo. 1795. History of Aerostation. In drawmg a sketch of the history of this interesting art, the reader will not expect that we can enter much into the detail of the numerous aerial voyages that have of late years been under- taken and successfully accomplished: indeed, if our limits would admit of such a detail, it would not afford the reader a proportionate | degree of interest, in consequence of the same- ness which necessarily attend exploits of this description. The terrestrial navigator gives us but little information till he sets foot on jand: when he begins to examine the pro- ductions of the country, the manners of its inhabitants, and the nature of its soil, then it is that our curiosity is awakened, and the re- cital of his adventure becomes interesting and important. But the aerial traveller has no such information to conyey: the harmonious silence of the upper regions, the grandeur and novelty of the prospect,—cities, towns, and villages, reduced by distance to the repre- sentation of a map,—the singular appearance of dense and heavy clouds rolling beneath his feet, illuminated by the solar rays, are cer- tainly subjects calculated to excite our ad- miration and astonishment; but when once related, they admit of no further variety than that which is supplied by the imagination of the writer. We shall therefore confine our attention principally to the first attempts of terial navigation, and a few of the voyages which have furnished the greatest information m this theory. The idea of flying or passing through the itmosphere, by means of wings and other con- rivances, was certainly entertained by the meients; and some accounts relate exploits if this kind having been performed ; but still here is every reason to suppose they are mere ctions, and that no means were ever pos- essed for accomplishing this undertaking till § he invention of balloons, which dates no fur- her back than the conclusion of the last cen- ary. Soon after Mr. Cavendish’s discovery f the specific gravity of inflammable air, it ceurred to the ingenious Dr. Black, that if ® bladder sufficiently light and thin were filled ‘ith this air, it would form a mass lighter van the same bulk of atmospheric air, and seimit. This thought was suggested in his os AER lectures, in 1767 and 1768; and he proposed; by means of the allantois of a calf, to try thé experiment. Other employments, however, prevented the execution of his design. The possibility of constructing a vessel, which when filled with mflammable air would ascend in the atmosphere, had occurred also to Mr. Cavallo, and to him belongs the honour of having first made experiments on this sub- ject, in the beginning of the year 1782, of which an account was read to the Royal So- ciety, on the 20th of June in that year. He tried bladders; but the thinnest of these, how- ever cleaned and scraped, were too heavy. In using China paper, he found that the in- flammable air passed through its pores like water through a sieve; and having failed of success by blowing this air into a thick solu- tion of gum, thick varnishes, and oil paint, he was under a necessity of being satisfied with soap balls, which, being inflated with inflam- mable air, by dipping the end of a small glass tube connected with a bladder containing the air into a thick solution of soap, and gently compressing the bladder, ascended rapidly in the atmosphere ; and these were the first sort of inflammable air-balloons that were ever made. But while aerostation seemed thus on the point of being made known in Britain, it was unexpectedly announced in France by two bro- thers, Stephen and John Montgolfier, natives of Annonay, and masters of a considerable paper-manufactory there, who had turned their thoughts to this project as early as the middle of the year 1782. Their idea was to form an artificial cloud by inclosing smoke in a bag, and making it carry up the covering along with it. In that year the experiment was made at Avignon with a fine silk bag; and by applying burning paper to an aperture at the bottom, the air was rarefied, and the bag ascended to the height of 70 feet. Various experiments were now made upon a large scale, which excited the public curiosity very greatly. An immense bag of linen, lined with paper, and containing upwards of 23,000 cubic feet, was found to have a power of lifting about 500 pounds, including its own weight. Burning chopped straw and wool under the aperture of the machine, immediately oeca- sioned it to swell, and afterwards to ascend into the atmosphere. In ten minutes it had risen 6000 feet; and when its force was ex- hausted, it fell to the ground at the distance of 7668 feet from the place it ascended.—Soon after this one of the brothers, invited by the Academy of Sciences to repeat his experiment at their expense, constructed a large balloon of an elliptical form; and in a preliminary ex- periment, this machine lifted from the ground eight persons who held it, and would have carried them all off, if more had not quickly come to their assistance. Next day the ma- chine was filled by the combustion of 50 pounds of straw and 12 pounds of wool, The balloon soon swelled and sustained itself in ihe air, together with the burden of between C2 AER 400 and 500 pounds weight. It was designed to repeat the experiment before the king at Versailles; but a violent storm of rain and wind happening to damage the machine, it became necessary to prepare a new one; and such expedition was used, that this vast bal- toon, near 60 feet in height and 43 in diameter, was made, painted within and without, and finely decorated, in the course of four days and four nights. Along with it was sent a wicker cage, containing a sheep, a cock, and a duck, which were the first animals ever sent on such a voyage. The full success of the experiment was, however, prevented by a violent gust of wind, which tore the machine in two places near the top, before it ascended. Still it rose 1440 feet; and after remaining in the air about eight minutes, fell ‘to the ground, at the dis- tance of 10,200 feet from the place ofits ascent, with the animals in perfect safety.—As the great powers of those aerostatic machines, and their very gradual descent, showed they were capable of transporting people through the air with all imaginable safety, M. Pilatre de Ro- zier offered himself to be the first aerial adven- turer, in a new machine constructed in a garden in the Fauxbourg of St. Antoine. It was of an oval shape, 48 feet in diameter and 74. in height, elegantly painted with the signs of the zodiac, ciphers of the king’s name, and other ornaments. A proper gallery, grate, &c. enabled the person who ascended to supply the fire with fuel, and thus keep up the ma- chine as long as he pleased: the weight of the whole apparatus was upwards of 1600 pounds. On the 15th of October, 1783, M. Pilatre placing himself in the gallery, the ma- chine was inflated and permitted to ascend to the height of 84 feet, where he kept it afloat about four minutes and a half; after which it descended very gently: and such was its tendency to ascend, that it rebounded to a considerable height after touching the ground. On repeating the experiment, he ascended to the height of 210 feet. His next ascent was 262 feet; and in the descent, a gust of wind having blown the machine over some large trees in an adjoining garden, M. Pilatre sud- denly extricated himself by throwing straw and wool on the fire, which raised him at once to asufficient height ; and on descending again, he once more raised himself to a proper height by the same means. Some time after he ascended with M. Girond de Vilette to the . height of 330 feet; hovering over Paris at least nine minutes in sight of all the inhabitants, and the machine keeping all the while a steady position. ‘These experiments having shown, that aerostatic machines might be raised or lowered at the pleasure of the persons who ascended, M. Pilatre and the Marquis d’Ar- landes, on the 21st of November, 1783, under- took an aerial voyage, which lasted about 25 minutes, and during which time they passed over a space of above five miles. From the account given by the Marquis, they met with _ Several different currents of air, the effect of which was to give a very sensible shock to AER the machine, and the directions of the motion seemed to be from the upper part downwards. It appears also, that they were in some danger of having the balloon burnt altogether; as the Marquis observed several round holes made by the fire in the lower part of it, which alarmed him considerably, and indeed not without reason. However, the progress of the fire was easily stopped by the application of a wet sponge, and all appearance of danger ceased. This voyage of M. Pilatre and the Marquis may be said to conclude the history of those aerostatic machines which were elevated by means of fire; these having been soon after superseded by balloons in which inflammable air was enclosed. This gas being consider- ably lighter than heated atmospheric air, pos- sessed many adva; »ges over the other. ‘The. first experiment was made by two brothers, the Messrs. Roberts, and M. Charles, a pro- fessor of experimental philosophy. The bag was composed of lutestring varnished over with a solution’ of the elastic gum called caoutchouc ; and was about 13 English feet in diameter. Many difficulties occurred in filling it with the inflammable air; but being at last set at liberty, after having been well filled, it was 35 pounds lighter than an equal bulk of common air. It remained in the at- mosphere only three quarters of an hour, dur- ing which it traversed 15 miles. Its sudden descent was supposed to have been owing to a rupture which had taken place when if ascended into the higher regions of the atmo- sphere. The event of this experiment, and the aerial voyage made by Messrs. Rozier and Arlandes, naturally suggested the idea of um dertaking something of the same kind with ¢ balloon filled with inflammable air. The ma chine used on this occasion was formed 0 gores of silk, covered with a varnish caout chouce, of a spherical figure, and measuring 272 feet in diameter. A net was spread ove) the upper hemisphere, and fastened to a hooy which passed round the middle of the balloon To this a sort of car was suspended, a few fee below the lower part of the balloon; and i order to prevent the bursting of the machine a valve was placed in it; by opening of whie some of the inflammable air might be ocea sionally let out. The car was of basket-wor. covered with linen and beautifully ornamented being 8 feet long, 4 broad, and 33 deep; i weight 130 pounds. Great difficulties agai occurred in filling the machine; but these ¢ last being removed, the two adventurers to their seats at three-quarters after one in afternoon of the 1st of December, 1783. 4 the time the balloon rose, the barometer wi at 30.18 inches; and by means of the pow of ascent with which they left the ground, fh balloon rose till the mercury fell to 27 ine from which they calculated their height to} about 600 yards. ‘Throwing out ballast oce sionally as they found the machine dese ing, by the escape of some of the inflammab air, they found it practicable to keep at pret - AER near the same distance from the earth daring the rest of their voyage, the quicksilver fluc- tuating between 27 and 27°65 inches, and the thermometer between 53° and 57°, the whole time. They continued in the air an hour and three quarters, and alighted at the distance of 27 miles from Paris; having suffered no incon- venience during their voyage, nor experienced any contrary currents of air, as had been felt by Messrs. Pilatre and Arlandes. As the bal- loon still retained a great quantity of inflam- mable gas, M. Charles determined to take another voyage by himself. M. Robert ac- cordingly got out of the machine ; which now being 130 pounds lighter, arose with such ve- locity that in 20 minutes he was almost 9000 feet high, and entirely out of sight of terrestrial objects. The globe, which had been rather flac- cid, soon began to swell, and the inflammable air escaped in great quantity. He also drew the valve, to prevent the balloonfrom bursting ; and theinflammable gas being considerably warmer than the external air, diffused itself all around, and was felt like a warm atmosphere. In ten minutes, however, the thermometer indicated a great variation of temperature ; his fingers were benumbed with cold, and he felt a vio- lent pain in his right ear and jaw, which he ascribed to the expansion of the air in these organs, as well as to the external cold. The beauty of the prospect which he now enjoyed, however, made amends for these inconve- niences. At his departure the sun was set on the valleys; but the height to which M: Charles was got in the atmosphere, rendered him again visible, though only for a short time. _He saw for a few seconds vapours rising from the valleys and rivers. The clouds seemed to ascend from the earth, and collect one upon the other, still preserving their usual form, | only their colour was grey and obscure, for want of sufficient light in the atmosphere. By the light of the moon, he perceived that the | machine was turning round with him in the , air; and he observed that there were contrary ‘ currents, which brought him back again. He ) observed also, with surprise, the effects of the wind, and that the streamers of his banners pointed upwards; which, he says, could not be the effect either of his ascent or descent, as he was moving horizontally at the time. , At last, recollecting his promise of returning ) to his friends in half an hour, he pulled the valve, and accelerated his descent. When within 200 feet of the earth, he threw out two | or three pounds of ballast, which rendered the balloon again stationary; but in a little time afterwards he gently alighted in a field about three miles distant from the place whence he set out; though by making allowance for all the turnings and windings of the voyage, he | supposes that he had gone through nine miles atleast. By the calculations made, it appears that he rose at this time not less than 10,500 feet; a height somewhat greater than that of Mount Etna. | The subsequent aerial voyages differ so little from those above related, that any particular AER description of them would be superfluous. We shall therefore pass over several of them in silence ; observing merely, that various at- tempts were made to render aecrostatic ma- chines manageable in the air, so as to pro- ceed in any direction at pleasure, but they all turned out abortive, as indeed might have been foreseen. In experiments of this kind, the whole machine is enveloped in a medium which is itself in rapid motion, tic wind some- times blowing at the rate of 60 or 70 miles per hour; and of course the balloon is carried for- ward with the same rapidity. It was there- fore not to be expected that any appendages, or apparatus, that could be constructed, would enable the aeronaut to proceed in an opposite direction, or even in a sideral one, at pleasure. The expense in filling a balloon with in- flammable gas is very considerable, and it therefore became an object of anxious in- quiry, how this might be avoided or reduced. The first attempt of this kind was made by the Duke de Chartres: it consisted in having a smaller balloon within the greater one, the smaller being filled with common air by means of a pair of bellows, when necessary; viz. whenever it was thought proper to descend, it being supposed that the machine would thus become heavier, and the air in the outer bal- loon condensed, and consequently, that the ascent or descent might be effected at plea- sure. ‘The circumstances of this voyage, how- ever, were so unfavourable, that it could not be ascertained whether or not the experiment would have succeeded, in a more serene state of the atmosphere, the weather being so bois- terous during this voyage, that the duke had a very narrow escape with his life. This scheme for raising or lowering aero- static machines by bags filled with common air, being thus rendered dubious, another me- thod was thought of; which was to put a small aerostatic machine, with rarified air, under an inflammable air-balloon, but at such a dis- tance, that the inflammable air of the latter might be perfectly out of the reach of the fire used for inflating the former; and thus by in- creasing or diminishing the fire in the small machine, the absolute weight of the whole would be considerably diminished or aug- mented. ‘This scheme was unhappily put in execution by the celebrated M. Pilatre de Rozier and M. Romaine. ‘Their inflammable air-balloon was about 37 feet in diameter, and the power of the rarified air one was equi- valent to about 60 pounds. ‘They ascended without any accident; but had not been long in the atmosphere, when the inflammable air- balloon was seen to swell very considerably, at the same time that the aeronauts wore ob- served, by means of telescopes, very anxious to get down, and busied in pulling the valve and opening the appendages to the balloon, in order to facilitate the escape of as much inflammable air as possible. Shortly after this the machine took fire at the height of about three quariers of a mile from the ground. No explosion was heard ; and che silk of the hal- AER loon seemed to resist the atmosphere for about a minute, after which it collapsed, and de- scended along with the two unfortunate tra- vellers so rapidly, that both of them were killed. Pilatre seemed to have been dead be- fore he came to the ground; but M. Romaine was alive when some persons came up to him, though he expired immediately after. The first aerial voyage in England was per- formed on the 15th of September, 1784, by Vincent Lunardj, a native of Italy. His bal- loon was made of oiled silk, painted in alter- nate stripes of blue and red: its diameter was 33 feet. From a net which went over about two-thirds of the balloon, descended 45 cords to a hoop hanging below it, and to which the gallery was attached. The balloon had no valve; and its neck, which terminated in the form of a pear, was the aperture through which the inflammable air was introduced, and through which it might be Iet out. The air for filling the balloon was produced from zinc by means of diluted vitriolic acid. M. Lunardi departed from the Artillery-ground at two o’clock; and with him were a dog, a cat, anda pigeon. After throwing out some sand to clear the houses, he ascended to a great height. ‘The direction of his motion at first was N.W. by W.; but as the balloon rose higher, it fell into another current of air, which carried if nearly N. About half after three he descended very near the earth, and landed the cat, which was almost dead with cold; then rising, he prosecuted his voyage. He ascribes his descent to the action of an oar; but as he was under a necessity of throwing out ballast in order to re-ascend, his descent was more probably occasioned by the loss of imflammable air. At ten minutes past four, he descended in a meadow near Ware, in Hertfordshire. The only philosophical instru- ment which he carried with him was a ther- mometer, which in the course of his voyage stood as low as 29°, and he observed that the drops of water collected round the balloon were frozen. The second aerial voyage in England was performed by Mr. Blanchard, and Mr. Sheldon, professor of Anatomy to the Royal Academy, being the first Englishman who ascended with an aerostatic machine. They ascended at Chelsea the 16th of October, at nine minutes past twelve o’clock. Mr. Blanchard having landed Mr. Sheldon at about 14 miles from Chelsea, re-ascended alone, and finally landed near Rumsey, in Hampshire, about 75 miles distant from London, haying gone nearly at the rate of 20 miles an hour. ‘The wings used on this occasion, it seems, produced no de- viation from the direction of the wind. Mr. Blanchard said, that he ascended so high as to feel a great difficulty of breathing ; and that a pigeon which flew away from the boat, la- boured for some time to sustain itself with its wings in the rarified air; but after wandering a good while, returned and rested on the side of the boat. Various voyages were now undertaken in AER England and other countries, which are omitted in the present article, as furnishing little or no additional interest ; but the last voyage of M. Garmerin in this country, being of a very novel kind, must not be passed over in silence. On the 2ist of September, 1802, this celebrated aeronaut undertook the singular and desperate experiment of ascending in a balloon, and then descending by means of a parachute. The ascent took place from St. George’s Parade, North Audley Street, and he descended safe in a field near the Small-pox Hospital, Pan- cras. ‘The balloon was of the usual sort, viz. of oiled silk, with a net, from which ropes pro- ceeded which terminated in, or were joined to, a single rope at a few feet below the balloon. To this rope the parachute was fastened in the following manner :—The reader may easily form to himself an idea of this parachute, by imagining a large umbrella of canvas, of about 30 feet in diameter, but destitute of the ribs and handle. Several ropes of about 30 feet in length, which proceeded from the edge of the parachute, terminated in a common join- ing, from which shorter ropes proceeded, and to the extremities of these a circular basket was fastened, and in this basket M. Garnerin placed himself. Now the single rope which has been said above to proceed from the balloon, passed through a hole in the centre of the parachute, also through certain tin tubes which were placed one after the other in the place of the handle or stick of an umbrella, and was lastly fastened to the basket; so that when the bal- loon was in the air, by cutting the end of this rope next to the basket, the parachute with the basket would be separated from the bal- loon, and in falling downwards would be na- turally opened by the resistance of the air, The use of the tin tubes was, to let the rope slip off with greater certainty, and to prevent its being entangled with any of the other ropes, as also to keep the parachute at a distance from the basket. The balloon began to be filled at about two o’clock. ‘There were 36 casks filled with iron filings, and diluted sul- phuric acid, for the production of the hydrogen gas. ‘These communicated with three other casks, or general receivers, to each of which was fixed a tube that emptied itself into the main tube, attached to the balloon. At six, the balloon being quite full of gas, and the pa-_ rachute, &c. being attached to it, M. Garnerin placed himself in the basket, and ascended majestically, amidst the acclamations of in- numerable spectators. The weather was the clearest and pleasantest imaginable; the wind was gentle, and about W. by S8.; in conse- quence of which, this machine went in the direction of about E. by N. In about eight minutes time the balloon and parachute had ascended to an immense height, and the acronaut, in the basket, could scarcely be per-. ceived. While every spectator was contem= plating the grand sight before them, M. Gar- nerin cut the rope, and in an instant he wa separated from the balloon, trusting his safety to the parachute. At first, viz. before the pa= AER rachute opened, he fell with great velocity; but as soon as this machine was expanded, which took place a few moments after, the lescent became very gentle and gradual, In his descent a remarkable circumstance was »bserved, namely, that the parachute, with the tppendage of cords and basket, soon began @ Vibrate like the pendulum of a clock; and he vibrations were so great, that more than mce the parachute and the basket seemed 0 be on the same level, or quite horizontal, vhich appeared extremely dangerous; how- ver, the extent of the vibrations diminished s he came pretty near the ground. On coming @ the earth, M. Garnerin experienced some retty strong shocks; and when he came out f the basket, he was much discompesed; but € soon recovered his spirits, and remained ‘ithout any material injury. Several other excursions have since been tade in England, by Mr. Sadler, from Ox- wd, Cambridge, and Hackney; but they pre- 2nt no novelty, and we therefore pass them y for the reasons before stated, and shall now onclude this article. with an account of the rocess employed for producing inflammable as, and the method of filling the balloon. ‘Inflammable air for ballaons may be ob- ‘ined in several ways: but the best methods ‘e, by applying acids to certain metals ; by whose log. is eS TOo? ’ ° ‘ 2 _ am a — °07918; therefore 07018 — 25% nearly, which is the number of strokes required. Some of the principal effects and phenomena of the air-pump are the following :—That, in the exhausted receiver, heavy and light bo- dies fall equally swift; so, a piece of metal and a feather fall from the top of a tall receiver to the bottom exactly together. That most ani- mals die in a minute or two: but, however, that vipers and frogs, though they swell much, live an hour or two; and after being seem- ingly quite dead, come to life again in the open air: that snails survive about ten hours ; efts, or slow-worms, two or three days; and leeches five or six. That oysters live for 24 That the heart of an eel taken out of the body continues to beat for good part of an hour, and that more briskly than im the air. That warm blood, milk, gall, &c. undergo a considerable intumescence and_ ebullition. That a mouse or other animal may be brought, by degrees, to survive longer in a rarefied air, than it does naturally. That air may retain its usual pressure, after it is become unfit for respiration. That the eggs of silk-worms hatch in vacuo. That vegetation stops. ‘That fire extinguishes; the flame of a candle usually going out in one minute; and a charcoal in about five minutes. That red-hot iron, how- \ ALB ever, seems not to be affected; and yet sul- phur or gunpowder are not lighted by it, but only fused. That a match, after lying seem- ingly extinet a long time, revives again on readmitting the air. That a flint and steel strike sparks of fire as copiously, and in all directions, as inair. That magnets, and mag- netic needles, act the same as in air. That the smoke of an extinguished luminary gra- dually settles to the bottom in a darkish body, leaving the upper part of the receiver clear and transparent; and that on inclining the vessel sometimes to one side, and sometimes to another, the fume pr&erves its surface horizontal, after the nature of other fluids. That heat may be produced by attrition. That camphire will not take fire; and that gun- powder, though some of the grains of a heap of it be kindled by a burning glass, will not give fire to the contiguous grains. That glow- worms lose their light in proportion as the air is exhausted, and at length become totally obscure ; but on readmitting the air, they pre- sently recover it again. That a bell, on being struck, is not heard to ring, or very faintly. That water freezes. But that a syphon will not run. ‘That electricity appears like the aurora borealis. With multitudes of other curious and important particulars, to be met with in the numerous writings on this machine, viz. besides the Philos. Trans. of most academies and societies, in the works of Torricelli, Pascal, Mersenne, Guericke, Schottus, Boyle, Hook, Duhamel, Mariotte, Hawksbee, Hales, Muschenbroeck, Grave- sande, Desaguliers, Franklin, Cotes, Helsham, Martin, Ferguson, Adams, Nicholson, Ca- vallo, Gregory, Hutton, &c. Pantoloiga. AiR-Vessel, in Hydraulics, is a name given to those metalline cylinders, which are placed between the two forcing-pumps in the im- proved fire-engines. The water is injected by the action of the pistons through two pipes, with valves, into this vessel; the air previously contained in it will be compressed by the water, in proportion to the quantity admitted, and by its spring force the water into a pipe, which will discharge a constant and equal stream; whereas in the common squirting engine, the stream is discontinued between the several strokes. AJUTAGE, (from the verb ajouter, to adapt) in Hydraulics, is a part of the apparatus of an artificial fountain, or jet d’eau, being a sort of tube fitted to the aperture or mouth of the vessel, through which the water is to be played. ALAMAC, the Arabic name for y Andro- meda. ALBANO (PETER), an Italian physician of the fourteenth century; author of a treatise on the Astrolabe. ALBEGALA, the Arabic name of the con- stellation Lyra. ALBERTAGNI, an Arabian prince and astronomer, who flourished about the middle of the ninth century. He composed a work on the science of the stars, comprising all parts of astronomy. This werk was printed ALF in Latin at Nuremberg in 1537, and again @ Bologna in 1645. | ALBERTUS (Magnus), a learned domini can, born in Suabia at the beginning of th thirteenth century, and died at Cologne ti 1280. He was author of a great many ma thematical works, which were printed a Lyons in twenty-one vols. folio, 1615. Bu he is not distinguished for any important dis coveries or improvements. ALBIREO, the name of 6 Cygnus. ALBUMASAR, an Arabian astrologer an astronomer of the ninth century. Two of hi works have been printed, viz. one, on Astre logy, at Venice in 1586; and the other, a Introduction to Astronomy, in 1489. | ALCYONE, the name of the brightest sta marked y in the Pleiades. ALDEBARAN, a star of the first magn tude, in the constellation Taurus or the B vulgarly called the Bull’s-eye. ALDERAIMIN. See ADERAIMIN. ALDHAFERA, a star of the third magn tude, in the constellation Leo. | ALEMBERT (JouN LE Ronp D’), an em nent French mathematician and philosophe was born at Paris 1717, and died in the san city in 1783. He was author of several if portant works on these sciences, as, { “Traité de Dynamque,” in 4to. Paris, 174 The 2d edit. 1758. - 7 “'Traité de ’Equilibre et du Moveme des Fluides,” Paris, 1744 and 1770. “ Reflexions sur la Cause générale d| Vents ;” which gained the prize at Berl 1746; published at Paris in 4to. 1747. | “ Recherches sur la Précession des Eq) noxes, &c.” Paris, 4to. 1749. | “‘ Essais d’une Nouvelle Théorie du Moy ment dés Fluides,” Paris, 4to. 1752. “‘ Recherches sur differens Points impi| tans du System du Monde,” Paris 1754 aj 1756; 3 vols. 4to. | “ Elémens de Phylosophie,” 1759. 9 ve i) “ Opuscules Mathematique,” Paris, in 4to, 1761 and 1778. “ Flémens de Musique,” 1 vol. 8vo. | Besides these complete works, d’Alemb] sent numerous important essays and pap to the Academy of Sciences at Paris, whi} are published in the Memoirs of that leari} institution; a complete list of which is giv} by Dr. Hutton in his Mathematical Dicti} ary, under the article d’Alembert. To t} author we are also indebted for the Theorg} Partial Differences ; as also for the Introd! tion to the “ Encyclopedia Methodique,” 4 various excellent articles in the body of tf work. | ALFECCA, a name given to the star e¢f monly called Lucida Corone. ALFRAGAN, a celebrated Arabian asi nomer, who flourished about the year & He wrote an astronomical work, which J been since translated into Latin; and | published at Ferrara in 1493; again at Tf remberg in 1537, with a preface by Melaj thon. A second translation was publishet A: LG rankfort in 1590; and a third, which is far 1e best, by Golius, at Leyden in 1669, ALGEBRA is a general method of re- ving mathematical problems, by means of juations ; or it is a method of computation ‘lative to all sorts of quantities, by means of ‘rtain indeterminate characters or symbols, hich have been invented for this purpose. /e have no records which enable us to de- rmine any thing with regard to the date or ithor of this very important science; having isen, in all probability, like most others, ‘such slow and imperceptible degrees, that ere we in possession of all the writings of e ancients, it would perhaps be difficult to aw a line, so as to determine the precise mmencement of the algebraic art. The talytical method of investigating problems ust have suggested itself very early to ma- ematicians. ‘Those unknown quantities uich are now represented by letters, were the infancy of this science probably ex- essed by their names at full length, and ery operation performed upon them, as ad- jon, subtraction, multiplication, &c. were pressed in the same manner. But it would mm be found, that in this way a useless and esome repetition of the same words would cur in the most simple problems, whence ‘ may easily conceive, how the idea of ex- *ssing quantities by letters first arose, which S, by substituting the initial of the word the word itself; the several operations re soon after represented in the same way; i thus by successive improvements arose s noble and comprehensive science, which its present state does honour to the inven- 2 genius of man. The term algebra is of Arabic original ; but etymology has been variously assigned by erent writers. Amongst the Arabians, from om it was immediately transmitted to us, 3 science was denominated algiabar almo- alah, which denoted the science of restitu- t and comparison, or resolution, and there- » expressed its nature with sufficient pre- on; and, accordingly, Lucas de Burgo, of the earliest European writers on alge- , calls it the rule of restoration and opposi- ; and affirms that we had both the name the art from the Arabs. Others, however, e derived the name from Geber, either the gic of a celebrated mathematician, to whom y have ascribed the invention of the nce; or from the word geber, which forms 1 the particle al, the appellation algebra. ious other derivations have been given to term, which it would be useless to mention ais place, many of them being both fanci- and improbable; leaving, therefore, this stigation, we shall proceed to give a brief jract of the principal features of the his- of algebra, with the names and improve- its of the most celebrated authors who > written on this subject. he origin of algebra, as we have before rved, like that of other sciences of ancient and gradual progress, is not easily ascer- ALG tained. The earliest treatise on that part of analytics, which is properly called algebra, now extant, is that of Diophantus, a Greek author of Alexandria, who flourished about the year 350, and who wrote thirteen’ books, though only six “ Arithmeticorum” of them are preserved ; which were printed, together with a single imperfect book on multangular numbers, in a Latin translation, by Xylander, fn 1575; and afterwards, in Greek and Latin, with a comment, in 1621 and 1670, by Gaspar Bachet and M. Fermat. These books do not contain a treatise on the elementary parts of algebra, but merely collections of difficult questions relating to square and cube num- bers, and several other curious properties of numbers, with their solutions. In his prefa- tory remarks, which are addressed to one Dionysius, and for whose use they are sup- posed to have been written, he recites the names and generation of the powers; the square he calls dynamis; the cube, cubus; the fourth power, dynamo dinamis ; the fifth power, dynamo cubus ; the sixth power, eubo cubus, &¢. 5 according to the sum of the indices of the pow- ers, Which powers he marks with the Greek initials ; expressing the unknown quantity by eeiwes, or the number; simply marking it in the solution by the final o;, and denoting the monades, or indefinite unit, by 4°. In his researches on the multiplication and division of simple species, he shows what powers they produce, and observes that minus multiplied by minus produces plus, and that minus mul- tiplied by plus produces minus; he employs the symbol +, or the inverted |, to denote minus; but has no mark to denote plus, Sup- posing his reader acquainted with the com- mon operations of addition, subtraction, mul- tiplication, and division of compound species, he proceeds to remark on the preparation of the equations that are deduced from the ques- tions, and which we now call reduction of equations, by collecting like quantities toge- ther, adding quantities that are minus, and subtracting those that are plus, being equiva- lent to what we term transposition, so as to bring the equation to its simplest terms, and then depressing it to a lower degree by equal division, when the powers of the unknown quantities are in every term; having thus ob- tained his final equation, he proceeds no farther, but merely says what the root is, without giv- ing any rules for finding it, or for the resolution of equations; thus intimating that rules for this purpose were to be found in some other work, either of his own or of some other per- son; and as all his final results are either simple equations, or simple quadratics, it does not appear how far his knowledge extended to the resolution of compound or affected equations. Diophantus is the only Greek author on algebra whose works have been handed down to us, though some traces of it appear in the writings of some authors much more ancient, as Archimedes, Euclid, Apollonius, &c.; and we a that the celebrated Hypatia wrote ALG a commentary on the work we have been describing. By what means the Arabs be- came possessed of this art is not known; some ‘have supposed they derived their knowledge of it from the Greeks, while others maintain that they owed both this science and their notation to the Hindoos; with regard to the latter, there scems to be little doubt, but the former is not so evident. However this may be, it is certain that the Brahmins possessed a knowledge of algebra, but whether prior to the Arabians, er subsequent to them, it is difficult to determine ; ; at all events, wherever algebra was invented or first cultiy ated, both the name and the science were transmitted to Europe, and particularly to Spain, by the Arabians or Saracens about the year 1100, or probably a little earlier. Italy, however, seems to haye taken the Jead in the cultivation of this science, after its intreduction into Europe; and Lucas Paciolus, or Lucas de Burgo, was one of the first who wrote on this subject, having pub- dished several treatises in- the years 1470, 1476, 1481, 1487, and 1509; but his principal work, “Summa Arithmetice et Geometriz Proportionumque et Proportionalitatum,” was published in Italian at Venice in 1494, and again in 1523. In this work he mentions se- veral writers, and particularly Leonardus Pi- sanus, who it seems flourished at the begin- ning of the thirteenth century. His proper name was Bonacci, an Italian merchant, who traded to the sea-ports of Africa and the Le- vant, whence he obtained his knowledge of this science, and to whom Italy was doubtless indebted for the first introduction of algebra. This Leonard must not be confounded with another Leonardus de Pesar, author, as it is said, of a book entitled “ Liber desideratus, &c.;” these being distinct persons. Montucla mentions two other authors who are supposed even to have preceded Leonardus Pisanus in this department of science, viz. Paul dell’ Abaco and Belmando, or Padua. However this may be, we know lit- tle of algebra amongst Europeans before the works of Lucas de Burgo, from whom we learn that the knowledge of the Europeans, in his time, (1500) extended no farther than to quadratic equations, of which they used only the positive roots, and they admitted only one unknown quantity ; that they had no marks or signs for either quantities or Opera- tions, excepting a few abbreviations of words or names; and that the art was merely em- ployed in resolving certain numerical pro- blems. After the publication of the books of Lucas de Burgo, the science of algebra became more generally known and improved, espe- cially by many in Italy; and about this time, er soon after, that is, about 1505, the first rule was there found out by Scipio Ferreus, for resolving one case of a compound cubic equation. But this science, as well as other branches of mathematics, was most of all cul- tivated and improved at this time by Hierony- "Beldomando, of ALG mus Cardan of Bonomia,a very learned author, whose arithmetical writings were the next that appeared in print, namely, in the year 1539, in nine books, in Latin, at Milan, whe he practised physic and read public lectures on mathematics; and in the year 1545 a tenth book was published, containing the whole doctrine of cubic equations, which had been in part revealed to him by Nicholas Tartalea, about the time of the publication of his first nine books. The chief improvements made by Cardam as collected from his writings, are stated a Dr. Hutton as follows: Tartalea having only communicated to him the rules for resolving these three cases of cubic equations, viz. vi x3 +bx=—=e +» he from thence raised a ve x= bz +e : large and complete work, lay- x3 +e mbx) ing down rules for all form and varieties of cubic equations, having all their terms, or wanting any of them, and having all possible varieties of signs; demon- strating all these rules geometrically; a tr eating very fully of almost all sorts of trang formations of equations, in a manner tota new. It appears also that he was well ace quainted with all the roots of equations, that are real, both positive and negative; or, as he calls them, true and fictitious ; and thathe made use of them both occasionally. He also showe that the even roots of positive quantities, ar of negative quantities, are real and negative but that the even roots of them are impossiblé or nothing, as to common use. He was als¢ roots of an equation, and that partly from the signs of the terms, and partly from the - nitude and relation of the co-effi cients. fe, equal to_ the number of changes of the si of the terms. That the co-efficient of th second term of the equation, is the difference between the positive and negative roots? Tha when the second term is wanted, the sum-@ the negative roots is equal to the sum of th positive roots. How to compose equation that shall have given roots: That chang si sible roots, were always in pairs. To chang the equation from one form to another, b taking away any term out of it. -'To increas or diminish the roots of a given equation. © appears also, that he had < arule for extrac Lint the cube root of such binomials as admit extraction ; and that he often used the lite notation a, b, c,d, &c. 'That he gave a ru for biquadratic equations, suiting all the cases; and that in the investigation of th rule, he made use of an assumed indeterml nate quantity; and afterwards found its vé lu by the arbitrary assumption of a relation Di tween the terms. Also, that he applied algebr to the resolution of geometrical problem and was well acquainted with the difficulty ALG What is called the InrEpuCIBLE Case; upon vhich he spent a great deal of time in attempt- ug to obtain a solution of it. And though he lid not fally succeed in this, he nevertheless nade many ingenious observations concern- ug it, laying down rules for many particular arms of it, and showing how to approximate ery nearly to the root in all cases whatso- ver. Nicholas Tartalea, or 'Tartaglea, of Brescia, ‘as contemporary with Cardan, and was pro- ably older than he was; but we do not know f any book of algebra published by him till 1e year 1546, the year after the date of Car- an’s work on cubic equations, when he print- Lhis “ Quesitaé Invenzioni diverse,” at Ve- ice; where he resided as a public lecturer on athematics.’ Tartalea made no alteration in ‘© notation or forms of expression used by ucas de Burgo; calling the first power of e unknown quantity, in his language, cosa; e second, censa; the third, eubo, &c.: and titing the names of all the operations in ords, at length, without using any contrac- ms, except the initial RK for root, or radi- lity. So that the only things remarkable this collection, are the discovery of the les for cubic equations, and the curious cir- mstances attending the same ; particularly > correspondence and conferences which k place between him and Cardan. See tic and algebra. From an examination of 3 work of Stifelius, it appears that the im- yements made by himself, or other Ger- ins, beyoud those of the Italians, as con- aed in Cardan’s book of 1589, were as fol- ies de introduced the characters +, —, V, for 8, minus, and root, or radix, as he called Also the initials 2¢, 3, ys, for the powers , 3, &c. He treated all the higher orders juadratics by the same general rule; and oduced the numeral exponents of the fers ——3, — 2, —1, 0, 1, 2, 3, &c. both po- ye and negative, as far as integral num- , but not fractional ones; calling them by name exponens, exponent; and taught the eral uses of the exponents in the several rations of powers, as we now use them in logarithms. He likewise used the gene- iteral notation A, B, C, D, &e. for so many ‘rent, unknown, or general quantities. hn Scheubelius wrote much about the of Cardan.and Stifelins; but as he takes totice of cubic equations, it is likely that ad neither seen nor beard any thing about 1. He treats pretty largely upon surds, gives a general rule for extracting the of any binomial or residual, a + b, where or both parts are surds, and a the greater ALG quantity; namely, that the square root of it is S tA a a VEE WER 1 9051 Ny 04) TORT ath which he illustrates by various examples, A few years after the appearance of these treatises in Italy and Germany, Robert Re- corde, a celebrated mathematician and phy- sician, born in Wales, proved, by his writings, that algebra was not altogether unknown in England. The first part of his Arithmetic was pub- lished in 1552, and the second part in 1557, under the title of “The Whetstone of Witte, which is the seconde part of Arithmetike : containing the Extraction of Rootes; the Cossike Practice, with the Rule of Equation ; and the Works of Surde Nombers.” The par- ticulars, which are new in the works of this author, are, The extraction of the roots of compound algebraic quantities; the use of the terms binomial and residual; the use of the sign of equality, or =. The first edition of Peletarius’s Algebra, was printed in 4to. at Paris, in 1558, under this title, “ Jacobi Peletarii Cenomaui de occulta parte Numerorum, quam Algebram vocant, Lib duo.” This is a very ingenious and mas. terly composition, treating in an able manner upon the several parts of the subject then known, except cubic equations. His real discoveries or improvements are these: That the root of an equation is one of the divisors of the absolute term. He taught how to reduce trinomial to simple terms, by mul- tiplying them by compound factors. He taught curious precepts and properties con-. cerning square and cube numbers, and the method of constructing a series of each by addition only ; namely, by adding successively their several orders of differences, Peter Ramus wrote his arithmetic and al- gebra about the year 1560. His notation of the powers is thus, , g, ¢, bq, being the ini- tials of latus, quadratus, cubus, biquadratus : but he treats only of simple and quadratic equations. Raphael Bombelli’s algebra was published at Bologna in the year 1579, in the Italian language: the dedication, however, is. dated 1572. In this work we meet with little im- provement or alteration, except in the nota- tion, where he uses W for the unknown quantity, with the numeral indices of Stifelius, The arithmetic of Simon Steven of Bruges, was published in 1585; and his algebra a short time afterwards. A general air of ori- ginality runs through the whole of this latter work of Stevenus; yet his more peculiar and remarkable inventions may be reduced te these: He invented a new character for the un- known quantity ; namely, a small circle O, within which he placed the numeral exponent of the power; and extended them to frac- tional and all other sorts, thereby denoting roots Same as powers by numeral expe- 2 ALG nents; an improvement hitherto thought to be of much later invention. Most of Vieta’s algebraical works were written about the year 1600, ora little before ; but some of them were not published till after his death, which happened in the year 1603. And his whole mathematical works were collected together by Francis Schooten, and elegantly printed in a folio volume, in 1646. The real inventions of this very ingenious author are stated in the following particulars: He introduced the general use of the let- ters of the alphabet to denote indefinite given quantities; which had only been done in some particular cases before his time; but the ge- neral use of letters for the unknown quan- tities, was before pretty common with Stife- lius aud his successors. Vieta uses the vowels A, E, I, O, Y, for the unknown quantities, and the consonants B, C, D, &c. for known ones. He invented and introduced many ex- pressions or terms, several of which are in use to this day: such as co-efficient, affirma~ tive and negative, pure and adfected, or affected, uncie, homogeneum, &c. and the line, or vinculum, over compound quantities, thus, A +B. His method of setting down equations, is to place the absolute known term on the right hand side alone, and on the other side all the terms which contain the unknown quantity, with their proper signs. He showed how to change the root of an equation in a given proportion; and derived, or raised, the cubic and biquadratic, &c. equa- tions from quadratics; not by composition, in Harriot’s way, but by squaring and other- wise multiplying certain parts of the qua- dratic. He extracted the roots of affected equations, by a method of approximation similar to that for pure powers; and gave the construction of certain equations, and ex- hibited their roots by means of angular sec- tions; before adverted to by Bombelli. Albert Girard, an ingenious Flemish ma- thematician, died about the year 1633. The work which entitles him to notice in this his- tory, is his “Invention Nouvelle en l’Algebre, &c.;” which was printed at Amsterdam, 1629, in small quarto, in sixty-three pages; viz. forty-nine pages on arithmetic and algebra, and the rest on the measure of the superfices of spherical triangles and polygons, by him then lately discovered. From this work of Girard’s we learn, That he was the first person who under- stood the general doctrine of the formation of the co-efficients of the powers, from the sums of their roots, and their products, &e. He was the first who understood the use of negative roots, in the solution of geometrical problems; and was the first who spoke of the imaginary roots, and understood that every equation might have as many roots, real and imaginary, and no more, as there are units in the index of the highest power. The celebrated Thomas Harriot flourished about the year 1610; near which time it is probable he wrote his algebra. His inven~ ALG tions and improvements were important though they may be comprehended in thre particulars. He introduced the uniform use of the smal letters a,b, c,d, &c.; viz. the vowels a,e, and ¢ for unknown quantities, and the consonant b, c, d, f, ke. for the known ones ; which h joins together like the letters of a word, t represent the multiplication of product of am number of these literal quantities, and prefix ing the numeral co-efficient as we do at pre sent, except being separated by a point, thu 5.bbe. For a root he sets the index of th root after the mark ./; as 3 for the cub root. He also introduced the characters > and <, for greater and less; and in’ the re duction of equations lie arranged the oper tions in separate steps, or lines, setting th explanations in the margin on the left han for each line. By which, and other mean he may be considered:as the introducer of th modern state of algebra, which quite change its form under his hands. In the next plac! he showed the universal generation of all th compound or affected equation, by the cor tinual multiplication of so many binomi roots; thereby plainly exhibiting to the ey the whole circumstance of the nature, my tery, and number of the roots of equation with the composition and relations of the ¢ efficients of the terms; and from which may of the most important properties have sin been deduced. He also greatly improved tl numerical extraction of the roots of ali-eq tions, by clear and explicit rules asd thods, drawn from the foregoing generatic or composition of affected equations of | degrees. Oughtred’s Clavis was first published 1631, the same year in which Harriot’s Alg bra was published by his friend Warn Oughtred chiefly follows Vieta in the notati by the capitals, the designation of produce power, &c. though with a few variations. | He was the first, as far as we can lea who set down the decimals without thf denominator; separating them thus, 21] In algebraic multiplication, he either joj the letters which represent the factors to} ther like a word, or connects them by > mark x, which is the first introduction of t character. He also seems to be the first ¥ used points to denote proportions, thus 7. 28.36; and for continued proportion he- this mark ++. He denoted roots by the e¢ mon radical sign, with q for square root, ¢ cube, gq for biquadrate. In equations, he gi express and general directions for the sev sorts of reductions. He uses the letter na / for universal, instead of the vinculun}} Vieta. “ in this work we likewise meet with first instance of applying algebra to geom . as to investigate new geometrical pro| €8. : In 1634, Herigone published, at Paris, | first course of mathematics, in 5 vols. 8H} in the second of which is contained. a g ALG treatise on algebra; in which he uses the notation by small letters, introduced by the algebra of Harriot, which was published three years before, though the rest of it does not resemble that work, and one would suspect that Herigone had not seen it. The whole of this piece bears evident marks of originality and ingenuity. Besides -++ for plus, he uses ‘7 for minus, and | for equality, with several other useful abbreviations and marks of his own. In the notation of powers, he does not ‘epeat the letters, like Harriot, but subjoins the numeral exponents to the letter, as Des- vartes did three years afterwards. And Heri- sone uses the same numeral exponents for ‘ots, as 38 for the cube root. Descartes’ Geometry was first published in (637, being six years after the appearance of darriot’s Algebra. This work of Descartes vas rather an application of algebra to geo- netry, than either algebra er geometry sepa- ately censidered. Still he made improve- nents in both; and with respect to the sci- mce of algebra, we may speak both of his im- rovements and his inventions. In the first, hat he might fit equations the better for their (pplication in the construction of problems, Descartes mentions, as it were by the by, aany things concerning the nature and re- uction of equations, without troubling him- elf about the first inventors of them, stating 1em in his own terms and manner, which is ommonly more clear and explicit, and often ‘ith improvements of his own. And under uis head we find that he chiefly followed Car- an, Vieta, Harriot, but especially the last; ad explained some of their rules and disco- erles more distinctly than they had done, ut varies a little in the notation. Descartes suppose d to represent the density of the osphere at the surface of the earth, we lhave A = 2 log. a F gain, in order to find x, let us take aknown ‘+ let us, for instance, take the density at surface, and at one foot above it, which, n the thermometer stands at 31°, and ba- eter at 291 inches, will be as follows; viz. a=0; d = 26057 oe ee Fe Ae is, at the surface the pressure will be ul to acolumn of air, of uniform density, 16057 feet; and consequently, one foot ve the surface, the pressure will be one less, or 26056 feet; and as the densities as the pressures, these numbers will also roper measures of the former; whence 26057 26056 aaking 26057 = n, we have 1 = x log. r g i n 1 1 1 n—1 =M io tan tsar 8$ | as in the present case, n is a great num- all the terms past the first may be neg- ed; and since also M = -43429448 in the mon system (see LOGARITHMS), we have 43429448 | 26057’ = 60000, very nearly : ; A—a=-1l=~x2 log. log. B=. SM 26057 *43429448 efore, the above formula is reduced to this: A = 60000 x log. € feet. r, since the height of the mercury in the meter is always proportional to the den- _ the fraction « = Me m and M being the ‘ht of the mercury at the earth’s surface, at the altitude A; also, since six feet is al one fathom, the same formula may be ‘rwise expressed thus : A = 10000 log. a fathoms ; /which is the general formula for measur- altitudes by the barometer. This simple tiplier is very convenient, and for which are indebted to our assuming the tem- iture at 31°; for with any other degree of thermometer we should have had a dif- nt multiplier: the particular degree, how- ', so far as relates to the accuracy of the ilt, is totally indifferent; because we may ays readily accommodate the result to any perature whatever. For by numerous ex- ments it has been found, that air expands ut _!-th part of its bulk for every degree of t, and contracts the same for every degree nce z = old, and consequently the result obtained a the above formula must be increased or ALT diminished by as many times z},th part of it+ self, as the temperature in degrees exceeds, or is less, than31°. Hence we have, according to the preceding principles, the following rule for measuring altitudes by the barometer, viz. Observe the height of the mercury at the bottom of the object to be measured, and again at its top; as also the degree of the thermometer in both these situations; and half the sum of these two last may be ac- counted the mean temperature. Then mul- tiply the difference of the logarithms of the two heights of the barometer by 10000, and correct the result by adding or subtracting so many times its 435th part, as the degrees of the mean temperature are more or less than 31°; and the last number will be the altitude in fathoms. Exam.—If the heights of the barometer at the bottom and top of a hill are 29°37 and 26:59 inches respectively, and the mean tem- perature 26°, what is the height? Log. 29°37 = 1:467904 Log. 26°59 = 1:424718 Diff. of Logs. —= 0°043186 Mult. by 10000 431°86 Now 31°—26° = 5° temp. below 31°; therefore 73, of 431°86 — 4:96 ; conseq. 431:86 — 4:96 = 426:90 fathoms, the altitude of the hill. Such was the method formerly employed in measuring altitudes with this instrument; but later observations have shown, that it requires certain modifications, on account of the diffe- rence in low and elevated situations, the ex- pansion of the column of mercury, and other circumstances; which the limits of our article will not admit of investigating. Before M. de Luc began his experiments with the baro- meter, a mean of the two temperatures shown by the thermometer attached to the barometer and the height of the barometer, at the bottom - and top of a hill, were thought sufficient for determing its altitude. De Luc, however, found that an additional or detached thermo- meter was also necessary (see his “‘ Recherches sur les Modifications de l’ Atmosphere’); and this has been since confirmed by the experi- ments of General Roy and Sir G. Shuckburgh. The formule for the height in fathoms, ac- cording to the two latter observers, are as follow : ROY. £10000 1 = -468d} x }1 + (f—32°) 00245 + SHUCKBURGH. 7 $ 100001 .440d} x} 1 + (f—82%) 00248 t Where (= diff. oflogs. of the heights ofbarom. d — diff. of degrees Farhenheit’s therm. — mean of the two temp. shown by the detached thermometer ex- posed for a few minutes to the open air in the shade of the two stations. The sign— takes place when the attached ther- AL I mometer is highest at the lower station, and the sign + when it is lowest at that station. Exam.—Vind the height of a mountain, from the following observations taken at the foot and summit: Barom. Attach therm. Detach. therm. Low. stat. 29°862 ..... 6875 3 es Hig. stat. 26°137 GSRi\ Bs. 55° Here we have, d= 5°, diff. detach. therm. Asad (0. eens. .. f = 63°, mean of attach. therm. Log. 29°862 = 1:475119 Log. 26187 = 1:417256 Diff. of Logs. = 0-057863 = 1. Then, by the first formula, f—82°=319, and 1 + (31 x -00245)— 1:07595 10000 2 = 10000 x 057863 = 578°63 468d— ‘468x5 ...... be e4. 576°29 Multiply by 1:67595 620 fathoms, SR ee see een the altitude sought; the decimals being re- jected as unimportant. Ramsden’s engraved table gives the height —= 3730 feet, or 6213 fathoms. This table is on a slip of paper, a foot long and about 31 inches wide ; the logarithmic differences, from 25 to 31 inches, are given to 500ths of an inch, and the corrections for the thermometer at both stations found by inspection. It may be observed, that in determining altitudes by the barometer it is best to make the observations at the upper and lower sta- tions at one and the same time, as nearly as ean be; but great care must be taken that the two barometers, and also the thermome- ters, are alike ; that is, they should precisely agree when together in all states of the air. It is also necessary that the specific gravity of the mercury be well ascertained, because it is not equally pure in ali barometers; which is the principal reason why different results have been so frequently obtained, from observations made with different barometers at the same stations. Other circumstances, however, not generally known, may contribute to such dis- agreement: thus Mr. Ramsden proved, by experiment, that the quicksilver in barometer tubes, made of different sorts of glass, will be suspended at different heights. See De Luc, Recherches, &c. Horseley and Maskelyne, . Phil. Trans. vol. lxiv. p. 214 and 158. Trembley and Saussure, vol.ii. p.616. Damen Dissertatio, Phys. et Math. &c. Gen. Roy, Phil. Trans. vol. Ixvii. (1777) p. 658, &e. Sir G. Shuckburgh, ibid.513. And La Place, Mechanique Celeste, tom. ii. p. 289. ALTITUDE of the Eye, in Perspective, is a right line let fall from the eye, perpendicular to the geometrical plane. ALTITUDE of the Pyramids in Egypt, was measured so long since as the time of Thales, which he effected by means of their shadow, and that of a pole set upright beside them, making the altitudes of the pole and pyramid ALT proportional to the lengths of their shade Plutarch has given an account of the mar of this operation, which is one of the geometrical observations we have an e account of. ALTITUDE, in Astronomy, is the arch \ verticle circle, measuring the height of. sun, moon, star, or other celestial obj above the horizon. . This altitude may be either true or appar The apparent altitude is that which appear; sensible observations made at any place on surface of the earth. And the true altitud that which results by correcting the appari on account of refraction and parallax. The quantity of the refraction is differen different altitudes; and the quantity of parallax is diflerent according to the dista: of the different luminaries: in the fixed s} this is too small to be observed ; in the su is only about 83 seconds; but in the moo} is about 58 minutes, at a mean. The altitude of a celestial object may very accurately determined, by measuring) are of an oblique great circle intercepted | tween the star and the horizon, and the in nation of the same great circle to the hori This may be put in practice by means of: equatoreal, thus: Let the sine of the estim i altitude of the object be s; elevate the eq toreal circle above the horizon to an angle, sine of which = ¥ s, rad. being =1. The clination circle being set to 0, direct the | of collimation to the star, by the equator and azimuth circles moved in their own plan observe the are of the equatoreal circle int cepted between the index and VI; if the si of this are = p, the sine of the observed a tude will be equal to p v s, radius being This indirect method has many advantage and is, in general, less exposed to errors th the direct method, in about the proportion 1 to 7. Atwood’s Lectures, p. 198, 227. _ Meridian ALTITUDE of the Sun, or any lestial object, is an arch of the meridian int cepted between the horizon and the cenire the object upon the meridian. The altitu of a celestial body is greatest when it com to the meridian of any place (the poles of th carth excepted, for there the altitude of a fixt body is subject to no Variation) ; and the al tude of any star which sets not, is least, aj the depression of any star which does set, greatest, when in the opposite part of 2 | meridian, . & ALTITUDE of the Pole, is an arch of : meridian intercepted between the horizon the pole: it is equal to the latitude of place. ’ ALTITUDE of the Equinoctial, is the eleva of that circle above the horizon, and is alwai equal to the complement of the latitude. ite Refraction of ALTITUDE, is an arch of} verticle circle, whereby the altitude of an he venly body is increased by refraction. Parallax of Altitude, is an’ arch of a vertic circle, whereby the altitude is decreased parallax, pall AM B ALTITUDE of the Earth’s or Moon’s Shadow, eclipses. See Eciipse. ALTITUDE Instrument, or Equal Altitude In- wment, one used to observe a celestial object en it has the same altitude on the east and st sides of the meridian. bservations of this kind are made for the pose of obtaining the true time of the sun’s sing the meridian: various modes of cal- ation have been recommended at different es; but we know of none (independent of les) that is preferable to the following me- 1 of deducing the true time of the sun’s sing the meridian, by the clock, from a iparison of four equal altitudes, observed two succeeding days. The rule was in- ted by the celebrated Dr. Rittenhouse, the erican astronomer. uppose there are four sets of altitudes ob- ed on two successive days, (vz. one set in morning, and one in the afternoon, of each ) the instrument being kept at exactly the e height both days; then the exact time of sun’s passing the meridian per clock, may eadily obtained by the following wile. — Take the difference in time be- en the forenoon observations of the two 3, and also between the afternoon obser- ons. | \half the diff. of the two differences, X; half the sum of the two differences, Y ; ithe half interval between the two observa- ‘tions of the same day, be Z. hen, if the times of the altitudes observed ae second day be both nearer 12, or both er from 12 per clock than on the first ..X will be the daily variation of the k, from apparent time, and Y will be the y difference, in time, of the sun’s coming 1e same altitude, arising from the change eclination. And the proposition will be th: Y::Z: KE, the equation sought; which ‘be found the same (without any sensible rence) as the equation obtained from the BS. ut of one of the observations on the second be nearer 12, and the other more remote 12, than on the first day, ...then Y will me the daily variation of the clock from went time, and X will be the daily diffe- ein time of the sun’s being at the same e. And the proportion will be ... 24°: Ne he equation E, thus obtained, is to be racted from the mean noon, if the sun’s dian altitude be daily increasing; but to idded, if it be daily decreasing. ‘The on of all this is very plain; and its mode »plication so obvious that it is needless to examples in this place: several, how- , may be seen in the first volume of the ‘rican Transactions, whence the rule was acted. LTITUDE, Circles of, Parallels of, Quadrant c. See the respective words. MBIENT, is nearly the same as encom- ing; thus the atmosphere, which encom- es the earth on all sides, is called the AMP ambient air; or, sometimes, the circumam- bient air or fluid. AMBIGENAL Hyperbola. BOLA. AMBIGUOUS Case, in Trigonometry, is that which arises in the solution of a problem, in which an acute angle and its opposite side are two of the given parts, and one of the sides about the given angle is the third part. With these data, the angle opposite the unknown side may be either obtuse or acute, because every sine answers to two angles which are supplements to each other; that is, the question admits of two solutions, and is hence denominated the ambiguous case. See SINE and TRIANGLE. Ameicuous Sign, in Algebra, is that in which both plus and minus enter; being writ- ten thus, +, and is read plus or minus. AMBIT, in Geometry, the same as Prri- METER. : AMBLIGON, or AMBLIGONAL, obtuse, an- gular. ; AMICABLE Numbers, are those pairs of numbers of which each of them is equal to all the divisors, or aliquot parts, of the other; thus, 234 and 220 are a pair of amicable num- bers; for See Hyper- 284 284 284 284 284 Por states 2 + FL + [a2 + 584 — 220 220 , 220 , 220 , 220 , 220 , 220 , 220 Tyr age binge hag) ppt cog tog &e, = 284. The only three pair of amicable numbers, at present known, are 284 and 220 17296 .... 18415 9363538 .... 9437056 Which were found by F. Schooten, sect. 9. of his *“ Exercitationes Mathematice ;” and who seems to have been the first author that distinguished them by the term amicable num- bers; though they were before treated of by Rudolphus, Descartes, and others. In order to find a pair of amicable numbers, we must first determine such a power of 2, as 2"; that the following formule may be prime numbers, viz. 3.2" —1 =a prime 6.2” —1—=a prime 18 .2?"— 1 =a prime then will 2"+* (18 .2*"—1), and 2”+*(3,2"—1) (6 . 2"—1), be the pair of amicable numbers required. See a demonstration of this theorem in Bar- low’s “ Elementary Investigations, &c.” AMONTONS (WILLIAM), an ingenious French experimental philosopher, author of a treatise “‘On Observations and Experiments concerning Barometers, Thermometers, Hy- groscopes, Hour-glasses, &c. ;” also of a “ New Theory of Friction.” He was born in 1663, and died in 1705. Many papers of this philosopher were published in the Memoirs of the Academy of Sciences for the years 1698, 1699, 1702, 1703, 1704, and 1705. AMPLITUDE, in Astronomy, is an arch ANA of the horizon, intercepted between the true east and west points, and the centre of the sun ora star at its rising or setting; so the amplitude is of two kinds; ortive or eastern, and oceiduous or western. These are also called northern or southern, as they fall in the northern or southern quarters of the horizon; and the compliment of the amplitude, or the distance of the point of rising or setting from the north or south point of the horizon, is called the azimuth. Lo find the sun’s amplitude, having the latitude and the sun’s declination given. Say, as the cosine of the latitude is to ra- dius, so is the sine of the sun’s or star’s decli- nation to the sine of the amplitude. Exam. Let it be required to find the am- plitude of the sun in the latitude of London (51° 32’); the declination being 23° 28’, As cosine 51°32’.......... 9°7938317 18 tO Tadsds... oe FS ke 10°0000000 SO is sin. dec. 23° 28'....... 9°6001181] To amplitude 39° 48’...... 9°8062864 And this is of the same name with the de- clination; viz. north, when the declination is north; and south, when it is south. AMPLITUDE, in Projectiles, the range of the projectiles, or the right line upon the ground, subtending the curvelinear path in which it moves. AMPLITUDE, Magnetical, is an arch of the horizon, contained between the sun or a star at its rising or setting, and the magnetical east or west point of the horizon indicated by the magnetical compass, or the amplitude or azimuth compass; or it is the difference of the rising or setting of the sun, from the east or west points of the compass. In order to ascertain this amplitude, place the compass on a steady place, from which the horizon may be clearly seen, and look- ing through the sight vanes of the com- pass, turn the instrument round till the cen- tre of the sun or star may be seen through the narrow slit, which is one of the sight vanes, exactly in the thread which bisects the aperture in the other sight vane; and when the centre of the celestial object, whether rising or setting, is just in the horizon, push the stop in the side of the box, so as to stop the card, and then read the degree of the card which stands just against the siducial line in the box; and this gives the amplitude re- quired. Then subtracting from this ampli- tude the known or true amplitude, and the difference will be the variation of the mag- netic needle. ANABIBAZON, in Astronomy, the Dra- gon’s Tail. ANACAMPTICS, the science of the re- flection of sounds, frequently in reference to echoes, which are said to be sounds anacamp- tically, or by reflection. ANACHRONISM, (from ava, higher, and xpoves, time) in Chronology, an error in the ANA computation of time, by which an even placed earlier than it really happened. ANACLASTIC Curves, a name given } Mairan to certain apparent curves formed the bottom of a vessel of water, to an e placed in the air; or the vault of the heaven seen by refraction through the atmosphere, ANACLASTIC Glasses, a kind of sonoror phials or glasses, chiefly made in German which have the property of being flexible an emitting a vehement noise by the huma breath. These glasses are a low kind of phia with flat bellies, resembling inverted funne whose bottoms are very thin, scarce surpas ing the thickness of an onion peel. bottom is not quite flat, but a little convey but upon applying the mouth to the orifice an gently inspiring, or as it were sucking out t} air, the bottom gives way with a prodigio crack, and from convex becomes concay¥ On the contrary, upon expiring, or breathi gently into the orifice of the same glass, #] bottom with no less noise bounds back to! former place, and becomes gibbous as %b fore. * ANACLASTICS, or Anacwatics, ( ave and xAaw, I break) an ancient name fi that part of optics which relates to refraeti light; being the same as is now denominatt} dioptrics. mi ANALEMMA, (from avercuCavw, DT tal backwards) a planisphere or projection of 4 sphere, on the plane of the meridian, orth} graphically made by perpendiculars fro} every point of that plane; the eye being su posed to be at an infinite distance, and int east or west point of the horizon In #} projection, the solstitial colure, and all i! parallels, are projected into co-centric cir pit equal to the real circles of the sphere ;_ / at t all circles whose plane pass through the € as the horizon and its parallels, are projee! into right lines equal to their diameters ; ] all oblique circles are projected into ellips having the diameter of the circles for transverse axis. See PROJECTION. Analemma is also used for a gnomon, astrolabe; consisting of the same projectit drawn on a plate of brass or wood, with} horizon fitted to it; and it is used for find the time of the sun’s rising or setting, f length of the longest day in any latitude, hour of the day, &c. It is also of consid able use to dialists. ah The most ancient treatise on this i stil ment now extant, was written by Ptolen|i and printed at Rome in 1562, with a coll! mentary by Commandine. Since that art many authors, as Aquilonius, Jacquet, Dy chales, &c. have written on the same subjet ANALOGOUS Quantities, are quantit of the same kind. St ANALOGY, in Mathematics, is the sai as PROPORTION. if ANALYSIS, (from ayaruw, to resolve) generally the method of resolving mathem col problems, by reducing them to equatiol and may be divided into ancient and moderi ANA The ancient analysis, as Pappus has de- ‘ibed itin his “*‘ Mathematical Collections,” . Vii. p. 157, ed. Commendini, Pisans, 1588, the method of proceeding from the thing ight taken for granted, through its conse- ences, to something that is really granted known; in which sense it is opposed to thesis, or composition, which commences h the last step of the analysis, and traces several steps backwards, making that in s case antecedent, which in the other was isequent, till we arrive at the thing sought, ich was assumed in the first step of the lysis. The principal authors on the ancient lysis, enumerated by Pappus ubi supra, Euclid, in his “ Data,” “ Porismata,” and e Locis ad Superficiem;” Apollonius, “De tione Rationes,” “De Sectione Spatii,” fe Tractionibus,’ ‘“ De Inclationibus,” fe Locis Planis,” and “ De Conicis ;” staeus, ‘““De Locis Solidis;’’ and Eratos- 1es, “‘ De Mediis Proportionalibus.” Pap- himself, who has given many examples ithe preceding writers, may also be added he above number. his analysis has also been cultivated by iy of the moderns; as Fermat, Viviani, taldus, Snellius, Huygens, Simson, Stew- Lawson, Hugo d’Omerique, &c.; particu- P by the latter, in his “ Analysis Geome- ‘ontucla, in his Histoire des Math. has mn an example, illustrating the method ued by the ancients in their analysis, *h at this day would be considered very yus; and for that reason we shall omit 1 this place, and barely explain the ern method of considering the same ct. The example above cited is as fol- ‘oB. 1. From the extremities of the base d B, of a given segment of a circle, it is ired to draw two lines AC, BC, meeting 1e point C, in the circumference, so that ‘shall have a given ratio to each other, ose that of F to G. ANALYSIS, ppose the thing done, wiz. that AC:BC:: F:G, let there be | n BH, mak- C eangle ABH \{f~ to the angle 3, and meet- B oe .C produced Then the A being also 10n, the two triangles ABC, ABH, are ngular; and, therefore, AC:BC: AB: BH given ratio; also A B being given, BH en in position and magnitude. SYNTHESIS, struction. Draw BH, making the angle {, equal to that which may be contained ANA in the given segment, and take AB to B H, in the given ration of F to G. Draw ACH and BC, Demonstration. The triangles A B C, ABH, are equi-angular, therefore AC:CB::AB:BH, which is the given ratio by construction. Pros. 2. To inscribe a square in a given triangle. ANALYSIS. Let ABC be the given trian- gle, and suppose DEFG to be a rib Square inscribed init:jomAEand 2 produce it till it A GHFB I meet CK, drawn parallel to the base A B in K, and demit HI perpendicular to A B, pro- duced if necessary. Then D EFG is a square by hypothesis, CK is parallel to DE, and HI to EF; and therefore by similar triangles AE:AK::DE >CK AF: AK ::EF=DE: KT; whence, since the three first terms in each set of proportionals are equal, the fourth must necessarily be so likewise; that is, KI —= CK—CH, or the figure CKIH, is a square, the side of which is equal to the per- pendicular of the triangle ABC. Hence the construction. SYNTHESIS. Construction. On the given perpendicular CH describe the square CKIH; join AK, and from the point E, in the side CI, draw EF perpendicular, and ED parallel to A B, draw also DG parallel to EF; so shall DEF G be the square required. Demonstration. For, by similar triangles, AK: AE:: CK >: DE and AK: AE:: KI=CK: EF; therefore, since the three first terms in each set of proportionals are equal, the fourth are also equal; thatis, DE EF, and DG=EF, and DE =GE by construction; and there- fore the figure DEF'G is equilateral, and is also equiangular by construction; it is there- fore a square, and it is inscribed in the given triangle as required. Modern ANALYSIS comprehends algebra, arithmetic of. infinites, infinite series, incre- ments, fluxions, &c.; for an account and illus- tration of each of which see the respective articles. To this branch of science we are in- debted for the noble inventions and improve- ments that have been made in mathematics and philosophy, for the last two centuries. It furnishes the most perfect examples of the manner in which the art of reasoning should be employed; it gives to the mind a wonderful skill for discovering things unknown by a few things given; and by employing short and easy symbols for expressing ideas, it presents to the understanding things which would otherwise be beyond its sphere. By means E ANA of this, geometrical demonstrations may be abridged; a long train of reasoning aided and facilitated by visible symbols. By this artifice, a great number of truths may be expressed in a single line; which, in the ordinary process, would occupy many pages: and thus, by con- templation of one line of calculation, we may acquire in a short time the knowledge of a whole science, which without this aid could scarcely be comprehended in several years. It is said, indeed, that Newton, who well knew the advantages of analysis in geometry and other sciences, frequently regretted that the study of the ancient geometry should be neg- lected and abandoned ; and it must be allow- ed that the method employed by the ancients, in their geometrical writings, is more rigorous than the modern analysis; and though it be greatly inferior to the moderns, in point of dispatch and facility of invention, it is never- theless highly useful in strengthening the mind, improving the reasoning faculties, and accustoming the youthful mathematician to a pure, clear, and accurate mode of investiga- tion and demonstration, though by a long and laborious process; to which he would reluc- tantly have submitted if his taste had been vitiated, if we may use the expression, by the modern analysis. On this circumstance were principally founded the complaints of New- ton; who feared, lest by too early use of the modern analysis, the science of geometry should lose that rigour and purity which cha- racterise its investigations; and the mind become debilitated by the facility of our analysis. He was, therefore, fully justified in recommending, to a certain extent, the study of the ancient geometricians, whose demon- strations, being more difficult and operose, afford greater exercise to the mind, accustom it to closer application, extend its views, and habituate it to patience and resolution, so necessary for making discoveries. This, how- ever, is the only, or principal, advantage re- sulting from it; and upon a comparison of both methods, the result seems to be that the analysis of the ancients is the best adapted to the commencement of our studies, as it serves to inform the mind, and to fix proper habits ; and that the modern should succeed, and is best suited to extend our views beyond the present limits, and to assist us in making new discoveries and improvements. Montucla’s Hist. des Math. tom. i. p. 166. Hutton’s Dict. vol. i. Rees’s Cyclop. vol. ii. Analysis is divided into finite and infinite, determinate, indeterminate, and residual; ana- lysis of Powers, of Curves, &c; for which see the respective terms. The principal early writers upon the ana- lysis of infinites, are Sir Isaac Newton, in his * Analysis per Quantitatum Series, Fluxio- nes, &e.” “ De Quadratura Curvaruin ;” Leibnitz, in Act. Eroditor. an. 1684; the Marquis de |’Hopital, in his “ Analysis des Infiniments petites,” 1696; Carre, in his ‘“* Method pour la Mesure des Surface, &c.”’ -“ De Constructione Aiquationum, &c,” 17% AND 1700; G. Manfredi, in a posthumous pie to which we may add, Mercator, Chey Craig, Gregory, and Walmsley. Those @) later date who have contributed to bring 1 science to perfection, are Euler, Simps Maclaurin, Lagrange, and La Croix ; to wh we might add a numerous list of other thors, of less celebrity. ) ANALYST, an algebraist, or one skil in analysis in general. Anaxyst is also the title of an ingeni tract, written by the celebrated Dr. Berkel against the doctrine of fluxions. ANALYTIC, or ANALYTICAL, any th partaking of the nature of analysis; or } formed by that method. ANALYTICS, the science, or doctri and use of analysis. ANAMORPHOSIS, in Perspective + Painting, a monstrous projection, or a rej sentation of some image, either on a p or curve surface, deformed or distorted ; | which, in a certain point of view, shall app regular, and drawn in just proportion. | Wolfius’s Catoptrics and Dioptrics, Pr ley’s History of Vision, and other opt authors. ANASTROUS, signs in Astronomy, ani given to the duodeeatemoria, or the twe portions of the ecliptic, which the signs ciently possessed, but have since deserted the precession of the equinoxes. ANAXAGORAS, one of the most ¢ brated philosophers amongst the anci who is said to have written on the Quadr of the Circle. He was born at Clazomem Tonia, about 550 years before Christ. ANAXIMANDER, a philosopher of letus, the disciple and successor of 'T He had a considerable knowledge of as nomy and geography, and first noticed obliquity of the ecliptic; he taught that moon received her light from the sun, that the earth is globular; to him is ase the invention of the sphere, the gnomon,, reographical charts. He lived about 600 y before Christ; at which period he makes obliquity of the ecliptic to be twenty-fo rees. : : ANAXIMENES, the pupil and suce¢ of Anaximander; to whom Pliny attri the invention of sun-dials. | ANDERSON (ALEXANDER), a celebr Scotch mathematician, who flourished al the beginning of the seventeenth century, published several works of his own, amt friend Vieta’s, as follows: . “ Supplementum Apollonii Redivivi,” Ghetaldus) Paris, 1612, 4to. * “ Prancisci Viete Fontenacensis de quationem, &c.” with an appendix by An son, Paris, 1615, 4to. a The Angular Sections of the same ail with the demonstrations by Anderson. — ANDROMEDA, a coristellation of northern hemisphere. See ConsTELLAT ‘al ANE ANELAR, the star marked «, in the head Castor. ANEMOMETER, (compounded of aysos, nd, and wergov, measure) in Mechanics, a ichine for measuring the force of the wind. 1e anemometer is variously contrived. The st of the kind seems to have been invented Wolfius in 1708; and an account of it first blished in his “ Aerometry” in 1709, and o in the “ Acta Eruditorum” of the same ar, and afterwards in his “ Mathematical ictionary,” and also in his “ Elementa Me- sseos,” vol. ii. p. 319. This machine of Wolfius’s was much im- oved by Martin (see fig. 3, plate II, Pneu- tics). Anopen frame of wood ABCD, &e. supported by the shaft or arbor I. In the 9 cross pieces HK LM, is moved a hori-~ ital axis QM, by means of the four sails em, of, xh, exposed to the wind in a proper nner. Upon this axis is formed a cone of d, MNO; upon which, as the sails move nd, a weight R or S is raised by a string ud its superfices, proceeding from ‘the iller to the larger end NO. Upon this ye end, or base of the cone, is fixed a et wheel K, in the teeth of which falls click X, to prevent any retrogade motion the depending weight. The construc- of this machine sufficiently shows that it y be accommodated to estimate the va- le force of the wind; because the force of weight will continually increase, as the ag advances on the conical surface, by ng at a greater distance from the axis of ion ; consequently, if such a weight be ed on the smaller part M, as will just keep machine in equilibrio in the weakest wind, weight to be raised as the wind becomes nger will be increased in proportion, and liameter of the cone NO, may be so large omparison of that of the smaller end at M, the strongest wind shall but just raise weight at the greater end. i the Philosophical Transactions for 1766, A. Brice describes a method which has 1 successfully practised by himself, of suring the velocity of the wind by means at of the shadow of clouds passing over surface of the earth. .d@’Onsen Bray invented a new anemo- r, which of itself expresses on paper not the several winds that have blown dur- the space of twenty-four hours, and at : hour each began and ended, but also lifferent strengths and velocities of each; Mem. Acad. Sciences, an. 1734, p- 169; also a description of some other instru- 's of this kind, in Encyclopedia Britan- and Rees’s Cyclopedia. NEMOSCOPE, (from avexos, wind, and v, I see) a term sometimes given to a -dial, which points out the course of the by an index that is connected with the lle on which the vane works. The term 30 applied to little machines which are ded to foretel the changes of the weather. AYGROSCOPE. ANG ANGLE Angulus, in Geometry, is formed by the opening ° f B or mutual incli- nation of two lines meeting in C a point; such is DP the angle BAC} or BAD. Note. When an angle is letters, that at the angular point must be read in the middle. But sometimes, for brevity sake, if there be but one angle at a point, that angle is denoted by the single letter standing at that point. ANGLES are of several different kinds or denominations, as rectilinear, curvilinear, sphe- rical, mixed, solid, &c. Rectilinear ANGLE, is that which is formed by the meeting of two right lines. Curvilinear ANGLE, is that which is formed by the meeting of two curve lines, Mixtilinear ANGLE, is formed by the meet- ing of a right line and curve. Spherical ANGLE, is that which is formed on the surface of a sphere, by the intersection of two great circles. See SPHERE, and SPHE- RICAL Trigonometry. Solid ANGLE, is formed by the mutual in- clination of more than two planes, or plane angles, meeting in a common point. See Solid ANGLE. Properties and Denominations of Rectilinear Angles. denoted by three Right ANGLE, is that which is formed by one line perpendicu- Jar to another; E % or that which is : subtended by a * y: quadrant of a cir- noe (ae cle; as the an- Ne gle BAC. All right angles are equal to one another. An Oblique ANGLE is that which is greater or less than a right angle; and these are dis- tinguished into two kinds, acute and obtuse. An Acute ANGLE is less than a right angle; as DAC. An Obtuse ANGLE is greater than a right angle; as EAC, Adjacent ANGLEs, are the two angles form- ed by one line meeting ano- ther, any where exceptatits ex- tremities; such p G A are the two an- gles BAD and BAC. These angles are said to be supplements to each other, their sum being equal to two right angles. Vertical, or Op- posite ANGLES, are such as have their legs mutual, conti- nuations of each other; as BAC and DAC. E2 B c ANG Vertical, or opposite angles, are equal to each other. Alternate ANGLES, are those made on the opposite sides of a line cutting two other — lines; as AFG and DGF. And if these | two lines are parailel, the alternate angles are equal. - External ANGLES, are those formed by the sides of any right-lined figure, and the adjacent sides pro- duced; such are the an- gles A, B, C, &c. The sum of all the ex- ternal angles of any figure is equal to four right an- gles. Internal ANGLES, are the angles within a figure, formed by the meeting of each two adjacent sides; as the angles a, 6, c, &c. The sum of all the inward angles of any right-lined figure, is equal to twice as many right angles, wanting four, as the figure has sides. An ANGLE at the Centre of a Circle, is that whose angular point is at the cen- tre; such is the angle ACB. An ANGLE at the Circum- ference, is that whose an- gular point is in any part of the circumference ; as the angle A DB. An angle at the centre is double an angle at the circumference, when both stand on the same are. An ANGLE in a Semi-Circle, is an angle at the circumference contained in a semi-circle, or standing upon a semi-circle or diameter. An angle in a semi-circle, is a right angle. An angle ina segment, greater than a semi- circle, is less than a right angle. An angle ina segment, less than a semi- circle, is greater than a right angle. ANGLE of Contact. See CoNnract. Angles of other less usual denominations are used by some authors, as Horned ANGLE, that which is formed by the circumference of a circle and a right line. Tnnular ANGLE, is formed by two curve lines, one concave and the other convex. Cissoid ANGLE, the inward angle, formed a the intersection of two spherical convex ines. & \A D PROBLEMS. 1. To bisect a given Angle, BAC. From the centre A, with any radius, de- scribe an are cutting off the equallines AD, AE; and from the two cen- tres D, E, with the same | radius describe arcs in- ANG tersecting in F, then draw AF; which wi bisect the angle as required. . 2. Ata given point A in the line AB, to mal an Angle equal to a given Angle C. From the centresA and C, with any one radius, C describe the | arcs DE, BG; then with the centre B, ar radius D E, describe an are cutting BG in{ Through G draw the line AG, and it w form the angle required. 3. To Measure the quantity of an Angle on Papt Apply the centre of a protracter to the ve tex of the angle, so that the radius may Col cide with one of the lines; and the degr shown by the other line, will give the me sure of the angle required. 7 Otherwise with the Line of Chords. With a radius equal to the chord of 6f describe an arc between the lines forming t angle; then apply the subtense of this a to the same scale of chords, and it will gi the measure sought. ANGLES, in Astronomy, receive the follo ing particular denomination: as ANGLE Commutation, of Elongation, of Position, & for which, see the respective terms. _y ANGLES, in Mechanics and Optics, also. ceive particular denominations, and are d tinguished into separate orders; as ANGI of Direction, Elevation, Inclination, Inflecti Incidence, Refraction, &c.; for which, see respective terms. i Optic ANGLE, is the angle included - tween two rays, drawn from the two extr points of an object to the centre of the pv of the eye. 4. ANGUINEAL Ayperbola. See Hye BOLA. y ANGULAR, something relating to, orl ing angles. M ANGULAR Motion, is that which is perfol ed by an oscillating or vibrating body, as ferred to the angle which it describes or pas over ina given time, the vertex of whie the point of suspension, or centre of me Hence all points in a pendulum have same angular motion, although their abso motions are different from each other, be greater or less, according to their dista from the centre of suspension. 4 ANGULAR Motion is also sometimes usel denote a motion which is partly curvilin and partly rectilinear ; as the motion ¢ ‘coach-wheel on a plane. 5 ANGULAR Sections, is a term used by Y to denote a species of analytical trigonome relating to the law of increase and deer of the sines and chords of multiple ares, W he first pnblished in his “ Canon Mathe ticus,” in 1579; a work which is now extret scarce, in consequence of its author hav soon after its publication, destroyed all copies that he could gain possession ol, ANN ccount of some inaccuracies which he dis- overed in the impression, Vieta demon- trated that if _ semi-circle as .B be divided ato equal arcs 3D, DE, EF, tee 'G, &c. andthe B a adius be taken qual to unity, and the supplemental chord JA — 2; then will E ABs AD=2 AE = 2z*—2 AF =2x*— 32 AG = 2*+— 427 + 2 AH = 2 — 523 + bx AI = 2° — 62+ + 9x? —2 &e. ke. Also, if the chord DB be taken equal to y, d the radius equal to unity as before; then ul BD y BE=2 — y BF=3 + ¥ BG=2 —4y’+y BH = 5y— 5y3 + 7 &c, These properties, in the present state of- ‘igonometry, are readily demonstrated; but 1 the state of the science in Vieta’s time, ieir investigation was attended with con- derable difficulty, and amply displays the yperior genius of their celebrated author. ee Montucla’s Hist. des Math. tome i. . 607. ANIMATED Needle, a needle touched with magnet. ANNUAL, in Astronomy, any thing which ates to the year, or which returns yearly. s ANNUAL Motion of the Earth, Argument of ongitude, Epacts, Equation, &e.; for which, @ EarTH, ARGUMENT, Epact, EQuaTIon, — — ~ re -ANNUITIES, signify any interest of mo- 2y, rents, or pensions, payable from time to ne, at particular periods. ‘The most general division of annuities is to annuities certain, and contingent annuities ; e payment of the latter depending upon me contingency; such, in particular, as the mtinuance of a life. ANNUITIES have also been divided into an- iities in possession, and annuities in reversion ; e former meaning such as have commenced, are to commence immediately; and the tter, such as will not commence till some wticular future eyent has happened, or till me given period of time has expired. ANNUITIES may be farther considered as sing payable yearly, half-yearly, or quarterly. The present value of an Annuty is that sum hich being improved at compound interest, ill be sufficient to pay the annuity. The present value of an Annuity certain, yable yearly, and the first payment of which | to be made at the end of a year, is com- ited as follows : ANN Let the annuity be supposed £100.; the present value of the first payment of it, or of a hundred pounds to be received a year hence, is that sum in hand, which being put to interest will amount to £100. in a year. In like manner, the present value of the se- cond payment, or of £ 100. to be received two years hence, is that sum which being put to interest immediately will amount to £€ 100. in two years; and so on for any number of years or payment; and the sum of the values of all the payments will be the present value of the annuity. Let the interest be supposed to be 4 per cent, ‘The sum which improved at 4 per cent. interest for the year, will produce £100. at the end of the year, is that sum which has the same proportion to 100 as 100 has to 104; the sum of the interest and principle. Say there- fore, as 104: 100:: 100: oe = 96°15, value of the first payment. And that sum which in two years will amount to £100. at the same rate of interest, is evidently that which in one year willameunt to £96°15. because we have seen that this sum will, in a year, amount to £100.; we have, therefore, on the same principle as above, 96°15 104: 100 :: 96°15: jo. = 92°45 val. 2d. pay. 104 : 100 :: 92°45: a = 85°48...... 3d. pay 104: 100 :: 85°48 : — = 82°19......4h. pay. &e ke. &e. But since the interest of money bears a constant ratio to principle, we may represent these proportions more conveniently by as- suming generally £1. as the yearly payment, andr as the amount of £1. for one year; and whatever results are thus produced, being multiplied by the yearly payment, will be the value of the annuity in such case. By this means the above proportions become YEN eg hoy ~ value Ist. payment hinted DSR gto, asda ; nt ? Cpr 2d. payme 1a 73-1 Bot oy eee 3d. payment &e.,._ dc. &e. The value, therefore, of an annuity of £1. for n years, is equal to the sum of the series. I — ri pe and therefore when the annuity is perpetual, or in perpetuity, the value of it is equal to the sum of the above series continued in in- jimitum. Now, in order to find the sum of this series, let us make it equal to v; then since 1 ee | < : + 3 + &c. , haa ye —] ye Vv 1 1 1 l or=lt+— tat st & aS ANN 1 whence, (*— 1)» = 1 hie 7 — J ] or Sees teeeseecees seoee v si Bcd —— x eo ; , ja 7” which is a general formula for the present ‘value of an annuity of £1. for n years. And since an annuity of a pounds per year wll be worth a times as much; we have generally m—) a r— 1 for the value of an annuity of a pounds per year, for » years. If the annuity is to be at s equal distant times in the year, then » instead of repre- senting the amount of £1. for a year, must be taken for the amount of £1. for the time of the first payment; also n, which is the number of payments in the above, will be- ‘ a come ns, and a will become ~— whence the n more general formula will be ea ymn zy . a era | ci a! —_— er — eer _— _———— -—— r—] mrn Poy yn Bolte ae therefore as above, = the annuity, or yearly rent ; n = the number of years, or payments ; , ry = the amount of £1. for a year, or for one payment; = the present value of the annuity; m = the whole amount; we shall have the several cases of annuities expressed by the following formule: rm” — 1 a eet ers, _ zs y—l + ees 1 Bete INS gy meee x vr” r'— ere hs =, SPs ivi ? ; xX @ 1 4 ie log. m—log. v S@ e668 — log. , ito Nae log. m—log.v py n 1 a Ge issi.. Be oe ae aie In this last theorem, R denotes the present value in reversion, after p years, or not com- mencing till after the first p years, being found by taking the difference between the two values 7? = | ¢ @ ond? ~ PES | re a r—} rn r— 1 ve Let us now illustrate these formule by a few examples. Required the present worth of an annuity of £100. per annum for 5 years, at yearly payments, half-yearly payments, and quarterly payments; the rate of interest being 4 per cent. in each case. In the first case,.* = 1°04, n= 5, anda 100; therefore, by formula 1, we have whence, by formula 2, we obtain ANN 04 00. | ° = 10d ee Res the second case, 7 = 1°02, »n = 10, an: 50; therefore 1:02?°— ] 50 arog — 1. * Tsar = 2450 1S In the third case, + = 1:01, n= 20, and = 25; therefore _ 1:017°— 1 25 = Tie. § ee eee Hence, it is evident that the greater th number of payments, the greater is the pre sent value of the annuity. What annuity, payable yearly, may be pw chased for 5 years with £2226.; taking th rate of interest at 4 per cent.? Here we have v = 2226, r = 1:04, n mé 1:04—1 shew inetaecind ks en (145i @ =o] X 2226 x 104 = £500. pe annum, nearly. . Required the whole amount of an annuil of £100. per annum, for 20 years, at 5 pe cent. per ann. yearly payments. Here we have r = 1°05, n= 20, a= 100. mmo n= x 100 = £3306. 12s. 1°05 — 1 Required the amount of the same annuit at half-yearly payments. Here r = 1:°025, n = 40, a = 50. Po} yr —] igh ( nt ere | . 1:0254° —1 -—————— X 50. == £3370, Oras 025 —1 a ) Required the amount of the same, at | terly payments. Here r = 1°0125, n = 80, a = 25; Ry Ope Sh ee BN Ee er ees 25 = £3402. 16 Whence again it follows, that the great the number of payments, the greater willl the amount of the annuity. To find the present value of an annuity the following table, we have only to find th amount for £1. at the given rate of interes and for the given time; which multiplied the given annuity, or payment, will be tt present worth. Exam. What is the present value of a annuity of £40. per ann. to continue 20 year at the rate of 4 per cent.? By the table, the amount of £1. for 20 year at 4 per cent. is 13°590326; therefore 13°590326 x 40 — £543. 12s. very nearly. For what relates to life annuities, see Lit Annuities and ASSURANCES; see also th works of Price, and Morgan, on this subjec and an excellent treatise on the Doctrine | Annuities, by F. Baily. ft sa therefore m= ANN A.N.N TABLE. 1e present Value of an Annuity of £1. per Ann. for any number of Years not exceeding 60, at any rate of Compound Interest from 3 to 6 per Cent. Pioudk tal Atay 2” per Cent. 5 per Cente At 3 per Cent. 34 per Cent. 4 per Cent. 6 per Cent, + OMNA A WW = 970874 1.913470 2.823611 3.716098 4.579708 5.417191 6.230283 7.019692 7.786109 8.530203 9.252624 9.954004 10.634955 11.296073 | 11.937935 12.561102 13.166118 13.753513 14.323799 14.877475 15.415024 15.936917 16.443608 16.935542 17.413148 17.876842 18.3270381 18.764108 19.188455 19.600441 20.000428 20.388755 20.765792 21.131837 21.487220 21.832252 22.167285 22,492462 22.808215 23.114772 23.4) 2400 23.701359 23.981902 24.254274 24.518713 24.775449 25.024708 25.266707 25.501657 25.729764 25.951 227 26.166240 26.374990 26.577660 26.774428 26.965464 27.150936 27.33 1005 27.50583 1 27.675564 966184 1.899694 2.801637 3.673079 4.515052 5.328553 6.114544 6.873956 7.607687 8.316605 9.001551 9.663334 10.302738 10.920520 11.517411 12.094117 12.65132] 13.189682 13.709837 14.2124038 14.697974 15.167125 15.602410 16.058368 16.481515 16.890352 17.285364 17.667019 18.035767 18.3892045 18.736276 19.068865 19.390208 19.700684 20,000661 20.290494 20.570525 20.841087 21.102500 21.355072 21.599104 21.834882 22.062689 22.282791 22.495450 22.700918 22.899438 23.091244 23.276564 23.455618 23.628616 23.795765 23.957260 24.113295 24.264053 24.409713 24.550448 24.686423 24.817800 24.944734 261 538 1.086095 2.775091 3.629895 - 451822 5.242137 6. 002055 6.732745 7.435032 8.110896 8.760477 e 385074 9.985648 10.563123 11.118387 11.652296 12.165669 12.659297 131 1338 3Q - 13.896326 14,029160 14.451 115 14.856842 15,246963 15,622080 15 982769 16.329580 16.663063 16.983715 17.292033 17.588494 17.873551 18.146674 18.41 1195 18.664613 18.908282 19.142579 19.367864 19.584485 19.792774 19.993052 20.185627 20.370795 20,548841 20.720040 20.884652 21.042936 21.195131 21.3841472 21.482185 21.617485 21.747582 21.872675 21.992957 22.108612 22,.219819 22.326749 22.429567 22.528430 22.623490 956938 _ 1.872668 2.748964 3.587526 4.389977 5.157872 5.892701 6.595886 7.268790 7.912718 8.528917 9.118581 9.682852 10.222825 10.739546 11,.234015 11.707191 12.159992 12.593294 13.007936 13.404724 13.784425 14.147775 14.495478 14.828209 15.146611 15.451303 15.742874 16.021889 16.288589 16.544391 16.788591 17.022862 17,.246753 17.461012 17.666040 17,.862240 18.049990 18.229656 18.401 584 18.866109 18.7235590 1§.874210 19,018383 19.156343 19.28337 1 19.414709 19.535607 19.651298 19.762008 19.867950 19.969330 20.066345 20.159181 20.248021 20.333034 20.414387 20.492236 20.566733 20.638022 952381 1.859410 2.723248 3.545950 4.329477 5.075692 5.786373 6.463213 7.107822 7.721785 8.506414 8.865252 9.393573 9.898641 10.579658 10.837770 11.274066 11.689587 12.085321 12.462210 12.821183 13.163003 13.488574 13.798642 14.093945 14.375185 14.643034 14:898127 15.141074 15.372451 15.592810 15.802677 16.002549 16.192904 16.374194 16.546852 16.711287 16.867893 17.01L7041 17.159086 17.294368 17.423208 17.5459 12 17.662773 17.774070 17.850066 17.981016 18.077158 18.168722 18.255925 18,.338977 18.418073 18.493405 18.165146 18,633472 18.698545 18.760519 18.819542 18.875754 18.929290 943396 1.883393 2.673012 3.465106 4.212364 4.917524 5.58235 1 6.209794 6.801692 7.360087 7.886874 8.383844 8.852683 9.294984 9.712249 10, 105895 10.477260 10.827603 1L.SS116 11.469921 11.764077 12.041582 12.303379 12.550358 12.783356 13.003 166 13.210534 13.406164 13,590721 13.764831 13.929086 14,084043 14.230230 14.368141 | 14.498246 14.620986 |. 14.736780 14,.846019 14.949075 15.046297 15. 158016 15.224543 15.306173 15.883182 15.455832 15,524370 15.589028 15.650027 15.707572 15.761861 15.813076 15,861893 15.906974 15,949976 15.990543 16.028814 16.064919 16.098980 16.131113 16.161428 ANO ANNULAR Eclipse, (from annulus, a ring) is an eclipse of the sun, in which the moon appears less than the sun; whence the eclipse which would otherwise have been total, be- comes an annular one; leaving a bright ring round the sun’s disc: such was the eclipse in 1764, which was observed in Spain, France, and England. The diameter of the moon is 29° 25’ in its apogee, and 33°37’ in its perigee; the diameter of the sun is 31°31 in its apogee, and 32° 36’ in its perigee; whence one may readily perceive that there ought to be fre- quent annular eclipses. The eclipses of 1737 and 1748, were annular in Scotland; the latter of which was observed by Le Monnier, who undertook a voyage from Paris to Edinburgh for that purpose; and to measure the diame- ter of the moon, as it appeared on the sun’s disc. The greatest duration of an annular eclipse is 12’ 24” of time; (Mem. de l’Acad. 1767) which exceeds the greatest duration of a total eclipse by 4’ 26”. See Ec ips. ANOMALISTICAL Year, in Astronomy, called also Periodical Year, is the space of time in which the earth, or other planet, passes through its orbit; which is longer than the tropical year, by reason of the precession of the equinox. For example, the tropical revolution of the sun, with respect to the CQUIMOK, Bein eticct Mi Ahh erssseec sh 3654 5h 48™ 45s but the siderial, or return to) , the same star...... .... 365 69 11 and the anomalistic emaTition as 365 6 15 20° because the sun’s apogee advances every year 652” with respect to the equinoxes, and the sun cannot arrive at the apogee till he has passed over the 653” more than the revolution of the year answering to the equinoxes. ANOMALOUS, (from « privative, Oars, plain, even) any thing deviating from the usual rule or method. ANOMALY, in Astronomy, is an irregu- larity in the motion of a planet, by which it deviates from the aphelion, or apogee; or it is the angular distance of a planet from the aphelion, or apogee ; that is, the angle formed by the line of the apsides, and another drawn through the planet. Kepler distinguishes three kinds of anomaly ; mean, eccentric, and true. Mean, or Simple ANoMALY, in the ancient astronomy, is the distance of a planet’s mean place from the apogee. But in the modern astronomy, in which a planet P is considered as revolying about the sun S, in an elliptic orbit as AP B, it is the time in which a planet moves from its aphelion A, to the mean place or point of its orbit P. Hence, as the elliptical area AS P is propor- tional to the, time in which / the planet de- j scribes the are AP, that area may represent the mean ano- iL or Gf ANT maly; or if PD be drawn perpendicular the transverse axis AB, and meet the circle D, described on the same axis; then the me; anomaly may also be represented by the ¢ cular trilineal ASD, which is always prope tional to the elliptic one ASP, by a knoy property of the ellipse and circle. Or dray ing SG perpendicular to the radius DC pr duced; then the mean anomaly is also pr portional to SG + the circular are AD, Ke Lect. Astron. Hence, taking DH =SG, #1 arc AH, or angle ACH, will be the me: anomaly in practice, as expressed in degre of a circle, the number of those degrees beit to 360°; as the elliptic trilineal area ASP» to the whole area of the ellipse; the degre of mean anomaly being those in the arc AF or angle ACH. é Eccentric ANOMALY, or of the Centre, is th arc AD of the circle ADB, intercepted bi tween the apis A, and the point D determine by the perpendicular DPE, to the line apsides, drawn through the place P of th planet; or it is the angle AC Dat the cent of the circle. Hence, the eccentric anomal is to the mean anomaly as AD to AD + S¢ oras AD to AH, or as the angle ACD { the angle ACH. a True, or Equated ANOMALY, is the ang ASP at the sun, which the planet’s distance AP from the aphelion appears under ; or th angle formed by the radius, vector, or lin SP, drawn from, the sun to the planet, wit the line of the apsides. : The finding of the true anomaly, when th mean anomaly is given, is a problem which ha engaged the attention of many able astron¢ mers. Dr. Wallis gave the first geometric solution of it, by means of the protracted ¢ cloid; and Newton did the same at prop. 3! lib. 1. Principia; other solutions, by means ( series, &c. may be seen in the works of diffe rent astronomical authors. ANS, a term given by Galileo to th prominent parts of the ring of Saturn, whie appears somewhat like handles, from whie the name is derived. ANSER, the name of a small star lyin) between Lyra and Aquila. ANTARCTIC Cirele, in Astronomy, is | small circle parallel to the equator, at the dis tance of 23° 28’ from the antarctic or southeri pole. The word is derived from ayrs, opposite and exo, the bear, or arctic pole. AnTARcTIC Pole, is the southern pole o the earth’s axis. ANTARES, the Scorpion’s heart; a star o the first magnitude, marked «, in the constel lation Scorpio. ; ANTECANIS, in Astronomy, the same ai Canis Minor. . ANTECEDENT of a Ratio, denotes first of the two terms of the ratio; thus in the proportion a: 6::c:d, a and ¢ are the twe antecedents, and 6 and d the two consequents. ANTECEDENTAL Calculus, a branch 0 analysis invented by J. Glenie, Esq. and pub: lished by him in 1793. The author professes iae | | ANT employ it, with advantage, instead of xions; but it has not been much attended by other mathematicians. ANTECEDENTIA, a term used by astro- ‘mers to denote a planet moving westward, } contrary to the order of the signs Aries, jurus, &c. As in the opposite case, or when {motion is eastward, they are said to move lvonsequentia. , ANTECIANS, or Anrocel, in Geography, se inhabitants of the earth which occupy ! same degree of latitude, but in different Jnispheres; the one being as much north as I other is south, ANTILOGARITHMS, the complement of logarithmic sine, tangent, &c. of an an- }; being the difference between them and ius. iNTINOUS, in Astronomy, a part of the stellation Aquila, or the Eagle. \NTIOCHIAN Sect, a class of ancient osophers, founded by Antiochus. ANTIOCHIAN Epocha, a method of computing 2, from the proclamation of liberty granted he city of Antioch, about the time of the Je of Pharsalia. . \NTIPARALLELS, in Geometry, are those ‘s which make equal angles with two other s, but in contrary order; that is, calling the er pair the first and second lines, and the br pair the third and fourth, if the angle le by the first and third lines be equal to langles made by the second and fourth ; on the contrary, the angle made by the ‘ and fourth be equal to the angles made he second and third; then each pair of s are antiparallels to each other; viz. the }and second, and the third and fourth. ; | ; i. ‘aus, if AB and AC are any two lines, ‘FC and FE be two others, cutting t so that term, viz. ars, against, and rodos, a foot. is obvious that antipodes must have the degree of latitude, but in a different sphere; and the difference in longitude !. It is therefore night with one, when i}day with the other; and summer with lio it is winter with the other. The A PI« antipodes to London lie a little south of New Zealand. ANTISCIANS, or Antiscu, in Geography, are the inhabitants of different hemispheres ; being a more general term than Antescians, who must also inhabit the same degree of latitude, which under the present term is not necessary, APERTURE, in Geometry, is the space formed by the mutual inclination or opening of two lines meeting in a point. APERTURE, in Hydraulics, is the hole through which a spouting fluid passes. See DISCHARGE of Fluids. APERTURE, in Optics, is the hole next the object glass of a telescope, or microscope, through which the light and the image of the object come into the tube, and are thence conveyed to the eye. APEX, the vertex, top, or summit of any thine. APHELION, or APHELIUM, (viz. aro, from, and Asoc, the sun) that point in the orbit of the earth, or any other planct, in which it is at its greatest distance from the sun: that is, at the extremity of the transverse diameter of the elliptic orbit, which is farthest distant from that focus in which the sun is placed. The aphelia of the planets are not fixed; for their mutual actions upon each other keep those points of their orbits in a continual mo- tion, which is greater or less in the different planets. For the aphelia of the planets, at present, see ELEMENTS of the Planets. APHELLAN, the name of «, Gemmini. APHRODISIUS, in Chronology, denotes the eleventh month in the Bythinian year; commencing on the 25th of July in ours. APIAN, or Appian, (PETER) called in Ger- man, Bienewitz, a celebrated astronomer and mathematician ; was born at Leipsic in 1495, and made professor of mathematics, at Ingol- Stadt, in 1524; where he died in the year 1552, at 57 years of age. Apian was the author of many works; of which the following are the most important: 1. “ Astronomicum Czesareum ;” folio, In- golstadt, 1540. 2. “Cosmography;” published in 1524, and again by Gemma Frisius, 1540; other editions are mentioned, as being published in 1530, 1550, and 1584. 3. “ Inscriptiones Orbis ;” 1534. 4. “Instrumentum Sinuum sive Primi Mo- bilis ;” 1540. Besides these, Apian was author of several other works; as his Ephemerides, from the year 1534 to 1570: a work on Shadows, Arithmeti- cal Centilogues, Arithmetic, Algebra, or as he calls it, the Rule of Coss: upon Gauging ; Almanacs ; Conjunctions, &c. He also gave the works of Ptolomy, in Greek: books of Kelipses: the Astrological work of Azoph: the works of Geber: the Perspective of Vitello ; of Critieal Days; and of the Rainbow : a new Astronomical and Geometrical Radius; with various uses of sines anid chords: a Universal Astrolabe of Numbers; Maps of the World, published in APO and of particular Countries, &c. Apian left a son, who was also an able mathematician. APIS Musca, the bee or fly, a southern con- stellation, containing four stars. See Con- STELLATION. APOCATASTATIS, in Astronomy, is the period of a planet, or the time employed in returning to the same point of the zodiac from which it set out. APOGEE, (from amo, from, yn, the earth) in the ancient Astronomy, that point in the orbit of the sun, or a planet, which is farthest dis- tant from the earth; and which, therefore, as applied to the sun, corresponds with the aphe- lion of modern astronomers. The ancients regarding the earth as the centre of the sys- tem, chiefly considered the apogee and pe- rigee: the moderns, making the sun the centre, change the apogee and perigee into aphelion and perihelion. Apogee is, how- ever, still used to denote the greatest distance of a body from the earth. The moon, for ex- ample, is in its apogee when farthest from the earth; and its aphelion, when farthest from the sun. APOLLONIAN Hyperbola and Parabola. See Hyperso.a, &e. APOLLONIUS, of Perga, a city in Pam- philia, was a celebrated geometrician who flourished in the reign of Ptolemy Euergetes, about 240 years before Christ; being about 60 years after Euclid, and 30 years later than Archimedes. ‘He studied a long time in Alex- andria, under the disciples of Euclid; and afterwards composed several curious and in- genious geometrical works, of which only his books of Conic Sections are now extant; and even these not perfect. For it appears from the author’s dedicatory epistle to Eude- mus, a geometrician in Pergamus, that this work consisted of eight books; only seven of which however have come down to us, From the collections of Pappus, and the commentaries of Eutocius, it appears that Apollonius was the author of various pieces in geometry, on account of which he acquired the title of the Great Geometrician. His Conics was the principal of them. Some have thought that Apollonius appropriated the writ- ings and discoveries of Archimedes; Hera- _ clius, who wrote the life of Archimedes, affirms it; though Eutocius endeavours to refute him. Although it should be allowed a ground- less supposition, that Archimedes was the first who wrote upon conics, notwithstanding his treatise on conics was greatly esteemed ; yet it is highly probable that Apollonius would avail himself of the writings of that author, as well as others who had gone before him; and, upon the whole, he is allowed the honour of explaining a diflicult subject better than had been done before; having made several improvements both in Archimedes’s problems, and in Euclid’s. His work upon conics was doubtless the most perfect of the kind among the ancients, and in some re- spects among the moderns also. The other writings of Apollonius, mentioned by Pappus, are,—l. The Section of a Ratio, or Propor- APO tional Sections, two books. 2. The Sectior of a Space, in two books. 3. Determinat Sections, in two books. 4. The Tangenciés in twa books. 5. The Inclinations, in twe books. 6. The Plane Loci, in two books. The contents of all these are mentioned by Pappus, and many lemmas are delivered rela tive to them; but none, or very few, of then have been transmitted to the moderns. Fron the account, however, that has been given o their contents, many restorations have bee made of these works by modern mathemati cians, as follow, vz. Vieta—Apollonius Gallus; the Tangenciet Paris, 1600, 4to. Sneldius—Apollonius Bats vius; Determ. Sect. 1601. 4to. a Rationes et Spatii, 1607. Ghetaldus—Apolle nius Ridivivus ; the Inclin. Venice, 1607, 4t Ghetaldus—Supplement to the above; Tanget cies, 1607. Ghetaldus—Apollonius Redivit lib.ii. 1613. Alex. Anderson—Supplem. Apo Red. Inclin. Paris, 1612,4to. Alex. Anderson Pro Zetetico Apolloniani, &c. 1615. Scho ten—Loca Plana restituta, 1656. Ferma Loca Plana, 2]ib. 1679, folio. Halley—Ape de Sect. Rationis, libri duo, Oxford, 1706, 8y Simson—Loca Plana, libri duo, Glasgow, 174 4to. Simson—Sect. Determinat. Glasgo 1776, 4to. Horsley—Apol. Inclinat. libri du Oxford, 1770. Lawson—The Tangencies, § two books, London, 1771, 4to. Lawson—D term. Sect. two books, London, 1772, 4 Wales—Determ. Sect. two books, Lond 1772, 4to. Burrow—The Inctinations, Li don, 1797, 4to. Hutton’s Dict. a APOSTERIORI, a term frequently € ployed in demonstrating a truth, wheth mathematical or philosophical ; when a call is proved from an effect, or a conclusion proved by something posterior, whether it an effect or only a consequent. 4 APOTOME, the remainder or differen between two lines or quantities, which only commensurable in power. Such ast difference between 1 and /2, or the di rence between the side and diagonal Of square. The word is derived from aor) I cut off. at Euclid, in his tenth book, treats of thi quantities, and distinguishes them under heads, viz. veh Apotomr Prima, is when the greater t is rational, and the difference of the squa of the two is a square number ; as, 3 — Ji APpotoME Secunda, is when the less tem rational, and the square root of the differe) of the squares of the two terms has to’ ereater term a ratio expressible in numb such is “18—4, because the difference the squares 18 and 16 is 2, and 2 is to” as /1 to 9, or as 1 to 3. M ApotomeE Tertia, is when both the te are irrational, and, as in the second, square root of the difference of their squi has to the greater term a rational ratio; ./ 24— 4/18, for the difference of their squi 24 and 18 is 6, and /6is to /24 as ¥' / 4, or as 1 to 2. yi APoToME Quarta, is when the greater ¥ 4 ‘ APP . rational number, and the square root of difference of the squares of the two terms not a rational ratio to it; as, 4— 3, ere the difference of the squares 16 and 3 3, and 13 has not a ratio in numbers t. L\POTOME Quinta, is when the less term is a onal number, and the square root of the erence of the squares of the two has not ational ratio to the greater; as /“6— 2, are the difference of the squares 6 and 4 is ind /2 to “6, or /1to /3, orl to 3, ot a rational ratio. .POTOME Sexta, is where both terms are tional, and the square root of the diffe- se of their squares has not a rational ratio he greater; as /“6— /2, where the diffe- se of the squares 6 and 2 is 4, and 4 to , or 2 to 6, is not a rational ratio. .PPARATUS, the appendages or utensils mging to machines; as the apparatus of uir-pump, electrical machine, &c.; mean- the various detached parts which are ne- sary for putting the machinery in action, for performing experiments, &c. ‘PPARENT, in Mathematics and Astro- y, is used to signify things as they appear is, in contradistinction from real or true; in this respect the apparent state of things ften very different from their real state: ; the case of distance, magnitude, &e. PPARENT Conjunction of the Planets, is n a right line, supposed to be drawn ugh their centres, passes through the eye he spectator, and not through the centre he earth.— And, in general, the apparent unction of any objects, is when they ap- ‘or are placed in the same right line with eye. PPARENT Diameter of an Object, is not the length of that diameter, but the angle sh it subtends at the eye, or under which ypears. This angle diminishes as the dis- é increases; so that a small object at a ll distance may have the same apparent aeter as a much larger object at a greater ance, provided they subtend the same or il angles at the eye. If the objects are lel to each other, their real diameters ‘in this case, proportional to their dis- ies. The epparent diameter also varies the position of the object; and of equal cts at equal distances, those which stand position most nearly perpendicular to the of their direction from the observer, will ar to have the greatest diameter: our of the apparent magnitude generally ing nearly as the optic angle. ; it although the optic angle be the usual ensible measure of the apparent magni- of an object, yet habit, and the frequent rience of looking at distant objects, by h we know that they are larger than they ar, has so far prevailed upon the imagi- m and judgment, as to cause this likewise we some share in our estimation of appa- magnitudes; so that these will be judged 2 more than in the ratio of the optic an- See Apparent Magnitude, [ APP APPARENT Distance. See DIsTANce. APPARENT Altitude of celestial Objects, is effected chiefly by refraction and parallax ; and that of terrestrial objects, by refraction, See those words. APPARENT Figure, is the figure or shape which an object appears under when viewed at a distance; and is often very different from the true figure. For a straight line, viewed at a distance, may appear but as a point; a surface, as a line; and a solid, as a surface. Also these may appear of different magni- tudes, and the surface and solid of different figures, according to their situation with re- spect to the eye: thus the arch of a circle may appear a straight line; a square, a trapezium, or even a triangle; a circle, an ellipsis; an- gular magnitudes, round; and a sphere, a circle. Also all objects have a tendency to roundness and smoothness, or appear less angular, as their distance is greater: for, as the distance is increased, the smaller angles and asperities first disappear, by subtending a less angle than one minute; after these, the next larger disappear, for the same reason; and so on continually, as the distance is more and more increased; the object seeming stiil more and more round and smooth. So, a triangle, or square, at a great distance ap- pears only as a round speck; and the edge of the moon appears round to the eye, notwith- standing the hills and valleys on her surface. And hence itis also, that near objects, as a range of lamps, and such like, seen at a great distance, appear to be contiguous, and to form one uniform continued magnitude by the intervals between them disappearing, from the smallness of the angles which they subtend. APPARENT JMotion, is either that motion which we perceive in a distant body that moves, the eye at the same time being either in motion or atrest; or that motion which an object at rest seems to have, while the eye itself only is in motion. The motions of bodies at a great distance, though really moving equally, or passing over equal spaces in equal times, may appear to be very unequal and irregular to the eye, which can only judge of them by the muta~ tion of the angle at the eye. And motions, to be equally visible, or appear equal, must be directly proportional to the distances of the objects moving. Again, very swift motions, as those of the luminaries, may not appear to be any motions at all, but like that of the hour-hand of a clock, on account of the great distance of the objects: and this will always happen, when the space actually passed over in one second of time, is less than about the 14000th part of its distance from the eye; for the hour-hand of a clock, and the stars about the earth, move at the rate of fifteen seconds of a degree in one second of time, which is only the 13751 part of the radius or distance from the eye. On the other hand, it is possible for the motion of a body to be so swift, as not to appear any motion at all; as when through the whole space it describes there constantly appears a continued surface APP And, because of the similar triangles ABC, GFC, it will be, as AB SOR ors OTs br; COE ae 82S a— sz: Hence multiplying extremes and means: we have... ax mba—bx; whence ax+bx= ba, ba ; Or ates ow sate the side of the a+JB’ square required. Cor. Hence it is obvious, that in all trian- gles whose bases and perpendiculars are con- stant, the side of the inscribed square will be constant also. Prob. 8. Giving the area or space of a rect- angle inscribed in a given triangle, to deter- mine the sides of the rectangle. Let ABC repre- sent the given trian- gle, and EF GH the inscribed rectangle, the area of which is given, and which let be represented by a; make CD=p, AB= 6, EF =2z, and ID =y; then will Cl—=p—y; and by similar triangles we shall have AB:CD:: EF: CI, or DE Chie pis a spy; whence ........ px = bp—by, and ....06 deeds Abdel fie any thelare a: ie — bp—by The first gives « = wy a the second .... x = -. y Whence again saad = oi bpy— by? =pa; te ny — —Pe The root of whichis y = £ = = J ¢ -E , the breadth required. Having thus found the breadth, the length may be obtained by dividing the area by the breadth; or otherwise we have, by finding the values of y instead of x, or. eeeattoetees seo by — y op? - px fa =i y el Eas whence Ld ew} ed = oy OF nswtn bpz—pzx* = ba; . which gives z*—bz = — af, whence ..... ek = ,/ (fronts ; 2 ah ap _Prob. 4. In a right-angled triangle, having given the lengths of two lines drawn from the acute angles to the middle of the opposite sides, to find the sides of the triangle. APP Let ABC repre- sent the proposed tri- angle of which the lines AD, CE, are given. Make AD= a, CE=b, AB= 2x, CB=—2y, AC te = 2z. Then we have . Poa | - . . : oo” e pe -* E 2 2 amm 2 t ‘ saad Apres t (Huclid, 47, 1 Now, eis . 427 + 16y* = 45" spain eh aie if” ih ay as therefore, 15y* = 46*—a’ | os oe and consq....2y = 2 / = 5 : )=@ And in the same manner we find , 2 consequently 2z =—9~/ (Rast Sk | BL Prob. 5. In a right-angled triangle, havin given the hypothenuse and side of the ij scribed square, to find the base and perper dicular. i ‘Let ABC be the pro- posed triangle, AC the given hypothenuse, and BD the given square. Make AC =A, FD = DES 4 AB = +c, BC ail . A v Then we have x” + y?= A? (Eucl. 47, 1.) B w and ..... (e—s): 8%: s:(y—s) by sim. trian Whence ...... LY —sSx—sy +5* ="; Mi Dr cscatineepeat xry—s(x+y). / By adding double this to the first equatio we obtain 27 + 22y + y? =h? + 2s (x + y),' (x+y)*—2s(a+y) =H; ; whence x+y=s+ Vv (h* + 87), t Now z + y being known, make it = n, the we have a i aml i afte we mt 4 Square the second, and subtract it fro double the first, and we obtain x — Qey + y* —2h?*—n?*. By extracting .... e—y = v(2h?—n?). POA cteccee re de c+y=n; | 2 therefore, x= mice A Eat —AB; N— / (2h* —n* 7 and Ad 4 y= VCE) = BC, ast quired. \ _ Prob. 6. Having given the sum of the thn sides of a right-angled triangle, and the pe pendicular, let fall from the right angle u the hypothenuse, to find the three sides of triangle. fy Let ABC be the proposed tri- angle, of which the sum of the sides and _ per- pendicular BD aregiven, Make .\ APP e sum of the sides = s, the perpendicular ip, ABx=2x, BC=y, AC=z. 1en we have x+y + 2=s, by the question. x* + y? = 2z* (Eucl. 47,11.) x:ziip:y, by sim. trians.; or Syma px. ow, add double this last equation to the se- nd, then a*7+2ry+y?*=2z* +2pz z>4+2xeyt+y? =s*—2sz +2’, ‘transposing z in the first equat.and squaring. Thence 2* + 2pz=s—Bsz + 2%, or s§ - — 2p + Qs — A C. Now z being known, make it =”; then the cond and third equations become eye = n* Mom] = Ne id and subtract double the last equation mm the first, and we have x +2xey+y? =n +2pn x*—Qrey + y* =n’ —2pn, e roots of which are, x + y= /(n* + 2pn) secvesscesseceseen LY eV (n?—2Zpn). Whence again, + Ae +2pn) +2 V(n?—2pn)=AB, an i= FV (n? + 2pn)—t V(n?—2pn) = BC. Wherefore the three sides AB, BC, AC, 2 determined. For the geometrical construction of these joblems, see ConsTRUCTION of Geometrical roblems, and ANALYSIS. The second branch of the Application of gebra to Geometry, or that which respects 2 higher geometry, or the nature and pro- rties of curve lines, was introduced by Des tes. In this way the nature of the curve expressed, or denoted by an algebraical ation, which is formed as follows :—A line conceived to be drawn to represent the meter, or some other principal line of the ia and upon this line, at any indefinite ints, are erected perpendiculars, which are led ordinates ; and the parts of the first line it off by them are termed abscisses. Calling (2 abscis x, and its corresponding ordinate y, i2 known nature of the curve, or the mutual Vation of the other lines in it, will furnish an ‘uation, involving x and y, with some other iter or letters which are known. And as 2 d y are common to every point in the pri- jury line, the equation, derived in this man- i will belong to every position or value of 2 absciss and ordinate; and may be pro- irly considered as expressing the nature of i2 curve in all points of it, and is usually ilied the equation of the curve. Hence cery particular curve will appear to have an propriate equation, differing from that of ‘ery other; either as to the number of the ims, the powers of the unknown quantities ‘and y, or the signs of the co-efficients of the ims of the equation. Thus the circle, the (ipse, the parabola, the hyperbola, and other ‘rves, have their peculiar and distinguishing (uations. | The geometry of curve lines has been much APP extended and improved by means of these al+ gebraical equations, and their roots are trans- ferred to the curve lines whose abscisses and ordinates have similar properties, and con- sequently, the known properties of curves are transferred to the equations that represent them. See CuRVE. APPLICATION of Geometry to Algebra, is the converse of the first of the two preceding cases; for as in that, algebra is employed in order to obtain the solution of a geometrical problem, so in this case geometry is made use of to obtain the solution of an algebraical problem. This relates principally to the find- ing the roots of an equation by a geometrical construction, which is explained under the article CONSTRUCTION. AppLicaTion of Algebra and Geometry to Mechanics. 'This is found on the same prin- ciples as the Application of Algebra to Geo- metry; and consists principally in represent- ing, by equations, the curves described by bodies in motion; as in the theory of Projec- tiles, &c. APPLICATION of Mechanics to Geometry, con- sists chiefly in the use that is sometimes made of the centre of gravity of figures, for deter- mining the contents of solids described by those figures. See CENTROBARY Method. APPLICATION of Geometry and Astronomy to Geography, principally consists in the three following articles; vz. In determining by geometrical and astronomical operations, the figure of the terrestrial globe; in finding the positions of places by their observed latitude and longitude; and in determining, by geo- metrical operations, the positions of places that are not very remote from one another. Astronomy and geography are again appli- cable to the theory of navigation. APPLICATION of Geometry and Algebra to Physics or Natural Philosophy. For this ap- plication we are indebted to Sir Isaac Newton, whose philosophy may therefore be called the geometrical or mathematical philosophy ; and upon this application are founded all the phy- sico-mathematical sciences. Hence a single observation or experiment will often produce a whole science. Having ascertained by ex- perience, that the rays of light, by reflection, make the angle of incidence equal to that of reflection, we hence deduce the whole science of Captoptics, which thus becomes purely geo- metrical, since it is reduced to the comparison of angles and lines given in position. ‘The case is also the same in many other sciences, APPLY, amongst mathematicians, some- times signifies the application of a line within a circle, or some other figure, so that the ex- tremities of the line shall be in the perimeter or circumference of the figure. See APPLI- CATION. APPOLLON, a name given to the star Castor. APPROACH. The Curve of equable Ap- proach was first proposed by Leibnitz, and has much engaged the attention of analysts. The curve is of such a nature, that a body de- APP scending by the sole power of its own gravity approaches the horizon equally in equal times. This curve has been found, by Bernoulli, Va- rignon, Maupertuis, and others, to be the se- cond cubical parabola, so placed that its point of regression, or vertex, is uppermost, and the descending body must commence its motion in it with a certain determinate velocity. Va- rignon rendered the question more general by investigating the curve which a body might describe in vacuo, so as to approach through a given point through equal spaces in equal times, according to any law of gravity. Mau- pertuis also resolved the same problem, in the case of a body descending in a medium, the resistance of which is proportionate to the square of the velocity. See Hist. Acad. R. Science, 1699 and 1730. Method of APPROACHES, a term used by Dr. Wallis, in his Algebra, to denote a method of resolving certain problems, relating to square numbers, &c.; which is done by first assigning certain limits to the quantities required, and then approaching nearer and nearer till a co- incidence is obtained. This method was in- vented by Dr. Pell, for the solution of equa- tions of the form a? —ay? =1; which problem was proposed by Fermat, as a challenge to all the English mathematicians of his time; viz. To find rational and integral values of x and y, in the above equation, for every value of a, except when it is a complete square. This rule is explained by Euler, in the 2d vol. of his Algebra. See also a general solution of these kind of problems in Barlow’s Theory of Numbers. APPROXIMATION, in Algebra and Arith- metic, is the method of approaching nearer and nearer to the quantity sought, when there is no method of obtaining the exact value: this is the case in all rules for finding the square or cube root of any number that is not an exact square or cube. Methods of approxi- mation in these cases, and for the roots of powers of all dimensions, will be found under the articles ExTRacTion and Root. APPROXIMATION to the Roots of Equations. As there is no direct method of determining the roots of equations beyond those of the fourth degree, and even in those of the third and fourth degree being very laborious by the direct rules, mathematicians have endeavoured to find approximating methods of determining the roots: of these, Newton’s rule is the most popular, and is founded on the following prin- ciples; If any two numbers, being substituted for the unknown quantity in an equation, give re- sults with opposite signs, an odd number of roots must be between these numbers. This appears from the property of the abso- lute term, and from this obvious maxim, that if a number of quantities be multiplied toge- ther, and the sign of-an odd number of them be changed, the sign of the product is changed. For when a positive number is substituted for az, the result is the absolute term of an equa- tion, whose roots are less than the roots of the APP : ~ given equation by that quantity. If the rest has the same sign as the piven absolute tert then from the property of this term, eith none, or an even number only, have had the) signs changed by the transformation; but the result has an opposite sign to that of th given absolute term, the sign of an odd numbe of the positive roots must have been changed in the first case, then, the quantity subst tuted must have been either greater than ea of an even number of the positive reots of th given equation, or less than any of them; i the second case, it must have been great than each of an odd number of the positiy roots. An odd number of the positive roo must, therefore, lie between them, when the give results with opposite signs. The sam observation is to be extended to the subst tution of negative quantities and the negati roots. This being premised, the reader wi readily follow the operation in the follo example: ; Let it be proposed to find an approxime value of x in the equation « x3 —2x—5—0. | Here we easily see, that one root is betwee 2 and 3; for these ntmbers being substitute for x, will give, the one a positive, and @ other a negative result. The root, therefor is greater than 2 and less than 3, and we mij therefore write x=2+/, or x=3—f 5 an by finding f, or an approximate value of it, either case, will obviously also give an A proximate value of x, by adding the value. f to 2, in the first case, and subtracting it fro| 3 in the second case. " Suppose x=2 +f, and substituting this y lue for x in the proposed equation, we have, 2? =8+12f+6f-+f3 —2xe=—4—2f | 26) 45 a$ 24-5 ——1 4 10f 4+ 6f" + f* therefore, to find f we have the equation | fi +6f? +10f—1=—0, i which is still a cubic; but since, from the : ture of our substitution, f must be less than the cube of it may be omitted as inconsidé able, and this will give A fi +if—2=0, or | pitas, 25 1) _ = 52 vg f= 4 tV/(2+3 = 6 ry which value of f, being added to 2, will gil an approximate value of x. ot But generally, when / is less than unity, | the higher powers off may be omitted, wil out any very sensible error; which being do in the present instance, will give 10f— 1 =} orf'=0'1, and therefore «=2-1 nearly. | Now as f= 0'1 nearly, letf—'1+g, a substituting this value for f in the ae equation, we have 5 ff? = 9001 + 003g +03 ¢7+23 Ff 6f*—006 + 12¢+ 6,7 T 10f= 1.5. + 0g —li=l f° +6f? +10f—1— 0°061 + 11:23¢46327 + ¢3 =0, APP id neglecting g? and g”, for the reasons ove stated, this equation becomes 0°061 + 11°23 g =0, or a OT 0084, pie i. Sige ence, then, f= -1 + &¢ = +1 —-0054 — 0946, arly ; and consequently x=2+f—=2 + -0946 — 2:0946. Phis operation may be extended to any gth at pleasure, by supposing again g = 0054 +h, and thence finding a value of A, 1 so on: this last substitution, if worked + Will give x =2:09455147. Another method of approximating towards roots of equations in general, is by the owing formula: set x" + aa 4 hb yr2 4 op gn-3 + &e.=w, any equation; and OM Ag g!M—1 4h y/n—2 + ex’ + &e. =, eing an approximate value of x; then hh (w—v) 2a (x—1) w + (n+ 1) 2" + (n— 1) ax"! n— 3) bx"? + &e.; or, (w—v) 2x‘ 1) a/™ + (n—l) ax’ nt (n—1l)v+(n+ n—3) ba’"-2 + &e, he first formula being applicable to the 's in which 2’ is greater than unity, and the md when 2’ is less than unity. hese formule are general for equations of imensions, but when reduced to particular s they become much more simple: thus, ‘cing them both into one, we have for (1.) Equations of the third Degree, (w —v) 2 : worv +223 + a27%’ (2.) Equations of the fourth Degree, (w—v) 22’ : + 3wordv+52'4+4+ 3az'3 + bx %—ex’ (3.) Equations of the fifth Degree, (w—v) x! t Sperde pax’ + 2a2'+ + bx’ 3—dr’ d in the same way we might find the cular formula answering to an equation y other degree. plication of the preceding Formua—The od of applying these formule will be at Obvious to those who are acquainted with ature of equations ; but to the student, a explanation becomes necessary. order to this it may be observed, that 2’ approximate value of x, found first by as hear the true root as possible, which cases ought to be true to the nearest in- Then substituting this value of a 3% approximate value of x will be obtained, will in all cases double the number of 3. Then, considering this as a new va- ‘2’, another still nearer value of x will ‘ermined ; and so on. stating the degree of approximation to tof doubling the number of figures in Sumed root, we are rather under than 16 real state of convergency ; for gene- x z' + U *. . APP rally, if the first supposition be teger, the first approximation ma y be obtained to three places, which, though it be not always correct in the last place, yet by employing it as & new approximation, the next value of x will, in most cases, be found true to six or seven figures, which is a degree of conver- gency that cannot be obtained by any other rule that we are acquainted with, at the same time that the operation is much more simple than by any other rule whatever. An example will sufficiently illustrate the preceding remarks. Example.—Vind the value of x in the cubis equation x3 +327 +32— 130 ; 4 simple in- Assume x' = 4, then by formula (1) 7 ee 22'3 = 128 ote 48 PE Ea come OS Meme w <== 130 » = 124 306 denominator. w = 130 : deere ed aa Se EC SRI TY 306) 24 (4°08 nearly. Therefore now x! = 4:08. Hence, adopting this new value, we have 23 == 67°9173 22'3 = 135:8346 32% —= 49-9392 ax’* = 49-9392 eo a eee ns" ty » = 130°0965 315°7738 w: = 130 w—v *0965 PN pees 4:08 315°77) -39372 (001247. Whence 4:08 — -001247 = 4-078753 Answer. This root is true in the seventh figure; and in the same manner the root of any other equation may be determined. The above method I lately published in Leybourn’s Mathematical Repository, and the investigation of it will be given in the intro- duction to my Mathematical Tables. In the same way the root of any equation may be obtained, and on similar principles the root of any number whatever, that is not a perfect power. Lia Grange has published a work on the subject of Numerical Equations, in which is given a method of approximation much more accurate than those above, or than any Which has yet appeared, and in which he has shown that all such rules are sometimes defective; for since there may be more than one root be- tween the limits first determined, the succeed- ing results will sometimes give one root and sometimes another. We should have gladly entered into an explanation of the method taught in this work; but it is impossible, in the limits to which we are under the necessity of confining this article, to give an adequate idea of it, and we must therefore refer the reader to the work itself—“ Sur la Résolution des Equations Numérique,” par La Grange ; as also to the article APPROXIMATION in the “ Encyclopédie Methodique.” | F AQU We ought to observe, however, that the method of La Grange, though extremely ac- curate, is more valuable as a theoretical than as a practical rule; it being too complicated for general use. plicable to the method of Budan, who has lately, viz..in 1807, published at Paris a work on this subject, entitled ‘‘ Nouvelle Méthode pour [a Résolution des Equations Numé- rique.” . Other ingenious methods of approximation to the roots of equation, may be seen in Clai- raut’s Algebra, and in those of Newton, Simp- son, Bonnycastle, Maclaurin, &c. On this sub- ject, too, the reader may consult J. Bernoulli, Opera, tom. iil. ; Taylor, in Phil. Trans. for 1717; Simpson’s Essays, and Select Exercises, &e. See also the articles ConTINUED I’RAC- TIONS, Roots, and EXTRACTION. APRIL, the fourth month of the year. APRIORI, Demonstration APRIORI, is that whereby an effect is proved from a cause, either a next, or remote one; OF a conclusion proved by something previous, whether it be a cause, or only an antecedent. APSES, in Astronomy, are the two points in the orbits of the planets where they are at their greatest and less distance from the sun or earth; the former being called the higher apsis, and the latter the lower apsis: but the higher apsis is more commonly called the a phelion, and the lower apsis the perihelion ; or, according to the ancient astronomy, the apogee and perigee. The diameter which joins these two points is called the Line of the apsides, and is supposed to pass through the centre of the orbit of the planet, and the centre of the sun or earth; in the modern astronomy, this line makes the longest or transverse axis of the elliptical orbit of the planet; and in its line is counted the eccentricity of the orbit. - According to the above definition, the Imes of the greatest and least distance are supposed to lie in the same straight line; which ts not always precisely the case, as the two fre- quently make an angle with each other; and what this angle differs from 180°, is called the motion of the line of the apsides: and when this is less than 180°, the motion is said to be contrary to the order of the signs; and when it is greater than 180°, the motion is said to be according to the order of the signs. The problem of determining the motion of the apses has been the subject of many inge- nious papers, and different solutions have been ziven to it by different authors ; vz. by Dr. Keil, in his Astronomy ; by Euler, in vol. vii. Acta. Petrop.; by Newton, in his Principia; &e. &e. APUS, or Apous, in Astronomy, a constel- lation of the southern hemisphere ; the num- ber of stars in which, according to the British catalogue, is 11. See CONSTELLATION. AQUARIUS, in Astronomy, one of the ce- lestial constellations, being the 11th sign in the zodiac beginning from Aries; its character is &,. representing a stream of water issu- ing from the vessel of Aquarius, or the water- And a similar remark is ap- . ARC pourer. The sun passes through this sign i the latter part of the month of January an the beginning of February. See ConsTe} LATION. | AQUEDUCT, q. d. ductus aqua, a condu of water, in Architecture and Hydraulics, is construction of stone and timber built on w even ground, to preserve the level of wate and to conduct it through canals from 0} place to another. Some of these aqueduc are visible, and others subterraneous: tho of the former sort are constructed ata gre height, across valleys and marshes, and sw ported by piers and ranges of arches; t latter are formed by piercing the mountal and conducting them below the surface oft earth. They are built of brick, stone, &e. a covered above with vaulted roofs, or flat ston serving to shelter the water from the sun aj rain: and of these, some are double and oth triple, that is, supported by two or three rang of arches. Of the latter kind, are the Po du-gard in Languedoc, supposed to have be built by the Romans to carry water to the ¢ of Nismes; that of Constantinople; and t which, according to Procopius, was construe by Cosroes, king of Persia, near Petra in M erelia,-and Which had three conduits in same direction, each elevated above the of] Some of the aqueducts are paved, others ¢ vey the water through a natural channe?} clay ; and it was frequently conducted by i of lead into reservoirs of the same metal into troughs of hewn stone. y The most modern and most extensive duct, is that built by Lewis XIV. near M tenon, for carrying the river Bure to Versai It is 7000fathoms long, and its elevation 4 fathoms; and contains 242 arcades. See] Trans. ap. Lowth. Abr. vol. i. p. 594. — for an inquiry into the nature and construc of the aqueducts of the Romans, see Pown “ Notices and Descriptions of Antiquities,” Ato. 1788. ie AQUILA, the Eagle, in Astronomy, is a stellation of the northern hemisphere, ust joined with Antinovs. See CONSTELLATE ARA, the Alter, in Astronomy, one 9} old constellations situated in the sout hemisphere. See CONSTELLATION. | ARAIMOMETER. See AREMOMETI ARC. See ARcH. i ARCAS, a name sometimes given to, TURUS, a bright star in the constell Bootes. 4 ARCH, or Arc, in Geometry, a par curve line; as of a circle, ellipse, &c. — Circular Arc, is any part of the are circle, and by which the magnitudes of a are compared; an angle being said to c@ so many degrees, minutes, &c. as are con in the are which subtends ft. Co-centric ARCS, are such as have the centre. Equal Arcs, are such ares of the sam cle, or of equal circles, which have the measure, or the same number of degre¢ nutes, &c. ; ARC Similar Arcs, are those which have the pune measure, but belonging to different rcles. The lengths of similar arcs are to ch other as the radii of the corresponding prcles )The length of circular arcs may be ex- j essed in terms of the sine, cosine, tangent, c. in the following manner : Let a represent any arc to radius 1.; s its ne, c its cosine, ¢ its tangent, ’s the secant, id v its versed sine; then will ba =t— Fe3 + £t5— 227 + 30° —&e. 5 ‘s? ‘99 Git) yO Oy) 17.0!) 9¢? Lit? 1.3.85 4.3.5, 87 + 245 7 24677 pel tf v 13v? | 13.503 . r] Pret) t 355+ ong5t aay ES § (@= sin. a, sec. La, sec. La, sec. La, &e. SI1GO; +> Pee Lua iso d = ‘01745329, &c. x d; where d represents the number of degrees, &c. contained in the given are. _ 8e—C alae mae chords of the are and half are. See Dr. Hutton’s Mensuration, and the ar- le RECTIFICATION. )And since similar ares are to each other as ir radii, it is obvious, that having the length }any are given to radius 1, the length of a Ailar are may be found for any other radius, | multiplying the length of the first are by / Siven radius. Or, since 01745329 is the gth of an arc of 1°, to radius 1; the length my arc, of which the measure is given, will }found by iultiplying the number of de- (es by °01745329, and that product again ithe given radius. or example; let it be proposed to find the isth of an arc of 30°, the radius being 9. Tere ‘01745329 x 30 x 9 = 4:7115, &e. the sth sought. ARCH, in Astronomy, has various denomi- ey according to the circle to which it is ied. Yiurnal AxcH of the Sun, is part of a circle allel to the equator, described by the sun lis Course between rising and setting. His (turnal arch is of the same kind, being that eribed between setting and rising. he latitude and elevation of the pole are Misured by arcs of the meridian, and the situde by an arch of a parallel circle. [ren ot Progression, or Direction, is an 1 of the ecliptic, which a planet seems to over, when its motion is direct, or ac- sling to the order of the signs. iRCH of Retrogradation, is an arch of the Hotic described when a planet is retro- le, or moyes contrary to the order of the 1s. RCH of Position, or Angle of Position, is Same with the hoary angle. RCH of Vision, is the sun’s depth below horizon at which a star, before hid in his , begins to appear again. his arch is a — nearly. Where C ande are the | ARC different for different planets; being for Mer- cury 10°, Venus 5°, Mars 114°, Jupiter 10°, Saturn 11°; a star of the 1st inagnitude 12°, 2d magnitude 13°, &c. This angle is not, how- ever, Constant in all cases for the same planet, but varies a little with the latitude and de- clination, &c. With respect to Venus, it is sometimes reduced to 0, as she is at times vi- sible even when the sun is some degrees above the horizon. ARCHES, in Architecture, are constructed of various forms, and are designated by va- rious names, according to their figure ; as cir- cular, elliptical, cyclodical, &e. Semicircular ARCHES, are those which con- sist of an exact semicircle, having their centre in the middle of the span or chord; such are the arches of Westminster-bridge, Skene ARCHES, are those that consist of a segment less than a semicircle. Arcu of Equilibrium, in the Theory of Bridges, is that which is in equilibrio in all its parts, and therefore equally strong through- out, having no tendency to break in one part more than in another; on which account it is generally preferred to any other. The arch of equilibration is not of any de- terminate curve, but varies according to the figure of the extrados ; every different extrados requiring a particular intrados, so that the thickness in every part may be proportional to the pressure. If the arch were equally thick throughout, the catenary curve would be the arch of equilibration; but as this can seldom or never happen, it is a mistaken idea to suppose this curve the best in all cases. The reader may find this subject fully treated of in Dr. Hutton’s “ Principles of Bridges,” and in his Tracts lately published; where the proper intrados is investigated for every par- ticular extrados, so as to form an arch of equi- libration in all cases whatever. It there ap- pears, that when Be o the upper side a use of the wall is a straight horizon- tal line, as in the annexed figure, the equation of the curve is thus expressed ; : “SOOT At mae m als Beet amen mee nnn abat J/(2ax + x7) log. —————____—_ a e° atr+ /(Qar +7’) : remy Se, log a where «=D P, ¥=PC, r=DQ, h=AQ, and a = DK; ‘and hence, when a, /, and r, are any given numbers, a table is formed for the corresponding values of x and y, by means of which the curve is constructed for any par- ticular occasion. And in a similar manner, if the curye of the intrados, and the depth of the key-stoue or central youssoir, be given, the equation of the extrados may be computed. ‘Thus, for ex- ample, in the case where the intrados is a circle, F2 ARC Let AGE be gp one half of the arch, C the cen- tre; AB the height or depth of the key-stone. D From the centre i, intersect C A, with a distance equal to C B, in shah’, D; and through D draw DH perpendicular to AC, and through G, any point in the in- trados, draw GC, intersecting DH in FP, Let EL be perpendicular to C E; join LB, and produce CG, till CK be equal to LB, so shall K be a point in the extrados; any number of which points may be found in the same manner. The line KC is always greater than CF, but approaches to it as the arch AG increases ; so that DH is an asymptote to the extrados, the equation of which is readily deduced from the above construction, being ave +P), he ii a ig ae Sn where CA =a, CD=b, CO=z,and OK =y. The extrados, in the case of a circular arch, is therefore a curve of the fourth order, very much resembling the conchoid of Nicomedes, having for its asymptote the line DH, and also a point of contrary flexure; so that it co- incides very nearly with the curve, in which a road is usually carried over a bridge. And this holds good, whatever portion of the circle the arch is supposed to consist of. On the subject of arches, see Hutton’s Principles of Bridges; Bossut, Recherches sur VEquilibre des Voutes, Mem. de l’Acad. des Sciences, 1774 and 1776; also Méchanique of the same author, edit. of 1802, p. 383; Prony. Arch. Hydraul. tom. i.; Atwood’s Treatise on the Const. and Prop. of Arches ; and Gregory’s Mechanics, vol. i. ArcH, Mural. See Murat Arch. ARCHER. See Sacitarius. ARCHIMEDES, of Syracuse, one of the greatest and most celebrated of the ancient mathematicians, was born in the above city about 240 years before Christ. The great and comprehensive genius of this author led him to the study of every branch of science ; arithmetic, geometry, mechanics, optics, hy- drodynamics, were alike the objects of his in- vestigations, and experienced alike the power- ful effects of his superior talents. To Archimedes we owe the first idea of the specific gravity of bodies, which arose out of the following circumstance. Hiero, King of Syracuse, having had reason to suspect that a goldsmith, who was employed to make him a crown of gold, had adulterated the metal by mixing with it a quantity of silver, re- quested Archimedes to endeavour to discover the cheat; which he did by procuring two masses of gold and silver of equal weight with the crown, which he immersed in a ves- sel full of water, and carefully noticed the ARC quantity of water which each displaced ; afte which he observed how much the crow caused to flow over; and on comparing thi quantity with each of the former, he wa able to ascertain the proportions of gold am silver in the crown. Sce Htero’s Crown Some ancient authors celebrate a glass ma chine made by Archimedes, which, accordim to them, represented exactly the motions ¢ the heavenly bodies; and he is also said t have made burning glasses, which destroye ships ata great distance. Archimedes becam most famous by his curious contrivances, hy which the city of Syracuse was so long dt fended, when besieged by the Roman cons} Marcellus; showering upon the enemy som times long darts and stones of vast weig and in great quantities; at other times lift their ships up into the air, that had come ne the walls, and dashing them to pieces by le ting them fall down again; nor could the find their safety in removing out of the reat of his cranes and levers, for then he contrivé to set fire to them with the rays of the st reflected from burning glasses. However, nt withstanding all his art, Syracuse was at leng taken by storm; and Archimedes was unfort nately killed in the assault. What gave Mé cellus the greatest concern, says. Plutar was the unhappy fate of Archimedes, who w at that time in his museum; with his mind intent upon some geometrical figures, that neither heard the noise of the Romans, 1 perceived the city to be taken. In this def of study and contemplation, a soldier cay suddenly upon him, and commanded him) follow him to Marcellus, which he refusing’ do till he had finished his problem, the sol in a rage drew his sword, and ran him throw Livy says, he was slain by a soldier not kno ing who he was, while he was drawing scher in the dust; that Marcellus was grieved at death, and took care of his funeral, and mé his name a protection and honour to thi who could claim a relationship to him. 1] death, it seems, happened about the 142d 143d Olympiad, or 210 years before the b of Christ. a The expression which Archimedes m use of to king Hiero is well known: “G me aplace to stand on,” said the philosop] ‘and I will move the earth.” ‘This aff¢ matter for a curious calculation, viz. to de mine how much time he would have reg ed to have moved the earth only one 0 Ozanam, after making proper allowan states the time at 3,653,745,176,803 centut See Hutton’s Ozanam, part 5. J A whole volume might be written upon curious methods and inventions of Archime} that appear in his mathematical writings | extant only. He was the first who squam curvilineal space; unless Hippocrates 1 be excepted on account of his lunes. Bt time the conic sections were admitted geometry, and he applied himself closel the measuring of them, as well as other figt Accordingly he determined the relatiol ARC ,eres, spheroids, and conoids, to cylinders #1 cones; and the relations of parabolas to tilineal planes, whose quadratures had long fore been determined by Euclid. He has -us also his attempts upon the circle: he ived that a circle is equal to a right-angled angle, whose base is equal to the circum- hence, and its altitude equal to the radius ; i 1 consequently, that its area is equal to the jtangle of half the diameter and half the sumference; thus reducing the quadrature ithe circle to the determination of the ratio yween the diameter and circumference; ich determination, however, has never yet jen effected, though we have very near ap- ximation. See CIRCLE. esides these figures, he determined the asures of the spiral, described by a point ving uniformly along a right line, the line the same time revolving with a uniform rular motion, determining the proportion of aréa to that of the circumscribed circle, as 9 the proportion of their sectors. Many of the works of this great man are lextant, though the greatest part of them lost. The pieces remaining are as follow: Two books on the Sphere and Cylinder.— The Dimension of the Circle, or proportion ween the diameter and the circumference. 3. Of Spiral Lines.—4. Of Conoids and 1eroids.—5. Of Equiponderants, or centres travity—6. The Quadrature of the Parabola. ", Of Bodies floating on Fluids.—8. Lem- ta.—9. Of the Number of the Sand. Che most complete edition now extant of works of Archimedes, is the magnificent in folio, printed at Oxford in 1792. This tion was prepared ready for the press by learned Joseph Torelli, of Verona, and in t state presented to the university of Ox- d. The Latin translation is a newone. At end of the whole, a large appendix is led, in two parts; the first being a Com- mtary on Archimedes’s paper upon Bodies t float on fluids, by the Rev. Abram Ro- tson; and the latter is a large collection various readings in the manuscript works Archimedes, found in the library of the » king of France, and of another at Flo- ice, as collated with the Basil edition. A -nch translation of the same has also been vly published by Peyrard. RCHIMEDES’s Screw. See SCREW. \RCHYTAS, a Pythagorean philosopher distinguished mathematician of Tarentum, » flourished about 400 years before Christ, . to whom Aristotle is said to have been ebted for his ethical principles and maxims. the ingenuity of Archytas, as a mathema- an, we owe, according to Eutocius, the ‘hod of finding two mean proportionals, hanically, between two given lines, with a w to the duplication of the cube; and we live from his skill in mechanics, the inven- 1of the screw, crane, and various hydraulic chines; to say nothing of his flying pigeon Vinged automaton. The astronomical and graphical knowledge of Arcbytas is cele- ARE brated by Horace in a beautiful ode, recording also his death, which was occasioned by a shipwreck, which Francis has rendered as fol- lows: ‘* Archytas, what avails thy nice survey Of Ocean’s countless sands, of earth and sea? In vain thy mighty spirit once could soar To orbs celestial, and their course explore ; If here, upon the tempest-beaten strand, You lie confin’d, till some more lib’ral hand Shall strew the pious dust in funeral rite, And wing thee to the boundless realms of light.” ARCTIC (from:aexjo;, the name given by the Greeks to the Little Bear) Circle, in Astro- nomy, a small circle of the sphere parallel to the equator, and distant 23° 28! from the arctic or northern pole. Arctic Pole, the northern pole of the world. ARCTOPHYLAX, a name sometimes given to the constellation Bootes. ARCTURUS, in Astronomy, a fixed star of the first magnitude, marked «, in the constel- lation Bootes ; and thought by some to be the nearest to our system of any of the fixed stars. ARCTUS, in Astronomy, a name given by some authors to the constellations Ursa Major and Minor, or the Great and Little Bear. AREA, in Geometry, is the superficial mea- sure or surface of any figure. The areas of similar plane figures are to each other as the square of their like sides, or other lineal di- mensions. ARENARIUS, the name of a celebrated work of Archimedes, principally written to show the power of numbers, which were not well known among the Greeks till the time of Archimedes. In this work he shows how to number the particles of sand that would be sufficient to fill a sphere, the diameter of which is equal to the distance of the fixed stars from the earth. An English translation of this work was published in London, by G. Anderson, in 1784. AREOMETER, (from aeasos, thin, and jas- teov, measure) an instrument for measuring the density or gravity of fluids. The invention of this instrument is, according to some authors, due to Hypatia, the daughter of Theon; but according to others, it was known and em- ployed by Archimedes. It is now commonly made of glass; consisting of a round hollow ball, which terminates in a long slender neck, hermetically sealed at top ; there being first as much running mercury put into it, as will serve to balance or keep it swimming in an erect position. The stem or neck is divided into degrees or parts, which are numbered, to show the specific gravity by the depth of its descent. Another instrument of this kind is described by Homberg, of Paris, in the Memoirs of the Acad. of Sciences for the year 1699; which is also described in the Phil. Trans. No. 262. By means of this instrument, its ingenious author deduced a table of the different weights of the principal chemical liquors, both in sum- mer and winter ; which varies a little, accord- ing as the liquors are more or less rarefiable ARI by heat, or condensible by cold. DROMETER. AREOMETRY, the science of measuring the specific gravity of fluids. See Phil. 'Trans. vol. Ixviii. ARGETENAR, a star of the fourth magni- tude in the constellation Eridanus. ARGO-NAVIS, the Ship, i in Astronomy, one of the old constellations in the southern he- mispliere, the number of stars in which, ac- cording to Flamstead, is'64. See CoNnsTEL- LATION. ARGUMENT, in Astronomy, is in general a quantity upon which another quantity or equation depends, or some circumstance re- lating to the motion of a planet; or it is an arch , whereby we seck another unknown arch, bearing some proportion to the first: hence ARGUMENT of Inclination, or ARGUMENT of Latitude, of any planet, is an arch of a pla- net’s orbit, intercepted between the ascend- ing node and the place of the planet from the sun, numbered according to the succession of the signs. Menstrual ARGUMENT of Latitude, is the dis- tance of the moon’s true place from the sun’s true place; by which is found the quantity of the real obscuration in eclipses. Annual ARGUMENT of the Moon’s Apogee, or simply Annial Argument, is the distance of the sun’s place from the place of the moon’s apogee; that is, the are of the ecliptic com- prised between those two places, ArGuMENT of the Parallax, denotes the effect it produces on an observation, and which serves for determining the true quantity of the horizontal parallax. ARGUMENT of the Equation of the Centre, is the anomaly, or distance, from the apogee or aphelion; because this equation is calculated in an elliptic orbit for every degree of ano- maly, and varies according to the variation of the anomaly. ARIES, or the Ram, in Astronomy, one of the celestial constellations, and the first of the old 12 signs of the zodiac: it is marked oy, in imitation of aram’s head. The sun enters this sign generally about the 20th of March. See CONSTELLATION. Artes, the denomination given to the bat- iering-ram of the ancients. Sce BATTERING Ram. ARIST AUS, an eminent geometrician of Crotonia, who lived betore Euclid, viz. 330 years before Christ. Pappus (Mathem. Collect. in prooent. lib. vii.) speaks of him as having been the author of five books on the Conic Sections, which, however, have never been transmitted down to us, though those of his contemporary (and, as some have said, his brother), Menechmus, have been preserved. ARISTARCHUS, acclebrated Greek astro- nomer and philosopher, was born at Samos, and flourished about the middle of the third century before Christ. Aristarchus is well known to have maintained the modern opi- nion with regard to the motion of the earth round the sun, and its revolution about its See HyY- ARI own centre or axis. He also taught, that fl annual orbit of the earth is but merely as point, compared with the distance of the fixe stars. His method of determining the di tance of the sun from the earth, was by mea) of the dychotamy of the moon (see DycH TAMY): and in this way he concluded, that contained at least 18, or 20 times, that of 1] moon from the earth. Aristarchus likewi found by methods, the detail of which won be too tedious, that the diameter of the mo¢ bears a greater proportion to that of the eart than that of 43 to 108, but less than that of to 60; so that the diameter of the moon, a cording to his statement, should be somewh less than a third part of the earth. He al estimated the apparent diameter of the st at the 720th part of the zodiac. Besides astronomical discoveries, Aristarchus invent a peculiar kind of hemispherical sun-dii mentioned by Vitruvius, b.ix. c.9. The on work of this ancient astronomer now extal is a treatise “‘on the Magnitude and Distane of the Sun and Moon;’ first published by V; lus, at Venice, fol. in 1498, afterwards ~ Wallis, with his own notes and Commanding version, at Oxford, in 1683, 8vo.; and aga in Wallis’s works, vol. iii. ARISTOTELIAN, any thing relating the doctrines or philosophy of Aristotle. ARISTOTELIAN Philosophy, the philosopl taught by Aristotle, and maintained by I followers. It is otherwise called the Perip tetic Philosophy, from their practice of teal ing while they were walking. "The principles of Aristotle’s philosophy a chiefly taid down in the four books De Ca| Tnstead of the more ancient systems, he inti duced matter, form, and privation, as the prij ciples of all things ; but it does not appear th] he derived much benefit from them in natu philosophy. His doctrines are for the ma part so obscurely expressed, that it has n been yet satisfactorily ascertained what | sentiments were on some of the most ii portant subjects. He attempted to. confy the Pythagorean doctrine, concerning the tw fold motion of the earth; and pretended demonstrate, that the matter of the heave is ungenerated, incorruptible, and not subje to any alteration ; and he supposed that f poe were carried round the earth in so orbs ARISTOTELIANS, a sect of philosophi so called from their leader Aristotle; and ¢ otherwise called Peripatetics. 4 ARISTOTLE, one of the most celebrat of the ancient philosophers, who flourish about 350 years before Christ. He is said have written 400 books, of which, howey only 20 have been handed down to then derns: these may be comprised under heads, viz. Poetry and Rhetoric ; Logi Ethics and Polities ; Physics ; and Metap sics: and such was the veneration paid. him, that his opinion was allowed to stand reason itself. Yet such is the altered state philosophy, that to say a philosopher of 1 ARI resent day were an Aristotelian, would be msidered a stigma upon his judgment and squirements. ARITHMETIC, formed fromeesmpos, nwm- v, the art and science of numbers; or that urt of mathematics which considers their oWers and properties, and teaches how to »mpute or calculate truly, and with ease and xpedition. The several rules relating to this ibject will be found under their respective sads, and we shall limit ourselves in this ace to giving a slight historical sketch of \e progress and improvement of this science, - least as far as any information on this head attainable at this day, which, though in me measure imperfect, will not, it is pre- imed, be found uninteresting to the curious vader. History of AritHMetic.—The early history ‘arithmetic, like that of many other branches ‘knowledge which have proved of the greatest ‘ility to mankind, is so perplexed by fabulous Jations, and a variety of uncertain traditions, iat though many attempts have been made » elucidate it, we have hitherto received but ttle satisfactory information on the subject; me authors attributing the invention to one ution, and some to another, according to the ncy or opinion of the writer, Thus Josephus asserts, that Abraham having tired from Chaldea into Egypt, during the me of a famine, was the first who taught the habitants of that country a knowledge both ‘arithmetic and astronomy, of which they ere before ignorant; while Plato and Dio- s¢nes Laertius, on the contrary, represent the rmer of those sciences, together with that of sometry, as having originated. among the gyptians themselves; to whom it is said they ere first communicated by their god Theut ‘ Thot, a deity whose character has been cought, by some, analogous to that of Mer- uy among the Greeks; and that, like him, 's office was to preside over the concerns of affic and numbers. Strabo, on the other hand, relates, that arith- etic, as well as astronomy, was generally msidered, in his time, as having been in- mted, or at least first cultivated, by the Phoe- cians; who it was supposed, from their being maritime and trading nation, could not have uformed the long voyages they were known have undertaken, or have managed their mmercial affairs, without some acquaintance ith the principles of these useful and neces- ry sciences. But this opinion, as far as it gards any claim to a priority of discovery, is rtainly erroneous; since it is to the Chal- ans, who are well known to have been a ore ancient people, that we are indebted for x knowledge of certain cycles, or astrono- ical epochs, which it is manifest could not ve been determined without their having ade considerable advances both in this sci- ce and that of arithmetic. On this obscure question, however, it is un- cessary to dilate; since it is evident, from e bare consideration of our natural wants, ARI and carliest impressions, that some knowledge of numbers, more or less perfect, must have been nearly coeval with human society: and though in the ruder ages of the world, when the mede of living was simple, and most things in common, the usual methods of reckoning must, in all probability, have been extremely limited, and founded on no fixed or general rules; yet as men advanced in civilization, and had oecasion to enter into frequent trans- actions with each other, it is reasonable to suppose, that the first simple notions of the art would be gradually extended, and various signs and processes devised for assisting the memory, and abridging the labour of long and. difficult computations. But by whom, or at what period, these improvements were first introduced into the science, it is now impos- sible to determine, as none of the early writ- ings on this subject have reached the present times, except the little that has been extracted from them, and employed by Proclus, in his Commentary on the First Book of Euclid’s Elements. We are greatly surprised to find, amid the uncertainties which attend this in- quiry, that almost all nations should have been led to fix upon the same numeral scale, as the basis of their arithmetic; for except the an- cient Chinese, and an obscure tribe mentioned by Aristotle, all the rest appear to have agreed in choosing the decuple division, or method of reckoning by periods of tens, as the most na- tural and commodious that could be devised. . This general conformity of various nations, in adopting the same idea with respect to their numeral scale, could evidently have arisen from no other cause than the custom. so uni- versally observed in childhood, of counting by the fingers; which being first reckoned singly from one to ten, and then successively over again, would naturally lead to the formation of the decimal scale, or decuple division of numbers, so commonly used ; and as the same cause must operate equally under similar cir- cumstances, there can be little doubt but a race of men having more or less than ten fin- gers, would have chosen that number, what- ever it might have been, as the radix or basis of their arithmetic. A people, for instance, with six fingers on each hand, would have certainly counted by periods of twelves ; which method, indeed, had it been originally adopted, would not only have sufficiently answered all purposes of calculation, but would likewise have been attended with some advantages which our present system does not possess.— Without entering, however, into farther re- marks of this kind, it will be sufficient at pre- sent to cbserve, that except in the common practice of dividing numbers into periods of tens, hundreds, thousands, &e. the ancient arithmetic was very different from that of the moderns, both with regard to the method of notation, and to the manner of performing the practical operations of the science. The He- brews and Greeks in particular, at a very early period, and after them the Romans, had re-_ course to the letters of the alphabet for the ARI representation of numbers, which they em- ployed in a way sufficiently commodious for the purposes of a confined numeration ; but, as they were ignorant of the method of giving to each of their characters a local, as well as simple value, the usual processes of multipli- cation and division must necessarily have been attended with considerable difficulty and em- barrassment. The marked superiority, indeed, of our pre- sent numeral system over that of the ancients, is so conspicuous, that since the time of its first introduction into Europe, nearly all know- ledge of the more imperfect and obscure me- thods before used is obliterated and forgotten ; and even the slight vestiges of these aban- doned monuments, which now remain, have become so rare and difficult to be traced, either in the original works, in which they might be expected to be found, or in the com- mentaries and translations of later writers, that except from the scanty relation that has been given of them by Wallis, and the more recent and ample detail of Delambre, but little farther information can be expected on the subject ; particularly as it is now well known, that the authors of most of the early perform- ances in which these methods were employed, have contented themselves with barely giving the results of calculations, without showing the nature of the process, or the different steps of the operation. Bonnycastle’s Arithmetic, 8vo. edition. As, however, the particular rules and modes of operations of the ancients, though now ob- solete, cannot fail of being interesting, we shall, in this historical sketch, give a short detail of the Grecian arithmetic. Arithmetic of the Greeks. We have before observed, that the Greeks divided all their numbers into periods of tens; but, for want of the happy idea of giving a local value to their numerical symbols, they were under the necessity of employing thirty- six characters, most of which were derived from their alphabet, and with which they con- trived to render their arithmetic very regular, and as unembarrassing as such a number of symbols would admit. ete of ae 1, 2s, 415, 6 7.8.9, Fe) 9 oeece eeee they employ- ed the chin ¢ a, B, Ys 8, fy $y e Ny 8. racters To represent ... 10, 20, 30, 40, 50, 60, 70, 80, 90, they mneey og Bath aged Vg of anny For the hun- dreds they} 0s > TT YU Osx bs w 2 TY ents iis But the thousands, 1000, 2000, &c. were represented by «a, B, y, 3, 4 5, 1, 0. That is, they had recourse again to the cha- racters of the simple units, with this difference only, that, in order to distinguish them from —S. ARI the former, they placed a small iota or dasi below the latter, With these characters, it is evident tha the Grecks ceuld express any number unde] 10000, or a myriad. Thus, 991 was represented by fa, DOU oii! sb ie8d. 1s fern 0249, 10: RRS oh Sakai ok era, SEMURE Pos woes cna can Shr tenes enviee iat: G40: cats. Cee ae as 4001 ....... EAL are eek Sw: and so on for others: whence it is evident that neither the order nor the number of cha racters had any effect in fixing the value o| any number intended to be expressed; fo) 4001 is expressed by two characters, 6420 by} three, and 7382 by four. Also the value o| each of those expressions is the same, in what ever order they are placed; thus 9448 is the same as 4490, or as 4003; and so on for any other possible combination but as regularity tended in a great measun towards simplicity, they generally wrote thi characters according to their value, as in th examples above. In order to express any number of myriads they made use of the letter M, placing aboy it the character representing the number o myriads they intended to indicate. Thus, — % Y M, M, M, M, Ke. Hi represented 10000, 20000, 30000, 40000. 'Thus| IN j ro, also, M expressed 370000, ze = 43720000) and, generally, the letter m placed beneatl any number, had the same effect as our an nexing four ciphers. This is the notation employed by Eutocius in his Commentaries on Archimedes; but iti evidently not very applicable to calculations. Diophantus and Pappus represented thei myriads by the two letters my placed after th number ; and hence, according to them, th above numbers would be written thus: @.Mv, §.Mv, y.Mv, Mv, &c. 370000 = ag.mu, and 43720000 = 3708.mu. Also 43728097 is expressed by 3708.mu nid, And 99999999 by 6446.mv O24! This notation in some measure resembk that which we employ for complex number such as feet and inches, or pounds and shi lings. . The same authors, however, employed still more simple notation, by dropping th Mv, and supplying its place with a point; thu instead of dro8.My ng, they wrote droB.ng2; and for 67949.My 640, they wrote 048.0219: SPOT Seer eee eeore eae ay Ne : Pe MOT Bae Ps a ne us. iy eer ten et tad bo Fu ie ane ET ere, ek, Pk Le. ea 7 ; eh j : taf nA: rrr # e ay ay at vee) ee r rene 1 heap ; ele bee yy Beare vat | Be ~. Xa Ws Oa ihe : a > 3 ihe 7] - ie aes Oba ae +s. ‘ ms j 2 a * eh . . - ? a ~ “pe ? -. "oy Oe Se an Re oe ae M : Aneent Characters. TS UWE Ga Characters ) uM of Boethtiis J oo gi \egaee BP ER NG Rs ee PPR EB Yo Cea | Of Alsephadt LB Bw G ORO Oe ee =" TS eee” he ee eel oe . ’ of Sacro Bosco Po PRB Re OS A OS: ae of hogerBacon Naprers Rods. SARASOTA “oe canes SSS EWA VAVAVAVAVAVATE ——— eee: Gee r | ASP | The Arch of Right AscEnston, is that por- ion of the equator, intercepted between the veginning of Aries and the point of the equa- ‘or, Which is the meridian: or it is the num- ver of degrees contained in it. This coin- ides with the right ascension itself, and vhich is the same in all parts of the globe. | Oblique AscENSION, is an arch of the equa- ‘or, intercepted between the first point of Aries, and that point of the equator which ises, together with the star, &c. in an oblique phere. The oblique ascension is counted rom west to east; and is greater or less, ac- vording to the different obliquities of the phere. » Arch of Oblique AscENsioON, is the arch on vhich the oblique ascension is measured; and vhich changes according to the latitude of he place. » ASCENSIONAL Difference, is the diffe- ence between the right and oblique ascen- ion of the same point on the surface of the phere. Or it is the time the sun rises or ets, before or after six o’clock. ' lo find the ascensional Difference of the Sun, | having the Sun’s Declination and the Latitude of the Place. | Say, As radius Is to the tangent of the latitude, So is the tangent of the sun’s declination , ‘To the sine of the ascensional difference. When the latitude and declination have he same name, the difference between the ight ascension, and the ascensional diffe- ence, is the oblique ascension: and their um is the oblique descension; but when they re of contrary names, the sum is the oblique Beension, and the difference is the oblique escension. | ASCII, are those inhabitants of the globe, tho, at certain times of the year, have no hadow; such are all those who inhabit the orrid zone. _ASCELLI, two fixed stars of the fourth i iagnitude, in the constellation Cancer. _ ASPECT, in Astronomy, is the situation of ae stars and planets with regard to each ther. There are five principal aspects; which, “ith their respective characters, are as fol- IWS? viz. §; Conjunction, when the angle con- | tained between any two pla-> 0 UEP M AAG WHA 55) ocsdiovnpeelecdesdswessy ss | ®, Sextile, when the angle is.............. 60° ' O, Quartile, when the angles is.......... 90° A, Trine, when the angle is.............. 120° &, Opposition, when the angle becomes 180° ' These terms were introduced for the pur- oses of astrology, but they are still retained 1 some cases in astronomical works; in the ormer case they are more numerous; but ach foolish distinctions it will be improper in ne present day to enumerate, ASS When the planets have exactly the dis- tances described above, they are called par- tile aspects; and when the distances have not precisely these measures, they are called platic aspects. ASPERITY, the roughness or inequality in the surface of bodies, ASSURANCE on Lives. By assuring a life, is meant obtaining security for a sum to be received should the life drop; in consideration of such a payment made to the assurer, as shall be a suflicient compensation for the loss and hazard to which he exposes himself. In estimating this cempensation, the amount of it will depend entirely on the rate of interest at which money is improved, and the proba- ble duration of the life of the individual. In order to illustrate this, let £100. be supposed to be assured on a life for a year to come, sup- posing money to bear an interest of 5 per cent. Now, first, if it were certain that the life would become extinct in the course of the year, the premium would be the present va- lue of £100. payable a year hence, which at 3 per cent. would be £95. 4s. 8d.; but if in- stead of the certain extinction of the life, it be known from the regular bills of mortality that out of 1000 persons of the age of the assured, only 250 falls in the course of a year, or which is the same, if wei or 4, represent 1000 P the prospect of its failing, then 4 of the above premium will be the fair compensation to be received by the assurer; and as there have been tables of the decrease of human life kept for many years, in various parts of thts and other kingdoms, the calculation of the fair premium to be paid for any age, for a year, is extremely simple, and needs no farther illus- tration. But the assurances most commonly practised, are those on single lives, either for a given term, or during their whole con- tinuance; and the premium is cither paid in a single present payment, or in annual pay- ments during the said term, or till the life fails; in both which cases the calculation be- comes more complicated. We will, however, endeavour to explain the principle of inves- tigation, as far as the limits of our article will admit. Let a be the number of persons living at the age of any given life A; let a’, a", a”, &c. be the number of persons that have died in the Ist, 2d, 3d, &c. year after the age of A; and let m be the amount of £1. for one year, and S the sum to be assured: which latter sum may be supposed also £1. for the sake of simplifi- cation; as it will only be necessary to mul- tiply the result by S, to find the premium in any other case. Now the probability that A dies in the first year is 7 the value therefore of the assurance for that year, for £1. will be i —_. because = represents the present value mea m of £1. a year hence, from the principles ex- plained under the article ANNUITIES: and on ASS “i the same principle, since ef is the probability of the life failing in the second year, therefore a’ m* a second year. In like manner the value of the assurance in the 3d, 4th, Sth, ....nth years will be will be the value of the assurance for the 7] iv Vv (nr) a a a a . yy ees ETL, respectively. ma ma mia m” a And therefore the whole value of the assur- ance for n years, will be a’ a!’ age ae ma me a m" a But this series may be otherwise express- ed; thus, eeeeceoen m* a é a—a ™ ~~ ma gla @ +2 m* m*a 1 a —(qa' + q!! +a") = ms ma + Ke. ..... ae MiOndessteeea 4O 1 a—(d@d +a" ta" +... acm) ie m 7 mM” a 1 a—a ~~ mn? m* a 1 a — (a + a’) ~~ m3 m3 a PO cies ddaed OCEduitstscece 10 1 a— (ait a! + al” bonis ae) ig m* + m” a / t Td Now the series “—% wha + / " ene CS idk Mra A is known to ex- m" a press the value of an annuity of £1. on the life of A for » years; and the series 1 by FTES. PRY AO an m m m m™ to express the value of an annuity of £1. cer- tain, for » years; calling therefore the first of these series M, and the second N: the whole will become equal to a—w meat ane et when wa’ + a” + a” +,&c. a oN MD) ont w m m mea Now since rat is equal to the perpe- tuity, or (p); therefore m—1_ 1 ah ge aay ON ai Pete He mm p+il- Hence the whole value of an assurance S, for any number years n, of the life A, will be a x {N—M4 22). pay; porcencteiom. OG P+i1 m” a where S is the sum assured, » the number of years tor which the assurance’is made, p the perpetuity, N the amount of an annuity cer- tain for n years, and M the amount of an an- nuity for x years on the life of A, also w the the life of A, in 1, 2, 3, &c. n years, will be ASS whole number of persons -that die in the years after the age of A, out of the number) living when the assurance was made. Se Life Annuities, and Propasixities of Life. ff the assurance be for the whole conti nuance of life, the fraction ae vanishes; }. becomes equal to the perpetuity; and M t the value of an annuity for the whole life ¢ A: so that in this case the formula become simply 7 x arian M Seeeeteetese merce Tau RA, If the assurance be that of an estate, ¢ perpetual annuity, the value of each payme of such annuity depending on the failure ¢ 1 a—da_ 1 a—(a+a') 1 Se Le eA ANT m ma ” m2 ma ms a —(a' +a" +a") 1 ow ys Ma ener ar bal vg veer ere wi m3 a m me a and the value of the fee simple, after » year, depending on the contingency of A havin dicd in the mean time, will be 2”; the whol am” value therefore of the assurance will be ; N—M + — x into the annuity [C]; m” or simply (p — M) multiplied into such a nuity, if the assurance is to be continued f the whole duration of the life of A. If the annual payment be required, thi shall be equivalent to the whole value of th assurance, found as above, formule (A) an (B), proceed as follows: wz. Divide th value of the assurance by the value of a annuity of £1. on the given life, for one yei less than the given term, increased by unit and the quotient will be the fixed annual pa) ment required. } Let us illustrate the above by an example supposing the term of years to be 27, the lil 39, the sum to be assured £100. and th interest 5 per cent. 4 ‘The value of an annuity of £1. certain, fi 27 years, at 5 per cent. is 14:643-—=N. The value of an annuity of £1. on the li of a person aged 39, for 27 years, accordil to the Northampton ‘Tables, is 11:191 — N, The present value of £1. payable 27 yea) hence, at the same rate of interest, is *267¢ 1 . _ = — 27 i valk m a The probability of the life failing g, accor ing to the same tables, is ae And the perpetuity is 20 = p. Also the sum assured being 100=S, _ These numbers, substituted in the form la [A], give S UP 2. 2) on gill a gy (N—M + 2?) —s1-976, present value, or payment. at Again, the value of an annuity of £1.0 the same life, for 26 years, = 11:019; whent ASS 31:276 8 __ 31'276 bold + 1 12019 ‘ment required. Nhen the assurance is for the whole con- 1ance of life, divide the present value, as ind in formula [B], by the value of the |, and the quotient will be the annual pay- pnt. yn the above principles, tables have been (aputed which exhibits at one view the an- ‘d payment that ought to be made in order vassure a life, at any age, for £100.; and ch, therefore, is equally applicable for any er sum, by a simple operation of multipli- ‘on and division; these annual payments y a little in different offices, but the most eral terms and conditions are as follows: “he person making the assurance is to de- e the place and date of birth of the person »se life is to be assured. Whether he have | the smail-pox. Whether subject to the t. And whether in the army or navy. — £2. 12s. the annual ditions of Assurance made by Persons on their own Lives. ‘he assurance to be void, if the person se life is assured shall depart beyond the Seven Years, | For the whole ‘ Life, at an 4 One Year. annual Pay- . f ment of ' d. — PON’ TEQDODHNDQNH Qe on Sli aallaed Ww RK Oe Woe oO Drs CaANIORW RK? — DOROSOKRHOHROCWOMIENNDWSWOD Leal BN DORCOK ONAN RE RK RK SK HO RAWSTAOOOMOW GA — i. 0 0 0 1 1 1 1 1 i 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 2 BOD DD et epee eet et ed ee WWWWWMNNNHNHONNVNYNYNYNVNVNONNNNHWY NNER | ne ; MORO RK OCONARWNRK SCOMNWAAA i or farther particulars relating to this in- ‘sting subject, see Simpson’s. “‘ Select Ex- eens D;, Price’s ‘‘ Treatise en Rever- ASS limits of Europe, shall die upon the seas (ex- cept in his Majesty’s packets passing between Great Britain and treland); or shall enter into or engage in any military or naval ser- vice whatever, without the previous consent of the company; or shall die by suicide, duelling, or the hand of justice; or shall not be, at the time tho assurance jis made, in good health. | Conditions of Assurance made by Persons on the Lives of others. The assurance to be void, if the person whose life is assured shall depart beyond the limits of Europe, shall die upon the seas (ex- cept in his Majesty’s packets passing between Great Britain and Ireland); or. shall enter into or engage in any military or naval ser- vice whatever, without the previous consent of the company; or shall not be, at the time the assurance is made, in good health. Any person making an assurance on the life of another, must be interested therein, agreeably to act of 14th of Geo. III. chap. 48. which prohibits wagering, or speculative in- surances. TABLE OF PREMIUMS assuring the sum of One Hundred Pounds upon the Life of any healthy Person, from the Age of Eight to Sixty-seven. Seven Years, | For the whole at an an- Life, at an One Year, nual Pay-] annual Pay- ment of ment of 1 I Os 11 10 1 eel “ — i=) =DAADONSCOSMAAARRAOCHW OD APA J _— — De wVoonmne owe eKWOKOOCcCHheEWWOOMAOADO:A OO PRR WO WWW WOWNNNHNNNNHNNNW NH WWr ik pd fom ped food ee KOSOOCrwe DWNwpoorntwarWe OVW SOWVHUOA Ww we QATAR RRO WWWWWWwWNnNNNnnnvnnunsn © DRBONMUVYDAHWDAA AAA aH A AHP PPA CS OW OF WH © iw) _ — He OOWORPKHODNOKHRONODAWOKHVNH KK CMMOr sionary Payments ;” Morgan’s “ Doctrine of Annuities and Assurances;” and an excellent Treatise on the same subject by F, Baily > AST see also the several articles, ANNUITIES, LIFE Annuities, SURVIVORSHIP, &c. ASTERION, the name of one of the dogs in the constellation Canes, Venatici. ASTEROIDS, (from asng, star, and sidos, like) in Astronomy, a name given by Dr. Herschel to the four new planets discovered by the foreign astronomers Piazzi, Olbers, and Harding; which are defined as “celestial bodies, either of little or considerable eccen- tricity round the sun, the plane of which may be inclined to the ecliptic in any angle what- ever. The motion may be direct or retro- gade; and they may, or may not, have con- siderable atmospheres, very small comas, discs, or nuclei:” a definition which seems to be invented for no other purpose than to de- prive every one but himself of the honour of having discovered a planet. ASTRISM, in Astronomy, an ancient term signifying the same as CONSTELLATION. “ASTRA, in Astronomy, a name given by some authors to the sign Virgo. ASTRAL, depending or belonging to the stars: as ASTRAL year, &e. ASTRODICTICUM, an astronomical in- strument invented by Mr. Weighel, by means of which several persons may view the same star at the same time. ASTROGNOSIA, signifies a knowledge of the fixed stars, their names, ranks, situa- tions, &e. ASTROLABE, (from asne, star, and Awu- Gays, I take; alluding to its use in taking, or observing, the stars) is the name of an an- cient astronomical instrument, very much resembling our armillary sphere. The first, and most celebrated instrument of this kind, is that mentioned by Hipparchus, which he made at Alexandria, the capital of Egypt, and which he lodged in a seeure place, where it served for divers astronomical observations. The astrolable has been treated of by Stofller, Gemma Frisius, Clavius, and several other early astronomers. ASTROLABE is likewise the name ofan in- strument formerly much used at sea for ascertaining the altitude of the sun, stars, &c. It consisted of a brass ring about fifteen inches in diameter, graduated into degrees and minutes, and fitted with an index move- able about its centre, and carrying two sights; the whole being attached to a small brass ring for suspending the instrument at the time of observation. ASTROLABE is also used by modern astro- nomers to denote a stereographic projection of the sphere, either upon the plane of the equator, the eye being supposed to be in the pole of the world; or upon the plane of the meridian, when the eye is supposed in the point of the intersection of the equinoctial and horizon. ASTRONOMICAL, any thing relating to astronomy; as Astronomical Day, Hour, Year, Observations, Tables; &c.: for which, see the respective substantives. ASTRONOMICALS, a name used by some AST q writers for sexagesimal fractions, on accoun of their use in astronomical observations. ASTRONOMICUS Radius. See Ranpuws, ASTRONOMY, (formed of asnp, star, ane youos, law or rule) is a mixed mathematica science, which treats of the heavenly bodies their motions, periods, eclipses, magnitudes &c, and of the causes on which they depend That part of the science which relates to th motions, magnitudes, and periods of revoly tion, is called Pure or Plain Astronomy ; anc that which investigates the causes and law; by which these motions are regulated, . called Phisical Astronomy. History of Astronomy.—The early historyo this science, like that of all others of ancien date, is too much disfigured by fabulous ani allegorical representations, to admit of anyag gular or satisfactory elucidation. It is pro bable, however, that some knowledge of th kind must have been nearly coeval with tl formation of society; for, besides motives mere curiosity, which are sufficient to hay invited men in all ages to examine the w e nificient and varying canopy of the heaver it is evident that some parts of the sciend are so connected with the common cons ern | of- life, as to render the cultivation . the " indispensably necessary. a Many traces of it have achhtingee 14 found amongst various nations, which shoj| that several of the most remarkable celest ic phenomena, at least, must have been obsen ed, and a knowledge of them disseminated i a very remote period. But in what age | (| country the science first originated, or | y whom it was gradually methodised and i 4 proved, is extremely uncertain; nothing m ; being known on this subject than what ¢ be obtained from the scanty and incide nti information of ancient writers, whose accon nh | are often too extravagant and ee 2 deserve much attention. © Amongst other relations of this kind, 1 be reckoned what is mentioned by Josehl in his Antiquities, who, in speaking of the) gress made in astronomy by Seth and his terity before the deluge, asserts thet they graved the principles of the science on ty pillars, one of stone and the other of brie called the pillars of Seth; and thatthe fo rn t of these was entire in his time. He also @ cribes to the. Antediluvians, a knowledge ¢ the astronomical cycle of 600 yeus; wh hic _ however, Montucla thinks, with mich great reason, was an invention of the Chaldeans; tl knowledge of which he, in all protability, ol tained cither from that people, or from som ancient writings which no longer subsist. — M. Bailly, in his elaborate history of al cient and modern astronomy, emleayours | trace the origin of this science among Chaldeans, Egyptians, Persians, [ndians, al Chinese, to a very early period, And he maintains, that it was cultivated in E and Chaldea 2800 years before Christ; Persia, 3209; in India, 3101; and in Chin 2952 years before that wzra. He also appt ' ‘ ; : ‘G: os y % a ; : Oe ad #1 > y — et Ay ¥ ba ra ‘ % ASTRON OMY. pam 2 ee re — dpparent Magnitude of the Sun as seen trom the several Planets. frem Mercury from the Earth from Mars from ° Jupiter from Venus from ° Saturn from °. Georgium Sidus Shu e tediis : Sylow , / Georgium Sidus or Uranus S&S it Mars ’ J 7 FE arth ° ° Venus ° ? Mercury ° 2 Engraved by Sam Laces | Ne London Liblithed March 31,7813. bvG&S Robinson, lidemoster how, & the rest of the Lroprtetors. = = ~ a R — a ~ po ne aS ann nS Pin ¥ ise oe A. Ce ae an 2 % wit ~e ise : 3 J Oa P hd har is eras - seo Paben AS TR ONO VI ane he. oe. Na a 7 Y ee ee \ | 7 ¢ ¥ a : ion pe } S'S : | aga = ri ee , rH) pe i ‘wi Terrestrial Glebe. ey SOT Engraved by Sam! Lacey. seri; Tendon, Published Septlg0,1813, by C&T Robinson, Paternoster Row, & the rest of the Proprietors. f. ; ; ‘i : . i Loe ae | | farallax . i : AST mds, that astronomy had been studied even ng before this distant period, and that we e only to date its revival from this’ time. In investigating the antiquity and progress | astronomy among the Indians, M. Bailly amines and compares four different sets of tronomical tables of the Indian philoso- ers; namely, that of the Siamese, explain- « by M. Cassini in 1689; that brought from idia by M. le Gentil, of the Academy of Siences; and two other manuscript tables, iid among the papers of the late M. de isle; all of which he found to accord toge-+ ier, and all referring to the meridian of jmares. It appears that the fundamental éoch of the Indian astronomy, is a remark- ale conjunction of the sun and moon, which tok place +t the distance of 3102 years be- fe Christ: and M. Bailly informs us, that by cr most accurate astronomical tables, such onjunction did really happen at that time. I: farther observes, that at present the In- dns calculate eclipses by the mean motions athe sun and moon commencing at a period 5)0 years distant. Che cycle of nineteen years is also used by tl: people, and their astronomy agrees with ndern discoveries as to the obliquity of the eiptic, an acceleration of the motion of the € unoctial points, and in many other respects. }om the researches made into the knowledge othe Indians, on these points, by Playfair, it he second volume of the Edinburgh Trans- mions, and several writers in the Asiatic Rsearches, it may be inferred that the fabu- lcs accounts of the antiquity of the world, bieved by the vulgar among the Hindoos, rer only to various periods assigned by their xs onomers for the commencement of diffe- ret calculations. ‘he solar year of the Brahmins of Terva- io is divided into twelve unequal months, 221 being equal to the time the sun occupies naoving through a sign; and in their calcu- aons for a day, they employ the time the su moves 1° in the ecliptic. Their sidereal yes consists of 3654, 6, 12m, 30%; and the riical, of 3654, 54, 50™, 35". They assign in- qalities to the motions of the planets, an- Mring very well to the annual parallax, and h equation of the centre. — fost authors, however, fix the origin of is onomy and astrology either in Chaldea, or nigypt; and accordingly, among the an- ‘i ts, we find the word Chaldean often used oistronomer, or astrologer. Indeed both of he nations pretended to a very high an- icity, and claimed the honour of producing h first cultivators of this science. The Chal- leis boasted of their temple, or Tower of 3cis, and of Zoroaster, whom they placed 10) years before the destruction of Troy: vile the Egyptians spoke with equal pride if reir colleges of priests, where astronomy vé taught ; and of the monument of Osyman- ly, in which, it is said, there was a golden i € Of 365 cubits in circumference, and one | | | AST eubit thick, divided into 365 equal parts, an- swering to the days of the year, &c. Diodorus Siculus informs us, that “ the southern parts of Arabia consist of sandy plains, of a prodigious extent; the travéllers through which directed their course by the Great and Little Bear, in the same manner as is done at sea.” He farther observes, that the Chaldeans made the annual motion of the sun oblique to the equator, and contrary to the daily niotion; and that they had thirty- six constellations; twelve in the zodiac, and twenty-four out of it. They also made an observation on Saturn in the year 228, A. C., which is preserved by Ptolemy; and it ap- pears to be the only one which they ever made on the planets, Mr. E. Barnard says, (Phil. Trans. No. 158.) that the Egyptians discover- ed, very early, that the stars had an annual motion of 50”, 9", 45”, in a year; and, ac- cording to Macrobius, the Egyptians made the planets revolve about the sun in the same order as we do: but it does not appear at what time the planets were discovered. From Chaldea and Egypt, the science of astronomy passed into Phenicia, which this people applied to the purposes of navigation, steering their course by the north polar star ; whence they became masters of the sea, and of almost all the commerce in the world. The Greeks, it is probable, derived their as- tronomical knowledge chiefly from the Egyp- tians and Phenicians, by means of several of their countrymen who visited those nations for the purpose of learning the different sciences. Newton supposes that most of the constellations were invented about the time of the Argonautic expedition; but it is more probable that they were, at least most part of them, of a much older date, and derived from other nations, though clothed in fables of their own invention or application. Several of the constellations are mentioned by Hesiod and Homer, the two most ancient writers among the Greeks, and who lived 870 years before Christ. Their knowledge in this science, however, was greatly improved by Thales the Milesian, and other Greeks, who travelled into Egypt, and brought from thence the chief principles of the science. Thales was born about 640 years before Christ; and was the first among the Greeks who observed stars, the solstices, the eclipses of the sun and moon, and predicted an eclipse of the sun; and the same was farther cultivated and extended by his successors, Anaximander, Anaximenes, and Anaxagoras; but most es- pecially by Pythagoras, who was born 577 years before Christ, and having resided a long time in Egypt, &c. brought from thence the learning of these people, taught the same in Greece and Italy, and founded the sect of the Pythagoreans. He taught that the sun was in the centre of the universe ; that the earth was round, and people had an- tipodes; that the moon reflected the rays of the sun, and was inhabited like the earth ; G AST that comets were a kind of wandering stars, disappearing in the farther parts of their orbits; that the white colour of the milky way was owing to the united brightness of a great multitude of small stars; and he supposed that the distances of the moon and planets from the earth, were in certain harmonic pro- portions to one another. Philolaus, a Pythagorean, who flourished about 450 years before Christ, asserted the annual motion of the earth about the sun; and not long after, the diurnal motion of the earth, on its own axis, was taught by Hicetus a Syracusian. About the same time flourish- ed, at Athens, Mcton and Euctemon, where they observed the summer solstice 432 years before Christ; and observed the risings and settings of the stars, and what seasons they answered to. Meton also invented the cycle of nineteen years, which still bears his name. Eratosthenes, who was born at Cyrene in the year 271 before Christ, measured the cir- cumference of the earth; and being invited to Alexandria, from Athens, by Ptolemy Euergetes, and made keeper of the royal library there, he set up for that prince those armillary spheres, which Hipparchus~ and Ptolemy the astronomers afterwards em- ployed so successfully in observing the hea- vens. He also determined the distance be- tween the tropics to be 44 of the whole me- ridian circle, which makes the obliquity of the ecliptic, in his time, to be 23° 514’. The celebrated Archimedes also cultivated astro- nomy, as well as geometry and mechanics; and constructed a kind of planetarium, or orrery, to represent the phenomena and mo- tions of the heavenly bodies. ‘To pass by several other of the ancients, who practised or cultivated astronomy, more or less, we find that Hipparchus, who flou- rished about 140 years before Christ, was the first who applied himself to the study of every part of this science ; and, as we are informed by Ptolemy, made great improvements init: he discovered that the orbits of the planets are eccentric, that the moon moved slower in the apogee than inher perigee, and that there was a motion of anticipation of the moon’s nodes; he constructed tables of the motions of the sun and moon, collected accounts of - such eclipses, &c. as had been made by the Egyptians and Chaldeans, and calculated all that were to happen for 600 years to come: he discovered that the fixed stars changed their places, having a slow motion of their own from west to east: he corrected the Ca- lippic period, and pointed out some errors in the method of Erastosthenes for measuring the circumference of the earth; he computed the sun’s distance more accurately than any of his predecessors: but his chief work is a catalogue which he made of the fixed stars, to the number of 1022, with their longitudes and latitudes, and apparent magnitudes ; which, with most of his other observations, are preserved by Ptolemy in his Almagest. AST But little progress was made in astronom from the time of Hipparchus to that of Pte lemy, who was born at Pelusium-in Egyp in the first century of the Christian era, an) who made the greatest part of his observe tions at the celebrated school of Alexandr in that country. Profiting by those of Ht parchus, and other ancient astronomers, h formed a system of his own, which, thoug erroneous, was followed for many ages by a nations: he compiled a great work, calle the Almagest, which contained the observi tions and collections of Hipparchus, an others of his predecessors in astronomy a performance which will ever be valuable the professors of that science. This wo was preserved from the grievous conflagrati¢ of the Alexandrine Library by the Saracen and translated out of Greek into Arabic, j the year 827, and again into Latin in 1230. During the long period, from the year till the beginning of the fourteenth centu the western parts of Europe were immer in the grossest ignorance and barbarity, whi the Arabians, profiting by the books they hé preserved from the wreck of the Alexandri Library, cultivated and improved all @ sciences, and particularly that of astronom in which they had many able professors at authors; the caliph, Al Manor, first intr duced a taste for the sciences into his er pire; and his grandson, Al Mamon, wi ascended the throne in 814, was a great @ courager and improver of the sciences, al especially of astronomy. Having construct proper instruments, he made many observ tions; determined the obliquity of the ecli to be 23° 35’; and under his auspices, a | eree of the circle of the earth was measur a second time, in the plain of Singar, on 7 border of the Red Sea. About the same tim Alferganus wrote elements of astronomy ; al the science was from that time greatly cul vated by the Arabians, but principally by All} tegnius, who flourished about the year 880, ai who greatly reformed astronomy by comp: ing his own observations with those of PI lemy; whence he computed the motion oft sun’s apogee from Ptolemy’s time to his ow settled the precession of the equinoxes at 0 degree in seventy years; and fixed the ¢ liquity of the ecliptic at 23° 35’; the tab) also which he composed for the meridian Aracta, were long esteemed for their accura by the Arabians. | After his time, though the Saracens many eminent astronomers, several centur| elapsed without producing any very valual| observations, excepting those of some eclip! observed by Ebn Younis, astronomer to” Caliph of Egypt; by means of which quantity. of the moon’s acceleration, si that time, has been determined. n Amongst other eminent Arabic astro mers, may be reckoned Arzachel, a Moor Spain, who observed the obliquity of 1 ecliptic, and who also improved trigonome 4 AS T constructing tables of sines, instead of ords of ares, dividing the diameter into irty equal parts; and Alhazen, his contem- rary, who wrote upon the twilight, the ight of the clouds, the phenomenon of the rizontal moon, and first showed the im- rtance of the theory of refraction in astro- my. The settlement of the Moors in Spain in- duced the sciences into Europe; from ich time they have continued to improve, d to be communicated from one people to ther, to the present time, when astronomy 1 all the sciences have arrived at a very inent degree of perfection. 'Che Emperor ederick II, about 1230, first began to en- wage learning, by restoring some decayed iversities and founding a new one in enna: he also caused the works of Ari- tle, and Ptolemy’s Almagest, to be trans- 2d into Latin; and from the translation of s work we may date the revival of astro- ny in Europe. Two years after this, John Sacro Bosco, or John of Halifax, compiled n Ptolemy, Albategnius, Alferganus, and er Arabic astronomers, his work “ De era,’ which was held in the greatest esti- ion for 300 years after, and was honoured commentaries by Clavius and other ned men. In 1240, Alphonso, King of stile, not only cultivated astronomy him- ; but greatly encouraged others; and by assistance of several learned men he cor- ted the tables of Ptolemy, and composed se which were denominated from him the honsine Tables. About the same time » Roger Bacon, an English monk, wrote eral tracts relative to astronomy, particu- y of the lunar aspects, the solar rays, and places of the fixed stars. And about the 1270, Vitello, a Polander, composed a tise on optics, in which he showed the of refraction in astronomy. ittle other improvement was made in this nce till the time of Purbach, who was in 1423. He composed new tables of *s for every ten minutes, making the radius y, with four ciphers annexed. He con- .cted spheres and globes, and wrote seve- astronomical tracts, as a commentary on lemy’s Almagest; some treatises on arith- ic and dialling, with tables for various ates; new tables of the fixed stars, re- ed to the middle of that century; and he ected the tables of the planets, making ‘ equations to them where the Alphonsine es were erroneous. In his solar tables, he ed the sun’s apogee in the beginning of cer; but retained the obliquity of the ntic 23° 333’, as determined by the latest ‘rvations. He also observed some eclipses, e new tables for computing them, and had finished a theory of the Shiota! when he -In 1462, being only thirty-nine years of fter Purbach, the subject of astronomy cultivated by John Muller, commonly ?d Regiomontanus; his labours were suc- AS TF ceeded by those of Bernard Walther; and Walther was followed by John Werner, 2 clergyman, at Nuremberg, &ec. he showed that the motion of the fixed stars, since called the precession of the equinoxes, was about 1° 10’ in 100 years. The celebrated Coper= nicus was the next who made any consider- able figure in astronomy. He very early con- ceived doubts of the Ptolemaic system, and entertained notions about the true one, which he gradually improved by a series of astronc~ mical observations, and the contemplation of former authors. By these he formed new tables, and completed his work in-the year 1530, containing these, and a renovation of the true system of the universe, in which all the planets are considered as revolving about the sun, as their common centre. After the death of Copernicus, the science and practice of astronomy were greatly im- proved by Schoner, Nonius, Appian, Gemma Frisius, Byrgius, &c.; and about the year 1561, William IV. Laudgrave of Hesse Cas- sel, applied himself to the study of this sci- ence, with the best instruments which could then be procured, made a great number of observations, published by Snelius in 1618, and preferred by Hevelius to those of Tycho Brahe. From these observations he formed a catalogue of 400 stars, with their latitudes and longitudes, and adapted them to the be- ginning of the year 1593. Tycho Brahe, a noble Dane, began his ob- servations about the same time with the Landgrave of Hesse, and observed the great conjunction of Jupiter and Saturn: but find- ing the usual instruments very inaccurate, he constructed many others, much larger and more exact. In 1571, he discovered a new star in the chair of Cassiopeia; which induced him, like Hipparchus on a similar occasion, to make a new catalogue of the stars; which he composed to the number of 777, and adapt- ed their places to the year 1600. In the year 1576, by favour of the King of Denmark, he built his new observatory, called Uraneburg, on the small island Huenna, opposite Co- penhagen, which he very amply furnished with many large instruments; some of them so divided as to show single minutes, and in others, the arch might be read off to ten seconds. Tycho invented a system to account for the planetary motions; but he is more to be noted on account of his accurate observa- tions, which tended much to the discovery of the real nature of the planetary orbits. While Tycho resided at Prague, with the emperor, he prevailed on Kepler to leave the University of Glatz and to come to him; and T'ycho dying in 1601, Kepler enjoyed the title of mathematician to the emperor; who ordered him to finish the tables of Tycho Brahe, which he published in 1627, under the title of Rodolphine. He died about the year 1630, at Ratisbon, where he was soliciting the arrears of his pension. From his own observations, and those of Tycho, Kepler dis- covered several of the true laws of nature, by G 2 AST which the motions of the celestial bodies are regulated. He discovered that all the planets revolve about the sun, not in circular but in elliptical orbits, having the sun in one of the foci of the ellipse; that their motions are not equable, but varying quicker or slower, as they are near to the sun, or farther from hin ; that the areas described by the variable line drawn from the planet to the sun are equal in equal times, and always proportional to the times of describing them. He also discovered, by trials, that the cubes of the distances of the planets from the sun were in the same proportion as the squares of their periodical times of revolution. By observations also on comets, he concluded that they are freely carried about among the orbits of the planets, in paths that are nearly rectilinear; but which he could not then determine. See Dr. Small on the discoveries of Kepler. About this time much was done by Wright, Napier, Bayer, &c. ‘To Napier we owe some excellent theorems and improvements in spherics, besides the ever memorable inven- tion of logarithms. Bayer, a German, pub- lished his Uranometria, or the figure of all the constellations visible in Europe, with the stars marked on them, and accompanied by names of the letters of the Greek alphabet ; a contrivance by which they may easily be referred to with distinctness and precision. About the same time too, astronomy was culti- vated abroad by Mercator, Maurolycus, Mag- nius, Homelius, Schulter, Steven, Galileo, &c.; and in England, by Thomas and Leonard Dieges, John Dee, Robert Hood, Harriot, &c. The beginning of the seventeenth century was particularly distinguished by the inven- tion of telescopes, and the application of them to the purposes of astronomy. The more distinguished early observations with the telescope, were made by Galileo, Harriot, Huygens, Hooke, Cassini, &c. Itis said that, from report only, Galileo made for himself telescopes, by which were discovered inequa- lities in the moon’s surface, Jupiter’s satel- jites, and the ring of Saturn; also spots on the sun, .by which be found out the revolu- tion of that luminary on its axis; and he dis- covered what was merely supposed by Pytha- goras, that the nebule and milky way were full of small stars. Harriot also, hitherto known only as an algebraist, made much the same dis- coveries as Galileo, and as early, if not more SO; as appears by his papers, not yet printed, in the possession of the Earl of Egremont. Hevelius, from his own curious observa- tions, furnished a catalogue of fixed stars, much more complete than 'Tycho’s. Huygens and Cassini discovered the satellites of Sa- turn, and his ring. And Gassendus, Horrox, Bullialdus, Ward, Ricciolus, Gascoign, &c. each contributed very considerably to the im- provement of astronomy. The immortal Newton demonstrated, from physical consideration, the great law that re- gulates all the heavenly motions, sets bounds to the planetary orbs, determined their AST greatest excursions from the sun, and ther nearest approaches to him. It was he wh first discovered whence arose that constan and regular proportion, observed by both pri mary and secondary planets, in their eiren lation round their central bodies; and thei distances, compared with their periods. Hi has also given us a new theory of the moon which accurately answers all her inequalities and accounts for them from the laws of gra vity and mechanics. . Mr. Flamstead was appointed the first as tronomer royal at Greenwich in 1675. H observed, for forty-four years, all the celestia phenomena, the sun, moon, planets, and fix ed stars; of all which he gave an improve theory and tables; viz.a catalogue of 300 stars, with their places, to the year 1689; 7 new solar tables, and a theory of the mooy according to Horrox; likewise, in Sir Jona Moor’s System of Mathematics, he gave) curious tract on the doctrine of the sphere showing how to construct geometrically eclip ses of the sun and moon, as well as occulta tions of the fixed stars by the moon. Andi was upon his tables that were constructe’ both Halley’s tables, and Newton’s theory ¢ the moon. Cassini also, the first French as tronomer royal, very much distinguished him self, making many observations on the sur moon, planets, and comets, and greatly im proved the elements of their motions. H also erected the gnomon, and drew the cel brated meridian line in the church of Petr nia, at Bologna. 1a In 1719, Mr. Flamstead was succeeded b Dr. Halley, the friend of Newton, and ama of the first eminence in all the departments ( literature and science ; who had been sent,é the early age of 21, to the island of St. Helen to observe the southern stars, a catalogue’ which he published in 1679; and a few yeal afterward he gave to the public his “ Synops) Astronomiz Cometice,” in which he vel tured to predict the return of a comet in 174) or 1759. He was the first who discovered th acceleration of the moon’s mean motion; all is the author of a very ingenious method fi finding her parallax, by three observed plact of a solar eclipse: he also showed the use th’ might be made of the approaching transit Venus, in 1761, in determining the distant of the sun from the earth; and recommend the method of determining the longitude f the moon’s distance from the sun and certa fixed stars, which has since been carried in execution at the instance of the late Astr nomer Royal. Dr. Halley also composed tabl of the sun, moon, and planets, with which] compared the observations he made of ti moon at Greenwich, amounting to near 150 and noticed the differences. About this tim an.attempt was made in France to measure degree of the earth, which was the occa of a warm dispute concerning its figure. — Cassini concluded, from the measurement Picard, that it was an oblong spheroid; } Newton, from a consideration of the laws AST ravity, and the diurnal motion of the earth, ad determined its figure to be that of an blate spheriod, flattened at the poles, and rotuberant at the equator. To determine his point, Louis XV. ordered two degrees of ‘he meridian to be measured ; one under or ear the equator, and the other as near as vossible to the pole : the result of this arduous undertaking was a confirmation of Newton’s savestigation. Messrs. Maupertuis, Clairaut, ce. were employed on the northern expedi- ion; and Condamine, Bouguer, Don Ulloa, \f Spain, &c. on the southern; who all fulfilled heir commissions with great credit to them- jelves and advantage to the sciences, making aany observations besides those immediately onnected with this subject. Among others, = Was found, by those who went to the south, hat the attraction of the mountains of Peru .ad a sensible effect on the plumb-lines of heir large instruments, which afforded an xperimental proof of the Newtonian doctrine f gravitation, that has since been completely .erified by the observations of Dr. Maskelyne, aade on the mountain Scliehallien, in Scot- fand. On the death of Dr. Halley, 1742, he vas succeeded by Dr. Bradley, who has ren- ) ‘ered himself celebrated by two of the finest liscoveries that have ever been made in astro- xomy,—the aberration of light, and the nuta- ion of the earth’s axis. Among other things xe also formed new and accurate tables of he motions of Jupiter’s satellites, as well 8 the most correct table of refractions yet -xtant. Also, with a large transit instrument, jnd a new mural quadrant of eight feet radius, onstructed hy Bird, in 1750 he made an im- ‘nense number of observations for settling the olaces of all the stars in the British catalogue, ‘ogether with near 150 places of the moon, he greater part of which he compared with Mayer’s tables. Dr. Bradley was succeeded ‘n 1762, in his office of astronomer royal, by Mr. Bliss; but who, being in a declining state of health, died in 1765, and was succeeded by Nevil Maskelyne, D. D. the late astronomer oyal, and who has rendered very considerable -ervices to the science, by his publication of he “Nautical Almanac,” the “ Requisite Tables,” &c.; and more particularly by his reat assiduity aud zeal, in bringing the lunar nethod of determining the longitude at sea nto general practice. _ In the mean time many other eminent ma- -hematicians, both of our own and other na- ions, were assiduously employed in endea- youring to promote the science of astronomy. Amongst these may be particularly distin- suished, Clairaut, d’Alembert, Euler, Simp- son, Walmsley, Mayer, de la Caille, Manfredi, Lambert, &c. Such was the state of astronomy when Dr. Herschel, by augmenting the powers of tele- scopes beyond the most sanguine expecta- dons, opened a scene altogether unlooked Jor, by the discovery of a new planet, 13th March, 1781, and which he named, in ho- | ‘ ; AST nour of his majesty, the Georgium Sidus ; but the planet is now more commonly called (at least by foreign astronomers) Uranus. This planet revolves about the sun in an orbit dou- ble in diameter to that of Saturn, and conse- quently the limits of the planetary system may be said to have been doubled by this disco- very. We are also indebted to the same cele- brated astronomer for a variety of observations on several other astronomical subjects ; such as the discovery of two additional satellites to Saturn, and those of his own new planet; and several other important and interesting re- sults, which the limits of our article will not admit of enumerating: they are mostly con- tained in the Philosophical'Transactions, where they will remain an honourable testimony of the learning, ingenuity, and industry, of this celebrated astronomer. On the Ist of January, 1801, another new planet was discovered between Mars and Ju- piter, by M. Piazzi, of Palermo, which is named Ceres; and since this time three others have been observed, revolving also between Mars and Jupiter, viz. Pallas, discovered by Dr. Olbers, March 28, 1802; Juno, first ob- served by Mr. Harding, at the observatory at Lilienthal, near Bremen, Sept. 1, 1804; and Vester, discovered by Dr. Olbers, 29th of March, 1807, being the second that we owe to this eminent astronomer. For the particular elements of these new planets, see the respec~- tive articles URANus, CERES, PALLAS, JUNO, and VESTER. Hence it appears, that within a few years the number of planets in our system have been nearly doubled, and many other important and interesting discoveries have been made during the same period: yet it must be acknowledged that we are still unacquainted with many par- ticulars, and which therefore still remain to exercise the talents of modern astronomers. We have not yet determined the times of ro- tation, and the proper figures of some of the planets and their satellites; nor do we know, with sufficient precision, the masses of those bodies. The theory of their motions also con- sists in a series of approximations, of which the convergence depends both upon the per- fection of the instruments and the progress of analysis, and which for that reason ought to acquire continually new degrees of exactness. We shall not, in this place, enter into any his- tory of the discoveries connected with what is termed the physical part of astronomy; not ihat it is less interesting, but because our limits will not admit of a farther extension of this article; besides this, being principally of modern date, abstracts of it will be found under various heads in this work; as CENTRAL Force, Density, Gravity, Kerver’s Laws, &c.&c. We shall therefore now conclude this article with a reference to such works as are best calculated to convey correct ideas to the student, both as relating to the history, prac- tice, and theory, of this interesting science; such are, Weilder’s History of Astronomy, AS Y brought. down to the year 1737; Bailly’s His- tory of ancient and modern Astronomy ; Mon- tucla’s Histoire des Mathematiques ; and the first volume of Lalande’s Astronomy. For the practical part of the subject, see Lalande’s and Vince’s Astronomy, in two vols. 4to. and Woodhouse’s Treatise, lately published, on the same subject: for the more familiar and introductory works, the reader may consult Bonnycastle’s, Gregory’s, Emerson’s, and Fer- guson’s works on Astronomy; to which may also be added, those of La Caille, Kiell, Lead- better, Street, &c.; and Vince’s Practical Astronomy, 8vo. And on the physical part of the science, see Dr. Gregory’s work, in two vols. published in 1702; Biot’s “ Tracte Elé- mentaire des Astronomie Physique ; and above all, the Mécanique Céleste of La Place. ASTROSCOPE (from ane, star, and cxo- orew, J consider), an astronomical instrument, composed of two cones, on whose surfaces are exhibited the stars and consteilations, by means of which they are both easily found in the heavens. This instrument was the invention of Schukhard, professor of ma- thematics at Tubingen, who published a trea- tise expressly on it in 1698. ASTROSCOPIA, is the art of observing and examining the stars. with the telescope, in order to discover their nature and pro- perties. ASTROTHESTIA, an ancient term, nearly synonymous with constellation. ASTRUM, or AsTRON, a constellation or assemblage of stars, the same as Aster de- notes a single star. ASYMMETRY (from «@ privitive, cuy, with, and penreov, measure), without measure, a want of proportion between the parts of a thing, as between the side and diagonal of a square, which are to each other as 1 : 2. ASYMPTOTE (from « privative, cvy, with, and wnlw, I fall; incoincident), is properly a right line, which approaches continually nearer to some curve, whose asymptote it is said to be, in such sort, that when they are both in- definitely produced, they are nearer together than by any assignable finite distance; or it may otherwise be considered as a tangent to the curve, when infinitely produced, or at an infinite distance. Two curves are also said to be asymptotical, when they thus continually approach indefinitely to a coincidence: thus two parabolas, placed with their axis in the same right line, are asymptotes to each other. Of lines of the second kind, or curves of the first kind, that is, the conic sections, only the -hyperbola has asymptotes, which are two in number. All curves of the second kind have at least one asymptote, but they may have three ; and all curves of the third kind may have four asymptotes, and so on. The con- choid, cissoid, and logarithmic curve, though not geometrical curves, have each one asymp- tote ; and the branch or leg of the curve that has an asymptote, is said to be of the hyper- bolic kind. sielib id to be ok the hyper # ASY The nature of an asymptote is very difficult to be conceived, by persons who are not ae- quainted with the higher geometry: they can- not comprehend how two lines should always continually approach each other, without the possibility of touching or coinciding; this mystery, however, may be elucidated, and the nature of these lines readily comprehended, by considering the generation of the conchoid of Nicomedes. which is as follows: . Let F K be any line inde- a finite towards K, and from the point P let therebedrawn yy the lines P A, PB, PC, PD, & c.making the several parts FA,GB, HC, Pees OS, a oe A all equal to each other: the curve ABC DE, &c. passing through all the extremities A, B, C, D, &e. is called the conchoid of Nicomedes; and the line F K produced, is the asymptote of the curve; and which it is obvious, fron the construction, can never coincide or touch the curve itself, although the latter continu- ally approaches towards the former. ah AsympTores, by some are distinguished into various orders. ‘The asymptote is said to! be of the first order, when it coincides with the base of the curvilinear figure; of the 2d order, when it is a right line parallel to the base; of the 3d order, when it is a right line! oblique to the base ; of the 4th order, when it is the common parabola, having its axis per- pendicular to the base ; and, in general, of the n +2 order, when it is a parabola whose or- dinate is always as the n power of the base. The asymptote is oblique to the base, when the ratio of the first flaxion of the ordinate to the fluxion of the base, approaches to an as- signable ratio, as its limit; but it is parallel to the base, or coincides with it, when this limit is not assignable. pd The areas bounded by curves and their asymptotes, though indefinitely extended, have sometimes limits to which they may ap- proach indefinitely near: and this happens in hyperbolas of all kinds, except the first or Apollonian, and in the logarithmic curve; as was observed above. But in the common hyperbola, and many other curves, the asymp- totical area has no such limit, but is infinitely great. Solids, too, generated by hyperbolic areas, revolving about their asymptotes, have sometimes their limits; and sometimes they may be produced till they exceed any given solid. Also the surface of such solid, when supposed to be infinitely produced, is either finite or infinite, according as the area of the generating figure is finite or infinite. ll The way of discovering whether any pro posed curves have asymptotes, and the manner of drawing them when they are inclined t _ AS Y 2 axis, may be easily derived from the me- nl of tangents, as in the following ex- iple ; M oe ee bat the curve be A DE, with the equation ae n 6 Xx n =2"(a+-x) the subtangent of which fend tobe TR et) eet er ma+(m+n)x ie intercepted line AT = (m +n) (ax + x*) Ma-+ (m+n) x NX we ma+(m+n)x un that the tangent TD will become an ymptote when, touching the curve at an finite distance, that is, when the absciss B= 2 becomes infinite, the intercepted line T (then = AM) shall remain finite. But, tting 2 infinite in the expression of AT, the st term ma of the denominator, is infinitely Jss than the other, and therefore vanishes. NOx (m+n) x 2%, that is Now it is [Bence, in this case, it will be ips which is a finite quantity: so that ie curve has an asymptote, which will pass 7 Ps Cp te rough the point M, making AM in ae" low, to draw it, let A H be raised perpendi- ilar to AB, and let the asymptote be,,. for ¢ample, MHP. This being supposed, if we ke x infinite, it will be«:y:: MA:AH; id on the supposition of 2 being infinite, the (uation of the curve above given, a being fen as nothing in respect of a, will be trans- a ym + 72 TR + 7 : 7 b ‘veting the root, and for convenience, making ° . t (tut, it will be yf a—=x/b; and, king the fluxions y x/ a= xr/b; so that ge Yi/a:/b Whence MA: AH:: fh. t na 1 a@:4/b, and because MA = —, we shall t imed into this, Or, ex- a —— t ' Wwe": AH ere? Wy b,.orsA Hi = fb / o If, therefore, we take AM = , and . lise the perpendicular AH = =. Ah oy the } a idefinite right line MHP, drawn through the »ints M and H, will be the asymptote of the irve ADE. If m and x be each = 1, the curve becomes ‘¢ Apollonian hyperbola, whose equation is ? = (a+) 2; then will t=2, and there- | | ! : ATM fore AM =a, and AH=4ax VS bap a /ab. That is, AM is half the transverse axis, and A H half the conjugate: results cor- responding with what are shown in treatises of conic sections. Again, suppose A D E in the same figure to be a curve whose equation is y?-——23 =azy, making AB =a, BD=y. By taking the fluxions we shall have 3y*4y—82*x=axy 3yi—axy, : an eft + ayx; as therefore y Taaean Be ay” and 3 ee 8 Ng len ad 3Yy 3x 2axy Or, y du” +ay instead Of 3 y3 —3 23, putting its value 3a2zy from the equation of the curve, it will be why (pOeY ees Sa*+ay suit the case of an asymptote, in which AT becomes A M, and the term ay is indefinitely small with regard to 32”; so that we shall ha — ey ve AM Setany But because, in the proposed equation, the indeterminates cannot be separated, nor con- sequently the value of AM determined, if we put AM= — = s (an expedient that may be x . Then, making zx infinite, to adopted in other such cases), we shall have yo ale which value, substituted for y in the a ‘ PY proposed equation, gives — —23 =3 527, 27 six as the last term will be as nothing with regard to 27 sia a or —x=—3s. But since z is infinite, the others, so that it will be —x=0; whence x = $a. Taking, therefore, AM — 4a, the asymp- tote must be drawn through the point M. Also, since it must be MA:AH?: 2: 4; the proposed equation will be reduced to #3 =y’, or xy, when z is infinite, and therefore «= y. Consequently, making MA — AH, the right line drawn through M and H, will be the asymptote sought. for more on this subject, see Agnesi’s In- stitutions, b. ii.; Maclaurin’s Flaxions, b. i. ch. 10; and Cramer’s Introduc. Analyse des Lignes courbes, art. 147. ATLANTIDES, a name sometimes given to the Pleiades, or seven stars. ATMOSPHERE (from eos, vapour, and oQasex, a sphere), that gaseous or aeriform ‘fluid which every where invests the surface of the terraqueous globe; and partakes of all its motions, both annual and diurnal. We have already considered the mecha- nical properties of this fluid, under the article Air; and it therefore only remains in this place to treat of it as forming one body, viz. its figure, altitude, pressure, &c. Figure of the Atmosphere—As the atme- ATM: sphere envelopes all parts of the surface of the earth, if both the one and the other were perfectly at rest, and were not endowed with a diurnal motion round their axis, then the atmosphere would be exactly spherical, ac- cording to the universal laws of gravity; for the parts of the surface of a fluid in a state of rest, must be equally remote from its centre. But the earth and the ambient atmosphere are invested with a diurnal motion, which car- ries them round their common axis of rota- tion ; and the different parts of both having a centrifugal force, the tendency of which is more considerable, and that of the centripetal force less, as the parts are more remote from the axis; and consequently the figure of the atmosphere must become that of an oblate spheroid, because the parts that correspond to the equator have a greater centrifugal force than the parts which correspond to the poles. Besides, the figure of the atmosphere must represent such a spheroid, in consequence of the sun striking the equatorial regions more directly than those about the poles: whence it follows, that the mass of air, or part of the atmosphere, about the polar regions, being less heated, cannot expand so much, nor reach so high; nevertheless, as the same force which contributes to elevate air, diminishes the pres- sure on the surface of the earth, higher co- lumns of it at or near the equator, all other circumstances being the same, will not be heavier than those of the lower belonging to the poles ; but, on the contrary, without some compensation they would be lighter, in con- sequence of the diminished gravity of the up- per strata. Mr. Kirwan (Irish Trans. for 1788, p. 61), stating the height of the mercury in the ba- rometer, on the level of the sea, indicating the natural state of the atmosphere to be 30 inches under the equator and under the poles, observes, that in order to produce this state, the weight of the atmosphere must be every where equal at the surface of the sea; and as the weight of the atmosphere proceeds from its density and altitude, this equality of weight requires that the atmosphere should be lowest where its density is greatest, and consequently highest where it is least. These extremes of density take place in the equatorial and polar regions. Under the equator, the centrifugal force, the distance from the centre of the earth, and the heat, are all at their maximum; and, on the contrary, in the polar regions they are all in their minimum state; and hence it follows, that the atmosphere must be highest under the equator and lowest under the poles, with their several intermediate gradations. Weight or Pressure of the Atmosphere—We have already had occasion to touch on this subject, in speaking of the specific gravity of air, and it will therefore be unnecessary, in the present place, to repeat again the history of the discovery, which we have seen is due to Torricelli. He found that the pressure of the atmosphere sustains a column of quick- ATM silver, of an ¢qual base and 30 inches heigh and as a cubical inch of quicksilver is fow to weigh near half a pound avoirdupois therefore the whole 30 inches, or the weig of the atmospbere on every square inch, nealy equal to 15ib. Again, it has been fom that the pressure of the atmosphere balane in the case of pumps, &c. a column of wat of about 344 feet high; and, the cubical fo. of water weighing just 1000 ounces, or 622) 344 times 624, or 215816. will be the weig of the column of water, or of the atmosphe on a base of a square foot; and consequent the 144th part of this, or 15Ib. is the weig of the atmosphere on a square inch; the sar as before. Hence Mr. Cotes computed, th the pressure of this ambient fluid on the whe surface of the earth, is equivalent to that ol globe of lead of 60 miles in diameter. Ai hence also it appears, that the pressure up the human body must be very considerabl for, admitting the surface of a man’s body to. about 15 square feet, and the pressure abo 15Ib. on a square inch, he must sust 32,4001b. or nearly 14% tons weight for his ¢ dinary load. And it might be easily show that the difference in the weight of air st tained by our bodies in different states of t, atmosphere, is often near a ton and a half, | Hence, it is so far from being a wond that we sometimes suffer in our health by; change of weather, that it is the greate wonder we do not suffer oftener, and mo) by such changes. For, when we consider th we are sometimes pressed upon by near a t and a half weight more than at another, a that the variation of the additional pressu of many pounds is often very sudden, it surprising that every such change does affect our constitutions much more consid ably. But the fact is, that.our bodies alwa) contain some elastic fluid, the spring of whi is just sufficient to counterbalance the weig of the atmosphere, / Height and Density of the Atmosphere.—Tl| is also a subject that has already been con! dered under the article ALTITUDE, at least) far as it is connected with the measureme) of altitudes by the barometer. We have se. under that article, that the densities of the)’ decrease in geometrical progression, as 1) altitudes increase in arithmetical progressio} and therefore, if no other cause existed,’ would follow that the atmosphere was of definite height. But this cannot be, in ec sequence of the other planetary bodies: o° atmosphere, for instance, cannot extend I) yond the common centre of attraction of t) earth and moon; for if in the first instance? conceive it to surpass this limit, it is obvio) that as the earth revolves on its axis, a thereby turns all its parts successively towal the moon, this body, in consequence of superior attraction beyond that point, wotl draw that part of our atmosphere towards If own centre, and either leave a vacuum | tween the terrestrial and lunar atmospher' ATM rhe limits of both would be the common ere of attraction of the two bodies. An- ter cause, viz. the centrifugal force, would { operate against an indefinitely-extended tosphere ; for as this fluid partakes of the inal motion of the earth, it is obvious, that eond that point where the centrifugal force qual to the force of gravity, the fluid would thrown off by the rotatory motion of the cy, and the limits of the atmosphere ter- ‘ated in that point. the air was every where of the same uni- yi density as at the earth’s surface, where sspecific gravity to that of water is about s$ to 2500; or where a cubic foot weighs Lyunces, it would follow, that its altitude 1d be about 53 miles: for we have seen that 1 whole atmospheric pressure is equal to but 33 or 84 feet of water; and the density of )' latter fluid being about 8334 times greater v1 that of air, we should have 8333 x 33 = 70 feet, for the height of a uniform co- vn; which is a little more than 5} miles. \ have seen, however, that this is not the a, but that the density decreases as the ltudes increase, the former in geometrical, ' the latter in arithmetical progression. ¥ have also, under the article ALTITUDE, hwn that the general formula for ascertain- valtitudes above the earth’s surface, at the 2 perature of 31°, is A = 10000 x log. oo 'eing the altitude, and m and M the heights the barometer, the former at the lower place 1 the latter at the top of the eminence, ch are also as the densities of the air at ise places, and therefore conversely to find density of the air corresponding to any ¢ticular altitudes, we may change the for- nainto A = 10000 log.m— 10000 log. M ; | A + 10000 log. M 10000 ych formula is deduced the following table, y.ch exhibits the comparative density of the iat the several corresponding heights, viz. vance log. m— . From xeight in Miles. No. of Times rarer. ot!) wea cease ght Tae 2 foe Bh ie oo ies on Mei avail Beason ls hppigaieded oa et ta 64 ea) ee . 256 ee ae Ec 1024 Bs ccc ice 4096 Se we 16384 | TR Oa GS so fee a 5 bessvevee, 262144 1 Up eee SOL Pas 1048576 \ by pursuing the calculation in this table, taight easily be shown, that a cubic inch of § air we breathe would be so much rarefied che height of 500 miles, that it would fill a } ere equal in diameter to the orbit of Saturn. ‘With regard to the extent of the atmo- sj} ere, we have already observed, that from ') principles upon which our calculation is (nded it is indefinite; but we haye likewise ATM seen, that it must of necessity have a certain limit, and this limit is generally estimated at much less than what was stated in the pre- ceding part of this article. For one of the principal effects of the atmosphere being the refraction of light, the particles of which are the smallest of any we know of in nature, it is reasonable to fix the boundary of the atmo- sphere where it begins to have the effect of bending the rays of light. Now Kepler, and after him La Hire, computed the height of the sensible atmosphere from the duration of the twilight, and from the magnitude of the terrestrial shadow in lunar eclipses, and fovnd that it was sufficiently dense at a height of between 40 and 50 miles, to reflect and inter- cept the light of the sun. So far, therefore, we may be certain that the atmosphere ex- tends; and at that altitude we may collect, from what has been already said, that the air is more than 10,000 times rarer than at the earth’s surface ; but how much farther it may be extended, is totally unknown. Cotes’s Hydrost. Lect. p. 128. See REFRACTION and TWILIGHT. Refractive and Reflective Powers of the At- mosphere.—That the atmosphere has a refrac- tive power, which is the cause of various phe- nomena, is unquestionable. This power is ascertained by the production of twilight, and by many other facts and experiments. It has also a reflective power, and this power is the cause of objects being so uniformly enlightened on all sides, and the want of it would render our situation here extremely unpleasant ; the shadows of objects would be so dark, and their enlightened sides so very bright, that probably we should only be able to see those parts of them which were absolutely exposed to the sun’s rays, if indeed the extreme light in this case did not even render them too powerful for the delicacy of the optic nerve. It should be observed, however, that though the atmo- sphere- greatly modifies the illumination of objects, yet it also obstructs the passage of a great deal of the solar light, as is obvious from the great difference that we experience between the light and heat of the sun in a morning or evening, and when it is shining in its meridian splendour, which difference is occasioned, in a great measure, by the diffe- rent extent of the atmosphere which it tra- verses in the two cases. See REFLECTION; and also Keill’s Astronomy, lect. 20. Temperature of the Atmosphere——The tem- perature of the atmosphere diminishes, as the distance from the earth increases, though ap- parently in a less ratio. M. de Saussure found, that by ascending from Geneva to Chamouni, a height of 347 toises, Reaumer’s thermometer fell 4° 2’; and that, on ascending from thence to the top of Mont Blanc, 1941 toises, it fell 20° 7’: this gives 221 feet English for a dimi- nution of 1° of Fahrenheit, in the first case, and 268 in the second. Nevertheless, from the accuracy which the rule for barometical measurement possesses, it may be inferred, that the decrease of heat for the greatest A TM heights which we can reach, is not far from uniform ; but that the rate for any particular case must be determined by observation, though the average in our climate may be stated at 1° for 270 feet of perpendicular ascent. On this subject, see La Grange, Mem. de Berlin, 1772, p. 206, &c. He thinks that the hypothesis of a uniform decrease of heat is the most conformable to appearances. Euler, in a volume of the same Memoirs, for 1754, p. 140, considers a harmonical pro- gression as the most probable. If the sole cause of the diminution of temperature were distance from the earth, and if it were ad- mitted that there is no current of air perpen- dicularly upwards, as there certainly is not, the diminution of temperature would follow the inverse ratio of the distance from the centre of the earth. Transactions of the Royal So- ciety of Edinburgh, vol. vi. p. 865. Professor Leslie, in the Notes on his Ele- ments of Geometry, p. 495, edit. 2d, has given a formula for determining the temperature of any stratum of air when the height of the mer- cury in the barometer is given. The column of mercury at the lower of two stations being 6, and at the upper 8, the diminution of heat, in degrees of the centigrade thermometer, is 2.5 ( FP uF , which seems to agree well with observation. The mean temperature of the atmosphere in any parallel of latitude remains nearly con- stant, but it decreases from the equator to either pole; and if t be made to represent the mean temperature of any parallel of which the latitude is L, M the mean temperature in the latitude of 45°, and M + E the mean tem- perature at the equator; then is ¢—=M+E.cos.2L; whence the mean temperature in any latitude is readily, ascertained. The mean tempera- ture in latitude 45° is 58° = M, at the equator it is 85°, whence 85° — 58° = 27° — E; therefore ¢ = 58° +- 27° x cos. 2 L, which, when 2 L > 90, the cosine being ne- gative, is less than 58°. See Mayer, “ Opera inedita,” vol.i. p.4; also Kirwan, “ Estimate of the Temperature of different Latitudes,” p. 18. But if the place is at any height above the surface of the sea, then the formula be-~ H Pant: E..cos. 2 L; M and E being still the same as above, and H the height of the place in English feet. On ascending into the atmosphere, at a cer- tain height in every latitude a point is found where it always freezes, or where it freezes more than it thaws, so that the mean tempe- rature does not exceed 32°, and the curve joining or passing through all those points is called the curve of perpetual congelation. The equation to which is found by making 32=M—4 4B. cos. 21, and this line, at the equator, is eleyated 15577 feet above the level of the sea. comes ¢ = M— ATM Whence ......... % = Hi E.cos.2 L + M—2 15577 un 27 + 58 — 327 be and conseq. ... H = 7642 + 7933. cos. 2, Which formula seems to agree very nea} with actual observation. ‘ See Playfair’s “‘ Outlines of Natural Phi p- 285; see also a different formula for e pressing the line of perpetual congelatic Leslie, “ Elements of Geometry,” 2d editi¢ p. 495. ATMOSPHERE, in Electricity, denotes th medium which is conceived to be diffus over the surface of electrified bodies, and some distance around them; by which oth bodies immerged in it become endowed wi an electricity, contrary to that of the body which the atmosphere belongs. On this su ject see Beccaria’s “ Artificial Electricity p. 179, Eng. edit.; Dr. Priestley’s “ Histo of Electricity,” vol. ii. sect. 5; and Cavalle Electricity, p. 241. a ATMOSPHERE of the Planets.—Since ¢ planets and their satellites are now universa. allowed to be bodies of a nature similar tot earth we inhabit, there are few, if any, p sophers of the present day, who attempt: deny that the planets are surrounded wi atmospheres analagous, in most respects, . that whose properties have been explained, the preceding articles. M. de la Place, in} Systeme du Monde, enters into a consideral detail respecting the atmospheres of the pl nets. “In all the changes to which the ¢ mosphere is subject,” says he, vol. ii. p. “the sum of the products of the particles the revolving body, and its atmosphere, m tiplied respectively by the areas they deseril round the common centre of gravity, the rat being projected on the plane of the equate remain the same in equal times. Supposin therefore, that by any cause whatever, the ¢ mosphere should be contracted, or that pé thereof should become condensed on the st face of the body, the rotatory motion of # body and its atmosphere would be accelerat for, the radii vectores of the areas described | the particles of the original atmosphere b coming smaller, the sum of the products all the particles, by their corresponding are cannot remain the same unless the veloei be augmented. The atmosphere is flatte towards the poles, and protuberant at # equator. But this oblateness has its limit and in the case where it is greatest, the rat of the polar and equatorial diameter is as to 3. The atmosphere cannot extend its! at the equator to a greater distance than the place where the centrifugal force is é actly equal to the force of gravity. Witht gard to the sun, this point is remote from} centre to a distance measuring the radius the orbit of a planet which would make revolution in the same period as that lumina employs in its rotation. The solar atmosphe cannot, therefore, extend to the orbit of M 4 A TT 1; and consequently it cannot produce the veal light, which appears to extend even .e orbit of the earth.” TMOSPHERIC Tides, are certain peri- jal changes in the atmosphere, similar, in » respects, to those of the ocean, and pro- hd in a great measure by the same causes, ithis subject see La Place’s “ Exposition iysteme du Monde,” liv. iv. ch. 12. JemosPHERic Stones. See ASROLITHS. (unt of solidity, hardness, and impenetra- *, Which preclude all division, and leave acancy for the admission of any foreign to separate or disunite its parts. VOMICAL Philosophy, is the doctrine of fas; a system which accounts for the in and formation of things from the hypo- fs, that atoms are endowed with weight pnotion. TRACTION (formed of ad, to, and tracto, j w), in Physics, a general term used to de- tthe cause, power, or principle, by which xdies mutually tend towards each other, ohere, till separated by some other power. laws, phenomena, &c. of attraction, form echief subject of Newtonian philosophy, « being found to obtain in alimost all the lerful operations of nature. he principle of attraction, in the Newto- sense of it, was first surmised by Coper- . “As for gravity,’ says he, “I con- (it as nothing more than a certain natural tence (appetentia) that the Creator has essed upon all the parts of matter, in order eir uniting or coalescing into a globular i, for their better preservation; and it is able that the same power is also inherent e sun and moon, and planets, that those (es may constantly retain that round form hich we see them.” (De Revol. Orb. @st. lib. i. cap. 9.) Kepler calls gravity a real and mutual affection between simi- odies, in order to their union. (Ast. Nov. trod.) And he pronounced more posi- 7, that no bodies whatever were abso- y light, but only relatively so; and con- onily, that all matter was subjected to the ry and law of gravitation. e first in this country who adopt@! the n of attraction, was Dr. Gilbert, in his De Magnete; and the next was the ce- ted Lord Bacon, in his Noy. Organ, lib. ii. r. 36, 45, 48. Sylv. cent. i. exp. 33; also 's treatise De Motu, particularly under particles of the 9th and the 13th sorts of Don. In France it was received by Fermat Roberval; and in Italy, by Galileo and ili. But till Newton appeared, this prin- / was very imperfectly defined and ap- 4. Ifore Newton, no one had entertained ' correct and clear notions of the doctrine niversal attraction as Dr. Hooke, who in ‘Attempt to prove the Motion of the 1,” 1674, observes, that the hypothesis ¢ which he explains the system of the ‘i is founded upon the three following a = ——— TOM, a particle of matter indivisible, on_ A/T T principles: 1. That all the celestial bodies have not only an attraction or gravitation to- wards their proper centres, but that they mu- tually attract each other within their sphere of activity. 2. That all bodies which have a simple and direct motion, continue to move in a right line, if some force, which operates without ceasing, does not constrain them to describe a circle, an ellipse, or some other more complicated curve. 3. That attraction is so much the more powerful, as the attracting bodies are nearer to each other. But Hooke was not able to solve the general problem re- lative to the law of attraction, which would occasion a body to describe an ellipse round another quiescent body placed in one of its foci; this admirable discovery, which requires the aid of the higher geometry, and does the highest honour to the human mind, being re- served for the genius of Newton. Attraction may be considered as it regards celestial bodies, terrestrial bodies, and the miuuter particles of bodies. ‘The first case is usually denoted by the word attraction, or universal gravitation, the second by gravita- tion, and the third by the words affinity, che- mical attraction, or molecular attraction. Many philosophers are now of opinion, that it is the same force contemplated under diffe- rent aspects, yet constantly subject to the same law. At a finite distance, all the bodies in nature attract one another in the direct ratio of the masses, and the inverse ratio of the square of the distance, which may be thus demonstrated: According to a law of Kepler, deduced from observation, the radii vectores of planets and comets describe about the sun areas pro- portional to the times; but this law can only have place so long as the force which inces- santly deflects each of these bodies from the right line is constantly directed towards a fixed point, which is the origin of the radii vectores. ‘The tendency, therefore, of the pla- nets and comets towards the sun, foliows ne- cessarily, from the proportionality of the areas described by the radii vectores to the times of description: this tendency is reciprocal. It is, in fact, a general law of nature, that action and reaction are equal and contrary: whence it results, that the planets and comets react upon the sun, and communicate to it a ten- dency towards each of them. oe The satellites of Uranus tend towards Uranus, and. Uranus towards his satellites: the satellites of Saturn tend towards Saturn, and Saturn towards them. The case is the same with regard to Jupiter and his satellites. The earth and moon tend likewise recipro- cally the one towards the other. 'The propor- tionality of the areas described by the satel- lites to the times of description, concur with the equality of action and reaction, to render these assertions unequivocal. All the satellites have a tendency towards the sun; for they are all animated by a regular motion about their respective planets, as if they had been immoyeable: whence it results ATT that the satellites are impelled with a motion common also to their planets; that is to say, that the same force by which the planets tend incessantly towards the sun, acts also upon the satellites, and that they are carried towards the sun with the same velocity as the plancts. And since the satellites tend towards the sun, it follows that the sun tends towards them, because of the equality of action and re- action. Observations have convinced us, that Sa- turn deviates a little from his path when he is near Jupiter, the largest of the plancts; whence it follows, that Saturn and Jupiter tend reciprocally the one towards the other. Saturn, as was observed by Flamstead, dis- turb the motions of Jupiter’s satellites, and draws them a little towards him; which proves that these satellites tend towards Saturn, and Saturn towards them. It is therefore true, that all the heavenly bodies tend reciprocally towards one another: but this tendency, or rather the attractive force which occasions it, appertains not solely to their aggregate mass; all their moleculz partake of it, or contribute to it. If the sun acted exclusively upon the centre of the earth, without attracting each of its particles, the undulations of the ocean would be incompa- rably greater, and very different from those which are daily presented to our view. The tendency of the earth towards the sun is, therefore, the result of the sum of the attrac- tions exerted upon all the molecule, which consequently attract the sun in the ratio of their respective masses: besides, every body upon the earth is attracted towards its centre proportionally to its mass. It reacts therefore upon it, the {attraction following the same ratio. Ifit were otherwise; if all the parts of the earth did not exert upon one another a reciprocal attraction, the centre of gravity of the earth would move by a constantly-acce- lerating motion, till at length it would be lost beyond the limits of the system. The attraction is therefore universal, reci- procal, and proportional to the mass. It re- mains to demonstrate, that this force is in- versely as the square of the distance. Observations have shown, that the squares of the periodic times of the celestial bodies are proportional to the cubes of the mean dis- tances. Farther, it is rigorously demonstra- ble, that when bodies circulate in such manner that the squares of the periodic times are pro- portional to the cubes of the distances, the central force which actuates them is in the inverse ratio of the square of the distance: therefore, supposing the planets to move in circular orbits (from which they, in fact, differ but little), they are solicited towards the sun by a force which varies inversely as the square of the distance. This supposition is not rigo- rous. But the constant relation of the squares of the periodic times, to the cubes of the dis- tances, being independent of the eccentricity, would doubtless subsist in the case where the eecentricity vanishes, that is, if the planets ATT | moved in circular orbits. Indeed, the 4; of the position may be readily estabhij with regard to elliptical orbits: but we. the demonstration, rather than protract) article to too great a length. If the planets revolve about the su virtue of a central force, which is recipro| as the square of the distance, it is natur) infer that the moon is retained in her orb| a central force directed towards the earth) which only differs from the gravity of te trial bodies, in the ratio of the diminy that is occasioned by the augmentation o] square of the moon’s distance. Now, it; be shown that the revolution of the 1 about the earth is a phenomenon of the 4} kind, and to be accounted for in the 4 manner (that is, by considering the joint ration of the projectile and gravitating fo as the curvilinear motion of a stone, bulk any other projectile near the surface o earth. If we had engines of a sufficient: to project a body in a right line parallel t horizon, with the velocity of 24326 Paris (nearly five English miles) in a second of: that body, setting aside the resistance ¢ air, would revolve about the earth like an For, 24326 is a mean proportional bet 39231600, the diameter of the earth, and the space described in the first second of by a heavy body falling from quiescene wards the earth. And the periodical tit such a projectile would be nearly equ 1 hour, 24 minutes, 27 seconds. If this could be carried to the distance of the n and projected in the same direction as th which the moon moves, with such a vel as would carry it through 188489 Paris fe a minute, it would revolve about the ear the same orbit as is described by the n) We know from experience, that the m with which a body near the surface 0 earth tends to its centre, is such as in a se! of time makes it descend through 157, feet. Supposing this motion to decreas! versely as the square of the distance; @ distance of the moon, which is equal 1 semidiameters of the earth, it would be 60! times less than at the surface of the earth iif therefore at that distance would be suffi to make a body descend through 15,4} feet in a minute of time. This is, in fact space through which the moon, at the dis of 60 semidiameters of the earth, desi from the tangent of its orbit, towards thet tre of the earth in a minute of time. Fol space is a third proportional to the diaillé of the moon’s orbit, and the arc describ! the same time. And 235389600 (the dia! of the moon’s orbit in Paris feet) is to 1 (the arc described in a minute), as 1884 to 15;;. Thus the motion agrees in qué as well as in direction, with the legitima! ferences from the motions of projectiles‘ the earth. And these phenomena are S() fectly coincident and similar, that they! be referred to the same principles, na? a projectile force and a gravitating fore’ ATT ig inversely as the squares of the dis- ces. n establishing this law of attraction, we :’e considered the centres of bodies, though 1 gravity is proper to each of their mole- ‘a because, in spheres, or in spheroids dif- éng but little from them, the attraction of } molecule most distant from the attracted ynt, and those of the nearest molecule, mu- vlly compensate in such manner that the (ul attraction is the same as if the molecule ve united at their centre of gravity. ‘his law of spheres suffers various modifi- ‘ions, when the bodies attracted are at the face or in the interior of the spheres. A » ly situated within a spherical shell, through- 1 of the same thickness, is equally attracted yjall sides; so that it will remain at rest in midst of the attractions it experiences. ' same thing obtains within an elliptical ll, whose interior and exterior surfaces similar and similarly placed. Supposing, refore, that the planets are homogeneous the the gravity in their interior diminishes the distance from their centres ; for the ex- equilibri- mn and. this ibriam must arise from an exact distri- , n of the weight of each arm and seale of e former; for on this depends the aeccu- 2)0f its action. Fofessor Playfair, in his ‘“ Outlines of Na- Philosophy,” has the following remarks {regard to the accuracy of the balance. I< It should rest in a horizontal position i loaded with equal weights. 2. It should great sensibility; that is, the addition of wul weight in either scale should disturb »/quilibrium, and make the beam incline uly from the horizontal position. 3. It d have great stability; that is, when dis- , it should quickly return to a state of t That the first requisite may be obtain- ie beam must have equal arms, and the i> Of suspension must be higher than the 1 of gravity. Were these centres to co- it, the beam, when the weights were i#@ would rest in any position, and the lion of the smallest weight would overset ilance, and place the beam in a vertical from which it would have no ten- ¢ to return. The sensibility, in this case, uj be the greatest possible; but the other »pquisites of level and stability would be ily lost. The case would be even worse, feentre of gravity were lower than the i} of suspension, as the balance, when mete would make a revolution of no less . semi-circle. When the centre of sus- isnis higher than the centre of gravity, hiweights be equal, the beam will be hori- a and if they be unequal, it will take ique position, and will raise the centre zivity of the whole, making the momen- ajn the side of the lighter weight equal to tin the side of the heavier, so that an i)»riam will again take place. ‘he second requisite is the sensibility of dance, or the smallness of the weight, ich a given angle of inclination is pro- If a be the length of the arm of the ave, and 6 the distance between the cen- suspension and the centre of gravity, P id in either scale, and W the weight of Tam, the sensibility of the balance is as Tew)’ it is therefore greater, the greater jagth of the arm, the less the distance | | jalance, as well as from the equal length- BAL between the two centres, and the less ths weight with which the balance is loaded. “ Lastly. The stability, or the force with which the state of equilibrium is recovered, is proportional to (2P + W) 6, the denominator of the preceding fraction. ‘The diminution of 6, therefore, while it in- creases the sensibility, lessens the stability of the balance. The lengthening of a will, how- ever, increase the former of these quantities, without diminishing the latter. The above formule are of great practical utility, because by means of them, one balance may be made having exactly the same sensibility and sta- bility with another; it is only required that the ratio of the lengths of the arms should be the same with that which is compounded of the ratios of the distances of the centres of gravity and suspension, and of the weights of the beams. “A balance made by Ramsden for the Royal Society, is capable of weighing ten pounds, and turns with half a grain, or a millionth part of the weight. Young’s Lec- tures, vol. i. p. 125. The descriptions of ba- lances, given by mechanical writers, are ge- nerally defective, as they do not give the values of a, b, and W, the quantities on which the merit of the balance depends, and by the knowledge of which, similar instruments might be constructed. In some of the nicest balances, 6 is made variable by means of a small moveable weight.” See La Hire, Traité de Mechanique, prop. 33; Muschenbroek, § 383, &c.; Euler’s Com- ment. Petrop. tom. x. p. 1; Magellan, Jour- nal de Phys. tom. xvii. (1781) p. 48. Compound BALaNncr, is a combination of several balances employed in weighing very heavy bodies, as anchors, great guns, &c. Danish BALance, is a sort of steel-yard, in very general use in various parts of the con- tinent of Europe; the principle of its action being as follows: ' It consists of a bar of iron, or batten of wood, having a heavy lump or knob at one end, and a scale or hook at the other. The goods to be weighed are placed in the scale, or suspended from the hook, and the whole is then balanced on a piece of cord, by sliding the latter about to and fro till the équilibrium obtains; and the weight of the body is then indicated by the graduated divisions of the instrument, which are thus computed : ¢ . WwW Let AB represent this balance, A the fixed weight, and W the body whose weight is te be ascertained, and C the moveable fulcrum, Make AB=d, AC =z; put also the weight A =a, and that at W —w; then, in case of equilibrium, xa:d—2x:: aiw by COMP. +00 ds riad+wia herefé ad. therefore...... She H2 BAL and hence by substituting w = 1, 2, 3, &e. Ib. the distance x or AC, for each of those weights, will be determined. Suppose, for example, the whole length — 30 inches, and the weight A —21b.; then, substituting these numbers for a and d, and the numbers 1, 2, 3, ke. for w, we shall have the following series: x == 20, 15, 12, 10, &c. oO 1, 2S, 4, me. We give this merely as the principle of computation; in real practice, of course, the weight of the bar, and the place of its centre of gravity, must enter into the calculation, which will render it alittle more complicated. See Gregory’s Mechanics, vol. il. Hydrostatic BALANCE, is an instrument con- trived for determining accurately the specific gravity of bodies, both solid and fluid; of which there are various constructions. See Hyprostatic Balance. Roman Baance, or Steel-yard. See STEEL- yard. BALANCE of a Clock, or Watch, is that part which by its motion regulates and determines the beat. : BaLAncr, in Astronomy, is the same as Lisra, which see. BALL, in a popular sense, is any spherical body, whether natural or artificial. Fire BAuts, in Meteorology, are luminous bodies generally appearing at a great height above the carth, and sometimes amazingly vivid and brilliant. See METEOR. Baw and Socket, is a joint frequently at- tached to surveying and astronomical instru- ments, whereby they may be moved horizon- tally, vertically, and obliquely. BALLISTA, (from Baaaw, to shoot) an an- cient military engine, which was used for throwing stones, darts, and javelins. BALLIsTA was formerly also the name given to the cross-staff, BALLISTIC Pendulum, an ingenious ma- chine, invented by Robins, for ascertaining the velocity of military projectiles, and con- sequently the force of fired gunpowder. It consists of a large block of s wood MN, suspended verti- tically by a strong horizontal iron axis at S, to which it is connected by a firm iron stem. Now to determine the velocity with which a ball is projected, the pendulum is so situated that the ball im- 7 pinges directly against it, and causes it to vibrate through a certain arc, which being . accurately observed, the velocity of projec- tion is computed as follows: Let the weight of the pendulum = w the weight of the ball......... = p And let O, G, and R, be the centres of oscil- lation, gravity, and gyration, and P the point of impact; also let g be a weight, which col- jected in the point P, a given force applied to P will generate the same angular velocity as » projectiles. ‘, Ne BAL if it were applied against the pendulum ifs in the point P. é ’ ; G This equivalent weight g = pen S 1X S$ b+ _wxSR’ : hy Re See GYRATION. bh Again, let v represent the velocity com nicated to the point P, and a the requi velocity of the ball. Then the block of wi being considered as non-elastic, the laws collision gives the following proportion ; # g + p:p::x:, whence ‘ xmvxX laa Lf See COLLISION. P And here, since p and q are known, it remains to assign the velocity v, which been communicated to the point P, ff having given the angle through which_ penduluin is impelled by the stroke. 4 For this purpose, make SO =s, SP& then the velocity acquired by the centt oscillation in a pendulum, which descril from rest any are of a circle has arrived a lowest point, is equal to that acquired heavy body, which has descended freely fi rest by the acceleration of gravity, et | space equal to the versed sine of the described by the pendulum. In like mai if any given velocity be communicated to ceutre of oscillation of a pendulum when escent, it will rise through an are whose sed sine is equal to the space through w a body falls freely from rest, in order t quire that velocity. Let, then, > represen versed sine of the angle described by the dulum te radius — 1, then will the ¢ of oscillation O, describe an arc durin motion, the versed sine of which = s; consequently, if we represent 167, feet! the central velocity of O = /4gbs fee second, (see ACCELERATION) and the vé z of the point P = / aS =e u This, therefore, will give us the velo¢i the ball the instant it struck the pendul that is, the required velocity 4 PSP daa ae Yeap asthe Bs: / sla) Pp Pp s | per second, ba | In this investigation, in which we hay lowed the method employed by Atwot his treatise “ On Rectilinear and Rotf Motion,” p. 215, the change in the cen} oscillation, which is caused by the ball ing in the point P, is omitted. But the aj ance for that change, and a few other quantities, are all considered by Dr. E in his Tracts, vol. i.; where also are giv} results of a great number of very act means of which the most useful and imp conclusions have been deduced in mq Dr. Hutton has also, since that pu plic | made many other experiments of the kind, by discharging cannon-balls atv BAR nees from the pendulum; from which is iced a complete series of the resistances ie air to balls passing through it, with all ees of velocities from 0 to 2000 feet per nd. : 1 this subject, see also Robins’s “ New wry of Gunnery,” and Mathematical Tracts. r, Antoni, Le Roy, Darcy, &c. have writ- m this subject. ALLISTICS, is sometimes used for pro- es, under which name this theory has treated of by Mersennus. ontucla distinguishes between projectiles ballistics, by defining the former to re- to bodies projected into free space, and atter into a resisting medium. \LLOON, in its general sense, signifies round hollow body; but it now more nonly implies an aerostatic machine. See ISTATION, ,RLOWE, an eminent English mathe- sian, who flourished about the beginning .e seventeenth century. He was par- rly distinguished for his accurate obser- ns relating to the magnet; on which ct he published an ingenious work, ed, “ Magnetical Advertisement, or divers aent observations and improved experi- $ concerning the nature and properties + loadstone.”’ London, 4to. 1616. “A brief Discovery of the idle Animad- mus of Mark Ridley, M.D. upon a trea- entitled Magnetical Advertisements.” on, 4to. 1618. side which, he was author of a work on ‘ation ; viz. “The Navigator’s Supply, containing ' things of principal importance belong- > Navigation, kc.” London, 4to. 1597. which latter work he gave a demonstra- f Wright’s, or Mercator’s division of the ian line; observing, that “This manner rde has been publiquely extant in print . thirtie yeares at least; but a cloude ;were) and thicke mist of ignorance doth Jit hitherto concealed. And so much nore, because some who were reckoned en of good knowledge, having by glaunc- xeeches (but never by any one reason of »nt) gone about what they could to dis- it.” fplcereR. (from Bapoo, weight, and , measure) an instrument for measuring eight of the atmosphere and its varia- in order principally to determine the ses of the weather, the heights of moun- and other eminences, &c. The baro- ‘is founded upon the Torricellian ex- 1ent, so called from its author Torricelli; in consequence of the previous sugges- f Galileo, with regard to the ascent of _in pumps, proceeded in 1643 to make iments with a tube filled with mercury, icturing that as this fluid was about thir- limes heavier than water, it would only at about one-thirteenth of the height to t the latter rises in pumps, or to about BAR thirty inches. He therefore filled a glass tube, about three feet long, with quicksilver ; and having sealed it hermetically at one end, he immersed the other in an open vessel of the same fluid. He found the mercury de~ scend in the tube, and finally settled at the height of about twenty-nine and a half Ro- man inches, and this, whether the tube was vertical or inclined, according to the known laws of hydrostatical pressure. This famous experiment was repeated and diversified in various ways, with tubes filled with other fluids, and the result being the same in all, except so far as relates to their specific gra- vilies, the weight and pressure of the air were established beyond the possibility of doubt, to any one who could divest himself of former prejudices. This, however, cannot always be done, and in consequence various objections were started, and new hypotheses framed in order to account for the suspension of the mercury, till the question was at length finally settled by the experiment of the Puy de Domme, which we have already mentioned under the article ALTITUDE. The real cause of the suspension of the mercury in the tube, and of water in pumps, was now generally admitted to be the atmos- pheric pressure, and repeated observations were made, connected with this subject; from which it was discovered that the column of mercury varied considerably in its height, at different times, and this variation was soon observed, to be followed by changes of the weather. This led to farther and more ac- curate observations, and various alterations and improvements were suggested in the form and construction of the barometer, or weather- glass, as it is sometimes called in consequence of its use in determining the changes in the weather. ‘The constructions of this instru- ment are now exceedingly varied and nume- rous, much beyond what our limits will admit of enumerating ; we must, therefore, content ourselves with describing the most popular and general forms, and refer the reader who wishes for farther information to the article BAROME- TER, in Rees’s Cyclopedia, where he will find a detailed account of nearly all the forms and uses to which this instrument has been applied. Common BAROMETER.—This is represented (at fig. 6, Plate If, Pneumatics) such as it was invented by Torricelli. ABis a glass tube open at one end, and hermetically sealed at the other, A, having its diameter about one-third or one-fourth of an inch, and its length thirty- three or thirty-four inches: this is filled with purified mercury so justly as not to have any air over it, nor any bubbles adhering to the sides of the tube, which is best done by means of a small paper or glass funnel, with a capillary tube. The orifice of the. tube, filled after this manner, so as to overflow, is then closely press- ed by the finger, so as to exclude any air be- tween it and the mercury; this done invert the tube, and immerse the finger and, end, thus stopped, into a bason of like purified quick- silver, and in this position withdraw the finger, BAR and the mercury will descend in the tube to some place as E, between twenty-eight and thirty-one Inches above the surface of the mercury in the bason at F; these being the limits between which it always stands near the surface of the earth or sea. Instead, however, of the detached vessel CD, the modern barometer tubes are curved at the bottom, and terminate in a bulb, which ought to be as big as it can be conveniently made, in order that the variation in the alti- tude of the mercury in the tube may affect, as little as possible, the depth of that in the bulb. The barometer tubes, under either of the above forms, are now to be enclosed in a wooden case or frame, to prevent their being broken, and the vessel or bulb, though open to the air, must be secured from dust; and thus far the construction will be completed. Next measure from the surface of the mer- cury at: }’, 28 inches to G, and 31 inches to H; dividing the spaces between them into inches and tenths, which are marked on a acale placed against the side of the tube; and these tenths are again subdivided into hun- dredth parts of an inch, by means of a sliding index carrying a vernier or nonius. In the common barometers called weather-glasses, the lowest limit at G is marked stormy, and the highest point, H, is marked on one side very dry for summer, and on the other, very hard frost for winter. To the next half-inch below the highest point are annexed set fair, on the one side, and set frost on the other. At the height of 30 inches, fay is marked on one side, and frost on the other. At the height of 293 inches is marked changeable, both for summer and winter; and at 29 inches, rain on one side, and snow on the other. At 281 inches, much rain on one side, and much snow ‘on the other; the lowest division being mark- ed stormy, as we have before observed. As the common barometer is the best, and most to be depended upon in accurate ob- servations, it may be proper to add some di- rections for preparing it: they are collected chiefly from the publications of Muschen- broek, Desaguliers, and De Luc, on this sub- ject. It appears from many experiments, that the mercury stands higher in tubes of a larger, than in those of a narrower bore; and there- fore, when observations are made with diffe- ‘rent barometers, attention should be paid to the difference of their diameters; and it would be desirable to have them constructed of tubes of the same diameter. The bore of the tube should be large (not less than one-fourth of an inch), in order to prevent the effects of the attraction of cohesion; but if they are one- third of an inch diameter, they are better. If a cistern be used as a reservoir for the stag- nant mercury, it should be large in proportion to the diameter of the tube, at least ten times greater; that the addition or subtraction of the mercury, contained between the greatest and least altitudes, may not sensibly affect its depth ; for the numbers marked on the scale BAR annexed to the tube show their distance f a fixed point, and cannot truly indicat height of the column above the mercu the cistern, unless its surface coincide w this point and be immoveable. In order m effectually to preserve the lower surface the same height from the divisions on) scale aftixed to the instrument, the fat | the late Mr. George Adams first appli the barometer a floating gage; by me which the same screw that renders the ba meter portable, regulates the surface 5 mercury in the cistern, so that it is alw the place from whence the divisions 0 scale commence. ‘The tube should be p served free from dust till it is used; and this purpose it may be hermetically sealed both ends, and one end may be opened ¥ a file, when it is to be filled. If this pree tion has not been observed, the inside sho be well cleansed, by washing it with alco highly rectified, and rubbing it with ali portion of shammy leather fastened to a W The mercury should be pure; and may purged of its air by previously boiling” a glazed earthen pipkin covered close; when the tube has been uniformly heated rendered electrical by rubbing it, the hot1 cury should be poured into it in a regi current, through a glass funnel with ak capillary tube, so that the air may not hi room to pass between the parts of the qui silver. M. de Luc directs, as Mr. Orme practised many years ago in the const nc | of his improved diagonal barometers, that, mercury should be boiled in the tube, as most effectual method of purging it of it and moisture. The process is briefly t He chooses a tube of 24 lines or 3 lines” and not exceeding half a line in thic q, then dividing the equation r?¢ = 79@ by 174, we have r@—-®94 — J, so that r@-? would like- wise be a root of the same equation, but this is impossible ; for since a is necessarily prime ton, because 7 is an imaginary root, and p —q being less than n, the whole (p—q)a is prime to n, and consequently r@-®« cannot be a root of the equation x” — 1 = 0; that is, no two of the above series of roots are equal! to each other. Ae With regard to the sum and continued pro- duct of these roots, this follows also from the known theory of equations; viz. that the sum of all the roots of any equation is equal to the co-efficient of the second term, which in the present case is zero:.and that the continued product of all the roots of an equation is equal to the absolute term, which is here —1; that is, introducing the real root x= 1 birt + r™ + 7) &e. re— Da — 0); OF 60%. 7% ep te 78? Sic, pW — Ve Again, 1 x r* x 1% x 9 x, &e. p@— Da —__] - OF 303 Pg eet Sage eee | as is evident. Another very remarkable and interesting property of these equations, is that the whole series of their roots, both real and imaginary, is expressible by means of certain general _ trigonometrical formule ; viz. All the roots of the equation x” — 1 =0 are contained in the general formula ane as Jf (cos.2 282 — 1) n x = cos. —— n k being equal to zero, or any integer number whatever, and 7 representing the semicireum-~ ference. | ~ And in equations of the form a” +4+1—0, the general formula, in which the roots are contained, is e=cos; eet / (cos.2 ELMS. 1)) P nr Me 7 k and x being the same as above. In the first of these formule, if k be divisible by x, it gives the real root; as does the second, when 2k + 1 is divisible by the same number, I ly BIN. Thus, in the following equations, viz. RO. Jn 22 608.0? 1 real root ane Mt | x = cos. 120° ck / (cos.* 120°— 1) ie x = cos. 180° R= 1 e+ 10) ing i ae x3—1—0<¢ taking = Ot Se €087 60" = ./(cos.” 60° — 1) k=-0......2=— ¢o0s.0°— ( } 1 real root 2 < Bi tie eae Sted — Oy taking + ./(cos.? 72° — 1) \n=2 Seles x — cos. 144° + /(cos.? 144° — 1) Ke. Ke, This very elegant theorem we owe to the celebrated Cotes, who applied it to various problems in his “ Harmonia Mensurarum ;” and it has since received other applications in different branches of analysis. By means of this theorem, and a table of sines, it is obvious that the roots of all binomial equa- tions may be determined ; but M. Gauss, in his ** Disquisiones Arithmeticz,” discovered the solution of these equations independently of any trigonometrical lines, by which means the latter became expressible in terms of the former; and he thus found the numerical values of several cosines, which were before considered as totally beyond the reach of numerical investigation. See Cores’s Theo- rem, and Gauss’s Theorem. Imaginary, or Impossible Binomia, is a binomial expression, of which one of the branches is imaginary; thus, a+ /—6b, and—at ¥—b, are both imaginary binomials. See IMAGINARY. BInomiat Surd, is used to denote a bino- mial, of which one or both of the branches are surd quantities; thus, a+ /b, /a +b, &e. are binomial surds. Euclid enumerates six kinds of binomial lines or surds, in the tenth book of his Elements, which are ex- actly equal to the six residuals or apotomes, there treated of also, and which have been already mentioned under the article APOTOME. Euclid’s six binomial lines are, yh al OS AE 2d..;. /18.4+- 4 “1 Menges A-7 Di Sine |: Sea 4th... 4+ /3 oth... / 6+ 2... ooo Oth... o/- 6 bi/D For the rules of Involution, Evolution, Mul- tiplication, &c. of binomial surds, see the re- spective articles; see also Surp. BINOMIAL Theorem, is a general algebraical expression or formula, by which any power or root of a quantity of two terms is expanded into a series. This is also commonly called the Newtonian theorem, or Newton's binomial theorem, on account. of his being commonly considered as the inventor of it, as he un- doubtedly was, at least in the case of frac- tional and negative indices; which includes all the other particular cases of powers, divi- sions, &e, This celebrated theorem, ‘as proposed in its _ therefore destroy each other. | BIN | most simple form, that is, is as follows; viz. n (n— 2) N ——e WN nt SER HET TG Srtoted ton, ae anima. aad iy ih. 925 | LN Oe . where the law of the series is immediately ob- | vious, and by means of which any binomial | quantity is readily raised to any proposed) power without the trouble of continual in-- volution ; for example, a for integral powers, a®— 2}? +) 3 3.2 3.2.1 ¥} 3— 93.2 gh 4 22“ gh? 2 ta cA Na A La we 6 | (a+b =a3 +307) + 3ab* + 63 Again, ys (a+b) =a’? +20°h+ wb + OOS ath, Ke. or (a+b)? =a? + 7a°b + 21 a5 b* + 35atd3, &e, | which series, if is obvious, must alway termie nate when the index 7 is an integer number, It must, however, be observed, that the signs of the several terms in the series are affected by the signs of the powers of b; that is, if 6 be negative, as in the residual quantity | a@—b, then all the odd powers of b will so likewise, and consequently all those terms into which the odd powers of 6 enter, will be preceded by a negative sign; but when 3d is positive, then all the signs of the series will be positive likewise. ‘Thus, Bet: 7 (Ai te 7.6.5 ‘a Tm a7 + Sh poe 5h? + ce a th3 C. Ce ie aa ee the signs being plus and minus aliernately in. the former case. 7% If we raise (1 + 1) to any power n, we shall have the co-eflicients only ; thus, i ok n , n(n—1) , n(n—1) (n—2) Vii ak 2 Tt. ee td eae +5 & 2". 4 whence the sum of all the co-efficients of a binomial are equal to the same power of 2, as. that to which the binomial is raised. ‘ B n—1__%,™n—1)_n(n—1)(n—2) Piet) brag ea 1.2.0 iter (= '05 i that is, the sum of the positive co-efficients is equal to the sum of the negative ones, vie If also we consider the latter part of t series, it will be found that the co-efficients from either extreme are the same, increasing from each end to the centre term, which i the greatest when there are an odd number of terms; or to the two centre terms, when there are an even number of terms, which are equal to each other, but greater than any other. of the co-eflicients. old It may be farther observed, that the num ber of terms is always one more than the ime dex of the power, and is therefore even when that is odd, and odd when that is even. “ This theorem for integral powers may be expressed in words, as follows; viz. Bo The index of the first, or leading quant is the same as that of the power, and in the succeeding terms it decreases always by 1y 1‘. BIN while that of the second part increases by 1, whereby the sum of the indices is always the same in each term. As to the co-efficients, the first is always unity, and the second the same as the index of the power, and for the rest multiply the co-efficient of each preceding term by the in- dex of the leading quantity in that term, and divide the product by the number of terms to that place, and the quotient will be the co efficient of the following term. And for the signs, they will be all plus when both terms of the root are plus, and alternately plus and minus when thessecond term is ne- gative. ' | : m. Lepr ; __ If we assume the index a i which form it may relate to roots as well as powers, it then becomes more general ; viz. }' :%9 ™ m m—-n ta, m—2n Ma+b)n—an +a n 5 rm) n O*-+,&c. nm Nh 2H Ses m 6 , m(m—n) b+ att 2) ne: xplee” n.2n ‘at 1, m(m—n)(m—2n) 3 . : \t n.2n.3n FS a } - mM m iy or (a + byn = ax x 1 ie Ae eo Oe n 2n ‘n—2n m—2n Ret Sag Pacts ke. i ‘vhere A, B,C, D, &c. are the preceding terms, wneluding their signs + or 1, the terms of the eries being all plus when b is positive, and dternately plus and minus when 6 is negative, adependently however of the effect of the 0-eflicients, made up of m and n, which may ‘e any numbers whatever, positive or nega- live. _ Or this theorem may be otherwise express- d as follows, and which is indeed its most mple practical form; viz. put a= P, and — PQ, then we shall have eS P4PQe =P + ™aQ4@—™ BQ4 | 7) 2n bine" cg 4. &e _ 3a ally is the index, P the first term, Q the cond term divided by the first, and A, B, » D, &c. the several foregoing terms, with ,eir proper signs. An example or two will sufficiently illus- ate this formula. 1. Let it be proposed to convert v (a? +b”), or | *457)*, into an infinite series. Here pghomey gg! eats 1 ; =, =; that Is, m= 1, and x= 2; a” n ®? arefore, m m ea), — (a*)* —a— A 1 b* b —) — — ea Q ined: are B —2 1: 1—2. Bb? _ b* _—1.54 ——. BQ = — =x — x —- = MNES sith 4 nic 2.4. de ae —2n 1-4, — 10+, 6? _ 3.1.5%.. eo 8-3 4.2.43 En64ae TD consequently, r }2 1. d+ 3.1.36 az bh 2 ae eee ped aber Soak le A (a* + 0°) : MOR. 4.2.43 6.4.2.a° 5.383.176 +, &e, 8.6.4.2a7 where the terms may be continued at plea- sure, the law of series being obvious. When the quantity to be expanded comes under the form of a fraction, the denominator must be placed under a negative index, and brought up into the numerator, thus: 1 di 1 hh —t A Ce ins PIR 55 (a+b) +, alsd pa = (a + b) (a? + x)-*; and so on. See Nrecative Index. 2. Let it, for example, be proposed to con- vert ; into an infinite series. ‘aba (a — b) (a—b) = P=a, Q=—-, m =—2, and n=1; whence (a — b) ie and therefore on 3 wr Whence 1 1 2b 5 b+ (a—b)*~ a aie hs Ms + rm aaisar The above theorem, which is of such exten- sive utility in analysis, we owe to the inven- tive genius of Newton; on whose monument, in Westminster Abbey, it is engraved, as being one of his many brilliant discoveries. In stating this theorem to be due to New- ton, however, it must be understood only as. relating to its most general form; for in the case of integral powers the theorem had been described by Briggs, in his “'Trigonometria Britannica,” long before Newton was born, and that by the general law of the terms, in- dependently of those of the preceding powers; although it may be fairly questioned whether even Briggs knew how, in the case of an in- tegral index, to exhibit the law of formation of the co-efficients under the form 453 a? 3b? a* n(n— 1) (n—2) (n—8), ....... yoo. COC .4 LevBou3 wdpuen. Ke, ane for though his method of forming the succes- sive co-eflicients amounts to nearly the same thing, yet the advancement in analysis de- pended on the cireumstance of the law which they observe, being expressed by means of the general symbol (7), without which its ex- tension would never have been made to ne- gative and fractional indices; so that Briggs, even in the case of integral powers, does not appear to be fully entitled to the honour of having invented the binomial theorem, pro- perly so called. 12 BIN But however this may be, it is universally agreed that no one before Newton had ever thought of extracting roots by means of infi- nite series. He was the first who happily dis- covered that by considering roots as powers having fractional indices, the same binomial series would equally serve for them all, whether the mdex should be fractional or integral, or the series finite or infinite; from which ex- tension of the theorem some of the most im- portant improvements, in the higher depart- ments of analysis, have arisen, particularly in the construction of logarithms and the doctrine of series in general, which have since been carried to the greatest degree of perfection. We cannot, in this place, enter into an ex- planation of the means that led Newton to this very brilliant discovery, which he himself has described at considerable length in a let- ter to Mr. Oldenburgh, dated Oct. 24, 1676, published in the “‘Commercium Epistolicum ;” from which it appears that it was one of the earliest flights of his genius, as he says it fol- lowed soon after he “had ventured upon the study of mathematics,” and while he was perusing the works of the celebrated Dr. Wallis, and considering the series of universal roots by the interpolation of which we exhibit the area of the circle and hyperbola. The binomial theorem was left by its author without demonstration, but being of such great importance in almost every analytical operation, this part of the subject was not long neglected, and one of the first demon- strations of it was that given by James Ber- noulli, which is to be found, among several other curious things, in a small treatise of his, entitled ‘‘ Ars Conjectandi,” which has been very improperly omitted in the collection of his works published by his nephew, Nicholas Bernoulli. But this is only applied to the case of integral and affirmative powers, and is nearly the same with that which was after- wards given by Stewart, in his Commentary on Newton’s Quadrature of Curves. It is founded on the doctrines of combinations, and the properties of figurate numbers, which are there shown to involve in them the gene- ration of these co-efficients, and in the in- stances before-mentioned, where the index of the binomial is a whole positive number, it is clearly and satisfactorily explained. Since that time, many attempts have been made to demonstrate the general case, or that where the index of the binomial is either a whole number or a fraction, positive or nega- tive; but most of these demonstrations having been conducted either by the method of in- crements, the multinomial theerem of De Moivre, or by fluxions, are commonly thought to be unsatisfactory and imperfect; and it should seem not without reason, as, indepen- dently of other objections, it appears contrary to the principles of science, as well as to just reasoning, to employ in a matter purely alge- braical, notions and doctrines derived from other branches, or from an’ analysis which is in some sort transcendental. BIN | For these reasons several eminent mathe- maticians have endeavoured to investigate this formula on pure analytical principles, in a more natural and obvious way; one of the first of these attempts being that of Landen, in his ‘‘ Discourse concerning the Residual Analysis ;’” and the next, that of Alpinus, in the eighth volume of the ‘ New Petersburg Memoirs.” But the legitimacy of the former may be objected to, as depending on vanish- ing fractions, and other considerations of too difficult and abstract a nature to be regarded as sufficiently convincing ; and the latter, though very ingenious, is not less difficult and embarrassing ; at least such is the opinion of Euler, who having himself first given a de- monstration of this theorem, in which, like Maclaurin, he employed the differential cal- culus, or method of fluxions, was afterwards led to deduce it from the principles of algebra_ alone; though he does not appear to have been much more successful than either of the” former. 7 S. L’Huillier of Geneva, perceiving the de-, fects and obscurity of these methods, hag, given a new demonstration of this formula in- one of the preliminary articles of his ae work, entitled “‘ Principiorum calculi differen- tialis etintegralis,” &c. whichis purely elemen- tary; and abating from itslength, and a fatigue ing detail of particulars, which the nature 0 the subject does not seem to require, he ap- pears to have accomplished his object, at least as far as the method he adopted would allow; for it must be confessed, that neither this, nor any other investigation that had hitherto ap- peared, has been attended with the simplicity and strictness which could be desired. The reason of which, as Woodhouse properly ob- sesves, in his “ Principles of Analytical - culation,” seems to be, that most mathemati- cians appear to have sought for some high origin of this theorem, distinct from the sim- ple operations of multiplication, division, ex- traction of roots, &e.: and instead of consider- ing the nature of the operations it was known to comprehend, hoped to supercede then by deductions drawn from abstruse and fine theo- ries; whereas it is clear, that with whatever imperfections these fundamental operations are attended, are also attached to the bino- mial theorem, which in a certain sense may be said to be a method of trial and conjec- ture. For as this theorem is only meant to express, in general terms, the algebraical rules above mentioned, it cannot possess a greater degree of certainty than is possessed by the simple operations themselves. a Having said thus much with regard to the demonstrations that have been given of thi celebrated theorem, it will be expected tha we should give one as free as possible from the objections that have been stated; but to do this in all its generality and detail, would far exceed the limits of this article; we must therefore, for the sake of abridging the inves tigation, observe, that in the series arisi from the expansion of (a+ 2)”, whatever BIQ operation is indicated by m, whether involu- tion, evolution, or division, the terms of the resulting series will necessarily arise by regu- lar and whole positive powers of x; also the first term of the series will always be of the same form or power as the whole binomial, and the co-eflicient of the second term will ibe equal to the same power; that is, the first ‘two terms are always a” + ma"—! x, W hatever ‘operation is indicated by the symbol m. See Woodhouse’s ‘Principles of Analytical Calcu- lation,” p. 21; see also the article f'UNCTION. This being admitted, it follows that the series may be put under the form, (a+a)"= a" +ma"—a + pa* + gus + 74%, &e. the letters p, q, 7, Ke. represeuting the un- known co-eflicients which are to be determined. For this purpose, make a = y + z, then the above binomial becomes (at(yt2))"= (a+y)+ z)™, which are identical expres- sions, and when expanded according to the proper forms, must be equal to each other. Now, (a + (y + z))™ oa et man"(y +2) +ply+z)*+q(y+z)3+,ke. =a" + ma" (y +z) + ply? + 2yz + &) +9 (y3 + Bye +, &e.) + &e. by omitting to set down the higher powers of z, these not being necessary in the demon- stration. And this again may still be further ‘simplified and w ritten, thus: ay $a tema 'y + py + gy? + ryt +, Ke. t ma”—z + Qpyz +3qy?z +4ry3z + ,&c. Again, ((a +'y) + ete = (a+y)" +n (a +y)" lz + p2* + qz7+rz44+, ke. which (by expanding (a+y)”,and ma tyr z, or mz(a +3)", and writing p’, q', 7, &c. for the co- -eflicients of this last, analogous to those above represented by p, g, r, &e. y becomes a+ manly + py’ + ‘ay ry? +, ke. any} ma”! z + m(m— ba” yz + ply*z + dy3z 4- , ‘yz; which two formule, I. and IL., being identi- _cal, the co-efficients involving the like powers of y and z must be equal; whence we have - _ m—=m; 2p = m(m—1), or. | m(m—1) »—» Did carlbBe" (m — 1) (m—2 En | foe Ase m3 viz.” In the same way we find _— m (m—1) (m- (m— 2) (m—3) | m~4 a toe ae which is the law of the co-efficients required to be found. See Manning’s Algebra. BIPARTIENT, is a number that divides ) another into two equal parts without a re- rs ie i] ; and consequently m—2 P= a / qa ——-+ « and so on: mainder. BIPARTITION, a division into two equal rts. in Algebra, is the square squared, or fourth wer of a quantity; thus 16is the biquadrate 2, or rat way i - BIQUADRATE, or BrauapRATIC Power, BIQ BIQUADRATIC Root, is the fourth root of any proposed quantity; thus 2 is the bi- quadratic, or fourth root of 16. The biquadratic root of a number is found by extracting the square root of it, and then the square root of that root; which last result will be the root sought. Thus the fourth root of 20736 is found as follows: 4/ 20736 = 144 the square root, /144 = 12, the 4throot sought; and so on for any other number. BIQUADRATIC ‘Equation, is an equation of the fourth degree, or in which the unknown quantity rises to the fourth power; thus, vt +ax3 + bx? +cex+d=0 is a biquadratic equation, im which a, d, e, and d, may be any numbers whatever, positive or negative, or any of them equal to zero or 0. A biquadratic equation is the highest order of equation that admits of a general solution ; all beyond this being resolvable only in par- ticular cases. For the solution of equations of the fourth degree, different solutions have been proposed by several authors. The first was by Lewis lerrari, a pupil of Cardan, and published by the latter author in 1540. The second was given by Descartes, in the third book of his Geometry. And others have since been proposed by Waring, Euler, Simp- son, and others. All these, however, though differing in form, are in principle essentially the same, as has been shown by Mr. Lea, in an ingenious tract lately published under the title of ‘Solution of Equations of the higher Orders.” 1. Ferrari’s Method (improperly called the Rule of Bombelli), as generalized by Simpson. Let there be proposed the following gene- ral equation, ef the fourth degree ; viz. vt + ax3 + ba* +ex + d=0; and let us sip Ase this equation to be the same as (x* + Fax + p)—(gx + 1° =0, where p,q, and r, are unknown, the values of which are to be, so determined as to make the latter equation, equal to that proposed. This is effected as follows : 3 4 T gry (x? -+ Lau+p)*= iy ane * penis — (gz +1) = — q?a2* = Qgra— r* by comparing which with the original, we shall have Fy? +.2p — q 6 By means of these three equations, the values of p,q, and 7, may be found in terms © of the known quantities a, 6, ce, and d; for we have 8p3 — 4bp* + @ac—8d)p—ad-+ 40d =20 which being a cubic equation the value of p becomes determined, and may now be con- sidered as known; and qg and r are also de- termined from the equations gq = Vv (ka® + 2p — db) ap—c. 2q Dre BiQ Having thus found the values of p, q, and r, the four values of x, in the proposed equa- tion, are also determined from the assumed equation {a* + fax +p) —(qxr+r)*=0, or z+ fax +p—t (gz +r); whence xu + (4a—q)x =r—p, or - +Ga+q)t=—7r—p, by taking the ambiguous sign by which (gz +r) is effected both + and —; whence the four required roots are ae ee r =e 4 +/ } fGs tat § /sat+ past pasltttey/ {(ieey 9) %. Descartes’s Method for Biquadratics. Descartes’s solution depended upon resolv- ing the biquadratic into two quadraties, by means of a cubie equation, in the following manner: | First, let the second term, or third power, be taken out of the equation, which reduces it to at+ pxa>+qer+r=0 Find the value of y in the cubic equation of je 1 ae econ y® + 2Qpyt—4rg4 q = 0, then the four values of x will be contained in the two quadratic equations rye thy + tp + oo eye + ty" + tp — 5 8 that is, c= iyev (—t—d) ss—iyt/ (iy — Ep ich ge 3. Euler’s Method for Biquadratics. {t having been observed, that the solution of quadratic equations depended upon the square root of a certain function of its co- efficients, and that of a cubic equation upon the cube roots of two such functions, it seem- ° ed to follow, by analogy, that the roots of a biquadratic would depend upon the fourth roots of three similar functions; or since the fourth roots of numbers ultimately depends upon the operations of the extracting of square roots, he began by supposing the roots of all biquadratics to be of the form r=Sp+ SQ+ V7; p,q, and r, being them- selves the root of some cubic which was to be determined. For this purpose he assumes a general bi- quadratic under the form a+ — ax* — bx —cec =O, and supposes its root to be EMP KARAT Shia ea Squared, becomes =ptqtr+2(vpqt+vpr+/49qr); or, erty p+qt+r=—f, it becomes Rin ie —f = 2(/pq + Vpr + v9"); BIR squaring this, we have ¢4(pq + pr + ar) ig xt —2fx? +f? = =) + 8lyale qr+/q" pr tm Making pq +prt+qr—g, “3 patting the latter part under the form ¥ 8Yprq(vp+vqatyv?), and substituting at the same lime pqr = h, we obtain AR +f? =4¢4+82vh OFS: seprateak xt —2fx*—B8 /h.x+(f?—4g)=05 so that of Siam, Kiribati > ee! a Rela rh=a ‘ f?—4g=— ¢, org—~0 +ie; @ and since......... PE a+ + =F, i Pa + pr t+qr =? qr — 7, % j » | it is obvious that p, qg, and r, are the three” roots of the cubic equation - | pe —fp* + gp —h=0; Ww hence the three quantities p,q and r, be come known, and consequently the roots } the proposed equation, at+—ax*—br—e= being as follows; viz. When b is positive, LSU. TOOt = Y¥ptrvyYqtvr Rie. ssiyus BS) A Pa Aa co Ghal baat Le=— VP +VE—-Vr 4th. 3..5. x=—Vp—Vqatvr When b ts negative, ' Ist. rootzx = Vp+vq—vr ada ys Ryall Oe een AD mat AY oh ae Z ele Sleek £=—J/ptV/qatV/r ’ 4th. ......% = — /p— Vq— vr. Let us illustrate this rule by an example. ‘ Given a* — 2527 + 602 — 36 = 0, to fin the four values of x. re oe b = — 60, c= 36; __ 625 769 225 = =— =—;h=— 7 therefore ice 398 = 5 +9, Ae “Ta Bp eat t cubic equation will be 525, 4 769, 225 2, the three roots of which are = 2 g=4r= >; s P = 4 q a= 5 — 4 ’ and the square ney of these are 5 Ap = = vg = Vr => Hence, as the value b is negative, the four roots are the following : ¥ 3.4 5 bi Ist. rootxz = = ~—-= | st.rootx = 5 + au dda: MEALS i, 2d. © -—H-4+-2= ro F 2-273 ae | 8°) 408 a 3d. .4...2=—=-+24+-= | ™ atats : 3.4 5 4th. ...... =e te = ° $° at a Waring’s rule for biquadratics may be seen : in chap. 3. of his “ Meditationes Algebraice,” or in Wood’s Algebra, p. 171; and man others might probably be found, more or | | BIS eommodious, in particular cases; but as we before observed, all these solutions, however different they may appear, are essentially the same, and are all reducible to the same rule and same form. See Lea’s “Solution of the higher Order of Equations.” Yor the construction of biquadratic equa- tions, see CONSTRUCTION. See also Descartes’s “ Geometry,” with the Commentaries of Schooten and others; Baker’s ‘“‘ Geometrical Key;” Slusius’s “‘ Mesolabium;” L’Hopital’s ‘* Conic Sections; Wolfius’s “ Elementa Ma- theseos;” Dr. Hutton’s “Course of Mathe- matics,” vol. iii. &e. Biquapratic Parabela. See PARABOLA. BIQUINTILE Aspect of the Planets, is when they are distant from each other 144°, or twice the fifth part of 360°. BISECTION, the division of a quantity mto two equal parts. BISSEXTILE, or Leap-Year, in Chrono- ogy, ® year consisting of 366 days, happen- ng once every four years, by reason of the iddition of a day in the month of February recover the six hours which the sun spends n his course each year, beyond the 365 days wdinarily allowed for it. The day thus added _ is also called bissex- ile; Caesar having appointed it to be intro- luced by reckoning the twenty-fourth of *ebruary twice; and as this day, in. the old iccount, was the same as the sixth of the valends of March, which had been long cele- rated among the Romans on account of the ‘xpulsion of ‘Tarquin, it was called “bis sex- us calendas Martii;” and from hence we iave derived the name bissextile. By the statute de anno bissextile, 21 Hen. Il. to prevent misunderstandings, the inter- alary day, and that next before it, are to be ccounted as one day. The astronomers conecrned in reforming he calendar, by order of Pope Gregory XIII. a 1582, observing that the bissextile in four ears ddded forty-four minutes more than the un spent in returning to the same point of he zodiac, aud computing that these super- vumerary minutes in 133 years would form day; to prevent any changes being thus in- ensibly introduced into the seasons, directed hat in the course of 400 years there should ¢ three bissextiles retrenched ; so that every entisimal year, which according to the Julian ecount is bissextile or leap-year, is a com- jon year in the Gregorian account, unless ae number of centuries can be divided by our without a remainder. Thus 1600 and 000 are bissextile ; but 1700, 1800, and 1900, re common. But with the exceptions of the ‘bove even centuries, any year which exactly ivides by four is leap-year; and when there ) any remainder, it indicates the number of ears since leap-year. | The Gregorian computation was received 1 most foreign countries ever since the re- arming of the calendar; and by act of par- ament, passed anno 1751, it commenced in the dominions under the crown of Great BLA Britain in the year following, ordering that the natural day following the second of Sep- tember, should be accounted the fourteenth ; omitting the intermediate eleven days of the common calendar. BLACK, an epithet applied to any thing opaque and porous, which imbibes the greater part of the light that falls on it, reflects little or none, and therefore exhibits no colour. Bodies of a black colour are found more inflammable, because the rays of light falling on them are not reflected outwards, but enter the body, and are often reflected and refracted within it, till they are stifled and lost. They are also found lighter, ceteris paribus, than white bodies, being more porous. The inflammability of black bodies, and their disposition to acquire heat beyond those of other colours, are easily evinced. Some appeal to the experiment of a white and black glove, worn in the same sun; the consequence will be, a very sensibly greater degree of heat in the one hand than in the other. The same thing appears from the phenomena of burn- ing glasses, by which black bodies are always found to kindle soonest; thus a burning glass, too weak to have any visible effect upon white paper, will readily kindle the same paper when rubbed over with ink. Mr. Boyle gives other proofs, still more obvious; he took a large tile, and having whited over one half of its superfices, and blacked the other, ex- ‘posed it to the sun; where having let it lie a convenient time, he found that whilst the whited part remained still cool, the black part was grown very hot. For farther satisfaction, the same author has sometimes left on the sur- face of the tile a part retaining its native red, and exposing all to the sun, has found the lat- ter to have contracted a superior heat in com- parison of the white part, but inferior to that of the black. So also on his exposing two pieces of silk, one white the other black, in the same window to the sun, he often found the latter considerably heated, when the former has remained cool. It is observable likewise, that rooms hung with black are not only darker, but warmer than others. Boyle’s Works abridged, tom. i. p. 144, and tom. ii. p- 36. To all which may be added, that a virtuoso of- unsuspected credit assured Mr. Boyle, that in a hot climate he had, by care- fully blackening the shells of eggs and expos- ing them to the sun, seen them thereby well roasted in a short time. Dr. Watson, the present Bishop of Landaff, covered the bulb of a thermometer with a black coating of Indian ink; in consequence of which, the mercury rose ten degrees. Phil. Trans. vol, lxiii. part i. p. 40. See more on this subject in Dr. Franklin’s “ Experiments, Observations, &c.;” Dr. Priestley’s ‘ Hist. of Vision,” p. 127—143. ; BLAGRAVE (JouHN), an eminent mathe- matician, who flourished about the beginning of the seventeenth century. He was author of the four following works; viz. 1. “ A Mathematical Jewel.” Showing the BOD construction and use of an instrument so called. London, folio, 1585. 2. “The Construction and Use of the fa- miliar Staff;’ for the mensuration of alti- tudes, &c. London, 4to. 1590. 3. “ Astrolabium Uranicum generale. ” London, 4to. 1596. 4, “The Art of Dialling, in two parts.” London, 4to. 1609. BLONDEL(Francis), acelebrated French mathematician and military engineer, was born in Picardy in 1617, and died in Paris in 1686. He was author of two distinct works on architecture, and another on fortification ; besides which, he published “ Course de Ma- thematiques;” Paris, 1683, 4to. “ L’Art de jetter des Bombes;” La Haye, 1685. ‘‘ His- toire de Calendries Roman ;” Paris, 1682, 4to. To which may be added, several ingenious pieces in the Memoires of the French Aca- demy of Sciences, particularly in the year 1666. BLUE, one of the seven primitive colours of the rays of light, into which they are di- vided when refracted through a glass prism. See CoLourRs and Prism. The blue colour of the sky is a remarkable phenomenon, which has been variously ac- counted for hy different philosophers. La Hire, after Leonardo de Vinci, attributed it to the effect which is produced by viewing a dark body through a white transparent one, which he observes always gives the sensation of blueness; and thus the sky being itself to- tally devoid of light, when viewed through the air illuminated and whitened by the sun, ap- pears of that blue colour so constantly ob- served. According to Newton, however, the phenomenon is to be accounted for on other principles. He observes, that all the vapours when they begin to condense and coalesce into natural particles, become first of such a bigness as to reflect the azure rays, before they can constitute clouds of other colours. Bouguer ascribes this blueness of the sky to the constitution of the air itself, being of such a nature that the fainter coloured rays are in- capable of making their way through any very considerable portion of it. See Newton’s Optics, p. 228; Bouguer, Traité d’Optique, p. 368; Priestley’s Hist. of Vision, p. 436; and Edinb, Ess. vol. ii. p. 75. BOB of a Pendulum, or Batu of a Pendu- lio, is the metallic weight which is attached to the lower extremity of a pendulum rod, by means of a tapped adjusting nut, at such a distance from the point of suspension as the time of a given vibration requires. See PEn- DULUM. BODY, or Soin, in Geometry, is that which has three dimensions, viz. length, breadth, and thickness. See Soin. Bony, in Physics or Natural Philosophy, is a solid, extended, palpable substance; of it- self merely passive, and indifferent either to motion or rest; but capable of any sort of mo- tion, and all figures and forms. Body is com- posed, according to the Pcripatetics, of matter, BOO form, and privation ; according to the Epicu- reans and Corpuscularians, of an assemblage of hooked heavy atoms ; according to the Car- tesians, of a certain quantity of extension; anil according to the Newtonians, of a system of association of solid, massy, hard, impenetra- ble, moveable particles, ranged or disposed in different manners, whence result bodies of different forms, which receive particular de nominations, according to the circumstances under which they appear. These elementary or component particles of bodies must be in» finitely hard, so as ever to remain unbroken and unchanged; which, as Newton observes is necessary, in order to the world’s remaini in the same state, and bodies continuing of the same nature and texture in several ages. Boptes are either hard, soft, or elastic. A hard Body is that whose parts do not yield to any stroke or percussion, but retains its figure unaltered. ie A soft Body is that whose parts yield te any stroke or impression, without restoring themselves again. Pt An elastic Body is that whose parts yield te any stroke, but immediately restore them: selves again, and the bedy retains the same figure as at first. ft: We know not, however, of any bodies thai are perfectly hard, soft, or elastic; but al possess these properties in a greater or les degree. + Bontiss are also either solid or fluid. ¢ A solid Body is that in which the attractive power of the particles of which it is composet exceed their repulsive power, and, conse quently, they are not readily moved one among another, and therefore the body will retai any figure that is given to it. } i A fluid Body is that in which the attractiv and repulsive powers of the particles are i exact equilibrio, and therefore yields to thy slightest impression. See SoLip and FLUID, Regular Bones, or Platonic Bovies, ar those which have all their sides, angles, am planes, similar and equal, of which there ar only the five following, viz? i 1. Tetraedron, contained under ; 4 equilatera triangles, 2, Hexanedrony iscsi’. os deeds avs .. 6 squares. — Se Ootaedrony, oysoes. iocdonbee ... 8 triangles, 4. Dodecaedron, ...........000000 12 pentagon: 5. Icosaedron, ....... teh ositiney .... 20 triangles. For the method of forming the five regula bodies, as also for finding their surfaces an solidities, see the respective articles. a BOILING. See EsuLvition. G BOOTES, a constellation of the norther hemisphere. See CONSTELLATION. a This constellation is called by various othe names; as Areas, Arctophylax, Arcturus M nor, Bubuleus, Canis-Latrans, Clamator, Ie: rus, Lycaon, Philometur, Plaustus-Casto Plorans, Thegnis, and Vociferator: by Hes; chius it is called Orion; and by the Ar Aramech, or Archamech. Schiller, instead | Bootes, makes the figure of Sylvester; Schrul hard, that of Nimrod; Weigelius, the thr BOR wedish Crowns. See Wolfius Lex. Math. . 266. BORDA (Cuartgs), formerly Chevalier de jorda, was born at Dax in France, May 4, 733, and distinguished himself in early life s an able mathematician. The Memoirs of ne French Academy of Sciences, from 1763 » 1767, contain excellent papers by Borda, on ne resistance of fluids, on water-wheels and umps, on the projections of bombs, on the rethod of determining curve lines, on the ropertics of maxima and minima, and on the est method of choosing by lot. In the years 771 and 1772 he performed a voyage by com- 1and of the French king, with Verdun de la ‘anne and Pingré, in the Flora frigate, to arious parts of Europe, Africa, and America, w the purpose of improving the sciences of eography and navigation, and of making ex- eriments with various nautical instruments, me-pieces, kc. An account of the result of lis expedition may be found in the Memoirs ‘the Paris Academy for 1773; and more uly in Voyage fait par Ordre du Roi, en 1771 t 1772; 2 vols. 4to. 1778. To Borda the ublic are likewise indebted for the best chart f the Canary Isles, which served as a model w the valuable map of those islands published 1 Spain in 1778. In 1787 Borda published a aluable work, entitled Description et Usage u Circle de Reflexion, in which he revived ud improved the use of the reflecting circle roposed by Tobias Mayer, in 1756. He was 1e first founder of the French schools of aval architecture ; and applied the principles if Euler to the uniform construction of ships, ) that all those of the French navy might be niform with respect to sailing. He likewise rought again into use Mayer’s method of neasuring terrestrial angles, after it had been me neglected; applied it to astronomical hservations; and invented a circle on a new onstruction, with moveable telescopes, toge- we: with other instruments, such as metallic jues for measuring bases, &c. He took the nost active part in the late reform of weights jad measures introduced in France; and he jaused to be calculated and printed, at his pwn expense, the logarithms of the decimal varts of the circle, according to the new divi- jon into 400 parts. In 1792 he invented in- truments for determining, with a precision efore unknown, the length of a pendulum winging seconds at Paris. He likewise com- uted new tables for gauging vessels ; and em- loyed himself in many other useful labours. « dropsy of the breast, however, terminated ris life, Feb. 20, 1799, in the 64th year of his ge. Lalande, Hist. Astron. 1799. Rees’s Hyelopedia. i BOREAL Signs, in Astronomy, are the orthern signs, or those north of the equator. ee eeALds Aurora, See Aurora Bo- ealis. BORELLI (JoHn ALPHONsUsS), a cele- ‘vated mathematician. He was born at Na- jles, in 1608, and became professor of philo- phy and mathematics at Florence and Pisa. | to BOS At length he settled at Rome, where he was greatly esteemed by queen Christina of Swe- den: he tanght mathematics in a convent in that city, and died i 1679. [is works are very numerous and valuable: besides several on medicmal subjects, which display much skill and judgment, we have the following mathematical ones: 1. Apollonii Pergei Co- nicorum, lib. 5, 6, et 7; Floren. 1661, fol. 2. Theorie Medicorum Planetarum ex causis physicis deducts; Vlor. 1666, 4t0. 3. De Vj Percussionis ; Bologna, 1667, 4to. This piece was reprinted, with his celebrated treatise De Motu Animalium, and another, De Mo- tionibus Naturalibus, in 1686. 4. Huclides Restitutus, &c.; Pisa, 1668, 4to. 5. Observa- tione intorno alla vistu ineguali degli Occi. This piece was inserted in the Journal of Rome, for the year 1669. 6. De Motionibus Naturalibus de Gravitate pendentibus; Regio Julio, 1670, 4to. 7. Meteroloyia Atnea, &e. ; Regio Julio, 1670, 4to. 8. Osservatione dell’ Ecclissi Lunare, 11 Gennaro, 1675; inserted in the Journal of Rome, 1675, p.34. 9. Ele- menta Conica A pollonii Pergeei, et Archimedis Opera, nova et breviori methodo demonstrata ; printed at Rome in 1679, in 12mo. at the end of the 3d edition of his Euclides Restitutus. 10. De Motu Animalium; Pars prima in 1680, and Pars altera in 1681, 4to. These were re- printed at Leyden, 1685, revised and corrected from many errors; with the addition of John Bernoulli’s Mathematical Meditations con- cerning the Motion of the Muscles. 11. At Leyden, 1686, in 4to. a more correct and ac- curate edition, revised by J. Broen, M. D. of Leyden, of his two pieces, De Vi Percussionis, et De Motionibus de Gravitate pendentibus. BOSCOVICH (RoGer JosePH), a very eminent mathematician and philosopher, was born May 11, 1711, at Ragusa. He studied Latin grammar in the schools which were taught by the Jesuits in his native city, until 1725, when, in consistence with a maxim of the Jesuits, to send their most eminent pupils to Rome for the completion of their educa- tion, he was remoyed to that city. After this he soon acquired very great reputation for his eminent attainments in divinity and science: at three successive periods he became pro- fessor of mathematics and astronomy at Rome, at Pavia, and at Milian. When the order of Jesuits was suppressed, he was invited to Paris, and received the place of director of the optical instruments of the marine. Previous to this, however, he had been employed, in conjunction with father Maire, in measuring a degree of the meridian in [aly, and in cor- recting the maps of the papal estate. He published, in 1755, av interesting account of the expedition in which these objects were effected. He had also been employed in ad- justing a disagreeable affair between the re- public of Lucca and the regency of Tuscany : and in a similar business between the republic of Ragusa and the court of Great Britain; which brought him to London, where he soon became acquainted with the most celebrated BOU British philosophers. He remained at Paris ten years, where he was much respected ; but being a foreigner, his celebrity was envied ; this, together with the irreligion which then prevailed among many of the French philoso- phers, was disagreeable to him, so that he obtained leave for two years absence to revisit his friends in Italy. The first place he tarried at in Italy was Bassano, where he printed five volumes in large octavo, containing a real treasure of optical and astronomical know- ledge. From Bassano our author went to Rome, and thence to Milan, where he took up his abode, being in the neighbourhood of his favourite observatory at Brera. Here he continued to enjoy the pleasures of study; and, occasionally, the society of many respected friends, until his two years of absence were nearly expired: his unwillingness to leave Italy, and at the same time a solicitude to avoid the charge of ingratitude from the French nation, occasioned great perplexity of mind, which was followed by deep melancholy, a disordered imagination, and, at length, di- rect insanity. He had, indeed, some lucid intervals, and once there were hopes of a re- covery; but he soon relapsed, and an impost- hume breaking in his breast, put an end io his mortal existence in Feb. 1787, in his 76th year. The works of Boscovich will demand and secure the esteem of posterity: they are, 1. Elements of Mathematics, with a Treatise on Conic Sections. 2. His many Dissertations published during his professorship in the Ro- man college. 3. His Account of the Survey of the Pope’s Estate. 4. A curious and ele- gant Poem on Solar and Lunar Eclipses. 5. The five volumes published at Bassano. 6. His Hydrodynamical Pieces. 7. A Theory of Natural Philosophy. This theory has, on the whole, been extremely well received in the scientific world; though it has certainly met with some powerful opponents. BOUGUER (PETER), a celebrated French mathematician, was born at Croisic, in Lower Bretagne, the 10th of February, 1698. His father was professor royal of hydrography, who dying in 1718, his son was appointed to suc- ceed him in his office of hydrographer, after a public examination of his qualifications ; being then only 15 years of age; an oceupation which he discharged with great success and dignity at that early age. In 1727, at the age of 29, he obtained the prize proposed by the Aca- demy of Sciences, for the best way of masting of ships. This first success of Bouguer was soon after followed by two others of the same kind: he successively gained the prizes of 1729 and 1731; the former, for the best man- ner of observing at sea the height of the stars; and the latter, for the most advantageous way of observing the declination of the magnetic needle, or the variation of the compass. In 1729, he gave an optical essay upon the gra- dation of light; a subject quite new, in which he examined the intensity of light, and deter- mined its degrees of diminution in passing through different pellucid mediums, and par- BOW ticularly that of the sun in traversing thi earth’s atmosphere. Mairan gave an extra¢ of this first essay in the Journal des Scava in 1730. In this same year, 1730, he waaiiat moved from the port of Croisic to that o Havre, which brought him into a nearer con’ nection with the Academy of Sciences, ir which he obtained, in 1731, the place of asso. ciate-geometrician, vacant by the promo b of Maupertuis to that of pensioner; and iy, 1735 he was promoted to the office of pen sioner-astronomer. ‘The same year he Way sent on the commission to South Ameri¢ with Messrs. Godin, Condamine, and Jeussi¢ to determine the measure of the degrees of th meridian, and the figure of the earth. In thi painful and troublesome business, of ten year) duration, chiefly among the lofty Cordelie mountains, our author determined many othe! new circumstances, beside the main object o the voyage; such as the expansion and con traction of metals and other substances, by th sudden and alternate changes of heat and col among those mountains; observations on thi refraction of the atmosphere from the tops6 the same, with the singular phenomenon | the sudden increase of the refraction, wh the star can be observed below the line of th level; the laws of the density of the air a different heights, from observations made a different points of these enormous mountains a determination that the mountains have ai effect upon a plummet, though he did not as sign the exact quantity of it; a method 0 estimating the errors committed by navigator in determining their rout ; a new constructiol of the log for measuring a ship’s way ; i | several other useful improvements. Ot inventions of Bouguer, made upon differen occasions, were as follow: The heliometer being a telescope with two object glasses affording a good method of measuring thi diameters of the larger planets with ease am exactness; a simple and ingenious anemo meter, for measuring the force of the wind experiments on the reciprocation ef the pen dulum, and those upon the manner of mea suring the force of the light, &e. &c. Thi close application which Bouguer gave t study, undermined his health, and terminate( his life the 15th of August, 1758, at 60 year of age. His chief works, that have been pub lished, are, 1. The Figure of the Earth, de termined by the observations made in Soutl America; 1749, 4to. 2. Treatise on Naviga tion and Pilotage; Paris, 1752, in 4to. Thi work has been abridged by M. La Caille, ii one vol. 8vo. 1768. 3. ‘Treatise on Ships, thei Construction and Motions; in 4to. 175€ 4, Optical 'Treatise on the Gradation of Light first in 1729, and a new edition in 1760, 1) 4to. His papers that were inserted. in th Memoirs of the Academy are very numerou and important. a BOW Compass, an instrument for drawin) arches of very large circles, for which th common compasses are too small. It consist! of a beam of wood or brass, with three lon, } BOE ‘ews that govern or bend a lath of wood of el to any arch. ‘This term is also some- res used to denote very small compasses riployed in describing archs, too small to accurately drawn by the common com- sses. 'IBOYLE (RoBERT), one of the greatest ilosophers of the 17th century, was born at Ismore, in Ireland, January 25, 1626, the ar in which the sciences were deprived of vir greatest ornament, Lord Bacon, whose yns of experimental philosophy our author yerwards so ably seconded and improved. Pyle was one of the first of those illustrious 'n who formed the Royal Society, in 1645, the purpose of improving experimental pwiedge, upon the plan laid down by Ba- 1; which society being, in 1654, removed Oxford, he went to reside there, where he 'y much improved the air-pump, which led ji to the discovery of several of the pro- ities of air. He also published, during his fidence at Oxford, several works relating the properties of air, and other philosophi- ) subjects; and in 1668 returned to Lomdon, flere he continued to reside till his death, ‘ich happened in the year 1691, in the 65th r of his age. soyle was author of a very great number mportan tworks, beautiful editions of which fe been published in London, in five vols. (0, and in six vols. quarto. Dr. Shaw also slished, in three vols. 4to. the same works bridged, methodized, and disposed under | general heads of Physics, Statics, Pneu- ies, Natural History, Chemistry, and Me- jne;”’ to which he has prefixed a short logue of the philosophical writings, ac- ding to the order of time when they were Plished, &c. as follows: . New Experiments, physico-mechanical, ching the Spring of the Air, and its Effects ; (0. 2. Sceptical Chemist, 1662; reprinted 679, with the addition of divers experi- its. 3. Certain Physiological Essays and w Tracts; 1661. 4. Considerations touch- the Usefulness of Experimental Philoso- ; 1663. 5, Experiments and Considera- Hs upon Colours; 1663. 6. New Experi- fats upon Cold; 1665. 7. Hydrostatical Pa- joxes ; 1666. 8. Origin of ’orms and Quali- according to corpuscular philosophy; 1666. ‘he admirable Refractions of the Air; 1670. ‘The Origin and Virtue of Gems; 1672. hi The Relation between Flame and Air; #2. 12. On the strange Subtilty, great Effi- iy; &c. of Effluvia; 1673. 13. The Saltness the Sea, Moisture of the Air, &c.; 1674. {On the hidden Qualities of Air; 1674. ®& The Excellence, &c. of the Mechanical othesis; 1674. 16. Porosity of Bodies ; ®t. 17. Naiural History of Mineral Waters; ft. 18. Experimenta et Observationes Phy- | t; 1691: which was the last work pub- #2d during his life. But two posthumous rks afterwards were published, viz. Natural tory of the Air, 1692; and Medicinal Ex- ‘ments, 1718, BRA The above are the principal philosophical works of this celebrated author, besides which he published various others on religion and other topics, which, not being connected with our subject, are not included in the above list of his writings. BRACHIA of a Balance, the arms, to the extremities of which the scales or weights are suspended. BRACHYSTOCHRONE, is the name which John Bernoulli gave to his celebrated problem of the ‘ Curve of swiftest Descent,” which he published in the Leipsic Acts, for June, 1696, under the following form: PROBLEMA NOVUM. Ad cujus solutionem mathematici invitantus “ Datis in plano verticali duobus punctis A et B, assignare mobili M, viam A M B, per quam gravitate sua descendens, et moveri incipiens a puncto A, brevissimo tempore perveniat ad ultrum punctum B.” That is, To find the curve along which a body would descend from a given point A, to another given point B, both in the same ver- tical plane, im the shortest time possible. At first view of this problem, it would be imagined that a right line, as it is the shortest path from one point to another, must likewise be the line of swiftest descent; but the atten- tive geometer will not hastily assert this, when he considers, that in a concave curve, de- scribed from one point to another, the moving body descends at first in a direction more approaching to a perpendicular, and conse- quently acquires a greater velocity than down an inclined plane; which greater velocity is to be set against the length of the path, which may cause the body to arrive at the point B sooner through the curve than down the plane. Metaphysics alone, therefore, cannot solve the question; in fact, it requires the utmost ac- curacy of mathematical investigation and cal- culation, the result of which shows, that the path required is a cycloid reversed, as we shall see in what follows, being at that time a new and remarkable property of this curve, which the researches of Huygens and Pascal had previously rendered so celebrated. See CycLolp. According to Bossut, Leibnitz resolved this problem the day on which he received it, but that he and John Bernoulli agreed to keep baek their solutions; but the fact of Leibnitz having obtained a correct solution, seems to be very doubtful; at all events, at the expira- tion of six months (the time allowed) no solu- tion was published, and the time was accord- ingly enlarged to one year, during which pe- riod the solutions of James Bernoulli, Newton, and the Marquis de |’Hopital, appeared. Ber- noulli’s aud Newton’s were both given in the Acta. Erud. Lips. for May, 1697, but the latter without a name; the real author of which, however, mathematicians had little trouble in divining ; for, as John Bernoulli observed on this occasion, ‘‘ ex unque leonem.” James Bernoulli, in the course of his inves- BRA tigations, had ascended to problems on isoperi- metrical figures, requiring still more profound speculations, which, after having resolved them, he proposed to mathematicians in ge- neral, at the conclusion of his solution of his brother’s problem. The rivalry in glory that had long divided the Bernoullis was fully displayed on this occasion. At first it was a little moderated by their habits of seeing each other, at least occasionally, and by the intervention of their common friends; but John having been ap- pointed professor of mathematics at Groningen in 1695, all private intercourse between them soon ceased, and they no longer corresponded except through the medium of periodical pub- lications, for the purpose of proposing to each other the most difficult problems ; and here it was that James Bernoulli, desirous of aveng- ing himself of the ingratitude of his brother, to whom he had been preceptor, challenged him by name to answer a certain problem there proposed, to which he added also the following; viz. To find among all the cycloids, which a heavy body may describe from a point to a line given in position, that cycloid which is described in the least possible time ; which propositions he concluded in nearly the fol- lowing words: “‘ A person for whom I pledge myself (Prodit NON NEMO, pro que Caveo) en- gages to give my brother, independently of the praise he will deserve, a prize of 50 florins, on condition that within three months he en- gages to resolve these problems, and within a year publishes legitimate solutions of them.” Adding, “ If at the expiration of this time no one shall have resolved them, I will make public my solutions.” ‘These propositions, as we have before ob- served, were sent with the solution of the Brachystockrone. And as soon as he had no- ticed the solutions to this (in doing which he bestowed great praise on that of Newton and De lHopital, and some slight censure on his brother’s), he undertook the solution of James’s problems, above mentioned; and, imagining that his theory of the line of swiftest descent was alone sufficient to solve them, the follow- ing expressions of ingenuous vanity escaped him: “ Difficult,” says he, “as these problems appear, I did not fail to apply to them the in- stant they came to my hands, and hear with what success: instead of three months, allowed me to sound their depth, and the remainder of the year to find their solutions, I have em- ployed only three minutes to examine, enter upon, and dive to the bottom of this mystery.” These high-sounding phrases were accompa- nied with the construction he gave of the problems, and the consequent demand of the prize, which he said he should give to the poor, as it cost him so little trouble to gain it. But the business was by no means so far ad- vanced as he supposed, as his solution only answered for particular cases, in consequence of his having made only two elements of the curve enter therein, while the general solution required three; and he therefore thus laid BRA | himself open to the keen and cutting } proach of his brother, who soon perceived what respect the solution was defective, a being at the same time perfectly sure of ] own, he published an advertisement in 16f in which he asserted that his brother’s n thod was defective. He still allowed geon tricians time to find the solution; and one gave it, he pledged himself for th things: 1st. To divine, with precision, 1 analysis of his brother. 2dly. Whatever might be, to point out fallacies init. 3dly,! give the true solution of the problem in all parts. -Adding, at the same time, that ifa person was sufficiently interested in the gress of science to venture a wager up these articles, that he would engage to forf an equal sum if he failed in the first; dow the sum, if he did not succeed in the secon and triple the sum, if he did not accomp the third. The singularity of this advertisement, a the reputation of the writer as a geometri@i a little staggered John Bernoulli’s confide in his method. He revised his solution, lowed that he had made a trifling mist which he ascribed to too great precipitan and sent a new result, but without assum a more modest tone, and again demanded | prize. m To these pretensions James Bernoulli la nically answered: “ I beg my brother to vise his Jast solution anew; to examine carefully in every point, and then to Tet know whether it be all right; as he must aware, that no attention can be paid to excuses of precipitancy after I have publis) my solution.” But John Bernoulli, whoy not aware of the radical defect of the 1 thod that he employed, felt an entire dence in his last result, and said, in re there was no necessity to revise what he done, and that his time would be much be spent in making new discoveries. To | confident assertion James ironically answer “1 never believed that my brother was ma of the true solution of my problem; an doubt it now more than ever, from the ¢ culty he makes of the revision of his solati if it cost him but three minutes, as he assé ‘to examine, enter upon, and dive to the | tom of the whole mystery,’ surely the rey could not require more: but suppose he s} double that time, how many new discoye would he be robbed of by the six mim thus employed?” A To this John again replied ; and the ma still remained undecided, till, in 1700, Ja Bernoulli printed at Basil a letter addre: to his brother, in which he invited him, % great moderation, to publish his method; | concluded by giving the formule of the | 4 9 eer brother ; the true solution, nor the defect of his | method, he at length gave it in a paper, W! was sent under a seal to the Academy of? BRA es at Paris, in the month of February, 1701; condition that it should not be opened yhout his consent, and after his brother had plished his analysis. 4s soon as James Bernoulli was informed » this, he had no longer any reason to keep ‘solution a secret: he accordingly made it blic, and maintained it by way of a thesis ‘Basil, in March, 1701, with a dedication to four illustrious mathematicians, De l’Ho- ul, Leibnitz, Newton, and [atio de Duil- |. He likewise printed it separately, under following title: “ Analysis magni Proble- jis isoperimetrici.” ‘This was considered ya prodigy of sagacity and invention; and eed, if the time be considered, it will not qtoo much to assert, that a more difficult \vlem was never resolved. The Marquis iP Hopital wrote to Leibnitz, that he had 1it with avidity, and that he had found it «y direct and accurate, which testimony } bnitz transmitted to John Bernoulli him- , though he was much prejudiced in his yur, having himself before examined and proved of John Bernoulli’s solution; the ner having submitted it to him for his opi- 1. .fter this publication, John Bernoulli main- aed a perfect silence, neither publishing his Wi solution, nor criticising that of his bro- tb. At length, in 1705, James Bernoulli il, and a short time after John Bernoulli tlished his solution in the Memoirs of the ademy for 1706. This, however, possessed ) Same radical defect that has been before jed, namely, that the author had consi- iad only two elements of the curve, instead irhich it is requisite to have three enter, or j2mploy an equivalent condition. In pro- ins of the same kind as that of the Line of Vitest Descent, where it is simply required palfil the conditions of the maximum or mi- ium, the applying of this condition to two Inents is sufficient to find the fluxional ation of the curve; but when, beside the nximum or minimum, the curve must pos- * a farther property of being isopemetrical jnother, this new condition requires that a id element of the curve shall have a certain fination with respect to the other two; | every determination, founded simply on i; first consideration, will give false results ; ept in those cases where a curve cannot »sfy one of the two conditions, without at i same time fulfilling the other; and of this n Bernoulli was at length so convinced, t he made it the basis of a new solution, me than thirteen years after his brother’s i th, confessing himself deceived in his first. ‘ai donner ici,” says he, “ pour reparer ‘e inadvertence une nouvelle maniére de pudre,” &e. This was a tardy avowal, but iwould still have done him honour, had he the same time acknowledged that his new ition was in substance the same as his ) ther’s, but given in a form which consi- ably abridged the calculation: instead of ch, he eyen in this seeks every occasion } { te BRA * to asperse his brother’s method, and this after a lapse of so many years, when, as Mr. Wood- house observes in his treatise on “ Isoperi- metrical Problems,” ‘‘ the recollection of his brother’s kindness, or zeal for a brother’s fame, ought to have assuaged and laid asleep all such angry passions.” We have been led into a farther detail of this dispute, which arose out of the Brachy- stochrone, than was at first intended; the im- portance, however, of the problem, both with regard to the properties which resulted from it, the celebrity of the disputants, and the im- pulse which it gave to the modern analysis, will be so many apologies for the length of this article, which we must now conclude with a solution of the original problem. See Rees’s Cyclopedia, article “ Isoperimetry ;” Bossut’s and Montucla’s “ Hist. des Mathe- matiques,” Woodhouse’s ‘ Treatise on Iso- perimetrical Problems,” and the article Isopr- RIMETRY. Let OGD be the curve; conceive a portion of it, CGD, to be divided into two parts, CG, GD; andtake g yy another element to the curve CLD, di- vided also into two parts, CL, LD, and indefinitely near to CGD: then since by the hypothesis the time through CG + B GD is to be a mi- nimum, and_= since K quantities at or near their state of minimum may be considered constant (their increments or decrements be- ing very small), we have t, CG,-+t GD=ate ChitttieD (t. CG abridgedly representing the time through CG) and therefore é CG—t.CL=t. LD—t.. GD. Again CE:CG::;t.CE:4CG, CG being considered as an inclined plane, BIEL dics CE, Clint CBRadiC.:; but.....MG: LG:: EG CG by the similar triangles LMG and GCE; therefore CE: CL :: EG x (t. CE): CG x (ti CG—t.CL) and EF: LG:: GF x @ EF):GD x @ LD—t.GD). Hence, equating the two values of LG, we have EG xt.CEx EF xGD= e GF xt. EF x CE x CG, or CG 3 wt EG x Tac EI eee ta +D GE x aE * CE x CG substituting ¢ EF for ¢. C E. Ora OE. OR: BF: CGPED, VHC’ VHE* a known property of the cycloid; which is, therefore, the Brachystochrone, or Curve of quickest Descent. BRADLEY (JAmgs), an eminent English astronomer, was born in Gloucestershire, in 1692, and very carly displayed great talents +* BRA for mathematical and astronomical pursuits, making many observations, of such a nature as laid the foundation of those discoveries for which he was so highly distinguished. These observations gained him the notice of Lord Macclesfield, Sir Isaac Newton, Dr. Halley, and many other members of the Royal Society, of which he soon after became a member. In 1721 he succeeded Dr. John Keill as Savilian professor of astronomy; on which he resigned his livings, agreeably to the rules of the foun- der. He now applied with great diligence to his favourite pursuit, and in the course of his numerous observations he discovered and settled the laws of the alterations of the fixed stars, from the progressive motion of light, combined with the earth’s annual motion about the sun, and the nutation of the earth’s axis, arising from the unequal attraction of the sun and moon on the different parts of the earth. ‘The former of these effects is called the aberration of the fixed stars, the theory of which he published in 1727; and the latter, the nutation of the earth’s axis, the theory of which appeared in 1737: so that in the space of about ten years he communicated to the world two of the finest discoveries in modern astronomy, which will for ever make a memo- rable epoch in the history of that science. See ABERRATION and NUTATION. On the death of Dr. Halley, he was ap- pointed astronomer royal, when he pursued his observations with unwearied diligence. However numerous the collection of astrono- mical instruments at that observatory, it was impossible that such an observer as Dr. Brad- ley should not desire to increase them, as well to answer those particular views, as in general to make observations with greater exactness. In the year 1748, therefore, he took the op- portunity of the visit of the Royal Society to the observatory, annually made to examine the instruments and receive the professor’s observations for the year, to represent so strongly the necessity of repairing the old in- struments, and providing new ones, that the society thought proper to make application to the king, who was pleased to order one thou- sand pounds for that purpose. This sum was laid out under the direction of our author, who, with the assistance of the late celebrated Mr. Graham and Mr. Bird, furnished the ob- servatory with as complete a collection of in- ‘struments as could well be desired. Being thus furnished, he pursued his observations with unwearied assiduity. These were regis- tered in perfeét order in thirteen folio volumes in MS., and presented to the University of Ox- ford in 1776, and it is much to be lamented that they have not yet been published. During the doctor’s residence at Flamstead House, the living of Greenwich hecame vacant, which the king presented to him; but he conscien- tiously declined accepting it, as its duties would interfere with his other engagements : on which his majesty settled upon him a pen- sion of two hundred and fifty pounds a year. He was elected a member of the Academy of BRA \ Sciences at Berlin, in 1747; of that at Pay in 1748; of that at Petersburgh, in 1754; of that at Bologna, in 1757. He was m; ried in the year 1744, but never had mo than one child, a daughter. He died in Jub 1762, in his 70th year; leaving behind him character of great talents and persevering it dustry, together with a remarkably placid an modest deportment. Nothing of his has ye been published, but a few papers in the Ph losophical Transactions. “at BRADWARDIN (Tuomas), archbishop(¢ Canterbury, was born at Hartfield, in Susse} lished by J. H.Savil. 2. De Geometria spt culativa, &c.; Paris, 1495, 1512, 1530. 3.0 Arithmetica practica; Paris, 1502, 1512. 4.0 Proportionibus; Paris, 1495; Venice, 15€ folio. 5. De Quadratura Circali; Paris, 149 folio. BRAHE (Tycuo), a famous Danish astr nomer, was born of a noble family, 7 Knudstorp, 1546. In 1574 he read lectu on astronomy at Copenhagen, by order of th king; who also built for him an sound tl toed \ in the isle of Huen in the Sound, — building being called Uranibourg. ‘The Ki added to the donation a pension and son lucrative places. In this situation he resid¢ about 20 years, pursuing his studies, mali observations, and receiving visits from most illustrious personages. On the death the king he lost his pension, in consequen) of which he left Uranibourg, and wert to penhagen, from whence he removed to Pragt where he was introduced to the emperor R dolphus: that prince treated him respectful gave him a magnificent house, till he cou procure one for him more fit for astronomi¢ observations ; he also assigned him a pensii of 3000 crowns, and promised him a fee | himself and his descendants. Here then) settled in the latter part of 1598, with] sons and scholars, and among them the cé brated Kepler, who had joined him. But did not long enjoy this happy situation ; f about three years after, he died, on the 24 of October, 1601, of a retention of urme, the 55th year of his age, and was interred a very magnificent manner in the princi church at Prague, where a noble monum¢ was erected to him; leaving, beside his wi two sons and four daughters. A The skill of Tycho Brahe in astronomy universally known, and his works are ve numerous, but the following are the mi valuable: 1. An Account of the new S$; which appeared Nov. 11, 1572, in Cassiope Copenh. 1573, in 4to. 2. Another Treatisé the new Phenomena of the Heavens. Int first part of which he treats of the restituti as he calls it, of the sun and of the fixed sta} and in the second part, of a new star whi had then made its appearance. 3. A Colli tion of Astronemical Epistles ; printed in 4) at Uranibourg, in 1596; Nuremberg, y | BRA at Vrankfort in 1610. It was dedicated laurice, landgrave of Hesse; because there in it a considerable number of letters of landgrave William, his father, and of Chris- ier Rothmann, the mathematician of that ce, to Tycho, and of Tycho to them. the Rudolphine Tables ; which he had not ‘hed when he died, but were revised and tlished by Kepler, as Tycho had desired. im accurate Enumeration of the fixed s; addressed to the emperor Rodolphus. . complete Catalogue of One Thousand of ixed Stars; which Kepler has inserted inthe jolphine Tables. 7. “ Historia Coelestis,” | History of the Heavens, in two parts: the econtains the observations he had made at nibourg, in sixteen books; the latter con- ‘the observations made at Wandesburg, venberg, Prague, &c. in four books. te apparatus of Tycho was purchased by ¢mperor Rodolphus, for 22,000 crowns. It ined, however, useless and concealed till roubles of Bohemia, when the army of Elector Palatine plundered them, and, ie true spirit of barbarism, breaking some em, and applying others to purposes for in they were never designed. The great tial globe of brass was preserved, carried Prague, and deposited with the Jesuits sia, in Silesia, whence it was afterwards f , in the year 1633, and placed in the hall » Royal Academy at Copenhagen. RANCH of a Curve, in Geometry, is a used to denote certain parts of a curve, fa are infinitely extended without return- spon themselves ; being called also infi- jranches: such are the legs of the para- und hyperbola. In order to understand etter the nature of these branches, let present absciss, and y the ordinate of a iE then y being expressed in terms of x, s latter be taken positive, y will have a ‘0 number of values, and the same if x ken negative; and the curve will have iny branches, as there are real values of esponding to both the negative and po- pvalues of x. ‘infinite branches of curves, are either i parabolic or hyperbolic kind. *abolic BRANCHES are those which may for an asymptote a parabola of a supe- t inferior order: thus, for example, the of which the equation is ave an infinite parabolic branch, which ave the common parabola for its asymp- z »f which the equation is y =<. For x supposed infinite, the last term vanishes, ® equation becomes simply y — 7? Which equation of the common parabola. fie same manner, if the equation was t meer i. 4 b3 Bet) epi e ald find that the infinite branch would BRA have for its asymptote the cubic parabola, the 3 equation of which is y = ~, a Hyperbolic BRANCHES, are those which have a right line, or an hyperbola of a superior or inferior degree for their asymptote. For ex- ample, the curve whose equation is x” b> Y= nes, a of which we have been speaking above, is re- 2 oy? duced when 2 = 0, to y =—, and its asymp- tote will be the infinite ordinate passing through its origin. It may also have for its asymptote the common hyperbola. In the same way, the curve whose equation a3 63 Sear will have for one asymptote the infinite ordi- nate which passes through the point where x = 0, and for the other the cubic hyperbola. It is evident that all the infinite branches of curves are either parabolic or hyperbolic ; for whatever may be the equation of the curve, y may be expressed in terms of x in a series of which all the terms are real; and it is ob- vious, that when x becomes either infinite or zero, the equation will be reduced to y =x , all the other terms being then regarded as nothing ; and therefore the branch will be of the parabolic kind, when m is positive and greater than 1; and of the hyperbolic kind, if m be negative, or 0, or less than 1. This equation, however, is not sufficient for determining the number of branches, nor the position of them. Thus, for example, if is ws ea P; + /ax, in making 2 infinite, we have 2Z y =~ , and we see that the branch is para- bolic, and at first sight we should be induced to consider it as having two infinite branches, the one corresponding to x positive, and the other to x negative. But this will evidently not obtain ; for in taking x negative, the ordi- rs nate y=— + /azx will be imaginary. For more on this subject see the article Curve; see also Cramer’s “ Introduction a l’Analyse des Lignes courbes,” chap. viii.; the Me- moires de l’Acad. Roy. des Sciences of Berlin, for 1746; and the Appendix to Maclaurin’s Algebra. BRANCKER (Tuomas), an English ma- thematician, was born, in Devonshire, in 1636, and died in 1676. Brancker wrote a piece on the Doctrine of the Sphere, in Latin, which was published at Oxford in 1662 ; and in 1668, he published at London, in 4to. a translation of Rhonius’s Algebra, with the title of “ An Introduction to Algebra;” which trea- tise having been communicated to Dr. John Pell, he received from him some assistance towards improving it, which he generously acknowledges in a letter to Mr. John Collins ; with whom, and some other gentlemen, pro-. - BRI ficients in this science, he continued a corres- pondence during his lite. BRIDGE, a work of carpentry or masonry, built over a river or canal, for the convenience of passing from one side to the other. See ARCH. BRIGGS (Henry), one of the greatest ma- ihematicians in the sixteenth century, was born at Warley Wood, in the parish of Hali- fax, in Yorkshire, in 1556. Briggs is deserv- edly celebrated for the great share he had in the improvement of logarithms. As soon as he was informed of this noble invention, he im- mediately set about the improvement of them ; and proposed the alteration of the scale from the hyperbolic form which Napier had given them, to that in which one is the logarithm of the ratio of 10 to 1; soon after, he had some correspondence with Napier, and indeed paid him a visit in Scotland, in order to confer with him upon this change. In the year 1622, he published a small tract on the “ North-west Passage to the South Seas, through the Con- tinent of Virginia and Hudson’s Bay;” the reason of which was probably, that he was then a member of the company trading to Virginia. His next performance was his great and claborate work, the Arithmetica Lo- garithmica, in folio; printed at London in 1624; a stupendous work for so short a time, containing the logarithms of 30,000 natural numbers, to 14 places of figures beside the index! Briggs lived also to complete a table _of logarithmic sines and tangents for the 100th part.of every degree, to 14 places of figures beside the index; with a table of natural sines for the same 100th parts to 15 places, and the tangents and secauts for the same to ten places, with the construction of the whole. ‘These tables were printed at Gouda in 1631, under the care of Adrian Viacq, and pub- lished in 1633, with the title of “ Trigonome- tria Britannica.” Soon after his going to Oxford, he was incorporated M.A. in that university; where he continued till his death, which happened on the twenty-sixth of Jan. 1630. Dr. Smith gives him the character of a man of great probity; a contemner of riches, and contented with his own station; prefer- ring a studious retirement to all the splen- did circumstances of life. His works, says Dr. Hutton, are more im- portant than numerous; some of which were published by other persons; the principal of them are as follows: 1. Logarithmorum Chilias prima. London, Svo. 1617. 2. Arithmetica Logarithmica. folio, 1624. 3. Trigonometrica Britannica,&c. Goudal, folio, 1623. 4, A Treatise onthe North-west Passage to the South Sea. London, 4to. 1622. 5. Euclidis Elementorum, vi. libri priores, &ec. London, folio, 1620. Beside a few other works of less importance ; amongst which is an English treatise of Arithmetic, Tables for the Improvement of London, BUR Navigation, &e. &c.; a complete list of whj may be seen in Dr. Hutton’s Math. Diet, Brices’ Logarithms, are those logarith now in common use. See LOGARITHMS, BRINAK, the Arabic name for a, Lyra, BROKEN Number, the same as FRAcTI which see. a Broken Ray, or Ray of Refraction, Dioptrics, is the line into which an incid ray is refracted, or broken, in crossing | second medium. > BROUNCKER, or BRouNKER (WILLIA Lord Viscount of Castle Lyons, in Lrelai was born about the year 1620, and died 1684, in the 64th year of his age. To L Brouncker we owe the first idea of the the of Continued Fractions, and the solution several ingenious problems relating to- Diophantine and Indeterminate Analysis, has several papers inserted in the Philoso) cal Transactions, the principal of which @ 1. Experiments concerning the recoilin Guns. vs 2, A Series for the Quadrature of the’ perbola. Several of his letters to Archbishop were also printed in Usher’s Letters, as’ as some to Dr. Wallis, in his “‘ Commere Fpistolicum.” ‘hy BULLIALDUS (IsHMAEL), an emil astronomer, was born at Laon, in the isl France, in 1605. He travelled in his yj for the sake of improvement, and afterw published several works, amongst we 1. Philolaus, sive de vero Systemati \ 2. Astronomia Philolaica, 1645; to whic added, 'T'abule Philolaica. ie 3. De Lineis Spiralibus Exerc. Geom. Paris, 1657. a 4, Opus novum ad Arithmeticam Infi rum. 1682. ‘y He also. wrote a piece or two upon metry and arithmetic. In 1661 he paid velius a visit at Dantzic, for the sake of ing his astronomical apparatus, and % wards became presbyter at Paris, when died in 1694, in the 89th year of his age. BURNING, the action of fire on be whereby the minute parts of them are. rated and thrown into violent motion, of them assuming the nature of fire selves, fly off in orbem, while the rest an sipated in form of vapour or reduced to BurNING Glass, a convex lens whicht mits the rays of light, but in their pa refract or incline them towards a cor point in the axis called the focus; and by combining together in a single point they of all the rays transmitted through the a very great degree of heat is accumula that point, which will fuse bodies that a fusible in the greatest culinary heat thi be produced. Burnine Mirrors, or Specula, are eo reflecting surfaces, commonly of meial, | reflect the rays of light falling upon then at the same time incline them towards termined point or focus, where their ace BUR ed effect operates in the most powerful (nner, burning and dissipating the hardest 11 most infusible bodies. ‘Those burning glasses which consist of re- icting convex lenses, though not entirely Hknown, were very imperfectly understood } the ancients; but the latter kind, the burn- * mirrors, they seem to have had in greater ‘fection than the moderns, at least if we y credit the relations of several eminent torians, who assert that Archimedes, by ans of such mirrors, burned and destroyed Roman fleet, which, under Marcellus, was ployed at the siege of Syracuse; and that clus in the same way destroyed the navy Vitellins, at the siege of Byzantium. “he ancient authors, who more particularly potion the burning mirrors of Archimedes, ¢ Lucien, in his “ Hippias ;” Gatien, “ De nperamentis,” lib. iii. cap. 2.; Zonaras An- . lib. ix.; and Tzetzes, chil. 2, hist. 35: to jm may be added the authorities of Dion, dorus Anthemius, Siculus, Eustatius, &c. se united testimonies would go a great if towards establishing the fact, were it not ft the silence of Polybius, Livy, and Plu- wh, have thrown considerable doubt on the ject, particularly as they have entered into ainute detail of the means employed by himedes in the defence of the city. Tzetzes 9 particular in his account of this matter, ‘his description suggested to Kircher the Y hod by which it was probably accom- led. “ When the fieet of Marcellus was iin bow-shot,” says Tzetzes, chil.2. hist. 35, e old man (Archimedes) brought an hex- baal mirror which he had previously pre- Pd, at a proper distance from which he placed other smaller mirrors, of the same 1, that moved in all directions on hinges, ch, when placed in the sun’s rays, directed n upon the Roman fleet, whereby it was iced to ashes;” and the other testimonies nave mentioned all tend towards establish- ‘the same fact; but the omission of this amstance by Plutarch, &c. as abeve stated, painly throws considerable doubt on the ‘fect as to the destruction of the fleet; gh there seems none whatever, whether part of the relation be true or not, that jhimedes was well acquainted with the itruction and properties of these reflecting ors, and that he possessed some which © very powerful in their effects. Indeed, urning property of reflectors is mentioned He treatise of Optics commonly ascribed to tid, theorem 31, and in some other places. here are also passages in some of the an- «ts, which seem to indicate that they also essed a knowledge of the burning power \fractors, although their maguifying powers ‘ar to have been wholly unknown to them. ts Aristophanes, in his comedy of “ The ids,” introduces Socrates as examining Upsiades about a method he had discovered sting clear of his debts. He replies, that *hought of employing a round transpa- stone or glass used in lighting fires, by | i i ‘ | : J | BUR which he intended to melt the bond, which ini those days was written on wax. The glass here used to melt the wax, De la Hire ob- serves, could not be concave, but convex, acting by refraction and not by reflection. Pliny also mentions globes of glass and crys- tal, which, being exposed to the sun, burnt the clothes on people’s backs; and Lactantius mentions the same thing, which incontes- tibly proves that the effects of convex glasses were not wholly unknown to the ancients. Among the moderns, one of the eartiesi who devised a burning mirror was the ccle- brated Lord Napier, the inventor of loga~ rithms, who in a paper containing Hints of secret Inventions, dated June 2, 1596 (the original of which is now among the MSS. in the Lambeth library, marked 658, anno 1596), says, “ First, the invention, proof, and perfect demonstration, geometrical and algebraieal, of a burning mirror, which, receiving of dis- persed beams of the sun, doth reflex the same beams altogether united, and concurring pre- cisely in one mathematical point, in the which point most necessarily is engendereth fire: with an evident demonstration of their error, who affirm this to be made a parabolic section. The use of this invention serveth for the burn- ing of the enemy’s ships, at whatsoever ap- pointed distance.—Secondly. The invention and sure demonstration of another mirror, which, receiving the dispersed beams of any material fire or flame, yieldeth also the former effect, and serveth for the like use.” Of the moderns, the most remarkable burn- ing glasses are those of Magine, of 20 inches diameter; of Sepatala, of Milan, near 42 inches diameter, and which burnt at the distance of 15 feet; of Settala, of Vilette, of Tschirnhau- sen, of Buffon, of Trudaine, and of Parker. That of M. de Vilette was 3 feet 11 inches in diameter, and its focal distance was 3 feet 2inches. Its substance is a composition of tin, copper, and tin glass. Some of its effects, as found by Dr. Harris and Dr. Desagutiers, are, that a silver sixpence melted in eae king George’s halfpenny melted in 16”, and ran in 84”; tin melted in 3”; and a diamond, weighing 4 grains, lost Zihs of its weight. That of Buffon is a polyhedron, 6 feet broad and as many high, consisting of 168 small mirrors, or flat pieces of looking-glass, each 6 inches square, by means of which, with the faint rays of the sun in the month of March, he set on fire boards of beech wood at 150 feet distance. Besides, his machine has the conveniency of burning downwards or horié zontally, as one pleases, each speculum being moveable, so as, by the means of three screws, to be set to a proper inclination for directing the rays towards any given point; and it turns either in its greater focus, or in any nearer interval, which our common burning glasses cannot do, their focus being fixed and deter mined. Buffon, at another time, burnt wood at the distance of 200 feet; and melted tin and lead at the distance of more than 120 fee: and silver at 50. K CAI Mr. Parker, of Fleet-street, London, was induced, at an expense of upwards of £700, to contrive, and at length to complete, a large transparent lens, that would serve the purpose of fusing and vitrifying such sub- stances as resist the fires of ordinary furnaces, and more especially of applying heat in vacuo, and in other circumstances in which it cannot be applied by any other means. After direct- ing his attention for several years to this ob- ject, and performing a great variety of expe- riments in the prosecution of it, he at last succeeded in the construction of a lens of flint glass, 3 feet in diameter, which, when fixed in its frame, exposes a surface of 32 inches in the clear; the distance of the focus is 6 feet 9 inches, and its diameter 1 inch. The rays from this large lens are received and transmitted through a smaller one of 13 inches diameter, its focul length 29 inches, and dia- meter of its focus 3 inch; so that this second lens increases the power of the former, as 8” to 37, or rather more than 7 to 1. From a variety of experiments made with this lens, the following are selected to serve as a specimen of its powers: Substances fused, with their Weight Weight in Time in and Time of Fusion. Grains. Seconds. Basie at SUE Acts anedes athens A SBE UAT Sey LIVOE, +, CEEON.. Vicaacmeacre MR. NIE, SPOUDET, MUILLO Sin scahgntss ost enn snes OH aienn - 20 RAEN GATE on caginicio anal be onag ST sce . 3 IN ERIC LL... oicitais'the purpose of making observations in the uthern hemisphere; and returned to France i 1754, after an absence of four years, during jnich period he was constantly employed in atronomical observations, by means of which, ;d a comparison of them with those made by qrtain other astronomers in Europe during ie same period, he settled the parallaxes of ‘veral of the heavenly bodies, making the jrallax of the sun 92”; of the moon 56’ 56"; « Mars, in his opposition, 36”; of Venus 88”. e also settled the laws by which astrono- cal refractions are varied by the different nsities of the air, by heat or cold, dryness moisture ; and showed an easy and practi- ble method of finding the longitude at sea means of the moon. He likewise had pro- sted another great work, but death termi- ted his useful labours before he had brought s plan to maturity. He died March 1, 1762, the 49th year of his age. CALCULATION, a reckoning, the result arithmetical operation. The word is de- ‘ed from calculus, in allusion to the practice ( the ancients, who used for this purpose cal- (fi, or little stones, as we now use counters, jures, &e. ‘CALCULATOR, a person who makes or forms calculations, which, in the plural, ns anciently written calculatores. Hes @ certain among mathematicians, de- tes a certain way of performing mathema- ‘al operations, investigations, &c. Thus we 'y antecedental calculus, differential calculus, xional calculus, &e. Antecedental CALCULUS. LL. ‘CALCUL des Derivations, or CALCULUS of erwations. See DERIVATIONS. ‘Catcutus Differentialis, Exponentialis, In- rralis, of Partial Differences, of Variations, » See DIFFERENTIAL, EXPONENTIAL, and | TEGRAL. | Fluxional Catcutus. See FLuxions. (Literal Cavcuxus is the same as Algebra. Numeral Catcutus is the same as Arith- etic. CALENDAR, CaLenparium, or KALEn- iR, a distribution of time accommodated to e uses of life, or a table or almanac con- ining the order of days, weeks, months, asts, &c. happening throughout the year. je word is derived from calende, anciently ‘itten in large characters at the head of ch month. The days in the calendar were originally vided into octades, or periods of eight days ; it afterwards, in imitation of the Jews, into riods of seven, agreeably to the Mosaic law; aich custom, Scaliger observes, was not in- dduced amongst the Romans till after the ne of Theodosius. There are divers calendars, according to the ferent forms of the year, and distribution time established in different countries; as See ANTECEDEN- CAL the Roman, Jewish, Persian, Julian, and Gre« gorian CALENDARS. The Roman CALENDAR was first formed by Romulus, who distributed time into several periods for the use of his followers and people. The year was first supposed to consist of only 304 days, which was divided into 10 months, some of these months containing 20 days, others 35 days, and some more: it began with the first of March and ended with De- cember. Numa reformed this calendar, and added the months of January and February, making it to commence on the first of January, and to consist of 355 days. But as this was evi- dently deficient of the true year, he ordered an intercalation of 45 days to be made every four years in this manner, viz. every two years an additional month of 22 days between February and March; and at the end of each two years more, another month of 23 days; the month thus interposed being called Mace- donius, or the intercalary February. Julius Cesar, with the aid of Sosigenes, a celebrated astronomer of those times, farther reformed the Roman calendar, from whence arose the Julian calendar, and the Julian or old style. Finding that the sun performed his annual course in 3651 days nearly, he di- vided the year into 365 days, but every fourth year 366 days, adding a day that year before the 24th of February; as stated under Bts- SEXTILE. This was farther reformed by order of Pope Gregory XIJI.; whence arose the terms Gregorian calendar and style, or new style; for the year of Julian being too long by nearly 11 minutes, it became necessary to omit three leap-years in the course of four centuries, which is done as stated under Bis- SEXTILE. ‘ A curious account of the calendars of Ro- mulus, of Numa Pompilius, and of Julius Cesar, with specimens, may be seen in M. Danet’s “ Dictionary of Antiquities,” article CALENDARIUM. Julian Christian CALENDAR, is that in which the days of the week are determined by the letters A, B, C, D, E, F, G, by means of the solar cycle ; and the new and full moons, par- ticularly the paschal full moon, with the feast of Easter, and the other moveable feasts de- pending upon it, by means of golden num- bers, or lunar cycles, rightly disposed through the Julian year. See CycLe and GOLDEN NUMBER. Gregorian CALENDAR is that which, by means of epacts rightly disposed through the several months, determines the new and full moons, with the time of Easter, and the move- able feasts depending upon it, in the Grego- rian year. This differs, therefore, from the Julian calendar, both in the form of the year, aud in as much as epacts are substituted in- stead of golden numbers. See Epacrt. Though the Gregorian calendar be more accurate than the Julian, yet it is not without imperfections, as Scaliger and Calvisius have fully usin nor is it perhaps possible to d¢- K 2 CAL vise any one that shall be quite perfect. Yet the reformed calendar, and that which is or- dered to be observed in England, by act of parliament made the 24th of George H., come very near to the point of accuracy: for by that act it is ordered, that ‘ Easter-day, on which the rest depend, is always the first Sunday after the full moon which happens upon or next after the 2ist day of March; and if the full moon happens upon a Sanday, Easter-day is the Sunday after.” Playfair, in his “ System of Chronology,” observes, that the method of intercalation used in the Gregorian calendar is not the most ac- curate. Ninety-seven days, or 100—3, are inserted in the space of four centuries. ‘This supposes the tropical year to consist of 365%, 5, 49’, 12”. On this supposition the interpo- lation would be exact, and the error would scarcely exceed one day in 268,000 years. But the reformers of the calendar made use of the Copernican year of 3654, 5", 49’, 20”. Instead, therefore, of inserting 97 days in 400 years, they ought to have added, at proper intervals, 41 days in 169 years, or 90 days in 371 years, or 131 in 540 years, &c. Recent observations have determined the quantity of the tropical year to be 365%, 5", 48', 452". Ad- mitting this to be the true quantity of it, the intercalations ought to be made as follow: + + + + +—-+ — A ETI 3s) Tee an, OWS.’ SOT MOLD #1057 TT?) BZ GAGs h MALS AILS 3 LOOMS ZS? » F562 +b acaite comeing eb en =t 1185 1313 441 2754 4067 9447 51302 287% 318° 349% 6672 985% 2288? T2za2zs? 6 Fy oP ©0749 172800. ‘ $0749, 172899; that is, one day ought to be intercalated in the space of 4 years, or rather 4 days in 17 years, or 8 days in 33 years, &c. If 41851 days were intercalated in 172800 years there would be no error. The signs + and — indicate that the number of intercalary days above which they are placed is too great or too small. Every succeeding number is more accurate than that which goes before. As this method of interpolation is different from that now in use, it is obvious that the Gregorian calendar must be corrected after a certain period of years. The correction, how- ever, will be inconsiderable for many ages, as it will amount only to a day and a half which is to be suppressed in the space of 5000 years. Reformed or Corrected CALENDAR, is that which, rejecting all the apparatus of golden numbers, epacts, and dominical letters, deter- mines the equinox, and the paschal full moon, with the moveable feasts depending upon it, by computation from astronomical tables. This calendar was introduced among the Pro- testant states of Germany in the year 1700, when 11 days were omitted in the month of February, to make the corrected style agree with ‘the Gregorian. This alteration in the form of the year they admitted for a time, in expectation that the true quantity of the tro- pical year being at length more accurately determined by observation, the Romanists ‘or arched legs, for the purpose of taking t ‘es - CAM would agree with them on some more cony nient intercalation. , ts French CALENDAR, 2 new form of calend: which commenced in France, Sept. 22, 179% but having been since laid aside, and the Gr) gorian calendar again adopted, it seems as less to enter into any particular deseriptic of this revolutionary measure. e CALENDS, or KALENDs, in the Rom: chronology, the first day of every month. word is formed from xaAew, I call, or proclain because, before the publication of the Rom: fasti, it was one of the offices of the pontifie to watch the appearance of the new moo and give notice thereof to the rex sacrificuly) upon which a sacrifice being offered, the po tiff summoned the people together in the C pitol, and there with a loud voice proclaime the namber of calends, or the day where the nones would be; which he did by repea ing this formula as often as there were da of calends, “ Calo Juno Novella.” When the name calende was given thereto, from e calare. And hence also our term calendar, CALIBER, or CALipER, properly denot the diameter of any round body: thus wes the caliber of the bore of a gun, the caliber, a shot, &c. + CALIBER or CALIPER Compasses, or sim Calipers, a sort of compass made with bow, | diameter of any round body. y CALIPPIC Period, in Chronology, a peri of 76 years, continually recurring, after whit it was supposed by Calippus, that the hm tions, kc. of the moon would return again the same order; which, however, is not € act, as it brings them too late by a day 225 years. i CALORIC (from calor, heat), a mode term introduced into philosophy, to den¢ that substance by the influence of whiché produced all the phenomena of heat, a which was formerly distinguished by the te igneous fluid, matter of heat, and other ana cous demonstrations. See HEAT. e CALORIFIC Rays. See Dark Rays. CAMELIGN. See ConsTELLATION. | CAMELOPARDALUS. See ConstEL) TION. ; # CAMERA Lucida, or Light CHAMBER} contrivance of Dr. Hook to make the imé of any object appear on a wall, in a Ii! room, either by day or night. Phil. Tra vol. xxxviii. p. 741, et seq. Hi Dr. Wollaston has recently invented a pt able instrument for drawing in perspective,! which he has given the name of camera luet See a description of this instrument, Rep tory of Arts, No. 57, N.S. Gregory’s tra} lation of Haiiy’s Philosophy, vol. ti. p. 387.) CameERA Obscura, or Dark CHAMBER, int tics, a machine or apparatus so construct} that principally by means of a convex gli or a convex glass and plane mirror, the ima of external objects are represented on a rou ground plane glass, white paper, or other § face, in the most vivid and distinct mat CAM th all their natural motions, colours, shades, . The first invention of the camera obscura 's been attributed to Baptista Porta. See 3; “* Magia Naturalis,” lib. xvii. cap. 6, first blished at Frankfort about the year 1589 1591. The first four books of this work we published at Antwerp in 1560, But Dr. feind, in his “ History of Physic,” (vol. ii. | 236) observes, that Friar Bacon, who flou- vhed in the beginning of the 13th century, éseribes the camera obscura, and all sorts of @.sses which magnify or diminish any object, ng it nearer to the eye, or remove it farther See also Bacon’s “ Opus Majus,” by Dr. »b, p. 286; and his epistle ‘“ ad Parisien- n,” and his “ Perspective,” cited by Dr. Pit, in his ‘* History of Oxfordshire,” p,215; m which we may conclude, that he had avery accurate and extensive acquaintance wh the properties of various kinds of glasses. Che use of the camera obscura is various: Ssists very much in explaining the nature nl rationale of vision; and hence by some it i been compared {to an artificial eye. It uibits. the most striking and entertaining ‘resentations of objects of all descriptions, iether near or distant, in their true perspeé- >; the colouring just and natural, their iat and shadows correct, and all their mo- Wis and relative positions according to the »sinal. By means of this instrument a per- ', however unacquainted with drawing, may Hineate objects with great facility and cor- (iness; and to the skilful artist it will be Gad indispensably useful in comparing his tches with the perfect representations bn in the camera, and by observing his -ctive imitations, he may correct as much »ossible his designs. ‘he theory of the camera obscura will be lily comprehended from the following figure, !which the ob- pe AB radiates | a small | fcture, C, upon |hite wall oppo- |. ii toit, andifthe | Ite or radiation | ind the aper- bCa be dark, the image of the object yi be painted on the wall, in an inverted (tion: for the aperture C being very small, trays issuing from the point B will fall on hose from the points A and D will fall ‘and d: wherefore, since the rays issuing ‘a the several points are not confounded, ‘will by reflection exhibit its appearance the wall. But since the rays A C and BC CAM tance of the image from the aperture is to the distance of the object from the same. To construct a permanent CAMERA Obscura.— 1, Darken a chamber, one of whose windows looks into a place set with various objects; leaving only a little aperture open in one Shutter. 2, In this aperture fit a lens, either plano-convex, or convex on both sides; so as to be a portion of a large sphere. 3. At a due distance, to be determined by experience, Spread a paper, or white cloth, unless there be a white wall for the purpose; for on this the images of the desired objects will be deli- neated invertedly. 4. If it be rather desired to have them appear erect, it is done either by means of a concave lens placed between the centre and the focus of the first lens; or by receiving the image on a plane speculum inclined to the horizon under an angle of 45° ; or by means of two lenses included in a draw- tube, m lieu of one.— Note. If the aperture do not exceed the size of a pea, the objects will be represented, even though no leas be used, That the images be clear and distinct, it is necessary that the objects be illuminated by the sun’s light shining upon them from the opposite quarter; so that, in a western pro- spect, the images will be best seen in a fore- noon; an eastern prospect, the afternoon; and a northern prospect, about noon; a southern prospect is the least eligible of any. But the best way is to have the lens fixed in a proper frame, on the top of a building, and made to move easily round in all directions, by a handle extended to the person who ma- nages the instrument; the images being then thrown down into a dark room immediately below it, upon a horizontal round plaister of Paris ground: for thus a view of all the ob- jects quite around may easily be taken in the space of a few minutes; as is the case of the excellent camera obscura placed on the top of the royal observatory at Greenwich, and with a very good one made by Holroyd of Leeds, in which the images are received on a table of more than six feet diameter. A very simple portable camera obscura is represented in the following figure: iirsect each other in the aperture, and the a} from the lowest points fall on the highest, hisituation of the object will of necessity be rted. Hence, since the angles at D and ‘Ye right, and the vertical ones at © are ful, B and 4, and A and a, will be also Gu; consequently, if the wall where the ‘et is delineated be parallel to it, ab: AB +C:DC, that is, the height of the image i be to the height of the object, as the dis- i ABCD is a small rectangular box, its length being about 20 or 24 inches, and its breadth about 10 inches. This box is closed on all sides, except the space EK I'G D, which is co- vered with a piece of ground glass, or trans- parent paper. In the other end, AB, is a moveable tube, L, with a proper lens, and EIHD is a plane reflecting mirror (set an angle of 45°) which intercepts the rays P p, Qq, &e, proceeding from the object PQ, CAN which are thence reflected upon the trans- parent skreen EFGD, where the image of the object will be painted in its natural co- lours, but in an inverted position, as QP; which, however, may be obtained erect, by the introduction of proper lenses in the tube L. Note. That a shade is necessary to keep off the external light from the skreen EF GD, which is commonly effected by having the box covered with a horizontal lid, under which are two wings which open at right an- gles to each side of it. Various other forms are given to this in- strnment, several of which may be seen in Rees’s Cyclop. CAMUS (CuHarLes STEPHEN LEwIs), a celebrated French mathematician, was born at Cressey in Brie, August 25, 1699, and died the fourth of May, 1768, in the sixty-ninth year ofhis age, The works published by this author are, 1. Course of Mathematics for the use of the Engineers, 4 vols. 8vo. 2. Elements of Mechanics. A work very valuable for the strictness and perspicuity of its demonstrations. 3. Elements of Arithmetic. Beside various papers which were published in the Memoirs of the Academy of Sciences ° at Paris; viz. 1. Of an accelerated Motion by Living Forces. 2. Of the Figure of the Teeth and Pinions in Clocks. 3. A Problem in Statics. 4, On the masting of Ships. 5. The Manner of working Oars, &c. &c. . CANCER, the Crab, one of the twelve signs of the zodiac, denoted by the character 9. See CONSTELLATION. Tropic of CancEeR, a small circle of the sphere parallel to the equinoctial, and passing through the beginning of the sign Cancer. CANES Venatici, the Hounds or Grey- hounds, a northern constellation. These two dogs are farther distinguished by the names Asterion and Chara. See CONSTELLATION. CANICULAR, the name formerly given to the constellation Canis Major; and the same was also sometimes applied to the star Sirius ; whence the Canicular Days. CANICULAR Days, or Dog-Days, denote pro- perly a certain number of days which precede and follow the helical rising of Canicular, or the Dog-star, in the morning; which were formerly the days of the greatest heat. The ancients imagined that this star, rising as above, was the cause of the hot sultry weathcr common at that season, as well as of the dis- tempers which usually followed them (Homer Il. lib. v. ver. 10; Virgil, Aln. lib. x. ver. 270); and somewhat of a similar notion still prevails in many of the remote parts of this country, even to the present day. The dog-days were commonly reckoned at about forty; wz. twenty before, and twenty after the heliacal rising of Sirius; and in the almanacs, were usually accounted absolutely from this circumstance, which in consequence CAP of the precession ofthe equinoxes brou cht they into the months of June and July, instead July and August: an alteration, howeve has been made in this respect, within a fe years; and they are now, without any regal to the rising of Sirius, accounted from July to August 11; these being considered tl hotest days of the year. sd CANICULAR Year, denotes the Egyptian n) tural year; which was computed from 01, helical rising of Canicular to another. wey CANIS Major, the Great Dog, a constell tion of the southern hemisphere. STELLATION. ; Canis Minor, a constellation of the southe) hemisphere. See CONSTELLATION. . CANON, in Arithmetic, Geometry, &@, general rule for resolving all cases of 1 kind: this word is now seldom used; wes instead of it, method, or formula. ‘Thus i quadratic equations, 2* + ax = 6, are solyi by the canon or formula % my Ae (a Tables of sines, tangents, &c. were al sometimes called canons. i CANOPUS, in Astronomy, a bright star the first magnitude, in the constellation Ay Canopus is also a name given by some the old astronomers, to a star under the seco, bend of Eridanus. i : CANTON (JoHN), an ingenious ha philosopher, was born at Stroud, in Gloue tershire, in 1718; well known for several . eenious philosophical discoveries, particula, for his experiment proving the compressiDl of water. About 1746, he made some mp tant improvements in electricity; and in 1% he presented to the Royal Society a meth of making artificial magnets, independent natural ones, for which he was elected member, and received the annual gold me In the same year he was honoured with degree of M.D. by the university of Ak deen; and afterwards chosen one of the Co cil of the Royal Society. His communica to that learned body, upon astronomical philosophical subjects, were very numer and important, but he never published } separate work. This ingenious and wot man died in March, 1772, in the fifty-fou year of his age. ; 1's CAPACITY of a Body, its solidity or € tent; but more commonly it denotes the | low or vacuity of bodies; thus we say, capacity of a vessel, meaning the quanti! will contain. ‘i CAPALLA, a bright star of the first n nitude, in the constellation Auriga. y CAPILLARY Tubes, in Physics, are % small pipes, whose canals or bores are ceedingly narrow, being so called from * resemblance to a hair in size; their usual meter being about one-twentieth or " thirtieth of an inch, some even are novf than one-fiftieth of an inch; and Dr.. assures us, that he drew some smaller | this, and resembling a spider’s thread. © | “| Pt » 4 ST x= Ca Mr. Atwood, in his “ Analysis,” proposes » following method for determining the meters of these very narrow tubes: Put o the tube some mercury, whose weight in tins is w, and let it occupy a length of the »e 1; then if 13°6 be the specific gravity of sreury, that of water being 1, the diameter | Al be = J 7 x 01923, as may be very adily demonstrated; and this will give the il bore of the tube much more accurately un by any other means whatever. Phenomena of Capillary Tubes.—One of the st singular phenomenon of these tubes is, it if you take several of them of different es, open at both ends, and immerse them a ‘le way into water, or any other fluid, it will mediately rise in the tubes to a considerable ight above the surface of that into which >y are immersed: these heights varying arly in a reciprocal proportion of the dia- oters, the greatest heights, according to Dr. ok, being about twenty-one inches. The heights, however, are not the same for fluids, some standing considerably higher or fwer than others; and with regard to quick- ver, it does not rise in the tubes at all, but, _the contrary, stands lower in the tube than the vessel into which the tube is immersed, d that so much the more as the diameter is raller. Another phenomenon of these tubes is, that ch of them as will naturally discharge water «ly by drops, when electrified yields it in a -rpetual stream. ‘Various hypotheses have been advanced to ae for the ascent of fluids in capillary bes; some attributing it to the attraction of i glass upon the upper surface of the fluid, hers again, to the diminished pressure of le air on the fluids in the tube, &c. It must, oadehbad be acknowledged, that at present Je cause of the phenomenon is by no means ‘tisfactorily accounted for. See Phil. ‘Trans. los. 355, 363 ; Muschenbroek, Intro. at Phil. at. tom.i.c. 20; Cote’s Hydrost. lect. ti. ; Par- hnson’s Syst. of Mech. and Hydrost. ch. v.; hd Cavailo’s Elem. of Nat. or Exp. Phil. dl. ii. ch. 5. | CAPRA, or the She Goat, a name sometimes ‘ven to the star Capalla. Capra is also the name of a small northern mstellation, consisting of three stars. | CAPRICORN, the Goat, a southern con- ‘ellation, and the tenth sign of the zodiac. | is denoted by the character V9, being in- nded for the representation of the goat’s orns. See CONSTELLATION. | Tropic of CAPRICORN, a small circle of the ‘a parallel to the equinoctial, passing trough the beginning of Capricorn, or the inter solstice, or point of the sun’s greatest uthern declination. -CAPUT Draconis, the Dragon’s Head, a ame given by some to the star «, Draconis, \ the head of that constellation. CARACT, or Carat, a term employed to *note the quality of gold. ‘The whole quan- | : CAR tity of metal is supposed to be divided into twenty-four parts, which are the carats, so that a carat is the twenty-fourth part of the whole quantity; which is farther divided in four equal parts, called grains of a carat, and these grains into halves and quarters. And the metal is said to be of so many carats fine, according to the quantity of pure gold that enters into its composition. Pure gold is, therefore, twenty-four carats fine ; and if there be twenty parts pure gold, and four of alloy, then it is said to be twenty carats fine, Ke. CARDAN (JERoM), a celebrated mathe- matician, physician, and astrologer, of the sixteenth century, was born at Pavia, in Italy, Sept. 24, 1501, and died at Rome, in Sept. 1575. In the capacity of physician he was invited to Scotland, to cure the Archbishop of St. Andrew’s of a grievous disorder, which he is said to have effected after the failure of the most celebrated physicians of that day. As an astrologer, he was invited to England in order to cast the nativity of Edward V. as he had already done that of his own; which latter not being precisely correct as to the time of his death, he is said to have starved himself for the honour of the science; viz. that his living might not discredit his art. Hudi- bras alludes to Cardan’s pretended astrologi- cal skill in the following lines: “ Cardan believes great states depend Upon the tip of the Bear’s tail’s end; And as she wisks it t’wards the sun, Strews mighty empires up and down.” Cardan was the author of numerous works on arithmetic, algebra, &c. which were pub- lished at Lyons in 10 vols. folio, 1663: and in the latter science he is supposed to have far excelled any of his own time; though the rule on which his fame more particularly depends, viz. for cubic equations, was not his own, but Tartaglia’s, who had communicated it to him under the strictest promise of secresy. See Cusic Equations and IRREDUCIBLE Case. CARDINAL Points, in Geography and Navigation, the four principal points of the compass; viz. east, west, north, and south. The term is derived from cardo, a hinge, being that on which all the others turn or depend. CarpINAL Signs, are those at the four quar ters, or the equinoxes and solstices; viz. the signs Aries, Libra, Cancer, and Capricorn. CARDINAL Winds, those that blow from the four cardinal points. CARDIOIDE, the name of a curve so de- nominated by Castilliani, from its resemblance to a heart, xzpdia: the construction of which is as follows: Through one extremi- ty A, of the diameter AB, of the circle AP B, draw a number of lines APQ, cutting the circle in PP, &c. upon these set off PQ — to the diameter AB; then the curve passing through all the points Q, Q, &c. is termed the Cardioide, ae * CAR ¥rom this generation of the curve is readily deduced its principal properties; viz, that every where PQ = AB CQorQQ=Aa =2AB AQ=zABtAP _ P always bisects QQ. “ The cardioide is an algebraical curve, the equation to which is, Putting a — AB the diameter, x = AD perp. to AB, y = DQ perp. to AD; then +_Gay3+- 2x) .—Gax*) + 2+ ve ¥ y aR sia or tye aay ate This equation evidently indicates a line of the fourth order; the principal properties of which may, however, be deduced from an equation of the form Rist bi Coss O ae es as may be seen in Euler’s ‘ Analysis Infini- torem,” vol. ii. p. 415, For the method of drawing tangents, and investigating the properties of this curve, see Phil. Trans. No. 461, sect. viii.; see also Me- moirs Acad. Scien. for 1705, where it was first treated of by Carré. _ CARL @Or, a gold coin of the Duchy of Brunswick, value 5 rix dollars, or 16s. 51d. sterling. CAROLIN d'Or, a Bavarian coin, value ten florins, or £1. Os. 9d. sterling. CARLINI, a coin of Naples, of the value of about 4d. sterling. CARRE' (Lewis), a French mathematician of considerable abilities, born in 1663, in the province of Brie, and died in 1711, in the forty-eighth year of his age. Carré had a great many papers published in the Memoires of the Academy of Sciences on various subjects ; a list of the principal of which may be séen in Dr. Hutton’s Mathematical Dictionary: he was also author of the first complete treatise on the integral calculus, under the title of “A Method of measuring Surfaces and Solids, and finding their Centres of Gravity, Percus- sion, and Oscillation ;” which was published at Paris in 1700. CARTES (RENE bDEs), an illustrious French philosopher and mathematician. He was born at La Haye, in Touraine, in 1596, and edu- cated under the Jesuits at La Fleche. At the desire of his friends, who wished to see him advanced in the military line, he went into the service of the Prince of Orange, in 1616, and during a truce between the Dutch and Spaniards wrote in Latin a treatise on music. After serving some time with the Duke of Bavaria, and in Hungary under the Count de Bocquoy, he resolved to quit the military life and devote himself to study: however, he travelled a great deal, particularly in the northern countries and ia England, improving his mind wherever he went. In 1634, he com- pleted several works, and among the rest his treatise of the World; in 1657, he published his books on Method, Dioptrics, Meteors, and Geometry, and while thus engaged upon mathematical and metaphysical pursuits, he xpplied assiduously to the study of anatomy. ? _ sibility of it in nature, but that the univers: CAR In 1647, the French King settled upon hip pension of 3000 livres, and the next year accepted of an invitation from the Queen Sweden, and went to Stockholm, where | obtained a pension and estate ; and died in 1650. i ‘‘ Nature,” says Voltaire, “had favou. Des Cartes with a strong and clear imagi, tion, whence he became a very singular p son, both in private life, and in his mannei reasoning. ‘This imagination could not concealed even in his philosophical writi which are every where adorned with y brilliant ingenious metaphors. Nature ] almost made him a poet; and indeed he wr a piece of poetry for the entertainment Christina, Queen of Sweden, which howe was suppressed in respect for his memory, extended the limits of geometry as far heye¢ the place where he found them, as New did after him; and first taught the metho¢ expressing curves by equations. He appl this geometrical and inventive genius to di trics, which when treated by him becam new art; and if he was mistaken in so things, the reason is, that a man who discoy a new tract of land, cannot at once know the properties of the soil. Those who co after him, and make these Jands fruitful, at least obliged to him for the discover Voltaire acknowledges that there are im merable errors in the rest of Des Cart works; but adds, that geometry was a gu which he himself had in some measure {0} ed, and which would have safely condue him through the several paths of natural } losophy: nevertheless he had at last aband ed this guide, and gave entirely into the} mour of framing hypotheses; and then philosophy was no more than an ingeni romance, fit only to amuse the ignorant. CARTESIAN Philosophy, or CARTESIANI the system of philosophy advanced by J Cartes, and maintained by his followers, | Cartesians. 3 The Cartesian philosophy is founded ont great principles, the one metaphysical, — other physical. The first principle of the C tesian philosophy is this, ‘‘ I think, theref I am: this is the foundation of Des Ca metaphysics: that on which his physies built is, “‘ That nothing exists but substanet Substance he makes of two kinds; the ( that thinks, the other is extended; so t actual thought and actual extension make essence of substance. The essence of mat being thus fixed in extension, Des Cartes € cludes that there is no vacuum, nor any p i *' as absolutely full: by this principle, mere spi is quite excluded ; for extension being imp! in the idea of space, matter is so too. Motion is defined to be the translation ¢ body from the neighbourhood of others t are in contact with it, and considered as rest, to the neighbourhood of other bodi by which the distinction is destroyed bet motion that is absolute or real, and that wh CAR elative or apparent. He maintains that same quantity of motion is always pre- ed in the universe, because God must be posed to act in the most constant and im- able manner; and hence also he deduces i three laws of motion. . Jpon these principles Des Cartes explains | oxida how the world was formed, and & the present phenomena of nature came drise. He supposes that God created mat- ‘of an indefinite extension, which he sepa- d into small square portions or masses, rof angles: that he impressed two motions this matter; the one, by which each part ved about its own centre ; and another, by ch an assemblage, or system of them, turn- round a common centre. From whence ie as many different vortices, or eddies, as ‘e were different masses of matter, thus ‘ing about common centres. he consequence of these motions in each ex, according to Des Cartes; is as follows: » parts of matter could not thus move and lve amongst one another, without having ir angles gradually broken; and this con- al friction of parts and angles must pro- e three elements: the first of these, an iitely fine dust, formed of the angles broken the second, the spheres remaining after he angular parts are thus removed; and .e particles not yet rendered smooth and erieal, but still retaining some of their les and hamous parts, from the third ele- it. ow the first or subtilest element, accord- to the laws of motion, must occupy the re of each system, or vortex, by reason of smallness of its parts: and this is the mat- which constitutes the sun, and the fixed s above, and the fire below. The second (vent, made up of spheres, forms the at- phere, and all the matter between the h aud the fixed stars, in such sort, that | largest spheres are always next the cir- iference of the vortex, and the smallest jt its centre. The third element, formed jhe irregular particles, is the matter that poses the earth, and all terrestrial bodies, ) ther with comets, spots in the sun, &e. -e accounts for the gravity of terrestrial es from the centrifugal force of the ether lying round the earth: and upon the same 2ral principles he pretends to explain the omena of the magnet, and to account for he other operations in nature. es Cartes infers the existence of God from principle, “ Cogito, ergo sum;” and thinks jnference just, because it satisfies his rea- so that he proceeds on a supposition that t satisfies his reason is true; and of course akes it for granted that his reason is not a cious but a true faculty: hence his argu- )t proceeds on a supposition that the point )2 proved is true. He therefore attempts to }e the truth of our faculties by an argu- )t which evidently and necessarily sup- }:s their ruth; and consequently his meta- jies is built on sophisms. As this philoso- CAS pher doubted where he ought to have been confident, so he is often confident where he ought to doubt. He admits not his own ex- istence till he thinks he has proved it; yet his system is replete with hypotheses taken for granted, without proof, almost without exami- nation. He sets out with the profession of universal scepticism; while many of his theo- ries are founded on the most unphilosophical credulity. Des Cartes erred materially when he concluded that the most perfect kind of science was to be arrived at by inferring effects from their causes; for, consisteutly with this conclusion, he endeavoured, from his know- ledge of the Deity (which must necessarily be imperfect), to deduce an explication of all his work, and thus produced a system almost in every respect repugnant to truth and experi- ment. Keill says, truly, that Des Cartes was so far from applying geometry and observa- tions to natural philosophy, that his whole system is but one continued blunder, on ac- count of his negligence in that point; which he could easily prove, by showing that his theory of the vortices, upon which his system is founded, is absolutely false, for that Newton has shown that the periodical times of all bodies that swim in vortices must be directly as the squares of their distances from the centre of them; but it is evident, from obser- yations, that the planets, in moving round the sun, observe a law quite different from this; for the squares of their periodical times are always as the cubes of their distances: and therefore, since they do not observe that law, which of necessity they must if they swim in* a vortex, it is a demonstration that there are no vortices in which the planets are carried round the sun. The Cartesian system has been often altered and mended; many ingenious men, for full a century, employing their talents in reforming one part and new-modelling another. But it is now acknowledged on all hands, that the foundation was faulty, and the superstructure erroneous; so that the fabric, though allowed to be a work of genius, is abandoned to neg- lect and ruin, and is pointed at as a memorial of philosophical presumption. CASTALI (Paut), a learned Jesuit, born at Placentia, in 1617. He was author of the following works ; viz. 1. Vacuum Proscriptum. 2. Terra Mecha- nis Mota. 3. Mechanicorum, libri octo. 4. De Igne Dissertationes. 5. De Angelis Dis- putatio Theolog. 6. Hydrostatice Disserta- tiones. 7. Optical Disputationes. Which latter work he wrote after he was blind, at more than eighty-eight years of age. CASSINI (Joun Dominic), an eminent as- tronomer, was born at a town in Piedmont, in Italy, June 8, 1625. In 1650, the senate of Bologna invited him to be their public ma- thematical professor, while he was yet but twenty-five years of age. And in 1652, a comet appeared, which he observed with great accuracy; and discovered that comets were not bodies accidentally generated in the at- CAS mosphere, as had been supposed, but of the same nature, and probably governed by the same laws, as the planets. The same year he resolved an astronomical problem, which Kep- ler and Bullialdus had given up as insolvable ; viz. to determine geometrically the apogee and eccentricity of a planet, from its true and mean place. In 1666, he printed at Rome a collection of astronomical pieces, and among others a ‘Theory of Jupiter’s Satellites.” Lewis XIV. of France desired leave of the pope for Cassini to come to Paris, which was granted ; but the time of absence was limited to six years, and at the expiration of the term he was commanded to return, and on his re- fusal his places were taken from him. Cas- sini was the first Professor of the Royal Ob- servatory, which was finished in 1670. Here he made numerous observations; and in 1684, he discovered four satellites of Saturn. In 1695, he went to Italy to examine the meri- dian line, which he had before settled in 1655 ; and in 1700, he continued it through France. Cassini died in 1712, in the eighty-seventh year of his age. Cassini (JOHN JAMEs), the son and succes- sor of the above; born at Paris in 1677, and educated at the Mazarine College, under Va- rignon, Professor of Mathematics. At the age of seventeen he was admitted a Member of the Academy of Sciences; and in 1696, he visited England, where he was chosen a Fellow of the Royal Society. He succeeded his father in 1712, and enriched the stock of science with many valuable discoveries. In 1740, he pub- lished his “ Astronomical Tables,” and “ Ele- ments of Astronomy ;”’ which were followed by other extensive works. Cassini did not confine himself entirely to astronomy, but made some curious experiments in electricity, and other subjects of natural philosophy. He also measured a perpendicular to the meridian of Paris, and from all his operations ventured to assert, in opposition to Newton, that the figure of the earth was an oblong spheroid. In consequence of these assertions, the King of France sent two companies of mathemati- cians, one to measure a degree at the equa- tor, and the other at the polar circle; and the comparison of the whole determined the figure to be that of an oblate spheroid, in contradic- tion to Cassini’s declaration. He died in con- sequence of a fall in 1756. Cassini DE THURY (C#SAR FRANCo!Is), the second son and successor of James Cassini; born at Paris in 1714, and at the age of ten calculated the phases of the great solar eclipse of 1727. He accompanied and assisted his father in several of his grand undertakings. In 1761, Cassini undertoek an expedition into Germany, for the purpose of continuing to Vienna the perpendicular of the Paris me- ridian; to unite the triangles of the chart of Trance with the points taken in Germany; to prepare the means of extending into this coun- try the same plan as in France; and thus to establish successively for all Europe a most useful uniformity. Our author was at Vienna CAS the sixth of June, 1761, the day of the trans of the planet Venus over the sun, of which | observed as much as the state of the weath would permit him to do, and published thea count of it in his ‘‘ Voyage en Allemagne.” M. Cassini published in the volumes of M moirs of the French Academy a prodigio. number of pieces, chiefly astronomical, t numerous to particularise in this place, } tween the years 1735 and 1770; consisting astronomical observations and question among which are observable, “ Research concerning the Parallax of the Sun, the Moe Mars, and Venus;’ on “ Astronomical R fractions,” and the effects caused in th quantity and Jaws by the weather; numero observations on the obliquity of the eclipt and on the law of its variations. In short, cultivated astronomy for fifty years, of | most important for that science that elapsed, for the magnitude and variety of « jects, in which he commonly sustained a pr cipal share. | M. Cassini died of the small-pox, the fou of September, 1784, in the seventy-first yi of his age; being succeeded in the Acade and as a Director of the Observatory, by only son, Count John Dominic Cassini; is the fourth in order by direct descent int honourable station. ‘ CASSINIAN Curve, or CASSINOID, an! liptic curve proposed by John Dominic € sini, as being the orbit of a planet. Inv curve, the product of two lines drawn fi its foci to any point in the curve shall equal to a given quantity, viz. to the rectar under the aphelion and perihelion distar of the planet. ‘The celestial observati however, by no means correspond with curve; and indeed it, in some cases, © breaches in its continuation, which are fectly incompatible with the motion 0 planet; so that it can by no means be mitted into astronomy. i CASSIOPETA, or Cassropea, a nortl constellation, situated near the north pol In 1572, there appeared a new star in constellation, which at first surpassed in nitude and brightness the planet Jupiter :b diminished by degrees, and finally disappea after having been visible for eighteen mo Dissertations were written on this subjec ‘Tycho Brahe, Kepler, Mausolycus, Lye Gramineus, &c. Beza, the Landgray Hesse, Rosa, &c. wrote in order to prove it was a comet, and the same which appe to the Magi at the birth of Christ, and th then came to declare his second coming. Several astronomers suppose this sta have a periodical return, and which has/ stated at about one hundred and fifty y This, however, is mere conjecture, and ai little satisfaction to philosophic minds, Herschell has given a statement of the! parative lustre of the stars in this cons tion, in the Phil. Trans. for 1796, vol. ba p. 463; see also vol. Ixxvi. p. 193. See STELLATION, . CAT ‘ASTOR, in Astronomy, a moiety of the istellation Gemini, called also Apollo. Also name of a bright star iu this constellation rked a, by Bayer. Yastor and Pollux. See GEMINI. JasTor and Pollux, a term applied by sailors ¢a certain kind of meteor frequently ob- ived at sea. SASTRAMETATION (from castrum, a wp, and metari, to measure or lay out), the ‘of encamping an army. ; DATABIBAZON. See Dracon’s Tail. DATACAUSTIC Curves (from xara, against, il xeiw, to bwrn), in the higher Geometry and Optics, are the species of caustic curves ned by reflection, which are generated in following manner: If there be an infinite nber ofrays AB, ', AD, &c. pro- ‘ding from the iating point A, reflected by any encurve BCDH, that the angles incidence be e- i to the angles -eflection; then curve EF G, to r ich the reflected G is BI, CE, DF, &c. are always tangents, at the points 1, E, F, &c. is the catacaustic caustic by reflection; being the same thing to say, that a caustic curve is that formed joining the points of concurrence of the eral reflected rays. ,f the reflected ray IB be produced to K, jthat AB = BK, and the curve K L be the jlute of the caustic EF G, beginning at the nt K; the portion of the caustic TE = 1B — AC) + (CE — B}), that is, the dif- once of the two incident rays added to the erence of the two reflected ones. When the curve BCD is a geometrical », the caustic will be so too, and will always rectifiable. The caustic of the circle is a ‘oid, or epicycloid, formed by the revolution a circle upon a circle. The catacaustic of -common cycloid, when the rays are pa- lel to its axis, is also a common cycloid, scribed by the revolution of a circle upon ‘same base. That of a logarithmic spiral uso the same curve. if the inside of a smooth bason, containing iit any white liquor, as milk, be placed in A : sun’s rays, or in a strong candlelight, will exhibit a very perfect catacaustic crve. \fhe principal writers on this subject are, Hopital, Carré, &c. Mem. Acad. Scien. an. 156 and 1705. \CATACOUSTICS (from xara, against, and ésw, I hear), called also CATAPHONES, the sence of reflected sounds or echoes ; being acoustics, what catoptrics is to optics. CATADIOPTRICAL Telescope.. See Re- ECTING Telescope. 4 . CATALOGUE of the Stars. See Star. CATAPULTA, the name of an ancient CAT military engine, used for throwing large stones. CATENARY, Catenania (from eiténa, a chain), in the higher Geometry, a mechanical curve, Which a chain or rope forms itself into, by its own weight, when hung freely between two points of suspension, whether those points be in the same horizontal plane or not. The nature of this curve was investigated by Galileo, who supposed it to be the same with the parabola; but though Jungius de- tected this mistake, its true nature was not discovered till the year 1691, when J. Ber- noulli published it as a problem, in the “ Acta Eruditorem.” Dr. D. Gregory, in 1697, pub- lished a method of investigating the properties before discovered by Bernoulli and Leibnitz ; Phil. Trans. ab. vol. 1. p. 48, &c. where he un- dertakes to show, that an inverted catenary is the best figure for an arch. Bernoulli, Opera, vol.i. p.48, &c. and vol. iii. p.491, &c.; Cotes’s Herm. Mens. p. 108. in order to conceive the general nature and properties of this curve, suppose a heavy and flexible line F D, having its two extremities G C ¥ and D firmly fixed at those points; then by its weight it will bend itself into the curve FAD, which is called the catenary. Let BD and bd be parallel to the horizon AB, its axis perpendicular to the horizon and to BD, and De parallel to AB; and the points B, 6, indefinitely near to each other. From the laws of mechanics, any three powers in equilibrio, are to each other as the lines pa- rallel to the lines of their direction (or inclined in any angle), and terminated by their mutual concourses: hence if Dd express the absolute gravity of the particle Dd, as it will, admit- ting the chain to be uniform throughout; then De will express that part of the gravity that acts perpendicular upon Dd; and by means of which this particle tends to reduce itself into a vertical position; and as it proceeds from the ponderous line DA, it is, ceteris paribus, proportional to the line D A, which is the cause of it. Farther, the line de will ex- press the force which acts against that co- natus of the particle Dd, by which it endea- vours to restore itself into a position perpen- dicular to the horizon, and prevent it from doing so. This force is constant, being no other than the resistance of the point A; and may therefore be expressed by any given right line a. Supposing, therefore, as before, the curve FAD, whose vertex, the lowest point of the catenary, is A, axis AB, and ordinate BD; _also the fluxion of the axis De = Bd; fluxion of the ordinate de; the relation of these two fluxions is expressed thus ; viz. De: ed::is curve DA: a “a CAT which is the fundamental property of the curve, and may be otherwise expressed, as follows: PutAB=2,DB=y, and A sah Sey eh Mads FAG RA RG OL SY 2, OF ¥ Hoo =z; then and from this equation the other property of the curve may be readily drawn; viz. Bf (x* + y?) (=z): Zi (a? + 27) zz / (a* + 27) the fluents of which give « = y(a? + z?). But at the vertex of the curve where -—0, and z — 0, this becomes 0 = (a? + 0) =a, and therefore by correction « = (a* -+27)—a, ora + x — /(a* + 2”); and hence also tye v((a + x)? — a*) = /(2aa + x?) z= g* whence x = Pane re Now any of these expressions will give the equation of the curve, in terms of the are and its absciss; in which it appears that a + 4x represents the hypothenuse of a right-angled triangle, whose legs are a and 2. So that if in BA, and HA, parallel to DB, and representing the tension as it acts, pro- duced there be taken AD=a, and AE — the curve Z or AD; then will the hypothenuse DE=a+~-2 or DB. And hence, any two of the three quantities a, x, z, being given, the third is given also. Again, from the first simple property ; viz. ULYRi Zid, oak =z zy by substituting the value of z above formed, it becomes ax ; Vv (2ax + x)’ the flueut of which equation is y = 2a x hyp. log. ; Vx+ /(2a + x) But at the vertex of the curve, Where x=0 and y=0, this becomes 0 —2a x hyp. log. 2a whence the correct equation of the fluent is gn vu + /(2a +2) y= 2a X hyp. log. Se aed enon which is also an equation to the curve, in terms of x and y, though not simply alge- braical. “oi fin This last equation, however, is reducible in another way to much simpler terms; that is, by first squaring the logarithmic quantity, and dividing its co-efficient by 2; then we shall have y =a xX hyp. log. of EET VCore) or d& = / (2ax+27) ¥; ory = ’ > y= x hyp. log. of aves Again multiplying both numerator and de- nominator by 2, and then squaring the pro- duct, and dividing the co-efficient by 2, it becomes, first Z+e y = 24a x hyp. loc. 4 Oh oF Vv (2* — 2?) , then CAT (z + a)? zt ee SO Ng tal y =a X hyp. Og. Swe CATHETUS (from xaber@,, perpendicu,: in Geometry, a line or radius falling pery| dicular upon another. Thus the catheti, right-angled triangle are the two sides include the right angle. a CATHETUS of Incidence, in Catoptrics, right line drawn from a radiant point, pery) dicular to the reflecting line, or the plan} the speculum or mirror. | CATHETus of Reflection, or of the right line drawn from the eye, or from | point of a reflected ray, perpendicular to| plane of reflection, or of the speculum. — CATOPTRICS (from XaTORTeOV, SPEC of xara and oTToucs, video, L see), the scie of reflex vision, or that branch of Optics wh} illustrates the laws and properties of li reflected from mirrors or specula. WH The principal authors who have treate this subject, amongst the ancients, are Euc Alhazon, and Vitellio. Euclid’s treatise is, first that is extant on this subject; it. published in Latin, in 1604, by John Pe and is included in Herigon’s “ Course of | thematics,” and in Gregory’s edition of works of Euclid, Ithas been, however, doi ed whether this piece was really the worl Euclid, although it is ascribed to him by }j clus, lib. ii. and by Marinus, in his preface! “ Buclid’s Data.” Alhazen was an Aralj author, and composed a large volume Optics about the year 1100; in which) treats pretty fully of catoptrics. V itellio | a native of Poland, and composed anot treatise on this subject about the year 127) Amongst the moderns, many authors hj directly, or indirectly, treated of this subj} Tacquet has demonstrated very much at ler! the properties of plane mirrors, in lib. i. off “Catoptrics,” printed in the collection of| works in folio. Fabri has also written on | subject, in his “Synopsis Optica;” Ja Gregory, in his “Optica Promota ;” and } ticularly Dr. Barrow, in his “ Optical It tures.” In the last mentioned work, its| thor has laid down the principles of | branch of optics with peculiar accuracy perspicuity, and have deduced from them) properties of spherical mirrors, both conc) and convex. We have also David Grego’ ** Elements of Catoptrics ;’ Wolfius’s “ ments of Catoptrics;’ Dr. Smith’s eld rate work on optics, in which he has an! discussed the laws of catoptrics; and ml others of later date, of which we may men! Wood’s “ Principles of Optics,” &e. &e. It does not enter into the plan of this wr to treat of catoptrics as a science, und Separate article; but all the principal pre sitions and properties relating to this subj? will be found under the_articles REFLECT) Mirror, Licut, and Vision, 4 CatorprrRic Cistulu, a machine wher. Ye t =a X hyp. log. CAV IL bodies are represented extremely large, near ones very distant, and diffused ugh a vast space, with other pleasing phe- 1ena; by means of mirrors disposed by the i of catoptrics, in the concavity of a kind hest. tf these there are various kinds, accom- lated to the particular views of the artist, r to multiply, others to magnify, deform, « some of which are described in Dr. Hut- ys Dictionary, and Rees’s Cyclopedia, and ‘ral others in the works above quoted. atoptric Dial, a dial which exhibits the rs, Ke. by reflected rays. ATOPTRIC Telescope, the same as Reflecting FESCOPE. AAVALIER[ (Bonaventura), an eminent ¢an mathematician, was born at Milan in 3}, and entered at an early age into the ¢r of the Jesuits of St. Jerome. Cavalieri a disciple of Galileo, and the friend of ‘icelli. In 1629, he communicated to the d his invention of the method of indivi- 's, Which obtained for him the honour of feeding Maginus, as Professor of Mathe- cs in the University of Bologna. This od of indivisibles ; answers to the objec- we Guldinus ; the use of indivisibles in & powers, or algebra, and in considera- | about gravity; with miscellaneous col- a of problems. Towards the close of ear, 1647, he died a martyr. to the gout, th had deprived him of the use of his fin- CEN gers. Montucla’s “ Hist. des Math.” vol. ii. p. 37, &e. CAUDA, the Latin for Twil; and is prefix- ed to the name of several constellations, to denote certain stars in their tails, thus: Caupa Capricorni, denotes the star in the tail of the constellation Bayer, marked y: Caupa Ceti, the same as B, Cetus. Caupa Cygni, the sames as a, Cygnus. Caupa Delphini, the same as ¢, Delphinus, Caupa Draconis, the Dracon’s Tail 3 which see. CauDA Leonis, the same as 6, Leo. CauDA Urse Majoris, the star marked n, in this constellation. CauDA Urse Minoris, marked «; which is the pole star. 3. CAUSTIC Curves. See Caracausric. CEGINUS, the name of y, Bootes. CELERITY, Creeriras, in Mechanics, is the velocity or swiftness of a body in motion, or that affection of a body in motion by which it can pass over a certain space in a certain time. CELESTIAL, relating to the heavens; as Celestial GLose, Celestial SPHERE, &c.; which see. CENTAURUS, the Centaur. See Con- STELLATION. CENTER. See Centre. CENTESM, the 100th part of a thing. CENTRAL, something relating to a centre. Thus, we have Centra Eelipse, CentRat Force, &e. CENTRAL Eelipse, is when the centres of the luminaries exactly coincide, and are directly in a line with the eye of the observer. CENTRAL Forces, are those forces which tends directly to or from a certain point or centre; or they are forces which cause a moving body to tend towards, or recede from, the centre of motion; and are hence divided into two kinds, according to their different relations to the centre, viz. whether it be to approach or to recede from it; being called, in the former case, centripetal force, and in the latter, centrifugal. [Sia The doctrine of central forces depends on the first Newtonian law of motion; viz. ““ Every body perseveres in its state of rest, or uniform motion in a right line, until a change is effected in it by the agency of some exter- nal force.” Hence, when a body at rest tends inces- santly to move, or when the velocity of any rectilinmeal motion is either accelerated or retarded continually, or when the direction of a motion is continually varied, and a curve line is described, these changes obviously in- dicate the action or influence of some exter- nal force, which operates without ceasing on the quiescent, or moving body. In the first case, this force is estimated by the pressure of the quiescent body against the obstacle which opposes its motion; in the case of a constant accelerated or retarded motion, the force is measured by the rate of acceleration or retar- dation; and in curvelineal motions, by the CEN flexure of the curve described, that is, by the constant deflection of the body from its rec- tinineal path, due regard being had in ail these cases to the time in which the effects are produced, and other circumstances, ac- cording to the principles of mechanics. Effects of the power or force of gravity, of each kind, fall under our constant observation near the surface of the earth; for the same power which renders bodies heavy while they are at rest, accelerates them when they fall, or retards them when they ascend or are pro- jected in any other direction than that of gra- vity. But we can judge of the forces or powers that act on the celestial bodies, by effects of the last kind only. And hence it is, that the doctrine of central forces is of so much use in the theory of the planetary motions. The doctrine of central forces for circular orbits, was first considered by Huygens; but Newton, who treated the subject more general- ly, and in book i. § 2, of his “ Principia,” has demonstrated this fundamental theorem ; viz. that the areas that revolving bodies describe, by radu drawn to animmovable centre, lie in the same immovable planes, and are proportional to the times in which they are described (Principia, lib.i. prop. 1). It has been observed, that this law, which Kepler first discovered from abso- lute observation, is the only general law in the doctrine of central forces; but since this law, as Newton himself has proved, cannot hold whenever a body has a tendency, by its gravity or force, to any other than one and the same point, there seems to be wanting some law that may serve to explain the mo- tion of the moon and satellites which have a gravity towards two different centres; and the law he lays down for this purpose is as follows: viz. that where a body is urged by two forces, tending constantly to two fixed points, it will describe, by lines drawn from the two fixed points, equal solids in equal times, about the line joining those two points. See Machin, on the laws of the moon’s mo- tion, in the postscript published at the end of the English translation of Newton’s “ Principia:” see also a demonstration of this law by Mr. W. Jones, Phil. Trans. vol. lix. art. 12, p. 74, Xe. The same subject has been elegantly inves- tigated, when the motion is directed to more than two centres, by many eminent mathe- maticians; and practical rules have been given for computing the places, &c. of the planets and satellites, by La Grange, La Place, War- ing, &c. See the Phil. Trans. and the Me- mois of Berlin and Paris. M. de Moivre, in his “ Miscel. Analyt.” p- 231, as well as in Phil. Trans. has treated on this subject, and to him we owe many ele- gant theorems relating to the doctrine of cen- tral forces. Varignon, Maclaurin, Simpson, Kuler, Emerson, and de l’Hopital, &c. have treated on this subject; to the latter of whom we owe the following general and comprehen- sive proposition ; viz. If a body of any determinate weight move CEN uniformly round a centre, with any given ye locity, its centrifugal force may be compute by this proportion. As the radius of the circle it describes, Is to double the height due to its velocity | So is the weight of the body, : To its centrifugal force. | So that if 6 represent the weight of th body, and 2g — 324 the force of gravity, vil velocity, and r the radius of the curve; shall have, from the laws of falling Ale 2 4g? sv% 31g: “a = the height due toi i} velocity. Whence, by the above proportion v bv } >—::b6:——_=f,t trifugal 2 r ae vee J, the centrifuga foree Consequently, when the centripetal Ht centrifugal forces are equal, the velocity, the body is equal to that which it woulda quire in falling through half the radius. — ~ 2. The central force of a body, moving the periphery of a circle, is as the versed a AM of the indefinitely small arc A E; or is as the square of that arc AE directly, a as the diameter AB inversely. For AM the space through which the body is dray, from the tangent | in the given time, and 2A. is the proper measure of the central forc But A E being very small, and therefore near equal to its chord, by the nature of the cir AE* 3 AB: AE:: AE: AM= AB F AD B 3. If two bodies revolve uniformly in di rent circles, their central forces are in 1 duplicate ratio of their velocities directly, a the diameters or radii of the circles inverse); ; ik a ml: vw that is, F: f:: Dian For the force, by the last article, is as A FE? a A E* AB D space A E uniformly described. 4. And hence, if the radii or diameters ¢ reciprocally in the duplicate ratio of the ve cities, the central forces will be reciprocallyn the duplicate ratio of the radii, or directly the fourth power of the velo that isif Vi tt cea) HL UG Rest se : R?:: > Viet 5. The central forces are as the diamets of the circles directly, and squares of the riodic times inversely. For ife be the ci ference described in the time ¢, witty the v ; and the velocity v is as city v; then the space ce =tv, or v = C5 he using this value of vin the third rule, it beco: CEN Mf :: ne te P: Ou He: Zs since DT Ret ATP eh TEA eM he diameter is as the priarel arid 6. If two bodies, revolving in different cir- Hes, be acted on by the same central force ; he periodic times are in the subduplicate atio of the diameters or radii of the circles’; Jor when F D '=f, then — pe and D:d:: mt: /D: /d:: ore SP. | 7. If the velocities be reciprocally as the listances from the centre, the central forces “ill be reciprocally as the cubes of the same istances, or directly as the cubes of the velo- fumes. That is, if V:v::7: R, thenis F: /f:: mes: V2: v3, . If the velocities be reciprocally in the ubduplicate ratio of the central distances, le squares of the times will be as the cubes f the Peau! for if Vs os: 7": Ky then ts met :: RS: 9. W etihve ‘if the forces be reciprocally as le squares of the central distances, the squares f the periodic times will be as s the cubes of ie ences or when F ; f:: 77: R’, then is -R3: Ton the arian theorems, we may de- uce the velocity and periodic time of a body volving in a circle, by means of its own tavity, at any given distance from the earth’s entre. Let ¢ “be the space through which a eavy body falls, at the surface of “the earth, ‘ithe first second of time, or 164, feet — AM ithe preceding figure; then Qe ‘will measure te force of gravity at the surface; and x be- ig assumed for the earth’s radius, AC; the locity of the body in a circle at its surface, one second, will be WAE = /(AB. AM) = v2rg = 2600 feet early; the radius of the enith being taken = 21000000 feet. .Again, putting ¢ = 3°14159, &e. we shall be ¥.2gr:2cr: Wies/ = Cee — 5078 seconds ee — = 1h 24m 38”; the eerie time at the reumference. _ Let now R represent the radius of another rele, described by a projectile, about the vuth’s centre; since the force of gravity va- _€s inversely, as the square of the distance, /€ shall have (theor. 8 — 9) { Bae Pear eal | . T?: t, or | ie yelocity ; in the Past whose radius is R. R? R? 3 Ast ce Reese tat Ps te periodic time in tha same circle. ‘Since we have found v = 26000, and t= these formulz become H feco/ — for the required velocity, att. — n078/ B * for the periodic time. CEN In the case of the moon R = 607, we have therefore 26000 / +, =3357 feet per second, the velocity. 5078 7216000 = 273, days, nearly the pe- riodic time. And in the same way the velocities of the planets, and their several periodic times, may be determined, their distances being given; or their periodic times being given, their dis- tances may be found by the converse opera- tion ; the periodic time of the earth’s revolu- tion, and its distance from the sun, being sup- posed known. It may be proper to observe, that though our first theorems related merely to circular motion, yet they are equally true for eliptic orbits, it having been satisfactorily demon- strated by the writers before mentioned, that the same law has place in the latter case, provided the revolution be made about one of the foci of the ellipse, as is the case in all the planetary motions; the semi-transverse axis being assumed as the mean distance. In the same way we may compute the cen- trifugal foree of a body at the equator, aris- ing from the earth’s rotation. For the perio- dic time when the centrifugal force is equal to the force of gravity, it has been shown above, is 5078 seconds, and 23 hours, 56 mi- nutes, or 86160 seconds, is the period of the earth’s rotation on its axis ; ater efore, by art. 5, as 861607 : 50787 :: 1: 42,5, the centrifugal force required ; which, therefore, is the 289th part of gravity at the ‘earth’s surface. Simp- son’s “ Fluxions,” p. 240, &c. Also, for another example, suppose A to be a ball of 1 ounce, which is whirled about the centre C, so as to describe the circle ABE, each revolution being made in half a second ; and the length of the cord AC equal to 2 feet: Here then ¢ — 3, r = 2, i it having been found above that ec uy ale — = T, is the perio- dic time, at the circumference of the earth, when the centrifugal force is equal to gravity ; hence then, by art. 5, as ate. as : F, or 1: f, which proportion becomes BE ta Sian MS he names AST — 9:819 Paget SNS ig ea ne & the centrifugal force, or that by which the string is stretched, viz. nearly 10 ounces, or 10 ten times the weight of the ball. Lastly, suppose the string and hall be sus- pended from a point D, and describes in its motion a conical surface D ABD; then putting DC =a, AC=r, andAD=A; and putting F — 1, the force of gravity as before ; then will the body A be affected by three forces, viz. gra- vity, acting parallel to DC; a centrifugal force in the direction C A, and the ten- sion of the string, or force & B by which it is stretched, in the direction DA; hence these three powers will be as the three CEN ' CEN sides of the triangle ADC respectively, and _ If more than two equal lines can be draw, Th from any point within a circle to the cireur, therefore, a5. 4/-D,onatA.D, cork 33 oi the ference, that point will be the centre. nh tension of the string, as compared with the CENTRE of a Conic Section, is that poi weight of the body. which bisects any diameter, or that point) 5 Q¢2r which all the diameters intersect each othe Also AC ora: AC or 7 ::1:——>, the ge- ‘This point in an ellipse is within the fig a ap Re ; if ei eet in the hyperbola without, and in the parahe neral expression for the centrifugal force above i, 5. 44 an infinite distance. ba found; hence 2 Centre of Conversion, in Mechanies, t= eae, ee a SE — 1108 /a term employed by M. Parent. Its signifi rf Ser (AE g tion may be thus conceived: if a stick be k the periodic time. on stagnant water, and drawn by a thre Those who wish to enter more minutely fastened toit, so that the thread makes alwa into the doctrine of central forces, may consult the same angle with it, the stick will be fom the Memoirs of the French Academy for the to turn about a certain point; which point years 1700, 1701, and 1710; Maclaurin’s called the centre of conversion. See Mem.i “ Fluxions ;’ Simpson’s “ Fluxions;’ Gre- Acad. Scien. abridged, vol. i. p. 191. ¥¢ gory’s “Mechanics ;” Leybourn’s “Reposi- Crnrre of a Curve, of the higher kind,) tory,” Nos. 4 and 5; and the Phil. Trans. the point where two diameters concur; a vol. xxviii. which contains a curious paper on when all the diameters concur in the sar this subject by Dr. Waring. point, it is called the general centre. On CENTRE, in a general sense, denotes a subject, see Abbé de Gua, “ Usages ¢ point equally remote from the extremes of a Analyse de Des Cartes ;” and Cramer, “] line, surface, or solid. The word is xs7pov, trod. 4l’Analyse des Lignes courbes.” which primarily signifies a point; being form- Centre of a Dial, is that point where ed from the verb x:vrew, pungero, to prick. gnomon or style, which is placed parallel) CENTRE of Aftraction of a Body, is that the axis of the earth, intersects the plane f point into which, if all its matter was col- the dial. oH lected, its action upon any remote particle Centre of an Ellipse. See CENTRE Off would still be the same, as it is while the Conic Section. | body retains its own proper form. Or it is CENTRE of Equant, in the old Astronomy that point to which bodies tend by their gra- point in the line of aphelion: being as © vity, or about which a planet revolves as a distant from the centre of the eccentric towais centre ; being attracted or impelled towards the aphelion, as the sun is from the centref it by the action of gravity. i the eccentric towards the perihelion. | The common centre of attraction of two or CenTRE of Equilibrium, is the same wy more bodies, is sometimes used to denote that respect to bodies immersed in a fluid, as fe point in which, if a particle of matter were centre of gravity is to bodies in free space; r placed, the action of each body upon it would it is a certain point on which if a body, or s- be equal, and where it will therefore remain tem of bodies, be suspended, they will rest, in equilibrio; having no tendency to move any position. one way rather than another. CENTRE of Friction, is that point int This is more properly termed, by some au- base of a body on which it revolves; ip thors, the point of equal attraction. The power which, if the whole surface of the base ail of attraction being directly as the masses of the mass of the body were collected, al the attracting bodies, and reciprocally as the made to revolve about the centre of the bit squares of their distances, we have the fol- of the given body, the angular velocity 4 lowing method of finding the common centre stroyed by its friction would be equal to ie of attraction of two bodies whose masses and angular velocity destroyed in the given bey distance from each other are given. by its friction in the same time. The distare Let M and m represent the masses of the of this centre from the centre of the bod's two bodies, and d their distance from each Sxs . : other. Put x for the distance of the point of base = WY hee where S = the sum of eit equal attraction from M, and y the distance of I the same from m; then by the laws of attrac- tion AMANO: | ae Ble eek J/m:i/Mi:y 32 particle into the square of its distance frp the axis of motion, T the sum of the produ of each part of the base, into its distance fri the centre of the base; s the area of the ba vm +vMivmiry +x(=d)ry and ¢ the solid content of the body. See? ve +/4M:/M:: aa a(=d sax ingenious paper on this subject by Mr. Vin Whence... y = DIY le Phil. Trans. for 1785, vol. Ixxv. p. 186. /m+ /M CENTRE of Gravity of any body, or syst# et Hephe: oe eM of bodies, is that point upon which thebe \ ~~ Sm + o/ M or system of bodies, acted upon only b ° CentTRE of a Circle, is that point in a circle force of gravity, will balance itself in all p ; which is equally distant from every point of tions; or it is a point which, when support the circumference, being that from which the the body or system will be supported, he circle is described, ever it may be situated in other respet CEN \Hence it follows, that if a line or plane passing yhrough the centre of gravity be supported, ‘he body or system will be also supported. And, conversely, if a body or system balance tself upon a line or plane, in all positions, the ventre of gravity is in that line or plane. Tn \: similar manner it will appear, that if a body est in equilibrio, when suspended from any woint, the centre of gravity of that body or ystem is in the perpendicular let fall from the entre of suspension; and on these principles depends the mechanical method of finding the ventre of gravity of bodies. To find the Centre of Gravity of bodies, me- chanically. | For this purpose, it is only necessary to dis- jose the body successively in two positions of \quilibrium, by the aid of two forces in ver- jteal directions, applied in succession to two ifferent points of the body, and the point of yatersection of these two directions will show he centre required. || This may be exemplified by particularising | few methods. Ifthe body have plane sides, \\$ a piece of board, hang it up by any point, hen a plumb-line suspended from the same Oint will pass through the centre of gravity ; herefore mark that line upon it: and after uispending the body by another point, apply he plummet to find another such line, then jill their intersection show the centre of \Wavity. } Or thus: hang the body by two strings from she same point fixed to different parts of the jody; then a plummet hung from the same point will fall on the centre of gravity. | Another method: Lay the body ou the edge jf a triangular prism, or such like, moving it dand fro till the parts on both sides are in )guilibrio, and mark a line upon it close by /ge edge of the prism: balance it again in nother position, and mark another line by jne edge of the prism; the vertical line pass- jag through the intersection of these lines, fill likewise pass through the centre of gra- yity. The same thing may be effected by lay- jag the body on a table, till it is just ready /9 fall off, and then marking a line upon it by jhe edge of the table: this done in two posi- ons of the body, will in like nianner point jut the centre of gravity. | t a | "0 find the Centre of Gravity of certain bodies, L ial geometrically. ) Prop. 1. To find the centre of gravity of two hiven bodies. | t A and B be the two given bodies, take AG: BG:: B: A; so shall G be the centre ravity of the two bodies, as is obvious om the principle of the lever; for the bodies ying suspended on the point G, they will femain in equilibrio. See Lever. Prop. 2. To find the centre of gravity of a gle, ABC, _be the badies col- CEN Bisect any two sides, Cc AC, CB, in the points D and E, joint AE, BD, and the point of intersec- . tion G will be the centre D E of gravity of the triangle. This is obvious, because the triangle would ba- lance itself on each of the lines A E, BD; for these 4 B&B bisecting the line BC, AC, bisect every pa- rallel section, and therefore the weight on each side is equal, and equally distant from these lines. _8. To find the centre of gravity of a trape- zium. Divide it iato two triangles, and find the centre of gravity of each; and then, by prop. 1, the centre of gravity of these two, which will be the centre of gravity of the traptzium. And _ in the same manner may be found the centre of gravity of any right-lined figure. General laws and determination of the Centre of Gravity. Prop. 1. To find the centre of any number of bodies placed in a right line. : Let A, B, C, D, &c. be the bodies collected into their respective centres of gravity; 5 any point in the straight line SAD; O the centre of gravity of all the bodies. S ae HE * PANNE D Then since the bodies balance each other upon O, we have, from the principle of the lever, ) AxAO+BxBO=CxCO+4+DxDO, that is, A x(SO—SA)+Bx(SO—SB)= C x (SC —SO) + Dx (SD—SO) and hence, by multiplication and transposition, AxSO+BxSO+CxSO0+DxSO= AxSA +BxSB+CxSC +DxSD and consequently, so _AxSA+BxSB+CxSC+DxSD He A+B+C+4+D If any of the bodies lie the other way from S, their distance must be reckoned negative ; and if SO be negative, the distance SO must be measured from S in that direction, which in the calculation was supposed negative. Prop. 2. If perpendiculars be drawn from any number of bodies to a given plane, the sum of the products of each body into its respective perpendicular distance from the plane, is equal to the product of the sum of all the bodies into the perpendicular distance of their common centre of gravity from the plane. Let A, B, C, &c. lected into their ae — respective centres Ex———__ [3 of gravity; PQ theBZ\Y given plane; draw 6\____ Aa, Bb, Ce, at right c angles to PQ, and consequently paral- lel to each other; Go L tt, CEN join A B, and sey: Meee os then E is the mike of eravity of A and B; through E draw Ee perpendicular to PQ, or pa- rallel to Aa, and x Ey perpendicular to Aa, or Bb; then in the similar triangles AEz, EB y Azv:AE:: By : BE Ag: By: AE; BE 3 BAS therefore A x ‘Ax — B x By; that is, A (wa — Aa) = B(Bb— yd). Or since Ea and E6 are parallelograms, A (Ee — Aa) = B (Bb — Ee) whence A x Ee + Bx Ee=AxAa+Bx Bb that is, (A + B)Ee=AxAa+Bx Bb. Again join EC, and take CG:GE::A + B:C, then G is the centre of gravity of the bodies A, B, C; draw Gg perpendicular to PQ; and it may be shown as before, that (A +B)x Ee+CxCe=(A+B+4+C)Ge or, Ax Aa+BxBb4+CxCcr=(A+B+4+0)G¢ and the same demonstration, itis obvious, may be extended to any number of bodies. Consequently G _AxAa+BxBb+Cx Ce+, ke. 5 — ATB+C + &e. and if the plane be drawn parallel to PQ, and at the distance Gg from it, the centre of gra- vity of the system lies somewhere in this plane. fn the same manner two other planes may be found, in each of which the centre of gravity lies, and the point where the three planes cut each other is the centre of gravity of the system. Now, from the preceding expression, for the centre of gravity of any system of bodies, may be derived a general method for finding that centre. For A, B,C, &c. may be considered as the elementary particles of a body, whose sum or mass is M=A+56+C +, c.; and A x Aa, B x Bb, C x Ce, &c., are the se- veral momenta of all these parts, viz. the product of each part from the plane. Hence then, in any body, find a general expression for the sum of the momenta, and divide it by the content of the body, so shall the quotient be the distance of the centre of gravity, from the vertex or any other fixed point, from which the momenta are estimated. But now, in order to find the general expression for the sum of the momenta, the problem divides itself into several cases, according as it is required to find the centre of gravity of a solid, of a plane surface, of a curve surface, as a thin hol- low shell of any form, or of a curve line of any description. We will investigate each case separately. ° Prop. 3. To find the centre of gravity ofa body considered as an area, solid, surface of a solid, or curve line. Let AL V be any curve, RL Er tf — the axis, in which the centre of gra- vity must lie; for as it bisects every ordinate IF in N, the parts on each ™: side .RL will al- CEN ways balance each other, and therefore the body will balance itself upon R L, and conse quently the centre of gravity must be some- where in that line. ns Put LN =2,IN = y,1L =z, and draw PQ parallel to Il’; then if we consider this) body to be made up of an indefinite number’ of corpuscles, and multiply each into its dis. tance from PQ, the sum of all the produets divided by the sum of all the corpuscles, oy by the whole body LG, will give us the dis/ tance of the centre of gravity from L; as. iy shown above in the preceding proposition, — Now, to get the sum of all the products, we must first have the fluxion of the sum, thd fluent of which will be the sum itself. | Put s for the fluxion of the body at the dis’ tance LN, or x, then will as be the fluxion| of the sum of all the products; also, s thi fluxion of the sum of all the corpuscles. Hen c taking G to represent the centre of gravi and p = 3.14159, &c., we > have 1. Now in the case of an area, s = 2y @, | ef fhe Zara 2) — flu. Yun id ue Dye | fit, Yr 2. In the case of a simple curve line, s ‘= 25 LG — flu. 2 uz _ flu. xz ‘ flu2z ~ flu z a 3. For the solid, we have, § = =p y” x, My cid __ flu. on Pa — fin. yeux Slu. p NO ud 4, Forthe simple sw yer thesolid, s = = py: LE lt py xz — flu. Saws “| Ue PY Z flu yz ® { | Let us illustrate these formule by an ex | ample in each case. Let the curve be the common par abola, th | equation of which is ax = y*, a being i parameter. 4 | 1. To find the centre of gravity of a par bolic area AL V. ‘| Here, y = / ax = a oe * theretors 4 | 7 ca x 2 2 2 x flu. a* x" ie flu, x2 x os a LGé = Sta cease tone Sis. a te 0 bs fladah ize 3% . 2a, =2LR, when x = LR. i 3. To find the centre of gravity of thep rabolic curve AL V. Ra Ce xt = (a? +H9 also, y= 4a? 2— | ots oe OF i, therefor, coe AE tinaiv (ttf <), conseqent iy as — flu. yuz yuz iis a m ‘(+a fly z flu. SoCal — flux / (4x + a) X "3 which being found, the quotient will give: distance ‘required. 3. To find the centre of gravity of the par holoid ALY. . Ea Sek UP a 4 2 CEN Here y? = ax, therefore flu yr xi _ flwax*x _ =x! uy cs fluarc ~ 32 LR, when « = LR. . To find the certre of gravity of the para- ¢ surface ALV. this differs from the second example, only entering into the numerator and denomi- = 3r= 2 : yw; and since y = a} x?, the final formule his case Is, 3 __ flu. x? J (4x +a) X x | flu. x? /(4x +a)xx ‘fluent of which being found, the quotient give the distance LG required. jasimilar manner the same point may be ymined in any other case, in which the ation of the curve is expressible in an al- aical equation ; thus, for example, we find centre of gravity of the following bodies ; putting generally a for the line joining the ex and middle of the base. ‘A plain triangle = 3 a. A right cone A circular sector, as arch: chord :: $ ra- , to the distance of, centre of gravity from centre. he altitude of the segment of a sphere, or roid or conoid, being represented by x, the whole axis by a, then the distance .e centre of gravity in each of these bodies i the vertex will be as follows: LG =a. | , 4a—3n : The sphere of spheroid ae Semi-sphere or semi-spheroid = 32x # Parabolic conoid = ........ apatite af : WAY 4a+32 | Hyperbolic conoid = ...... wm CT or a the above subject the reader may con- | Vince’s “ Principles of Fluxions,”’ Simp- }\ Maclaurin, Lyons, Emerson, and most tors on the doctrine of fluxions. See also sory’s “ Mechanics.” ne position, distance, and motion of the ere of gravity of any body, is a medium of 1 0sitions and distances of all the particles body; which property of this centre has ced some authors to call it the centre of jion, others, the centre of mean distance, And on this principle it is, that it be- Hes so important, in all mechanical ques- ii, to determine the centre of gravity of (es; for this being found, the whole body msidered as condensed into this single tt, by which means the greatest possible ilicity is obtained. ther uses of the centre of gravity will be ‘d under the article Cenrro-Baryc. ENTRE of Gyration, of a body, or system idies, is that point into which, if the whole ls were coilected, a given force applied at ven distance from the axis of suspension, (ld produce the same angular velocity in i iame time, as if the bodies were disposed Heir respective distances. This point dif- ‘ifrom the centre of oscillation only in this, CEN that in the latter case the motion is produced by the gravity of the body or of its particles, but in the case of the centre of gyration the body is put in motion by some other force acting at one place only. To determine the Centre of Gyration. Let A, B, C, &e. be particles of a body, or the bo- dies which together form a system; P the given force ap- plied at D; R the centre of gyration. Then the force which accelerates Da D, while these bodies are at their respective distances, is P x SD” He A x SA*+B x SB?+C x SC*+’ ~ Now let the whole mass be collected in R; and the accelerating force upon D is P xs De (A+ B+ C +,&c.) x 8 R* But since P, and the angular velocity of D, are by the definition the same in both cases, the absolute velocity of D is the same, and therefore the accelerating force upon D must be the same; that is, Pix: D? we AxSA*+B x SB* +, &c. P x SD’ (A +B+C +, &e.) x S R* Whence ; A xX SA* + B x SB? 4, &e. SK= a) v ALB LOy ke: and consequently, if s be the fluxion of the body at the distance zx from the axis, a R= Jie. 2 8s. sR=\/ es 1. This inthe case if a right line becomes he U2 . sR=/ Bi toe= V5 2. The plane of a circle, or cylinder, revoly- ing about the axis, SR = radius x v3. 3. The periphery of a circle, about the dia- meter, SR = radius x V3. 4, A wheel with a very thin rim, revolving about its axle, SR = radius. 5. The plane ofa circle, about the diameter, SR = £ radius. 6. The surface of a sphere, about the dia- meter, SR = radius Xx v §. 7. A globe, about the diameter, SR = ra- dius xX / %. : 8. In a cone, about the axis, SR = radius Sa ae Se ; The distance of the centre of gyration from the axis of motion, is a mean proportional be- tween the distance of the centre of gravity and oscillation from the same axis. Hence, when any two of these distances are known, the third may be readily determined. CenTRE of Inertia. See CENTRE of Gravity. Centre of Magnitude, is the point which is L. 2 CEN equally distant from the similar external parts of a body. CENTRE of mean Distances. Gravity. CENTRE of Motion, that point which remains at rest, while all the points of a body move about it. And this is the same in uniform bodies of the same matter throughout, as the centre of gravity. CentRE of Oscillation is that point in the axis of suspension of a vibrating body, in which if all the matter of the system were collected, any force applied there would ge- nerate the same angular velocity In a given time, as the same force at the centre of gravity, the parts of the system revolving in their re- spective places. Or, since the force of gravity upon the whole body may be considered as a single force, equivalent to the weight of the body, applied at its centre of gravity; the centre of oscilla- tion is that point in a vibrating body, into which if the whole mass were concentrated, and attached to the same axis of motion, it would then vibrate in the same time the body does in its natural state. The problem of finding the centre of ascil- lation of bodies of different forms, particularly of circular sectors on different points of sus- Praga was first proposed by Mersennus to uygens; and to the latter celebrated mathe- matician we are indebted for the first complete solution, although some partial cases had been before considered by Des Cartes, Fabry, &c. Since the invention of the fluxional calculus it has become a common problem in almost all works on. this subject; but the curious reader who wishes to see the early methods made use of in the investigation of this prob- lem is referred to the Leipsic Acts for 1691 and 1714, where the subject is treated ina very masterly manner by the Bernoullis; see also, ‘“ Herman De Motu Corporum Solido- rum et Fluidorum,” and Huygens’s “ Horo- logium Oscillatorium.” See CENTRE of To determine the Centre of Oscillation. Let several bodies oscillate about the point S, as if the mass of each were concentrated into the points A, B, C; referred to the same plane perpendicular to the axis of motion merely to facilitate the investigation. The action exerted by gravity of each of them may be decomposed into two forces, of which one whose direction passes through the centre of suspension is de- stroyed by its resistance ; the other is perpendicular in direction to the former, and is alone efficacious: in moving the body or system. Now gravity tends to im- press the s same velocity upon the painte,A B,C, in the vertical direction; which velocity we shall denote by §, and by m, n, p, the sines of CEN the angles, which the supposed inflexible b SA, SB, SC, &c. form with the vertical § Drawing A M, BN, CP, parallel to A L, aj each equal to g, they will represent the | celerating forces of the points A, B, C, or} spaces which they would desenibes in the fij unit of time, if they were left to themseh But because of the obliquity of these for; lA SA, SB,SC, ifthe rectangles am, 4 , be constructed, the spaces run overt Be ‘sty Aa, B 6, Ce; and as the angles Al 1 taken ate move with different veld ties. But if we suppose them connected gether in an invariable manner, so that t) all perform their vibrations in the same fi; the velocity of some will be augmented W that of others will be diminished ; and a aggregate of the forces which solicit the | tem is always the same, it is necessary the sum of the motions lost should be e¢ to that of the motions gained; or that the of the motions lost should be equal to 4} considering the former as positive, the la as negative. { Let us represent by A, B, C, the masse the three bodies, by a, °, c, their distak from the point of suspension, and by a, | the initial velocities which they lose; quantities of motion lost will be Aa, BBs which must be in equilibrio, therefore the } of their moments taken with regard tol point Sis nothing; and as these respe(y distances from that point are a, }, ec, we id have sal Aaw + BbB + Cey= 0. Let fbe the velocity which the point A | jected to the laws of the system would re \ in the first unit of time; as all the point scribe similar arcs, their initial velocities proportional to the distances from the et aM of suspension; therefore that of B will I and that of C will be of Now the veli lost by each body is equal to the velocity wi it would have had minus that which it: ul *) be gp — “ef, whence, by substituting ig lues in the preceding equation, we have he(em—f) + 28 (en — 22) =. Multiplying by a to this eyuation of fractions, and: finding! value of f, we have . hen ec(Aadm + Babn + Caep it Aa + Bb* + Ce? From the points A, B, C, let fall thp pendiculars AI, BK, CL, upon SL, ancl the centre of gravity, H, of the system 2 CEN G perpendicular to the same line. The sum the moments of the weights A, B, C, refer- 1 to the point S, is equal to the moment of sir resultant which passes through the point ; therefore AI+B.BK4+C.CL=(A+B+4+C)HG. The triangles SAI, SBK, SCL, SHG, fe, putting SH — A, and the sine of the gle HS G equal to r, AS sin. ASI=am, BK=BS sin. BSK=dn, u=CS sin. CSL=ep,AG—SG sin.GSH=hr. Substituting, therefore, instead of these lines ir values in the preceding equation, we ‘ve m+ Bobn+ Cepr=(A+B + O)jhr; wence there results im 7st A + B+ 0) hr L ~ Aa + Bb* + Cc?” fo ascertain the actual position of the point Pose invariable connection with the system *8 not change its velocity, let x be its dis- ice from the centre of suspension, and S the e of the angle which the inflexible rod that ains it to that point makes with the vertical; accelerating force when it moves singly is ; in the contrary case it is proportional to || distance from the point S, and of conse- wnce is equal to = f; but these two forces, a ithe initial velocities they produce, must be tial; therefore = = gs, or, putting the pre- be 3 ling value of f for it, there arises , (A+B4Og¢hrex _ oP }) Aa*+ Bb* + Ce? mae ac 'n which we find _s Aa* + Bd + Cc? That the point sought may be the centre of Lillation, it is not merely necessary that these ) velocities be equal in the first instant, they Ist continue so in every instant of the de- nt; therefore x remaining the same, this sation should have place whatever be the bition of the point sought, and that of the wine of gravity, relatively to the vertical, i ag tis to say, whatever be s and r, the ratio — :- Sherefore constant ; and consequently we heat the same time r — 0, s = 0; which | ravity, and the point of suspension are in ). and the same right line. Hence it results ‘$= 7, and that ; = og —Ae + Bo + Co? re (A+B+C)A~ he same kind of reasoning applies exactly, Hever many the- number of particles may } therefore, to find the centre of oscillation # system of particles or of bodies, we must i tiply the weight of each of them by the Ce of its distance from the point of sus- (sion, and divide the sum of these products jhe weights multiplied by the distance of ‘centre of gravity from the centre of sus- “3i10n; this quotient expresses the distance /t@ centre of oscillation from the point of ws that the centre of oscillation, the centre . CEN suspension measured on the continuation of the line joining the centre of gravity and that point. In order to render the above expression similar to those in the preceding articles: Let us Call S the point of suspension, O the centre of oscillation, or S O the distance of the centre of oscillation from the point of suspension ; also let s be the fluxion of the body at the distance 2; then the above formula becomes so — flu. x? 8 flu. xs As an example, let it be proposed to find the centre of oscillation of a right line, or cylinder, suspended at one end. In this case 4 z e I oe j Si a tak Oe GE Ju. 2 a iz . that is, the centre of oscillation is 4 of the whole length from the point of suspension. If the centre of oscillation be made the point of suspension, the point of suspension will become the centre of oscillation. The centres of oscillation for various figures, vibrating flatways, are as expressed below, viz. Nature of the Figure. When suspended by Vertex. Tsosceles triangle 3 of its altitude Common parabola _— 3 of its altitude 2m+1 3m 41 As to figures moved laterally or sideways, or edgeways, that is about an axis perpendi- cular to the plane of the figure, the finding the centre of oscillation is difficult; because all the parts of the weight in the same hori- zontal plane, on account of their unequal dis- tances from the point of suspension, do not move with the same velocity; as is shown by Huygens, in his “‘ Horol. Oscil.” He found, in this case, the distance of the centre of oscil- lation below the axis, viz. Any parabola X its altitude. EY 8 CIC OS sala soen¥s 3 of the diameter Inarectangle susp. , p by one angle ee § 3 of the diagonal In a parabola sus ; € . 5 , I P . 3 axis + 4 param. by its vertex...... The same susp. by) mid. of base....... § 7 axis + 7 param. : 3 are X radius In a sector of a circle 4 chord : radius base? Ina CONG}. erisiys vee $ axis + —_——— 5 axis 2 rz . Tn a Sphere :..0ss0.beess g&+ =—-wherg ris the radius, and g = a + 7, the rad. added to the lengths of the thread by which it is suspended. See Gregory’s “ Mechanics,” book x1. ch. 4; also Simpson’s “‘ Fluxions,” art. 183, &c. Pan- tologia. ~Stone in his “ Fluxions,” and Emerson in his “ Mechanics,” make the centre of oscilla- tion of a cone to be at #ths of its axis from the vertex, proceeding upon the erroneous supposition, that every particle in the cone’s base moves with equal velocity; but it ap- pears, from what is done in the Gentleman’s Diary for 1805, p. 34, that the eentre of oscil- CEN lation cannot fall within the cone at all, unless the altitude be greater than the semidiameter of the cone’s base; and when the altitude and semibase are equal, the centre of the base is the centre of oscillation; but when the semidiameter of the base exceeds the alti- tude, it always falls below the base. The same conclusions may also be readily deduced from the above expression, for the centre of oscillation of a cone. Centre of Percussion, in a moving body, is that point where the percussion or stroke is the greatest, in which the whole percutient force of the body is supposed to be collected ; or about which the impetus of the parts is ba- lanced on every side, so that it may be stop- ped by an immoveable obstacle at this point, and rest on it without acting on the centre of suspension. 1. When the percutient body revolves about a fixed point, the centre of percussion is the same with the centre of oscillation; and is determined in the same manner, viz. by con- sidering the impetus of the parts as so many weights applied to an inflexible right line void of gravity ; namely, by dividing the sum of the products of the forces of the parts mul- tiplied by their distances from the point of suspension, by the sum of the forces. And therefore, what has been above shown of the centre of oscillation, will hold also of the centre of percussion, when the body revolves about a fixed point. For instance, that the centre of percussion in a cylinder is at 3 of its length from the point of suspension; or that a stick of a cylindrical figure, supposing the centre of motion at the hand, will strike the greatest blow at a point about Zds of its length from the hand. 2. But when the body moves with a parallel motion, or all its parts with the same celerity, then the centre of percussion is the same as the centre of gravity. For the momenta are the products of the weights and celerities ; and to multiply equiponderating bodies by the same velocity, is the same thing as to take equimultiples ; but the equimultiples of equipouderating bodies do also equiponderate, therefore equivalent momenta are disposed about the centre of gravity, and consequently in this case the two centres coincide, and what is shown of the one will hold in the other. CENTRE Phonic, in Acousties, is the place where the speaker stands in polysyllabical and articulate echoes. CENTRE Phonocamptic is the place or object that returns the voice. CENTRE of Position, in Mechanics, denotes a point of any body, or system of bodies, so selected, that we may properly estimate the situation and motion of the body, or sy stem, by those of this point. It is evident that, in mechanical discussions, the point, by the po- sition of which we estimate the position and distance of the whole, must be so determined that its position and distance of the whole, estimated in any direction whatever, shall be the average of the positions and distances of ’ single sail whose position and magnitudel CEN every particle of the mass estimated in fh same direction. Accordingly this will be th case, if the point be so selected that when’ plane is made to pass through it in any dire tion whatever, and perpendiculars are draw, to this plane from every particle in the bod or system, the sum of all the perpendicula on one side of this plane is equal to the su of all the perpendiculars on the other sid On this subject the reader may consult th article Position, Centre of, in the Suppleme, to the Encyclopedia Britannica. | CENTRE of Pressure, or Meta centre ofaf against a plane, is that point against whic i force being applied equal and contrary tot whole pressure, it will sustain it, so as th, the body pressed on will not incline to eith side. This is the same as the centre of pe cussion, supposing the axis of motion to be} the intersection of this plane with the surfa) of the fluid; and the centre of pressure up| a plane parallel to the horizon, or upon a plane where the pressure is uniform, is t) same as the centre of gravity of that pla | Emerson’s ‘‘ Mechanics,” prop. 91. a | CENTRE Of spontaneous Rotation, is uf point which remains at rest the instant body is struck, or about which the body \ gins to revolve. Segnes, in a short paper ( titled “ Specimen Theorie Turbunim,” - shown, that if a body of any size or form, af rotatory or gyratory motions, be left enti) to itself, it will always have three He axes of rotation, that is, all the rotatory tions by which it is effected may be consi reduced to three, which are performed rou three axes perpendicular to each other, pé-j ing through the centre of gravity, and alv) preserving the same position in absolute sp ‘ while the centre of gravity is at rest, or mu uniformly forward in a right line. This 4) ject is farther illustrated in a paper in, Mem. of Acad. of Scien. for 1761, on (@) ‘“‘ Stowage of Ships,” by Albert Euler, sol)! the celebrated Leonard Euler. The latter i also treated on the same subject i in the Ma of Berlin, for 1759; as also in his “ me Motus Corporum rigidorum.” See also lember, “ Opuscula, ” vols. i. and iv. | CENTRE Velic, Velique, or Velie Point, isi centre of gravity of an equivalent sail, or aM } such as cause it to be acted upon by the wi : when the vessel is sailing, so that the 0 tion shall be the same, as that which t place while the sails have their usual posit is Bouguer in his “ Treatise on Ships, ” yds lished in 1746, examines the best positions the masts, the extent that should be ep ay the sails, and the various motions of and pitching that may take place, accoriy to the changes of the velie point ; and practical skill, which he combined wit found theoretical knowledge, enabled bint throw such light on this subject, as, if a | : would be of great utility to the practice vigator. “Meta CENTRE. See CENTRE of Presi) CEN _ CENTRIFUGAL Force, is that by which a body revolving about a centre or about another body, has a tendency to recede from it. CENTRIPETAL Force, is that by which a body is perpetually urged towards a centre, and thereby made to revolve in a curve in- stead of a right line. See Centra Forces. CENTRO Baryco, from xev]pov, centre, and | Baupup, heavy, centre of weight, the same as the centre of gravity, which see. CENTRO BARYC Method, in Mechanics, is a method of measuring or determining the / quantity of any surface or solid, by consider- ing it as generated by motion, and multiplying the generating line, or surface, into the path of its centre of gravity, viz. _ Krery figure, whether superficial or solid, generated by the motion of a line or surface, is equal to the product of the generating magnitude into the way of its centre of gravity. This curious property of the centre of gra- ity may be demonstrated as follows: Let us suppose a lever loaded with two Weights, and a fixed point in this lever about which it revolves. It is well known that the sum of the products of each weight, by its distance from the point, is equal to the pro- duct of the sum of the weights, by the distance of the centre of gravity from the same point. Then if we consider the lever to revolve about ‘this fixed point, the circumference will be proportional to the radii, and the, sum of the products of each weight by the path or cir- cumference which it describes, will be equal to the product of the sum of the weights by the circumference described by the centre of Bravity. And the same reasoning may be ‘equally applied to any number of bodies, and the principle thus rendered universal. Hence it appears that the centre of gravity of any line or figure being given, the content of the surface or solid generated by it may be determined ; and conversely the content of a body being given, the place of its centre of vity may be found. _ For example, we have seen, under the ar- \ticle Centre of Gravity, that ina triangle this ‘centre is two-thirds dis- ve \fance from the vertex, mea- \Sured on the line which is ‘drawn from that point to the middle of the base, that isin the above figure, AG— ; i G being the cen- tre of gravity of A BC, the ns surface, the base of BC is bisected jmp. . NowasAD:AG:: Ls S.25.3,2 erefore BD: OG::3:2 ut BD=1 EC, therefore OG — $BD—1EC. zp EC = the circumference described by the point G sAgain, BC.AB =i EC.AB —area of the generating surface ABC. Hence by the Miles p EC x £EC,AB=I2EC™AB CER which is the known rule for the content of a cone. This elegant theorem was first noticed by Pappus, in the preface to the 7th book of his ‘“* Collections ;” but the Jesuit Guildimus was the first who set it in a proper point of view. Since his time it has been used by many ma- thematicians in measuring the surfaces and capacities of solids of rotation, and it may often be employed with facility when the in- tegral calculus is very difficult. For, as Leib- nitz has remarked, the method will hold, though the axis or centre be continually changed during the generative motion. CENTRUM. See Centre. CENTURY, any thing divided or ranged into periods of hundreds. Thus we say, such a century of the christian era, meaning so many hundred years since the commence- ment of that wra; and which is necessarily . one more than the number of years mentioned in the date, the present is the 19th century, which commenced on the Ist of Jan. 1801. CEPHUS, one of the old northern constel- lations. See CONSTELLATION. f CERBERUS 4Hercules, in the Britanni¢é Catalogue. See CONSTELLATION. CERATOID. See InrLection. CERES, the name given by Piazzi of Pa- Jarmo, to a planet which he discovered on the Ist of January, i801. M. Piazzi, in a brief account he has pub- lished of the discovery of this plant, states, “That having been engaged for nine years in verifying the positions of the stars as collected in the catalogues of various astronomers, he was searching, on the Ist of January, 1801, among many others, for the 87th in the cata- logue of the zodiacal stars of the Abbé de la Caille, when he observed that this star was preceded by another, which, according to his usual custom, he wished to observe also; es- pecially as it did not interrupt the principal observation. Its light was somewhat faint, and its colour resembled that of Jupiter; but Jike many others which, in regard to their magnitude, are usually placed in the eighth class. At that time no doubt arose respecting the nature of it; but on the evening of the 2d, having repeated his observations, and finding that they did not correspond either in time or zenith distance, he suspected that some error had been committed in his observations on the preceding day. He then began to enter- tain some idea that it might perhaps bea new planet. In the evening of the 3d his conjec- ture was confirmed, and he assured himself it was not a fixed star. However, before he would speak of it, he waited till the evening of the 4th, on which he had the satisfaction of finding that it had moved according to the same laws which it had observed on the pre- ceding days. At this time the motion was retrograde; but on the 10th of January it be- came direct. He continued to observe the planet till the 13th of February, when he was obliged by illness to discontinue his observa- tions. M. Piazzi then transmitted accounts CHA of his observations to several celebrated as- tronomers, in order that they might calculate the orbit of the new star, and trace out its progress in the heavens: but it eluded every search that was made for it, until December 7th, when it was re-discoyered by the assidu- ous Dr. Zach, of Saxe-Gotha; and soon after it was observed by Dr. Olbers, at Bremen; by Mechain, at Paris; by the royal astronomer, at Greenwich; by Dr. Herschel, at Slough; and by various other persons both in England and Scotland. Elements according to Burckhardt. Inclmation of orpit....... -scsansb ans 0s 10° 37’ TEES fo cchesciisaocogsens ss 0ntneraseceeaae » HS ie Ma | SEOQOGR. OF LOU Yc. ons osus5 can epee nema 2-17 19 PATINMRLOEE Sey sk eaters: oosanene en se tase sncae 10 26 69 Passage of the aphelion ) ; January 1, yen} greene 8 hours Eccentricity of orbit............ 0.0791 Mean’ distante.:\.. .-c.cas..ccs 2.7677 Revolution ..............s0s0ee000. 4.606 years Elements according to Gaiiss. Inclination of orbit ............. Os 10° 36’ 57” NOGG ves 4 .ccctencoieeeemees rithgatee ee wae a et Epoch of 1801 .......05... Net ls 2 16 28 0 Mean anomaly ..............0000 8 15 55 O Aphelion .i..iSia a ae akaeee .10 26 27 38 Eccentricity: icicegascgiastses ot 0.0825017 Equation 02.332. Gaya 9° 28! Distance ............. been eirkast ine 2.7395 Revoliition \..7Ascotasse.. Ge 16814 12% gm CETUS, the Whale, one of the old southern constellations, See CONSTELLATION. In the neck of Cetus is a remarkable star, which becomes brighter and fainter alternate- ly, owing, as is supposed, to its rotation on its ee the period of its change is about 312 ays. CEULEN (Lupotpen Van), an eminent Dutch mathematician in the early part of the 17th century, particularly celebrated for his ap- proximation towards the true circumference of the circle, which he carried to 36 places of figures; a work of immense labour, having been performed by the continual bisection of an arc. See CIRCLE, These numbers, in imitation of Archimedes, were engraven on Van Ceulen’s tomb, though it must be acknowledged, that they are rather a monument of his patient labour than of his genius. CHAIN, an instrument used in surveying, of which there are different kinds, but that which is most commonly employed for this purpose, is the Gunter chain, so called from the name of its inventor. _This chain is 4 poles, or 66 feet long, and is divided into 100 square parts or links, each link being 7.92 inches in length; 1 square chain = 10000 links = 16 poles 10 square chains = 100000 links — 160 poles =. aere. Hence we have the following easy method of converting links or chains to acres. From the number of links point off 5 figures ” b ay L, CHA to the right-hand for decimals, and those on. the left will be-acres ; multiply the decimals by 4, and point off again 5 places for decimals, and those on the leit will be roods ; ae these decimals by 40, and point off the des cimals as above, and the figures on the left will be poles, or perches, which is commonly the lowest denomination. See SURVEYING, _ CHALDRON, an English dry measure of capacity mostly used in measuring coals. The chaldron contains 36 bushels, and it weighs, about 28 ewt. By act of parliament the Newcastle chaldron is to weigh 524 ewt., | and this is to the London chaldron in the ratio” of 15 to 8, which gives 28 cwt. for the London chaldron as above stated. Ty CHANCES, an interesting branch of the modern analysis, which treats of the proba=: bility of certain events taking place, by con- templating the different ways in which they may happen or fall. (ihe The doctrine of chances is a subject of which the ancients seem to have had no idea; the discovery of it is wholly due to the mo- derns; but, like most other theories, it has’ grown into a science by such imperceptible degrees, that we can scarcely say to whom! we are indebted for the first invention. Th It seems that M. Mere, a friend of the ce=) lebrated Pascal, who was himself no mathe=¢ matician, proposed to the latter ade , about the year 1654, two problems relating to} probabilities, vz. 1. “'T'wo gamesters wanting} each a certain number of points in a game,; agree to desist from playing ; how ought the} money to be parted between them?” 2, “ Tn} how many throws may a person undertake to’ throw a ceriain number of points with two! dice?” These problems, which were quite off a novel kind, were the subject of several let ters between Pascal, Fermat, Roberval, and) other mathematicians of that day; and from) t this we may date the origin of this branch of analysis. ! af Pascal answered both these questions, bat they were not published till after his death, in) his work, entitled, “ Triangle Arithmetique,”| which contains a variety of very curious theo. rems relating to figurate numbers, combina tions, chances, &e. b ¥ ; In 1657, Huygens answered the second ¢ the above problems, and published his solution) in a small tract, entitled “ Ratiocinia de Luc ¢ Alez,” which is given by Schoolen at the ene of his work, “ Exercitationes Geometric.) inthis part Huygens proposed five problems ot, the doctrine of chances, as challenges to ma: thematicians; the novelty of which, and th celebrity of their author, soon excited parti cular attention, and several papers were now published on the subjectin the areisteae ‘ different learned societies. . LAI Dr. Halley applied this theory to the doctrine of life annuities, and gave a table of the pro_ babilities of life for every 5 years, from 1 yea) to 70 years of age. Phil. Trans. No. 196 The same application of it was also made by Hudde, and the celebrated De Witto Craig, i pe CHA cotch mathematician, applied it to the esti- ation of moral evidence, particularly as re- ting to the christian religion, in a.work en- tled, “ Philosophie Christiane Principia laithematica,’”’ London, 1699, republished at eipsic in 1755, In 1685, James Bernoulli proposed, in the Journal des Scavans de France,” two pro- lems relating to the doctrine of chances, and hich not being answered, he himself gave ie solution of them in the Leipsic Acts for (390; and afterwards undertook a work in \hich this subject is treated more at length, titled, ““ De Arte Conjectandi,” but the au- yor died before he had completed it. It was jpwever afterwards published, viz. in 1713; ad has since been republished by Baron }laseres, with copious notes and commen- sries. )In 1708, Montmort published his “ Essai nalyse sur les Jeux de Hazard’; and in "13, a second edition of the same work ap- jared, much enlarged and improved, and mtaining several letters that passed between ym and Messrs. John and Nicolas Bernoulli ji this subject. He here mentioned two jorks which appeared in the interval between 'stwo editions, the one a Latin Thesis of N. ermoulli, and the other that of De Moivre, ititled, “ De Mensura Sortis,” published )stin the Phil. Trans. for 1710. This work /ntained some reflections on the Analyse of yontmort, which called forth a reply from the yiter in his second edition; and which was jain replied to by De Moivre, in the preface y his “ Doctrine of Chances,” published in 18, republished again in 1738, and lastly in 80; of these the two latter editions may be Msidered as complete works on this subject. ‘ter the period of De Moivre’s publication, in 18, the “ Doctrine of Chances” became very pular, and several works have since been \blished either wholly or in part on this hject, viz. 1. Simpson, “ On the Nature and ws of Chance,” 1740. 2. Clark’s “ Law of jaance,” 1758. 3. A work on the same sub- 4 by Condorcet in 1781, beside occasional | ey in the works of various other authors, D’Alembert’s “ Opuscula, &e.” Dodson’s Mathematical Repository,” vol. ii. Price, iil. Trans. 1762; Waring, Phil. Trans. 1791. also, Montucla, “ Histoire des Mathema- } ues,” tom. iil. p. 380. Laws of Chance. The various circumstances 'd limitations under which events may hap- 4 8, render it impossible to reduce the laws yehance to a few determined rules and prin- tiles, as is done in various other branches of talysis; much must necessarily be left to the figment of the analyst, and no snbject re- fires more his care and attention. Pefinition 1. The probability of an event is )? fatio of the chance for its happening to all chances, both for its happening and failing. Def. 2. The expectation of an event, is the [sent valne of any sum or thing which de- Jads either on the happening or on the failing ‘such an event. Def. 3. Events are independent, when the -n. (1 —1) (n—2)...(n —m CHA happening of any one of them neither in- creases or lessens the probability of the rest. Prop. 1. Ifan event may take place in n diffe- rent ways, and each of these be equally likely to happen, the probability that it will take place in a specified way is properly represented by 1 ; d : me certainty being represented by unity; or, which is the same thing, if the value of cer- tainty be unity, the value of the expectation that the event will happen in a specified way is iy 2% For the sum of all the probabilities is cer- tainty, or unity, because the event must take place in some one of the ways, and the pro- babilities are equal, therefore each of them is -. And if the certainty be a, the value of the , j a expectation will be —. we Prop. 2. If an event happen in a ways, and fail in b ways, all being equally probable, the ; ° . . a . chance of its happening is GLE and of its | failing : . This follows from definition 1, ath Thus the probability of casting an ace with , Be ee : a single die in one throw is e’ of casting an Os uae ace or deuce is = and so on. Again, the probability of drawing an ace . 4 1 out of let k of cards is — or —. ut of a complete pack of cards is ota For there are four aces and 52 cards, or 4 chances for the event happening, and 52 for its happening and failing. Example. Let there be » balls a, b, e, d, &e. thrown promiscuously into a bag, and let it be required to find the probability of any specified number of them in being drawn, in taking out m balls. Here we must find the number of combina- tions that can be formed out of n, different | things taken m at a time; and so many ways are there for the event happening and failing. But of all these combinations only one an- swers the condition required, therefore the probability of that happening will be unity divided by the number of combinations. Now the number of such combinations is £2) inet Sone WE She Seer “ BINATIONS. 1,2.3,.4&e.m m (7 — 1) (ww — 2).. (mn —m +1) is the probability required. Suppose, for example, it is required to de- termine the probability of drawing out of the 52 cards in a pack the four aces in four draws. Here m = 4 and n = 52, whence the 1.2.3.4 ropability is 444. Prony 3) Bl. aD. 40 Hence CHA Exam. 2. Let still the number of balls be », and the number to be drawn m; to find the probability that out of the m balls p of them shall be specified ones. First, the number of combinations that can be formed out of n things taken p at a time, «2% — 1) (~—2) (n—3).. ..(n—p + 1) ‘i TST and so many ways are there in which the event may happen and fail. And the number of combinations that can be formed out of m things taken p ata time, is m (m — 1) (m — 2)...(m— p 4+ 1) bi. BoD ep and so many ways are there in which it may happen, and consequently the last expression, divided by the former, will be the probability sought; that is, onlin 1) (ram 2)...(m—p +1) n(n—1) (n—2)....@—p +1) will express the probability required. Thus for example, let a pack of cards be divided into two equal parcels required, the probability of the four aces being found in one of the parcels in particular. Here » = 52, m = 26, and p — 4, whence 26 .. 25.24.23 52.51 .50.49 Prop. 3. If two events be independent of each other, it the probability that one will the probability is happen be— ; and the probability that the other will happen be : , the probability that , nbeet | they will both happen is nig For each of the ways in which the first can happen or fail may be combined with each of the n ways in which the other can happen or fail, and thus form mn combinations, and there is only one in which both can happen, therefore the probability is we mn Cor. 1. And thus, if there be any number of independent events, the probability that they will all happen is equal to the product of all the separate probabilities ; a is, if the BF Bary | several probabilities be - sn pac oe NT then ae the probability that iiss rill all happen is mMnrs’ Cor. 2. The same has also place with re- gard to the failure of an ev at an if the pro- bability of hanfehing be — we —, &c. the n’ r probability of failing will Be m—1l n—l1 r— 1 m 4 the probability of all of them failing will be (m— 1) (n — 1) (r—1) &e. m.n.7 KC. This proposition and corollaries are of ex- tensive use and application in the doctrine of chances, as will appear in the following ex- amples. , &c.; and consequently CHA Exam. 1. Required the probability of cast ing two aces in one throw with two dice, 0 in two successive throws with one die. The probability of casting one ace with: single die is = and with the other 5 henee the probability of both aces coming uy) he ee ee | | 6 6 36 | Exam. 2. Required the probability of throw ing at least one ace with two dice. ‘Here it will be best to find the probaly of not throwing them, which is for each die ; ly an, —— and therefore that we fail in both is 2 x 3. =. and this taken from unity will be the pre bability of succeeding; that is, the Ai of throwing one ace is 1 — sae Po meters i Exam. 3. If there be 7 balls, ue E C,: thrown promiscuously into a bag, required 4 probability of drawing out in three draws tl three a, b, ec. The probability of drawing out one of 7 i the first draw, is 2 ; of drawing one of the tw n remaining ones the second time is 2 n— 1 1 of drawing out the third, the third time ise — -——— therefore the probability that these events ¢ happen is > x —. 2 x le a ae a ying Pe 3 And in the same way it will be found, th if there were m specified ones to be drawn t probability would be m m—t1 m— 2 1 ie | yes pa ee BG) a aml * 2? am But if the order in which they are to} drawn be also specified, then the probaly is expressed by 1 1 . 1 1 , ap nn n—'1 n—2* "7 —m +n The first of these cases is the same 5 exam. 1, prop.2, andis merely proposed to shi’ the dependence of the different methods. | Exam. 4. Required the probability of thro- ing an ace and then a deuce with one die. The chance of throwing an ace is e al the chance of throwing a deuce in the a trial is a3 therefore the chance of both h- 1 pening is — I 36 Exam. 5. If 6 white and 5 black balls & thrown promiscuously into a bag, what is. probability that a person will draw out firs@ white, and then a black ball? 7 he probability of drawing a white ball bt is “ and this being done, the probability! CHA ‘drawing a black ball is = or 5? because there are 5 white and 5 black balls left; therefore the probability required is— x —~ —=—. Or e probability required is— x 5 7 Or we mayreason thus; unless the person draws a white ball first, the whole is at an end ; there- fore the probability that he will have a chance of drawing a black ball is s and when he has this chance, the probability of its succeed- 1 a? ~ therefore the probability that ing is 5 or a 10 : Ca ‘both these events will take place is 7 ~~ 3 om Exam. 6. The same supposition being made, ‘what is the chance of drawing a white ball and then two black balls? The probability of drawing a white ball and 2” or aes ‘then a black one is it ; when these two are removed, there are 5 white and 4 black balls Jeft; and the probability of drawing a black ball, out of these, is 53 therefore the proba- ; Bat 4 4 bility required is — -_=-— bil y required is=— x 5 = 35 Exam. 7. Required the probability of throw- jing an ace, with a single die, in three trials. i The chance of failing the first time is 2, at ao and the chance of failing the next is 3 and the same for the third; therefore the chance & Meo ses lof failing thrice is a and the chance of not 125 _ 91 ¥ 216 216 ~Exam. 8. In how many trials may a person vundertake, for an even wager, to throw an ace with a single die? Let x be the number of trials; then, as in the last Art. the chance of failing a times to- i both times is 1— : : | gether is ( 2 ),and this by the question is equal | x , to the chance of happening, or (2) mae ; ; hence « X log. - — log. = or x X log. 5 — log. 6 ook __ log. 1 — log. 2 __ : | Jog. 1 — log. 2, and « — 28 98 = — Ne Beate: 300d + log. 5 — log. 6 log. 2 y ies >. as ite. 6 —log.8’ since log. 1 — 0; Le. = 38, nearly. \aerrop. 1. If the probability of an event a a+b to find the probability of its happening once, | twice, three times, &c. exactly, in 7 trials. happening in one trial be represented by The probability of its happening in any one 4 G HA a F } oye . icing the probability of its failing in all the other » — 1 trials is bri happening in one particular trial, and failing particular trial being ; therefore the probability of its au*" (a + 6)” trials, the probability that it will happen in some one of these, and fail in the rest, is x ; nabe—! times as great, or alae The probability in the rest, is ; and since there are n ofits happening in any two particular trials, a* §r»—2 ——_. (a +6)” — ways in which it may happen and failing in all the rest, is and there are 2. twice in 2 trials and fail in all the rest ; there- fore the probability that it will happen twice n— 1] ek ee n.-——— in n trials is In the same (a + by" manner, the probability of its happening ex- actly three times is Te Nee ee ee and the probability of its happening exactly ¢ n—1 n—2 OE ba os al vie: t (a + 6)" Cor. 1. The probability of the events failing exactly ¢ times in x trials may be shown, in the same way, to be times is n—1l n—2 n—t+1 eee & eoeoe pings cont (its * BF eae” 3 i (a + b)” Cor. 2. The probability of the events hap- pening at least ¢ times in x trials is a®*+-na*"b+n. — a”—*h* ,...tan—t + lterms (a + by For if it happen every time, or fail only once, twice, .... » — é times, it happens ¢ times; therefore the whole probability of its happening at least ¢ times, is the sum of the probabilities of its happening every time, of failing only once, twice, .... 2 —¢times; and the sum of these probabilities is a*+na*—b+n “= a2 hb? ,...tan—t + 1 terms (a + by" Exam, 1. What is the probability of throw- ing an ace, twice, at least, in three trials, with a single dic? In this case, » = 3, ¢ = 2,a=1, b=5; and 1+ 3.5:2096 a2 6.6.6 2167 27° We have been indebted to Wood’s “ Alge- bra” for several of the preceding examples, to which work the reader is referred for further application of the above principles. See also De the probability required is CHA Moivre, and the several works mentioned in the preceding part of this article. CHANGES, in Mathematics, denote the va- rious arrangements that may take place in the order or situation of a given number of things ; and is distinguished from the more general term permutations, in this; that in the latter there may be any number of things, and any number taken at a time; while in the former, the whole number is always supposed to enter. The whole number of changes that a given number of things 2 admits of, is equal to the continued product1.2.3.4.... 3 thus the number of changes of 6 things = 1.2.3.4.5.6 720 7 things = 1.2.3.4.5.6.7 = 5040 See PERMUTATION and COMBINATION. CHARACTERISTIC of a Logarithm, the same as index, or exponent. See INDEX and LoGARITHM. CHARACTERISTIC Triangle of a Curve, is the differential, or elementary right-angled trian- gle, whose three sides are, the fluxions of the absciss, ordinate and curve ; the fluxion of the curve being the hypothenuse. So if pq be parallel, and indefinitely near, to the ordinate PQ, and Qr pa- rallel to the absciss AP; then Qr is the fluxion of the absciss AP, and qr gay the fluxion of the ordinate 4 PP PQ, and Qq the fluxion of the curve AQ; hence the elementary triangle Q qr is the cha- racteristic triangle of the curve AQ, the three sides of which are x, y, 2. This term was introduced into the theory of curves by Dr. Barrow, prior to the inven- tion of the fluxional calculus; and was the means of greatly simplifying the methods de Maximis et Minimis, as then practised, and ap- proached extremely near to the general idea of the doctrine of fluxions. CHARACTERS, in Mathematics, are cer- tain symbols introduced in order to represent either quantities or operations. Algebraical CHARACTERS, are those used to denote operations, equalities, proportions, &c. See ALGEBRAICAL Definitions. Astronomical CHARACTERS, are those used to denote the Aspects, PLANETS, S1Gns, &c. See the respective articles. 3 Geometrical and Trigonometrical CHARAC- TERS. See GEOMETRY and TRIGONOMETRY. Numeral CHARACTERS, are those used to represent numbers. See the Arabian, Greek, Hebrew, and Roman numeral characters, un- der the article NOTATION. CHARGE, in Electricity, in a strict sense, denotes the accumulation of the electric mat- ter on one surface of an electric, as a pane of glass, Leyden phial, &c. whilst an equal quan- tity passes off from the opposite surface. See LEYDEN Phial. CHARGE, in Gunnery, is the quantity of powder and. ball, or shot, put into a piece of erdnance, in order to prepare it for execution. Diflerent charges of powder, with the same — qQ 4. he has found the length of the charge, pro- CHA weight of ball, produce different velocities in the ball, which are in the subduplicate ratio of the weights of powder; and when the weight of powder is the same, and the ball varied, the velocity produced is in the reciprocal sub- duplicate ratio of the weight of the ball: and thus corresponds both to theory and practice, See Dr. Hutton’s paper on Gunpowder, in the Phil. Trans. for 1778, p. 50, and his “Tracts,” vol. i. p. 266. This, however, is on a sup- position that the gun is of an indefinite length; whereas, on account of the limited length of guns, some variation from this law occurs in practice, as well as in theory; in consequence of which it appears, that the velocity of the | ball increases with the charge only to a cer- tain point, which is peculiar to each gun, | where the velocity is the greatest: and that, by farther increasing the charge, the velocity | is gradually diminished, till the bore is quite full of powder. By an easy fluxionary pro- cess it appears, that calling the length of the | bore of the gun b, the length of the charge | producing the greatest velocity ought to be 5718281898" or about 3ths of the length of the bore. But for several reasons, says Dr. Hutton, in practice, the length of the charge, producing the greatest velocity, falls short of that above mentioned ; and the more so as the ) gun is the larger. From many experiments, ducing the greatest velocity, in guns of va- rious lengths of bore, from 15 to 40 calibres, as follows: Length of Charge for Length of Bore in Calibres. greatest Velocities. 15 See reereceseowseetoe PPOROOT SHH EH SER EH HEHE Ere 3 20 eee eeeeee Soe eees COST TET Eee FEF Hee eeseetenere =e 30 eeetreeerees OCCA e eee eee FOF SHB eH SeHEeEee ee + 40 see ereeroereseres Cee ree en ePosesesreee toeretes as In practice, however, the charge which pro- duces the greatest velocity, is not that whieh; produces the greatest effect, at least in batter}, ing down the gates, &c. of fortified places, andin naval actions; for the balls, in these cases, pe- netrate and pass quite through, and therefore! communicate to the objects they strike against, only a part of their momentum. See Robins’s) “Tracts,” vol. i. p. 290; “True Principles of| Gunnery,” &e. p. 129, 266; Dr. Hutton’s) “'Tracts,” vol. i. p. 266; see also Moore’s! “Theory of Rockets and Naval Gunnery.” — CHARLES’s Wain, a name given by somet old astronomical writers to the constellation) Ursa Major. CHART, or Sea Chart, a hydrographical or sea map for the use of navigators ; being al projection of some part of the sea in plano,! showing the sea-coasts, rocks, sands, bear! ings, &c. Fournier ascribes the invention of sea charts to Henry, son of John, King offi Portugal. These charts are of various kinds, 4 the plain chart, Mercator’s or Wright’s chart,| the globular chart, &c. . In the construction of charts, great o D should be taken that the several parts of them preserve their position to one another, in the | CHA same order as on the earth; and it is probable that the finding out of proper methods to do this gave rise to the various medes of projection. There are many ways of constructing maps and charts; but they depend chiefly on two principles. First, by considering the earth as _alarge extended flat surface; and the charts made on this supposition are usually called plain charts. Secondly, by considering the earth as a sphere; and the charts made on _ this principle are sometimes called globular charts, or Mercator’s charts, or reduced charts, or projected charts. _ Plain Cuarts have the meridian, as well as the parallels of latitude, drawn parallel to each other, and the degrees of longitude and _ latitude every where equal to those at the equator. And therefore.such charts must be ' deficient in several respects. For, first, since in reality all the meridians meet in the poles, it is absurd to represent them, especially in | large charts, by parallel right lines. Secondly, as plain charts show the degrees of the seve- ral parallels as equal to those of the equator, and west must be represented much larger | than they really are. And, thirdly, in a plain | chart, while the same rhumb is kept, the ves- ‘ sel appears to sail on a great circle, which is not really the case. Yet plain charts made | for a small extent, as a few degrees in length {and breadth, may be tolerably exact, espe- | cially for any part within the torrid zone; and | éven a plain chart made for the whole of this Zone will differ but little from the truth. Mercator’s Cuart, like the plain charts, has ithe meridians represented by parallel right lines, and the degrees of the parallels, or lon- ‘gitude, every where equal to those at the equator, so that they are increased more and More, above their natural size, as they ap- 'proach towards the pole; but then the de- ' grees of the meridians, or of latitude, are in- 'ereased in the same proportion at the same part; so that the same proportion is preserved i between them as on the globe itself. This ‘chart has its name from that of the author, Girard Mercator, who first proposed it for use ‘in the year 1556, and made the first charts of this kind; though they were not altogether (on true or exact principles, nor does it appear that he perfectly understood them. Neither, ‘indeed, was the thought originally his own; ‘We mean, the thought of lengthening the de- grees of the meridian in some proportion ; for ‘that was hinted by Ptolemy, near two thou- sand years ago. It was not perfected, how- ever, till Mr. Wright first demonstrated it about the year 1590, and showed a ready way of constructing it, by enlarging the meridian ‘line by the continual addition of the secants. ‘See his “‘ Correction of Errors in Navigation,” ‘published in 1599; and the article MERIDIONAL Parts. Globular CHART, is a projection so called ‘from the conformity it bears to the globe it- ‘self; and was proposed hy Messrs. Senex, therefore the distances of places lying east. “ CHO Wilson, and Harris. This is a meridional projection, in which the parallels are equi- distant circles, having the pole for their com- mon centre, and the meridians curvilinear and inclined, so as all to meet in the pole, or common centre of the parallels. By which means the several parts of the earth have their proper proportion of magnitude, distance, and situation, nearly the same as on the globe itself; which renders it a good method for geographical maps. Hydrographical Cuarts, are sheets of large paper, on which several’parts of the land and sea are described, with their respective coasts, harbours, sounds, flats, rocks, shelves, sands, &c.; also the points of the compass, and the latitudes and longitudes of the places. Selonographic CHARTS, are particular de- scriptions of the appearances, spots, and ma- cule of the moon. Topographic CHARTS, are draughts of some small parts only of the earth, or of some par- ticular place, without regard to its relative situation, as London, York, &e. CHEYNE (GrorcGe), an English mathe- matician, in the early part of the last century. He was author of a treatise on the inverse method of fluxions, entitled ‘“ Fluxionem Methodus Inversa,” &c. 4to. 1703. De Moivre wrote some animadversions on this book, in an 8vo. vol. 1704; which were replied to by Cheyne, in 1708. CHILIAD (from yirsao, mille, a thousand), an assemblage of several things ranged by thousands. The term was particularly applied to tables of logarithms, which were at first divided into thousands. : CHILLEDRON, a solid figure of 1000 faces. CHILIAGON, a plane figure of 1000 sides and angles. CHORD (Latin, chorda, as applied to the string of a bow), in Geometry, is the E right line joining the ex- tremities of any are of a circle; such are the lines B Cc A, bse 1. A line drawn from the centre to bisect a chord, is perpendicular to the chord; or if it be perpendicular to the AWD no) chord, it bisects both the chord and the arc of _ B. the chord. 2. Chords which are equally distant from . sh the centre of a circle, are equal to each other; or if they are equal to each other, they are equally distant from the centre. 3. The chord of an are is a mean propor- tional between the diameter and versed sine of that arc. Line of Cuorpbs. SECTOR. CHoRD, or Corp, in Music, denotes the string or line, from the vibration of which the sensation of sound is excited; and by whose divisions the several degrees of time are determined, See Plane ScaLe and OT i To divide a Chord A B, in the most stmple man- ner, so as to exhibit all the original concords. CED A 8 Divide the given line into two equal parts at C; then subdivide the part C B equally in two in D, and again the part C D into two equal parts in E. Here AC to AB is an oc- tave; AC to AD, a fifth; AD to AB, a fourth; AC to AE, a greater third; AE to EB, a greater sixth; and AE to AB, aless sixth. The Length and Weight of a String being given, together with the Weight that stretches the String ; to find the Time of a single Vibra- tion. Let / be the length of the chord in feet, 1 its weight, or rather a small weight fixed to the middle and equal to that of the whole chord, and w the tension, or the weight by which the chord is stretched; then will the : . : 1] l time of one vibration be aca ——, and 7 32ZW the number of vibrations per second —— 11 : wh ”, See Dr. Hutton’s “Select Exer- cises,” prob. 21. On this subject, the reader may also consult Euler’s ‘“‘ Tentam, Nov. Theor. Mus.;” Taylor’s ‘ Methodus Incremen- torum;”’ Smith’s “‘ Harmonics ;”’ Maclaurin’s “EF luxions;’” Malcolm’s “ ‘Treatise of Music ;” and Cavallo’s ‘ Philosophy,” vol. ii. CHOROGRAPHY, the art of drawing maps of particular provinces, or districts. The word is derived from xweos, region, and yeahw, £ describe. Chorography is distinguished from geogra- phy, as the description of a particular country is from that of the whole earth. And from topography, as the description of the same country from thai of a single place, town, or district. CHROMATICS (from yxewua, colour), is that part of optics which explains the several pro- perties of the colours of light, and of natural bodies. See CoLours. CHROMATIC is also a species of music, which proceeds by semi-tones and minor-thirds ; be- ing thus denominated in consequence of the small intervals of this scale easily blending with each other, as is the case with the inter- mediate shades of colours. CHRONOLOGY (from xeovos, time, and Aoyos, discourse), is the art of measuring time, distinguishing its several constitutent parts, such as centuries, ages, years, months, weeks, &c. by appropriate marks and characters; and adjusting these parts in a methodical and orderly manner, to past transactions, by means of eras, epochs, cycles, &c.; which see, under the respective articles. See also TimE, YEAR, c. CHRONOMETER (from ypovoc, time, and Metpoy, Measure), in general, denotes any in- CIR strument or machine for measuring time ; such | as dials, clocks, watches, &c. It is, however, now more commonly used to denote a kind of © clock, so contrived as to measure very small | portions of time with great accuracy; which, — by some of those instruments, may be done to | the sixteenth part of a second. | A watch also of very accurate construction” | is commonly called a chronometer. i * CHRONOSCOPE (from xpov0s and sxorew, to see), a name sometimes applied to a pen- dulum, or machine, for measuring time. CIPHER, or CyPHer, one of our numerals, — viz. 0. This word is said to be derived from the Hebrew, 15D, saphar: but as these peoplé ; were unacquainted with the use of this sym- bol, the above derivation is very doubtful, | The Italians write this word zifra; the French, chiffre; the Latins, cipha: whence it follows | that it is more properly, in English, written cipher, than cypher. Of the great advantage | attending the introduction of the 0 into the theory of notation, see some remarks under the articles ARITHMETIC and NOTATION, bi CIRCLE, in Geometry, a plane figure bound. — ed by acurve-line, every where equally distant from a point within it, called the centre. The periphery, or circumference of a circle, is sometimes called the circle, though impro- perly so, as that name denotes the space con- | tained within the circumference, and not the circumference itself. a a Lo find the Area, or Circumference, of a Circle, | the Diameter being given. # 1. Multiply the diameter by 3°14159, and | the product will be the circumference. ye) 2. Multiply the square of the diameter by | °7854, and the product will be the area. ‘I Or general, if we put diameter = D, cir-) cumference = C, area = A, and 3:14159 = P, we have the following relations between ; those four quantities ; viz. we 4A A ¥ e seeees = AB into the pro- sed number of mal parts at the ints S, T, V, &c.; jen on one side ofthe ameter describethe ni-circles on the meters AS, AT; ‘V; and on the other side of the diameters 'V, BT, BS; so shall the parts 17, 35, 53, ‘|, be all equal, both in area and periphery. To inscribe and circumscribe circles about ‘gular figures, as Triangles, Squares, Penta- ins, Hexagons, &c. see the respective ar- , | les. Quadrature and Rectification of the Circle — iis is a problem which, of all others, have lst engaged the attention of mathematicians, least amongst the ancients; for being the lst regular and best known of any curveli- wfigure, it soon became a question to de- mine its area; and as no reason could be n why this should not be found, the greater tof the ancients seems to have attempted t solution of it, as have likewise several of ‘moderns. In the present day, however, ‘ugh the problem has not been demonstrat- ‘to be absolutely impossible, yet few trials how made, and the solution is given up ‘hopeless. But though neither the rectifi- “ton nor quadrature of the circle has been Cnpletely determined, yet approximations them have been found, and some of these ‘such a degree of accuracy, that in a circle '0se diameter is equal to that of the orbit of | | st . CIR Saturn, we should notin assigning its circum- ference differ from the truth, in the breadth of a single hair. We cannot, in the present article, enter into investigations of the several approximat- ing rules that have been given for finding the circumference and area of a circle; what the limits of the work will admit of on this head, will be found under the articles QUADRATURE and RECTIFICATION; and in the present in- stance, we must content ourselves with merely stating the several results, without entering at all into the methods made use of in deter- mining them. 1. The simplest and most ancient approxi- mation is as follows: As 7 to 22, so is the diam. to the circum. This is the ratio given by Archimedes, in his book “ De Dimensione Circuli.” Other approximations, nearer than the above, but in larger numbers, are as follows: viz. as 106: 333 113 : 355 1702 : 5347 1815 : 5702 which are each more accurate than the pre- ceding one. Vieta, in his “Universalum Inspectionum ad Canonum Mathematicorum,” published in 1579, by means of the inscribed and circum- scribed polygons of 393216 sides, has carried the ratio to 10 places of figures; showing that if the diameter of the circle be 1000, the cir- cumference will be greater than 3141°5926°535 but less than 3141'5926°537 and this ratio has been successively carried to a greater and greater degree of approxi- mation. Van Ceulen, in his work “ De Circulo et Adscriptis,” has, by the same means, carried it to 36 places of figures; which were also re- computed and confirmed by Willebrord Snell. These were still farther extended by Mr. Abraham Sharp: this indefatigable calculator having carried them to 72 places of figures, in a sheet of paper published about the year 1706, by means of the series of Dr. Halley, from the tangent of an are of 30°. They were afterwards carried to 100 places, by Machin ; and, finally, to 128 places, by De Lagny, in the Memoirs of the Academy of Sciences, at Paris, for 1719; where he shows, that if the diameter of a circle be 1, the cir- cumference will be 31415,92653,58979 ,32384,62643,38327,95028, 84197, 16939,93751 ,05820,97494,45923,07816, 40628,62089,98628,03482,53421,17067,98214, 80865, 13272,30664,70938,446 + or 447 — Beside the above approximations, a few others of a different kind may be added; such as those of Brounker, Wallis, &c. Lord Brounker found the ratio of the square of the diameter, to the area of a circle ina continued fraction; showing it to be as so is diam. to the circum. CIR 1 2 + 25 “2 +49 ey, which is the first instance of the ratio bemg * given by means of an infinite series of any kind. Wallis, in his “ Arithmetic of Infinites,” shows, that the area of the circle is to the square of its diameter, 3x3x5x5x7 x 7, &e. 2x4x4x6~x 6, &e. lt 9, 25 | 49 S: 81 ee 8: BA ae RO Which series he shows to be identical with the above, given by Brounker. 3. J. Bernoulli has shown, that if the dia- meter of a circle is denoted by 7, the 4 log. /—1 eT which is a finite but imaginary expression. Since the invention of fluxions, a great va- riety of series have been found for expressing the circumference of a circle, of which we have selected a few of the neatest; others may be seen under the articles QUADRATURE and RECTIFICATION, If the diameter of a circle be 1, and ¢ be taken to represent the circumference ; then &e. as 1 to Ke. circumference — Kr Pg ny C= hina A(t Bre as 7 Ke.) 13 ek ed 1 ¢ =. V8(I ae s-scetst 7 &e-) eavRA(I—s5+ Sage t phe) e=8 (5 + oe te) A variety of other forms might here have been given; but as we shall have occasion to enter on this subject under the article Qua- DRATURE, we have omitted them in this place, and refer the reader for further information to the above article. For an entertaining account of the different attempts made to obtain the quadrature of the circle, see the new edition of Ozanam’s “Re- creations,” vol. i. p. 358, &c.; or Montucla’s “ Histoire des Mathematiques,” vol. iv. p.619, &c. And for the more useful properties of the circle, consult Euclid’s “ Elements,” book iii. and the best authors on geometry; as Simp- son, Emerson, Hutton, Bonnycastle, Legen- dre, &e. And for several curious properties of this figure, see Stone’s ‘Mathematical Dictionary.” ‘ 3 CircLes of the higher Orders, are curves, the properties of which are expressed by the following equations: vey” 33 ys a— 2X, or yt! = a™ a—x a” sy 33" 3 a= 2)",.or y™ 3" = 2" a— x)" where ais the axis, « the absciss, and y the ordinate. Curves defined by this equation CIR will be ovals, when m is an odd number. But when m and 7 are each equal to 1, the equa. tion becomes that of the common circle. Circe of Curvature, in Geometry, that cir. cle, the curvature of which is equal to that o any curve at a certain point. It is also calle¢ the circle of equi-curyature. See Rapius o Curvature. i Circties of the Sphere, such as cut the a dane sphere, and have their circumference i its surface. They are either moveable, or fix ed. The first, are those whose peripheries are the moveable surface, and which therefo revolve with its diurnal motion; as the meri dians, &e. The latter having their peripher, in the immoveable surface, do not revolve; a. the ecliptic, equator, and its parallels, &c. 7 | The circles of the sphere are either gree or little. ; A great Circ.E of the Sphere, is that whit divides it into two equal parts or hemisphere) having the same centre and diameter with i as the horizon, meridian, &c. . A little, or less C1RcLE of the Sphere, divid¢ the sphere into two unequal parts, havin) neither the same centre nor diameter with sphere; its diameter being only some choi of the sphere less than its axis. Such as tl parallels of latitude, &c. iy Circies of Altitude. See ALMUCANTARS, Circes of Declination, are great circlesi tersecting each other in the poles of the worl Diurnal Circ es, are parallels to the equ noctial, supposed to be described by the sta, and other points of the heavens, in their apy rent diurnal rotation about the earth. It may here be observed, that most cire} of the sphere are transferred from the heave to the earth; and have thus a place in g graphy, as well asin astronomy; all the pons of each circle being conceived as let fl perpendicularly on the surface of the terr- trial globe, and hence tracing out circlesipy fectly sumilar to them. Thus the terrestl equator is a circle conceived precisely un) the equinoctial line, which is in the heaves and so of the rest. eh CircLes of Excursion, are circles paralle 0 the ecliptic, and at such a distance from it that the excursions of the planets towards | poles of the ecliptic may be included will them; usually fixed at ten degrees. Circies, horary, in Dialling, are the lik which show the hours on dials; though thy be not drawn circular, but nearly straight CIRCLE, horary, on the Globe, a brazen cle fixed to the north pole, and furnished an index, showing the difference of meridili and serving for the solution of many probla On globes of late structure, this circle iso? placed on the equator, and the index is nt to slide on a brass wire running parallel ti equator, and above it. : CircLe of ILlimination, a circle through the centre of the earth or moon, é pendicular to a line drawn from the su! the respective body. This is supposet separate the illuminated part of the # a rom the darkened part; which it does very early. Cincies of Latitude, or Secondaries of the liptic, are great circles perpendicular to the dlane of the ecliptic, passing through the poles hereof, and through every star and planet. Chey are so called, because they serve to mea- ure the latitude of the stars, which is nothing sat an arch of one of these circles intercepted vetween the star and the ecliptic. Circles of Longitude, are several less cir- es, parallel to the ecliptic ; still diminishing, n proportion as they recede from it. On the rehes of these circles the longitude of the tars is reckoned. Circe of perpetual Apparition, one of the ess circles, parallel to the equator; described ly any point of the sphere touching the north- tm point of the horizon; and carried about vith the diurnal motion. All the stars in- luded within this circle never set, but ar CircLe of perpetual Occultation, is another ircle at a like distance from the equator; and mtains all those stars which never appear in ur hemisphere. The stars situated between 1ese circles alternately rise and set at certain mes. | Polar CincLEs, are immoyeable circles, pa- allel to the equator, and at a distance from ae poles equal to the greatest declination of we ecliptic. See Arctic and ANTARCTIC. ‘Circies, Reflecting. See CircuLaR Instru- vents. Circ es, Vertical. See AZIMUTH. CIRCULAR, any thing relating to the cir- e@; as CircuLar Arcs, Instruments, Lines, farts, Ring, &c. “CircuLaR Are, any part of the circumfe- mee of a circle. - To find the Lengths of Circular Ares. | Let x represent the radius, d the diameter, lithe circumference of the circle, s the sine ) the arc, and v the versed sine of the half arc, d m its measure, in degrees, &c.; then 1. The arc = rm x ‘0174533. “cing v 3u7 ae } ai 2.3d xf 2.4.5d~ + 3.5v3 & / rae Cc. 2. The are = A 2.4.6.7 d 14 4 3Gp 4 29 ast ta 33° P2 4.5 t 6.7 &e. here g = @t and A, B, C, &c. are the Ist, 1, 3d, &c. terms. f s 3st | 25x } dick 3.3 r2 o 5.2.4 r+ + | 3.53° ' ———. &c. > or 6 '8.Thearc =4 72467 32 52 mde apd & vo Me u'” ‘toa? Vag) tego t 7°q Lyk, i gpa ~~ ver visible above the horizon. ore CIR Z : where ¢ tort and A, B, C, D, &c. the pre- ceding terms. To which, may be added the following ap- proximations: 4, The arc = adx/ ae 3q—9 Be ssssseuns 2X jody/ Pt _ + 4vdu} nearly. nearly. FES et Gn Wetsdepernperns Bernd nearly. where C’ is the chord of the whole are, and ec’ the chord of half the are. CIRCULAR Instruments, or Reflecting or Mui- tiplying Circles, are instruments which possess all the advantages of accuracy attending large instruments, in diminishing the errors of divi- sion and eccentricity at pleasure, by means of reflection, though the instruments themselves are small and portable. Circular instruments may be considered as improvements of Hadley’s octant, and the marine sexant, which are for the same pur- pose; wz. for observing the altitudes, dis- tances, &c. of the heavenly bodies, extremely useful for navigators in finding the moon’s distance, and other nautical purposes. The best instruments of this kind, are those of Mayer, Borda, and Rios; descriptions of which may be seen in Montucla’s “ Histoire des Mathematique,” tom. iii. p. 527; Swan- berg’s “ Exposition des Opérations faite en Lapponié pour la Determinations d’un Arc du Meridian,” p. 29—34; see also Mackey’s “* Navigation ;” and for a catalogue of papers relating to such instruments, see Dr. Young’s “Natural Philosophy,” vol. ii. p. 350—362; and the article CircLg, Rees’s Cyclopedia. CIRCULAR Lines, lines relating to the circle; as sines, tangents, &c. CIRCULAR Numbers, a name sometimes ap- plied to numbers whose powers terminate in the same digits as their roots, an old and use- less distinction. CircULAR Parts (Napier’s), are five parts of a right-angled, or a quadrantal spherical triangle; they are the legs, the complement of the hypothenuse, and the complements of the two oblique angles. Concerning these circular parts, Napier gave a general rule, in his “ Logarithmorum Canonis Descriptio,” which is this: “The rec- tangle, under the radius and the sine of the middle part, is equal to the rectangle under the tangents of the adjacent parts, and to the rectangle under the cosines of the opposite parts. The right angle or quadrantal side being neglected, the two sides and the com- plements of the other three natural parts are called the circular parts; as they follow each other, as it were, in a circular order. Of these, any one being fixed upon as ‘the middle part, those next it are the adjacent, and those farthest from it the opposite parts.” Lord Bacon’s “ Life of Napier,” p. 98. This ae contains within itself all the par- Oh ticular rules for the solution of right-angled spherical triangles, and they were thus brought into one general comprehensive theorem, for the sake of the memory; as thus, by charging » the memory with this one rule alone, all the «ases of right-angled spherical triangles may be resolved, and those of oblique ones also, by letting fall a perpendicular, excepting the two cases in which there are given either the three sides, or the three angles. If the reader attend to the circumstance of the second let- ters of the words, tangents, and cosines, being the same with the first of the words adjacent and opposite, he will find it almost impossible to forget the rule. And the rule for the solu- tion of the two cases of spherical triangles, for which the former of itself is insufficient, may be thus expressed: of the circular parts of an oblique spherical triangle, the rectangle under the tangents of half the sum and half the difference of the segments at the middle part (formed by a perpendicular drawn from an angle to the opposite side), is equal to the rectangle under the tangents of half the sum and half the difference of the opposite parts. By the circular parts of an oblique spherical triangle, are meant its three sides and_ the supplements of its three angles. Any of these six being assumed as a middle part, the oppo- site parts are those two of the same denomi- nation with it; that is, if the middle part is one of the sides, the opposite parts are the other two, and if the middle part is the sup- plement of one of the angles, the opposite parts are the supplements of the other two. Mr. Walicr Fisher has given, in the Trans- actions of the Royal Society of Edinburgh, rules somewhat analogous to these of Napier’s, which will serve for the solution of all the cases of plane and spherical triangles. “A principal recommendation of these rules is, that they clear trigonometry of all am- biguity, so far as it can be effected, and that they are particularly useful in the solution of spherical oblique triangles. To apply them to plane triangles, instead of the sine or tangent of a side, take the side itself. That they may be more easily remembered, the following words, formed from the abbreviation of the terms.of the properties, should be committed to memory ; sao satom, tao, sarsalm.” THEOR. i. Given two parts and an opposite one. S.A:8.Q::S.a:8.0 TuHeor. 2. An included part given or sought. ) A 29° A+ @ : Oo—0 ,.™M Ss. 5 3 We 5) a0 Me 5 {P.> THEOR. 3. A—a Ata , O—O0 O+0 T. - i. ay he at ke avy: 2 : oe 2 Teor. 4. Given the three sides or angles of an-oblique-angied triangle. aia ea bri S.AxS.a: R18 ATE 3 Ste ts Me “M denotes the middle part of the triangle, and must always be assumed between two Ga given parts. It is either a side or the supple ment of an angle, and is sometimes given sometimes not. “A and a are the two parts adjacent to th middle, and of a different denomination from ity “© and 0 denote the two parts opposite ta the adjacent parts, and of the same denomi- nation with the middle part. “f “‘Zis the last or most distant part, and of @ different denomination from the middie part.” CircuLar Ring, is the measure or space included between two concentric circles, and the area of it is consequently equal to the dif ference of the areas of the two circles. Or the area may be found thus: Multiply the sum 0} the two diameters by the difference of them; and that product again by °7854, which wil give the area required. ‘pel CrrcuLar Sailing, the method of naviga’ ing a ship upon a great circle of the globe, See SAILING. zl CIRCULAR Sectors, are the areas boundec by any are of a circle and two radii; the mea) sure of which is found as follows: " Let / represent the length of the are of th sector, and m its measure, in degrees, mi nutes, &c.; then “te 1. Area of sector = irl _ Pe, Oy) PF OUNCE ee Se Pe BOE re 360 | CIRCULAR Segments, is the space bounde: by any are and its chord; the area of whic! may be found by the following formula: — | If A’ represent the area of the circular se¢ tor, and C’ the chord of the arc, then 7 1. Area of segment = A’ — £C (r — v) f,) 2s ae 2vu/dv X vem 33d? —_ — &e. t or ‘A 2. Area = . ‘ 3y 5v | 2 ! 2 — —— A — ——a yo hd peed a, a __ 737 q, _ 9.5v pt | 9.6d 11.8d A,.B, C, &c. being the preceding terms. Area’ = 40 UO 4 3. Area fvvyv +7 A 7von 3v Dv — —- cc. W = (d— »). ay © iy 2 &c. where V = ( v). | 4, Area = 2re" — +5 gA— ba q i = q? C, &e. where ¢” represents the cosine uM half the are, and q =<. A, B, C, &e. bei the preceding terms. a ‘To which, may be added the following aj proximations ; viz. 5. Avea=$} v(dv—v*) +3 vdv | neat) 6. Area = 4v (dv — 30”) nearly. an If C’ be made to represent the chord of #1 whole are, and ec the chord of half the at then | 7. Area = 4,u (C’ + 4c") nearly. CIR 8. Area = tu (v(4C? + 40?) nearly. 9. Area = d* x by tabular n° answering to in the table of circular segments. Note. ‘The area of circular zones will be d by finding the difference of the two seg- ‘ents. And the area of circular rings, by iding the difference of the areas of the two rcles. Or by making D and d the diameters, n 10. Area of the ring = (D + d) (D — d) | 7854. Hutton’s “ Mensuration.” CIRCULATING Decimals, or Recurring ecimals, are those that consist of a repetition a small number of digits, as "646464, &c. 27127127, &e.; in fact, every decimal that ‘not finite, is a circulating decimal, or is ch, that if continued far enough the same ‘ures will again recur; but it is only those, which the periods of circulation consist of few figures, that receive generally the defi- tion of circulating decimals. hen the circulation consists of the same sit repeated, it is called a Simple Circulate, d is distinguished by a point placed over it; 111; Se.) °333. 3, &c..,, When 2 period of circulation consists of more than e digit, it is called a Compound Circulate, dis distinguished thus, °234234234, &c. = $4. A Mixed Circulate is that which has ner figures in it that are not repeated, as 348484, &c.; and these are represented thus, 34. All operations, as multiplication, division, . of these numbers, may be performed by 2same rules as common decimals; and in it, there are very few cases in which those ‘es are not to be preferred, though some thors, as Brown, Cunn, Malcolm, Emerson, mn, and particularly Henry Clarke, have lated at considerable length the theory of culating decimals; as also Dr. Wallis, who ms to have been the first author that distin- ished them under a separate head. | REDUCTION OF CIRCULATING DECIMALS. . e.1. To reduce a Simple or Compound Cir- _, ulate to its equivalent Fraction. Rule. Take the figures in the given deci- il, considered as a whole number, for the merator; aud as many 9’s as there are wes in the circulate for the denominator ; serving only, that when there are any in- jral figures in the circulate, as many ciphers st be annexed to the numerator, as the thest place in the repetend is distant from i point; thus, EXAMPLES. imeanhe circulate; “6— § = #2 ; of — 36 — 4. PP eee wearer seeeese 36 od 09 —_— ee Ba Peer erene. cesereees ‘O9 aa wil Mars Tr (B Peteeservees teteree 2.063 pars 2335 chats 2+ 8e2. To reduce a Mixed Circulate to its equi- valent Fraction. Rule. Subtract the finite part of the expres- Q, Considered as a whole number, from the | CIR whole mixed repetend, taken in the same man- ner for the numerator; and to as many 9’s as there are repeating places in the circulate, annex as many ciphers as there are finite decimal places for a denominator; thus, EXAMPLES. 1. The circulate 138 == ns = $45 =e . ae vee 2. Again... 2°418 = awa = 2414 — 223 And so on in other examples. Having thus shown how to reduce any cir- culate, simple, compound, or mixed, to its equivalent fraction; all the rules of addition, subtraction, multiplication, division, &c. of circulating decimals, may be performed by the rules that are given for the same: opera- tions in fractions; which see, under their respective heads. CIRCUMFERENCE, CircumMrereENTIA, in a general sense, denotes the line, or lines, bounding any figure ; but it is commonly used, in a more limited sense, to denote the peri- phery of a circle, which is every where equally distant from a certain point within, called the centre. See CIRCLE. CIRCUMFERENTOR, aninstrument used by surveyors in taking angles; it consists of a brass circle and index, in one piece, commonly about seven inches in diameter, and index about fourteen inches long, and one and a half inches broad. On the circle is a card or com- pass, divided into 360 degrees; the meridian line of which answers to the middle of the breadth of the index. There is also soldered on the circumference a brass ring, on which screws another ring with a flat glass in it, so as to form a kind of box for the needle, sus- pended on the needle in the centre of the cir- cle. There are also two sights to screw on, and slide up and down the index, as also a ball and socket screwed on the under side of the circle, to receive the leg of the three- legged staff. The above description answers to the most simple form of the circumferator; byt an im- proved instrument of this kind is sold by Jones, of Holborn, which in some measure answers the purpose of a theodolite. CIRCUMGY RATION, the whirling mo- tion of a body about a centre. CIRCUM-Polar Stars, are those situated near the north pole of the heavens, or those which revolve about it without setting. CIRCUMSCRIBED Figure, is that which is circumscribed or drawn about another fi- gure, so as to touch it on every side. A right-lined figure is said to be circum- scribed about another, when all the angles of the latter fall in the sides of the former. A right-lined figure circumscribes a curve- linear one, when the periphery of the latter touches all the sides of the former, A circle, or other curvilinear figure, cir- cumscribes a right-lined one, when all the angles of the latter are in the periphery of the former. M 2 CL A. _ CIRCUMSCRIBED Hyperbola, one of Newton's hyperbolas, of the second order, which cuts its asymptotes, and contains the parts cut off within itself. CISSOID of Dvocles (from the Greek xiccodor, probably from its branching figure), in the higher Geometry, is a curve line of the second order, invented by Diocles, an ancient treek geometrician, for the purpose of finding two continued meay proportionals between two other given lines. The generation of this curve is as follows: At the extremity B of the diameter AB of the circle AOB draw perpendicular to it the indefinite line C BD, to which from the other extremity A draw several lines cutting the cir- cle in I, O, N, &e. and upon these lines set off the corresponding equal distances, viz. HM =SAL«FO— AO, CLAN, &cra cen the curve line drawn through all the points M, O, L, &e. is the cissoid. Other methods of constructing this curve may be seen in Newton’s “ Universal Arithmetic,” and Emer- son on Curve Lines. The cissoid has the following properties: i. The curve has two infinite legs AMOL, Amol, meeting in a cusp A, and tending con- tinually towards the indefinite line CBD, which is their common asymptote. 2. The curve passes through O and 0, points in the circle equally distant from A and B; or it bisects each semi-circle. 3. Letting fall per- pendiculars MP, IK, from any corresponding points I, M; then is AP = BK, and AM = HI, because AI = MH. 4 AP: PB:: MP?: AP*. So that if the diameter A B be —a, the absciss AP = x, and the ordinate PM=y; then is x: a—wa:: y?:2*, or 3 =y”. (a — x), which is the equation of the curve. 5. The whole infinitely long cissoidal space, contained between the infinite asymptote BCR and the curves LOAol, &e. of the cissoid, is equal to triple the generating circle AOBoA. 6. All cissoids are similar figures. See other properties of this curve, in the works of Dr. Wallis, vol. i. p. 545. CIVIL Day, Month, Year. See the articles Day, Month, YEAR. CLAIRAULT (Atexts-CLaupe), a cele- brated French mathematician, born at Paris, May 13, 1713, and died May 17, 1765, aged fifty-two. He was author of the following works: i 1. On Curves of a Double Curvature. 1730, “« Oi 2. Elements of Geometry. 1741, 8vo. p 3. Theory of the Figure of the Earth. 1743, vO. GG E 4, Elements of Algebra. 1746, 8vo. — ~ 5. Tables of the Moon. 1754, 8vo. Besides a great number of papers in th Memoirs of the Academy of Sciences, fror the year 1727 to 1762. + | CLAVIUS (CuHristoPHER), a German Jé suit, born at Bamberg, excelled in the know ledge of mathematics, and was one of th! chief persons employed to rectify the caler! dar; the defence of which he also undertoo against those who censured it, especially Sed liger. He died at Rome, in 1612, aged seve i five. His works have been printed in fiy) volumes, folio; the principal of which is hj} “Commentary on Euclid’s Elements.” CLEOSTRATUS, a celebrated astron( mer, born in Tenedos, was, according to Plin) the first who proposed the signs of the zodiay others say, that he only invented the sigi Aries and Sagittarius. He also corrected th errors of the Grecian year, about the 306 before Christ. ae CLEPSYDRA, an instrument or machi! serving to measure time by the fall of a ce) tain quantity of water. The word comes fro xre@lw, condo, vdwp, aqua, water; though the} have likewise been clepsydre made with me cury. The Egyptians, by this machine, mej sured the course of the sun. Tycho Brabj in later days, made use of it to measure tf] motion of the stars, &c.; and Dudley used tf same contrivance in making all his maritin observations. The use of clepsydre is ve} ancient; they were invented in Egypt, und the Ptolemies; as were also sun-dials. Thy use was chiefly in the winter, as the sun-dii) served in the summer: but they had two gré defects; the one, that the water ran out wi a greater or less facility, as the air was mc or less dense; the other, that the water? more readily at the beginning than towai the conclusion. a The Construction of a common Clepsydra} To divide any cylindrical vessel into parts,) be emptied in each division of time, the Hi) wherein the whole, and that wherein any ph is to be evacuated, being given. Supposi cylindrical vessel, whose charge of wal flows out in twelve hours, were required} be divided into two parts, to be evacua! each hour. 1. As the part of time 1 is to € whole time 12, so is the same time 12Tt fourth proportional 144. 2. Divide the @ tude of the vessel into 144 equal parts: bi the last will fall to the last hour; the three ns above to the last part but one; the five nf to the tenth hour; lastly, the twenty-th last to the first hour. For since the times® crease in the series of the natural numlh 1,2,3,4,5, &c. and the altitudes, if the nu? ration be in a retrograde order from the twel! hour, increase in the series of the unequal bers 1,3,5,7,9, &c. the altitudes computed i the twelfth hour will be as the squares of times 1, 4, 9; 16, 25, &c. Therefore the squie of the whole time, 144, comprehend all 1 parts of the altitude of the vessel to be ‘a cuated. But a third proportional to 1 an? | ! CLO the square of 12, and consequently it is the umber of equal parts in which the altitude to be divided, to be distributed according the series of the unequal numbers, through e equal interval of hours. There were many ads of clepsydrae among the ancients; but ey all had this in common, that the water a generally through a narrow passage, from e vessel to another, and in the lower was a see of cork or light wood, which, as the ssel filled, rose up by degrees, and showed e hour. Clepsydre have been much improved of @ years in their construction; but as their e is now superseded by the accuracy of our ydern time-pieces, we shall not dwell longer on them here, but shall refer to vols. i. and 1 of the machines approved by the French vademy, and to vol. xliv. of the Philoso- ical Transactions, for descriptions of the st instrument of this kind with which we 2 acquainted. f \CLIMATE, or Ciime, in the ancient Geo- phy, a part of the surface of the earth, or ae; bounded by two lesser circles parallel the equator; and of such a breadth, as that » longest day in the parallel nearer the pole yeeeds the longest day in that next the equa- i, by some certain space, as half an hour, van hour. Vulgarly, the term climate is bestowed on y country or region differing from another, Laer in respect of the seasons, the quality ithe soil, or even the manners of the inha- yants, without any regard to the length of i longest day. LOCK, a well-known instrument for mea- ving time; it is regulated by means of ‘a ULUM, the laws of which will be found tier that article. ‘The nature of this work does not require t we should enter into any particular de- siption of the mechanism of this instrument; \ ‘Shall, therefore, merely refer the reader, \0 is desirous of information on this subject, those works which have been written pro- \Sedly on the theory and practice of horo- Y; of which a very numerous list is given Gregory’s “ Mechanics,” vol. ii. p. 140. ILOUD, a visible aggregate of minute drops (water suspended in the atmosphere. The word is probably derived from the )glo-Saxon, Lehlod, covered, hidden; the face iaeaven being so in those parts where clouds year. The same aggregate, which in this ‘tation is called a cloud, obtains the name mist, when seen to arise from the earth or ‘fers ; and fog, when it envelopes and covers Observer. Yet the two latter, viewed na greater distance or elevaiion, present ate appearances of clouds; while those, in ir turn, become mists and fogs, in propor- 1 aS we approach and penetrate them. a) i 4 | bis, that the particles of which a cloud con- )8; ave always more or less clectrified; and His fluid has hence been considered as the tis concluded, from numerous observa-. COE cause of the formation of all clouds whatever, whether of thunder, hail, rain, or snow. The hypotheses which assumes the exist- ence of vesicular vapour, and makes the par- ticles of clouds to be hollow spheres, which unite and descend in rain when ruptured, however sanctioned by the authority of seve- ral eminent philosophers, does not seem ne- cessary to the science of meteorology in its present state; it being evident that the buoy- ancy of the particles is not more perfect than it ought to be, if we regard them as mere drops of water. In fact, they always descend, and the water is elevated again only by being converted into invisible vapour. COASTING is that part of navigation which is carried on near the coast or shore, without losing sight of land, except occasionally for a short period. COCHLEA, the same as screw; being thus called in consequence of its resemblance to the spiral shell of a snail, called by the Latins, cochlea. CO-EFFICIENTS, in Algebra, are num- bers or letters prefixed to other letters, or un- Known quantities, into which they are sup- posed to be multiplied; and therefore, with such letters, or the quantities represented by them, making a product, or co-efficient pro- duct; whence the name, which was given to them by Vieta. When a quantity stands alone, without be- ing preceded by any number or letter, it is always supposed to have unity for its co-effi- cients: thus, the co-eflicients of the terms z, ax’, by?, &e. are 1, a, b, &e. In equations the absolute term is some- times classed under the general term co-effi- cient, in which it is supposed to be prefixed to x° or y°, &c. all such quantities being equal _ to unity: thus, in the formula 23 + ax? + ex + d, the co-efficients are 1, a, c, and d; the first 1 being understood, and the last being supposed to precede 2° = 1. In equations we have the following remark- able property of the co-eflicients; viz. 1, 'The co-efficient of the second is equal to the sum of the roots of the equation with their signs changed. 2, The co-efficient of the third term is equal to the sum of all the product, taken two and two. The co-efficient of the fourth term is equal to the sum of all the pro- duct, taken three and three together, with their signs changed, and so on; and, finally, the absolute term is equal to the product of all the roots, with their signs changed, if the number of terms be even, but without being changed, if the number of terms be odd; this term being here supposed to stand on the left- hand side of the equation. Thus, for example, in the cubic equation, a3 + ax* + bx + ¢e=0; supposing the roots to be p, g, and r, we shall have a=—(p +q +7) b= pat+pr+aqr c=—pqr »° and the same for every order of equations. ® COH The sum of all the co-efficients of the bi- nomial (a + a)" = 2”, and of (a— 2)" —0"—0; that is, the sum of the positive co-efficients is equal to the sum of the negative ones; and conseguently their sum is equal to zero. See BrnomiaL Theorem. COFFER Dam, a term applied by engi- neers to denote the enclosures formed for lay- ing the foundation of piers and other works in water, to exclude the surrounding fluid, and thus forming a protection both to the work and workmen. COHESION, that species of attraction which, uniting particle to particle, retains to- gether the component parts of the same mass ; being thus distinguished from adhesion, or that species of attraction which takes place between the surfaces of similar or dissimilar bodies. Whatever may be the cause of cohesion, its effects are evident and certain, and the diffe- rent degrees of it is what constitute bodies of different forms and properties: at least, this is the opinion of Newton, and other eminent philosophers. Various experiments have been made by different authors to ascertain the power of co- hesion, in bodies of different textures; as woods, metals, glass, &c. These experiments have generally been made on cylindric bars, an inch, or certain parts of an inch in diame- ter; and as it has been proved that the direct cohesive strength of a body is in the joint ratio of its primitive elasticity, toughness, and area of its section, it follows that, having as- certained the cohesive power with any given section, the same may be found for the same substance for any other section; and thus the experiments of different authors, compared with each other. The following table shows the weights which were necessary to tear asunder rods of diffe- rent substances, whose bases were each a square inch, the weights being applied in the direction of their length. METALS. lbs. Avoirdupoise. 1 TO) aR RN ear ine GRE .. 135000 ME els 5 6 68 PN aa) Site beep 74000 SOMALI cc Ras5, sch susxp'socsphiseknn 50100 PEDO UVOE DASE wcsnpaaeseesdesigens his 28600 oy ET ARMT TA Fi ge ORO 41500 RRM MILD on 5 fda sav cna ts xan'ess 22000 PUM LEU oa pinet peck sae nnaxe 4000 RS MMERTEIN FC os on a dons | dean vaccone 2900 YAY eee aks tet ha's ie cca i 2600 PONTE bia ven tisee ears Altre 5; 1000 RM RE Mavens ta rise pw se sip g'te 860 WOODS, Ibs. Avoirdupoise. Pea Dalen §. Aiidetts iesvin vides 17300 Aldet \c. 355 ae yO ee ee? 13900 Elm ..6..0.00. ihe {done Sin wtcacds de das gene LORUO DCS i on Mes cakivavexevnie edo toce 12500 PW TT OW eythane. oes 12500 PB roc uiec ons > AAV pee 12600 PUM Ake ois sa ee .... 11800 Per Reif iy hte hiots deed wee 10000 LS FFP eee Bape” sph tp oitis shh OEE _ to the want or decrement of the other; sot y -— COL Walhut 2s SANRIO SS ae + Pitch, Pine ...... ptaSi deta aemant « 7656 Cypress CO AR eee 6000, Poplar....... ARS oa ALE My RR 5500 | OF) LARA SEP COP BEE a tb Retake 4880 | Other experiments have been made to a certain the strength of cohesion in bodie when placed horizontally, and loaded wif weights in different parts; but as we sha have occasion to return to this subject ai the article STRENGTH of Materials, we shi in this place merely state the result of Mu chenbroek’s experiments, on a few of the mo) common woods. For this purpose he fix pieces of wood, by one end, into a squa, hole in a metal plate, and hung weights t wards the other end, till they broke at f hole. The weights, and their distances fro the point of support, are shown in the follo ing table. | Weight in ox WOODS. Dist. in Inches. Pine WAS DE Poi vie Redes 365 Rar beste Wo, RIE Os. AE 40 Beechyeut.uekts YEP SST fet F 56} Bland a. Az DO) hie aeene 44 Oeeieia Soi ce aks «- 48 5 ldéritn ncauaiit i DF Nec hed ae In the above table the rods were rectang- lar parallelopipedons, and the side of square section .26 of an inch. Coulomb found the lateral cohesion of bré and stone only =; more than the direct co sion; which, for stone, was 215\b. for a squi inch; for good brick, from 280 to 300. Count Rumford found the cohesive streni of a cylinder of iron, an inch in diamef 63466, or 631731b.; the mean, 63320; whl is only 54, more than Emerson’s result. Sickingen makes the comparative cohes) strength of gold, 150955; of silver, 190771) platina, 262361; of copper, 304696; of f iron, 362927; of hard iron, 559880. Guyi makes platina a little stronger. ! In Buffon’s experiments, b, d, and J, beg the breadth, depth, and length of a bean) oak, in inches; the weight which broke ii (= — 10). See I pounds was = bd* ther the article STRENGTH. Vor farther information on this subjee reader may consult Ritter on Cohesion, # bert’s Journal, iv. 1; Benzenberg on Cohe ! Gilbert, xvi. 76; Fontana on Solidity i! Fluidity, Soc. Ital. 1.89; and Dr. T. Young the Cohesion of Fluids, in the Phil. Trans} 1805, or in the second volume of his ‘“ Nat‘ Philosophy.” COLD, in common language, denotes! sensation which is felt, or the effect whii! produced, by the abstraction of heat; thi! heat and cold are opposite to each other,! the existence or increment of the one, is el the same degree of temperature may be ei hot or cold, according as it is compared 4 COL a colder or hotter temperature. Thus the cli- mate of Great Britain is a cold climate, in comparison with that of the West India islands; and a hot climate, in comparison with that of Siberia. If a man warms one of his ‘hands near a fire, whilst he cools his other ‘hand by means of ice; and if afterwards he ‘plunges both his hands in a bason of water of ‘the common temperature of the atmosphere, ‘that water will feel cold to the hand that has Men heated, and hot to the other hand. From this, it appears that cold is not any thing real, but merely a privation of heat; so that instead of saying that a body has been cooled to a bertain degree, it may with equal truth and ‘oropriety be said that the body has been de- mived of heat to that certain degree. | Notwithstanding the simplicity of this theory, and the conviction which seems to accom- any it, philosophers have often entertained loubts concerning it; and they have endea- woured to inquire into the real state of the natter, by devising experiments capable of lemonstrating whether the cause of heat was ny thing real, and that of cold only a priva- jon or diminution of the former; or, v2se versa, whether the cause of cold was any thing real, and that of heat a diminution; or, lastly, whe- ther the production of heat, and the produc- ‘ion of cold, were not owing to two distinct brinciples or elements. On the supposition that the cause of one of those effects only is ‘eal, it is much more natural to suppose that he cause of heat is the real principle or ele- nent; since its effects, viz. enlargement of he bulk of bodies, the separation of their yarts, &c. are such as must be produced by he introduction of something real; and the hstraction of this principle may naturally woduce the effects of cold, such as contrac- ‘jon of the bulk of bodies, agglutination, &c. ; whereas it would be unnatural to suppose hat a body contracts its bulk, as its parts ‘ome into closer contact, because something Ase has been introduced amongst them. With respect to the last supposition, wz. whe- her the effects of heat and those of cold be ‘tot owing to two distinct principles, a few rguments, and the equivocal result of a few ‘-xperiments, have, at times, been adduced in upport of it. But the general and preyailing /pinion among philosophers is, that a single lement, called caloric, produces heat, or the ‘fleets of expanding bodies separating their skew &e.; and that cold is only a relative ‘/xpression ; that is, meaning only the decre- ‘nent of heat; so that real or absolute cold onsists only in the total abstraction of calo- ‘ie; and, that such a point, viz. the zero of eat may be determined, has been shown by ‘he experiments, the discoveries, and the cal- eons of some late eminent philosophers, iz. Irvine, Black, Crawford, and others. It 3 impossible, in this plaee, to enter into an avestigation of the metheds made use of in letermining this remarkable point; we must, herefore, on this head, refer the reader to the works of the above-mentioned authors, or to EF COL the article CoLp, Rees’s Cyclopedia; where the whole is luminously investigated and illus- trated. COLLIMATION (from ecollimo, aim), Line of, on a telescope, is a line passing through the intersection of those wires that are fixed in the focus, and the centre of the object glass. COLLINS (Joun), an English mathemati- clan, was born near Oxford, in 1624. He first published, in 1652, an ‘“ Introduction to Mer- chants’ Accounts,” in folio, which was re- printed, with additions, in 1665. 'This work was followed by others on practical mathe- matics; and some papers in the ‘Transactions of the Royal Society, of which he was chosen a member. He was also made accomptant to the Royal Fishery Company; and wrote seve- ral commercial tracts. He died in November, 1683. Mr. Collins was a very useful man to the sciences, keeping up a constant correspon- dence with the most learned men, both at home and abroad, and promoting the publica- tion of many valuable works, which, but for him, would uever have been seen by the pub- lie; particularly Dr. Barrow’s ‘“ Optical and Geometrical Lectures ;” his “ Abridgment of the Works of Archimedes, Apollonius, and Theodosius ;” Branker’s “Translation of Rhronius’s Algebra,” with Dr. Pell’s ‘‘ Addi- tions,” &ce.; which were procured by his fre- quent solicitations. COLLISION (from colliso, dashing together), in Mechanics and Physics, is the meeting and mutual striking of two or more bodies, one of which, at least, is in motion. The most simple of the problems relating to collision was that of a body proceeding to strike against another at rest, or moving before it with less velocity, or approaching towards it. Des Cartes, misled by his metaphysical principles, which had in-. duced him to suppose that the same quantity of absolute motion always exists in the world, concluded that the sum of the motions after the impact was equal to the sum of the mo- tions before it. But the proposition is truc only in the first and second of these cases: it is false when the two bodies meet each other ; for in that case, the sum of the motions after the impact is equal to the difference of their motions before it, not to their sum. Thus Des Cartes discovered only part of the truth. In 1661, Huygens, Wallis, and Sir Christopher Wren, all discovered the true laws of per- cussion separately, and without any commu- nication with each other, as has been. com- pletely proved. ‘The basis of their solutions is, that in the mutual percussion of several bodies, the absolute quantity of motion of the centre of gravity is the same after. as before the shock. Farther, when the bodies are elas- tic, the relative velocity is the same after as before percussion. All this, however, is upon the supposition that bodies are either per- fectly hard, or perfectly elastic; but as there do not exist in nature any bodies which we know of either the one or the ether of these COL kinds, the usual theories are of little or no service in practical mechanics. Yet the laws of collision, under the above hypotheses, being demonstratively deduced from the general laws of motion, which are justly assumed as axioms, determine the velocity that would take place if those suppositions were real: and it is afterwards more easy and practi- cable to estimate, and to allow for those de- viations of practical results from the abstract theory, which arise from the imperfect hard- ness or elasticity of bodies, from resistances of mediums, frictions, &c. We cannot, however, even under the above simplification of the theory, enter at any length into the investigation of the several different cases under which collision may take place ; we must, therefore, limit ourselves to the state- ments of the results of some of the most po- pular propositions, and refer the reader who. wishes for farther information to the several treatises on mechanics, in which this subject is investigated; such are the mechanics of Emerson, “Wood, Gregory, &c. 1. Ifthe impact of two perfectly hard bodies be direct, they will, after impact, either remain at rest, or move on uniformly together with different velocities, according to the cireum- stances under which they met. o——_—_—_—__—__¢,-—_—_ B Z Cc Let B and 6 represent two perfectly hard bodies, and let the velocity of B be represented by V, and that of b by », which may be taken either positive, negative according as 6 moves in the same direction as B, or contrary to that direction, and it will be zero when 6 is at rest. This notation being understood, all the cir- cumstance of the motion of the two bodies, after collision, will be expressed by the for- BV +b» B+6’ which being accommodated to the three cir- cumstances under which » may enter become mula velocity = _BV + 6» § when both bodi d velocity —2Y + 5? § w ies move a RORY B+6 ; in the same direction . BV — bv ¢ whenthe bodiesmovedin perscntye= B+ } contrary directions -_. BV__ ¢ when the body 6 was at velocity =——_—_ Bub ; boats These formula arise from the supposition of the bodies being perfectly hard, and conse- quently that the two after impact move on - uniformly together as one mass. In cases of perfectly clastic bodies, other formule have place which express the motion of each body separately; as in the following proposition. If the impact of two perfectly elastic bodies be direct, their relative velocities will be the same both before and after impact, or they will recede from each other with the same ve- igcity with which they met; that is, they will _ be equally distant, in equal times, both before and after their collision, although ‘the absolute velocity of each may be changed. The cir- cumstances attending this change of motion on Coy in the two bodies, using the above notation, are expressed in the two following formule :_ 2b6v+(B—b) V 7 B ao b y" 2BV+(B—b) v 2 B+6 which needs no modification, when the motion of b is inthe same direction with that of B. In the other-case of b’s motion, the senerab formulas become —2bv + (B—))V the velocity of B the velocity of 5 the velocity of B ” the velocity of b when 5 moves in a contrary direction to that’ of B, which arise from taking v negative. And Se the velocity of B a Pan) the velocity of b ‘iy when d was at rest before impact, thatis, when v=o. Ifa perfectly hard body B, impinge obliquely. upon a perfectly hard and immoveable plane, AD, it will after collision move along the, plane in the direction CA. And its velocity before impact 4 Is to its velocity after impact ‘ As radius ( ) Is to the cosine of the angle ewan | plane in the direc- tion CE, with the But if the body " be elastic it will rebound from the a a i same velocity, and . at the same angle with which it met it, that is, the angle AC E will be equal to the nat BCD. COLOUR, in natural philosophy, that pro perty of bodies which affects the sight only, or that property possessed by the elementary, rays of light, separated by any means what, ever, of exciting in us different sensations ac cording to their differentrefrangibility. Thy colour may be considered in two respeets, a it regards bodies in general, and as it is Pia duced by solar light. 4 Before the time of Newton, the ideas con cerning colour were very vague and unsatisface tory. The Pythagoreans called colour the su perficies of bodies: Plato said thatit was aflami issuing from them: according to Zeno, it wa the first configuration of matter: and Aris: totle said it was that which made bodies ac tually transparent. Des Cartes accounte colour a modification of light, and he imagine that the difference of colour proceeded from th’ prevalence of the direct or rotatory motion ¢ the particles of light. «Grimaldi, Dechales and many others, imagined that the difference of colour depended upon the quick or slov_ vibrations of a certain elastic medium wi which the universe is filled. Rohault con ceived, that the different colours were mad by the rays of light entering the eyeat differer - COL ngles with respect to the optic axis. And jr. Hooke imagined that colour is caused by 1¢ sensation of the oblique or uneven pulses flight; which being capable of no more than wo varieties, he concluded there could be no nore than two primary colours. | Philosophers were also formerly of opinion, ‘nat the solar light was simple and uniform, vithout any difference or varicty in its parts, nd that the different colours of objects were aade by refraction, reflection, or shadows. jut Newton taught them the errors of their ormer opinions; he taught them to dissect a wngle ray of light with the minutest precision, nd demonstrated that every ray was itself a omposition of several rays all of different co- ours, each of which when separate held to its wn nature, simple and unchanged by every xperiment that could be tried upon it. Or, ) be more particular, light is not all similar ad homogeneal, but compounded of hetero- eneal and dissimilar rays, some of which in ke instances being more refrangible, and thers less refrangible, and those which are ost refrangible are also most reflexible ; and seording as they differ in refrangibility and flexibility, they are endowed with the power fexciting in us sensations of different co- vurs. Sir Isaac Newton’s theory of light and co- yurs is striking and beautiful in itself, and educed from clear and decisive experiments. Ast, That lights which differ in colour, dif- x also in degrees of refrangibility. 2d, That the light of the sun, notwithstand- ig its uniform appearance, consists of rays ifferently refrangible. 3d, That those rays which are more refran- ble than others, are also more reflexible. 4th, 'That as the rays of light differ in de- \rees of refrangibility and reflexibility, so they ‘so differ in their disposition to exhibit this * that particular colour; and that colours are ot qualifications of light derived from refrac- ons or reflections of natural bodies, as was enerally believed, but original and connate roperties, which are different in different rays, mie rays being disposed to exhibit a red co- : oe no other, and some a green and no /her, and so of the rest of the prismatic ' lours. 5th, That the light of the sun consists of jolet-making, indigo-making, blue-making, ‘een-making, yellow-making, orange-mak- ‘g, and red-making rays; and all of these are fierent in their degrees of refrangibility and flexibility ; for the rays which produce red stm are the least refrangible, and those ‘atmake the violet the most; and the rest more or less refrangible as they approach ‘ther of these extremes, in the order already ned: that is, orange is least refrangible ext to red, yellow next to orange, and so on; that to the same degree of refrangibility /€re ever belongs the same colour, and to the |me colour the same degree of refrangibility. 6th, Every homogeneal ray, considered ‘part, isvefracted according to one and the Dh z COL same rule, so that its sine of incidence is to its sine of refraction in a given ratio; that is, every different coloured ray has a different ratio belonging to it. 7th, The species of colour, and degree of refrangibility and reflexibility, proper to any particular sort of rays, is not mutable by re- flection or refraction from natural bodies, nor by any other cause that has been yet observed. When any one kind ofrays has been separated from those of other kinds, it has obstinately retained its colours, notwithstanding all en- deayours to bring about a change. 8th, Yet seeming transmutations of colours may be made, where there is any mixture of divers sorts of rays ; for, in such mixtures, the component colours appear not, but, by their mutually alloying each other, constitute an intermediate colour. 9th, There are therefore two sorts of colour, the one original and simple, the other com- pounded of these; and all the colours in the universe are either the colours of homogeneal, simple light, or compounded of these mixed together in certain proportions.. The colours of simple light are, as we observed before, violet, indigo, blue, green, yellow, orange, and red, together with an indefinite variety of in- termediate gradations. The colours of com- pounded light are differently compounded of these simple rays, mixed in various propor- tions: thus a mixture of yellow-making and blue-making rays exhibits a green colour, and a mixture of red and yellow makes an orange; and in any colour the same in specie with the primary ones may be ‘produced by the com- position of the two colours next adjacent in the series of colours generated by the prism, whereof the one is next more refrangible, and the other next less refrangible. But this is not the case with those which are situated at too great a distance ; orange and indigo do not produce the intermediate green, nor scar- Jet and green the intermediate yellow. 10th, The most surprising and wonderful composition of light is that of whiteness ; there is no one sort of rays which can alone exhibit that colour: itis ever compounded, and to its composition all the aforesaid primary colours are requisite. 11th, As whiteness is produced by a copious reflection of rays of all sorts of colours, when there is a due proportion in the mixture ; so, on the contrary, blackness is produced by a suffocation and absorption of the incident light, which being stopped and suppressed in the blaok body, is not reflected outward, but re- flected and refracted within the body till it be stifled and lost. Newton’s method of accounting for the dif- ferent colours of bodies, from their reflecting this or that kind of rays most copiously, is so easy and natural, that his system quickly over- came all objections, and to this day continues to be almost universally believed. It is now commonly acknowledged, that the light of the sun, which to us seems perfectly homogeneal and white, is composed of no fewer than seven COM different colours, viz. red, orange, yellow, green, blue, purple, and violet or indigo. A body which appears of a red colour, hath the property of reflecting the red rays more pow- erfully than any of the others; and so of the orange, yellow, green, &c. A body which is of a black colour, instead of reflecting, absorbs all or the greatest part of the rays that fall upon it; and, on the contrary, a body which appears white, reflects the greatest part of the rays indiscriminately, without separating the one from the other. COLUMBA-Noachi, Noah's Dove, a south- ern constellation. See CONSTELLATION. COLURES, in Astronomy and Geography, two great circles supposed to intersect each other, at right angles, in the pole of the world, and to pass through the solstitial and equinoc- tial points of the ecliptic; that which passes through the former point being called solsttial colure, and the other the eqguznoctial colure. The word is derived from xora@-, mutilus, or tunicatus, and sea, tail; appearing with the tail cut; because never seen entire above the horizon. COMA-Berenices, Berenice’s Hair, a con- steilation of the northern hemisphere. See CONSTELLATION. COMBINATIONS, in Mathematics, denote the different collections that may be formed out of any given number of things, taken a certain number at a time, without regard to the order in which they may be arranged ; and are thus distinguished from permutations, or changes, which have reference to the order in which the several quantities may be disposed. In order to find the number of combinations that a given number of things will admit of, Jet'us take the continued product of the fac- tors, (a + a) (a + b) (a + ¢) (aw + d), &e. viz. Q) 2 +a x +5 st ahibe + ab (2) x +e ai+ta ab ik + vets + abe Cc. be (3) a +d ab atta «<) abe : x3 +4 be ary thd x +abed d bd bed cd Now without pursuing this developement any farther, it is obvious, that in each of these formule the coefficient of the second term is equal to the sum of all the quantities a, b, ec, &c. that enter into the expression; the third is the sum of all the possible combinations of every two of them; the fourth of every three; the Jifth of every four, and so on; that is, the number of combinations that may be formed out of any number of things (»), taken a cer- tain number at a time (m), will be expressed by the coeflicient of the m+ 1 termof the above _ pressing the number of combinations that may COM expanded function, carried to n factors; or making a, b, ce, d, &c. each equal to 1, the same will be expressed by the m + 1 coefficient of the expanded binomial (2 + 1)"; which from the known law of the binomial theorem‘ is equal to wf n (n—1) (n—2) (n—3)........ (n—m—lI)| 1... 2 ..a3 se ems wee which is, therefore, a general formule for ex-! be formed out of n things taken any number m at a time. ; Suppose, for example, it were required to} find how many combinations may be formed) out of 13 cards, all different from each other, taken 4 at a time? 13.12.11.10 1.2.3.4 The above formule leads us also to the me- thod of finding the whole possible number o} combinations that may be formed out of ¢ given number of things n, by taking 1, 2,3, 4 &e. at a time, to » at a time; for this it is ob vious will be equal to the sum of all the coef ficients of the expanded binomial (@ + 1)” wanting the first term. But the sum of al these coefficients is equal to 2”, (see Bino| MIAL); therefore the number of possible com binations of m things is equal to 2”—1. Thi single quantities a, 6, c, &c. being classed un der the general term combination, for the saki of analogy, if these be excluded the formula will be 2” — (nm + 1). Thus the number of all the combination! that may be formed out of the five letters a, c, d,e = 25— 1 =31, or excluding the sing] terms, the number of combinations is 25—6= 26, which are the following, viz. Comh. of two, Comb, of three. | Comb. of four. Here we have 715 combinations) (ae att ab bd abe ade abed ac be abd bed abce ad cd abe bee abde ae ce acd bde acde bede | be de ace cde abede of five | In the above propositions, the things ar supposed to be all different from each other but if there be several things of one sort an several of another, the problem requires / different solution. *y To determine the number of combinatior that may be formed out of a given number! things, in which there are m things of on sort, » of another sort, p of another sort, &e by taking 1 at a time, 2 ata time, 3 at a tim &ec. to any given number of things at a time Rule. Place in one horizontal row m + units, annexing ciphers on the right hand, t the whole number of units and ciphers e€ ceeds the greatest number of things to 1 taken at a time by unity. | Under each of these terms write the sum © n + 1, left-hand terms, including that as o1 of them under which the number is place and under each of these, the p + 1 left-har terms of the last line; then the q + | tern of this, and so on through all the number | ie COM different things, and the last line will be the answer. Exam. Given a number of the form a', 65, ct, d+, e+, £3, g, to find how many different di- visors it has, that are the product of 6 factors, of 5 factors, 4 factors, Kc. Here m— 5, n= 5, p—4, g¢=—4, r = 4, $—3,¢—1; then by the rule | Oe ie | 1 1 O—m +1 units m2 3 4 5 6 5—= u+1terms mo. °66lh610.CUS 20, SSH p t+ i terms 1 410 2 35 54 74—=q-+1 terms mp 15 35 7O 323 193 —r-+l1terms mG 21 56 125 243 421 —s + lterms 17 #27 #77 #«+(181 368 664—¢ + Iterms That is, the number has 7 prime divisors, 27 that are composed of two factors, 77 hav- ing three factors, &c. See an investigation of this rule, Nicholson’s Phil. Journ. vol. xxiii. p. 203-205. The theory of combinations is of the great- est use in the doctrine of chances and proba- bilities, for the probability of an cvent happen- ing or failing depends generally upon the num- ber of combinations that may be formed, or that may take place amongst the circum- stances on which the event ultimately depends. Thus if it were required to find the chance of drawing any four specified cards out of a pack vof 52, we must find how many combinations of four may be formed out of 52 things, and ‘as we are as likely to draw one of these com- binations as another, the probability of draw- jing that particular one specified, is expressed by 1 divided by the number of such combina- 52.51.50. 49 1 BE ULaeie Beet, seen Os Se ei dBi ee a 1.2.3.4. 270725 See the articles PERMUTATIONS and CHANCES. COMBUSTIBLE Bodies, Latin combus- tum, from comburo, to burn, are those bodies which are susceptible of fire; as they are ‘bodies which when once set on fire will con- (tinue to burn without the farther accession of ‘fuel. _ COMBUSTION, or Combustum, in the an- cient astronomy, denoted the situation ofa planet when it appeared on or very near the un’s disc. ‘The word is derived from com- duro, to burn; because the planet was then Supposed to be completely enveloped in the ‘sun’s rays. Combustion, in Natural Philosophy, denotes the decomposition of certain substances ac- companied by light and heat. COMET, in Astronomy, from coma, hair, a evenly body in the planetary regions, ap- pearing at uncertain periods, and which, during “the time of its appearance, has a motion in /some respects similar to those of the planets. ‘The orbits of the comets however differ from those of the planets in their being more eccen- dato, and being inclined to the plane of the _ecliptics in angles of various magnitudes, the plane of some of them being nearly coincident, and others nearly perpendicular to the plane | of the ecliptic. The motions of comets are | i COM also some of them direct, and others retrograde, whereas those of the planets are all direct. Comets are popularly divided into three distinct classes, viz. bearded, tailed, and hairy comets ; though in fact this distinction relates rather to the circumstance under which they are seen, than to any difference of the bodies themselves. . Thus when the comet is eastward of the sun, and moves from it, it is said to be bearded, because the light precedes it in the manner of a beard. When the comet is westward of the sun, and sets after it, it is said to be tazled ; because the train follows it in the manner of a tail. Lastly, when the sun and the comet are in opposition, the train is hid behind the body of the comet, excepting a little which appears round it in the form of a border of hair or coma, whence it is called hairy, and whence the name of comet is derived. A comet is also divided by astronomers into distinct parts, as the nucleus, the head, the coma, and the tail. 'The nucleus is the dense part ef the comet which is supposed to be of a similar nature to the other planetary bodies, and much resembling them in appearance when viewed through a good telescope. The head of the comet is that part of it in which the nucleus is involved, and which appears with ‘a fainter light than the nucleus in its centre, but brighter than that of the coma and tail. The coma is a faint light surrounding the head of the comet; and the taz/is the long train of light by which these bodies are com- monly attended, as above described. With regard to the nature of comets va- rious conjectures have been advanced, both by ancient and modern astronomers; some of the former, particularly Aristotle, considering . them merely as accidental fires or meteors which were kindled in the atmosphere, while others more rationally maintained that they were permanent bodies like the planets, and only differing from them in their appearances, which latter fact was first positively ascertained and demonstrated by Tycho Brahe, from his observations on the comet of 1577; since which period no doubt has been entertained as to the permanency of their bodies, but all beyond this is, even in the present day, mere matter of conjecture and hypotheses, and as such is omitted in the present article. The Appearances of Comets, viewed through a telescope, as well as to the naked eye, are liable to apparent changes, which Newton as- cribes to certain changes in their atmosphere, arising from some unknown cause; and this opinion seems, in some measure, to have been confirmed by subsequent observations. Hevelius observed, of the comet of 1661, that its body was of a yellowish colour, very bright and conspicuous, but without any glit- tering light: and in the middle was a dense ruddy nucleus, nearly equal in apparent mag- nitude to Jupiter, encompassed with a much fainter and thinner matter. ‘The same astro- nomer made observations on the comet of COM 1665, and particularly informs us, that he could discern the shadow of its head cast dis- tinctly upon the tail; and the same appear- ance was observed in the comets of 1680 and 1744; it was also very obvious, even to the naked eye, in the late brilliant comet of 1811, though it seems very questionable as to its being the shadow of the head; its form, when viewed through a telescope, having been very different from that which a shadow in such circumstances would have assumed. Dr. Herschel made a great number of ob- servations on the comet of 1807, in order to ascertain whether or not it shone by retlected light, or from the borrowed rays of the sun, and finally concluded that it shone by its own light, because had it been borrowed, it ought in that part of its orbit where it was during these observations to have appeared gibous, whereas its disc was full and well defined, which could only arise from the comet shining with its own light. It is however extremely doubtful, how far the most accurate observations may be de- pended upon in such a very delicate case as that of determining so small an obscuration of light; and with regard to the comet of 1811, Dr. Herschel himself considered it as shining by reflected light. Now it seems a deviation from the admirable simplicity of the laws of nature, to suppose bodies, apparently so much alike in all other respects, should be so very different in this, viz. that the one should shine with its own proper light, and the other by the borrowed rays of the sun; we must therefore conclude, that Dr. Herschel deceived himself in his observations in 1807, and that comets shine by means of the solar rays reflected by them, the same as the other planetary bodies. Of the Magnitude of Comets. 'The estimates that have been given by Tycho, Hevelius, and some others, of the magnitudes of comets, are not sufficiently accurate to be depended upon ; for it does not appear that they distinguish between the nucleus and the surrounding at- mosphere. Thus Tycho computes that. the true diameter of the comet in 1577, was in proportion to the diameter of the earth, as 3 is to 14. Hevelius made the diameter of the nucleus of the comet of 1661, and also that of 1665, at the commencement of their appear- ance, to be less than a tenth part of the dia- meter of the earth; and from the parallax and apparent magnitude of the head of the comet of 1652, on the 10th of December, he com- puted its diameter to be to that of the earth, as 52 to 100. By the same method he found the true diameter of the head of the comet of 1664, to be at one time 12 semi-diameters of the earth, at another time not much above 5 semi-diameters. The diameter of the atmos- phere is often ten or fifteen times as great as that of the nucleus; the former, in the comet of 1682, was measured by Flamstead, and found to be 2’, but the diameter of the nucleus only 11 or 12". Some comets, from the ap- parent magnitude and distance compared, have been judged to be much larger than the COM moon, and even equal to some of the primary planets. The diameter of that of 1744, when at the distance of the sun-from us, measured about 1’, and therefore its diameter must be | about three times the diameter of the earth; | at another time the diameter of its nucleus was nearly equal to that of Jupiter. ST Hence it has been coujectured, that some | of the solar eclipses recorded in history, that | cannot be yerified by calculation from tables © of the sun and moon, have been occasioned by the interposition of comets between the sun and the earth. 'The eclipses of the sun j mentioned by Herodotus (I. vii. c. 37, and | 1. ix. c. 10) have been thus accounted for, and | also the eclipse that happened a few days be- fore the death of Augustus, mentioned by | Dion; and it is observable, that Seneca saw | a comet the same year. History records some | comets.that have appeared as large as the sun | (vid. Seneca, N. Q. 1. vii. c. 15); and there | fore if such a comet, near its perigec, were to | come between the sun and our earth, it would eclipse him for a time. The darkness which | is recorded tohave happened at the crucifixion, | has also been accounted for in the same way. | Dr. Herschel observed several comets which | seemed to have no nucleus, as was the case| with that remarkable one of 1811, but in the comet of 1807, the nucleus was very visible} with a good telescope; and its diameter, | deduced from a mean of several observations, | appeared to subtend an angle of about 14,’ which at the distance it then was from the earth, gives for its lineal diameter about 538) miles. This result, though probably con-) siderably far from correct, may still be con- sidered the most accurate that has been given) of the real dimensions of a comet. Va The Tails of Comets. The principal pheno-| mena which have been observed with respect; to the tails of comets are as follow: 1. Their tails appear the largest and brightest imme-| diately after the passage through their peri-) helion, or nearest approach to the sun. 2. The) tail of a comet always declines from a just op- position to the sun, towards those parts whieh the body or nucleus pass over, in its progress through its orbit. 3. This declination is the smallest when the head or nucleus approaches nearest the sun, as is still less near the nu-) cleus of the comet, than towards the extremi y of the tail. 4. The tails are somewhat brighte and more distinctly defined in their convex than in their concave part. 5. They are als broader at the upper extremity than near the centre of the comet. 6. Their tails are always transparent, and the smallest stars are seem through them without any sensible diminution) of their light, and without those effects of re- fraction, which might be expected from view- ing them through a visible medium, which ef cumstance seems to indicate that the tails art composed of extremely rare and attenuatec matter; but with regard to their formation ¥ re may be said to be totally ignorant, notwith; standing the numerous hypotheses that have i COM been advanced on this subject. The reader, however, who is desirous of indulging in such speculations, may consult Euler, in the “ Me- moirs of Berlin,” vol. ii.p.117; and Hamilton’s “ Philosophical Essays,” p. 91. The Jengths of the tails of comets are very various, both as to reality and appearance ; and with regard to the latter, it depends ona variety of circumstances. Longomontanus mentions a comet, Dec. 16, 1618, which, had its tail appearing under an angle of 104°; that fof 1680, appeared under an angle 70°, accord- ing to Newton very brilliant; the comet of 1744, had a tail which at one time appeared to extend 16° from its body; and which, al- lowing the sun’s parallax to be 82”, must have feeen above 46 million miles in length; and the diameter of its nucleus was nearly equal 'o that of Jupiter. The tail of the comet of (759, according to Pingré, subtented an an- tle of 90°, but the light was very faint; and he tail of the late brilliant comet of 1811, vhen at its maximum, subtended an angle of it least 16 degrees, and was then computed 0 be at least 23 million miles in length. The Distances of Comets. The analogy be- ween the periodical time of the planets, and heir distance from the sun, discovered by Kepler, takes place, of course, also in the somets, at least in those which revolve in el- iptic orbits ; and consequently, if the periodic ‘ime of a comet were known, its mean dis- “ance might be easily computed. Now the tomet of 1759 is known to perform its révo- ution in 76 years, nearly, whence it appears hat its mean distance is about 18 times that the earth, or a little less than the mean dis- ance of Uranus; but in consequence of the reat eccentricity of its orbit, i's aphelion joint, or greatest distance from the sun, is early double that of the above planet. The serihelion distance of this comet is about ‘6 f the mean distance of the earth, which being aken from 36, the mean transverse axis of its mbit, leaves 35:4 for its aphelion distance, vhich is nearly double the greatest distance f Uranus, and about four times that of Saturn. | The above is the only comet whose periodic turn has been ascertained, and consequently he only one whose mean distance can be shown, but with regard to the perihelion dis- ance of these bodies, this may be determined ’y observations ; and accordingly we have an €count of this element-of the orbits of about 90 comets, which have been observed with sonsiderable accuracy. The greater number ff these have had their perihelion point fall within the terrestrial orbit, and many of them \t less than half the mean distance of the yarth; but the comet of 1680 is that of all thers which approaches the nearest to the sun, Ie perihelion distance being only ‘006 of the erihelion distance of the earth, that is, about 40000 miles from the sun’s centre, and must herefore, according to Newton, have been \nvolved in its atmosphere. This comet also assed very near the terrestrial orbit, having : COM been, according to Dr. Halley’s calculation, on the 1lith of November, 1 h. 6 min. P.M. not more than one semi-diameter of the earth, or about 4000 miles to the northward of the earth’s orbit, at which time had we been in this part of our track, the comet would have -had a parallax exceeding that of the moon; and the mutual gravitation of the two bodies must have caused a change in the incli- nation of the earth’s orbit, and in the length of the year; at the same time the waters on the earth would have been so elevated from the same cause, as would in all probability have caused a universal deluge, and reduced this beautiful frame to its original chaos. The limits of a comet’s distance may be easily ascertained from its tail, it being sup- posed to be directed from the sun. Let S be the sun, E the earth, ET the line in whichthehead of the comet appears, EW the linein which the extremity of the tail is observed, and draw ST parallel to EW ;thenthe comet is within the distance ET. For if the comet were at T, the tail would be- directed ina line parallel to EW, and therefore could never appear in that line. Now TEW is known from observation, and consequently its equal, ETS, together with TES, the angular dis- tance of the comet from the sun, and ES to find S'T the limit of the comet’s distance. The Orbits of Comets, and their periodical Revolutions. a {t is extremely difficult to determine, from computation, the elliptic orbit of a comet, to any degree of accuracy; for when the orbit is very eccentric, a very small error in the ob- servation will change the computed orbit into a parabola, or hyperbola. Now, from the thickness and inequality of the atmosphere with which the comet is surrounded, it is im- possible to determine, with any precision, when either the limb or centre of the comet pass the wire at the time of observation. And this un- certainty in the observations will subject the computed orbit to a great error. Hence it happened, that M. Bouguer determined the orbit of the comet in 1729 to be an hyperbola. M. Euler first determined the same for the comet in 1744; but having received more ac- curate observations, he found it to be an el- lipse. The period of the comet in 1680 ap- pears, from observations, to be 575 years, which M. Euler, by his computation, deter- mined to be 1664 years. The only safe way to get the period of comets, is to compare the elements of all those which have been com- puted, and where you find they agree very well, you may conclude that they are elements of the same comet, it being so extremely im- probable that the orbits oftwo different comets should have the same inclination, the same COM erihelion distance, and the places of the peri- helion and node the same. ‘Thus, know- ing the periodic time, we get the major axis of the ellipse; and the perihelion distance be- ing known, the minor axis will be known. When the elements of the orbits agree, the comets may be the same, although the periodic times should vary a little; as that may arise from the attraction of the bodies in our system, and which may also alter all the other ele- ments in a small degree. It has been well ascertained, that the comet which appeared in 1759 had its periodic time increased considerably by the attraction of Jupiter and Saturn. This comet was seen In 1682, 1607, and 1531, all the elements agrec- ing, except a little variation of the periodic time. Dr. Halley suspected the comet in 1680 to have been the same which appeared in 1106, 531, and 44 years before Christ, when Julius Cesar was murdered; and that its period was 575 years. Mr. Dunthorne, however, in the Philosophical Transactions, vol. xlvii. has endeavoured to show from a MS. in Pembroke Hall library, that the comet of 1106 could not be the same with that of 1680; but M. de la Lande adopts the opinion of Dr. Halley. He also conjectured, in the first edi- tion of his Synopsis, without repeating it in the second edition, that the comet observed by Apian in 1532, was the same as that ob- served by Hevelius in 1661 ; if so, it ought to have returned in 1789 or 1790; but it has. never been observed. The interval between the passages of the comet by the perihelion in 1532 and 1661, is 128 years, 89 days, 1 hour, 29 minutes; (32 of the years being bis- sextile) which, added to the time of the peri- helion in 1661, together with 11 days to re- duce it from the Julian to the Gregorian style, which we now use, brings out the time of the next perihelion, to be April 27th, 1" 20’, in the year 1789. But M. Mechain having col- lected all the observations in 1532, and cal- culated the orbit again, found it to be sensibly different from that determined by Dr. Halley, which renders it very doubtful whether this was the comet which appeared in 1661; and this doubt is increased, by its not having appeared in 1790. The comet in 1770, whose periodic time M. Lexell computed to be five years and seven months, has not been observed since. There can be no doubt but that the path of this comet, for the time it was observed, be- longed to an orbit whose periodic. time was that found by M. Lexell, as the computations for such an orbit agreed so very well with the observations. But the revolution was proba- bly longer before 1770: for as the comet pass- ed very near to Jupiter in 1767, its periodic time might be sensibly increased by the action of that planet; and as it has not been observed since, we may conjecture, with M: Lexell, that having passed in 1772 again into the sphere of sensible attraction of Jupiter, a new disturbing force might probably take place, and destroy the effect of the other. = “yy COM y According to the above elements, the comet would be in conjunction with Jupiter on Au- gust 23, 1779, and its distance from Jupiter would be only 23, ofits distance from the sun; consequently the sun’s action would be only zigth of that of Jupiter. If the comet re- turned to its perihelion in March 1776, it. would then not be visible. See M. Lexell’s account in the Phil. Trans. 1779. The ele- ments of thé orbits of the comets in 1264 and 1556, were so nearly the same, that it is very probable it was the same comet; if so, it ought to appear again about the year 1848. “y Mr. Cole, in his “ Theory of Comets,” ad- vances a hypothesis, which in some cases may perhaps be accurate. He supposes that the orbit of a comet is not an ellipse ; but that, when it passes its perihelion, it has acquired so great a velocity, that its centripetal force is overcome by its centrifugal, and that conse- quently the comet continues to fly off in a pa- rabola, or hyperbola, till it come within the attraction of some fixed star; that this attrae- tion may give it a new direction, and increase its velocity till it come to an apsis below that star, when it may again fly off, either in a pa- rabola or hyperbola, and proceed till it fall within the attraction of another star; and thus visit many different systems. Dr. Halley has given us a table of the as- tronomical elements of twenty-four comets, on the supposition that they moved in para- bolas, though he thought it extremely pro- bable that they really moved in very eccentri ellipses, and consequently returned after long periods of time. This table commences with the year 1337, and closes with 1698. By means of this table, and others similar to it, if may be determined whenever a new come shall appear, by comparing it therewith, whe ther it be one of those which have already ap peared; and consequently its period and t axis of its orbit be ascertained, and its retur foretold. See his “Synopsis of the Astrono1 of Comets,” annexed to Gregory’s “ Astron my.” ‘This was first published in the Philoso: phical Transactions in 1705, and re-publishe¢ with his Astronomical Tables in 1749, M. de la Caille changed this table into another of: more convenient form, by putting the areaj for the times. Another table has since been compute from the observations contained in the Philo sophical Transactions; M. de la Caille’s “ Astre nomy;” and De la Lande’s “ Histoire de Ii Comets,” de 1759; and “‘ Connoissance de Movemens Celestes,” 1762 and 1764. In th table are seen the elements of twenty-five othe comets, from the year 1264 to 1762. Th most extensive table for calculating the mo tions of comets, was computed by M. d Lambre ; it is inserted, as we have alread mentioned in Mr. Vince’s “ Astronomy,” vol. | Another table on an extensive scale, compute! by Mr. Lee, an ingenious and excellent astre tronomcr, and an attentive observer of th heavens, is given in Rees’s Cyclop. ‘i COM The number of comets that are stated in the ost accurate aecounts to have appeared, ice the commencement of our era is about 0; and before that zra, about 100 others é recorded to have been seen, though it is obable that not more than half of them were mets. The elements of the comet of 1770, and the jectory of its path, may be found in the ety, vol. of the American Philosophical ciety, vol. i. In Whiston’s Solar System, the orbit of veral comets are delineated, and the periods as many of them as were then known ex- essed. Yo determine the course of a Comet mechanically. e following ingenious method, by a thread, owe to Longomontanus: observe four stars md the comet, such, as that the comet may in the intersection of the right lines that a the two opposite stars; which is easily nd by means ofa thread placed before the eye, l extended towards the stars and comet. juppose, v.g'r. the net’s place in the 2 ee wre wens to be be- eg en the four stars, ee B, C, D,where the wo Pe: 2 joining the stars Qe ie yand D, passes #p ee ough the body of comet; as does also the line passing ough A and C. Jn a globe, whereon these four stars are nd, extend a thread B and D, and another ough A and C; the point of intersection (give the place of the comet E. This prac- ‘being repeated for several days, the comet’s tse will be had on the globe, which course _be found to be a great circle ; if this great le, drawn through three distant places, and Wing its path among the stars, be continued it intersect the ecliptic, it will show nearly iplace of the node, and the inclination of | orbit to the ecliptic. The plane of the (e and inclination of the orbit being thus iid from several triplets of places, inde- (dent of each other, a medium of the ae may be considered as tolerably ac- lite. . ’n the subject of comets, see Newton’s tincipia,” lib. iii.;, Halley’s “ Synopsis of iets ;”” Sejour, “ Essai sur les Cometes,” ; M. Pingre’s “ Cometographio,” 2 vols. 11781; Sir H. Englefield’s work “ On the ermination of the Orbits of Comets; M. e’s “ General Considerations” on the si- Jons of the orbits of all the planets and ets which have hitherto been calculated, tted in the Memoirs of the Academy of (nees of Berlin; Dr. Gregory’s “ Astro- Cy ;” Cole’s “ Treatise on Comets,” 1803 ; la Lande, “ Theorie des Cometes,” 1759, “ Astronomie,” vol. iii.; “ An Account fie Discoveries concerning Comets, with Way to find their Orbits, &c.” by Thomas cer, 1757; Vince’s “ Astronomy,” &e, &e. COM COMETARIUM, a curious machine, for conveying an idea of the revolution of a comet about the sun. It is contrived in such a man- ner, as by elliptical wheels to show the un- equal motion of a comet in every part of its orbit. The comet is represented by a small brass ball, carried by a wire, in an elliptic groove about the sun in one of its foci; and the years of its period are shown by an index moving with an equable motion over a gra- duated silver circle. See a particular descrip- tion, with a plate, in “ Ferguson’s Astron.” Svo. p. 412. The construction of the cometarium has been recently improved by Mr. W. Jones, optician, Holborn. COMMANDINE (FRrepeErick), a celebrat- ed mathematician and linguist, was born at Urbino in Italy, in 1509; and died in 1575, in the 66th year of hisage. Toa great depth, and just taste in mathematics, he joined a critical skill in the Greek language ; a happy conjunction which made him very well quali-_ fied for translating and expounding the writ- ings of the Greek mathematicians. He is greatly applauded by Bianchanus, and other writers; and he justly deserved their encomi- ums. Of his own works Conimandine pub- lished the following: 1. Commentarius in Planispherium Ptolomei, 1558, in 4to. 2. De Centro Gravitatis Solidorum, Bonon. 1565, in 4to. 3. Horologiorum Descriptio, Rom. 1562, in 4to. He translated and illustrated with notes several works of Archimedes, Apol- Jonius, Euclid, and Ptolemy; most of which were beautifully printed in 4to. by the cele- brated Aldus. COMMENSURABLE, among geometri- cians, an appellation given to such quantities as are measured by one and the same common measure. COMMENSURABLE Numbers, whether inte- gers, surds, or fractions, are such as can be measured or divided by some other number without any remainder: such are 12 and 18, as being measured by 6 and 3: also 2 2, and 3 / 2, being measured by 2. COMMENSURABLE in Power, is said of right lines, when their squares are measured by one and the same space or superficies. COMMENSURABLE Surds, those that being reduced to their least terms, become true figurative quantities of their kind; and are therefore as a rational quantity to a rational one. COMMON Measure or Divisor, in Arith- metic, is that number which will divide two other numbers without leaving a remainder ; and the greatest of such divisors is called the greatest common measure, or greatest common divisor. See Common Divisor and FRac- TION. COMMUNICATION of Motion, that act of a moving body by which it gives motion, or transfers its motion to another body. See MoTION. COMMUTATION, in Astronomy. Angle CO M of commutation is the distance between the sun’s true place seen from the earth, and the place of a planet reduced to the ecliptic; and is, therefore, found by subtracting the same jongitude from the heliocentric longitude of the planet. COMPANY, Rule of. See FELLowsHIP. COMPASS,: or Mariner’s Compass, an in- strument used at sea by mariners to direct and ascertain the course of their ship. The invention of this instrument is com- monly ascribed to Flavio Gioia, or Flavio, of Malphi, about the year 1302. Others ascribe the invention to the Chinese, the knowledge of which, according to Gilbert, in his book pote Magnete,” was brought from thence into Europe about the year 1260, by Marcus Paulus, a Venetian; and this account seems probable, from the circumstance of the early European navigators having used the compass in the same way as the Chinese do at present, viz. by letting it float on a small piece of cork in a bason of water, instead of suspending it on a pivot as now practised. The Chinese divided their compass into 24 points, and Flavio Gioia, in 1302, into only 8; but how long the present division of it into 32 points has been adopted is not precisely known. ‘There are other cir- cumstances attending the history of the com- pass, which seems to indicate that it was known in Europe even before the date as- cribed to its introduction by Gilbert. T'au- chette relates some verses of Guoyot de Pro- vence, who lived in France about the year 1200, in which there is an obvious allusion to the mariner’s compass. In fact there appears to be no distinct account of the time when the directive property of the magnet was dis- covered; but as it is well known that its at- tractive power had been known to the most ancient philosophers of Greece, it seems highly improbable that its directive quality should have remained unknown for so many ages. The common construction of the mariner’s compass is extremely simple. It consists of a circular brass box which contains a paper card, on which is drawn the 32 points of the compass ; and this card is fixed on a magnetic needle, which always turns to the north, eXx- cept a small deviation which is variable at different places, and at the same place at dif- ferent times. See VARIATION. The compass, however, like all other instru- ments of acknowledged importance, have un- dergone.considerable improvements; and its construction, as well as the method of sup- porting it at sea, so as to protect it against the motion of the ship, are very various, but the principle of their structure is nearly the same in all; one of the common form is drawn at fig. 5, Plate VI. Azimuth Compass. This differs from the common sea compass only in this, that the circumference of the card or box is divided into degrees; also to the box is fitted an in- dex with two sights, which are upright pieces of brass placed diametrically opposite to each a COM other, having a slit down the middle of thei through which the sun or star i is to be view at the time of observation. Fig. 4, Plate V The use of this instrument is to -take f bearing of any celestial object, when it is or above the horizon, in order to find from t magnetical azimnth, or amplitude, the vari tion of the needle. Land Compass. The structure of thisinstr ment is upon the same principle as the ma ner’s compass above described. It consis like that of a box and needle, but instead this being fixed to a card, and playing wi it on a pivot, it here plays alone, the ¢a being drawn on the bottom of the box, and cirele divided into 360 degrees on the limb. Compass Dials, are small pocket sun-dia fitted to a box carring a magnetic needle, order to place them in the “meridian at Ny time of observations. Points of the Compass. These are thirty-two principal points of divisions dray on the compass card ; and are otherwise call Rhumbs ; each of which has a particular ¢ nomination expressed by means of the init of the four first points, North, East, Sou West, viz. North. East. South. West. _ N. by E. E.byS. S.by W. W. by N N.N.E. E.S.E. S.S.W. “W. Nigg N.E.byN. S.E.byE. S.W. by S. N.W.by N. E. S. E. S. W. N. WL N.E. by E. S.E.byS. S.W.by W. N.W. by K.N.E. S.S.E. W.S: W.. N. NW E.by N. S.by E. W.byS. N. by Each of these points contain 11° 155¢ are again divided into 1 points, - contain 2° 485". See Ruums. Compasses, or Pair of Compasses, a mat matical instrument for describing cire} measuring and dividing lines, or figures, & The common Compasses consists of sharp-pointed branches or leg's of iron, st brass, or other metal, joined together at | top by a rivet, about which they move as ¢ centre; and are too well known to need | ih particular description. Compasses of three Legs, or Triang Compasses; the construction of which is | that of the common compasses, with the ail tion of a third leg or point, which has a mo} every way. ‘Their use is to take three po at once, and so to form triangles, and} down three positions of a map to be copie the same time. Beam Compasses consist of a long stral beam or bar, carrying two brass cursors } i of these being fixed at one end, the other if ing along the beam, with a screw to fast on occasionally. 'T'o the cursors may be sei) ed points of any kind, whether steel, penl or the like. To the fixed cursor is sometit applied an adjusting or micrometer screw? which a considerable extent is obtained! very great nicety. Bow Compasses, or Bows, are a small of compasses, that shut up in a hoop, wé COM serves for a handle. Their use is to describe ares or circumferences with a very small ra- dius. Caliber Compasses. See CALIBER. Cylindrical and Spherical COMPASSES, con- sist of four branches, joined in a centre, two of which are circular and two flat, a little bent at the ends: their use is to take the diameter, thickness, or calibre of round or cylindric bodies; such as guns, pipes, &e. Elliptic Compas- SES. ‘These are used for drawing ellipses, orovals ofany kind; ‘they consist of a¢ beam A B, about a foot long, bearing three cursors, to one of which may be screwed any point or pencil; to the bottom of the other two are Tiveted two sliding dove-tails, adjusted in grooves made in the cross branches of the instrument. These having a motion every Way, by turning about the long branch, go backwards and forwards along the cross; so that when the beam has gone half way about, one of these will have moved the whole length of one of the branches, and when it has got quite round, the same dove-tail has got back the whole length of the branch; and the same may be repeated on the other side. The distance between the two sliding dove- fails is the distance between the two foci of the ellipsis ; so that by changing that distance, any proposed ellipse may be described. iN German Compasses, have their legs a little sent outwards, towards the top; so that when shut the points only meet. fTairy CoMPAssEs, are so contrived within ide, by a small adjusting screw to one of the legs, as to take an extent to a hair’s breadth. Proportional Compasses, are those in which he joint lies, not at the end of the legs, but be- ‘ween the points terminating each leg. These we either simple or compound. In the former ort the centre or place of the joint is fixed; 0 that one pair of them serves only for one roportion. In the compound ones the joint way be set at any distance, and consequently ny proportion whatever easily obtained. Spring Compasses, or Dividers, are made {hardened steel, with an arched head, which y its springs opens the legs; the opening eing directed by a circular screw fastened to ne of the legs let through the other, and vorked with a nut. Geometry of the Compasses, a species of eometry invented by M. Mascheroni, of Milan, Y which all the elementary problems of plane ‘eometry are performed by the compasses nly, without the use of the ruler ; this geometry ‘as published by the above author at Milan, ‘nder the title of “ Geometria del Compasso,” ©. 179-, 8vo., and translated into French, ‘nder the title of “ Géométrie du Compas,” by 1. Caretti; Paris, 1798. It is, however, as Montuclar observes, more ingenious than pro- / finite arc Ar B; COM found, and may be considered rather as a sub- ject of curiosity than of real utility. We will give here the construction of two or three of the problems, leaving their demonstra- tion to the ingenuity of the reader ; and referring him for farther information to the work itself. 1. 'To divide a circle into four equal parts the centre of the circle being given. Apply the radius BA on Opes round the circumfer- Pan tnt ence to C, D, E, &e. From the point B with radius BD describe an arc, and from E with | the same radius de- scribe another arc, cut- ting the former in L; apply the distance A L from E to m, and from 7 B to n, so shall the circumference be divided into four equal parts in the points B, m, E, x. Cor. The circumference is divided into two equal parts, by the points B and E; into three equal parts by the points B, D, I’; into four equal parts by the points B, m, E, n; and into six equal parts by the points B, C, D, E, F, G. 2. To bisect any arc. ofa circle of which the centre is given. Withtheradius TEs al by AB, of the given G arc, and from the extremities ofthat arc B and C let there be described the two ares AD, AE; take AD and r A E, each equal to BC; and from D and E as centres, with the radius D C, or B E, deseribe two arcs cut- ting each other in F. Then with the distance AF, and centres D and E, describe two arcs which will intersect each other in the given are, and their point of intersection, G, will bisect the arc as required. Cor. By means of this and the preceding theorem the circumference may be divided into 8, 16, 32, &c. equal parts, as also into 12, 24, 48, &c. equal parts. 3. To find a third proportional to two given distances, Q7, MN, of which the first Qr is greater than the second MN. From the cen- 8) tre Q, and ra- dius Q yr, de- scribe the inde- Dp A é H ' and from the centre rv, and radius MN, let Af therebedescrib- |: ed the semi- circumference S BAS, and the distance AS will be the third proportional required. 4. To divide a given distance AB inte ex- treme and mean ratio. COM From the centre A and radius AB, de- scribe the circleBDA; and in its circumfe- rence make AB=BC =CD =DH hd; make also = Ba BD — Ka, (by prop.i.)and againAa=Db=—db, and the distance A B will be divided in mean and extreme ratio in the point 6; that is, we shall have BA:Ab3:;:Ab: OB. 5. To divide a right line or distance A B into two equal parts. From the point A as a centre, and ra- dius A B, de- scribe the semi- circumference BCDE, which B may be found by applying the radius as in prob. 1. From the points B and E as centres, and with the same radius AB, describe two indefinite arcs CP, DQ; from the same points B and E as centres; and with the radius BE, let there be described the two arcs EQ, BP; from the centre P, and radius P B, de- scribe the arc BM; and lastly from the point fas a centre, and radius P Q, describe an arc cutting the arc BM in the point M, so shall M be the point of bisection required. COMPLEMENT of an Arch or Angle, is what it wants of 90 degrees; thus the com- plement of 50° is 40°, and the complement of 40° is 50°. Arithmetical COMPLEMENT, of a logarithm, is what the logarithm wants of 10°00000, &c. ; and the easiest way to find it is, beginning at the left hand, to subtract every figure from 9, and the last from 10. So the arithmetical complement of 8°2501396 is 1°7498604. It is commonly used in trigonometrical calculations, when the first term of a proportion is not radius; in that case, add together the loga- rithms of the 3d, 2d, and arithmetical comple- ment of the Ist term, and subtract 10 from the result. COMPLEMENT, in Astronomy, denotes the distance ofa star from the zenith; or the arch comprehended between the place of the star above the horizon and the zenith, being the same as is otherwise called the co-latitude. CoMPLEMENT of Life, a term much used in the doctrine of life annuities, by De Moivre: according to him it denotes the number of years which a given life wants of 86, this being the age which he considered as the utmost probable extent of life. So 56 is the comple- ment of 30, and 30 is the complement of 56. See De Moivre’s “ Treatise on Annuities,” p. = ; and Price on “ Reversionary Payments,” p. 2. ComPLements of a Parallelogram, are the two smaller parallelograms made by drawing fo right lines through a point in the diagonal ; p 2 COM and parallel to the sides of the parallelogram. In every parallelogram these complements are equal to each other. Thus P and Q are the A .——- B yg complements in the pa- ; tek, Laks 7 rallelogram A BC D, Ne i, iB a each other. it and they are equal to COMPOSITE Number, is that which ig produced by the multiplication of two or mor numbers or factors, and is thus distinguished from a prime number, which cannot be so pro= duced ; thus 12 =3 x4, 20 =4x5, 26=5xsd are composite numbers. a § COMPOSITION, as understood by logi cians, is a species of reasoning by which we proceed from things that are known and given | step by step till we arrive at others, which were before unknown. See SYNTHESIS. iy Composition is also used to denote a par ticular kind of permutation, which consists im the taking of a given number of quanti out of as many equal rows of different quam. tities, one out of every row, and combining. them together. > ComPosITION of Forces, in Mechanics, is the method of finding the quantity and direction. ofa single force, equivalent to two or more, forces of which the quantity and direction are: given; and is thus distinguished from Rese lution of Forces, whichis the method of resoly= ing a given force into two or more forces, the| combined effect of which shall be equivalent) to the single given force. See PARALLELOGRAM of Forces. =| Composition of Proportion, is when ef! four proportionals, the sum of the Ist and) 2d is to the 2d, as the sum of the 3d and 4th) is to the 4th. ra Thusif @ .:bi:e 2d) aM By comp. a +6: 6:: ‘ c+d:d @ See Euclid, def. 15, book 5. 2): Composition of Ratios, in Arithmetic and) Algebra, is performed by multiplying the quan- tities or exponents of two or more ratios t0- gether, which product is then said to be com- | pounded of all the other ratios whose ex ponents were multiplied; thus if the ratios € a:6,ce:d;e:f be multiplied together, we shall have Ky Bi, ys) 6 Ae ; ma d icv Oa ia and the ratio of ace to bd f, is then said to be compounded of the several ratios of a to 6 eto d,etof, &c. And hence it follows tha) in any continued proportion the ratio of” lirst term to the last is compounded of all thi intermediate ratios.' This is by some author called addition of ratios. See Ratio. COMPOUND Jnterest, is that which arise from a sum of money put out at interest, b increasing the principal each payment by th: amount of that payment. See INTEREST. | CompounD Motion, is that which arise from the effect of several conspiring forces) which may render it either rectilinear 0 curvilinear, according to the nature of th . CON orees and the circumstances under which hey act. See Motion and PARALLELOGRAM f FORCES. Compound Multiplication, Division, Sub- raction, &c. See the respective terms. CompounD Numbers. The same as CompPo- irE Numbers. CompounD Pendulum. See PENDULUM. - Compound Quantities, in Algebra, are those onnected together by the signs + and —; hus a + 6, a + b—c, &c. are compound quan- ities; and these are distinguished into bino- nials, trinomials, &c. according to the number f terms of which they are composed. OMPOUND Ratio, is that which arises from he composition of ratios. See Composition. COMPRESSIBILITY, in Philosophy, that ality of a body or fluid, by which it yields 9 the pressure of another body or force, so as ) be brought into a narrower compass. The compressibility of water was for a long me doubted by philosophers, and the famous Jorentine experiment seemed for a time to ecide the question in the negative. See LORENTINE Experiment. But Mr. Canton, y a very delicate experiment, found that rater was really compressible into less than s natural bulk, and that it was more so in ‘inter than in summer; but with spirits of “ine and oil of olives the contrary had place ; aese being most compressible in the latter pason. | i The following table shows the quantity of ompression of these fluids and mercury, when ie thermometer was at 50° and barometer 22 inches. _ Compression of Milionth pts. Spec. grav. | Spirits of wine ...... BOIS ~ «OGRE . 846 Oil of olives............... BE As hsanks 918 BREW BLET f.ccssenssnvatae OR i ake 1000 ML WATE cs .is..cioocceeee BD ckedeunt 1028 Eg a + WRSpEns 13595 Whence Mr. Canton infers that these fluids e not only compressible but elastic; but at their degree of compressibility is not, as ight have been expected, in the inverse ratio their densities. See Phil. Trans. vol. lii. , and vol. liv. 1764. COMPUTATION, the method of estimat- g time, weights, measures, kc. See CALcu- ATION. CONCAVE, an expression used to denote é curvilinear vacuity of hollow bodies. NCAVE Lenses, or Mirrors, have either de or both sides concave. See Lens and IRROR. | CONCAVITY, from coneave, the hollow vacuity of bodies. /CONCAVO-Concave Lens, is that which is -Jneave on both sides. ) Concavo-Convex Lens, is that which is con- "GON one side, and convex on the other. | CONCENTRIC, having a common centre, concentric circles, ellipses, &c. ~ CONCHOID, or ConcuiLts (from the Latin , shell), the name of a curve invented 7 Nicomedes ; and hence commonly called ae! CON the Conchoid of Nicomedes, which was much used by the ancients in the construction of solid problems. It is thus con- structed: AP and BD being two lines intersecting at right angles, from P draw a number of other lines PF DE, &e. en which take al- . ways DE = DF =AB or BC; so shall the curve line drawn through all the points E, BE, E, be the first conchoid, or that of Nichomedes ; and the curve drawn through all the other points FF’, F, is called the second conchoid ; though, in reality, they are both but parts of the same curve, having the same pole P, and four infinite legs, to which the line DB D is a common asymptote. . To determine the equation of the curve; putA B=BC = DE=DF =a, PB=), BG = EH =a, and GE= BH=y; then the equa- tion to the first conchoid will be x x (b+ x)” + a y* =a* x (b+ 2), or xt + 2bx3 + bx? +27 y* = a*b? +20? bu-+a*x?; and, changing only the sign of a, as being ne- gative in the other curve, the equation to the second conchoid will be x* x (b— 2)” X x y* =a? X (6—x)’, or + —2b x3 +b x? 4+ x? y? = ab? —2a* bx ax. Of the whole conchoid, expressed by these’ two equations, or rather one equation only, with different signs, there are three cases or species ; as, first, ‘ when BC is less than BP, the conchoid will have a punctum conjugatum, as at C in the above figure ; when BC is equal to BP, the conchoid wil? take the following form, having a cuspis at P ; A and when BC is greater than BP, the coui- choid will have a nodus, as in the following figure ; The equation to this curve may, however, - be expressed far more simply than above. For, if in addition to the preceding notation we put the variable angle BP D = @, and the va~ riable line PE or PF =z, we shall have PB b ene 2 Oe and consequently PE= N2 CON b 4h Ae EA tees ony 3 + a, while Tit cies ge 5 a; or gene cos b rally, z—— +a. COS. The conchoid admits of a very simple me- chanical construction, as follows: In the first figure above, let DD, &c. repre- sent a ruler, in which is cut a channel or grove, So that a pin firmly fixed in the move- able rod A B P in the point B may slide freely within it. Then if there be fixed another pin at P for the moveable rod to press against, whilst the point B is carried along in the grove D, D, &e. it will by its motion describe the two curves EE E, &c. FF F, &c. by two pen- cils fixed for that purpose in the points A and C. Newton approves of the use of this curve for the trisection of angles, finding two mean proportionals, and in the construction of pro- blems, for which purposes it was employed by the ancients. See Newton’s “ Universal Arith- metic.” CONCRETE Numbers, are those which are made to denote any particular thing, as three pounds, three guineas, &c. and are thus distin- guished from abstract numbers, which have reference to no particular subject or thing, as 3, 4, &e. CONDENSATION, the act whereby abody isrendered more dense, compact, and heavy. Condensation is, by most writers, distinguished from compression, by considering the latter as performed by some external violence ; whereas the former is the action of cold. CONDENSER, a pneumatic engine, or syringe, whereby an uncommon quantity of air may be condensed into a given space; so that sometimes ten atmospheres, or ten times as much air as there is at the same time, in the same space, without the engine, may be thrown in by means of it, and its egress pre- vented by valves properly disposed. It consists of a brass cylinder, wherein is a moveable piston; which being drawn out, the air rushes into the cylinder through a hole provided on purpose ; and when the piston is again forced into the cylinder, the air is driven into the receiver through an orifice, furnished with a valve to prevent its escape. CON DORCET (JEAN ANTOINE NICOLAS Ca- RITAT DE), a French philosopher and mathema- tician. He was born at Ribemont, in Picardy, in 1743, of a noble family, from whom he de- rived the title of marquis. He was educated at the college of Navarre, where he showed a strong predilection for the mathematics, and his genius being indulged, he soon distinguish- ed himself among the geometricians. In 1765 appeared his first work, “ Sur le Calcul In- tegral,”’ which was received with approbation by the Academy. In 1767, he published a treatise, “ Of the Problem of the Three Bo- dies ;” and the year following, his “ Analytical Essays.” In 1769, he was chosen member of the Academy, and continued to publish nu- through it almost as freely as through th CON merous memoirs and essays on mathematical and philosophical subjects, He was also author of an elementary book on arithmetic, and an ingenious work, en. titled a “ Sketch of the Progress of the Hum Mind.” Condorcet was a conspicuous cha- racter in the French revolution, and died i prison, as is supposed, by poison, which he administered to himself, March 28, 1794, the fifty-first year of his age. 4 CONDUCTOR, in Electricity, a term first in- troduced into this science by Dr. Desaguliers, and used to denote those substances which are capable of receiving and transmitting elee- tricity ; in opposition to electrics, in which the matter or virtue of electricity may be exci and accumulated, or retained. The forme are also called non-electrics, and the latter non-conductors. And all bodies are ranker under one or other of these two classes, thougl none of them are perfect electrics, nor perfect conductors, so as wholly to retain, or freely and without resistance to transmit the electric fluid. = To the class of conductors belong all metal; and semi-metals, ores, and all fluids, (excep’ air and oils), together with the substances containing them, the eflluvia of flaming bodies ice (unless very hard frozen), and snow, mos saline and stony substances, charcoals, 0 which the best are those that have been ex posed to the greatest heat; smoke, and _ the vapour of hot water. Mi It seems probable that the electric flui passes through the substance, and not mere} over the surfaces of metallic conductors ; be cause, if a wire of any kind of metal be covere with some electric substance, as resin, sealin wax, &c. and a jar be discharged through if the charge will be conducted as well as with out the electric coating. y It has also been alleged, that electricit will pervade a vacuum, and he transmitte substance of the best conductor; but Mb Walsh found that the electric spark or shoel would no more pass through a perfect vacuur than through a stick of solid glass. In othe instances, however, when the vacuum ha been made with all possible care the exper ment has not not succeeded. eS It has also been observed, that many of th forementioned substances are capable of beit electrified, and that their conducting pow may be destroyed and recovered by differe! processes: for example, green wood is a C01 ductor; but baked, it becomes a non-condui tor; again, its conducting power is restore by charring it; and lastly, it is destroyed b) reducing this to ashes. P| Again, many electric substances, as glas) resin, air, &c. become conductors by bein) made very hot; however, air heated by gla: must be excepted. See on this subject, Priestley’s “ History | Electricity,” vol.i.; Franklin’s “ Letter,” p. 96 and 262, edit. 1769; Cayallo’s “ Co © CON ete Treat. of Electr.” chap. ii.; Henley’s Exper. and Obser. in Electr.” also Philos. rans. vol. Ixvii. p. 122; and elsewhere in the fferent volumes of the Transactions. ‘Conpuctor, Prime, is an insulated con- tetor, so connected with the electrical ma- ine, as to receive the electricity immediate! y m the excited electric. Convuctoks of Lightning, are pointed me- ‘lie rods fixed to the upper parts of build- gs, to secure them from strokes of lightning. ese were invented and proposed by Dr. anklin for this purpose, soon after the iden- y of electricity and lightning was ascertain- ; and they exhibit a very important and eful application of modern discoveries in sscience. ‘This ingenious philosopher hay- ; found that pointed bodies are better fitted receiving and throwing off thé electric fire mm such as are terminated by blunt ends or ‘surfaces, and that metals are the readiest 1 best conductors, soon discovered that itning and electricity resembled each other this and other distinguishing properties: therefore recommended a pointed metalline , to be raised some feet above the highest tofa building, and to be continued down » the ground, or the nearest water. The atning, should it ever come within a certain ance of this rod or wire, would be attracted it, and pass through it in preference to any er part of the building, and be conveyed ) the earth or water, and there dissipated, hout doing any damage to the building. ny facts have occured to evince the utility this simple and seemingly trifling appa- is. And yet some electricians, of whom Wilson was the chief, have objected to pointed termination of this conductor, erring rather a blunt end; because they ceive a point invites the electricity from clouds, and attracts it at a greater distance 1a blunt conductor. Philos. Trans. vol. p. 234; vol. Ixiil. p. 49; and vol. Ixviii. 32, ONE, is a solid body having a circular , and its other extremity terminated in aigle point or vertex. nes are either right, or oblique. | Right Cone, is that in which the right | joining the vertex and centre of the base, a to the plane of the base; as | Cc. % Ama } Right Cone may be conceived to be ‘rated by revolution of the right-angled igle BDC, about its perpendicular BD. | thus, Euclid defines a cone to be a solid ‘e whose base is a circle, and is produced yhe entire revolution of the plane of a | CON right-angled triangle about its perpendicular, being called the axis of the cone. Right cones are distinguished into classes, according to the magnitude of the angle at the vertex, made by a plane passing through that point perpendicular to the base. An Acute-angled Cons, is that in which the angle ABC is acute, or less than a right angle. An Equilateral Cont has its side AB equal to the diameter of the base AC. An Obtuse-angled Cony, is that in which the angle A BC is obtuse, or greater than a right angle. A Right-angled Cone, is that in which the angle ABC is a right angle. An Oblique Cone, is that in which the line joining the vertex and centre of the base is not perpendicular, but oblique, to the plane of the base; as LMN. This solid, which is not treated of by the ancient geometricians, is evidently not included in the pre- ceding definition, that hav- ing reference to the right cone only. It has there- fore been an object with 1. the moderns to render the Oe above more general, so as to include both cases under one and the same general defi- nition or description, which is as follows: If a line VA continu- ally pass through the point V, turning upon that point as a joint, and the lower part of it be carried round the circumference A BC ofa circle; then the space enclosed between that cir- cle and the path of the line is a cone. The cir- cle ABC is the base of the cone, V the vertex, and the line VD the axis; ~ D being the centre of the Cc circle. Properties of the Cone.—1. Every cone, whe- ther right or oblique, is equal to one-third of a cylinder of equal base and altitude. And therefore the solidity of a cone is found by multiplying the area of its base by one-third. of its perpendicular altitude. 2. The curve surface of a right cone is equal to a circular sector, having its radius equal to the slant height of the cone, and its arc equal to the whole circumference of the cone’s base. And therefore this surface is equal to half the product of the slant side into the circumfe- rence of the base. 3. The surface of an oblique cone is not quadrable ; indeed no rule has yet been found that will even lead to a practical approxima- tion of its area, notwithstanding the attempts of several ingenious and able mathematicians. See a Memoire on this subject by Euler, in the Nouv, Mem. de Petersburgh; see also M \ N c Ce] R CON Agnesia’s “ Institutions,” and Leybourn’s «“ Mathematical Repository,” new series, vol. 1. 4, The solidity of a cone with an elliptic base, forming part of a right cone, is equal to the product of its surface by a third of one of the perpendiculars, drawn from the point in which the axis of the right cone intersects the ellipse; and itis also equal to one-third of the height of the cone multiplied by the area of the elliptic base. Consequently the above perpendicular is to the height of the cone, as the elliptic base is to the curve surface. See Dr. Barrow’s “ Lectiones Geometrice.” Frustrum of a Cong, is that which is formed by cutting off the upper part of a cone, by a plane parallel to its base; as the figure ABCD. To find the Curve-Surface of the Frustrum of a Cone. Multiply half the sum of the two periphe- ries AB, DC, by the slant height DA. To find the Solidity of the Frustrum of a Cone. Add the square of the two diameters and their product into one sum; then multiply this sum by the perpendicular altitude, and again by °7854, and one-third the product will be the solidity. Or thus, put the greater dia- meter — D, the less diameter = d, the height = h, and ‘7854 = p; then Surface = 2p (D + d) Solidity = $ph(D* + Dd + d’) Fi aes D3 — d3 Solidity = $ph x D Centre of Gravity and Oscillation of a Cone. See the respective articles. Conss of the higher Kinds, are those whose bases are circles of the higher kinds; and are generated by supposing a right line fixed in a point above, though conceived capable of being extended more or less, on occasion; and moved or carried round a circle. Cone of Rays, in Optics, includes all the aeveral rays which fall from any point of a radiant, on the surface of a glass. Cone, or Spindle (Double), in Mechanics, is a solid formed of two equal cones joined at their bases. If this be laid on the lower part of two rulers, making an angle with each other, and elevated in a certain degree above the horizontal plane, the cones will roll up- wards towards the raised ends, and seem to ascend, though in reality the centre of gravity is descending lower. CONFIGURATION, the exterior surface or shape that bounds bodies, and gives them _ their particular figure. CONGELATION, the transition of a liquid into a solid state, in consequence of an ab- CON + straction of heat: thus, metals, oil, water, & are said to congeal when they pass from — fluid into a solid state. With regard to fluids congelation and freezing meaning the sam thing. Water congeals at 32°; and there a few Jiquids that will not congeal, if the ter perature be brought sufficiently low. Eve particular kind of substance requires a diffé rent degree of temperature for its congelatioy which affords an obvious reason why parti lar substances remain always fluid, whil others’ remain always solid, in the comme temperature of the atmosphere ; and wi others are sometimes fluid, and at others so according to the vicissitudes of the season’ and the variety of climates. a CONGRUITY, in Geometry, is the same; identity, those lines and surface being 601 gruous which will coincide or fill the san space. CONGRUOUS Quantities, are those w are of the same kind, and therefore admit comparison; and quantities which cannot so compared, are incongruous quantities. abstract numbers are congruous; but crete numbers are not congruous, unless t quantities they represent be so. Thus, 3 a 4, as abstract numbers, are congruous;_ if they denote 3 pounds and 4 miles, theya incongruous. Hence it follows, that them thod commonly given in books of arithme for stating questions in the Rule of Three, improper; because it supposes a compatt between quantities which are ihcongruous. We cannot say properly, that 3 pounds men :: 6 pounds : 8 men; but that 3 pom : 6 pounds :: 4men: 8men. See Bor castle’s ‘ Arithmetic,” 8vo. edition. ; CONICAL, any thing of a conical form, relating to the cone. i Conical Ellipse, Hyperbola, Parabola, \ note those figures, under their most sim form, as cut from the cone, to disti gu them from the same figures of a higher ore See the respective articles. ConicaL Ungula, or Conic Ungula, is lid formed by a plane passing through thes and base of a cone; as the figure ECBF y «| To find the Solidity of a Conic Ungwllal In a table of circular segments caleulek to radius 3. a Put P for the tabula segment whose ver" sine = BD Hd AB ; @ for that whose versed sine is BD—AB4 —ABFT! GC Put also diameter AB =D, diameter | = d, height CL = h, and °78539, &c. = CON ‘Then we have i P D3 Q d3 ovens anetebe x Sissel ctilaninnh Whisoed Dlaelt 4 poz — the content of the ungula EF CB. Dz ears} 3 eee ee fro d3\—P.D3+Q.d ot x t = — the content of the ungula EFCGA. e And these, when the cutting plane. passes through the extremity of the hase of the frus- trum, : so that BD = BA reduces to the more _ Lo find the Curve-Surface of Conic Ungula. The same notation remaining, we have in apart of the base, /(4l? + (D —d)’) D—d (4 (D + d)—AD as = mal aide eran o ( d— AD ) x * V 7=aD S58 of the circle AB, whose height is a (dr AD) face of the frustrum, will be the surface of the other ungula. And here again, when BD = AB the above formula reduces to by (4° +(D—d)?) D—d ‘= surface ABC. py (4h? + (D—d)’) D—d = surface ACG. See an investigation of these formule, Dr. ‘Hutton’s “ Mensuration,” p. 163, et seq. third edition.- See also Bonnycastle’s “ Mensura- tion.” | _CONICS, the theory: of the conic sections. “Contc Sections, as the name implies, are | such curve-lines and plane figures as are pro- ‘ duced by the intersection of a plane with a j Cone. From the different positions of the cutting pete, pie arise five different sections; viz. angle, circle, ellipse, parabola, and hyper- r only the three latter are particu- : larly denominated conic sections. ‘1. If the secant, or cutting plane, pass through the vertex of the cone, and any part of the base, the section will evidently be a ‘triangle; as AV B. P. nié Ris ee Pa oS x (as vDd—d*) simple form. | a x 0'2618 Dh = greater ungula r aa P* Ee (D v Dd—d’) x 02618 dh = less ungula D—d x) 3 ae x 0:2618h = difference of the angula. the first case, where the cutting plane cuts off 2 x } seg. FBE —o = the curve-surface FE BC. And the diffe- rence between this and the whole curve-sur- see Bo 2. If the plane cut the cone parallel to the circular base, the section will be a circle; as ABD. | Re A af DON 3. Ifthe plane pass through theside and base of the cone parallel to the other side; that is, if the cutting plane make the same angle with the base as the side of the cone makes,the sec- tion is a parabola; as DAB. E 4. When the plane cuts both sides of the cone; that is, when it makes with the base a less angle than the side of the cone makes, the section is an ellipse; as BAD. A B 5. When the plane passes through the side and base of the cone, making a greater angle with the base than the side of the cone makes, the section is an hyperbola ; as DAF. e a B E And if the plane be continued to cut the opposite cone, this section is called the oppo- site hyperbola to the former; as Bed. _ 6. The vertices of any section are the points where the cutting plane meets the opposite sides of the cone; as A and B in the preced- ing figures. Hence the ellipse and hyperbola CON have each two vertices, but the parabola only one, unless we consider the other as at an infinite distance. 7. The axis, or transverse diameter, of a co- nic section, is the line joining its vertices; as AB; therefore the axis of an ellipse is within the figure, of the hyperbola without it, and in the parabola it is infinite in length. 8. The centre of a conic section is in that point which bisects the axis. Hence the cen- tre of an ellipse is within the figure, of the hyperbola without the figure, and in the parabo- la it is at an infinite distance from the vertex. The definition of the other lines, in and about the conic sections, will be found under their respective heads; and the principal pro- perties of the different sections, under the articles ELLIPSE, HYPERBOLA, and PARABOLA. The conic sections are of themselves a sys- tem of regular curves allied to each other, the doctrine of which is of the greatest use in physical astronomy, as well as in the physico- mathematical sciences, and has been much cultivated by both ancient and modern ma- thematicians. The first of the ancients who is mentioned as having written on the conic sections is Aristeus, who composed five books relating to this subject; but they have not been handed down tous. The most ancient treatise that has reached our time is that of Apollonius, containing eight books, the first four of which have been frequently re-published; but Dr. Halley’s edition hasall the eight. Pappus, in his “Collect. Mathem.” lib. vii. says, that the first four of these were written by Euclid, though perfected by Apollonius, who added the other four to them. Among the moderns, the chief are “ Mydorgius de Sectionibus Conicis;” Gre- gory St. Vincent’s “ Quadratura Circuli et Sectionum Coni;” “De la Hire de Sectioni- bus Conicis;” “Trevigar Elem. Section. Con.;” De Witt’s “Elementa Curvarum ;” Dr. Wallis’s “Conic Sections ;’? De ?Hopi- tal’s' “ Anal. Treat. of Conic Sections ;” Simson’s “Section. Con.;” Milne’s “ Ele- menta Section. Conicarum ;” Muller’s “Conic Sections;” Jack’s “Conic Sections ;” Emer- son’s “Conic Sectious;” Steel’s “Conic Sec- tions;” Dr. Hamilton’s elegant ‘Treatise ;” Dr. Hutton’s, Mr. T. Newton’s, Mr. Vince’s, Mr. Walker’s, and Dr. Abraham Robertson’s. ‘There are three methods of investigating and demonstrating the various properties of the conic sections. The first is to consider them as they are really cut from the cone itself, which is the way adopted by all the ancients ; and is pursued very elegantly by Dr. Hamil- ton, Dr. Hutton, and others among the mo- derns. In the second method, the properties are deduced from arbitrary descriptions of the curves in plano; the properties belonging to these curves being shown to apply to those actually cut from the cone: this method has lead to able treatises from Boscovich, Simson, and Dr. Robertson. In the third method, the chief properties are inferred from the different modifications of the general alge- CON braic equation of lines of the second order, and the established analogies between the pro-_ perties of equations and those of curves. ‘This — method has been admirably elucidated by) Euler, Prony, and Lacroix. ‘To which works _ the reader is referred for farther information. — CONJUGATE Azis of an Ellipse and Hy~ perbola. See Axis. o ConJuGATE Diameter of a Conie Section. See DIAMETER. , ConsuGATE Hyperbola. See HYPERBOLAY — CONJUNCTION, in Astronomy, the meet- ing of two or more stars or planets in the same degree of the zodiac. %: CONJUNCTION may be considered as either true or apparent. When the two bodies meet in the same point of both longitude and lati- tude, the conjunction is true; when they agree in longitude, but differ in latitude, the con-) junction is apparent. fis CoNnJUNCTION is either heliocentric or geo= centric. Heliocentric conjunction is that which would appear to an observer at the sun; geo- centric, that which would appear to one upon the earth. Geocentrie CONJUNCTIONS are either superior | or inferior: thus, when a planet is seen on the same circle of latitude with the sun, but be- yond him, the conjunction is called superior; when the planet is seen between the earth and sun, the conjunction is znferior. Grand CoNJUNCTIONS, are those wherein several of the planets are seen near together. M. de la Lande informs us, that on May 22, | 1702, Jupiter and Saturn were within 1° 4’ of) each other; on February 11, 1524, Venus): Mars, Jupiter, and Saturn, were very near each other, and Mercury not above 16° from them; on November 11, 1544, Mercury, Ve- nus, Jupiter, and Saturn, were within the’ space of 10°; on March 17, 1725, Mercury, Venus, Mars, and Jupiter, were so near each other as to be all seen through the same tele- scope without altering its position; and on December 23, 1769, Venus, Mars, and Jupi- ter, were within 1° of each other. ee The Chinese have a remarkable record of a conjunction of five planets, which happened in the time of their emperor, 'Tehuen-hiu; who, according to the Chinese annals, reign ed from the year 2514, before Christ, to the year 2436, B.C; and which is thought to prove the great antiquity of this empire, and of astronomical science amongst these peoples A record so singular could not fail to engage} the attention of modern astronomers, who have made laborious calculations in order to ascertain whether or not such phenomenon really took place about this period ; Cassini’s computation seemed to discredit the fact, but) others have since been made by M. Muller, Desvignoles, Kirch, &c. more favourable to) the Chinese history. Muller found that such a conjunction must have happened about the year 2459; Desvignoles, in using the tables of Laniberg, found that the moon and the four planets, Saturn, Jupiter, Mars, and Mercury, were within 14° of each other im the constel- CON ion Pisces, February 28, 2459, B.C.; which culation was afterwards confirmed by Kirch, the “ Miscellane Berolinensia,’ tom. v. her calculations have been since made by we correct tables, which make the con- iction to have happened, February 8, 2461, C.; and therefore, upon the whole, there ms reason to conclude that such an event dly happened about the time recorded by » Chinese; and their little skill in astrono- eal calculations will not allow of our sup- sing, as some have done, that their know- ige of the fact was derived from backward mputation. See Montucla’s ‘ Histoire des athematiques,” tom. i. p. 452. CONOID, a solid figure generated by the yolution of any conic section about its axis ; d hence receives particular denominations, cording to the seetion from which it is pro- eed; as ELLipticaL Conoid or Spheroid, Pa- BoLic Conoid, and HyPERBOLIc Conotd ; for lich, see the respective articles. CONON, an ancient Greek mathematician, y friend of Archimedes. He was well skilled geometry and astronomy, but is not cele- ated for any particular discovery. CONSECTARY, the same as COROLLARY ; ich see. CONSEQUENT, is the latter of two terms 'a ratio, or that to which the antecedent referred and compared; thus, in the ratio _b, bis the consequent, and @ the antece- nt. CONSEQUENTIA, a Latin term, com- mly employed by astronomers to denote > real or apparent motion of a planet or met, when it is moving from west to east, according to the order of the signs; and is as opposed. to anteeedentia, which denotes a ntrary motion. See ANTECEDENTIA. CONSISTENT Bodies, aterm partially em- yed by philosophers, particularly by Boyle, denote firm, or fixed bodies, in opposition fluid ones. CONSTANT Quantities, in Algebra, are ose whose values are known, or which re- ain constantly the same. These are com- only denoted by the leading letters of the phabet, a, b, e, &e. to distinguish them from ie variable and unknown quantities, which 2 represented by the final letters, z, y, x, Ke. ert At ION, in Astronomy, an as- mblage or system of several stars, expressed drepresented under the name and figure of me animal or other emblem. ‘The ancients portioned out the firmament 0 several parts or constellations; reducing ertain number of stars under the represen- jtions of certain images, in order to aid the )agination and the memory, to conceive and tain their number and disposition, and even , distinguish the virtues which astrologers rece to them; in which sense a man is sid to be born under a happy constellation, ». under a happy configuration of the hea- ynly bodies. 'The division of the heavens into constella- yns.is very ancient, and probably as old as CON astronomy itself; at least, it was known to the most ancient authors extant, whether sacred or profane. In the book of Job, men- tion is made of the names ofsome of them; as in chap. ix. ver.9, “ Which maketh Acturus, Orion, and the Pleiades, and the chambers of the south.’ By the “ chambers of the south,” some have understood the constella- tions near the south pole, which are invisible to the inhabitants of the northern hemisphere. From the manner in which Job speaks of commerce, we may infer that he lived in a country frequented by merchants, who im- ported thither the rarieties of the south. ‘To this purpose, Sir Isaac Newton suggests (* Chronology,” p. 157), that Job, who lived in Arabia Petraea among the merchants, might have derived from them his knowledge of the constellations. And again, mention of them occurs in that sublime expostulation, chap. XXXViii. 31, 32, ‘“‘ Canst thou restrain the sweet influence of the Pleiades, or loosen the bands of Orion? Canst thou bring forth Mazzaroth (by which, some understand the twelve signs of the zodiac), or canst thou guide Acturus with his sons?” In the Prophecy of Amos, who is supposed to have lived 790 years B.C., we have the fol- lowing exhortation (chap. v. 7, 8): “‘ Ye who turn judgment into wormwood, and leave off righteousness in the earth; seek him that maketh the seven stars and Orion, and turn- eth the shadow of death into the morning, and maketh the day dark with night; that called for the waters of the sea, and poureth them out upon the face of the earth: the Lord is his name.” In this passage, the seven stars and Orion are mentioned as being well known, both by Amos, who was a herdsman of Tekoa, and the common people to whom this exhor- tation was addressed ; and we may hence in- fer, that the constellations had been invented for some time before that period. Some of the constellations are also occasionally men- tioned by Hesiod and Homer, who flourished above 900 years B.C.; and Aratus of Tarsus, the astronomical poet, who lived about 277 years B.C. in his “ Phenomena,” professedly treats of them all, except some few which were invented after his time; showing how each constellation is situated with regard to those that are near it, what position it bears with respect to the principal circles of the sphere, and what other constellations rise or set with it. Hipparchus, the Bithynian, has shown that Aratus followed the descriptions of Eudoxus, who flourished about 366 years B.C.; and it is very probable that the Greek astronomers who succeeded him, continued to use the same figures of the constellations till the time of Ptolemy, though not without some varia- tions and additions. “Ptolemy’s “ Almagest” has been in such esteem among astronomers, that almost all who have written since his time have agreed in drawing the figures of the constellations, or supposing them to be drawn, so as to answer his description, as far as possible ; and indeed this is necessary, in \ a con” ; erder fo avoid confusion, when ancient and modern observations are compared. The divi- sion of the ancients only took place in the visi- _ ble firmament, or so much as came under their notice: this they distributed into forty-eight constellations; those being reckoned ancient constellations which have been received from the Greeks, and particularly from Ptolemy ; twelve of these took up the zodiac. The other stars, on the northern side ofthe zodiac, were disposed into twenty-one constellations ; and those on the southern side were distri- buted into fifteen constellations: the names of which will be seen in the following table. The other stars, not comprehended under these constellations, yet visible to the naked eye, the ancients called informes, or sporades ; some of which the modern astronomers have since reduced into new figures, or constella- tions. Thus, Hevelius forms between Leo and Ursa major, Leo minor; and between Ursa minor and Auriga, over Gemini, he makes Lynx; under the tail of Ursa major, Canis venatici, Cerberus, Vulpecula, and Anser, Scutum Sobieski, Lacerta, Camelopardalus, Monocerus, and Sextans. In these constellations, the stars are com- monly distinguished by that part of the image wherein they are found. Bayer distinguishes them farther by the letters of the Greek alpha- bet: and many of them again, have their pe- culiar names; as Arcturus between the knees of Bootes; Gemina, or Lucida, in the Corona Septentrionalis; Palilitium, or Aldebaran, in the Bull’s eye, Pleiades in the neck, and Hyades in the forehead of the Bull; Castor and Pollux in the head of Gemini, Capalla in the shoulder of Auriga; Regulus, or Cor Leonis ; Spica Virginis in the hand, and Vin- demiatrix in the shoulder of Vi irgo; Antares, or Cor Scorpionis; Fomelhaut in the mouth of Piscis-Australis; Regel in the foot of Orion; Sirius in the mouth of Canis major; Procyon in the back of Canis minor; and the Pole- star the last in the tail of Ursa minor. But as the stars, grouped in the constella- tions, are capable of being reduced to very different figures, those of the ‘Chinese and Japanese are very different from ours; and some superstitious Arabians, though they re- t ceived their astronomy from the Greeks, changed all such on the celestial ele deities and heroes, have proposed from sw ed the ancient figures, to refer them tos 4 to represent the blessed Virgin, &c. the zodiac, substituted those of the twel the constellations in the heavens. Thus Ar the manger of Christ; Hercules, into | University of Jena, made a new order of € stellations. Thus, Ursa major he transform cross of Cologne; the Triangle, into co : However, the more intelligent among a 4 fore, are still retained; both because be et CON given some of their constellations differs figures, because they thought it unla draw any human figure; and therefore # some other form. Some Christian astro) mers displeased to see the heavens of t fixed stars occupied by the fabulous heath stitious zeal, but withont a due regard to { science of astronomy, to introduce a re mation in this respect; and whilst they ret 4 scripture history. With this view, they wou have Aries, or the Ram, to be a memori ial that which was offered instead of Isaac; V ir venerable Bede, instead of the profane nz a and figures of the twelve constellation $ apostles; whose example being followed Julius Schillerius in 1627, he completed t reformation, and gave scripture names t0 or the Ram, became converted into St. Pet Taurus, or the Bull, into St. Andrew; And meda, into the sepulchre of Christ; Lyra, i Magi coming from the East; a ; David, &e. Weigelius, professor of mathematics in’ stellations; converting the firmament int celum heraldicum; and introducing the ai of all the princes in Europe, by way of into the elephant of the kingdom of Di mark; the Swan, into the Ruta, with swo of the house of Saxony ; Ophiuchus, into’ passes, which he calis the symbol of artifice and the Pleiades, into the Abacus Pytha ricus, which he calls that of merchants, & nomers never approved of innovation; as ing no purpose but to introduce con into astronomy. ‘The old constellations, could not be substituted! and likewise to ke up the greater correspondence and uniform! between the old and new astronomy. | Northern Constellations. “é Zodiacal Constellations. CON TABLE CON OF THE ANCIENT AND MODERN CONSTELLATIONS. Aries.. Names. eee men tee eeeee @eeeeee eee ee eeeses fie Ley laa Ri aarti ate 7 riy oiaae Gemini...... eee ieee an Cras ote Cancer TOPE OST OEE ee Cee eee eee Tee EEE e Tee) See eee eee) aes eb) Le rate ttiey ean 3g same, SU AWICUATIUS « .tueo eter cnonceapaosneie cts CapricornuS..........cupsessesneasas : bgt Tt Ra pity aE A a mery ee hiaae Aquarius.......... Pee gehts 2 Ye Teles SEAEOR SA bade lac ceases "ese0 Ursa major Perseu woes cover eeecceceececesces Ss Cec atondceseses oe es easBbicesccvee Se OAR Oe eae Heese BET EHESSHH EHH EE Pe eee eeer ais eseenseteerereeReseete Cepheus ..........seseesssecerecarcees *Canis ae as i.e. Asterion et Char Cor. Crcroli.. ae cd civ okttee sd Feet Triangulum........0.cecsssereee stees Triang Musca Lynz.. Leo mi WULUNE MANUS .ceccccecsccvceees PORE ewe Se SESHTS SHEESH HEHEHE ES ae e eee re eee Se SSH eoeesereersenees TLOP «.os0008 YT eee ree ere eT eee Coma Berenices .....cccserseccecsees , Camelopardalus.......ccserereseseeees Mons Menelaus .......cccccessercsces TSOPOMA TROTEGRNUS)..,.cbcc-swercceeeed Serpen Sccncorecccccccenccsccsecssceetace Seutum Sobieshi...... We ci nae heed Hercules cum Ramo et Cerbero.. Serpen Taurus Lyra.. tarius s sive Ophiuchus.. Poniatowshi .. Seem eee et eee HHO e THE Hee sereeeeeeee Vulpeculus et ARSCR oTWeise ctleees Sagitta Aquila eee eee eer eeeee Pee rsevesvea tse eoe CUM ANEINOUS .....0.0000s 2 Delphinuss .........ceseeseesesstesenene RIS oi carta chicas shivams yen ashen Cassi0pea....ce.cseeee jasekdhiea \avoeans Equulus...... I UMEDA Sas DPT | is eT) EP) Seer) ee Pegasts.......e-.e0 NF ie RR? ANndromed@.........essds.scrsevneress * Those in Italics are modern constellations. Number of the Stars. Ptolemy. Tycho. Hevelius. British. LS sep: heey WR a: 6) 4A es 40 ap DL n.. LSI Bb i Rah oh, ETE anes 19"). 26.9 202.00) 68 ee een | Berge | See ne, toh ay, Meee, BLD EF sO ee aa th ogg sh > ak) eenkey ee rs 3 Nee Se ee) a 6 8. os esa ee an ee BS A AT nck LOB DD 540,-GO Nags ae 8 kW Edis 24 BS e280 FSe > OF 29 ... 29 ... 46 .., 69 1A 009 D040 04.66 23 18 ... 52... 54 B31... B2ii.6 40... 80 13> en 4 cD .i0)| do — ... 23... 25 _ —..—.. «6S 4 4h cA Siu 3G — ~~ — 10 — — — 6 — —- 19 ... 44 _- — — 53 —~ 14.°%...20.. 48 — — 39 0 08 — — — ll 8 is het ocak 13 6. TS 22 OF ooo —.. Ts 8 20 si) SS AO ae Lk 29 Fi 1S. ...: 40 74 10 sa PE, LLG oe 2S as yy ME Lee ge OF B she Oday see LO — .. 12 ... 23 Citak oer oe 10... 10... 14... 18 100 eke ie Se oes OL 13 wel ae ean sot xs.) OD 457) 4) acre... 10 a —..—.. 16 OY ss Wie OO. 04 se Ps We ee: ier OPA Principal Stars. Mag. Aldebaran... add | Castor and Pollux. 1,2 FROguias,, ne whAns-.. 1 Spica Virginis......... 1 Zubenich Meli........ 2 Antares....... Se 1 Schengen ipod «cscs 3 Pole-Btani, ss .cesscoss 2 MPLS. Best sk eb cus < ecceaet Agen CT iisss sae 2 | A APTOS). eed 8h. vues i Arcturts........ Wt. sake ag | Ragtaber cent... s.5.. 3 Alderamin............0. . Ras Algiatha.......... 3 Ras Alhagus........... 3 VOSA icc eeas cbawerevecaes 1 PA EATT is ctish ote thle rere 1 Deneb Adige.......... ] Markab....:...sccsceees re AIMAac ..tisbscccceseers 2 Southern Constellations. CON yy 3 TABLE OF THE ANCIENT AND MODERN CONSTELLATIONS. ‘a Names. | Number of the Stars. Principal Stars. | Ptolemy. Tycho. Hevelius, British. Mag. ORME cel society ceed ohh a articrv'ess ba8 — oo — 7 Officina Sculptorid .....ccceeeeeeees — see — ene ove 12 | Beri AWS. i ilok desc cdecce concvsdeecos une. she BLU buted (kpc te of SOIR se once 1] FLY Mrs ....0cceesvesssssevee ceases erecee — un ee oe 10 é' | CRE a -s hee tdas sheen ote e¥n paSenoes tise | Meee ise ed 49%,.. 97 | Menkat: wsncasceanses » 2) Fornax Chemicd ....ceccccseesess cose | ae — ... 14 a) EXOFOLOR GUI RR 05s RG Gave ives on cde — a — ... 12 | spat Rhomboidalis ........04 — — — ... 10 i Atphias Dorado... sab. .cWce-tens — —. — 7 g } Colapraxitellas ....1. :<.sussssescesses — — — ... 16 | Lepus svdeeh os eaatwin Canes UkGGua Spee sees 6 ae t, ls CE OALED vay 19 a Ookumba Woacht..i.. ii. desthecdecken — — — ... 10 j OLION . i, 0+0ks 000 avs onhes GsGvdgsetsphtee |) OGu pe, ee sd Oe (ke D) | ASO LELOOUME 5 5... cceres 1 ATED INGVIS$-isconansencpnspareaeies . | 45 3... 4... 64 | Canopus........0 See 1] Calis MApOMs. vildie sc kbipweavnae ne 29 1S 1.0 21. | secon aWiiset + te lft Equuleus Pictoriwus......00. 00000006 — ..— ww. — «8 1 Monoceros .......00+ dodek cat entieaman hs — — 19 ... 31 Canis minor,..i. css aes 2 2 13... 14 | Procyon...... CAlasekie aes } Chameaclon see cess seer sateadtabas — — — 10 Pyais, Nauticand...cpseeanins iter ee ee 4 Piscis Volans ....... ethgentnpsuatipecgsh. wetlaby. Cor ikes, “> sabe’ O. RAWULA ..0netuns ces a davaaes Tae caepibed S beta LO te. 81 2. GO”) Coriimeens ats onbh dc iele Welds 2. cess esassed Fee iss ecetshartl :| —As LD Robur Carolinum ...... bent sant ease — 6. — — 12 Machina Pnewmatiea....cccccccoses |) —— oe. — —. 38 A CMler ss Ness ecesnss Gveeeeicis Teahbs es 7 3 10, oc..ST ATR not coe cine teen COT VINE cnc cactechecuonhuseeain treme 7 Ang nee OD | ATR GEMINI is coscus tins tosceuen Crosters el Cruzero.seccccecccerseces — — — 6 PAPE LMSC Re cuaekenoha te saaknage s ee ee 4 Apus, or Avis Indica............004 — .. — ae we DM s CUrcinus...corcesvvee Speedin ctsbes A. mee: — —.. 4 Centaurus... reer ee eee Sy a — ... 35 PSUPUSs< ccc coubenceectsendtmmares es teses 19 ... — —- ... 24 Quadra Buclidis ......c0cceccsseceese — — — ... 12 Triangulum Australe .....4... were | meee a 5 AG. 052. ceeen oe AS eee Aare 2 , — 9 Telescopnusns petieet ses cosberes tein, a ee 9 Corona Australis........ccceeeeeeee 13 1. — — 12 * DUPRE RG a is Pe gal wh a — ww. — vo 14 Tidus. ,, ieee Oe her ee — — ... 12 MRCrOSCOpiUme.... -ripesdeets: Seessnnes — a ee ee 10 Octans Hadleianus .....cc.0ecseceees — — 1. — ... 43 RPPUS Sy cS sadoe thee hi RE cae pois — — ... 14 Taher re ee Reece cid ae <— —. 9 Piscis Australis ....... Rustscvtyeok 18 .... — — ... 24 | Fomelhaut........ Sa ee Const Magnitudes. Total lef. 20s Bde). ath. 5th. 6th. } Zodiac British Catalogue ........ 12 onc... 6,016,494, F205 163; 646"... 1014 — Northern ....... nan rab lak pene cee ap BA. Aas vs 6,124," 95," 200, 2OL, Goo) cca sssce 1251 Southern...... aPacanscivewbedertiaeeens BEND Gi eeavave’': 95/0; 4.” EON, eek, s Oe lsteecuace 863 20, 76,223, 510, 695, 1604 3128 | CON a “CON CONSTRUCTION of Equations, in Alge- va, is the finding the roots of equations by ieans of geometrical figures, which is effected y lines or curves, according to the order or egree of the equation. To construct a Simple Equation—This is ‘one by resolving the equation into a propor- on thus: the general form for simple equa- ons being az =), or ax = 1.4, it is only ‘ecessary to find a fourth proportional to the iree quantities a, 1, and 6; which fourth pro- ortional will give the value of x. To construct a Quadratic Equation.—TIf it be simple quadratic, it will reduce to this form 2—ab; whence a:x::2:6, or x= ab, mean proportional between a and 6; which a well known geometrical theorem. 2. If the quadratic be affected, let it be a? -2ax = 6b’; then form the right-angled jangle whose base AB is a, and perpendi- ilar BC is 6; with the centre A and radius C, describe the semi-circle DCE; then DB id BE will be the two roots of the given aadratic ; x being = V(b? + a*) ta. OY eaipe tein ‘Me, ®. A B 3. If the quadratic be x? — 2ax = b*, the mstruction will be the same as that of the receding one, x —a + (b> + a’) having ie positive and one negative value. 4. But if the form be x* — 2ax = — Bb’, or tz — x* — b*; construct a right-angled tri- igle whose hypothenuse FG is a, and per- ndicular GH is 6; with the radius FG d centre F, describe a semicircle IGK; en shall IH and HK be the roots of the ven equation, x being =a + /(a* — Bb’), this form, if a be greater than 6, the equa- m will have two positive roots; but if a be 3s than 4, the solution is manifestly impos- dle. rt “Gs w ms | i as \ 1 PRE ’ Construction of Cubic and Biquadratic quations. The roots of a biquadratic equa- /m may be determined by the intersection of 0 Conic sections ; for the equation by which © ordinates, from the four points in which jase conic sections may cut one another, can | determined, will arise to four dimensions, 4d the conic sections may be so assumed as ‘make this equation coincide with any pro- |sed biquadratic; so that the ordinates from on four intersections will represent the roots of the proposed biquadrate. CON Let M M'M’M” be a parabola whose axis is AP, and M M'M’M” a cir- cle whose centre is C and radius CM, cutting the para- bola in the points M M’ M’ M”; and from these points draw the ordinates to the axis MP, 1} 4 sll AP’, MP”, and from C draw CD perpendicular to the axis, and CN parallel to it meeting PM in N. Let AD =a, DC =}b, CM = 2, the pa- rameter of the parabola =p, AP =z, PM=y; then px = y*. Also CM? = CN? + NM?, Hoe 4b)? 5 or v* — 2ax + a + y? —2Qby + b% =n, 2 a ‘ Y and substituting for x its value Y and arrang- or n* = (x — a)* ing the terms according to the powers of y, this equation becomes y*—(2pa—p*)y*? —2 bp*y + (a* +b*—n*) p= 0 which is a biquadratic equation, whose roots are PM, P’M’, P’M”, and P”“M”. Which equation may be made to coincide with any biquadratic equation whose second term is wanting. Wood’s “ Algebra,” p. 295, If one of the intersections of the conic sec- tion fall upon the axis, then one of the ordi- nates vanishes, and the equation by which these ordinates are determined will then be only of three dimensions, or a cubic, to which any proposed cubic equation may be accom- modated, so that the three remaining ordi- nates will be the three roots of the proposed cubic. Vieta, in his “ Canonica Recensione affec- tionum Geometricarum,”’ and Getaldus, in his “Opus posthumum de Resolutione et Com- positione Mathematica,” and also Des Cartes, in his “ Geometria,” have shown how to con- struct simple and quadratic equations. Des Cartes has also shown how to constuct cubic and biquadratic equations, by the intersec- tion of a circle and parabola; and the same has been done by Baker, in his “ Clayis Geometrica,” or Geometrical Key. But the genuine foundation of all their constructions was first explained by Slusius, in his ‘‘ Meso- labium,” part ii. This doctrine is also treated of by De la Hire, in a small treatise, entitled “Ta Construction des Equations Analytiques,” annexed to his “ Conic Sections.” Newton, at the end of his “ Universal Arithmetic,” has given the construction of cubic and biqua- dratic equations mechanically; and also by the conchoid and cissoid, as well as the conic sections. See also Dr. Halley’s construction of cubic and biquadratic equations, and Col- son’s, in the Philosophical ‘Transactions ; the Marquis de l’Hospital’s “'Traite Analytique CON des Sections Coniques,” lib. iv.; and Maclau- rin’s “ Algebra,” part iii. chap. 3. CONSTRUCTION of Geometrical Problems. What relates to the construction of the sim- ple elementary problems, such as inscribing and circumscribing polygons in circles, the bisection of arcs, angles, and lines, and va- rious other cases of this kind, will be found under the respective articles in this Dictionary ; we shall, therefore, limit our remarks in this place to a rather higher order of problems, in which from certain parts of .a figure being given, others are required to be found, and the whole figure to be constructed. direct method of arriving at the construction in problems of this kind is by the Geometrical ANALYsIs, which is illustrated under that article; but when this is once discovered, it is commonly put into the more elegant form of synthesis, of which we mean to give an idea in the present article. It will be observed, with regard to what is said above of the geo- metrical analysis being the only direct me- thod, that we do not mean to assert that it is the only one that can be employed; in many cases, the geometrican discovers the construc- tion from a slight examination of the data, and in these cases the analysis becomes use- less, and he immediately puts it into the syn- thetic form. All we mean to say is, that with- out the analysis the whole is left to the inge-_ nuity of the student, and much will depend upon practice, but more particularly upon an accurate knowledge of the elements of geo- metry. Pros. lL, Giving the base of a right-angled triangle, and the difference between the hy- pothenuse and perpendicular, to construct the triangle. Const. Let AB be the fy given base, and BD the difference of the other 8 twe sides. From the extremity B, of the given base AB, demit the perpendicular BD, equal to the given difference, and produce it 5 indefinitely towards C. Join AD, and from the point A draw AC, making the angle DAC = ADC, and meeting BD produced in C; so shall ABC be the triangle required. Demons. Since the angle DAC = ADC, the side AC = CD (Eue. vi. 1), therefore the difference between AC and CB is equal to BD, but BD is equal to the given diffe- rence by construction, and AB is equal to the given base; also the angle Bis a right angle: consequently ABC is the right-angled tri- angle required. if the difference BD is equal to, or exceeds the base AB, the problem is impossible; for in either of those cases the line AC would not meet the continuation of B D, which is neces- sary for the construction. This limitation arises from a known geometrical problem, though it is not found in Euclid’s “ Elements ;” Bo gare! A B The only” ‘I draw IH perpendicular to AB, meeting CON be viz. “The difference of any two sides of a } triangle is less than the third side.” a4 Pror. 2. To inscribe, in a given triangle, a rectangle equal in area to a given rectangle. Let ABC be the ant given triangle, and ABK a reciangle, equal to the given rectangle; if is re- quired to inscribe, in the triangle ABC, a Ss rectangle equal to the a rectangle ABK. i Const. rom C let fall the perpendicul CK, cutting the base AB in D; on CD ar CK, as diameters, describe the semicire] CHD, CIK, the latter cutting AB in I; fro lesser semi-circle in H. Through H drs EHF parallel to AB, and draw EL, F perpendicular to E}’; so shall EF LM bet rectangle required. Demons. By Euclid, xiii. 6, we have HG* =CG.GD, and 1D? =CD.DKy Now since HG = ID, we have also CG. GD wl OR EINE BES a | \ 1, " 1 | | consequently....... CG: CVURDE: GD but by sim. trian... CG: CD:: FF: AB | therefore ............ DK:GD:: EF: AB Whence, by taking the rectangle of the ex tremes and means, we have DK.AB=EF.GD. iy But DK .AB is the measure of the reet angle ABK, and EF .GD is the measure the rectangle EF ML; that is, the rectangh EEFML is equal to the giyen rectangle A BK and it is scribed in the Bven triangle ABC as required. e if the line HI produced, cut the semicirél CHD in two places, then two such rectal gles may be inscribed; if it touches the cirel only, then there can be but one rectangle and that will be the greatest possible; ane HI fall beyond the circle, then the problet is impossible. a: In the following problems we shall give onl the construction, leaving the demonstrati@ of them for the amusement of the reader. — Pros. 3. Giving the lengths of two li drawn from the acute angles of a righ angled triangle, to the middle of the opposit sides, to construct the triangle. % Const. Let AD and FB be the two given lines, on one of which, as AD, describe the semi-circle ACD; bisect AD A in E, and on AE 3——————__-¥". @ describe the semi-circle AF E. Take | = ZED, and FG = $F B, and apply By from G to the semicircle A FE, cutting it in| Through F draw the line AFC, and throug D and G the lines CDB and FGB, meeti each other in B; join AB, and ABC will the triangle required. a G a CON nos. 4. Giving the hypothennse of a right- led triangle, and the side of its inseribed wre, to construct the triangle. jonstruc. On the m hypothenuse | describe the le ACD, and ‘athe centre G iit the perpen- iar GD ; join }, and perpen- ilar to it, at the . it B, draw BE = to half the diagonal of given square ; join DE, and produce it to making EF — BE. Then from D as a hire, and with DF as a radius, describe the bEC cutting the circle in C; join AC, BC, | ABC will be the triangle required. or more on this subject, see Simpson’s bra,” “Geometry,” and his “ Select reises ;’ and for the algebraical solution 1ese problems, see APPLICATION. ONTACT, the relative state of two things touch each other, but without cutting or ring, or whose surfaces join to each other hout any interstice. The contact of a ile and right line, or of two circles, or of spheres, is only in a single point. nele of Contact, is that made by a curve- aud a tangent to it, at the point of con- b 5 as the aigle THK, ign a | proved by Euclid, that angle of contact be- len a right line and a ile, is less than any ¢t-lined angle whatever; gh it does not therefore ww that it is of no mag- ide or quantity. This M #been the subject of great disputes amongst fnetricians, in which Peletarius, Clavius, )quet, Wallis, &c. bore a considerable re ; Peletarius, Ozanam, and Wallis, con- Ming that it is no angle at all, against ius, who rightly asserts that it is not Holutely nothing in itself, but only of no ‘nitude in comparison with a right-lined tle, being a quantity of a different kind or Wire ;.as a line in respect to a surface, or a ace in respect to a solid, &c. And since time, it has been proved, by Sir Isaac yton and others, that angles of contact « be compared to each other, though not to j t-lined angles, and what are the propor- i| s which they bear to each other. Thus, | circular angles of contact LHK, IHL, to each other reciprocally as the square (softhe diameters HM, HN. And hence circular angle of contact may be divided, ae intermediate circles, into any aber of parts, and in any proportion. 1 if, instead of circles, the curves were jabolas, and the point of contact H the 4imon vertex of their axis; the angles of a would then be reciprocally as the | nce are roots of their parameters. But in ytical and hyperbolical angles of contact, se will be reciprocally as the square roots | CON of the ratio compounded of the ratios of the parameters, and the transverse axis. CONTENT, a term commonly used to de- note the surface or solidity of bodies; so that to find the content of a body, is to find the measure or quantity of its surface or solidity. CONTIGUITY, the actual contact of two bodies. CONTIGUOUS, the state of two bodies or things which are in absolute contact. Contiguous Angle, the same as adjacent angle. See ANGLE. CONTINGENT Line, in Dialling, is the intersection of the plane of the dial and equi- noctial. CONTINUAL Proportionals, are a series of quantities, in which the ratio between each two adjacent terms is equal; as in the series Ly 2, Ay dr By ¥ 16, &e, b, ab, a*b, a3 b, ath, &c. See PROGRESSION. CONTINUATION, in the Fluxional Ana- lysis, or the finding of Fluents by Continuation, is a method of finding one fluent from another ; that is, the fluent of one fluxion being given, ihe fluent of another fluxion is thence to be determined. Few general rules can be given in these operations, but they principally consist in assuming some quantity y, in the form of a rectangle, or product of two factors, which are such, that the one of them drawn into the fluxion of the other, may be of the form of the fluxion proposed; then taking the fluxion of the proposed rectangle there will be deduced a value of the original fluxion, in terms. that will, frequently, admit of finite fluents. This will be better illustrated by a few ex- amples. 1. Let it be proposed to find the fluent of xs 2 P(xe™ Le d*\- / Here it is evident that by assuming y = a /(x” + a*), we shall have one part of the fluxion of this product; thatis, x x F v (a 4+-a7) of the same form as the proposed fluxion. Now putting the assumed rectangle into flux- : es oF x* a ions, wehave y =a /(x* +a") + VY (a? +a?) _ ata a x Rirrern ct Miee aay NY eas : petal doe me a* x T cad wees s/s J (x? +a?) / (a + a) Re z Whence ——— ther CLANS By 1 y—ha> x pba Sars | PROBE aOR. “EERO CONOR now the fluent of = hyp. log. x + v (x*-+0") therefore flu ae ee Ee x f(x” + a’) 1€ Ta pay fi stone J x” +-a*) —t a’ ; hyp. log. « + /(x? + a”). ; x32 / (x* +”), Assume y == 2 (a? +a”); then 2, Required the fluent of CON a 4 : asa y= Wx J (x tt GE ea awe mE oe = y — Wer /(a* +a’). 3 Now the fluent of Qa x / (x? 4-a”) = 3 (a* +a")? therefore flu. x3 x 3 TEE ROVE +O) 3 + For more on this subject see Dealtry’s “ Plux- ions.” f CONTINUED Fractions, are a_ certain species of fractions extremely useful in va- rious arithmetical and indeterminate problems, in the reduction of ratios, the extraction of roots, &c.; a description and illustration of which is given in the subsequent part of this article. Continued fractions seem first to have been introduced to the attention of mathematicians by Lord Brounker, about the middle of the seventeenth century, he having employed them in order to express the ratio of the area of a circle to that of its circumscribed square. Wallis, in “ Arithmetica Infinitorum,” has also some slight researches on this subject, and shows in an indirect but ingenious man- ner the identity of Brounker’s expression and his own for the area of the circle; but neither of these able mathematicians appear to have had any knowledge of the most important properties and advantages of the theory they were investigating. Huygens was the first who applied conti- nued fractions to any useful purpose, which he did in the construction of his planetary automaton. In order to have a machine that would represent with precision the planetary motions, it is obvious that the number of teeth in the several wheels should bear an exact pro- portion to the times of the planets’ revolutions ; but this, even to the nearest day, would re- quire many more teeth than could be intro- duced in a machine of this deseription, and it therefore became an important question, what number of teeth should be made use of to approximate to the true ratio as near as pos- sible. The theory of continued fractions was admirably adapted to this purpose, an advan- tage of which Huygens readily availed him- self; he investigated the nature of these num- bers much more minutely than had been done by Brounker and Wallis, and to him we may consider ourselves indebted for most of the advantages that arithmetic and analysis have since derived from these principles. See Huy- gens’s “ Descrip. Autom. Planet. in Opera. Reliq.” p. 173, edit. Amster. 1728. J. Bernoulli afterwards made a useful ap- plication of continued fractions, in his “ Recueil pourles Astronomers,” vol.i., where he gives a new species of calculation for facilitating the construction of tables of proportional parts. The theory then passed into the hands of Euler, who has on this subject several very profound and elegant essays and disquisitions in the Petersburgh Transactions ; viz. in vols. ix. and xi. of the old, and vols. ix. and xi. of CON the new Acta. Petro. as also in his “ Analy Infinitorum.” La Grange also, in the Memoirs of Ber for 1767 and 1765, has two papers on tl! subject, in which he first gave a general so) tion of allindeterminate problems of the seco degree by means of continued fractions, a he has treated the same subject still more| large in his additions to the “‘ Elements of / gebra” of Euler. ; The same subject has been still fart pursued by Legendre, in his “ Essai sur Théorié des Nombres,” in which the doctri of continued fractions is clearly developed,a its application to several interesting proble perspicuously illustrated. A concise inves gation ofthe principal properties of these nw bers is likewise given in Barlow’s “ Theory Number,” and which, it is presumed, may consulted with advantage, by those who 2 not acquainted with the analytical works foreign authors. “4 Theory of Continued Fractions. 1. fraction of the form a b ict: va d + is called a continued fraction, and the numb of its terms will be finite or infinite, accordi as the quantity which it expresses, or 3 which it is deduced, is rational or irrationé The above is the most general form thate be given to continued fractions, but there¢ few cases in which it is necessary to hayea| others than those whose numerators are unl) and signs +, as in the following, viz. Me pose to consider them under the latter fo only. ie 2. The series of fractions formed of the fi term, the first two terms, the first three tern &c. of any continued fraction, are called e verging fractions ; thus, : a | 1 1 ko 1 Legs Ct ms Sc er c when reduced to the forms ) be+1 &c. aa 1 b a’ ab+1’ a(ab+1)+e sl are converging fractions. a , Prop. 1. To reduce any proposed rational Fr tion to a continued Fraction. e Divide the denominator by the numerat: then the divisor by the remainder, and so ¢ as in finding the greatest common measure two numbers; and the successive quotiel will be the several denominators of the eq) valent continued fraction. ra If the numerator of the proposed fracti be greater than its denominator, the continu fraction will be preceded by an integer. CON Exam. Reduce IW) toacontinued fraction. 9743 1171)9743(8, 375)1171(3 46)375(8 7)46(6 4)7(1 3)4(1 1)3(3 herefore the proposed fraction, 1171 1 743 8 — 1 9743 8 + 3 4 me fh 1 os Y gl i And in the same way any other rational frac- ion may be reduced to a continued fraction. tis not, however, in this form that they are pplied to any useful arithmetical purpose. In mder to their being thus employed they must ie reduced to a series of converging fractions, 8 in the following proposition. *rop. 2. To reduce any given continued Fraction to a Series of converging Fractions. Write all the denominators of the succes- ive terms of the continued fraction, in a line, hus: a, b, ec, d, &e. hen the first converging fraction will have nity for its numerator, and first term a, of the uotients for its denominator. The second raction will have 8 for its numerator, and for ts denominator ab + 1; and the numerators f all the succeeding fractions will be found y multiplying the numerator, last found, by he corresponding term in the above series, nd adding to the product the preceding nu- aerator. And the denominators are obtained aexactly the same manner, viz. by multiply- ag each denominator by the corresponding erms in the above series, and adding to the roduct the preceding denominator. Thus if it were required to reduce the receding continued fraction to a series of onyerging fractions, the operation would itand thus: uotients 8, 3, 8, 6, i. 1, 3, ee ee Me AY Se tee ke | 8” 25’ 208’ ) And this series of fractions has this remark- ble property, that each approximates to the ‘ue ratio expressed by the original fraction, earer than any preceding one, and nearer jaan any dther fraction whatever of less terms. nd hence, the great use of these numbers in 1¢ reduction of ratios. See RATIO. _ ' Another singular property of them is this, nat the difference between the product of le numerator of any one of those fractions, Me the denominator of the adjacent one ; and ony. frac. 1€ product of the denominator of the former, ito the numerator of the latter, is always qual to unity; thus, 1b 3.8 = 1; 3.208 n 25,25 =1, ke. | CON and hence their use in the solution of indeter- minate equations of the first degree. All indeterminate equations of the first degree may be ultimately reduced tothe form az — by = + 1, and since the last fraction of the converging series is the original fraction itself; it follows, that if we take «=the denominator of the preceding fraction, and y = the nume- ‘ rator of the same, we shall have ax — by = eft Cage For ‘example, if the proposed equation were 1171 « — 9743 y = 1, we should have x = 2754, and y = 331, which gives 11712754 — 9743°331 = 1, as required. See INDETERMI- NATE Analysis, a Prop. 3. To extract the Square Root of any given integer Number, not an exact Square, by con- tinued Fractions. It is obvious here, in the first place, that since the quantity that we have to represent by a continued fraction is irrational, the frac- tion which expresses it must consist of an indefinite number of terms; but still it has this property, that after a certain period, the denominators of the several fractions recur again in the same order as at first ad infinitum, and consequently after having arrived at that period, the operation of extraction may be thence discontinued, and the series carried on to any length at pleasure. The method of extracting the roots of num- bers by continued fractions will be best shown by a partial example. . Let it be required to extract the square root of 19 by continued fractions. Here, /19—4 3 i hee _——_—_— = _— Kiros o 1 + PIPES 4% 1 /19 +4 3 /19+4 /19—2 5 ——— = —2+4+———_————2 3 sic 3 + i942 a4 1 /19+2 2 /419+2 /19—3 2 qraitinetenclicock, ON aeeepsens a wet 5 ai: 5 I+ Fiops t 1 /19+3 2 419+3 __ ¥19—3_. wi Aig TRE ME RG Tey 19 Fa 1 /19 +3 : 3 /194+3_ Y19—2_, =e mages Sok meen hh ae 1 JS/19+2 : 4 1 ) CON Rad) /19 +4 I J19 44g 4 VID 4 gs, And here, since we have obtained the same expression as we began with, we may discon- tinue our extraction, as the quotients 4; 2, 1, 3, 1, 2,8; 2, 1,3, 1, 2,8; 2,1, &e. must ne- cessarily always recur in the same order ad in- finitum. Now if we substitute for the frac- tions, VI9+4 YINF2, VIDTS y r 1 es ge, OC, 3 ) 2 their respective values as foundin the preced- ing operation, we shall have, /19 4 --+- ——-—— 1 194+4—744+- —-————= —_ +3 4.71972 3 ] 1 A Anges re B9 +344 4. VATS 1 2 pe 1 oS ane | Lt 54 sod 1 CT ee aan and therefore the square root of 19 has been transformed into a continued fraction as re- quired. And hence it is obvious, that it may now be converted into a series of converging fractions, as in the preceding example ; thus, quotients 4; 2, 1, 3, 4, 2, 8, &e. 4 9 13 48 61 160 Ke fractions | 3 3’ Tl’ i’ 39” each of which fractions expresses the square root of 19 nearer than any preceding one, and nearer than any other fraction containing smaller terms; and as the series of quotients is periodical, it may be continued at pleasure, and the approximation carried on to any de- gree of accuracy required. The operation in this partial example is ob- vious; we first find the greatest integer con- tained in 19, whichis 4; therefore, “19 = 4+ we, or multiplying this last fraction /19+ 4 ‘ 19—16 by ————, we obtain v 1944 ———— = Y V9 +2 A aA ee 3 A+ ———— = 4 + —___—_—— + /19+4 + /19+4. We next find 3 the greatest integer in vee which is 2, whence “a —2 + — ; and pro- ceeding as before, /19+4 /19—2 19—4 ee 4 Yt 3 eae i 3 (4/19 — 2) 5 2+——.., &e. + Tio pe *° and in the same way we may proceed with any other integer not-a square. CON. “al But this may be more readily effected by means of the following formula; where N is. the given number, @ the greatest integer con. tained in /N, and u, uw’, w', &e. the greatest integers contained in the respective fractions to which they correspond ; viz. : ve tema + &cjl.a—o=m; a= =n vem mu + &e.|n.u— m =m’; Noi wid ty Rated ol m + elm! —m’=m"; nN ne | LD fen + &eln” u! —m" =m"; == &e. —n" &e. &e. ; Thus in the preceding example where N= 19, and a — 4, the calculation will stand thus; 419+0_ _,, 19—4*_, Meet a4 4&0. | 1.4—-0=4; —T— Se VID PA Se _., 19—27me a — Zz VTP =1+&e. |5.1—258; <= ; a> | ; YISTS —3 4 &e,|2.8—3=3; > St &e. &e. &e. i By this latter formula the square root of am irrational square may be easily extracted, par ticularly as the computation on the right h is very readily supplied without setting tow the steps of the operation as is done in thi above example, which is merely for th purpose of explanation. It is not, however for the sake of the arithmetical extraction ¢ the square root, that this method has bee devised, but in consequence of its applicatio) to indeterminate equations of the second de gree, which admit of no other general methe of solution; as was first shown by La Grange in the Memoirs of Berlin for 1767 and 176 See INDETERMINATE Analysis. i ContTINUED Proportion, is that which th consequent of the first ratio is the same @ the antecedent of the second. Thus 2; 6%:| : 18 is a continued proportion. | CONTINUITY, an uninterrupted connet tion. - Law of Continuity, is that by which vi riable quantities passing from one magnitu to another, pass through all the intermedia’ magnitudes, without ever passing over any them abruptly. Many philosophers and met) physicians have asserted the probable conf mity of natural operations to this law; bi father Boscovich goes farther, and proves th’ the law is universal. Thus we see that t distances of two bodies can never be changy without their passing through all the interm) diate distances. We see the plancts rm} each with different velocities and directio) in the several parts of its orbit; but still 0) serving the law of continuity. In heavy bod? projected the velocity increases and decreas} through all the intermediate velocities: ay the same happens with regard to electric CON less dense without passing through the inter- mediate densities. The light of the day in- creases in the morning and decreases at night, _ through all the intermediate possible degrees, _ And thus, if we go through nature, we shall, if all things be rightly contemplated, see the _ law of continuity strictly to take place. We sometimes, it is trae, make abrupt transitions in our minds; as when we compare the length ‘of one day with that of another immediately , following, and say that the latter is two or » three minutes longer or shorter than the for- mer, passing all at once, in the common way of speaking, completely round the globe ; but ‘if we consider the several intervening longi- tudes, we shall find days of all the interme- diate lengths. Sometimes also we confound a quick motion with an instantaneous one: thus we are apt to imagine that a ball is thrown abruptly out of a gun when fired ; but, in truth, some space of time is required for the gradual inflammation of the powder, the rarefaction of the air, and the communication of motion to the ball. In like manner all the other apparent objections to the law may be satisfactorily solved. CONTOUR, a French word signifying the yutline or periphery of a figure. CONTRA-Harmonical Proportion, is, when f three terms, the difference of the Ist and ‘dis to the difference of the 2d and 3d, as he 3d is to the Ist. CONTRACTED Vein, in Hydraulics. See TENA Contracta. | CONTRACTION, in Arithmetic, signifies, enerally, the shortening or contracting of any peration ; thus, CONTRACTION in Multiplication of Decimals, t Contracted Multiplication of Decimals, is ve method of shortening the operation, by re- ining only a certain number of decimals in le product, when the usual, or common rule, ould give many more. /It is for this reason very useful to persons ho are in the habit of performing long cal- lations; but to those who have but seldom ich operations to perform, itis of but little vice, generally requiring more time than to ‘ceed by the common rule. Neither can it be ybmitted to the same proof as any example orked by the common method. Rule. Set the units place of the multiplier, tder that decimal in the multiplicand, whose ace is to be retained in the product, and spose of the rest of the figures in an inverted contrary order to that in which they are ually placed: then, in multiplying, reject all ies to the right hand of the multiplying sit, and arrange the products so, that the “ht hand figure of each may fallin a straight € directly below each other; observing, to ease the first figure of every line with lat would arise by carrying 1, from 5 to 14; from 15 to 24; 3, from 25, 34, &c., from the duct of the two preceding figures when 1begin to multiply; and the sum of them \1 be the product required. - tind magnetism. No body becomes more or. CON EXAMPLES, 1. Multiply 34:17165 by 78-3338, retaining only 5 places of decimals in the product, 341716 50 multiplicand 3333°87 multiplier inverted ° 2392015 50 273373 20 10251 50 1025 15 102 51 10 25 2676°778 11 answer. 2. Again multiply ‘546768 by °671686, re- taining only 7 decimals in the product. ‘046768 0 multiplicand 6861760 multiplier inverted 3280608 382738 5468 3281 437 33 °3672565 answer. It should be observed here, that the last figure of the product is not always correct; it is, therefore, best always to retain one more decimal place than is required; which may be cancelled when the operation is finished. CONTRACTION in Division of Decimals, or Contracted Division, is the method of perform- ing this rule, so that the divisor may be short- ened one figure for every place obtained in the quotient; by which means the whole operation is much contracted, and in many cases considerable labour is avoided. Rule. Take as many of the left-hand figures of the divisor, as are equal to one more than the number of integers and decimals to be retained in the quotient; and find how many times they will go in the first figures of the dividend as usual. Then, consider each re- mainder as a new dividend, and in dividin it, leave out one figure to the right hand of the divisor ; remembering to carry for the increase of the figure cut off, as in the preceding rule for multiplication. | Note. When there are not so many figures in the divisor as are required to be in the quo- tient, begin the operation with all the figures as usual, and continue it, till the number of figures in the divisor, exceed by 1, the num- ber remaining to be found in the quotient; then begin the contraction. EXAMPLES. 1. Divide 2508'9280 by 92°41035 ; 6 places being sufficient to be retained in the quotient, 9241035)2508'9280(27°1498 660 7210 13 8485 4 6075 9111 794 55 O02 CON 2. Divide 746 by 843641, retaining 5 places in the quotient. 84'3641)746'0000(8°8426 710872 . 3 5959 2213 526 20 Most authors, indeed all that I remember to have seen, direct that only as many figures of the divisor should be taken, as are equal to those to be retained in the quotient, but in that case, the last figure of the quotient cannot be depended upon. ConTRACTION in the Extraction of the Square Root, is of a similar nature to the preceding case, and is performed by the following Rule. Proceed in the extraction, after the common method, till there be found half, or one more than half, the required number of figures in the root; and for the rest, divide the Jast remainder by the corresponding divisor, as in the foregoing rule of division. EXAMPLE. Required the square root of 14876:2357, and find the root to 9 places of figures. 14876°2357(121'96 I 22! 48 2| 44 241| 476 1} 241 2429|23523 9}21861 24386) 166257 6| 146316 24392) 1994100(8175 4274 1835 128: 6 Whence 121°968175 is the root required. If there had been 10 figures required in the root, then the division would have stood thus : 24'392°8)1994100(81753 42676 18283 1208 88 15 Which gives the root = 121°9681753 true to 10 places. ContTrRAcTiION, in Physics, the diminishing the extent or dimensions of a body, or the causing its parts to approach nearer to each other, in which sense it stands opposed to di- latation or expansion. Water and all aqueous fluids are gradually contracted by a diminution of temperature, until they arrive at a certain point, which is about 8° above the freezing point; but below that point they begin to expand, and continue to do so according as the temperature is lower- ed; and similar effects have been observed with regard to some metals. In speaking of - real value of a thing. * s CON contraction, a remarkable phenomenon, of considerable importance in manufactures, ob- trudes itself on our notice ; which is, the hard- ness that certain bodies acquire m conse- quence of a sudden contraction, and this is particularly the case with glass and some of the metals. Thus, glass vessels, suddenly cooled after having been formed, are so very brittle that they hardly bear to be touched with any hard body. ‘The cause of this effect is thus explained by Dr. Young. ‘ When glass in fusiom is very suddenly cooled, its ex- ternal parts become solid first, and determine the magnitude of the whole piece, while it still remains fluid within. The internal part, as it cools, is disposed to contract still further, but its contraction is prevented by the resistance of the external parts, which form an arch or vault round it, so that the whole is left ina state of constraint; and as soon as the equili- brium is disturbed in any one part, the whole aggregate is destroyed. Hence it, becomes necessary to anneal all glass, by placing it in an oven, where it is left to cool slowly; for, without this precaution, a very slight cause would destroy it. The Bologna jars, some- times called proofs, are small thick vessels made for the purpose of exhibiting this effect; they are usually destroyed by the impulse ofa small and sharp body; for instance, a single grain of sand, drepped into them, and a small body appears to be often more effectual than a larger one; perhaps because the larger one is more liable to strike the glass with § obtuse part of its surface.” Rees’s Cyelo On this subject see also Dalton’s “ Ch mistry.” ‘ie CONVERGENT, or Converging, the te dency of different things, variously disposed to one common point. It is also sometime used to denote an approximation towards th See Curves. i, See def. 2, Co CONVERGING Curves. CONVERGING Fractions. TINUED Fractions. CONVERGING Lines, those which tend to common pomt. e CONVERGING Rays, those which tend to common focus. v CONVERGING Series, those series wh terms continually diminish. See SERIES. CONVERSE, in Mathematics, commo signifies the same thing as reverse. Thus, proposition is called the converse of anoth when, after a conclusion is drawn from son thing supposed in the converse proposifi that conclusion is supposed; and then, t which in the other was supposed, is now dra as a conclusion from it: thus, when two si of a triangle are equal, the angles under th sides are equal ; and, on the converse, if th angles are equal, the two sides are equal. Converse propositions are not necess: true, but require a demonstration; and Eu always demonstrates such as he has occat for. An instance or two will show this. two right-lined figures are so exactly of a COP and form (both respecting their sides and an- gles) that being laid one on the other, their boundary lines do exactly coincide and agree, then noone doubts that these figures are equal. Now try the converse. If two right-lined figures are equal, then if they be laid the one off'the other, their boundary lines exactly co- incide and agree. It is manifest that this proposition, though the converse of the former, jis by no means true. A triangle and a square smay have equal areas; but itis impossible the ,sides of the former can all coincide with those ,of the latter. Again, if two triangles have )their sides respectively equal, their angles will jalso be respectively equal, by Euc.i. 8. But, \if two triangles have their angles respectively equal, it does not follow that the sides will be respectively equal: this may or may not be true, according to circumstances. Converse propositions, therefore, need a proof, notwith- standing this has been termed superfluous and impertinent by Emerson and some others. » CONVERSION of Proportion, is when of four proportionals it is inferred that the Ist, is to its excess above the 2d, as the 3d, to its excess above the 4th. Conversion, Centre of, in Mechanics. CENTRE of Conversion. CONVEX, round or curved, or protuberant outwards, as the outside of a globular body. Convex Lens Mirror. See Lens Mirror, See Ke. CONVEXITY, the exterior or outward surface of a convex or round body. ~ CONVEX0O-Concave Lens, is one that is convex on one side, and concave on the other. Convexo-Convex Lens, is one that is convex on both sides. CO-ORDINATES, in the theory of Curves, signify any absciss, and its corresponding or- linate. See Agsciss, ORDINATE, and CURVE, (| COPECK, a Russian coinage, of about the iame value as our farthing. | COPERNICAN System, is that system of he world in which the sun is supposed at rest; and the earth and the several planets to re- olye about him as a centre, while the moon ind the other satellites revolve about their espective primaries in the same manner. The heavens and stars are here supposed it rest, and that diurnal motion that they ap- year to have from east to west. is imputed to he earth’s motion from west to east. This system was asserted and taught by nany of the ancients, particularly by Ecphan- jus, Seleueus, Aristarchus, Philolaus, Clean- yhes, Samius, Nicetas, Heraclides, Ponticus, lato, and Pythagoras; from the last of whom it was anciently denominated the Pythagorean System. _ It was also held by Archimedes in his book ‘De Granorum Arenae Numero,” but after lim it became neglected, and even forgotten ormany ages; it having been only again re- ‘ived in the beginning of the 16th century by Jopernicus, from whom it took the new name if the Copernican System, | = COP COPERNICUS, (Nicoras), an eminent astronomer, was born at Thorn in Prussia, Jan. 10, 1472. He very early discovered a great bias for mathematics, which he pursued through all its various branches; and soon ac- quired so great areputation, that he was chosen professor of mathematics at Rome, where he taught for a long time with great applause. He also made some astronomical observations there about the year 1500. Returning to his own country some years after, he began to apply his vast knowledge in mathematics to correct the system of astronomy which then prevailed. He set himself to collect all the books which had been written by philosophers and astrono- mers, and to examine all the various hypo- theses they had invented for the solution of the celestial phenomena; to try if a more symmetrical order and constitution of the parts of the world could not be discovered: of all the hypotheses of the ancients none pleased him so well as the Pythagorean, which made the sun the centre of the system, and supposed the earth to move not only round the sun but round its own axis also. After much profound contemplation, and many care- ful calculations, he removed the obscurities of this old system, and, in fact, much improved it. His discoveries and improvements he com- prised in a book, the publication of which he prudently suppressed, till he had a powerful patron, pope Paul IIT. a lover of astronomy, to protect him. Alluding to the admonition of the poet, he tells the pontiff, “‘ he had suf- fered that fruit of his labours to ripen, not nine years only, but four times nine.” At length he committed the care of the impression to two friends, Schoner and Osiander, in a distant city. The work was printed in 1543, under ‘the title of “ Revolutionibus Orbium Coelestium ;” and the author received a copy of it a few hours before his death, on the 24th of May 1543, he being then 70 years of age. Few works have destroyed more errors, or established more important truths, than this great work of Copernicus. His noble theory was at first coldly received, or utterly re- jected: but the labours of future astronomers at length obtained it a complete triumph. Copernicus was also the first who demon- strated the double orbit of the moon; her menstrual motion about the earth, and her an- nual about the sun. Nor did this great man stop here: for, after laying a solid foundation of the celestial physics, he began the super- structure, by surmising a principle of attrac- tion to be inherent in all matter. Copernicus also wrote a tract on Trigonometry; and has exhibited tokens of the versatility of his talents, as it is acknowledged that he had considerable skill as a painter, and was extremely well acquainted with the Latin and Greek lan- guages. CopERNIcuS, the name of an astronomical instrument, invented by Whiston, to show the motion and phenomena of the planets, both primary and secondary, It is founded upon COR the Copernican system, and therefore called by his name. COPFSTUCK, a coinage of Germany, value $id. sterling. “COR, the Latin word for heart, and hence used as a distinguishing appellation of several stars supposed to be situated in the heart of the respective constellations; as Cor Caroli, Charles’ Heart, a term first ap- plied to a single star, but in ‘the Britith cata- logue it now is a constellation, which consists of three stars. See Northern CONSTELLATIONS. Cor Hydrae, the Hyrda’s Heart, a star of the second magnitude. Cor Leonis, ‘the Lion’s Heart; the same as REGULUS. Cor Scorpii, the Scorpion’s Heart. ANTARES. " COROLLARY, a consequent truth, which follows immediately from some preceding truth or demonstration. Thus, when it is de- monstrated, that “ If two sides of a triangle are equal, the angles opposite to them are also equal”’—it may be drawn as a corollary, “ That if the three sides ofa triangle are equal, the three angles are equal also.” CORONA, Crown, in Opties. See Hato. Corona Borealis, or Septentrionalis, the Northern Crown. See CONSTELLATION. Corona Australis, or Meridianalis, the Southern Crown. See CONSTELLATION. CORPUSCLE, the diminutive of corpus, is used to denote the minute particles that con- stitute natural bodies. CORPUSCULAR Philosophy, that scheme or system of physics, in which the phenomena of bodies are accounted for, from the motion, rest, position, &c. of the corpuscles or atoms of which the bodies consist. CORRECTION of a Fluent, in the Inverse Method of Fluxions, is the determination of the constant or invariable quantity which belongs to the fluent ef a given fluxion; and which does not naturally arise from that fluxion, but depends wholly on the nature of the problem whence the fluxion was deduced. That such correction is frequently neces- sary will be obvious, if we consider, that the fluxion of x is x, and the fluxion of 2 + c¢ is also a, whatever be the value of the constant part ec, and with whatever sign it is connected with the v ariable a. And hence, conversely, the fluent of « may be either 2, or «+ ¢; and this, as well as the precise value of ihe quan- tity c, must be determined from the nature of the problem, which may be done by the fol- lowing rule. First, take the fiuent according to the pro- per. rules for that purpose, and then observe whether this fluent becomes equal to zero, or to some constant quantity, when the nature of the problem requires that it should; if it do, it is the complete fluent, and no correction is necessary, but if not, it wants a correction, and this correction is the difference between the two general sides when reduced to that particular state. Hence, connect the con- stant, but indeterminate quantity e, with one See COS 7 side of the fluxional equation, found by he proper rules as above; then in this equa substitute for the variable quantities, ail values as they are known to have at any par- ticular state, place, or time, and then from that particular state of the equation find the valne of ce, which will be the correction. = quired. ie Thus to find the correct fluent of z = ant oe The general fluentis z = 4 a «+, or adding ¢; according to the above rule it is z = 5 az* +e, Now, if from the nature of the problem it ap pears, that when z= 0, x = 0 also, then write ing o for z and x, we have o =0 + ¢, ore=o and the fluent found as above is the correct fluent. But if when x = 0, z =a, then writing these values, we have a =o + c, or e =4, which is the required correction, whence z =% axv*+a, Again, if when z = 0, x = b, any constant quantity, we have 0 = 1ab++c,ore=—§ ab+; whence z = azt— ¥ abt. Suppose, for the sake of farther illustration) that a body is projected with a velocity ofa feet per second from a given place, to deter mine its velocity after it has passed over a certain space 2, it having been acted upon by some law which rendered it subject to the fluxional calculus; and suppose the resulting fluxional equation to be " vm=aen+ 3az* 2x ¥ the fluent of this is . vmiar+axrite ce being the corr ection. Now from the nature of the equation, when x — 0, or before the body had passed over any space, the velocity was a; therefore, Bs t, x =o, v =a; and writing these values we have ae amo+c. ore =a, ea it the correct fluent is ds vmseartaxi+a. @ Whereas if the body had in the first instance been only solicited by gravity, then when x =@, v would also have been equal to 0; and we should have found e = 0, or the gene fluent would have been the correct fluent r quired. ae CORVUS, the Crow, a southern conste si tion. See CONSTELLATION. ry COSECANT, Cosine, Cotangent, Coversed Sine, are the secant, sine, tangent, and versed sine of the complement of the arch or angl Co being merely'a contraction of the word complement; first introduced by Gunter. Se : SECANT, SINE, &c. By COSMICAL (from xoruos), relating to the world, an astronomical term anciently uct express the rising or setting of the sun, ora star, but more commonly to denote a star’s rising or setting with the sun. : COSMOLABE (from xocjz0s, bork ahd AwpuBavw, I take), an ancient instrument fo measuring distances, pois celestial and ter- restial. Oe | COSMOLOGY. (from xocj405, and Aoyos, di course), the science of the world in genera viz. its formation, construction, laws, &e. fi iT COT | COS, Rule of (Regola de Cosa), the term ap- plied by the Italians to denote the science of algebra, on its first introduction into Europe. See ALGEBRA. COTES (RoGerR), an eminent English ma- wt oats born in 1682, at Burbach in eicestershire. After receiving an excellent preparatory education he was removed to Trinity College, Cambridge ; of which society he was chosen fellow. He was at the same time tutor to the sons of the Marquis of Kent, to which family he was allied. The next year he was chosen Plumian professor of astronomy and experimental philosophy, and the same year took his degree of M.A. In 1713 he was ordained, and published a new edition of Sir Isaac Newton's “ Principia,” with a learned preface. He died in 1716, and was buried in the chapel of Trinity College. Mr. Cotes left behind him some very ingenious, and indeed admirable tracts, part of which, with the “To- gometria,” were published in 1722, by Dr. Robert Sinith, his cousin and successor in his professorship, afterward master of Trinity Col- lege, under the title of “‘ Harmonia Mensura- rum,” which contains a number of very in- genious and learned works. He wrote also a “‘Compendium of Arithmetic;” of the “ Re- solution of Equations ;” of “ Dioptries ;” and of the “ Nature of Curves.” Besides these pieces, he drew up, in the time of his lectures, a course of “ Hydrostatical and Pneumatical Lectures,” in English, which were published also by Dr. Smith, in 8vo. 1737, and are held in great estimation. So high an opinion had Sir Isaac Newton of our author’s genius, that he used to say, “If Cotes had lived, we had known something.” COTESIAN Theorem, in.the higher Geo- metry, an appellation distinguishing an ele- gant property of the circle discovered by Mr. Cotes, and of great use in the integration of - differentials by rational fractions. 'The theo- rem is this: If the factors of the binomial a” + 2x” be 7) required, the index.” being an integer number. With the centre O and radius AO =a, describe a circle, and divide its cir- -cumference into as many equal parts 4 as there are units in 2n, at the points A,B,C, D,&c. then in the radius, pro- duced if necessary, i al ‘take OP = x, and from the point P, to all ‘the points of divisiou in the circumference, . draw the lines PA, PB, PC, &c. so shall these lines taken alternately be the factors sought;viz. eo PBxPD x PF, de, =a" + 2%, ‘and PA x PC x PE,&c. =a" na", viz. a®@— a or x” — a”, according as the point P is within or without the circle. Por instance, if n = 5, divide the circum- COT ference into 10 equal parts, and the point P being within the circle, then will OAS +OP5=BPxDPxFPxHPx KP, and OAS‘—OPS=APxCPxEPxGPxIP. Or, supposing OA = a, and OP = x, since the circle is divided into ten equal parts, we shall know the sines and cosines of each of the ares, as AB, AC, AD, &c. and shall obtain the expressions of PA, PB, PC, &e, or PB’, PC?, PD*, &c.. We have, in this individual example, PA — a— x, which is the first divisor sought of a5—25. The arc AC being 72°, its sine Ce is given as well as its cosine Oc, and consequently Pe will be x—cos, 72°. Therefore PC? = Ce? + 2” — 22 cos. 72° +. cos.*72°, But the squares of any sine, and its corresponding cosine, are together equal to the square of the radius; so that the expression becomes transformed into this, PC* = a’ — 2x cos. 72° + x. In like man- ner, we shall have PE? =a* +-22 cos. 144° 427; but cos. 144° = — cos. 36°, which will give PE? = a — 22 cos. 36° + 2”. The three fac- tors of a5 — x5 are, therefore, a— x, a” — 2x COs. 72° + a”, and a* — Sa cos. 36° + 2”. And generally assuming the radias equal to 1, and power = n, in which case the ex- pression becomes 1—2”, or x”— 1; the general factors will all be contained in the formula, x” — 2cos. esis et Lie '0 k being any integer not divisible by n, and + representing the semi-circumference, which may be thus demonstrated. It is a known trigonometrical property, that if 2cos.ymu+ i 2 cos. ny = a" 4 ~n, from which two equations are readily deduced the following, viz. x° -— 2 cos. yaw - 1 =) xn —'2 cos. ny. 2" +1=—0 which must necessarily have one common root, being both derived from the same value of x; and since they are both reciprocal equa- tions, if x be one root, t will be another (see x Reciprocal Equations); they have, therefore, two commen roots; that is, the two roots of the first equation are also roots of the second ; and consequently from the known theory of equations, the former is a divisor of the latter. Qhor If now we make y = Fie or ny = Zho, these equations become x” — 2Cos. cau nm x* —2cos. Z2kaoxz™?+1—0 But the cos. 2k7 — 1, because 27 repre- sents the whole circumference; therefore the latter equation now reduces it to | x* — 22" +120, or (a*— 17 = 0 having still for its divisor the other formula, 6° S@omlenee +1=0; nm z+1=0 that is, the roots of the equation (a"— 1)* =9, CRA or xe —1=0, are all contained in the above formula. Hence by giving to & the successive values k=1, 2,3, &e. 1 (n—1) the following for- mul will be obtained, wiz. pri Bisa Ee +1—0 n ie a Gag FE +1-—0 nm St: PPM aca DR a a. | o4) n which contain amongst them all the factors or imaginary roots of the proposed equation x"—1—0. : Qk And if instead of making y = —, we as- sume ny — 2ka + m our second formula a?2—2cos.ny.a*+1=—0 will be reduced to 2?” + 22” +1= 0, or (a” + 1)? —0, because cos. (2ha +2)=—1, And the general factor of this formula is = 2ie0s, CF EE x—1=—0, which is the other branch of the Cotesian Theorem. Baxlow’s “ Theory of Numbers,” p. 480. Other demonstrations of this theorem may be seen in Dr. Pemberton’s “ Epist. de Cotesii;’” Dr. Smith’s “ Theoremeta,” and added to Cotes’s ‘“‘ Harm. Mens.;” De Moivre’s “ Miscel. Analyt. ;” and Waring’s ‘ Letter to Dr. Powell.” See also Nicholson’s ‘“ Journal,” vol. xxiv. p.278; and ‘“ Legons des Fonctions,” by LaGrange. : The preceding theorem is of very extensive use in various branches of analysis, but no application of it has more interested the minds of mathematicians than that which has been made by M. Gauss, in his celebrated theorem relating to the division of the circle. See Gauss’s Theorem and Polygon. CounteR-Harmonical. See Contra-Har- monical. COUNTERPOISE, any weight which placed in opposition to another weight pro- duces an equilibrium, but it is more commonly used to denote the weight used in the Roman balance or steel-yard. COURSE, in Navigation, the point of the compass, or horizon, which the ship steers on, or the angle which the rhumb line on which it sails makes with the meridian; and is some- times reckoned in degrees, and sometimes in points and quarter points of the compass. CRAIG (Jonn), a very respectable Scotch mathematician, author of a very singular work, entitled, “ YTheologiz Christiana principia Mathematica,” which was printed at London in 1699, and reprinted at Leipsic in 1755, in which he maintains by mathematical calcula- tion that christianity will last only 1454 years from the date of his writing. Besides his numerous papers in the London Philosophical Transactions, and the work above mentioned, he published ‘‘ Methodus Figurarum Quadratus,” &c. An. 1685; “ De CRO Quadraturis et Locis,” 1693; “ De Calculo Fluentium,” 1718. This author is well known by his general and commodious formule for the construction of local equations of the third and fourth degrees. His treatise “ De Cur- varum Quadraturis,”’ contains many ingenious inventions, and especially some very general series, Which, by the comparison of the coef- ficients of those series with the exponents of the equation of the proposed curve, will im~ mediately give the area in finite terms, when that is possible; which method he farther pur- sued, and improved in his treatise “ De Cal- culo Fluentium.” of either the birth or death of this author. CRAMER (GasrRIEL), a mathematician of Geneva, born in 1694. He became professor of mathematics, and a member of most of the learned societies in Europe. He died in 1782, He wrote some ingenious original pieces, among which the principal is entitled, “ In- troduction a l’Analyse des Lignes courbes algebraiques,” in 4to. which was published at Geneva in 1750. This work is equally de- serving of recommendation for the profound- — ness of its doctrine, and the perspicuity of its ' developements. Cramer also edited the works © of John and James Bernoulli. CRANE, in Astronomy, a southern constel- lation. SEE GRUuss. CRATER, the Cup, a southern constella- tion. See CONSTELLATION. CREPUSCULUM, in Astronomy, twilight; the time from the first dawn or appearance of © the morning to the rising sun; and again, be- tween the setting of the sun and the last re- mains of day. Pappus derives the word from creperus; which, he says, anciently signified — uncertain, doubtful, viz. a dubious light. The crepusculum is usually computed to begin — and end when the sun is about 18 degrees be- low the horizon ; for then the stars of the sixth — magnitude disappear in the morning, and ap- pear in the evening. It is of longer duration in the solstices than in the equinoxes, and longer in an oblique than in a right sphere. _ The crepuscula are occasioned by the sun’s rays refracted in. our atmosphere, and reflected from the particles thereof to the eye. See TWILIGHT. . CREUTZER, a coinage of Germany, 20 of which is 1 Copfstiick. CRONICAL. See ACHRONICAL. CRONOS, a name anciently given to the planet Saturn. CROSIER, a southern constellation. See CONSTELLATION. It is by means of four of the stars in this constellation that navigators find the south pole. CROSS, an instrument used in surveying for the purpose of raising perpendiculars. It consists merely of two pair of sights set at right angles to each other, mounted on a staff, of afconvenient height for use. The method — of using it, is to get in a line with one pair of sights, the marks at the extremities of the base line, or that on whieh the perpendicular We have noexact account — CUB ; to be raised, and through the other pair the iark er object to which the perpendicular is » be measured; and the line or distance be- veen the foot of the staff and the object will e the perpendicular required. See Sur- EYING. - Cross-Multiplication. See DuopEcIMALs. Cross-Staff, or Fore-Staff, an instrument wmerly used by mariners for taking the me- dian altitude of the sun or stars. CROWN, in Astronomy. See Corona. Crown, in Geometry. See CircuLar Ring. _ Crown, is also a silver coinage of England ad many other countries; the English crown _ equal in value to 5 shillings; the Danish crown is 2s, 8id.; Germany, 4s.73d.; French, s. 91d. sterling. CRYSTALLINE Heavens, in the old as- onomy, two orbs imagined between the pri- um mobile and the firmament, in the Ptole- aic system, in which the heavens are sup- »sed solid, and only susceptible of a single otion. These orbs are said to have been troduced by Alphonsus, in order to explain hat the ancients called the motion of trepi- ition or titubation. CUBATURE, is the finding the solid con- nt of any proposed body, the same as quad- ture signifies the finding the — superticial ea. CUBE, in Solid Geometry, is a regular solid ody, consisting of six equal square sides or ces, and is otherwise called a HEX£pDRON, hich see. |The solidity of a cube is equal to the cube the length ofits side; and the surface equal six times the square of the same DupLica- ON of the Cube. See Dup.icaTion. CUBES, or Cube Numbers, in Arithmetie, d the Theory of Numbers, are those whose be root is a complete integer; or they are imbers produced by multiplying a given imber twice into itself, or by the multiplica- in of three equal factors, thus, lx1xl—1; K2xXx2—8;3X3x3—=27;4x4x4=—64, +. are cube numbers. There are several remarkable properties longing to cube numbers, the principal of iich are as follow: 1, All cube numbers are of one of the forms 1,0r4n + 1, that is, all cube numbers are (her divisible by 4, or when divided by 4 ne 1 for a remainder. 2, All cube numbers are of one of the forms 1, Or 9n + 1; that is, they are either di- sible by 9, or when divided by 9, they leave ‘aremainder = 1. 3. Cube numbers divided by 6 leave the me remainder as their root, when divided 6. And consequently the difference be- ‘een any integral cube and its root is di- ‘tible by 6. \* Neither the sum nor difference of two bes can be a cube, that is, the equation + = y§ = 2? is impossible. 5. The sum of any number of consecutive ‘bes beginning with unity is a square, whose ; J | q CUB root is equal to the sum of the roots of all the cubes; thus 1 4+23=—9 — 3? 13 + 23 + 33 = 36 — 62 13 + 23 + 33 + 43 — 100 — 102 &e. &c. 6. The third differences of consecutive cubes os equal to each other, being each equal to 6; thus Cubes 1 8 27 64 125 Ist diff. 7 19 37 61 2d diff. 12 18 24 3d diff. Gr =, 6: = &ec, See the demonstration of these and various other properties of cube numbers, in Barlow’s “ Theory of Numbers,” part 1, chap. v. Cust Root of a Number, is that number which being multiplied twice by itself wiil produce the given number. To find the Cube Root of a given Number. 1st Method.—1. Separate the given number into periods of three figures each, by putting a point over the place of units, another over the place of thousands, and so on over every third figure, to the left hand in integer, and to the right in decimals; and then find the nearest cube root of the first period and set it in the quotient. 2. Subtract the cube of the figure of the root, thus found, from the first period, to the left hand, and annex the following period to the remainder for a dividend. 3. Divide this dividend by 3 times the square of the figure of the root above deter- mined, and the first figure of the quotient will be the second figure of the root. 4. Subtract the cube of these two figures of the root from the first two periods on the left, and to the remainder annex the following pe- riod, for a new dividend, which divide as be- fore; and so on till the whole is finished. And finally point off as many figures for integers as there are periods of integers in the proposed number. Ex. Required the cube root of 41278°242816 44 278:242816(84'56 root 33 27 37>x3 = 27)14278(4 2d figure of the root 41278 ist two periods 343 39304 34° X 33468) 1974:242(5, 3d figure 41278242 1st three periods 3453 41063°625 345* x 3357075) 214'617816(6-4th figure and thns the operation may be carried on till the root be obtained to any degree of accuracy required. This method, however, is extremely laborious, and is seldom or never employed ; as other methods have been found in which the approximation is carried on much more ra- idly. : od Method. Find by trials the nearest ra- tional cube to the given number, and call it the assumed cube. Then, as double the as- CUB , sumed cube added to the given number, is to double the given number added to the as- sumed, cube, so is the root of the assumed cube to the required root, nearly. Or, as the first sum is to the difference of the given num- ber and assumed cube, so is the assumed root to the difference of the roots, nearly.—By taking the cube of the root thus found, for the assumed cube, and repeating the opera- tion, the root will be had to a still greater degree of exactness. Ex. Required the cube root of 21035'8. By a few trials the root is found to be between 27 ‘and 28. Taking, therefore, 27, its cube is 19683, which is the assumed cube.- Then, 19683 21035°8 2 2 39366 42071.6 21035°8 19683 | As 604018 : 61754°6::27;27°'604.7 the root nearly. Again, for a second operation, the cube of this root is 21035°318645155823, and the pro- _ cess by the latter method will be’ 21035'318645 &c. 2 42070°637290 21035°8 21035'8 21035°318645 Ke. As 63106°48729 : diff. °481355::27°6047: the dif. 000210834 conseq. the root req. is 27°604910834 The above rule, which is as simple as can be expected, seems to have been derived from Dr. Halley’s rational formula, and was first given in the above form by Dodson. 3d Method, Write the given number under the form a3 = 6, then by the binomial theorem, V(e+tb)=at sal Lo ORY SO 1s Stree Sesh 9.8 a te O as 2.5.8. b+ a3 cz &e.; in which formula sub- stituting the values of a and 6, the root may as above be found to any degree of accuracy required. It should be observed, however, that this formula can only be advantageously employed when the given number is only a little more or less than an exact cube. Ex. Extract the cube root of 1001. Here 1001 = 103 +1, therefore a = 10 and b= 1; consequently 71001 — 10 Noth ID & / 1001 =10 + 3557 — 9105 * Srioe — &* a/ 1001 — 10 + 003333 — 000001111 + &e. — 10.0033322. 4th Method, by logarithms. Divide the loga- rithm of the given number by 3, and the quo- tient will be the logarithm of the root; the na- tural number of which will be the root itself, Thus to find the cube root of 7867. log. 7867 — 3)3°S97404 root 19715 = 1°299135 Such are the principal rules for extracting the cube root of numbers, of which the last by CUB logarithms is certainly by far the simplest but here the root cannot always be found t that degree of accuracy that might be desire¢ However, with a good table of logarithms, | or 7 places of figures may be depended upor which is enough for most practical purposes, CUBIC Equation, in Algebra, is that i which the highest power of the unknow quantity rises to the third degree, as 23 - ax* + bx + co; if only the third powe enters, it is called a semple cubie equation, th solution of which presents no difficulty, bein found by means of the rules given under th article SIMPLE Equations. When other poy ers beside the third enters, the roots are foun in the following manner. | Having first reduced the equation to tl form 23+ ax—=b. See Equation. Make a: p +4, then Me ei va ie rae cy 1 ante a(p + q) . Whence by addition, since p + ¢g =2,¥ have - vo +ax—p?+ Bpqtart+geirsb If, therefore, we assume a = — 3 pq, ¥ have p? + q? = c, that is, we obtain the t following equations: fpP+e=sd t Spq’=—“~a. | The second gives p =F7 which cube and substituted, in the first gives, 3 i hl —a 27 g3 or, by reduction, 7 | Whence, @ =Zb+ v(kb* + 4 a3) q=/\tb+vGr4+ za in the same way, | and, f P =4/ } Hb VER +09) Whence x in the proposed equation, becon! wal (Eb+ 30 taba) ta) Eb—v Eb | Which formula is commonly, though | roneously, called Carden’s Rule for Qi Equations. See History of ALGEBRA. | In the above formula £ 6* is always positi} being the square of 4 6, but zy a will be gative, or positive, according as a, in the} posed equation, is negative or positive, beg the cube of 4a; in the latter case, each brail of the root has a real value, which may bes certained by the proper rules; and in the ‘ mer case, the same will have place, if 4,43 ED, each branch taken separately becomes i ginary, and the root cannot be found by it rule; notwithstanding that the two toget are equal to a real quantity. This is whi termed the Irreducible Casein Cubic Equati , the solution of which, by means of forma has bid defiance to the attempts of man the ablest mathematicians of modern tit’ See IRREDUCIBLE Case. = CUB EXAMPLES, 1. Giving the equation «3 + 6x = 4, to ad x. Here a = + 6, andé6 = + 4, therefore the rmula —v(E0* + a3) sxVv(2+ v44+8)+ ¥(2— v4+4+8) a= (24+ v12)+ Y(2— v12). 2. Giving the equation x3 +18 2—6, to find. Here a = + 18, b = 6, therefore 1 b = 3, > = 9, and 51 a3 = 216; whence iz V (3+ v9 +4 216)+¥ (3—Vv9 +216) (@>= (3 4+ 7225) + (3 — 225) (e=¥ 18+ 3/ —12—3, 1g— 3, 12—331313. 3. Giving the equation x3—15 «—4, to find x. Here a = — 15, b = 4, therefore 1 6 = 2, = 4, 3 a3 = — 125; whence = ¥ (24+ V4—125)+- 7 (2— 4— 125) e= (2+ lly—1l) + Y2Q—1lyvy—1) ich is therefore of the Irreducible form, both aches of the root being imaginary. But \this particular case, as also in some others, |, root of each may be extracted. See Surps. ym Which it will be seen that ¥(2 + MNy—1)=2 + vy—1 ¥ (2— llv—1) = 2—/—1. Nhence by addition, x = 4 is one of the (ts of the proposed equation. Another method of solving cubic equations 'y certain trigonometrical formule, the first \t of which was given by Bombelli in his ‘ebra; but the formule were first exhibited )Vieta. Next, Albert Girard, in his Inven- | Nouvelle en 0 Algebra, shows how to re- ‘e the irreducible case by similar means; l:e which time these formulz have become ty generally known; in order to an enu- ation of which, it will be necessary to di- ) the general equation «3+ ax = b, into tfour following distinct forms; according tand 6 are positive or negative, and as 3, is greater or less than £ 67, in the latter ; oe | h Le i x3+axr—b6b—0 Zinc t? + ax +56—0 NE seas a —axr—b—0 ae x—ax+b—0 dlution of Form 1...x3 + ax —b = 0. ut : : (- )i=tanz ; and ¥ tan.(45°—12z)— \) Ue hen...2 = a,/ = x cot. Qu, olution of Form 2.23 +ax+b6—0. “aC; y= tan. z, and ¥ tan.(45°— iz) — Te, yaen will...2— — ar/ A xX cot. Zu. Vetex of Form 3...03 —ax— b= 0. ‘Us form resolves itself into two cases, | J fib+ VER + aber’ fs CUL i 2 om according as 3 (=) Fis less, or greater, than1, In the first case, put ; ( 5)" = cos. 2, and ¥ tan. (45° — 12) = tan. x. Then will...2 = anf t X cos. 2 u. then has x the three following values, viz. \ ae cfm nd ye A, 3 X cos 5 Qi cesen VISA bei =X cos. (60°+ a) RE ee Lge 70k 44 pe x / 3 X cos. ( 60 =) Solution of Form 4,..x3— ax +b — 0. This form has also two cases, according as Te i Se ; (5)? is less, or greater, than 1. 2 E In the first case, put < ( a)F = cos. z, and V tan. (45° — 1 z) = tan. u. Then will...2 = —2/ * X cos, Zu. 3 In the second case, put 4 ( S yz =) COS. z, then has x the three following values: Lied aie i a DIA Lt ee ofA x vs xX cos 3 Dok das eS 24/6 X cos. (60° a =) BES ene oe ar/ $ X cos. (60° — =) The two latter cases of forms 3, and 4, belong to the irreducible case, each of which gives three real roots, or values of x, whereas the other forms have each only one real root. See IRREDUCIBLE Case. Beside these two methods of solving cubic ~ equations, others have been given by different authors; and various series have been con- trived for this purpose. See a paper on this subject by Dr. Hutton, in the Philosophical Transactions for 1780; also the tracts of the same author. Legendre has shown how to find the roots of cubie equations by Continued Fractions, which may be seen in Part 1, of his Essai sur la Theorie des Nombres. And an ad- mirable ehapter on the same subject, in Bon- nycastle’s *“‘ Algebra,” vol. ii. Construction of Cusic Equations. See Con- STRUCTION. . Cusic Hyperbola and Parabola. See Hy- PERBOLA and PARABOLA. . CUBING a Solid. See Cupature. CUBO-Cube, the 6th power. Cuno-Cubo-Cube, the 9th power. CULMINATE, (Culmen, Latin), to be ver- tical, or on the meridian. CULMINATING Point, that point of a circle of the sphere that is on the meridian. CULMINATION, in Astronomy, the pas- sage of any heavenly body over the meridian, CUR er its greatest altitude during its diurnal re- volution. ; In order to find the time of a stai’s culmi- nating, we must estimate the time nearly, and find the right ascension both of the sun and star corrected for this estimated time; then the difference between these right ascensions, converted into solar time, at the rate of 15 de- grees to the hour, gives the time of southing. See White’s “ Ephemeris,” p. 45. CUNEUS, the Latin term for wedge. See WEDGE. . CUNITIA, a celebrated lady of Silesia, authoress of a mathematical work entitled “Urania Propitia,” published in 1650, in Latin and German; in which are contained various accurate astronomical tables caleulat- ed upon Kepler’s hypothesis. CURRENTS, (from curro, I run), in navi- gation, are certain settings of the stream, by which ships are obliged to alter their course, in order to arrive at the destined port. 'The . setting of the current is that point of the com- pass to which the waters run; and the drift of the current is the rate it runs per hour. For the method of determining their course and drift, see Robertson’s ‘‘ Navigation,’ vol. ii. book 7, sect. 8, where the sailing in currents is largely exemplified. CURSOR, a Runner, or Messenger, a small sliding piece of brass, in some mathematical instruments; as the piece in an equinoctial ring dial, which slides to the day of the month, the point that slides along the beam com- pass, &e. CURTATE, Distance, (from curt, short), in Astronomy, is the distance of a planet’s place, from the sun or earth reduced to the ecliptic ; or, the interval between the sun or earth, and that point where a perpendicular, let fall from the planet, meets with the ecliptic. CURTATION, is the interval between a planet’s distance from the sun, and the cur- * tate distance, CURVATURE of a Line, is its bending or flexure; by which it becomes a curve of any particular form and properties. Thus the na- ture of the curvature of the circle is such, that every point in the periphery is equally distant from a point within called the centre ; and so the curvature of the same circle is everywhere the same, but the curvature of all other curves is continually varying. The curvature of a circle is so much the more, as its radius is less, being always reci- procally as the radius and the curvature of other curves is measured by the reciprocal of the radius of a circle having the same degree of curvature as any curve has, at some certain point. Every curve is bent from its tangent by its curvature, the measure of which is the same as that of the angle of contact formed by the curve and tangent. Now the same tangent 1s common to an infinite number of circles, or other curves, all touching it, and each other in the same point of contact. So that any curve may be touched by an infinite number a» CUR of different circles at the same point: and some of these circles fall wholly within it being more curved, or having a greater curva ture than that curve; while others fall withow it near the point of contact, or between th curve and tangent at that point, and so, bein less deflected from the tangent than the cury is, they have a less degree of curvature there Consequently there is one, of this infinite num ber of circles, which neither falls below it no above it, but, being equally deflected from th tangent, coincides with it most intimately ¢ all the circles ; and the radius of this circlej called the radius of curvature of the curye also the circle itself is called the circle of em vature, or the osculatory circle of that cury because it touches it so closely that no othe circle can be drawn between it and fl curve. | if . ’ 5 iy To determine the Radius and Circle of Ona ¢ Let AEe be any curve, concave towards its axis A.D; draw an ordinate DE to the point E where the curve is required to be found; and suppose EC per-. pendicular tothe curve, and equal to the radius of the circle B Ee of . curvature sought; last- B G @ Jy, draw Ed parallel to AD, and de paral and indefinitely near to DE; thereby maki Ed the fluxion or increment of the abse A.D, also de the fluxion of the ordinate D. and Ee that of the curve AE. Now puta AD, y=DE,z= AE, andr= CE they dius of curvature; thenis Ed = x, de= and Ee =z. Now, by sim. tri. the 3 lines.. Ed, de \ E OF... ©, Ys * are respectively as the three...GE, GC, € therefore AR. ainicccea oe alee GC. when y¥ is constant. 1x For example, suppose it were required to nd the radius or circle of curvature to any oint of a parabola, its vertex being A, and xis AD. Now the equation of the curve is «= y*; hence ax = 2yy, and ax = zy”, ipposing y constant, also a*a* = 4y*y?; ence r, or Sor EWE 5, (2 +4y | (a4) x YX 2a 2/a ie general value of the radius of curvature r any point E, the ordinate to which cuts I the absciss AD = x. Hence, when z or the absciss is nothing, a 2/ 2 =z, for the radius of curvature at the ver- \x of aparabola; that is, the diameter of the ‘rele of curvature, at the vertex of a para- da, is equal to a, the parameter of the axis. For other valuable particulars relating to arvature, the radii of curvature, &c. see Mac- urin’s “ Algebra,” Maclaurin’s “ Fluxions,” mpson’s “ Fluxions,” Stone’s “Fluxions,” uler’s “ Analysis Infinitorum,” &ce. CURVE, in Geometry, is a line, the several rts of which proceed in different directions, td are successively posited towards diffe- nt points in space. A plane Curve, is that of which the several ‘ints in it lie in the same plane; and when is is not the case, the curve is said to be se of double curvature. See CuRVE of Double irvature. Curve Lines are distinguished into alge- aical or geometrical, and transcendental or echanical. Algebraical or Geometrical Curves, are ose in which the relation of the abscisses to e ordinates can be expressed by a common \sebraical expression. Transcendental or Mechanical CURVES, are ,eh as cannot be defined or expressed by _ algebraical equation; or when they are pressed by an equation, one of its terms is Variable quantity, or a curve line. See RANSCENDENTAL. Thus, y = log. x, y= A. hl. 2, y — A. cos. 2, &e. vt doctrine of curve lines, in general, as le last expression becomes barely oe ee noted by algebraical equations, was first in- rduced by Des Cartes, who called algebrai- Leurves geometrical ones; admitting none ‘© into the construction of problems. But 2wion, and after him, Leibnitz and Wolfius, ? of another opinion; they think that in the | CUR construction of a problem, one curve is not te be preferred to another, for its being defined by a more simple equation, but for its being more easily described. As in algebra we class equations into orders, according to the number of their dimensions, so also lines are distinguished into orders, ac- cording to the dimensions of the equation by which they are expressed; so that a line re- presented by the general equation O=a + bx + cy is a line of the first order; but this equation comprises the right line only, which is the most simple of all lines; and since, for this reason, the name curve does not properly ap- ply to the first order, it is usual not to distin- guish the different orders by the name of curve lines, but simply by the generic term of lines; and hence the first order of lines does not com- prehend any curves, but solely the right line, Lines of the second order, and which are sometimes called curves of the first order, are comprised in the general equation ' Ua + bx + cy + dx* + exy + fy"; that is to say, we may class among lines of the second order all the curve lines which this equation expresses, x and y denoting the rectangular co-ordinates; which order of lines are more commonly called conic sections, be- cause they all result from sections of the cone, and consists of four species only, viz. the circle, ellipse, parabola, and hyperbola. The par- ticular equations of these, when the abscisses begin at the vertex, and the ordinates perpen- dicular to the respective abscisses, are as fol- lows; viz. Circle... y? = dx —2x’, d being the diameter. 2 Ellipse.. y? = Ps (ta — x) where ¢ and ¢ are the two axis of the ellipse. Parabola... y* = px, where pis the parameter. 2 Hyperbola y* = = (tx + x?), ¢ and ¢ repre- senting the two axis of the hyperbola. Lines of the third order, or curves of the second order, are comprised in the general equation O=a+t bx + cy + dx* + exy + fy? + egaxs + hay + kay* + ly}, Lines of the fourth order, or curves of the third order, are comprised in the general equation O=a + bx + cy + dx + exy + fy* + gx + hazy + kay + ly3 + met + nury + px y” + quy? + ry*; taking always a and y for rectangular co- ordinates. Hence, it appears that in a general equa- tion of the third order there are 10 constant quantities, and in that of the fourth order 15, which may be determined at pleasure; whence it results that the kinds of lines of the third order, and much more those of the fourth and higher orders, are considerably more numerous than those of the second. _ According to Newton, there are 72 species of lines of the third order; but Sterling dis- CUR covered four more species of redundant hy- perbolas ; and Stone two other species of redundant hyperbolas, expressed by the equa- tion xy* = bx* + cx +d; viz. in the case when ba* + ex +d=—0 has two unequal negative roots, and in that where the equa- tion has two equal negative roots. So that there are at least 78 different species of lines of the third order. Indeed Euler, who classes all the varieties of lines of the third order under 16 general species, affirms that they comprehend more than 80 varieties. And the same author has shown that lines of the fourth order comprize more than 5000 varieties ; and those of the fifth and higher orders, must of course be much more numerous. The theory of curves forms a very consi- derable branch of the mathematical science ; of which, however, it is impossible to give any idea adequate to its importance and in- terest, in the limits to which we must confine this article; we must, therefore, confine our observations to some of the most popular curves, and refer the reader who is desirous of farther information, to the works mentioned at the conclusion of this article. To find the Equation of a Curve, its description or characteristic properties being given. 1. Of the Circle——Here the characteristic property is, that all lines drawn from the cen- tre to the circumference are equal to each other. yy Let therefore C P=r, MP=y, MA =a, then CM = x — x, and C P?— CM’? = PM? (Euclid, xlvii. 1); that is, ye — (7 — x)” = y?, ony? =2ra—z?, which is the equation to the circle. 2. Of the Parabola.—Let a point S be taken without the right line C B, and let the infinite line SM revolve about the point S in the plane SBC; also let CM, which is perpen- dicular to CB, cut SM in M; then if SM be always equal to CM, the point M will describe a parabola. ie Me aA Se Through S draw BSP at right angles to CB, and if SB be bisected in A, the curve will pass through that point. Draw MP per- spendicular to BP; and let AP =a, PM= AS = a; then y CUR SP? + PM*=SM*=CM*= BP’, or (a — a)* + y* = (x + a)’; that is, x* —Qax + a® + y* =x” 4+ 2ax + a*, On y” = 4ax, which is the equation the parabola. Other examples relating to equation. curves, whose description or whose char teristic property is given, will be found une the articles Cissoip, ConcHoip, and ofl curves treated of in different parts of ft work. Having given the Equation of a Curve, to scribe it and trace its principal properties, The method of tracing the curves by val is extremely obvious; it is only necessary; assume any point in a right line, as the on of the abscisses, on which having set off a number of abscisses, find the correspond) ordinates from the given equation to the om and the line passing through all their ext: mities will be the locus of the proposed eq tion. There are many cases, however, | which the locus may be described by the 1: tion of lines, angles, &c.; as is the case in the conic sections, and many other curves, the cissoid, conchoid, &c.; and in these e it is obvious we may use the mechanical ¢} struction, to find the roots of the equation; example of which, for a biquadratic ¢ cubic, is given under the article Constr’ TION, and several others may be seen in ch). ii. part 3, of Maclaurin’s “ Algebra.” Before we proceed to show the princi properties of a curve, from the given equa by which it is expressed, it will be pron define some of the terms commonly emplo: in this research. 5 | | : Cc | ‘1. The line BD, which continually approae tls towards the curve PF, without touching, except at an infinite distance, is called nt asymptote. See ASYMPTOTE. et 2. If the curve cuts itself by passing twé through the same point, as at P, then Is called a punctum duplex. Dp. 3. If the curve cuts itself twice in the sae point, it is called a punctum triplex, &e. | 4. The oval contained between C and ]is called a nodus. | 7 f 5. If the oval be indefinitely small, so ttt the nodus vanish, the point P is called a pi tum conjugatum. 1 6. That point in which two of the brane’ of the curve terminate, is called a cupis. l These terms will be better understood referring to the article ConcHoip. See i the following examples. a Exam. 1. Let the proposed equation? ay x=. CUR gg Cc B tad ) Assume the point A in the right line CAB; ‘4 set of AP =x, AP’ =—z2z; now in ith cases y will be positive, and y increases x increases ; the curve, therefore, has two jinite branches on the same side of the line (3. This curve is the common parabola. Exam. 2. Let now the proposed equation | oa x. Tere, since the power of x is an odd num- , when z is negative y is also negative; the ve has, therefore, two infinite branches, on erent sides of the line CB. This curve is ne of the third order, and is called the ical parabola. IxaM. 3. Let the proposed equation be G—2>) x (x—b)* =x" y*; ory = (a?— x?) t— ere, if x = 0 then y becomes infinite, and efore the ordinate at A is an assymptote to curve. If AB — 3, and P be taken be- *n A and B, then shall PM and PM’ be pl, and lie on different sides of the absciss If x — b, then the two values of y va- , because x — b = 0; and consequently surve passes through B, and has there a tum duplex. ; AP be taken greater than AB, then shall (2 be two values of. y, as before, having O/rary signs; that value which was positive ere, being now become negative, and the "| tive yalue being become positive. |, then the two values of y vanish, because *—a7) = 0. And if A Pis taken greater ‘ AD, then a*— x* becomes negative, the ¥alues of y impossible ; and therefore hate does not pass beyond D. atif AD be taken equal to a, and P comes. CUR If now a be supposed negative, we shall find y= AW (a? — 2) x +2) x ~ and if x vanish both these values of y become infinite, and consequently the curve has two infinite ares on each side of the asymptote, As x increases y diminishes, and when «x =a, y¥ = 0; and consequently the curve passes through E, if AE be taken equal to AD on the opposite side. If x be supposed greater than a, then y becomes impossible ; and therefore no part of the curve can be found beyond E. This curve is the conchoid of Nicomedes. If a =b it will have a cuspis in B, the nodus between B and D vanishing. And if a be less than 6, the point B will be a punetum conju- gatum. Our limits will not admit of entering further on this subject; the reader, therefore, who is desirous of further information on this inte- resting subject, is referred to Cramer’s “ In- troduction 4 Analyse des Lignes Courbes Algebraiques;” which is'a most admirable work, composed by its author for the use of students. The Appendix to Maclaurin’s Al- gebra, entitled ‘De Linearum Geometrica- rum,” &c. (which is translated into English in some of the editions of the above work) ; also the “‘Geometria Organica,” of the same author. Stirling’s “ Linez tertii ordinis New- tonianz,” and Euler’s “Introductio in Ana- lysis Infinitorum,” vol. 2. Other writers on this subject, beside the several treatises on the conic sections, are Archimedes’s “De Spiralibus ;” Des Carte’s “Geometria;” Bar- row’s “ Lectiones Geometrice ;” Newton’s “ Enumeratio Linearum Tertii Ordines ;” Brackenridge’s “‘ Descriptio Linearum Curva- rum.” See also a dissertation on this subject by Carnot, in his “Geometrie de Position.” Newton shows that curves may be generated by shadows. He says, if upon any infinite plane, illuminated from a lucid point, the sha- dows of figures be projected, the shadows of the conic sections will always be conic sec- tions; those of the curves of the second kind, will always be curves of the second kind; those of the curves of the third kind, will always be curves of the third kind; and so on, ad infinitum. And, like as the projected-shadow of a cir- cle generates all the conic sections, so the five diverging parabolas, by their shadows, will generate and exhibit all the.rest of the curves of the second kind: and thus some of the most simple curves of the other kinds may be found, which will form, by their shadows upon a plane, projected from a lucid point, all the other curves of that same kind. And in the French Memoirs may be seen a demon- stration of this projection, with a specimen of a few of the curves of the second order, which may be generated by a plane cutting a solid formed from the motion of an infinite right line along a diverging parabola, having an oval always passing through a given or fixed point above the plane of that parabola. The CUR above method of Newton has also been pur- sued and illustrated with great elegance by Mr. Murdoch, in his treatise entitled ‘“ New- toni Genesis Curvarum per umbras, seu Per-" spective Universalis Elementa.” Mr. Maclaurin, in his ‘‘ Geometria Orga- nica,” shows how to describe several of the species of curves of the second order, espe- cially those having a double point, by the mo- tion of right lines and angles; but a good commodious description, by a continued mo- tion of those curves which have no double point, is ranked by Newton among the most difficult problems. Newton gives also other methods of description, by lines or angles revolving above given poles; and Mr. Brac- kenridge has given a general method of de- scribing curves, by the intersection of right lines moving about points in a given plane. See Phil. Trans. No. 437, or Abr. vol. viii. p. 58; and some particular cases are demon- strated in his “‘Exere. Geometrica de Curva- rum Descriptione.” Family of Curves, is an assemblage of seve- ral curves of different kinds, all defined by the same equation of an indeterminate degree ; but differently, according to the diversity of their kind. For example, suppose an equation of an in- determinate degree, a"—! # = y™: ifm = 2, then will ax=y’; if m=3, then will aay; if m — 4, then is a3x = y+, &c.; all which curves are said to be of the same family or tribe. Curve of Double Curvature, is used to de- note the curve-line, all the parts of which are not situated in the same plane. A curve which can only be traced upon a curve surface, and not upon a plane surface, is called a curve of double’curvature. ‘These kinds of curves may be considered as gene- rated by the track of a poimt which is moved upon a curve surface, the direction of its mo- tion being continually deflected either towards the right, or towards the left hand; thus it happens that the line so described is curved, in two senses; for in effect, it partakes of the curvature of the curve surfaces, and of the continual and successive deflections of the describing point. Two curve surfaces which mutually pene- -trate each other, form also, in general, by their intersection, a curve of double curvature. Such, for example, is the curve which is form- ed by the mutual penetration of a right cylin- der and a sphere, supposing that the axis of the cylinder does not pass through the centre of the sphere. We have said in general, for it may happen, on account of particular cir- cumstances, that the intersections of two curve surfaces is a plane curve. Thus, in the preceding example, if the axis of the cylinder passed through the centre of the sphere, the curve of intersection would be the common circle. Ingenious disquisitions on curves of double curvature, have been given by. Euler, Cramer, Bossut, and Lacroix. Curve of Equable Approach. Seg APPROACH. ve "OVC Curve, Exponential, is a curve defined | an exponential equation. . Curve of quickest Descent. See Braci. STOCHRONE. Curve, Reflectoire. See ANACLASTIC. | CURVILINEAR, any thing relating, curves; as curvilinear angle, figure, surfé &e.; being such as are formed or bounded) curves. i CUSP, Cusris, in Astronomy, properly ; notes the point of a spear, but is used to} press the points or horns of the moon, or ot} luminary. vl Cusp, in Geometry, is used for the point) corner, formed by two parts of a curve mf ing and terminating there. “yf CYCLE (from xvxdos, @ circle of time} certain period or series of numbers, proed ing orderly from first to last, then returi} again to the first, and so circulating per tually. | Cycles have chiefly arisen from the inet mensurability of the revolutions of the e[ and celestial bodies to one another. The} parent revolution of the sun about the e| has been arbitrarily divided into 24 he which is the basis or foundation of alli mensuration of time, whether days, years, But neither the annual motion of the sun,f that of the other heavenly bodies, cath measured exactly, and without any rem) der, by hours, or their multiples. That ol sun, for example, is 365 days, 5 hours, 48 nutes, nearly; that of the moon, 29 ¢ 12 hours, 44 minutes, nearly. Hence, in order to express these frac} in whole numbers, and yet in numbers wi only express days and years, cycles have | invented; which, comprehending several} lutions of the same body, replace it, afl certain number of years, in the same poil| the heaven whence it first departed; or, Wi is the same thing, in the same place 0 civil calendar. | thii q i J The most remarkable of these are lowing: The Cycle of the Sun, or Solar Cyele, is volution of 28 years, in which time thea of the months return again to the same of the week; the sun’s place to the same and degrees of the ecliptic on the samer and days, so as not to differ one degree if years; and the leap-years begin the ¢ course over again, with respect to the dis the week on which the days of the monti The Cycle of the Moon, commonly callet Golden Number, is a revolution of 19 yea} which time the conjunctions. opposition# other aspects of the moon, are within anit and a half of being the same as they we the same days of the months 19 years bd The indiction is a revolution of 15 years only by the Romans for indicating the! of certain payments made by the suabjes the republic. It was established by Coil tine, A. D. 312. : AM The year of our Saviour’s birth, ace! to the vulgar era, was the ninth year’ CYC slar cycle, the first year of the funar cycle ; ad the 312th year after his birth was the first ear after the Roman indiction. Hence, to find the Year of the Solar Cycle, id 9 to any given year of Christ, and divide 1e sum by 28, the quotient is the number of yeles elapsed since his birth, and the remain- er is the cycle for the given year: if nothing ‘mains, the cycle is 28. 1Zo findethe Lunar Cycle.—Add one to the ven year of Christ, and divide the sum by ?; the quotient is the number of cycles _apsed in the interval, and the remainder is le cycle for the given year: if nothing re- ains, the cycle is 19. Lastly, subtract 312 ‘om the given year of Christ, and divide the mainder by 15, and what remains after this vision, is the indiction for the given year: if thing remains, the indiction is 15, The Cycle of Easter, also called the Diony- mm Period, is arevolution of 532 years; found ‘multiplying the solar cycle 28 by the lunar cle 19. If the new moons did not antici- te upon this cycle, Easter-day would always ‘the Sunday next after the full moon which lows the twenty-first of March; but, on count of the above anticipation, to which ) proper regard was had before the late alte- ‘tion of the style, the ecclesiastic Easter has Shs times been a week different from the te Easter within this last century; which sonvenience is now remedied by making e table, which used to find Easter for ever, | the Common Prayer Book, of no longer ie than the lunar difference from the new le will admit of. The earliest Easter pos- ile, is the 22d of March; the latest, the 25th April. Within these limits are thirty-five ys, and the number belonging to each of iim is called the. number of direction; be- ase thereby the time of Easter is found for y given year. CYCLOGRAPH, (from xvxdros, a circle, d ypadw, L describe), an instrument, as its me unports, used for describing arches of cles, commonly in those cases which require sreater radii than can be obtained with a x of compasses; and in some cases, where $s simple instrument cannot be employed. The most simple cyclograph, next to the npasses, is that commonly used by artificers ‘describing arches for the tops of doors, adows, &c. iS is made to move, keeping both legs in istant contact with those points; so will t angle C describe the arch of a circle, as is nts because all angles in the same seg- er - nee D E DCE represent two rods, forming any de- mined angle at C; A and B are two fixed is, Or nails, between which the cyclograph nts are equal to each other; and therefore, | aversely, if the angles be equal the curve is . CYC the arch of a circle. The description of other Instruments of this kind may be seen in Adams’s “ Geom. and Graph. Essays,” p. 151. CYCLOID, (from xuxr0- and «doc, like), or TROCHOID, a mechanical or transcendental curve, possessing several very curious pro- perties; the generation of which will be un- derstood from the following figure, I A's B Conceive the circle EPE to rotate along the right line AB, in the same plane with the circle, ‘until a fixed point P in the circumfe- rence, which at first touched the right line at A, touches it again at B, after an entire revo- lution; then the curve AC B, traced upon the plane by the point P, is called a cycloid. The line A B is called the base, and DC the axis of the cycloid; and the circle EP E the generating circle. D If the point P be without the circumference of the generating circle, then the curve de- scribed by that point is properly called a tro- choid, but commonly also a curtate cycloid. And if the point P be within the circum- ference of the generating circle, then the curve is called the prolate cycloid. If the generating circle, instead of revolv- ing along a right line, as in the above figures, is made to revolve along either the concave or convex circumference of another circle, then the curve so generated is called an Epicy- cLoIp; the principal properties of which will be found under that article. The cycloid is a modern curve, the inven- tion of which has been attributed by Wallis to one Bovillus, who wrote about the year 1501; and Cardinal Cusa, who lived about fifty years earlier, is also said to have dis- covered this curve ; but neither of them seems to have been acquainted with any of its pro- perties, and the simple description of it does not appear to entitle them to be considered as its inventor. Galileo, about the year 1615, and other ma~- P COYC thematicians of that date, made some re- searches relating to this curve; but they were not attended with success. Roberval was the fixgst who found the area of the cycloid, and the same was afterwards done by Torricelli; and the drawing of tangents to it was effected both by Des Cartes and Fermat. Pascal, under the feigned name of Detton- ville, in 1658, proposed some problems con- cerning this curve to the mathematicians of that day, offering a prize to any one who should answer them within a given time. These problems related to finding the areas of certain cycloidal spaces, the centre of gra- vity of the figure, and of certain of its seg- ments, and the dimensions of solids generated by the revolution of different parts of the curve. Dr. Wallis undertook these problems, and claimed the prize; but it was refused, in con- sequence of some mistake that he had made, which was not however in principle, but merely in the caleulation. Huygens also in- vestigated the nature of the cycloid, and en- riched it with several curious properties. Leib- nitz and John Bernoulli discovered certain quadrable spaces; and the latter found it to be the curve of swiftest descent. We cannot in this place enter into an in- vestigation of the several curious properties which belong exclusively to the cycloid, as this would occupy much more space than can with propriety be allotted to this article; we must, therefore, content ourselves with enu- merating the principal of them, and refer the reader for their demonstrations to the works mentioned at the conclusion of the article. Properties of the Cycloid—Draw any right ordinate FGH, fig. 1. above, join CG, and from H draw HI parallel to CG, meeting the axis DC produced in I. Then, 1. LH is a tangent to the cycloid at the point H. 2. The circular are CG is equal to the right line HG. ; 3. The semi-circumference CGD = semi- base DB. 4. The cycloidal are CH = double the chord CG. 5. The semi-cycloidal are C B = double the diameter C D. 6. The area of the eycloid AC BA = triple the area of the circle CGD. 7. The three spaces ACD, the circle CGD, and C BD, are equal to each other. 8. The upper segment of a cycloid cut off by a line parallel to the base, at + of the axis from the vertex, is equal to the regular hex- agon inscribed in the generating circle. 9. The solid generated by the revolution of the cycloid about its base AB, is to its cir- cumscribing cylinder as 5to 8. 10. The solid generated about the tangent parallel to the base, is to its circumscribing cylinder as 7 to 8. 11. And the solid generated about the tan- gent parallel to the axis, is to its cireumscrib- ing cylinder as 6 to 8. ‘a CYG 12. The centre of gravity of the whole cloid, is }ths of the axis from the vertex. cy 2 oO “ IK 13. The evolute of a cycloid is another equa cycloid. Hence, if two equal semi-cycloid OP, OQ, be joined at O, so thatOM = MK the diameter of the generating circle, and th string of a pendulum be suspended at O, hay ing its length = OK = OP; then by plyin the string round the curve O P, to which iti equal, and the ball of the pendulum be let g¢ it will describe and vibrate in the other oye PKQ. And all its vibrations, whether i great arcs or in small ones, will be performe in the same time. And the time of one vibri tion is to the time of falling perpendicular through the axis MK, as the circumference of a cirele to its diameter, or as 3°1416: 1, 14. The cycloid is the curve of swiftest di scent; that is, a body will fall from Q to} through the arc OK, in less time than by a other rout. See BRACHYSTOCHRONE. e Other properties of the cycloid, beside tho; above mentioned, may be found in the folloy ing works: Schooten’s ‘‘Commentary on Di Cartes ;” Torricelli’s Appendix “De Dime sione Cycloids,” subjoined to his treatise “T Dimensione Parabole:’” Fabri’s “ Synops Geometrice:’ Wallis’s “Treatise upon # Cycloid:” and Huygen’s “ Horolog. Oscillai Other authors have also written incidental upon this subject; as Reinau, Newton, Lei nitz, de la Loubere, Roberval, Wren, de Hire, Cotes, &c; and a number of authors a later date. : CYGNUS, the Swan, a northern constell tion. See CONSTELLATION. ms CYLINDER, is a solid having two equ circular ends parallel to each other, and ey plane section parallel to the ends is also a ¢ cle, and equal to them. . Cylinders are either right or oblique. A Right CYLINDER, is that whose si perpendicular to the plane of its base; ast figure ABCD. i ACB L CC g>D0 . An Oblique CYLINDER, is that whose si not perpendicular, but oblique to the plan its base; as LMNO. A right cylinder may be conceived Iv to CEN generated by the revolution of a rectangle, as PBDQ, about one of its sides PQ, which remains fixed, and which is called the axis of the cylinder. Or we may otherwise con- eeive it to be generated by carrying a right line parallel to itself, about the circumferences of two equal and parallel circles; which an- swers as well for the oblique, as for the right oylinder. To find the Surface and Solidity of a Cylinder, whether right or oblique. 1. Multiply the circumference of its base xy its length, and the product will be the area. . 2. Multiply the area of its base by its per- yendicular height, and the product will be the olidity. -CYLINDRIC, or CytinpricaL, any thing elating to the cylinder. Cy.inpric Ring, is a solid which may be sonceived by supposing a cylinder to be bent ound into a circular form, so as to return pon itself. lo find the Surface and Solidity of a circular Ring. ns | 1. Multiply the circumference of a perpen- ticular section of the ring, by half the sum of he inward and outward diameters, and that rroduct again by 3°1416, and the last product vill be the surface of the ring. 2. Multiply the area ofa perpendicular sec- ion by the same quantities, and the last pro- uct will be the solidity. DAALDE, a Dutch coin, value 2s. 7d. ster- ng. D’ALEMBERT. See ALempert D’. ~DARCEY (Cownt), an ingenious philoso- her and mathematician, born in Ireland in 725, but educated in France, where he con- tantly resided after the age of fourteen. larcey was author of several essays; viz. “ An ‘ssay on Artillery,” published im 1760; and aother “On the Duration of Light,” 1765; sside numerous papers published in the Me- \oirs of the Academy of Sciences, from the ear 1742 to 1765; as also in tome i. of the Savyans Etrangers.” This author died in 779, in the fifty-fourth year of his age. “DARK Chamber. See Camera Obscura. ) Dark Rays, in Philosophy, are certain ema- ore from the sun, that have been recently iscovered, which are not perceptible to our ‘yes, and are only manifested by their effects ; the thermometer. Dr. Herschel and Mr. Ritter are the dis- | CEN Cy tinpric Ungula, is a solid formed by a plane passing obliquely through the side and base of a cylinder; as EDG, in the figure below. SY D ‘Bb ea, “ye G To find the Surface and Solidity of a cylindrie Ungula. Put h = the height of AD, v — the versed sine AF, s — the right sine FG, ec — the cosine = LAB — AP, b = the base or area of the seg. GAE, d = the diameter AB, a — the arc GAE. Then, 1. Convex surface = ‘h (ds — ac) v 2. Solidity........... it (383 — be) See Hautton’s and Bonnycastle’s ‘“‘ Mensura- tion.” ; CY LINDROID, is a solid resembling the common cylinder, except having elliptical in- stead of circular ends. CYNOSURA, a name given by the Greeks to the constellation Ursa Major. CYPHER. See CIPHER. D coverers of these emanations. 'The former of those gentlemen observed, that when the rays of the sun are refracted by a glass prism, and form the coloured spectrum upon any surface, a thermometer placed beyond the spectrum is elevated by the heat of certain rays, or ema- nations, which are by no means visible; and which have thence been denominated calorific rays. DATA, in Mathematics, denote certain quantities which are given or known, and by means of which other quantities, which are unknown, are to be determined. Euclid, in his book of ‘‘ Data,” uses this term to denote such spaces, lines, angle, &c. as are given, or to which others may be found equal. See Simson’s edition of Euclid’s “ Data,” given at the end of ‘ Elements of Geometry.” DA'TUM, the singular of data, DAY, in Astronomy, is that portion of time which elapses between two successive transits of the sun over the same meridian; and the L2 DAY hours are counted from one period to another, from one to twenty-four. In common conversation, the term day im- plies only that portion of time while the sunis above the meridian, in contradistinction to night, or that time while the sun is below the meridian; and the commencement of the day is reckoned from one midnight to an- other, being divided into two periods of hours, counted from one to twelve, twice over. See Cril Day. Mean Solar Day. The solar day is the time between one transit of the sun and another ; but this period is not of equal duration at all seasons of the year, some days exceeding, and others being less than the mean solar day, which as the term imports, is amean between the lengths of all the days, or that portion of time which would be accounted a day if they were all of eaual duration. And the accumu- lated difference between the mean solar day and the successive transits of the sun, is what gives rise to what is called the equation of time. See Equation of Time. Sidereal Day, is that period in which »the earth makes one complete revolution on his axis, and is measured by the interval between two successive transits of any fixed star over the same meridian. The sidereal day is about four minutes less than the mean solar day, the difference arising from the motion of the earth in its orbit. Hence, the number of sidereal days exceed the number of mean solar days; the year consisting of 366 of the former, and only 365 of the latter. The sidereal day is the most uniform of all astronomical periods, neither observation nor theory having yet de- tected in it the least variation, though some conjectures as to a want of uniformity have been advanced. Euler (in Phil. Trans. vol. xlvi. p. 358) says he has some reasons, deduced from Jupiter’s action on the earth, to think that the earth’s revolution upon its axis becomes continually more and more rapid. But M. Laplace (“ Mechanique Celeste,” tom. iii.) proves both from theory and from a’computation of eclipses that took place more than 2000 years ago, that the mean length of the day has undergone no change; thus establishing the invariability of the most essential measure in all astronomical observations. Civil Day, is the time allotted for day in civil purposes, and begins differently in diffe- rent nations, but still including one whole rotation of the earth on its axis; beginning either at sun-rise, sun-set, noon, or midnight. First, at sun-rising, among the ancient Babylo- nians, Persians, Syrians, and most other eastern nations, with the present inhabitants of the Balearic Islands, the Greeks, Xe. Secondly, at sun-setting, among the ancient Athenians and Jews, with the Austrians, Bo- hemians, Marcomanni, Silesians, modern Ita- lians, and Chinese. Thirdly, at noon, with astronomers and the ancient Umbri and Ara- bians. And, fourthly, at midnight, among the ancient Egyptians and Romans, with the mo- DEC dern English, French, Dutch, Germans, Spa* niards, and Portuguese. . % The day is divided into hours; and a cer- tain number of days makes a week, a month, ora year. ‘The old Latin names for the days in the week are still retained in the journals” of parliament and of medical men. They are as follow: dies Solis, dies Lune, dies Martis, dies Mercurii, dies Jovis, dies Veneris, and dies | Saturni. e. DECAGON (from 3:xa, ten, and yone, angle) a plane geometrical figure of ten sides and» ten angles. When all the sides and angles. are equal it is a regular decagon, and may be inscribed in a circle ;-otherwise, not. iy If the radius of a circle, or the side of the inscribed hexagon, be divided in extreme and mean proportion, the greater segment will be the side of a decagon inscribed in the same circle. And, therefore, as the side of the de- cagon is to the radius, so is the radius to the sum of the two. Whence, if the radius of the, circle be r, the side of the inscribed decagon will be a x r, and if its side be 9 its area — s* x 7°694209., ¥ To inscribe a regular decagon in a circle, See PENTAGON. - DECEMBER, the last month of our year, but the tenth month of the Romans, whene its name; viz. from decem, ten. DECHALES, an excellent mathematician, born at Chambery in Savoy, in 1611; and die at Turin, where he was professor of math matics, in 1678, in the sixty-seventh year of his age. His principal works are: 1. An edi- tion of Euclid’s Elements. And, 2. A Dis- course on Fortification; which, with some others, were collected and printed in 3 vols. folio, under the title of “‘ Mundus Mathom ticus;” and afterwards enlarged in 4 vols folio; Lyons, 1690. t DECIMAL Arithmetic (decimal tenths), in a general sense, denotes the common arithme- tic, in which we count by periods of tens; and is otherwise, and more properly, called Denary Arithmetic, to distinguish it from the Binary, Duodenary, and other scales of arithmetic. See NOTATION. 4 Decimal Fraction, is a fraction having al- Ways some power of 10 for its denominator which consists of either 10, 100, 1000, &e. de- noting the number of equal parts into which the integer or whole is supposed to be divide aS 5, zs, rs, Ke. But, for the sake brevity, the numerator only is expressed, li a whole number with a point on the left of i as ‘2, -02, 002, &c. and which must alwa consist of as many figures as there are ciphers in the denominator; the places between t significant figures and the point being suppliec with ciphers, when necessary, as above. Con- sequently the same number of figures on the right of the decimal point has always the same deiominators Thus, the denominator of the fractions ‘5000, °0746, -0005, is 10000. And hence it appears, that the value of & . uf 4 DEC decimal fraction is not altered by ciphers on the right hand ; for ‘5000 (or ~999;) when re- duced to its lowest terms is the same as ‘5, each being equal to 4. In mixed numbers the decimals are se- parated from the integers by a point; thus, 25,4, is written 25°02. It is also evident that the value of decimals decreases in the same tenfold proportion from the point towards the right hand, as that of integers Hicreases to- wards the left. For the several operations of Mouttieuication, Diviston, &c. of Decimals, see the several articles. _ DECLINATION, -in Astronomy, the dis- tance of the sun, a star, planet, or other point of the sphere of the world, from the equator, either northward or southward; and is the same with latitude, in geography. Declina- tion is either real or apparent, according as the real or apparent place of the ebject is considered. - The declination of any heavenly body, as of a star, may be easily found by the following rule: take the meridian altitude of the star, at any piace where the latitude is known; -the complement of this is the zenith distance, and is called north or south, as the staris north or south at the time of observation. ‘Fhen, 1. When the latitude of the place and zenith distance of the star are of different kinds, namely, one north and the otker south, their difference will be the declination ; .and.it is of the same kind with the. latitude, when that is the greatest of the. two, .otherwise it is of the contrary kind. 2. If the latitude and the zenith distance are ”% the same kind, 7.e. both north, or both south, their sum is the declination; and it is of the same kind with the latitude. Accurate tables of the sun’s deelination are yublished regularly in the nautical almanacs. Circles of DECLINATION, are great circles of the sphere passing through the poles of the vorld, en which the declination is measured. . Parallels of DECLINATION, are small circles of the sphere parallel to the equator. Parallax, or Refraction of DECLINATION, is uch an arch of a meridian as is equal to the change produced in the declination by parallax w refraction, respectively. Dec ination of the Compass or Needle. See VARIATION. DECLINATION of a Vertical Plane or Wall, in Dialing, is an arch of the horizon, compre- 1ended either between the plane and the wime vertical, when it is counted from east 0 west, or between the plane and the meri- ian when it is counted from north to south. - DECLINATOR, an instrument for deter- ‘nining the declination or inclination of reclin- ‘ng planes. * DECLINERS, or Dec.inine Dials, are hose which cut either the plane of the prime vertical circle, or plane of the horizon ob- ‘iquely. ' DECLIVITY, a sloping or oblique de- scent. | DECREMENTS, are the small parts by hich a variable quantity decreases, and are DEF thus opposed to increments, which are the small parts by which a variable quantity is successively increased. See INCREMENTS, DE- CREMENT Of Life, and CompLement of Life. DECUPLE (decuplo, tenfold), a term of relation between quantities involving a ten- fold scale of proportion, or one thing ten times as much as another. | ‘DECUSSATION, DecussaTIo, in Geome- try and Opties, the same as cutting or cross- ing. DEE (Jonny), an English mathematician, but more celebrated for his pretended know- ledge in astrology and magic, on which sub- jects he published some works ; ‘but the only mathematieal performances of his, of any im- portance, are a ‘“ Mathematical Preface to Billingsley’s Euclid,” and “ Annotations” on the-tenth book of the same work; both pub- lished in 1570. These are very elaborate, and shows him to have been an acute geometrician. Bee was-bern in London, 1527, and died at Mortlake in 1608, at the age of eighty-one. A catalogue of the printed and published works of Dee are given in his “‘ Compendius Rehearsal,” together with a great number of his other works which never were pub- lished. DEFERENT, or DEFERENS, in the Ancient Astronomy, is the same as ECCENTRIC. DEFICIENT Ayperbola. See HYPERBOLA. DEFICIENT Numbers, are those the sum of whose divisors is less than the number itself. DEFINITE Quantities, in Mathematics, are those which are of a certain and determined magnitude. DEFINITION, an enumeration or specifi- cation of the simple ideas of which a com- pound idea consists, in order to ascertain its nature and character. Definitions are of two kinds; the one nomi- nal, or of the name; the other real, or of the thing. Definition of the name, or nominal defini- lion, is that which explains the sense or signi- fication appropriated to a word; or, as Wol- fius more accurately considers it, it is an enu- meration of certain marks, or characters, suf- ficient to distinguish the thing defined from any other thing; so as to leave it out of doubt what the subject is that is intended, or de- noted, by the name. Such is the definition of a square, when it is said to be a quadrilateral, equilateral, rect- angular figure. 7a DEFLECTION (from deflecto, to bend, oF turn aside), the turning any thing aside from its former course by some adventitious or ex- ternal cause. The word is often applied to the tendency of a ship from her true course, by reason of currents, &c. which deflect or turn her out of her right way. DerLectIon of the Rays of Light, is a pro- perty which Dr. Hook observed in 1674-5, and read an account of before the Royal Society, March 18, the same year. He says, he found it different both from reflection and refra¢- DEG tion; and that it was made towards the sur- face of the opacous body perpendicularly. This is the same property which Sir Isaac Newton calls inflection. It is called by others diffraction. DEFLECTIVE Forces, are those forces which act upon a moving body in a direction different from that of its actual course, in con- sequence of which the body is deflected, or turned, or drawn aside, from the direction in which it is moving. Such is the attractive force of the sun upon the earth in its orbit. DEGREE, in Algebra, a term applied to equations, to distinguish the highest power of the unknown quantity. Thus, if the index of that power be 3 or 4, the equation is respec- tively of the 3d or 4th degree. Decree, in Geometry or Trigonometry, is the 360th part of the circumference of any circle. Every circle being considered as di- vided into 360 parts, called degrees; which are marked by a small © near the top of the figure ; thus 45° is 45 degrees. The degree is subdivided into 60 smaller parts, called minutes, meaning first minutes ; the minute into 60 others, called seconds; the second into 60 thirds, &c. Thus 45° 12’ 20” are 45 degrees, 12 minutes, 20 seconds. The magnitude or quantity of angles is ac- counted in degrees ; for because of the uniform curvature of a circle in all its parts, equal angles at the centre are subtended by equal ares, and by similar arcs in peripheries of dif- ferent diameters; and an angle is said to be of so many degrees, as are contained in the arc of any circle comprehended between the legs of the angle, and having the angular point for its centre. Thus we say, an angle of 90°, or of 45° 24’. It is also usual to say, sucha star is elevated so many degrees above the horizon, or declines so many degrees from the equator; or such a town is situate so many degrees of latitude or longitude. A sign of the ecliptic or zodiac contains 30 degrees. DecrReE of Latitude, is the space or dis- tance on the meridian through which an ob- server must move, to vary his latitude by one degree, or to increase or diminish the dis- tance of a star from the zenith by one degree ; and which, on the supposition of the perfect, sphericity of the earth, is the 360th part of the meridian. The quantity of a degree of a meridian, or other great circle, on the surface of the earth, is variously determined by different observers ; and the methods made use of are also various. Eratosthenes, 250 years before Christ, first determined the magnitude of a degree of the meridian, between Alexandria and Syene on the borders of Ethiopia, by measuring the dis- tance between those places, and comparing it with the difference of a star’s zenith distances at those places; and found it to be 6944 Stadia. ' f Ptolemy fixes the degree at 683 Arabic miles, counting 71 stadia to a mile. The Arabs themselves, who made a computation of the diameter of the carth, by measuring the distance of two places under the same meri- DEG dian, in the plains of Sennar, by order of Al- mamon, make it only 56 miles. Kepler, deter- : mining the diameter of the earth by the dis- - tance of two mountains, makes a degree 13 | German miles ; but his method is far from . being accurate. to Snellius, professor of mathematics at Ley-_ den, and our countryman, Mr. Norwood, | measured each of them a meridional degree: the former in Holland, in 1620, the extent of | which he made 66'91 English miles; and the | latter, in 1635, between London and York, » and made it 69°545 miles. In 1644, Riccioli | also measured a degree by three separate. methods between Mount Parderno and the. Tower of Modeno in Htaly, and obtained a, mean length of 75°066 English miles. N oni of these, however, appear to have drawn any conclusion from the measurements relative to the figure of the earth. But when the. construction of the telescope had received considerable improvement, the planet | piter was found to differ materially from a sphere; and experience proved that the vibra- tions of the pendulum were slowest at the equator, and quicker towards the poles. These: two circumstances first suggested to Huygens that the earth was not spherical; and its rota-: tory motion about its axis naturally led him: to conclude that it was flattened at the poles}, and from the combination of its centrifugal force with that of gravity, he calculated that the diameter of the earth at the poles, was to its diameter at the equator as 578 to 579. But as he regarded the force of gravity as residing in the centre of the earth only, his solution was: not founded upon accurate principles. New- ton, about this time, also undertook the solu- tion of this problem upon a mere hypothesis. He supposed the earth’to be constituted of an infinite number of particles, mutually and equally acting upon each other; and from this he calculated that its figure was an ellipsoid, having its polar and equatorial diameters in the ratio of 229 to 230. } No farther investigations respecting the fi gure of the earth would, perhaps, have been undertaken, had not the trigonometrical ope- rations in France, conducted by M. Picard in 1669, and revised by M. Cassini in 1718, tend- ed to show that the earth was not an oblate, but a prolate spheroid. M. Picard obtained for the length of a degree 68°945 English miles; and M. Cassini, 69°119. A circumstance sé unexpected as this, naturally excited a con- siderable degree of inquiry and controversy among the philosophers of Europe; and the French government, at the instance of th academy, in 1735, sent out two companieé of mathematicians to determine the point, by measuring two meridional degrees; the on at the equator, and the other as near the north pole as convenient. Accordingly, Messrs, Godin, Bouguer, and La Condamine, from France, with Messrs. Juan and Ulloa, fa Spain, proceeded to Peru; and Messrs. Mau- pertuis, Clairaut, Camus, Le Monnier, and the Abbe Outhier, with M. Celsus, a Swedish astronomer, went to Lapland. After encoun: DEG tering many unforeseen difliculties and delays, which it required great address and ingenuity to overcome, both parties accomplished: the object of their mission aud returned to France: the company from Lapland in 1737, and that from Peru in 1744. The degree in Lapland, the middle point of which was in latitude 66° 20’, was found to contain 69°403 English miles. 'The Spanish astronomers published a ‘separate account of the measurement taken in Peru, and assigned for the length of the de- gree at the equator 68°157 English miles. M. ‘Bouguer made the same degree to contain 68°732; and M. Condamine states it at 68°713 miles. In 1740, during the time that the ope- rations were carried on near the equator, Messrs. Cassini and La Caille examined the former measures in France, and after correct- ‘ing several considerable errors, they found the dJengths of two separate degrees, the middle points of which were in latitude 49° 22’ and 45°, to be 69121 and 69:°092 English miles, respectively. The results of all the measures were now decidedly in favour of the oblate figure of the earth; and the only remaining difficulty was -to reconcile them to each other; for when taken by pairs, they gave different degrees of ellipticity, or compression, at the poles. ‘The measures of France aud Peru gave 313 to 314; while those of France and Lapland gave 128 :to 129, and those of Peru and Lapland 212 to 213, for the proportions of the two diameters. Some time afterwards, M. Clairaut publish- ed his elaborate treatise on the figure of the earth; in which he shows, from the Newtonian theory of gravity, the form of which a body of the mean density of the earth, and revolving with the same velocity, would acquire from its rotatory motion. He proves that there are two figures (oblate spheroids), and only two, in which the equilibrium would be pre- served; and that these are, when the two diameters of each are one to another, as 1 to ‘680, and 231 to 233. Each of these is equally possible ; but the former is evidently not the figure of the earth. With this ellipticity, how- ever, the vibrations of the pendulum do not agree ; since the mean, deduced from a great ‘umber of experiments with this instrument, ‘is 34,. In 1752, M. La Caille found the length of a degree at the Cape of Good Hope, in south »latitude 33° 182’, to contain 69°076 English miles. Boscovich, in 1755, determined a de- gree in Italy, latitude 43°, to be 68°998 miles. In 1764, I’. Beccaria completed the measure- ‘ment of a portion of a meridian near Turin, and deduced from it 69°061 English miles for the extent of a degree, in latitude 44° 44’ north. Near Vienna, in north latitude 47° 40’, | the length of a degree was found to he 69°142 miles, by Leisganig, in 1766. Likewise ) Messrs. Mason and Dixon, in 1768, determin- » ed the extent in Maryland and Pennsylvania, in North America, and found it equal to 68°893 miles; its middle point being in latitude 39° 12’ north. From the trigonometrical sur- _ vey of England and Wales, completed in 1802, } DEG under the direction of Lieutenant-Colonel Mudge, four separate degrees of the meridian were determined; the mean latitude of which was 51° 29'542”, and the mean length 69°1457 English miles. ‘The re-measuremeut of a de- gree in Lapland, as a correction of the pre- vious French operations, was carried on dur- ing the years 1801, 1802, and 1803, by Messrs. Ofverboom, Swanberg, Holinquist, and Pa- lander, Swedish mathematicians: and from the account of their operations, published by M. Swanberg, the length of a meridional de- gree, north latitude 66° 20/ 10" (the centre of the arc) is 69°2689 English miles. From a com- parison of this result with those from the mea- surements taken in Peru, the East Indies, and France, M. Swanberg deduces a mean of sxzoe7 for the ellipticity, and 3963°26 miles for the equatorial radius of the earth. In 1803, Colonel Lambton also measured a degree in the East Indies, north latitude 12° 32’, which he made 68°7445 miles. But the latest mea- surement of this nature is that of the meridian comprised between Barcelona in Spain, and the Balearic isles. This meridian had pre- viously been measured by Messrs. Mechain and Delambre, from Dunkirk to Barcelona ; and the two French mathematicians, Messrs. Biot and Arago, with Messrs. Chaix and Ro- driguez, Spanish commissioners, were ap- pointed to continue the line to the small island of Formentera. From their report to the French Board of Longitude it appears, that the latitude of Formentera is 42:961777 erades;, that of Dunkirk, as observed by De- lambre, is 56°760652; and the distance be- tween the two places 1374438°72 metres. Hence, according to this statement, a degree is equal to 68°769 English miles; and its mid- dle latitude 44° 522’. Mr. Swanberg also observes, that occulta- tions of the fixed stars by the moon is another method by which the figure of the earth may be determined; and Mr. 'Treisnecker, after comparing a great nuinber of these occulta- tions, concludes that the ellipticity of the earth is 335: The illustrious mathematician and astrono- mer, M. Laplace, in his “ Mécanique Celeste,” has calculated the ellipticity of the earth from the effect of precession and nutation, to be zig; which is nearly an arithmetical mean between those obtained from occultations of the stars, and the vibrations of the pendulum. The theory of the moon also gives 33, for the ellipticity. The difference between M. Swanberg’s re- sult and that of the French mathematicians, taken in the same place in 1737, is accounted for by M. Swanberg, in observing that the latter had omitted to allow for the difference of the level in the measurement of their base. And the variety in the other results present rather an interesting circumstance, as it in- dicates something in the figure and conforma- tion of the earth which is not yet understood, and naturally gives rise to the following im- portant and intricate questions; to which all the accumulated knowledge of the present DEG times has not been able to obtain complete solutions. Is the earth a spheroid of revolu- tion? Are the northern and southern hemis- pheres equal and similar to each other? What is the ratio of an are of the meridian, mea- sured in a given latitude, to the whole meri- dian? It is perhaps no very hazardous con- jecture to observe, that, after all, the effects of precession and nutation, with the vibrations of the pendulum, will be found among the most accurate means of determining, at least, the general outiines of our globe. We are indebted for the preceding account of the mo- dern measurements to Myer’s “ Geography,” in which work the reader will also find a like accurate description of some earlier attempts of the same nature. We have observed that the length of a de- gree of the meridian increases as we proceed from the equator towards the poles, which may probably be thought to stand in need of some illustration, being contrary to what at first sight appears to be the necessary conse- quence of the spheroidical figure of the earth ; TABLE DEG for the poles being nearer the centre of the. earth than the equitorial regions, it seems to follow that the are subtending a degree in the former place must be less than the are subs. tending a degree in the latter. But this is on a supposition that a perpendicular line is ak ways directed towards the earth’s centre, or that a plumb-line takes this direction when freely suspended and left solely to the action of gravity, which, however, is not the case, for a plumb-line so circumstanced settles into a direction perpendicular to the earth’s surface, or to a tangent at that point; and therefore in| order to have a difference of one degree of lati- tude the places must be at such a distance that the tangents at those places, when pro= duced, shall cut each other in an angle of one degree, and consequently where the curvature is the least the distance between the two places must be so much greater; and hence it is that degrees of latitude are always less and less as we approach towards the equator, where they are the least of all, the curvature being there the most sensible. ; oft the different lengths of a degree, as measured in various parts of the earth, the tvme of ite measurement, the latitude of its middle WAX &e. Extent in Eng. ; ; Date. Latiinde, miles and dec. | Measures. Countries. : Wak ne A EAS ate Peal ait en, LRA CAE RNR Solute ea AL aie) bk 19) ,I/ ja ‘ Lees 1525 | 49° 2ut' N OS°763 7) IVE Ternel..... 3. je0e.sese esse France 1620 52° 4 N. 66° 91 SOTIGLIUUS. 5 sctor ccs ce Pere Holland ‘41635 | 53 15 N.' | 69°546 "| Norwood... 0... England Wh a Rigeiolie ee eee Italy 1669 ) S OPED i PUN ad ieeocesccesrarttse ine Post eae? vie Dea a Bowrap league tt hee {| France 737 | 66 20 N.| 69°403 | Maupertius, &c........... - | Lapland 49220 NN. | 697121] Gags a. ; ‘ 1740 | 45 00 N 69. og2 5 | Cassini and La Caille.... | France su 8°751 | Juan and Ulloa............ 1744 0 90 Vise =} DSOULUIEN seater severe tee } Peru dies: 713° |’ Condaminé. ::.,.......5.2... : 1762 7 3S 18S St 69° 076". 11a Caille oor. Aen. Cape of Good Hope | 1765” | 43-0. Ne} 68*998"") “Boscovith:. 8.00. ? 17640") 44 944-°-N, 1 69-061"! | Betearia ote, va | ttaly 1766 | 47 40 N.} 69°142 | Leisganig ................... Germany 1768 | 39 12 N. | 68°893 | Mason and Dixon........ America 1802 | 51 29547N.] 69°146 | Lt. Colonel Mudge ...... | England 1803 | 66 203 N.| 69-292 Swanberg, &c...,.......... Lapland 12 32° N.’| 68°743 | | ‘Lambton 0000000... Misore 3 1808 | 44 52f N. 68-769 Biot, Arago, &c............ France Ellipticities of the earth, expressed in parts of its equatorial diameter. Authors. Ellipticities, EP SR EU Sree Huyghens Newton Maupertuis, &c. Swanberg........ Clairault......... Treisnsoker...... Laplace.......... . Principles. Theory of gravity Mensuration of arcs Rotatory motion Vibrations of the pendulum Occultations of the fixed stars Precession and nutation Theory of the moon. DEN Decree of Longitude, is the space between o meridians that make an angle of 1° with ch other at the poles; the quantity or length whieh is variable, according to the latitude, jing every where as the cosine of the Jati- Mie; viz as the cosine of one latitude is to - cosine of another, so is the length of a de- ee in the former latitude to that in the latter, , the supposition that the earth is spherical. iit taking the earth as a spheroid, the degree | longitude may be found in any given lati- de L, by saying, 1. As the equatorial diameter to the polar, fis tang. 9O-—L, to tang. of an angle A; on, 2. As radius to sine of A, sois the length 1a degree of the equator to the length of a gree on the parallel of the given latitude. te LONGITUDE. DEINCLINERS, in Dialing, are those als, which both decline and incline or re- ine at the same time. DELIACAL Problem. See DUPLICATION the Cube. DELPHINUS, the Dolphin. See CoNnsTEL- TION. DEMETRIUS, a celebrated Cynic philo- ioher, who lived in the time of the Emperor ospasian. ‘DEMOCRITUS, one of the greatest phi- jophers of antiquity, who flourished about (0 years before the christian era. DEMOIVRE (ApraAHam), en eminent ma- ematician, was born in France, May 1667, t settled in England at the age of eighteen, d where he continued to reside til! his death, nich happened in Nov. 1754, in the eighty- ist year of his age. ‘Demoivre was author of several interesting pers published in the Transactions of the byal Society, in different volumes, from the aeteenth to the forty-third, beside several mplete works ; as, 1. De Mensura Sortis, 1711, relating to the uws of Chance. 2. Doctrine of Chances, 4to. 1718, 1738, (50. ‘The two latter editions are much es- jemed, and contain a complete investigation | this theory. |3. Miscellanea Analytica de Seriabus, &c. 30, 4to. 4. Annuities on Lives, 1724, 1742, 1750. DEMONSTRATION, acertain or convinc- ig proof of some proposition. | /'DENDROMETER, (from dedpoy, a tree, and spew, ZL measure), an instrument used in easuring trees and standing timbers. DENEB, an Arabic term signifying taz/, and nce applied to the names of certain stars in e tails of some of the constellations, as Deneb dige in the tail of the Swan. DENOMINATOR of a Fraction, is that imber written below the line expressing the umber of parts into which the unit is sup- 5 12’ } sed to be divided; thus in the fractions ; 12 and 8 are the denominators. DEN DENOMINATOR of a Ratio, is sometimes used in the sense of exponent of a ratio. DENSITY, from the Latin densitas, strictly speaking, denotes vicinity or closeness of par- ticles; but in mechanical science it is used as a term of comparison, expressing the pro- portion of the number of equal molecule, or the quantity of matter in one body to the num-~ ber of equal molecule in the same bulk of another body; density, therefore, is directly as the quantity of matter, and inversely as the magnitude of the body. Since it may be shown experimentally that the quantities of matter, or the masses in dif- ferent bodies, are proportional to their weight, of consequence, the density of any body is directly as its weight, and inversely as its mag- nitude; or the inverse ratio of the magnitudes of two bodies, having experimentally equal weights (in the same place), constitutes the ratio of their densities. No body is absolutely or perfectly dense, that is, no space is perfectly full of matter, so as to have no vacuity or interstices; on the contrary, it is the opinion of Newton, that even the densest bodies, as gold, &c. contain but a small portion of matter, and a great por- tion of vacuity, or that they contain a great deal more pores or empty space than real sub- stance. Density of the Air. See ATMOSPHERE. Density of the Earth. The determination of the density of the earth, as compared with that of water, or any other known body, is a subject which has excited considerable interest amongst modern mathematicians, and nothing can at first sight seem more beyond the reach of human science than the due solution of this problem, yet this has been determined, and on such principles, that if it be not correctly true, it is probably an extremely near approx- imation. The first idea of determining the density of the earth was suggested by M. Bouguer, in consequence of the attraction of Chimboraco, which affected his plumb-line while engaged with Condamine in measuring a degree of the meridian, near Quito, in Peru. This led to the experiments on the mountain Schehallien, , in Scotland, which were carried on under the direction of Dr. Maskelyne, and afterwards submitted to calculation by Dr. Hutton, who determined the density of the earth to be to that of water, as 44 tol. But in consequence of the specific gravity of the mountain being assumed rather less than it ought to have been, the above result is less than the true density, as has since been shown both by Dr. Hutton and Professor Playfair, the former of whom makes it, in his corrected paper, as 99 to 20, or nearly as 5 to 1. The same problem has been attempted on similar principles, but totally in a different manner, by the late Mr. Cavendish, who found the density of the earth to be to that of water, as 5'48 to 1. Taking a mean of all thesc, we have the density of the earth to that of water, DEN as 5°24 to 1, and which, as we before observed, is probably an extremely near approximation. Dr. Maskelyne’s account of his experiments is given, Phil. Trans. Ixv. nos. 48 and 49 ; Dr. Hutton’s “ Computation,” vol. Ixviii. His corrected paper is given in his Tracts lately published; Playfair’s, in the Trans. for 1811 ; and Cavendish’s, in the same, for 1798. See ATTRACTION of Mountains. Density of the Planets. We have seen,*in the preceding part of this article, that the density of a body is directly as its mass, and reciprocally as its magnitude ; therefore, any two of these being given the third may be de- termined. Now the magnitude of the several planets, as also’of ihe sun, being supposed known from observation, if we can determine their masses, their densities will thus also be- come known. The determination of the masses of certain of the planets is very easy, but in others it is attended with considerable difficulty ; we will explain the former method, but for the latter we must refer the readers to the several works on physical astronomy; particularly those of Biot and Laplace; see also Vince’s ‘ Astro- nomy,” vol. ii. The power of attraction with which any central body acts upon, another body revolv- ing about it, is directly as the mass of the central body, and reciprocally as the square of the distance of the revolving body; and this power may be measured by the deflection of the revolving body in a given time from the direction of its tangent. the earth as the central, and the moon as the revolving body, the deflection of the latter ni one second is known to be =48, ofa foot, that As, it deviates so much from the direction of its tangent in one second of time, as may be readily ascertained by computation, being equal to the versed sine of the arc described in one second. If the distance between the earth and moon had been double what it is, this de- flection would have been only 4 of the above, ifthe distance were only half what it is, this deflection would be four times as much, and so on, because the power of the central body is reciprocally as the square of its distance. Again, if the distance of the bodies remain the same, and the mass of the central body be doubled, the deflection will be doubled; if tripled, the deflection will be tripled, and so on; because the power of the central body is directly as its mass. Now Jupiter’s first satellite revolves about that planet at the same distance as the moon does about the earth, but its deflection in one second is 256 times that of the moon ; whence, the distances being equal, it follows, that the mass of Jupiter is 256 times greater than that of the earth. Had these distances not been equal, we must have found what the deflec- tion of the moon would be at the given dis- tance; then comparing this with that of the satellite, the mass in any other case might be determined. Having thus found the mass, and the mag- Now if we consider: DEP nilude béing known, we shall have, assumin the mass of the earth as unity; as 1 divide by the magnitude of the earth, is to the ma; of any planet divided by its magnitude ; so 1, the density of the earth, to the density the planet; and in this way the density of an planet haying a satellite may be readily con puted. In other cases recourse must be had | the disturbing force, which is a very laborio computation. ? The density of the sun, and the sever planets, as deducted from Laplace’s “ Syste du Monde,” latest edition, are as follows; v Sun......00. pet ees 0°25226 °°" WIGHOULy 2. s20re- +3: 2°58330 OMe ee NB ONS 1°02400 4 Earth....... Seton hase 1°00000 NiaTe oe ele 0°65630 WUpIESE ccoe ste -haseoky 0°20093 SUMLULENE sole to ccepbves ves 0°10349 Uranus ..,..s: PRD dele! 0°21085 DEPARTURE, in Navigation, is the ea ing or westing of a ship in respect of the m ridian it departed or sailed from; or it is tl difference of longitude in miles either east west, between the present meridian the sh is under, and that where the last reckoning observation was made. ’ DEPRESSION of Equations, in Algebr is the reduction of an equation to one of low dimensions, an operation, however, that.¢ only be effected in particular cases, viz. why a certain relation has place between the roo of the equation ; thus if an equation has equ roots with cither like or contrary signs, or the equation be a reciprocal one, having ea\/ pair of roots the reciprocal of each other, al) in short under almost any known relation | the roots the equation may be depressed one of lower dimensions. If an equation ha) two equal roots it may be depressed one degre if three equal roots it may be depressed ty degrees, and so on: and in reciprocal equ: tions of even dimensions may be depress to half the original degree; and if they are | odd dimensions, to half the original degn minus 1. Sce RecrerocaL Kquations. Generally if the original equation 2”— al 4+. ga"—*— &e. = 0, have m, equal roo} the equation na*—! —(n—1) pa"—? + (n= qx"— — &e,. =o, has m—1 of those roots, may be readily shown. Hence, when the are m equal roots, the two equations have common measure of this form («#—a)"7! being one of the roots) which may be ‘ra tl in the usual way, and m roots of the origi equation may thus be known. Divide equation by («—a)”, and the resulting equatit of n —m dimensions contains the other root Thus, let the cubic 23 — pa* + qx—re have two equal roots ; then 32*—2px+q= has one of them; and the two equations have common measure, which is a simple equatio the quantities 323 — 3pa* + 3qx--37, a 32*°—2px'+q, have also the same comm : , 6q—-2 measure, which being found, we have T= | DEP 3 (te for a divisor of the equation 5 > 4 qu —r =o; this divisor being likewise ‘ r— t = 0, we obtain = Tete Thus two ots of the equation are discovered ; and since s the sum of all the roots, the third root is > difference between p and the sum of the o equal roots. If two roots of an equation be of the form ‘a, —a, differing only in their signs, change 2 signs of the roots, aud the resulting equa- m has two roots + a4,—a; thus we have (0 eqiations with a common measure, «*— a’, nich may be found, and the equation de- essed, as in the preceding case. Sce War- rs “ Med. Alg.” cap. 3. See also Bonny- stle’s “ Algebra,” vol. il. Depression of the Pole, is the quantity of approximation towards the horizon as we oceed towards the equator. Depression of the Sun, or of a Star, is its stance at any time below the horizon, mea- red by an arc of a vertical circle. . Depression of the Visible Horizon, denotes - sinking or dipping below the true horizon- { plane; whether caused by some variation the atmosphere, or by the different height the observer’s eye above the surface of the sea. f : Thus, in the above figure, where the ob- yrver is situated above the earth at A; with- at refraction the visual ray would be AE, ud in that case E is the most distant point jhich could be scen; but by refraction, the 1y FG, coming from the point G, may be yen at F, so as to go on from thence in the ae F A; and then the view is extended as ras G, and the depression of the horizon of xe sea is in the line AT’, which points higher xan A E, but extends the view farther. From inspection of the figure it is evident, that _ the refraction were greater, the view would > extended still farther, as to M; though the apression of the horizon of the sea would 1en be less, as is shown by the line ALM: hence also it appears, that, by reason of the ifference of refraction in the air, our horizon sometimes more extensive than at others. ' c The depression, or dip, may be readily cal- | DER culated, for, in the above figure, the eye being at B, the sensible horizon is FG, and the de- pression is theangle F BA; but this angle, as well as ACB, is the complement of ABC; therefore the angle C is equal to the depres- sion. Hence,wehave CA:CB:: rad.:sec. of C ; or as CB: CA:: rad. : cosin of C. By either of which analogies a table may be readily constructed, for any heights BD. The depression by calculation is in all cases greater than that by observation; but the difference is variable, as is the refraction which causes it. Tables of the depression are given in the Re- quisite Tables for the Nautical Almanac. DEPTH is the third dimension of a body, or it is the distance of one object below another object or place. DERHAM (WILLIAM), an eminent English philosopher and mathematician, was born at Worcester in 1657, and died at Upminster in 1738, being in the 78th year of his age. He was author of several valuable papers in the Philosophical Transactions, which are con- tained in the several volumes from the 19th to 39th, both inclusive ; beside these he pub- lished a small work, entitled the “ Artificial Clockmaker,” which has been several times reprinted. He revised the ‘‘ Miscellanea Cu- riosa” in 3 vols. 8vo.; and published the Phi- losophical Experiments of Dr. Hooke, and other ingenious men of his own time. DERIVATIONS, the Calculus of, a general method of considering quantities, deriving themselves one from another, particularly de- veloped and illustrated by M. Arbogast, pro- “fessor of mathematics at Strasburgh, in a trea- tise entitled “‘ Du Calcul. des. Derivations,” &c. 4to. Strasburgh, 1800. “To form an idea,” says the author, “ of these derivations, it is to be observed, that quantities or functions that” are deduced the one from the other, by any uniform process of operations, are derived quantities, as are the successive differentials, these being all derived from each other ina uniformsmanner. And this idea may be ex- tended by considering quantities that are de- rived one from another, not in themselves, but solely in the operation by which they are con- nected with each other, the quantities them- selves being any whatever arbitrary and in- dependent. Thus on the supposition that, out of many different letters, the first enters solely into a function, while the two next enter into the derivative of that function, that the first three by the same law, enter into the derivative of that derivative, and so on; we shall have the derivative in the extended sense that I have given to them. In my theory, the quantities designated by the different let- ters, are not derived one from another, and the derivatives I consider are less the deriva- tives of quantities, than of operations, as al- rebra is less a calculus of quantities than of operations, arithmetical or geometrical, to be performed on quantities.” Derivation is the operation by which a de- rivative is deduced from that which precedes it, or from the function. The method of de- DER rivations, in general, consists in discovering the law by which different quantities are con- nected with each other, and in making use of this law as a method of calculation, for passing from one derivative to another. We cannot attempt in this place to enter into a very minute illustration of the calculus of derivations, we shall, therefore, simply en- deavour to explain the algorithm of this species of analysis; and must refer the curious reader for further information to the work itself, or to Woodhouse’s ‘ Principles of Analy tical Calculation,” p. 22. It is a fact which has been repeatedly de- monstrated, and now generally admitted, that in the dev elopment ‘of (a + x)”, whatever operation is indicated by m, whether involu- tion, evolution, division, &c., the terms of the series involve the successive integral powers of x, and that the second term of the series is always of the form ma”—'x, so that the coef- ficient of the term involving a is always found by multiplying the index m of the first term, into that term when its index has been di- minished by unity. Let then D be assumed to denote that operation; then, Da” represents ma”—', Da*’-'= (m — 1) ahi8. Das®=—mar"= 1, D.1m =m 1"~!, or m, D.147= 4.172, so also DD a” = m Da”—! = m (m— 1) a™-? n»— § MDDa®—=m (m— 1) Da”? = pe ao m.(m — 1) (m — 2) a™—3, Whence it readily follows that DDD....D, supposing there to be » D’s will represent m such operations as those above, and ex- pressing these x» D’s by the abridged ex- pression D”, we shall have D* a" =m (m—1) (m— 2) (m—8), &e, (m—n +1) a"™—” Ding 18 a dae, Dat, A B20 D? 1" = m (m — 1) (m — 2) 1" -3 D? a-"= — m — (m + 1) a-™-? &e. &e. Whence it follows that the binomial (a + a) according to this notation becomes D2a™ az D3 a” 43 a+2)"=a"+ Da” ay ee te (44 2)e—a" + Dae +——, 1.203 Or if D* be und dt _ ri D* be un erstoo o represent 73477, D2 D3 euehas te2e ly tio What 35 = wee on: Oa ke. then the above is more simply expressed (a+a)"=a"+ Dax + D?a"x’? + Dax? + Ke. € c Such is the nature of the notation employed by M. Arbogast, in his theory of derivations, and which he applies in the first place for as- certaining the form of the development of the function of any multinomial a + bx + ca? + dx, &c.; and which being general, in- cludes, besides the multinomial of De Moivre given in his “ Miscellanea Analytica,” p. 87, all such expressions as sin. (a+ba-+e2* +e. ) log. (a + bx + cx* + &c.), &e., and this in the most commodious manner, by means of the DER general symbol D; and when the operati indicated are to be actually performed, | coeflicient easily result in terms of the pc nomial quantities. After having exhibited the general form the function ? (a + bx + cx* + &c.) he she how from one term to deduce the next s ceeding one, and likewise how to calcul any term whatever of the development in pendently of all the others; but the length the operation precludes the possibility of « plaining his method. In the latter part of: first article, the author applies his method derivations to assign the sum of the power; the roots of an equation, in terms of the e¢ ficients of the equation ; and the form which he deduces is remarkable, for the si plicity of the law by which the coefficie, are expressed. Vandermonde’s “ Mem. de Acad. Scien.” 1771, p. 373; Euler’s “ Com Petro.,” vol. xv.; ‘La Grange’s ‘ Mem. Berlin, dé 1768 ; and Waring’s ‘‘ Meditatiox Algebraice,” p. 1; having given gene formule, for the sum of the powers of roots of an equation, M. Arbogast compa his own with the demonstration of those, a shows in what manner they follow from it. His next object is, the development) functions of two or more polynomials, arrang according to the powers of the same lette and hence he forms the product of any n ber of series arranged in the same order. | Then follows the development of functi¢ of one or more polynomials, arranged relati to. the powers and to the products of two more different letters, into series arranged the same manner ; accordingly he reduces to the following general theorem: «| ** Any function whatever of one or mo simple, double, or triple polynomials beii given, to write immediately the series of t development of this function; and farth¢ to write immediately the development of ai term whatever of this series, independents the other terms.” He then proceeds to various applications derivations to recurring series, simple, doug triple, &c. of any order whatever. The authors who have treated of recarg series are, De Moivre in his “ Miscellan Analytica,” and ‘“ Doctrine of Chances: Euler’s “ Introductio en Anal. Infinitorum La Grange’s “ Melanges de Turin,” and) the ‘‘ Memoirs of the Academies of Paris al Berlin ;” and La Place in the ‘‘ Memoirs” pr sented to the Academy of Paris, and in @ “* Memoirs” of the Academy. To the subje: of the researches of these celebrated math maticians, M. Arbogast applies his method | derivations, and certainly obtains expressiot very admirable for their simplicity, as well: for the facility with which they can be @ panded. M. Arbogast determines the sim and cosine of any multiple arc, in cosines | the simple arc, and his demonstrations ft the forms expressing the sines, cosines, & are clear and rigorous. This next arti¢l contains applications of the calculus of der wr % DES ois to the general reversion of series, a ) of the work which is executed with sin- rability. He then shows the use of his ry in the differential calculus, which he ysiders as a particular case of the calculus f erivations. ‘or a farther account of this subject we it refer to the work itself. See also fonthly Review ; new Series,” vol. xxxvi. 24-532; Montucla’s ‘“ Hist. des Math.” iy. p. 659, and the other works above erred to. JESAGULIERS (JOHN THEOPHILUS), an nent experimental philosopher, was born Rochelle in France in 1683, but educated mingland, where he afterwards continued to fide. He was a member of several learned geties, and made numerous communica- i's on the subjects of optics, mechanics, &c. (1 to the Royal Society of London, and to j Academy of Sciences at Paris; the former {hich are contained in the volumes of the imsactions, from the 29th to the 42d. jeside these he published a _ valuable ourse of Experimental Philosophy,” in 2 ». 4to. 1734. He also edited an edition of bgory’s “ Elements of Catoptrics and Diop- s,”’ with an “ Appendix on Reflecting Te- opes,” 8vo. 1735. Desaguliers died in 29, in the 66th year of his age. VESCENDING Latitude, is the latitude of janet in its return from the nodes to the ator. YESCENSION, in Astronomy, is cither fit or oblique. Right DEscENSION, is the arch of the equa- DW ich descends with the sign or star below horizon of a direct sphere. Yblique Descension, is the arch of the ator which descends with the sign or star yw the horizon of an oblique sphere. See |:ENSION. )ESCENSIONAL Difference, is the dif- ance between the right and oblique descen- i: of the same star, or point of the heavens. YESCENT of Bodies, in Mechanics, is motion or tendency towards the centre he earth, either in a direct or oblique di- (ion. The laws of the descent of bodies in i space are given under the article Ac- ERATION, their descent along inclined aes under the article INCLINED PLANE, and iluids under RESISTANCE. iN e shall, therefore, under the present ar- de, merely refer to a singular phenomenon necting falling bodies, which is their de- Vion from the perpendicular occasioned by i rotation of the earth on its axis. When idea of the earth’s motion was started by ernicus, it was strongly objected to it, Ht if the earth reaily revolved, a stone let ‘from the top of a tower ought to fall con- erably to the westward of the foot of it, ng, according to their notion, left behind by , motion of the earth; and the supporters he doctrine were not then sufficiently ac- j inted with the composition of motion to > lain away the difficulty. However, when ' laws of motion became better under- } : —— ~_—m- DES stood, it was discovered that the body, in- stead of falling to the westward of the tower, ought to fall a little to the eastward of it, in consequence of the velocity of rotation being greater at the top than at the foot of the tower ; and this deviation is said to have been really detected by M. Guglielmini and M. Benzen- berg, the former finding it to be equal to 8 lines in falling 241 feet, and the latter 5 lines for a fall of 262 feet. But how far experiments on such delicate subjects may be depended on, may be a mat- ter of doubt, but of the truth of the theoretical deflection no doubt can be entertained. La Place gives the following theorem for the com- putation of this deflection: Let 4 = the height the body falls, g = double the space a body will descend in the first second from the action of gravity, n — the angle of the earth’s rota- tion in the same time, at the rate of 360°— 0°99727 in a day, and 6 the colatitude of the re also A the deviation towards the east, then eee 2nh sin. Renee 8 (Bulletin des Science, No. 75.) DESCRIBENT, in Geometry, is the line or surface, from the motion of which, a sur- face, or body, is Supposed to be generated or described. DESCRIPTIVE Geometry, the name given to a species of geometry almost entirely new, and which we owe in great measure to M. Monge. When any surface. whatever penetrates another, there most frequently results from their intersection curves of double curvature, the determination of which is necessary in many arts, as in groined vault-work, cutting arch-stones, wood-cutting, for ornamental work, &c. the form of which is frequently very singular and complicated: it is in the solu- tion of problems appertaining to these subjects that descriptive geometry is especially useful. Some architects, more versed in geometry than persons of that profession commonly are, have long ago thrown some light on the first principles of this kind of geometry. There is, for example, a work by a jesuit named Father Courcier, who examined and showed how to describe the curves resulting from the mutual penetration of cylindrical, spherical, and co- nical surfaces: this work was published at Paris in 1663. P. Deraud,.Matheurin Jousse, Frezier, &c. had likewise contributed a little towards the promotion of this branch of geometry. But Monge has given it very great extension, not only by proposing and resolv- ing various problems both curious and diffi- cult, but by the invention of several new and interesting theorems. We can only mention in this place one or two of the problems and theorems. Thus, among the problems: Ist, Two right lines being given in space, and which are neither parallel nor in the same plane, to find in both of them the points of their least distance, and the position of the line joining these points. 2d, Three spheres DIA, being given in space to determine the position of the plane which touches them. There are ’ also curious problems relative to lines of dou- ble curvature, and to surfaces resulting from the application of a right line that leans con- tinually upon two or three others given in position in space. Among the theorems, the following may be mentioned ;: if a plane sur- face given in space is projected upon three planes, the one horizontal, and the two others vertical and perpendicular to each other, the square of that surface will be equal to the sum of the squares of the three surfaces of projec- tion. This theorem is as interesting in the geometry of solids, as Pythagoras’s theorem (Euc. 1. xviii.) is in plane geometry. But for more on this subject, we must refer to Monge’s and Lacroix’s ingenious works, enti- tled ‘“‘ Geometrie Descriptive.” DETERMINATE Problem, in Geometry and Analysis, is that which admits but of a certain and determinate number of solutions, being thus opposed to indeterminate problems, the number of whose solution is unlimited. DETERMINATE Section, is the title of a tract or general problem of Apellonius’s, which have never been handed down to the moderns; but from the account given of that work by Pap- pus, various attempts have been made at a - restoration of it by Snellius, Lawson, Wales, &e. See APOLLONIUS. DEVELOPMENT, is a term in frequent use amongst modern analysts ; to denote the transformation of any function into the form of a series: thus the development (a + 6)”, is the expansion of it into the form a” 4+ ma”~b + sees ¥} a®—*h + &e. | Ra DEVIATION, in the ancient Astronomy, a motion of the deferent either towards or from the ecliptic. DEVIATION of a falling body. See DESCENT. DEW, a thin light insensible mist, or rain, which ascends from the earth with a slow motion, and falling again while the sun is below the horizon. DE WIT (Joun), the celebrated Dutch pensionary, was born at Dort, in 1625; and murdered by the mob in a civil commotion. De Wit was equally esteemed both as a politician and mathematician, and wrote on both subjects; his work ‘ Elementa Cur- varum Linearum,” which he published in 1650, was one of the most profound mathematical works of that period. DIACAUSTIC Curve, or Caustie by Re- fraction, is a species of caustic curves, the genesis of which is in the following manner: See FUNCTION. Imagine an infinite number of rays B A, BM, DIA BD, &c. issuing from the same lumine point B, refracted to or from the perpendiew MC, by the given curve AMD; and so, t} CE, the sines of the angles of inciden C ME, be always to CG the sines of the ; fracted angles CMG, in a given ratio: th the curve HEN, that touches all the ; fracted rays AH, MF, DN, &e. is called t diacaustic, or caustic by refraction. ‘T, word is derived from dia, through, and xaiw, burn. e4 DIACOUSTICS, or Diaphonies, (omg through, and axew, I hear), the considerati, of the properties of sound refracted in paul through different mediums. r DIAGONAL, (from dia, through, and yuy| angle), in Geometry, a right line drawn aet a quadrilateral or other figure, whether pla} or solid, from the vertex or summit of © angle to that of another; and is by some; thors called the diameter, and by others }; diametral of the figure. Af It is demonstrable, 1. That every diago)| divides a parallelogram into two equal pai, 2. That two diagonals drawn in any paral. ogram bisect each other. 3. A line passi, through the middle point of the diagonal o, parallelogram, divides the figure into tp equal parts. 4. The diagonal of a square incommensurable with one of its sides. 5.7; sum of the squares of the two diagon sf every parallelogram is equal to the sum of § squares of the four sides. 6. In any tra zium, the sum of the squares of the four sii is equal to the sum of the squares of the t diagonals, together with four times the squé of the distance between the middle points the diagonals. 7. In any trapezium, the s of the squares of the two diagonals is dout the sum of the squares of two lines bisect} the two pairs of opposite sides. 8. In @ quadrilateral inscribed in a circle, the rect gle of the two diagonals is equal to the st of the two rectangles under the two pair#) opposite sides. 9. In every parallelopiyi of the solid, is equal to the sum of the squat of its twelve edges. 10. In every hexaed regular or not, the sum of the squares of) twelve edges, plus the sum of the squares the twelve diagonals of the faces, is equal three times the sum of the squares of the fil diagonals which cross the solid, plus four tin) the sum of the squares of the six right li which join, two by two, the middle point) those four latter diagonals. | polygon, and in every polyédron, the sum the squares of the lines which join, two} two, the middle points both of sides and é gonals, is the quarter of the sum of the squa of those sides and diagonals; multiplied the number of summits of the polygon polyédron, diminished by two units. 12) farther generalization of the latter prope leads to the most celebrated property of centre of gravity. | DtaGonat Scale. See SCALe. a DIAGRAM, (from die, and ypadw, £ scribe), is a scheme for the explanation) } DIA ymonstration of any figure or of its pro- orties, DIAL, or Sun-Dial, an instrument serving ‘measure time, by means of the shadow of e sun. The word is formed from dies, day, »eause indicating the hour of the day. The acients also call it sciathericum, from its doing by the shadow. DIA is more accurately defined, a draught, » description, of certain lines on a plane, or rface, of a body given, so contrived, as that 'e shadow of a stile, or ray, of the sun, passed ‘rough a hole therein, shall touch certain »ints at certain hours. The antiquity of dials is beyond doubt. yme attribute their invention to Anaximenes jlesius ; and others to Thales. Vitruvius entions one made by the ancient Chaldee storian Berosus, on a reclining plane, almost rallel to the equinoctial. Aristarchns Sa- ius invented the hemispherical dial. And ere were some spherical ones, with a needle cagnomon. The discus of Aristarchus was horizontal dial, with its limb raised up all jund, to prevent the shadows stretching too It was late ere the Romans became ac- iainted with dials. The first sun-dial at ome was set up by Papirius Cursor, about le year of the city 460; before which time, lys Pliny, there is no mention of any account (time but by the sun’s rising and setting: it jas set up at or near the temple of Quirinus; it being inaccurate, about 30 years after, other was brought out of Sicily by the con- il M. Valerius Messala, which he placed on {pillar near the Rostrum; but neither did sis show time truly, because not made for jat latitude; and, after using it 99 years, jartius Philippus set up another more exact. /The diversity of sun-dials arises from the erent situation of the planes, and from the erent figure of the surfaces upon which yey are described; whence they become de- Vminated equinoctial, horizontal, vertical, polar, wect, erect, declining, inclining, reclining, &c. br which see the respective articles. \Universal DIAL, there are several kinds of 4als called universal, because they serve for + latitudes. One of a very ingenious con- vuetion has lately been invented by Mr. G. Yright of London. The hour circle, or arch } and latitude arch C, (plate ix. fig. 3.) are ‘ie portions of two meridian circles; one ‘ied, and the other moveable. The hour or fal-plate S EI at top is fixed to the arch C, ‘?d has an index that moves with the hour an E; therefore the construction of this (al is perfectly similar to the construction of je meridians and hour-circles upon a com- ‘Dn globe. The peculiar problems to be per- med by this instrument are, 1. To find the bitude of any place. 2. The latitude of the jice being known, to find the time by the $n and stars. 3. To find the sun or stars \itude and azimuth. But the dial being properly adjusted, a leat variety of astronomical and trigonome- DIA trical problems may be wrought upon it, which however our limits will not allow of detailing ; we must therefore refer the reader to Jones’s “ Instrumental Dialling,” where he will find ample information on this subject. DIALLING, the art of drawing dials on the surface of any given body, whether plane or curved. Dialling is wholly founded on the first motion of the heavenly bodies, and chiefly the sun; or rather on the diurnal motion of the earth: so that the elements of spherics, and spherical trigonometry, should be under- stood before a person advances to the theory of dialling. The edge of the plane by which the time of the day is found is called the stile of the dial, which must be parallel to the earth’s axis; and the line on which this plane is erected is called the substile. The angle included be- tween the substile and stile is called the elevation or height of the stile. The principles of dialling may be aptly il- lustrated by the phenomena of a hollow or transparent sphere, as of glass. Thus suppose aP Bp to represent the earth as transparent ; and its equator as divided into 24 equal parts by so many meridian semicircles a, b, c, d, e, &e. one of whichis the geographical meridian of any given place, as London, which itis sup- posed is at the point a; and if the hour of 12 were marked at the equator, both upon that meridian and the opposite one, and all the rest of the hours in order on the other meri- dians, those meridians would be the hour cir- cles of London: because, as the sun appears to move round the earth, which is in the cen- tre of the visible heavens, in 24 hours, he will pass from one meridian to another in an hour. Then, if the sphere had an opaque axis, as PE p, terminating in the poles P and p, the shadow of the axis, which is in the same plane with the sun and with each meridian, would fall upon every particular meridian and hour, when the sun came to the plane of the op- posite meridian, and would consequently show the time at London, and at all other places on the same meridian. If this sphere were cut through. the middle by a solid plane ABCD in the rational horizon of London, one half of the axis EP would be above the plane, and the other half below it; and if straight lines were drawn from the centre of the plane to those points where its circum- DIA ference is cut by the hour circles of the sphere, those lines would be the hour lines of an horizontal dial for London; for the shadow of the axis would fall upon each particular hour line of the dial, when it fell upon the like hour circle of the sphere. If the plane which cuts the sphere be up- right, as AFCG, touching the given place, for exam. London, at F, and directly facing the meridian of London, it will then become the plane of an erect direct south dial; and if right lines be drawn from its centre E, to those points of its circumference where the hour circles of the sphere cut it, these will be the hour lines of a verticat or direct south dial for London, to which the hours are to be set in the figure, contrary to those on an horizontal dial; and the lower half E’p of the axis will cast a shadow on the hour of the day in this dial, at the same time that it would fall upon the like hour circle of the sphere, if the dial plane was not in the way. If the plane still facing the meridian, be made to incline, or recline, any number of degrees, the hour circles of the sphere will still cut the edge of the plane in those points to which the hour lines must be drawn straight from the centre; and the axis of the sphere will cast a shadow on these lines at the re- spective hours. The like will still hold, if the -plane be made to decline by any number of degrees. from the meridian towards the east or west; provided the declination be less than 90 degrees, or the reclination be less than the co-latitude of the place; and the axis of the sphere will be the gnomon: otherwise, the axis will have no elevation above the plane of the dial, and cannot be a gnomon. Thus it appears that the plane of every dial represents the plane of some great circle on the earth, and the gnomon of the earth’s axis ; the vertex of a right gnomon the centre of the earth or visible heavens; and the plane of the dial is just as far from this centre as from the vertex of this stile. The earth itself, com- pared with its distance from the sun, is con- sidered as a point; and therefore, if a small sphere of glass be placed upon any part of the earth’s surface, so that its axis be parallel to the axis of the earth, and the sphere have such lines upon it, and such planes within it, as above described; it will show the hours of the day as truly as if it were placed at the DIA earth’s centre, and the shell of the earth we) as transparent as glass. (ferguson, lect, x, To describe an Horizontal Dial, geometrically, Draw a meridian line BK.L, and from any part C, erect the perpendicular C.D, and make the angle CAD = to the lati- tude of the place, and draw the line dE, meeting AB in E. Make EB=ED, and from the centre B, with radius EB, describe a quadrant EBF, which divide into six equal part Through E draw GH perpendicular to A] From the centre B through the several subd visions of the quadrant E F, draw the lines B Bd, &c. meeting the line G H in the points b, ec, &e. ‘ From E draw the line EA, and set off E Ff, Eg,&c. respective equal to Ea, Eb, Ee,& Through the point A draw the linesa XJ, b c[X, &e. also, e I, f II, g UI, &c. which w be the several hour lines required. Then A erect the gnomon or stile at an angle equ to the elevation of the pole, or latitude of # place, and the dial will be complete. | To compute the Angles of the Hour Lines trigonometrically. : As Radius ; : Is to the sine of the latitude of the place, So is tangent of sun’s distance from 1 meridian . To the tangent of the angle from the me dian on the dial. . The preceding construction and comput tion for the case of an horizontal dial may || said to include the whole theory of dialin for there is no plane, however obliquely | tuated with respect to any given place, b what is parallel to the horizon of some oth] place; and therefore, if we find that oth place by a problem on the terrestrial globe, | by a trigonometrical calculation, and co struct a horizontal dial for it, that dial appli, to the plane where it is to serve will be a tr} dial for that place.—Thus, an erect dire! south dial in 514° .N, lat. would be a horizo} tal dial on the same meridian, 90° southwai) which falls in with 382° S. lat.: but if the » right plane declines from facing the south! the given place, it would be a horizontal pla! 90° from that place; but for a different lony tude, which would alter the reckoning of t) hours accordingly. at We cannot in this place enter into a m¢ minute explanation of this subject, and mi} therefore refer the reader, who is desirous’ farther information, to the works written e pressly on dialing, of which the following 4 the principal ones, viz. Vitruvius, in his “ 4 chiteciure,” cap. 4. and 7. lib. ix.; Sebastil Munster, in his “ Horolographia ;” John Di A DIA nder, “de Horologiorum varia Composi- ione ;” Conrade Gesner’s “ Pandectz ;” An- rew Schoner’s “ Gnominice;” Fred. Com- 1andine, “ de Horologiorum Descriptione ;” oan. Bapt. Benedictus, ‘de Gnomonum -Imbrarumque Solarium Usu ;” Joannes Geor- ius Schomberg, “ Exegesis Fundamentorum momonicorum;” Solomon de Caus, “ Traité tes Horologes Solaires ;” Joan. Bapt. Trolta, Praxis Horologiorum;’ Desargues, ‘“ Ma- iere Universelle pour poser |’ Essieu et placer bs Heures et antres Choses aux Cadrans jolaires;” Ath. Kircher, “ Ars magna Lucis » Umbre ;” Hallum, “ Explicatio Horologii : Horto Regio Londini;” ‘“ Tractatus Horo- giorum,” Joannis Mark; Clavius, ‘“ Gno- onices de Horologiis ;’ in which he demon- rates both the theory and the operations ter the rigid manner of the ancient mathe- aticians; Dechales, Ozanam, and Schottus, ive much easier treatises on this subject; did also Wolfius in his “ Elementa.” M. card gave anew method of making large als, by caleulating the hour lines; and M. e la Hire, in his “ Dialling,” printed in 1683, ve a geometrical method of drawing hour ies from certain points, determined by ob- rvation. Everhard Walper, in 1625, pub- hed his “ Dialling,” in which he lays down ‘method of drawing the primary dials on a iry easy foundation; and the same founda- mis also described at length by Sebastian unster, in his “ Rudimenta Mathematica,” blished in 1651. In 1672, Sturmius pub- hed a new edition of Walper’s “ Dialling,” th the addition of a whole second part, con- ming inclining and declining dials, &c. In 08, the same work, with Sturmius’s addi- ns, was re-published, with the addition of ourth part, containing Picard’s and De la re’s method of drawing large dials, which ikes much the best and fullest book on the yect. Paterson, Michael, and Muller, have th written on dialling, in the German lan- age: Coetsius, in his “ Horologiographia a,” printed in 1689; Gauppen, in his i gets DIA “ Gnomonica Mechanica; Leybourn, in his Dialling ;” Bion, in his “ Use of Mathema- tical Instruments ;” Wells, in his “ Art of Shadows.” There is also a treatise by M. Deparceux, 1740. Mr. Ferguson has also written on the same subject in his “ Lee- ‘tures on Mechanics,” above quoted; and Emer- son, in his “ Dialling ;” Leadbetter, in his “ Mechanic Dialling ;’’ Mr. W. Jones, in his “ Instrumental Dialling ;” and the learned bi- shop Horsley, in his ‘‘ Mathematical Tracts.” Dr. Brewster-has described, in the appendix to his valuable edition of ‘“‘ Ferguson’s Lec- tures,” an analemmatic dial which sets itself; and many ingenious constructions of dials are given in Dr. Hutton’s translation of ‘“* Mon- tucla’s Recreations.” DIAMETER (from de, through, and perpoy, measure), in Geometry, the line which passing through the centre of a circle, or other curvi- linear figure, divides it or its respective ordi- nates into equal parts. Conjugate DiAMETER of a Conie Section. See CONJUGATE. Transverse DIAMETER of a Conic Section, is a line drawn through the foci to the curve. Diameter of the Planets, &c. in Astronomy, are either real or apparent. The real diameters are the absolute mea-~ sure of them in miles, &c. and their apparent diameters are the angles under which they appear to spectators on the earth, and are therefore different under different circum- stances; viz. according as they are nearer or more remote from the earth. As these angles are very small the diameter of the planet may be considered as part of a circle of which the earth is the centre, and they are said to con- tain so many minutes, seconds, &c. as is mea- sured by that arc; and consequently the ap- parent diameter of a planet is in the inverse ratio of its distance. And hence again the apparent diameters and distances being known their real diameters thence become deter- mined. TABLE OF THE MEAN AND APPARENT DIAMETERS OF THE PLANETS. Apparent mean diam./4yyarent diameter Mean diameter in|APparent diam. of the sun seen from] Proportional diam. § hag from the/as seen from the sun.| English miles. ie inoue ieee cat Oe ee PON ee, ; 882269 ree 110°000 (Mercury ........ 11”8 16” 3124 80! “4 a de 579 30" 7702 46' 9 EE PL eMsa een 17-4 7916 32’ 1:0 Se 894 10” 4328 21’ 5 Supiter........... 39” 37” 91522 6" 1 116 Saturn ...... ..., 18” 16” 76068 34 9°8 (Utanus:.......... 3°54 4” 30112 16 4°25 WVEOOn’............). 31! 265 4"6 2160 32’ hd The diameters of the new planets have not been accurately determined. Yor the diameters of the rings of Saturn, see SATURN. Q DIF DIAPHONICS. See Diacoustics. DIFFERENCE, is the remainder, .after taking the less of two quantities from the greater. Ascensional DIFFERENCE. See ASCENSION. DIFFERENCE of Longitude of two places, is an are of the equator contained between the meridians of those places. DIFFERENTIAL, in the higher Geometry, is an infinitely small quantity, or part of quantity so small as to be less than any as- signable one, and is thus denominated, be- cause it is frequently considered as the differ- ence of two quantities, and as such is the foundation of the differential calculus. DIFFERENTIAL Calculus, is a method of dif- ferencing quantities, or of finding an infinitely small quantity, which being taken infinite times shall be equal to a given quantity; or 7t is the arithmetic of the infinitely small dif- ferences of variable quantities ; and thus dif- fers in its metaphysics from the fluxional cal- culus, those quantities which in the former are considered as infinitely small differences, being in the latter supposed to be the infinitely small or momeutary increments of variable or flowing quantities. This difference in laying the foundation of the two methods has also given rise to a dif- ference in the notation, the fluxion of any quantity as x being represented by x, and the differential of it by dx; in other respects the two methods are precisely the same, and we shall therefore refer the reader for the several rules and principles to the article FLUXIONs, where also is given a sketch of the history of the differential and fluxional calculus. It will be proper, however, to give in this place a view of the differential notation and metaphy- sical principles on which it is treated by the most eminent analyst of the present day. The following tables exhibits the differen- tial notation, corresponding to the several cases of the fluxional, which will be sufficient for the reader’s comprehending the former without entering into any particular explanation. Fluxional z, z, mre aE AT ee z ul u du d?u du d?u— d3.u a(2 dz Differential Be ae? a? devdy aay The -abeve notation being understood, the principles on which La Grange has treated the differential calculus will be comprehended from the following illustration. Let wu =f be any function of x, then if for x we substitute x 4+ h, the development of this will be of this form: TT] 2 f@th=fetfrh+T*e + &e. : ti Fluxional —, —, x x vt the coefficients f’x, f= being derived from in primitive function fx, and independent of i. Hencef(x +h)—fx=f' xh 4s + &e. ~~ DIF which quantity represents the difference tween fx, and what fx becomes when zi increased by /, or when it becomes a + A, — Let the first term of this difference bh called the differential, and be denoted by th: expression’d.fx; then we have d. fx =f ah d.fx wt i D> Hence to find f’a, divide the difference between two successive values of fx by th increment of x; but since f’x is indepe dent of A, h must disappear by this divisior and may be represented by any symbol ¢ pleasure. ia Therefore, for the sake of uniformity, let be represented by dx; thenf’a 4 ~. Hence dx ; in order to find dfx or the differential of fi write in the function fx, x + dx instead of: then develop f (2 + dx) as far as the term effected with the first power of dz, and sul tract fx from it. ii Hence f («& + dx)—f xr=—f'xdz ¥ f (@ 4+'dz)—f' x =f sdx” ae St! (a + dx) —f'«x =f" axdz therefore f/x = &e. &e. Also f’xdz — d. fz, f’adz = d.f' x, Sem . tee ae gia Ns e therefore f’x = af et at ) ou, “dies dz dx / dz jet But since dz is invariable, Ps S (a + dx) dx — f'adx = f"xdxu™ hence df zdz.= f were . but df’adx = ddfx = d*fx, d’ representil the second difierential of « according to preceding notation. Whence it appears, since dfx. sr a fT SE Vik ih oa ii a envied (42) 7 also fx = das a(afs) 1 fe dx J dais dx " and thus the derived functions f’x, f’ 2, f” ¥ d. &e. so that the development © } Sor alin i > &c. may be represented by the quantities fe Bfx da?’ dx’ (« + dx) takes the following form; viz. he a ile d*fx dfx S@tds)\—fe* Te Toast Taadele which is the celebrated theorem of Bre ‘Taylor. See FUNCTIONS. * DIFFERENTIO Differential Calculus, the method of differencing differentials, as the preceding article.

luminary is wholly covered, the digits fipsed are precisely 12; and when it is more im covered, as is frequently the case in tar eclipses, then more than 12 digits are sd to be eclipsed. DILATION. See Expansion. DIMENSION, in Geometry, is either length, yadth, or thickness. Thus a line has only y2 dimension, length; a surface two, length \ breadth; and a body or solid, length, adth, and thickness. /JIMENSION of an Equation, or any other qimtity in Algebra, is used with regard to h hest power that enters into its composition: is an equation is said to be of one, two, ee, &c. dimensions, according as it involves simple quantity, the square, cube, &c.; so ta simple equation is of one dimension, a yidratic of two, a cubic of three, &e. MMETIENT has been sometimes used for ameter. JINOCRATES, a celebrated Macedonian paaect who flourished in the time of Alex- er. JIONYSIUS, an ancient geographer, sup- ed to have flourished in the time of the Augustus Cesar. JONysiaN Period. See PEeRiop. MOPHANTINE Analysis, or Problems, {igebra, are certain questions relating to sare, cube, &c. numbers, and rational right- ‘led triangles ; the properties of which were it discussed by Diophantus, in his “ Arith- : c. bs er Diophantus, the subject was taken I ( DIO up and extended by his translators and com- mentators, particularly by Bachet, Fermat, and De Billy; the two former having added a variety of notes to ihe original text, and the latter composed an entire and ingenious work on the subject, entitled “ Diophantus Redi- vius.”” The subject has since been pursued by Ozanam, Prestet, Kersey, Saunderson, and Kuler. A number of curious questions of this kind are commonly given in our mathematical periodical works, and particularly in Ley- bourn’s “ Repository,”, in which are several ingenious essays on the Diophantine analysis by Mr. Cunliffe: the same subject is also treated of by professor Leslie, in vol. ii. of Edin. Phil. Trans.; and in chap. vi. part ii. of Barlow’s ‘“ Theory of Numbers.” We cannot enter at any great length upon the solution of these problems; the following, however, will throw some light on the sub- ject. 1. To divide a given square number into two other square numbers. Let a’ represent the given square, and 2? and y* the required squares; then we have only to satisfy the equation poate Hy", OF a~ ee y” = Pea In order to which, assume Pe aty—* q rnin bare a—yomte : P From which we readily deduce Og PE Is Gees eee _ Pa. ay = 2" 9% — PT 1) ti, P PY Whence gr EE pict g Pi tale eg x (2P44 Vee Dk 2pq pr rig p+q” Where the indeterminates p and g may be as- sumed at pleasure. Cor. Ifa be the sum of two squares, p and g may be so assumed that p® + 9? =a, or any factor of a; in which case the above expres- sions will be integral; and as many different integer values may be found for x and y as there are different ways of resolving a into the sum of two squares, or any of its factors) Exam. 1, Resolve 657 into two other squares, Here x = 2P9 x 69 and y 5 (ee p* of gq ? p* -- q* And since 65 = 87 + 17 = 7* + 47; we may take p 8, and g=1; which gives x = 16, and y — 63; p=7, and g = 4; which gives «= 56, and y = 33; Also since 13 = 3” + 2*is a factor of 65, we may take p — 3, andg —=2; which gives x = 60, and y = 2; Q2 f DIO And again, 5 = 2* + 1* is a factor of 65, therefore we may take p = 2, and ¢=1; which gives x = 52, and y = 39; So that 65% = 167 + 637 — 56* +337 = 60% + 257 = 52? + 39°. Which are the only integral solutions that the equation admits of; but fractional answers may be obtained, ad libitum. 2. To divide a number that is equal to the snm of two given squares into two other square numbers. Let a* and 6? represent the two given squares, and 2? and y* the two required ones; then we must solve the equation, a” + b* = 2* + y", or Sinaia 2 For which purpose, let us assume a¢e= Pt Pt Aaah Di, p Sag + gz =py + pb Whence..... ap — px = qy — 4b Or. att = pb —aq occecncenase pe +qy—qb + ap Now multiplying these equations by p and q, so as to eliminate x and y, we have by the common rules § pqx — p'y = p (pb — aq) Upge + Py = 4 (Gb — ap) bq + 2apq — bp* p + qr And in the same manner, OE ih Ld Brom ag” ms Pp + q In which expressions p and q are indeter- minates, and may be assumed at pleasure. Cor. If the given sum a* + 6° be such as to admit of a resolution into two other in- tegral squares, it will be better to resolve the given number, or sum, into its factors; which in that case will also be the sums of two squares; then their product will give the re- quired squares: thus, if a” + b* = (m* + n*) ~(m™ + n”); then by a + 0* = (mm & = m'n)* Whence, y = nn’) -+ (mn Ca nin one ly =mn = mn Exam. 1. It is required to resolve 85 = 9” -+ 2? into two other integral squares, Here 85 =—5 x 17 = (2? + 1’) x (47 +1); whence m = 2, » = 1; also m’ = 4, and vn’ = 1. noe, eee. bt om OS Ore Whence > —2.144.1=2, or6. that is, 85 = 9° +2 =—7° +6. * But if the given number be not resolvable into factors of the form we have supposed, then it isin vain to seek for integral solutions; therefore aeaeeeeee and we must then proceed according to the . foregoing proposition. Exam. 2. It is required to resolve 5 = 2* -+- 1* into two other squares. Here a = 2, and 6 = |; and by taking p = 2, ‘ 3) and q = 3, we have x = =, and y = 75 DIO 3. To find three square numbers in arith metical progression. un Let a’, y*, and 2’, represent the three rm quired squares; and it will then be necessan| to solve the equation 45} 2 2” = Qy?. In order to which, let x = m + n, and z: m —n, then 2* + y* = 2m + Zn? = 2y?, And it only remains to find ‘ m + at y*. We have, therefore, ; = p Ff q" n = 2pq ‘ which values being substituted for m and | in the equations z=m-+n, and y=m— give ;: =—p—_ + 2pq ial z—P 7 — Ang pe Pe ae , In which formula p and g may be assum) at pleasure. al If, for example, we take p = 2,g = 1, three squares will be 7*, 5”, and 1*. Again, assuming p =3, and q = 2, we ha 177, 137, 7, for the required squares. — 2d Method. The equation 2* + 2* =} may be put under the form + Qy? — 2” = 2°; and this again may be represented by (2y + 2)? —2(y +a az Therefore, making $2 Sane eda f ' - i y + v= 2pq we have, by subtraction, y = p* (+ 2q°— 2pq «= 4pq—p* — 2q* ceo, gi 32 These results are apparently different fi the former, but they are readily reduciblé the same, and will in their present fi equally answer the required conditions. Exam. Assuming p=3, and q=1, wel y = 5, x = 1, andz = 7; which is the s as one of the preceding solutions. 4. To divide the sum of three square ni bers, in arithmetical progression, into t other squares which shall also be in arith tical progression. Ay Let s = a* + 6? + c*, represent the | of three square numbers, in arithmetical gression, and let 2”, y*, z%, be the requ! squares; then it will be necessary tos} the equations 4 sf dake are e | x* 4 2 = Dy* ay And here we soon see that y* = 48,! also 6 = 4s, therefore y” = 67; substitt! this value of y*, the equation is reduce! that of finding the values of x and z il! equation x* + 27. = 2b; py the solution of which has been give’ prob.2; where we find (by making 4 =) that article) | _— _. bp? + 2bpg—bq U mm 2 2 = Bi: Fe tias y= bg? +2bpq—bp* _ p* + g* 77; DIO In which expressions p and qg may be as- imed at pleasure; and thus any number of ts of squares may be obtained, which shall in arithmetical progression, and their sum jual to the given sum. ; Exam. 1. Find three square numbers, in ‘ithmetical progression, whose sum shall be mal to re yee F = 75: Here, since = 5, we shall have by assum- 'g successively he Ree, 85 2 FY (= os i (=8,9=2forthesg.( = ) +574 = ) =9o (oii 31 \? ne Sy ( : ) +5°+( -) —75 sr 245) 4 a4 (109) ag j-9,q—4 frsnsteiee (= +5 “P oe jenna ia '&e. &e. sral square numbers, in arithmetical pro- pasion, that should be resolvable into other re squares having the same property; : er If it were required to find three in- *n 6 must be so assumed that 6? may be solved into two integral squares; which may done by making 6 equal to the product of -25 0 13 27 804 3 4 40 30 34 9) A hana | air glam. 33. 21 17 57 rabe.26 34 9 15 as ist South & below 16 45 29 28 119 28 41 0 21 8 39 =o 1777 | 35 65 45 37 1774 41 6 63 49 1777 145 47 5 1773 70 Some have endeavoured to find the latitude and longitude of places by means of the dip- ping-needle, particularly Burrowes, Gilbert, Bond, Whiston, &c.; but as nothing of im- pou followed from their attempts, it would e useless to enter upon any explanation of their particular methods in this place. DIRECT, a term frequently used in arith- metic, astronomy, optics, &c. to denote some 7] DIR peculiar condition or circumstance; as Dire Proportion, Ratio, Rays, Vision, &e.; for whic see the respective substantives. y Direct, in Astronomy. See CONSEQUENTIA, Direct Dial, is one which points direct} to any one of the four cardinal points, and - hence called direct east, west, north, or sou dial, according to the point towards which is directed. Direct Sphere. See Ricut Sphere. DIRECTION, in Astronomy, the motio and other phenomena of a planet;when direc Line of Direction, in Gunnery, is the direy line in which the piece is pointed; and, ; Mechanics, the same term implies the line} which a body moves, or in which a force is @ plied. When two conspiring forces act on body at the same time, the angle included b tween the lines of their direction is ‘all : “F ef angle of direction. (% Number of Direction, in the Calenda the number of days that Septuagesima Sund; falls after the seventeenth day of January. Quantity of Direction, in Mechanics, is term sometimes used to denote what is mo commonly called Momentum. r DIRECTLY, is used in nearly the sar sense as direct; thus we say, quantities @ directly proportional, which is only anoth way of stating them to be in direct propt tion; and, in Mechanics, one body is said impinge directly upon another, when t former strikes the latter perpendicular to surface. - DIRECTRIX, in the Conic Sections, is certain right line perpendicular to the axis the curve, and frequently referred to in tre ing of the properties of those figures, fp the description of them in plano. ty The indefinite right x line D X is the direc- trix; and it is such that if any point S be assumed without it, and whilst the right line SPrevolves about S as acentre, a point P moves in it m such 2 manner that its dis- tance from 8 shallal- — / s ways be to PE, its distance from the DX, ina given ratio; the curve described the point P is a conic section, and it 19 ellipse, parabola, or hyperbola, according SP is less than, equal to, or greater than] The ratio of SP to P E is called the de | nate ratio; hence it follows, that in the ely AD, the distance of the directrix from) vertex is greater than AS; the distance? the vertex to the focus, in the parabola A} equal to AS, and in the hyperbola it is le See the description of the ellipse, parab and hyperbola in plano, under the respec articles. ie DirEctTRIX, or DiriGent, is also that or plane along which another line or plar supposed to moye,.in the generation of @ face or solid, i e DIS . Thus the surface B ABCD may be sup- posed to be generat- ed by the motion of the line AB along the line AD; in which A D case AD is called the Dirigent, and AB the Describent. DISC, or Disk (from dicxos), in Astronomy, the body or face of the sun, moon, &c. as they appear to the naked eye. DISCHARGE of Fluids. The discharge of fluids through apertures in the sides and bot- toms of vessels, is a subject which has engag- ed the attention both of theorists and experi- mentalists; and it may, therefore, be consi- ered under either point of view. In the present article it will probably be best to keep these cases totaily distinct from eachother. We shall, in the first place, state the principal pro- positions in the theoretical part of the subject, nd then give a slight view of some of the most remarkable and accurate experiments, Py Bossut, Venturi, Eytelwein, Vince, Young, o. 1. If a fluid runs through a pipe or tube f an uniform shape, equal quantities of it will pass through every parallel section of the tube. . This is very evident, because the same quan- ity of fluid must pass through the same sec- tion in the same time. It must be observed, nowever, that though the water runs with the jame velocity through every section of the jube, yet it does not run with equal velocity through every part of the same section. Its a being swiftest towards the middle, Cc nd slower towards the sides of the tube, vhere it is retarded by friction. 2. If a fluid runs through a tube or pipe, cept constantly full by means of a proper ‘upply, but which is not of uniform shape; hen the velocity of the fluids, in different sec- jons, will be inversely as the areas of the ‘ections. | Since the tube is always full, the same quan- ity of water must pass through every section id it in the same time; but if the area of one ection be half as great as the area of another ection, the same quantity of water cannot ass through both sections at the same time, less it passes through the former section vith double the velocity with which it passes wough the latter. If the area of the former ection be one-third, or one-fifth, of the area the latter, the same quantity of water must through the former with a velocity which three times, or five times, the velocity with hich it passes through the latter. Hence, in eral, the velocities must be inversely as the eas of the sections. '3. If a fluid, flowing through a small ori- ze in the bottom of a vessel, be kept-con- antly full by means of an uniform supply at ip, the velocity of the effluent fluid will be ys to that acquired by a heavy body in ling freely through the height of the surface. ‘the fluid above the orifice. DIS Let MNOP represent a vessel filled with a non- elastic fluid up to the level of IK; MP, the bottom in which is the aperture CD (very small in compa- rison with MP), CIKD the column of the fluid stand- ing directly above the aper- ture, and CABD the low- ~ est plate of the fluid imme- diately contiguous to the aperture. Also let v denote the velocity which a heavy body would acquire in falling freely through BD, the height of the plate; and V the velocity acquired by the same plate during its descent through the same space until it is discharged by the pressure of the column CIKD. If we suppose the lowest plate of fluid ACBD, to fall as a heavy body through the height BD, its moving force will be its own weight. Again, suppose it to be accelerated by its own weight, together with the pressure of the am- bient fluid, about the column CLIK D; viz. by the weight of the column CIKD through the same space; that is, while it is accelerated from quiescence until it is actually discharg- ed: then (agreeably to the established laws of dynainics) the velocity in the former case will be to that in the latter, as the moving forces and the times in which they act directly, and the quantities of mattcr moved inversely. But the moving forces are to each other, as the heights BD and KD; the times in which they act are inversely as the velocities, the space through which the body is accelerated being given; and the quantities of matter moved are BD, KD equal; therefore v: V :: 3 whence Vv v2: V2:: BD: KD, orv: V:: yBD: y BD. Now v is the velocity which a heavy body would actually acquire in falling through the space BD; consequently V, the velocity of the effluent fluid, is that which a heavy body would acquire in falling through KD, the whole altitude of the fluid above the orifice. 4, In the same man- 1 Gilg ie | ner it may be shown that if a pipe be insert- ed horizontally in the vessel MN OP, the plate of fiuid ACB will be dis- charged with the same velocity as before (if its centre of pressure be of yy the same depth), what- ever be the thickness of the plate; this velo- city not depending upon a continual accele- ration through the length of the tube, other- wise the effluent fluid could not attain its full velocity, until a column had been discharged whose base is equal to the orifice, and height equal to the length of the tube; whereas we find by experience, that this full velocity can be attained by the thinnest plate which can be let escape from the aperture. & The velocities and quantities discharged v : ' " H 1 ' pod YE [ oe DIS at different depths, are as the square roots of the depths. 6. The quantity run out in any time is equal to a cylinder, or prism, whose base is the area of the orifice, and its altitudes the space de- scribed in that time by the velocity acquired by failing through the height of the fluid. So that if h denote the height of the fluid, a the area of the aperture, g — 324 1th feet, or 386 inches, and é the time of efflux, we shall have, for the quantity discharged, Q = at V 2h. Cir when @ and A are expressed in feet, Q = 80208 at Vh feet. When a and / are expressed in inches, Q = 27°7387 at Yh inch. If the orifice is a circle whose diameter is d, then 0°785398d* must be substituted for a; And when d and A are expressed in feet, Q = 6-29952d Wh feet. When d and A are expressed in inches, Q = 21°78592 d1* h inches. _And from either of these it will be easy to find either a, ¢, or A, when the other three quantities are given. 7. The force with which the effluent water impinges against any quiescent body, is pro- portional to the altitude of the fluid above the orifice. For the force is as the velocity mul- tiplied by the quantity of matter; but the quan- tity discharged, in a given time, is as the velocity ; therefore the force is as the square of the velocity; that is, by the demonstration of the proposition, (art. 3.) as the height of the fluid. 8. The water spouts out with the same ve- locity whether it be downwards, or upwards, or sideways; because the pressure of fluids is the same, in all directions, at the same depth. 9. When a vessel is left to discharge itself gradually, through an orifice in its bottom, if the area of the section parallel to the bot- tom be every where the same, the velocity of the surface of the fluid, and consequently the velocity of the efflux, will be uniformly retarded. . For (by prop. 2) the velocity of the descend- ing surface is to the velocity at the surface, as the area of the orifice to the area of the surface, which is a constant ratio; conse- quently, the velocity of the descending sur- face varies as the velocity at the orifice, or as the square root of A; that is, the velocity of the descending surface varies, as the square root of the space which it has to describe: so that this exactly corresponds with the case of a body projected perpendicularly upwards, where the velocity is as the square root of the space to be described: whence, as the retard- ing force is constant in the instance referred to, it must also be constant in the case before us, and the retardation uniform. From this comparison we deduce the following obvious corollaries. The qnantities of water in a prismatic ves- sel, discharged through an aperture in the bottom, decrease in equal times; as the series DIS q| of odd numbers 1, 3, 5, 7,9, &c. taken in an! inverted order. : The quantity of water contained in upright prismatic vessel, is half that w would be discharged in the time of the entire gradual evacuation of the vessel, if the wate, be kept always at the same altitude. £4 10. If upon the altitude of the fluid in a vessel, as a diameter, we describe a semi. circle, the horizontal space described by the fluid spouting from a vertical orifice, at point in the diameter, will be as the ordinate of the circle drawn from that point; the hori zontal space being measured on the planeo the bottom of the vessel. ow When the aperture is vertical, and indefi nitely small (as supposed here), the fluid wil! spout ont horizontally with the velocity duc to the altitude of the fluid above the orifices! (by prop. 3); and this velocity, combined wil the perpendicular velocity arising from the action of gravity, will cause every particle and consequently the whole jet, to describe the curve of a parabola. Now the velocity A » ; ‘ cerns meee, ees & = Cc i) 5) in ¥ with which the fluid is expelled from any h as G, is such as if uniformly preserved woulk carry a particle through a space equal to 2BG in the time of the fail through BG; but, afte quitting the orifice, it describes the parabolic curve, and arrives at the horizontal plane C] in the same time as a body would fall freeh through GD; so that to find the distance DE since the times are as the roots of the spaces we have this analogy “GB: YGD:: 2BG! RS AC Nc Bc sae | DE=i— =2,/BE 6D = 260 by the nature of the circle. And the same wil hold with respect to any other point in BD. | If the apertures be made at. equal dis tances from the top and bottom of the ves sel (kept full of fluid), the horizontal distance to which the water will spout from these ape tures will be equal. For when Dg = BG) we shall have2 / Be. ¢D = 2vBG.GD and consequently DE the same in both cases. When the oritice is at the point bisectin, the altitade of the fluid in the vessel, # fluid will spout to the greatest distance the horizontal plane ; and that distance, i measured on the plane of the bottom of the ve ; sel, will be equal to the depth of the fluid in if For IK, the ordinate from the centre I is th’ greatest which can be drawn in the semi circle, and DF, which is = 21K, is then = 2BI=BD. a The reader who wishes to examine th theoretical part of the present subject mor DIS it large, may consult the following works: Gravesande’s ‘‘ Nat. Phil.;” J. Bernouilli m the Motion of Water in Pipes, Act. Petro. x. x.; Euleron the Motion of Water in Pipes, (ect. Ber. 1752; also his ‘Principles of Hy- lranlies,” ibid 1755; on the Re-action of Vater in Pipes, Act. Petro. N.S. vi.; Raccolta li autori che trattano del moto delle Acque, ‘lor. 1725, which contains the investigations of Archimedes, Albici, Galileo, Castelli, Micheli- Mi Borelli, Montanari, Viviani, Cassini, Gue- ielmi, Grandi, Manfredi, and Picard; Borda im the Discharge of Fluids, Act. Petro. 1766; uagrange on the Motion of Fluids, Act. Berl. 781; Young on Spouting Fluids, Irish Trans. 1781; Prony’s “Archit. H ydraulique;” Lorgna im Spouting Fluids, Society Italy iv; Venturi ur la Communication laterale de Mouve- nent dans les Fluides, Par. 1798; Bossut’s pe rodyn. Gregory’s ‘‘ Mechanics,” b. 4, i. &e. We have already extended this article be- ‘ond what was first intended, and therefore ‘ome of the principal experiments that have een made in order to illustrate this interest- ag inquiry. _ With respect to the quantity of fluid dis- hharged, Bossut’s experiments, which were aade with peculiar accuracy, demand parti- ular notice. They are expressed in the fol- owing table, which exhibits the quantity of uid discharged through orifices pierced in hin plates; in measures of the Paris foot pyal, which is to the English foot, as 1066 to | Bossut’s “ Hydrodyn.” t.ii. p. 47. Theoretical dis- charges in one minute througha circular aperture of tin, expressed in cubic inches. Constant altitude of the water in the reservoir ahove the aperture, in Paris feet. Real discharges in the time through the same orifice, expressed also in cubic inches, —_—_— 4381 6196 7389 8763 9797 10732 11592 12592 13144 13855 14530 15180 15797 16393 16968 SaN oon | The following is another set of experiments aade by the same author with different aper- ares; in which the water was kept constantly ‘t the altitude of eleven feet, eight inches, on lines, from the centre of each aperture. I Cubie Inches furnished in -iperi One Minute. iments. 1. With an horizontal circular aper- ture, 6 lines in diameter.......... 2311 / aust be very brief in stating the results of’ DIS Cubic Inches : furnished in Experiments, "One Minute, 2. With a circular horizontal aper- ture, 1 inch in diameter........... 9281 3. With a circular horizontal aper- ture, 2 inches in diameter . 37203 4, With a rectangular horizontal aperture, 1 inch by 3 lines....... 2933 5. With a square horizontal aperture, the side 1 inch...... Ley sbethe dbus 11817 6. With asquare horizontal aperture, the sade. 2 Inches... s-ese0se A000) Constant height of water 9 feet. . Lateral circular aperture, 6 lines TRG IAINALGT A sap: ceppnnvdbdchnoee anaeid 8. Lateral circular aperture, 1 inch in diameter Constant height of water 4 feet. 9. Lateral circular aperture, 6 lines in diameter . Lateral cireular aperture, 1 inch in diameter Constant height of water 7 lines. Lateral and circular orifice, 1 inch in diameter ~J eee Pee ee eee eee eee eee eee. ee ee ee eee ee eee ee eee eee eee ee 11. 628 From these experiments we may derive the following deductions ; viz. 1. The quantities of fluid discharged, in equal times, from different sized apertures, the altitude of the fluids being the same, are to each other nearly as the areas of the aper- tures. 2. The quantities of water discharged, in equal times, by the same aperture, with diffe- rent altitudes of water in the reservoir, are nearly as the square roots of the correspond- ing altitudes of the water in the reservoir above the centre of the aperture. 3. That, in general, the quantities of water discharged, in the same time, by different apertures, and under unequal altitudes of the water in the reservoir, are to each other in a compound ratio of the areas of the apertures, and the syuare roots of the altitudes. 4, That on account of the friction, the smallest aperture discharge proportionally less water than those that are larger and of a si- milar figure, the water in the respective reser- voirs being at the same height. 5. That of several apertures whose areas are equal, that which has the smallest circum- ference will discharge more water than the other, the water in the reservoirs being at the same altitude, and this because there is less friction. Hence, circular apertures are most advantageous, as they have less rubbing sur- face under the same areas. To this we can only add, that if instead of the orifice being pierced in a plate of tin or other thin plate, a cylindrical pipe or tube be inserted in the vessel whose length is from two to four times the diameter of the orifice, then a greater quantity of water will"be dis- charged through it than through the simple aperture, in an equal portion of time, all other circumstances remaining the same; the quan- tity of the fluid discharged, in the two cases, being to each other as 133 to 100 nearly. An Sete eee HHO eH TEE eEE eH ee DIS account of several other experiments connect- ed with this subject may be seen in Gregory’s “‘ Mechanics,” vol. i. DISCHARGER, or DiscHarGinG Fod, in Electricity, is arod used for the purpose of discharging a jar or battery of its contents, without injury or pain to the operator. Dif- ferent forms have been given to this instru- ment, which are described by Cavallo and other authors on electricity. DISCOUNT, or REBATE, is an allowance made on a bill or any other debt not yet be- come due, in consideration of present pay- ment. Bankers, merchants, &c. allow for discount a sum equal to the interest of the bill for the time before it becomes due, which however is not just; for as the true value of the discount is equal to the difference between the debt and its present worth, it is equal only to the interest of that present worth, instead of the interest on the whole debt. And therefore the rule for finding the true discount is this: As the amount of £1. for the given rate and time Is to the given sum or debt, So is the interest of £1. for the given rate and time To the discount of the debt. So that if p be the principal or debt, 7 the rate of interest, and ¢ the time;;} rt c F then as 1 + ré: p:: Do eres 2 which is the true “discount. Hence also p — re A 1l+prt Boe “~ L+ rt received. Or, making a to represent the present worth, we have is the present worth, or sum to be Ase = the present value P Ll + rt p —a(1 +~rt)= the sum due cor ee = the number of years i Large at A table of discounts may be seen in Smart’s “Tables of Interest ;” the same is also given in Barly’s valuable “ Treatise on Interest and Annuities.” DISCRETE, or Dissuncr Proportion, is that in which the ratio between two or more pairs of numbers is the same, and yet the pro- portion is not continued, so as that the ratio may be the same between the consequent of one pair and the antecedent of the next pair. Thus 3: 6 :: 5: 10 is a disjunct proportion; but 3: 6:: 12: 24 is a continued proportion. DIscRETE Quantity, is such as is not con- tinued and joined together. Such, for in- stance, is any number; for its parts, being distinct units, cannot be united into one continuum ; for in this there are no actual determinate parts, as there are in a number composed of several units. a = the rate of interest. ae , DIS q DISJUNCT. See Discrete. wy DISPART, in Gunnery, is setting a mark upon the muzzle-ring, or thereabouts, of a piece of ordnance, so that a sight taken upon the top of the base ring, against the touch. hole, by the mark set on or near the muzzle may be parallel to the axis of the gun. ry DISPERSION, commonly signifies the s tering or dissipating the parts of any body; and. hence, in Optics, it denotes the same as divergency. a Point of DisPERSIoN, in Dioptries, the po from which the refracted rays begin to verge when their refraction renders them div ergent. is DISPERSION of Light, in Optics, denotes the enlargement of a pencil or beam of light which is produced by its passage from one medium to another; and this enlargemen arises from the nature of the medium. pt Dr. Wollaston’s mode of examining refrac tive and dispersive powers is described in Phil. Trans. for 1802, or Nicholson's Journal vol. iv. p. 89. Extensive tables of refractive and dispersive powers are given by Dr. T Young, at p. 296, 299, vol. ii. of his a Philosophy. DISSIPATION, in Physics, an insenaie loss or consumption of the minute parts of ¢ body, or that apeney whereby they fly off ane are lost. hi Circle of Dissipation, in Optics, denote the circular space upon the retina, which ¥ taken up by one of ,the extreme pencils ¢ rays issuing from any object. 4 Radius of Dissipation, is the radius of th circle of dissipation. DISSOLVENT, any thing which dissolves DISSOLUTION, the separation of ab into its most minute parts. DISTANCE, according to the common ceptation of the word, is space consider barely in length between any two objects 0 bodies, and is the shortest line that can D drawn between them. mh, DISTANCE, in Astronomy, as of the sun) planets, &c. is either real or proportional ; i is also farther distinguished into mean, pé helion, and aphelion distance. 4 Aphelion Distance of the Planets, is wh 4 they are at their greatest distance from M sun. Perihelion DIsTANCE of the Planets, is whe they are at their least distance from the sum : Mean Distance of the Planets, is a mé between their aphelion and perihelion dis tances. Proportional Distances of the Plenstoly ar the distances of the several planets from th sun, compared with the distance of any one ¢ them considered as uhity. &eal DisTAaNcss, are the absolute distanee of those bodies, as "compared with any terre trial measure, as miles, leagues, &c. ee The proportional distances of the olaai from the sun, any one of them (as for exampl that of the earth) being considered as unit) are readily determined from Kepler’s th DIS aw ; viz. that the squares of the periodic times »f revolving bodies about the same central ody, are as the cubes of their respective dis- -ances:.and hence also the real distance of any one of the planets being known, the ab- solute distance of all the others may be de- ‘ermined. Now the real distance of the earth yom the sun has been determined, by means of the transit of Venus, to be about 93,000,000 miles. See PARALLAX and Transit. And ence we have the following tablet of the real nd proportional distances of the several pla- acts. Proportional Real mean Dist. mean Dist. *3870981 ... 86 million miles "7233323 ... 10000000 ... 1'°5236935 ... 22373000 ... 2°667 1630 ... 2°7674060 ... 2°7675920 ... Jupiter...... 5°2027911 ... Saturn...:.: 9°5387705 . - Uranus..... 19°1833050 For the distance of the Moon, and the other econdaries from their several primaries, see iy SATELLITES. Distance of the fixed Stars from the Earth or Sun, has never yet been determined; we nly know it is so great, that the whole dia- meter of the earth’s orbit, which is near two hundred million miles, is but as a point com- pared with their distance, and therefore forms o sensible measure whereby it may be esti- mated. DisTANce of the Sun from the Moon’s Node r Apogee, is an arch of the ecliptic inter- cepted between the sun’s true place and the moon’s node or apogee. Curtate Distance. See CURTATE. Accessible DISTANCES, are such as may be measured by the application of any lineal measure. Tnaccessible DistANCES, are those which can- not be measured by the application of any tineal measure, but by means of angles and trigonometrical rules and formule ; these, however, are too numerous to be treated of in this place, we must refer the reader for infor- mation on this subject to the first part of Bonnycastle’s “Trigonometry,” where a great variety of questions of this kind are perspi- cuously illustrated. The distance of objects may also be ascer- tained by means of sound; for as this has been found by experiment to travel at the rate of about 1142 feet per second, if the time which -elapses between the firing of a gun and the ‘report of the same be duly observed, the dis- ‘tance in feet will be found by multiplying the umber of seconds by 1142; and in this way ime may estimate the distance of a thunder cloud, by the number of seconds being ob- sserved that elapses between the flash of light- ning, and the clap of thunder by which it is succeeded. | Apparent DisTANCE, in Optics, is that distance Mercury... Venus Earth....... Mars........ VW esta.,..... BiiN0. ee: Ceres ..:... Pallas....... seeee . ) DIS at which we judge an object to be placed, when seen afar off, and which is usually very different from the true distance; because we are apt to think that all very remote objects whose parts cannot well be distinguished, and which have no other object in view near them, are at the same distance from us, though perhaps one of them is thousands of miles:nearer than the other; as is the case with the sun, moon, and planets. Dr. Potersfield gives a very distinct and comprehensive view of the natural methods of judging concerning the distance of objects. The most universal, and frequently the most sure means of judging of the distance of ob- jects is, he says, the angle made by the optic axis. For our two eyes are like two different stations, by the assistance of which distances are taken; and this is the reason why those persons who are blind of one eye so frequently miss their mark in pouring liquor in a glass, snuffing a candle, and such other actions as require that the distance be exactly distin- euished. ‘To convince ourselves of the use- fulness of this method of judging of the dis- tance of objects, he directs us to suspend a ring in a thread, so that its side may be to- wards us, and the hole in it to the right and left hand; and taking a small rod, crooked at the end, retire from the ring two or three paces; and having with one hand covered one of our eyes, to endeavour with the other to pass the crooked end of the rod through the ring. This, he says, appears very easy; and yet upon trial, perhaps once in 100 times, we shall not succeed, especially if we move the rod a little quickly. Our author observes, that by persons recol- lecting the time when they began to be subject to the mistakes above mentioned, they may tell when it was that they lost the use of one of their eyes; which many persons are long igno- rant of, aud which may be a circumstance of some consequence to a surgeon. ‘The use of this second method of judging of distances, De Chales limits to 120 feet; beyond which, he says, we are not sensible of any difference in the angle of the optic axis, A third method of judging of the distance of objects consists in their apparent magni- tudes, on which so much stress was laid by Dr. Smith. From this change in the magni- tude of the image upon the retina, we easily judge of the distance of objects, as often as we are otherwise acquainted with the mag- nitude of the objects themselves ; but as often as we are ignorant of the real magnitude of bodies, we can never, from their apparent mag- nitude, form any judgment of their distance. From this we may see why we are so fre- quently deceived in our estimates of distance, by any extraordinary maguitudes of objects seen at the end of it; as, in travelling towards a large city, or a castle, ora cathedral church, or a mountain larger than ordinary, we fancy them to be nearer than we find them to be. This also is the reason why animals, and alk small objects, seen in valleys contiguous te large mountains, appear exceedingly small. DIS For we think the mountain nearer to us than if it wereesmaller; and we should be surprised at the smallness of the neighbouring animals, if we thought them farther off. For the same reason we think them exceedingly small when they are placed on the top of a mountain, or a large building, which appear nearer to us than they really are, on account of their ex- traordinary size. Dr. Jurin clearly accounts for our imagin- ing objects, when seen from a high building, to be smaller than they are, and smaller than we fancy them to be when we view them at the same distance on level ground. Itis, says he, because we have no distinct idea of dis- tance in that direction, and therefore judge of the things by their pictures upon the eye only; but custom will enable us to judge rightly even in this case. -Let a boy, says he, who has never been upon any high building, go to the top of the Monument, and look down into the street; the objects seen there, as men and horses, will appear so small as greatly to surprise him. But ten to twenty years after, if in the mean time he has used himself now and then to look down from that and other great heights, he will no longer find the same objects to ap- pear so small. And if he were to view the same objects from such heights as frequently as he sees them upon the same level with himself in the streets, he supposes that they would appear to him just of the same magni- tude from the top of the Monument, as they do from a window one story high. For this reason it is, that statutes placed upon very high buildings ought to be made of a larger size than those which are seen at a nearer distance; because all persons, except archi- tects, are apt to imagine the height of such buildings to be much less than it really is. The fourth method by which Dr. Porterfield says that we judge of the distance of objects, is the force with which their colour strikes upon our eyes. For if we be assured that two objects are of a similar and like colour, and that one appears more bright and lively than the other, we judge that the brighter object is the nearer of the two. The fifth method consists in the different appearance of the small parts of objects. When these parts appear distinct, we judge that the object is near; but when they appear confused, or when they do not appear at all, we judge that it is at a greater distance. For the image of any object, or part of an object, diminishes as the distance of it increases. The sixth and last method by which we judge of the distance of objects is, that the eye does not represent to our mind one object alone, but at the same time all those that are placed betwixt us and the principal object, whose distance we are considering; and the more this distance is divided into separate and distinct parts, the greater it appears to be. For this reason, distances upon uneven sur- faces appear less than upon a plane: for the inequalities of the surface, such as hills, and holes, and rivers, that lie low and out of sight, -lie behind them from appearing; and so th DIV either do not appear, or hinder the parts whole apparent distance is diminished by th parts that do not appear in it. This is th reason that the banks of a river appear conti guous to a distant eye, when the river is loy, and not seen. On this subject the reader may consul Smith’s “ Optics,” vol. i. p. 52; Robins’ “Tracts,” vol. ii. p. 230; Porterfield on th) Kye, vol. i. p. 105, and vol. ii. p. 387; an: Priestley’s “ History of Vision,” p. 205. Othe important reflections are made on this curiow subject by M. Bouguer; by Haris, in hi “Optics;” by Dr. Young and M. Haity, i) their respective Courses of Philosophy. Distance, in Navigation, is the number ¢ miles or leagues that a ship has sailed fror, one point to another. . Line of Distance, in Perspective, is a righ line drawn from the eye to the principal poir of the plane. rs Point of Distance, in Perspective, is the point in the horizontal line which is at th same distance from the principal point as th eye is from the same. | Distinct Base, in Optics, is the same wit, what is otherwise called the focus. DITTON (HumpuHrey), a mathematicia’ of the seventeenth century, was born ¢ Salisbury, May 29, 1675. He was author ¢ several tracts, which were published in th Phil. Trans. ‘The principal of these we on the Tangents of Curves, vol. xxiii.; Spher cal Catoptrics, in the Phil. Trans. for 170: He also published a treatise on the “ Gener: Laws and Nature of Motion,” 8vo. 1705; “A Institution of Fluxions,” &c. 8vo. 1706; tk “Synopsis Algebraica” of John Alexande in 1709; and a “Treatise of Perspective,” i 1712; beside other works on theology, i Ditton died in 1715, in the fortieth year | his age. ' DIVERGENT, tending to various par’ from one point; thus we say, diverging liné rays, &c.; meaning those lines or rays whic issuing from one common point, go off fro) that point in various directions. DiverGcinG Hyperbola and Parabola. TiyPERBOLA and PARABOLA. | DivERGING Series, in Analysis, are thos series, the terms of which increase more ar more the farther they are continued. a DIVIDEND, in Arithmetic, is that numb which is to be divided by some other numb called the divisor. DIVING, the art of descending under w ter to considerable depths, and remainit | 1 Se there for some time in order to recover thi which have been sunk; as also for the purpo of bringing up corals, pearls, sponge, &e. Divine Bell, an apparatus used for the pu pose of diving. It is most commonly mat in the form of a truncated cone, the smalle end being closed and the larger one open. is weighted with lead, and so suspended th: it may be sunk full of air, with its open ba! downwards, and as near as may be parallel’ the horizon, so as to close with the surface ; DIV the water. For a particular description of his instrument, under various forms, see tees’s Cyclopedia, article Divine Bell; see jlso Phil. Trans. abs. vols. iv. vi. and viii; nd Desagulier’s ‘ Exper. Phil.” vol. ii. Divine Bladder, an instrument invented by 3orelli for the purpose of diving, in order to »bviate some of the inconveniences attending the common diving bell. | DIVISIBILITY, that quality of a body by which it admits of separation into parts. The livisibility of quantity is a subject which has ‘iven rise to various arguments amongst phi- losophers, some contending that this separa- ion may be carried on ad infinitum, while thers maintain that it cannot be extended veyond certain limits. We shall not, in this vlace, enter upon the subject in dispute, but nerely state some of those facts which have been ascertained from experiment, and which if they do not show the absolute infinite divi- ibility of matter, they at least demonstrate, ia the most evident manner, that it may be arried on to an extent which is truly asto- ishing. Boyle mentions a silken thread 300 ards long, that weighed but two grains and | half. He also measured leaf gold, and found 4y weighing it that fifty square inches weigh- id but one grain: if the length of an inch be ivided into 200 parts, the eye may distin- ‘uish them all; therefore there are in one quare inch 40000 visible parts; and in one erain of it there are 2000000 of such parts: which visible parts no one will deny to be iarther divisible. | Again, a whole ounce of silver may be gilt vith eight grains of gold, which may be after- vards drawn into a wire 13000 feet long. » In odoriferous bodies we can still perceive greater subtilty of parts, and even such as ire actually separated from one another; se- eral bodies scarce lose any sensible part ‘f their weight in a long time, and yet conti- jmally fill a very large space with odoriferous articles. veal particles, furnish another surprising in- tance of the minuteness of some parts of matter. A lighted candle placed on a plane vill be visible two miles, and consequently 41a sphere, whose diameter is four miles, ith Juminous particles, before it has lost any nsibite part of its weight. And as the force ff any bedy is directly in proportion to its mantity of matter multiplied by its velocity, nd since tlie velocity of the particles of light } demonstrated to be at least a million of jmes greater than the velocity of a cannon all, it is plain. that ifa million of these par- oles were round and as big as a small grain f sand, we dt!'st.no more open our eyes to he light, than t4 expose to sand shot point jlank from a cannon, ' By help of microscopes, such objects as vould otherwise esCapne our sight appear very jarge : there are soMe small animals scarcely visible with the best microscopes ; and yet hese have all the parts necessary for life, as . . The particles of light, if light consists of ’ DIV blood and other liquors. How wonderful, therefore, must be the parts which constitute these fluids! DIVISION, is one of the principal rules in Arithmetic and Algebra, it consists in finding how often a less number is contained in a greater. The number to be divided is called the dividend, the number by which the divi- sion is made is the divisor; the number of times that this is contained in the former, is called the quotient, and if any thing remains, after the operation is finished, it is called the remainder. Division is either simple or com- pound. Simple Division, is when both the divisor and dividend are integral numbers. Rule. Draw a small curve-line on the right and left of the dividend, and write the divisor on the left; then find how many times the divisor is contained, in as many of the left- hand figures of the dividend as are just neces- sary, and place that number on the right. Multiply the divisor by this number, and place the product under the figure of the dividend above mentioned. Subtract this product from that part of the dividend under which it stands, and bring down the next figure of the divi- dend, or more if necessary, to the right of the remainder. Divide this number, so increased, as before, and so on till the whole is finished. Note 1. When it is necessary to bring down more than one figure to the remainder, a cipher must be placed in the quotient for every figure thus brought down. Note 2. If the divisor do not exceed 12, the quotient may be written down as it arises, immediately under the dividend. Proof of Division. Multiply the divisor and quotient together, and add to this product the remainder, which ought to be equal to the dividend if the work be right. EXAMPLE. 5)674346(1 7)643287(1 134869 quotients 91891 5 7 674346 proof 643287 See meee 346 ) 7486716 ( 21637 - 692 346 "566 129822 346 86548 ‘ 5007 «64911 2076 314 1311 7486716 proof. 1GSSes ee CL Sometimes, for the sake of abridging the operation, the successive products are omitted, and the subtraction is made figure for figure as the work is carried on; by this method, the foregoing example would stand as follows. Division may also be proved by the cross, the same as multiplication; by casting out the DIV nines from the divisor and quotient; and again out of the product of their remainders; and this last remainder ought to be the same as that arising from the dividend, after the re- mainder, arising in the operation, is subtracted from it; thus, 346)7486716(21637 566 2207 4 i311 proot} <4 2736 A: 314 This is generally called the Italian method of division. AY) Compound Division, is when the dividend is a compound quantity. Vf 2) Rule. Divide the highest denomination of the dividend by the divisor, as in the former rule. Reduce the remainder, if any, to the next inferior denomination, and divide as be- fore ; reduce this remainder again, and divide as before, and so on till the whole is finished. Note. If the divisor exceed 12, and be a composite number, divide by its factors suc- cessively, instead of the whole number at once. EXAMPLES. £. 8... ae Bf 3)4 13 9( 7)9 13 11( 111 3 quotients 1 7 93 3 7 413 9 _ proof 9 13 11 3. Divide £214. 14s. 10d. by 24. £.: 8. d 4)214 14 10( 6) 53 13 84( 8 18 lit 4 24—-4x6 Division of Fractions, is performed by the following rule. Reduce all mixed numbers to improper fractions, then invest the terms of the divisor, and multiply the numerators and the denominators together as in multiplication ; observing that such factors, as are common in the numerators and denominators, may be cancelled. EXAMPLE l. 1 (lees bast 4 _ 28 Meenas +O | Brit QF Bin Me ot 9_ 27 Panes Sad 4 Ad bake 1408 7 7 Se Segre Bg sg 1 ) tocar A is a O53 § cs 27 Division of Decimals, is performed the same as in the simple rule of division, observing only to point off in the quotient as many decimal places, as those in the dividend exceed those in the divisor; and if there be not so many the defect must be supplied by prefixing ci- phers. Another way to know the place for the deci- mal point is this; the first figure of the quo- tient must be made to occupy the same place EV either in integers or decimals, as does thai figure of the dividend that stands over thre unit’s place of the first product. Note. 'The division may be carried on to any extent required by annexing ciphers to the remainders after all the given figures have been used. EXAMPLE, 486'4)784'6840(1°615 quotient. 298 28 7444 2.5800 1480 Division of Cireulating Decimals, is per- formed by converting the repetends into their equivalent fractions, and ther proceeding as in division of fractions. See CircUuLATING Decimals. ** Contracted Division. See CONTRACTION. Division, in Algebra, is the method of find- ing the quotient arising from the division of. one indeterminate quantity by another, which may be considered under two Cases. Case 1. When the divisor and dividend are both simple quantities. . Rule. Divide the co-efficients, as in arith- metic, and to the quotient annex the result arising from the division of the indeterminate quantities. Note. When the divisor and dividend have like signs, the sign of the quotient is plus +; and when they are unlike, the sign of the quo- tient is minus —, as in multiplication. EXAMPLES. 6 ab)24a* b* 7x*)35x3 y3 6x? /y)72x3y 4a b 5x y3 l2xvy — 4ab*)32abc* — 4xy)— 1627 ¥3 —se + 4x y* 4.a7 b) —24a3 b3 c* — 6a b*c* y Case 2. When the dividend is a compound quantity, and the divisor either simple or com pound. Rule. Set the divisor on the b dend, and proceed ¢ operation the ssality as in division of numbers, observing stall ti same rule as above with recard to the signisy’ ~ . . | ft of the divul Amr LES a—b)a?—2ab + b4a— a*— ab — ab+ } f aarp seaman oe nce x + y)x? + y' ay + y* | x? 4 ay fe - iH ¢ y° | yey a xy? erent nana 3 af or ag +y + xy + y3 ee ed = DIV ae — 2Qy)x*— 16 y4(23 + Quy + Ary”? +8y3 at+— Qarsy + 2a%y —1 6 y* + 2a3y— 4277” + 4x7y* — 16 y+ + 47y*? — 8xy3 + 8xry?—16y* + 8ay3— 16y* Note. When there is a remainder after the division, it must be written over the divisor, and annexed as a fraction to the quotient. Thus for example, if the foregoing dividend had been a+ + 16y*, instead of x+— 16y4, then the remainder would have been 3274, and the answer would in that case have been, eer 16y* 2 2 34, 32y* ey =r? 4+ 2x*y + 4axy” + 8y erat — Division of Algebraic Fractions, is per- formed the same as in the case of simple frac- tions; viz. reduced all mixed expressions to improper fractions, then invert the terms of the divisor, and multiply the numerators and denominators as in the rule above quoted, observing to cancel all factors that are common to the numerators and denominators. EXAMPLES. 1, 5a, Ba _ Sa Set 2502 "46 * 5c* ~ 4b ~ 3a* — 12ab Sry | 9wz _ 3xy | 7x3? _ 7x7yt S272 ° Fa3yi— 52*z”~ Qwz — bzw at—yt ety? xt— yt r xr a ee hee t+ xy - ae+y) «x+y aha be ed — —x—y. zr+y Division of Surds, is the method of ascer- aining the quotient arising from the division of ‘ne irrational quantity by another. Rule. Reduce the given surds to their sim- lest form, and the radical parts thus arising 0 like radicals; then divide the co-efficient ‘ifthe dividend by the co-efficient of the di- visor for the new co-efficient, and one surd art by the other for the required surd, which jeing annexed with its proper radical sign to he co-efficient before found, will be the an- Wer required. | Note. If the radical signs be not the same, ut the quantities under them be equal, the ivision will be effected by subtracting the adex representing the radical of the divisor vom the index of that representing the radical f the dividend. } EXAMPLES. Rive ks fi 105. ey 736 2! aie et i" V7 qo eee | Vo4aty 32 /6y BY _ 2. ——__-L = ‘a vy — Me l6x — Syae tt Vg TY otY | 3, ¥108 874 _%4_%16_ | . eT LT 8 oP) | J/18 Pf BOB RS v ™m n i—n DIV Note. When the proposed divisor is a bino- mial surd, consisting of the sum or difference of two square roots, it may be rendered ya- tional by multiplying both numerator and de- nominator, or which is the same both divisor and dividend, by the same two quantities, but connected with a contrary sign to that by which they are connected in the denominator; that is by + when that is—; and by —, when that is + ; because the product of the sum of two quantities multiplied by their difference, is equal to the difference of their squares. Thus for example, required the quotient arising from the division of “5 + /7 by ¥13 + /10. By the rule, V5+V7 _ V54+V7 | vI8—V10 /13 + /10 4/13 + 410° Y18— 10 — V65+ /91—5 v2— 70 3 : DUCTION of Surds. Division by Logarithms. See LOGARITHMS. Division of Ratios, or Divided Ratio,is when of four proportional quantities, the differences of the antecedents and consequents are com-. pared either with the antecedents or conse- quents ; thus if GMb Od a—b:a::ce—dic then eal >b::ce—d:d Division of Lands and Surfaces. See Gro- DASIA. DIVISOR, is that number or quantity, which exactly divides another number or quan- tity, without leaving a remainder. Divisor also signifies that number by which another number is to be divided, without regard to See Re- ‘what may remain after division. Common Divison, in Arithmetic, is that number which will exactly divide two, or more, given numbers; and the greatest of all such di- visors is called the greatest common divisor, or the greatest common measure. The following theorems are frequently use- ful in finding the divisors of numbers. 1. If the last digit of any number be divi- sible by 2, the whole number is divisible by 2. Ifthe two last digits be divisible by 4, the whole number is divisible by 4. If the three last digits be divisible by 8, the whole number is divi- sible by 8. And, generally, if the last n digits of any number be divisible by 2", the whole number is divisible by 2”. 2. If the sum of the digits of any number be divisible by 3 or by 9, the whole number is divisible by 3 or 9; and if also the last digit be even the whole number is divisible by 18. 3. If a number terminate with 5, it is divi- sible by 5; and if it terminate in 0, it is divi- sible by either 10 or 5. 4. If the sums of the alternate digits be equal, or if one sum exceed the, other by 11, or by any multiple of 11, the whole number is divisible by 11. See vol. i. p. 24, Euler’s “ Al- gebra,” 2d English edition. By means of these theorems we are enabled frequently to ascertain the divisors of numbers, but when the greatest common divisor of tw@ DIV numbers is required we must proceed by the following rule. To find the greatest common Divisors of two given Numbers. Rule. Divide the greater number by the less, then divide the divisor by the remainder ; and thus continue always dividing the last divisor by the last remainder till nothing re- mains, and the last divisor will be that re- quired. Note. If the greatest common divisor of three or more numbers be required; find the common divisor of two of them first, then of this gommon divisor and another of the given num- bers, and so on to the last; so shall the last di-’ visor be the greatest common divisor required. EXAMPLES. Required the greatest common diviser of 7631 and 26415. 7631)26415(3 22893 3022)7631(2 7044 greatest com. diy. = 587)3522(6 3522 0 which number will divide both the given num- bers, for 34 26415 _,, aay = 13; and aay 45. Common Divisor, in Algebra, is any alge- braical formula or expression, that will exactly divide two or more other algé@braical formule, without leaving a remainder; and the greatest of such divisors is called the greatest common divisor. To find the greatest common Divisor of two given Algebraical Expressions. Rule. Arrange the terms according to the dimension of one of the letters, and then di- vide one of the proposed quantity by the other; then the last divisor by the last re- mainder, and so on, the same as in finding the greatest common divisors of two numbers, and . the last divisor will be that required. Note. All the letters or figures, which are common to each term of the divisor, nvust be thrown out of them before they are used in the operation. EXAMPLES. 1. Find the gréatest common divisor of x23 — b?a, and x7 + 2bx + b. Here x being common to both terms of the quantity 23 —6’x, it may be thrown out; when it becomes x” — b*; then by the rule x* — b*)a* + 262 +0701 a — b* 2bx + 2b’ Again, cancelling 26 in this remainder, we have x + b)x*— b*(x—b x* + bx, —ba— b* —ba— hb? DIV therefore x + bis the greatest common div! sor of the two expressions proposed. 2. Again, required the greatest commo divisor of «+ — b+, and 25 + 6? x3, Here by cancelling x? in the latter quax tity, we have | x* + b*)\at+— bt (2* — D* at + 5? 2? bz? — bt — b*a:* — b+ and since the divisor obtains, without leavin a remainder, we have x” + 6* for the greate; common divisor required. Note. If it be required to find the greate: common divisor of three or more quantitie we must first find the greatest common divis¢ of two of them, then of this common divisc and a third quantity, and so on to the last, an the last divisor will be that required. ! Divisors, Theory of, relates to the invest gation of certain properties of divisors, wit regard to their forms, powers, sums, &c¢ which forms one of the most important an interesting researches in the theory of nun bers. This branch of mathematical invest gation has, within these few years, engage the attention of several eminent analyst who have brought it to a considerable degre of perfection, and deduced many very curiot results. The first germ of this theory is fou in Fermat’s edition of “ Diophantus,” insert in marginal notes, several of which propertie however, were barely mentioned without ar demonstration; and thus they remained #7 Euler undertook their demonsiration ; he kb wise considerably extended them by theorel of his own; the same was also done by BF grange ; but their papers on this subject bem scnt to the Memoires of Petersburg and Bem were not very commonly read ; but lately L’ gendre published a work professedly on theory of numbers, which contains many i teresting theorems on the subject of divisor and M. Gauss, about the same time (180 published his “* Arithmetical Disquisitions which also contain many curious remarks rf lative to this theory. See also Barlow’s “ El mentary Investigations,” &c. The limits of the present work will not a mit of any minute researches on this subjec we shall therefore only state a few of principal theorems, and refer the reader f their demonstrations to the works above tioned. at Let N represent any number, and a, 6, &c. its prime factors; then may N be expre: ed by N=a™b" ec, &c.; thus. 1728 = 2°83 8640 = 2°. 33.5, &e. a THEOREMS. “& 1. Now any number-N, being redncegs the. above form, or N = a” &” e&, &c., then W the number of all the divisors of N be exp ed by the formula, (m +1) (n +1) (p + 1), &e. Suppose, for example, it be required to fi how many divisors belong to the number 361 Virst, 860 = 23.3%. 5': therefore, m= DIV n=2,p—1; whence(3 +N 2 +1) + 1) —4 x3 x 2= 24, the number of divisors re- quired, | Again, let it be required to find how many numbers there are by which 1000 is divisible. Here we have 1000 = 23.53, or m = 3, n=3, therefore, (8+1)(8+1)=4x4=—16; there are, therefore, 16 numbers by which 1000 is divisible. 2. 'To find a number that shall have any given number of divisors. Let w represent the required number of divisors ; resolve w into any factors, as w= xy xz, &c. Take m=x—l, n=y—l, pa=z-—1, &e. so shall a” b»c?, &c. be the number sought, a, b, and c being taken any prime numbers whatever, providing they be not equal to each other. ¢ Exam. 1. Find a number that has 30 di- Visors. | Here we have 80 =2 x3 x 5; whence m 1, n=2, p = 4, and consequently a.b’.c*, is the required form. Ifnow a= 2, b=3, c=5; then 2.3*.5*=11250 a= 5,65, ¢ = 2; then 5.3* .2*= 720 {,@2-5,6-22e 3; then 532*.3+—=1620 ‘Each of which numbers has 30 divisors, and it is obvious that various others might be ob- tained that would have the same property. 3. If Na” b" ce”, &c. as before, then will the sum of all the divisors of N be expressed by the formula amti_y br+i__] scone ge aA tiie a tabicnnmistenn Nop seer eet a—l b—1 - ec—l For example, if the sum of all the divisors of 360 was required. Since 360 = 23,37.5, we have A Ripa Fellas pled £ Silay GA _ a Sonate —15x13x6—=1170 OR aR enmae ke Spas teeter vhich is the sum of all the divisors of 360, tself being included as one of them. vach other, then every divisor of the formula 24 u*, is itself also of the same form ; thatis, t number that is the sum of two squares, rime to each other, can only have for divisors iumbers that are also the sums of two squares. _ Thus for example, 65 = 8* + 1*, can only ve divided by 13 and’5, both of which are the ums of two squares; for 13 = 37 +2’, and y=2* +17. Also 50 —7* + 1’, can only ave for divisors 2,5, 10, 25; now 2 = 1? + 1? p= 2? + 17, 10 — 3? + 1’, and 26 = 4 4+ 37; md it isthe same for all other numbers in- ‘luded in the formula ¢? + u*, ¢ and wu being wime to each other, without which latter con- lition there can be no limitation given to the orms of the divisors of this formula or any ther. 5. Every divisor ofthe formula ¢* + 2w’, and u being prime.to each other, is of the ame form. Or, the divisors of the sum of a Ze and double a square, are also the sum fasquare and double a square. | For example, 99 is a number of this form, or 99 = 1 +4 2.77; the divisors of which are | 4. If t and u be any two numbers prime to. DIU 3, 9, 11, and 33. Now 8 —12+4+2.17; 9— 17 + 2.27; 113? + 2.17, and 33 = 17 + 2.4%, And the same is true of every number contained in the formula ¢* + 2u?, providing ¢ and « be prime to each other. 6. Every divisor of the formula t? —2y?, is also of the same form. Or the divisors of the difference between a square and double a square, is also the difference of a square and double a square. For example, 98 is of this form, or 98 = 10*— 2.17%, and its divisors are 2, 7,14, and 49, Now 2 = 2*— 2.17; 7=3*—2.17, 14=4? — 2,1*, and 49 = 9* — 2.4”. 7. Every odd divisor of the formula ¢ + 3u”, is-also of the same form. Thus for example, 133 = 5? +- 3.67, and its divisors are 7 and 19. Now 7 = 2? +. 3.17%, and 19 = 4* +. 3.17. 8. Kvery odd divisor of the formula ¢* — 5u? is itself also of the same form. For example, 95 = 10* —5. 1’, its divisors being 5 and 19. Now 5=5*—5.2* and 19 = 7-—5.2*. ‘The above properties of divisors, which are highly curious, are due to Lagrange, being first given by him in the Memoirs of Berlin, with various others, which have been since considerably extended by Legendre ; the latter author having given, in his “ Essai sur la Theorie des Nombres,” a very comprehensive table, containing the quadratic and lineal forms of divisors belonging to a variety of different algebraical formule, which are very useful in ascertaining whether a given number be prime or not, of which application there are several examples in part iii. of the above work. For other properties of divisors, see Barlow’s “Theory of Numbers,” and the article PoWERs. Divisors, Method of, is a method given by Sir I. Newton for discovering the roots of an equation, the principles of which may be briefly stated as follows: Since the absolute term of every equation is equal to the product of all the roots; and farther, since an equation having integral co- efficients, must have its roots either integral, irrational, or impossible ; or, which is the same, since such an equation cannot have a frac- tional root, it follows that if it have any ra- tional root, it must be found amongst the factors of this absolute quantity, having either a po- sitive or negative sign prefixed. In order, therefore, to find a root, we have only to try for the unknown quantity, the different factors of this number, and if by such trial none of them obtain, then we may conclude with cer- tainty that the equation has no rational root. Newton, indeed, gives a method of shortening or abridging the trials, but upon the whole it is of little or no advantage, we shall there- fore not enter into an explanation of it in this place, but refer the reader to Newton’s “ Uni- versal Arithmetic,” p. 206. See also Simpson’s “ Algebra,” section 12. DIURNAL, in Astronomy, any thing relat- ing to the day, in opposition to nocturnal, relating to the night. R DOD DiurNAL Are, is the apparent are described by the heavenly bodies in consequence of the rotation of the earth. DiuRNAL Motion of a Planet, is the number of degrees, minutes, &e. which a planet moves in 24 hours, Diurnat Motion of the Earth, is its rotation round its axis, the duration of which consti- tutes the natural day. This motion has ge- nerally been considered as uniform, though there have been some astronomers who have suspected a trifling irregularity; and the late investigations of the French astronomers tends rather to confirm the latter hypothesis. DODECAGON (from dudexe, twelve, and youe, angle), a regular polygon of 12 equal sides and angles. To inscribe a Dodecagon in a Circle. round the circumference, which will divide it into six equal parts; then bisect each of those parts, which will divide the circumference into 12 parts for the dodecagon required. NG ~~ Lo find the Area of a Dodecagon. i . ageIf the side of a dodecagon be 1, its area is equalto 8 x tan. 75°=3 (2+ /3)=11:1961524 nearly: and as the areas of polygons are to each other as the square of their like sides, we have as* © . 17; 11°1961524 :: s* : s* x 11°1961524 = the area of a dodecagon whose side is s. Lo find the Side of a Dodecagon, the radius of ats circumscribed circle being given. Divide the radius in extreme and mean ratio, or so that the less part may be to the greater, as this last is to the whole line or radius; then will this greater part be the side required. Therefore when the radius is 1, putting x for the side required; we have x* = 1 — 32, or 2* + a — 1; whence x =——__ *~ aS the sides of similar polygons are as the dia- meters, or radii, of their circumscribing cir- rJ/ia—r ; and as cles, therefore — the side of a dode- cagon, the radius of whose circumscribing circle is r, DODECAHEDRON (from Jdodexe, twelve, and idpa, seat), in Geometry, one of the regular Platonic bodies, comprehended under 12 equal sides or faces, each of which is a regular pen- tagon. Or, a dodecahedron may be conceived to consist of twelve equal pentagonal pyramids, whose vertices or tops all meet in one com- mon point, which will be the centre of the sphere circumscribing the dodecahedron. Lo find the Surface and Solidity of a Dodeca- hedron, the side of one of its equal faces being given, Let s represent the given side, then, will Surlace=15 v (1+24/ 5) s*=20°64577885257 Solidity=5 // (“A237 603118965" « Apply the radius of the circle six times; “gr DOL The Radius of the Sphere circumscribing a Dodecahedron being given, to find its side or lineal edge, surface, and solidity. Let R represent the given radius, then will Side =R (4a) Surface — 10 R*v(2—2¥ 5) OO 20 3 38+yY5 lidity,= — ( Soli ity 3 ‘R “75 30 The Radius of the inscribed Sphere being given==. r, we have again for the side, surface, and so- lidity, : Side —=ry(50—22¥5) Surface = 3 or 7 (1380 — 58y 5) ; Solidity = 1 or (180 — 58 / 5) The Side of a Dodecahedron being given, to find athe Radii of the circumscribing and inseribed “® Spheres. Here putting s for the side; we have Rad. cireum. sphere = (rvs) $ J (250 + 110¥ 5) ‘ 20 . The side of a dodecahedron, inscribed in @ sphere, is equal to the greater part of the side of a cube inscribed in the same sphere, when cut in extreme and mean proportion. Ifa line be cut in extreme and mean ratio, and the less part of it be taken for the side of a dodecahedron, the greater part will be the side of a cube inscribed in the same sphere. — The side of the cube is equal to the right line that subtends the angle of the pentagon, which forms one of the equal sides of the do- decahedron, inscribed in the same sphere, See the demonstration of the above, and other curious properties of this solid, in Hutton’s “ Mensuration.” See also the article Re- GULAR Body. i" DODECATEMORY (from dwdexa, twelve, and jsep0s, part), the twelfth part of a circle. — The term is chiefly applied to the twelve houses, or parts, of the zodiac, of the primum mobile, to distinguish them from the twelve signs; though some authors use the same term when speaking of the 12 signs of the zodiac, because they each contain a twelfth part of the whole circle. y DOG, in Astronomy. See Canis. e DOLLAR, a silver coinage of Spain and of the United States, the former being worth 4s. 54d. of the coinage before 1772; and 4s. 44d, since that date; which latter is also about the value of the American dollar. Y DOLLOND (Jonn), a celebrated optician, was born of French parents in Spitalfields, London, June 1706. His attention was very early directed to philosophical subjects, and particularly to the theory and practical part of optics; and in 1753 he presented a paper to the Royal Society on this subject, which was published in vol. xlviii. of the Phil. Trans. ; another paper of Mr. Dollond’s, de- scribing his improvement of Savery’s micro- meter, is given in the same volume at p. 178, SSE Rad. inscri. sphere = DOM and a third paper at p. 278. In 1757 he began a regular course of experiments, which led to the construction of his celebrated re- fracting telescopes; an account of which ex- periments is published in Phil. 'Prans. vol. 1. and for which communication he was pre- sented with Sy’Godfrey Copley’s medal. In the beginning of the year 1761, Mr. Dol- lond was elected a fellow of the Royal Society, and optician to his majesty, but did not live long to enjoy those honours; for on the 30th of November of the same year, he was seized with a fit of apoplexy which rendered him speechless, and occasioned his death a few hours after. DOLPHIN, in Astronomy. See DELPHINUS. DOME, in Architecture, is a roof, or vault, rising from a circular, elliptic, or polygonal base, or plan; with a convexity outwards, or a concavity inwards, in such a manner, that all the horizontal sections made by planes will be similar figures round a vertical axis. Domes are denominated by the figures of the basis on which they are erected; and are therefore called polygonal, circular, or elliptic domes. Circular domes are of several kinds, as spherical, spheroidal, or ellipsoidal, hyperbo- loidal, paraboloidal, &c. Domes that rise higher than the radius of the base are called surmounted domes; and those which rise less than this dimension, are termed diminished or surbased domes. Domes that rise from cir- cular basis are called also cupolas. DOMINICAL Letter, in Chronology, pro- perly called Sunday letter, one of the seven letters of the alphabet ABC DEF G, used in almanacs, ephemerises, &c. to denote the Sundays throughout the year. In our almanacs, the first seven letters of the alphabet are commonly placed to show on what days of the week the days of the months fall throughout the year. And -because one of those seven letters must.necessarily stand against Sunday, it is printed ina capital form, and called the dominical letter; the other six being inserted in different characters, to de- note the other six days of the week. Now, _ since a common Julian year contains 365 days, if this number be divided by 7 (the num- ber of days in a week) there will remain one day. If there had been no remainder, it is ob- vious the year would constantly begin on the same day of the week: but since one remains, it is plain that the year must begin and end on the same day of the week; and therefore the next year will begin on the day following. Hence, when January begins on Sunday, A is the dominical or Sunday letter for that year: then, because the next year begins on Mon- day, the Sunday will fall on the seventh day, to which is annexed the seventh letter G, avhich therefore will be the dominical letter for all that year: and as the third year will degin on Tuesday, the Sunday will fall on the sixth day; therefore F will be the Sunday letter for that year. Whence it is evident, that the Sunday letters will go annually in retrograde order thus, G, I’, E, D, C, B, A. DOU And, in the course of seven years, if they were all common ones, the same days of the week and dominiecal letters would return to the same days of the mouths. But because there are 366 days in a leap-year, if the num- ber be divided by 7, there will remain ‘two days over and above the 52 weeks of which the year consists. And, therefore, if the leap- year begins on Sunday, it will end on Mon- day; and as the year will begin on Tuesday, the first Sunday whereof must fall on the 6th of January, to which is annexed the letter F, and not G, as in common years. By this means, the leap-year returning every fourth year, the order of the dominical letters is in- terrupted; and the series cannot return to its first state till after four times seven, or 28 years; and then the same days of the months return in order to the same days of the week as before. The dominical letter may be found uniyer- sally, for any year of any century, thus: Divide the centuries by 4; and take twice what remains from 6; then add the remainder to the odd years above the even centuries, and. their 4th. Divide their sum by 7, and the remainder taken from 7 will leave the number answering to the letter required. Thus, for the year 1878 the letter is I’. For the centuries 18 divided by 4, leave 2; the double of which taken from 6 leaves 2 again; to which add the odd years 78, and their 4th part 19, the sum 99 divided by 7 leaves 1; which taken from 7, leaves 6, an- swering to F', the 6th letter in the alphabet. DONN (BENJAMIN), an English mathema- tician, born at Biddeford, in Pevonshire, in 1729. He kept a school in that town for some years, and while there made a complete survey of the county, for which he received a premium of £100. from the Society for pro- moting Arts and Commerce. He also pub - lished his ‘“‘ Mathematical Essays,” in octavo, which had a favourable reception, and pro- cured him the office of keeper of the library at Bristol; where he also kept a flourishing academy for some years. In 1771 he printed an “ Epitome of Natural and Experimental Philosophy,’ 12mo. ; and in 1774, a work en- titled “The British Mariner’s Assistant,” being a collection of tables for nautical pur- poses. In 1796 he was appointed master of mechanics to the king.. He died in 1798, leaving behind him the character of an in- genious and worthy man. Besides the hoals above mentioned, he wrote ‘ ‘Treatises on Geometry, Book-keeping, and Trigonome- ”? try. “DOPPIA, or Pistole, a coinage of differes.i countries, and of different values. £. 8. d. 1s 7 1 3 92 coinage bef.17 1 27 since 1786 of Rome.......+ mz (AB nbs DORADO. See XIPHIAS. DOUBLE Horizontal, Dial, a dial having a double gnomon. ! R2 Doppia of Milan= of Piedmont...= ; DUC ’ Dovuste Ounce, a coinage of Naples, value £1. 1s. 3d. sterling. Dous.e Point. See Punctum Duplex. DRACHM, or Dram, a small weight, the 8th part of an onnce in apothecaries weight, and the 16th part of an ounce avoirdupoise. DRACO, the Dragon, a northern constel- Jation. See CONSTELLATION. DRAGON. See Draco. Dracon’s Head and Tail, are the nodes of the planets, but more particularly of the moon, being the points in which the ecliptic is in- tersected by her orbit, in an angle of about 5°18’. One of these points is to the northward, the moon beginning then to have north lati- tude; and the other southward, where she commences south latitude; the former point being represented by the knot Q for the head, and the other by the same reversed, or %& for the tail. And near these points it is that all eclipses of the sun and moon happen. See Moon. DRAM. See Dracum. DRIFT, in Navigation, denotes the angle which the line of the ship’s motion makes with the nearest meridian, when she drives with her side to the wind and waves, and is not governed by the power of the helm; and also the distance which the ship drives on that line, so called only in a storm. DUCTILITY, the extensibility and cohe- sion of particles which enables a metal to be drawn out into wire without breaking. There is but a slight shade of difference between this property and that of malleability. The great ductility of some bodies, espe- cially gold, is very surprising: the gold-beaters and wire-drawers furnish us with abundant proofs of this property; they every day re- duce gold into lamelle inconceivably thin, yet without the least aperture or pore dis- coverable, even by the microscope: a single grain of gold may be stretched under the hammer, into a leaf that will cover a house, and yet the leaf remain so compact as not to transmit the rays of light, nor even admit spirit of wine to transude. Dr. Halley took the following method to compute the ductility of gold: he learned from the wire-drawers, that an ounce of gold is sufficient to gild, that is, to cover or coat a silver cylinder of forty- eight ounces weight, which cylinder may be drawn out into a wire so very fine, that two yards shall weigh only one grain; and conse- quently ninety-eight yards of the same wire, only forty-nine grains: so that a single grain of gold here gilds ninety-eight yards; and, of course, the ten-thousandth part of a grain is here above one-third of an inch long. And since the third part of an inch is yet capable of being divided into ten less parts visible to the naked eye, it is evident that the hundred- thousandth part of a grain of gold may be seen without the assistance of a microscope. Proceeding in his calculation, he found, at length, that a cube ofgold,whose side is the hun- dredth part of an inch, contains 2,433,000,000 visible parts; and yet, though the gold with DUC which such wire is coated is stretched to suck a degree, so intimately do its parts cohere, that there is not any appearance of the colour of the silver underneath. Mr. Boyle, examining some leaf-gold, found that a grain and a quarter in weight took up an area of fifty square inches; supposing therefore the leaf divided by parallel lines 100th part of an inch apart, a grain of gold will be divided into five hundred thousand minute squares, all discernible by a good eye: the same author shows, that an ounce of gold drawn out in wire would reach 155 miles and a half. But M. Reaumur has carried the ductility of gold to a still greater extent. What is called gold-wire, every body knows, is only a silver one gilt. The cylinder of silver, covered with leaf-gold, they draw through the hole of an iron, and the gilding still keeps pace with the wire, stretch it to what length they can. Now-M. Reaumur shows, that in the com- mon way of drawing gold-wire, a cylinder of silver twenty-two inches long, and fifteen lines in diameter, is stretched to 1,163,520 feet, or is 634,692 lines longer than before, which amounts to about ninety-seven leagues. To wind this thread on silk for use, they first flatten it, in doing which it stretches at least one-seventh farther, so that the twenty-two inches are now 111 leagues; but in the flat- tening, instead of one-seventh, they could stretch it one-fourth, which would bring it to 120 leagues. This appears a prodigious ex- tension, and yet it is nothing to what this gentleman has proved gold to be capable of. Ductitity of Glass. We all know that, when well penetrated with the heat of the fire, the workmen can figure and manage glass like soft wax; but what is more remarkable, it may be drawn, or spun out, into threads exceedingly long and fine. Our ordinary spinners do not form their threads of silk, flax, or the like, with half the ease and expedition the glass-spinners do threads of this brittle matter. We have some of them used in plumeés for children’s heads, and divers other works, much finer than any — hair, and which bend and wave like hair with every wind. Nothing is more simple and easy than the method of making them. There are two work- men employed; the first holds one end of a piece of glass over the flame of a lamp; and, when the heat has softened it, a second ope- rator applies a glass hook to the metal thus in fusion; and, withdrawing the hook again, it brings with it a thread of glass, which still ad-. heres to the mass: then, fitting his hook on the circumference of a wheel about two feet and a half in diameter, he turns the wheel as ‘fast as he pleases; which, drawing out the thread, winds it on its rim, till, after a certain number of revolutions, it is covered with a skein of glass-thread. _ ‘The mass in fusion over the lamp diminishes insensibly, being wound out like a clue of silk upon the wheel; and the parts, as they DUO recede from the flame, cooling, become more coherent to those next to them, and this by degrees: the parts nearest the fire are always the least coherent, and, of consequence, must give way to the effort the rest make to draw them towards the wheel. The circumference of these threads is usually a flat oval, being three or four times as broad as thick; some of them seem scarcely bigger than the thread ofa silk-worm, and are sur- prisingly flexible. If the two ends of such threads are knotted together, they may be drawn and bent, till the aperture, or space in the middle of the knot, does not exeeed one- fourth of a line, or one-forty-eighth of an inch, in diameter. Hence M. Reaumur advances, that the flexibility of glass increases in proportion to the fineness of the threads; and that, pro- bably, had we but the art of drawing threads as fine as a spider’s web, we might weave stuffs and cloths of them for wear. Accord- ingly, he made some experiments this way; and found that be could make threads fine enough, viz. as fine, in his judgment, as spi- der’s thread, but he could never make them long enough to do any thing with them. DUCAT, the name of a coin common in many European nations, but of different value in different places, viz. s.d. Hungary .... = 9 5 sterling. Switzerland = 9 5 Denmark.... = 7 6 Germany ... = 9 42 Holland...... —=9 45 Sweden...... ='9 4 Naplésiait' 3°77 Venice ...2.. —3 4t DUCATOON, a coinage in Holland and Flanders; in the former country its value is . 5s. 6d., and in the latter 5s. 24d. DUODECIMALS, or Cross Multiplication | (from duodiem, twelve), is a rule used by work- men and artificers in computing the content _ of their work; dimensions are usually taken in feet, inches, and parts; but of the last, all those less than 1 of inches are frequently omitted as of little or mo consequence, and the same is always done in casting up the contents, where they are of still less import- ance. Rule. Set down the two dimensions tobe multiplied together under each other, so that feet may stand under feet; inches under inches; and parts under parts. Then mul- tiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each directly under its corresponding term, observing to carry 1, for every 12, from the parts to the inches, and from the inches to the feet. In like manner multiply all the multipli- cand, by the inches and parts of the multi- plier, and set the result of each one place removed to the right-hand of those in the multiplicand, and the sum of these successive | products will be the answer. DUO Or, instead of multiplying by the inches and parts, such parts of the multiplicand may be taken as these are of a foot. EXAMPLES. Multiply 6 feet, 4 inches, 3 parts; by 10 feet, 3 inches, 9 parts. Te Gt 6 4 3 fo, 63 6 6 1 Diy a 4 9 feet 65 6 3 ae 3 Otherwise: Ste tm 3in.—i6 4 3 10 3 9 ts) 2 ee SE ee Ae 4 9% 6G feet 65 +z as above. It may not be amiss to observe here, that the feet in the answer are square feet, but the numbers standing in the place of inches are not square inches, as one might at first infer, but 12th parts of square feet, each part being equal to 12 square inches. In like manner the numbers standing in the third place, or place of parts, are so many 12th parts of the preceding denomination, these therefore are square inches; and in the same manner, if the operation be carried farther, every suc- cessive place will be a 12th part of that pre- ceding it. See DUODENARY Arithmetic. DUODECIMO, is used to denote the size of a book when the sheets are folded into twelve leaves. DUODECUPLE, consisting of twelves. DUODENARY Arithmetic, is that in which the local value of the figures increases, in a twelvefold proportion from right to left, in- stead of the tenfold proportion, in the common or denary arithmetic. ‘Thus 1111, in duode- nary scale, expresses 123 + 127+ 12+1= 1885 in the common scale. Every number may be converted from one scale to the other, and in many cases the duodenary system (particularly after practice has rendered it familiar) possesses considerable advantages over that in common use. In the duodenary scale of notation, there must be introduced two new characters for expressing 10 and 11, and these may be represented by 9, and , ° that is 10 = 9; 11 = «; so that the digits of this system becomes, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, %, x, with which characters any number whatever may be expressed according to the duodenary system, the same as in common arithmetic, by the nine simple digits. See Notation. To perform Duodecimal Operations by means of the Duodenary Scale. Rule. Transform the feet, if above 12, into the duodenary scale, and set the inches and DUO — patts as decimals; then multiply a8 in com- mon arithmetic, except carrying for every 12 instead of for every 10 as in common opera- tions; and finally transform the integral part of the product into the denary scale. EXAMPLES. 1. Multiply 17 feet, 3 inches, 4 parts; by 19 feet, 5 inches, 11 parts. 15°34 — 17 feet, 3 = inches 17'5¢ = 19 feet, 5 +5 inches 13908 - Ay ' 7248 00 4 <2 153 4 240°96 88 — 336 ft. 9’ 6” 8” Sv One great advantage of this method of per- forming duodecimal operations is, that it may be submitted to proof the same as common multiplication, by casting out the elevens, in- stead of casting out the nines, as is done in ordinary operations. 2. Find the solidity of a cube, whose side is 18 feet, 7 inches, 7”. egy se tote 77 1177 Vor dl proof Tr ol N3 1177 | PRL 1177 1359761 11:77 4 fs 9049867 aN 9049 867 135 976°1 1359 761 1571281417 = 2588 ft. 2’ 8” "arg 3. Giving the area of a rectangle équal to 174 feet 11 inches, and its length 15 feet 7 inches; to find its breadth in feet and inches. 174 ft. LLin, = 1267, and 15 ft. 7 in, = 13°7, | 13°7) 126 +3 (72841 123 5 3 60 272 $90 048 540 524 ~ 180 187 45 = The breadth is therefore 11 ft. Qin. 8’4’’8'". proof 0 em And thus we may perform any arithmetical operations with the same ¢ase as We dé in common numbers; except, that continued practice in the latter renders the rules a little more familiar. DYN 4, Having given the area of a’squafe equal to 17ft. 4fin.; required the length of its side; 15°46 (4:2020 | ~ bo , proof 146 5 144 7 eit Die Ese 14804 732800 674404 47378 Ns bee Therefore the side is 4ft. 2in. 0” 2” 10". See Barlow’s “Theory of Numbers,” p. 275; see also the article NoTATION. DUODENARY Scale of Notaticn, that in which the local value of the digits increase in a twelvefold proportion, as in the preceding article. DUPLE, or Dovus_e Ratio, is that in which the antecedent term is double the consequent. Sub-Dup_Le Ratio, is that in which the con- sequent term is double the antecedent. DUPLICATE Ratio, is the square of a ratio, or the ratio of the squares of two quan- tities; thus a? : b* is in the duplicate ratio of a:b. DUPLICATION, the doubling of a quan- tity, or multiplying by 2. DuPLicaTION of a Cube, is the finding of the side of a cube which shall be double in soli- dity to a given cube. This cannot be eflected | geometrically, as it requires the solution of a “eubic equation, or requires the finding of two mean proportionals, viz. between the side of the given cube and the double of the same, the first of which two mean proportionals is the side of the double cube, as was first ob- served by Hippocrates of Chios. This celebrated problem was often atiempt- ed by the ancient geometricians ; being first proposed by the oracle of Apollo at Delphos: this oracle, being consulted as to the manner. of stopping a plague which then raged at” Athens, returned for answer, that the plague should cease when Apollo’s altar, which was | cubical, should be doubled. Hence this ob- tained the name cf the Delian problem. For more on this subject, we refer to Hutton’s ““Ozanam,” vol. i. p.340. See also Valerius Maximus, lib. vili.; Eutocius’s ‘Com. on lib, ti. ;” Archimedes’s “ De Sphera et Cylin- dro;” and Pappus, lib. iil, prop. 5; lib. iv. prop. 22. ' DYNAMICS (from duvapss, power), is the science of moving powers, or of the action of forces on solid bodies, when the result of that_ action is motion. Mechanics, in its most ex-— tensive meaning, of quantity, of extension, and of motion. Now that branch of it which considers the state of solids at rest, such as their equilibrium, their weight, pressure, &e. is called Statics; and that which treats of their motion, Dynamics: So when fluids, instead of solids, are the sub= is the science which treats | : EAR jects of investigation, that branch which treats of their equilibrium, weight, pressure, &c. is called Hydrditaties; and that which treats of their motion, Hydrodynamics. An investiga- tion of the several laws of dynamics, as OscIL- LATION, GYRATION, PERCUSSION, PROJEC- TILES, &c. will be found under the respective articles. DYNAMOMETER (from duvepss, power, and yerpew, J measure), a measure of power, is the name of an instrument intended for measuring the muscular strength of men, horses, and other animals. Various forms EAR have been given to this instrument; but, upon the whole, the common. spring steel-yard, though not intended expressly for the pur- pose, is as well, or perhaps better, adapted to it, than any that has been yet contrived. For this purpose it has a ring at one end, and a hook at the other; by the latter of which it is fixed to any wimovable post, and by means of the other the person or animal is enabled to draw, and the. strength exerted is measured by the graduated rod which is thus drawn out. See Spring STEEL-Yard. 10) EAGLE. See AQuiLa. EaG Ler, a coinage of the United States, va- jue 10 dollars, or £2. 3s. 8d. sterling. EARTH, in Astronomy, is a primary planet | belonging to the solar system, the third in _ order from the sun, performing its revolution about that body, between the two planets Venus and Mars. In geography, we may de- fine it that terraqueous globe which we in- | habit, composed of land and water. Figure of the Earth.—V arious opinions were formerly held as to the figure of the earth; some of the ancients maintaining that it was cylindrical, others that it was an extended plane; while others, as far back as Pythagoras, gave to it that globuiar form which the voy- ages and discoveries of the moderns have since rendered evident to demonstration. It would be useless in the present state of the science, to enter into any formal proof of the spherical figure of the earth; this is a truth which is now generally admitted, and - stands in need of no confirmation beyond that which is drawn from the voyages of the seve- | ral circumnavigators. It was, however, for a long time supposed | that, abstracting from the small inequalities on the terrestrial surface, the form of the earth was that of a perfect globe; and who it was that first suggested the idea of its devia- tion from this figure is not precisely known; no doubt, however, now remains of its being an oblate spheroid, the proportion of its two diameters being to each other in about the . ratio of 331 to 332. Picard, in his measurement of the earth, published in 1671, speaks of a conjecture pro- posed to the academy, that supposing the diurnal motion of the earth, heavy bodies should descend with less force at the equator - than at the poles, and observes, that for the same reason there should be a difference in the length of the pendulum vibrating seconds in different latitudes. Tt was in the same year that Richer was sent to Cayenne; and among other objects of his voyage, he was charged by the academy to observe the length of the pendulum vibrat- ing seconds. He returned in 1672, and men- tions his observations of the pendulum as the most important he had made. ‘The same mea~ sure which had been marked at Cayenne on a rod of iron, according to the length which had been found necessary to make the pen- dulum vibrate seconds, being brought back and compared with that marked at Paris, the difference was found to be a line and a quar- ter, that at Cayenne being the shortest. ‘The vibrations of the pendulum on which the ex- periment was made were-very small, and con- tinued sensible for fifty-two minutes of time, and were compared with an excellent clock which vibrated seconds. Moreover, the clock which Richer took to Cayenne having been adjusted to beat seconds at Paris, retarded two minutes a day at Cayenne; so that no doubt remained of the dimimution ef the force of gravity at the equator. This was the first direet proof of the diurnal motion of earth. ‘Huygens was then led to suspect that the same cause might produce a protuberance of the equatorial parts of the earth, and a corresponding depression of the poles. Cassini had already observed the ob- late figure of Jupiter, which analogy strongly favoured the supposition of a similar pheno- menon on the earth, and the most obvious method of ascertaining the fact being by direct measurement, astronomers were sent to various parts of the world to measure the lengths of different degrees. See DEGREE. Huygens was certainly the first person who attempted to determine the figure of the earth by direct calculation; but he assumed an hy- pothesis which, since the discovery of the law of yniversal gravitation, has been found to be inadmissible: this hypothesis supposes that the whole of the attractive force resides in the EAR centre of the earth, and that its power varies in- versely as the square of the distance. Upon this supposition, Huygens computes the ellipticity of the earth to be ~4,, the centrifugal force at the equator being 1, of the force of gravity. Newton, upon the supposition of the earth being an homogeneous fluid, estimates the ellipticity at .4,. But Clairaut was the first mathematician who gave a general solution of this problem, adapted to the hypothesis of a variable density. The result which he ob- tained from his investigation was as curious as unexpected; it appeared that if the den- sity of the strata, of which the earth is com- posed, increases towards the centre, the ellip- ticity will be less than in the hypothesis of Newton, and greater than on that of Huy- gens; and moreover, that the fraction ex- pressing the ellipticity being added to the fraction expressing the augmentation of gra- vity at the poles, will together always make a constant quantity, which is equal to 3 of the fraction which expresses the proportion which the centrifugal force at the equator bears to that of gravity. The same problem has since been inyes- tigated by several eminent mathematicians, - and particularly by Laplace, in his ‘ Mecha- nique Celeste ;” but it must be acknowledged that considerable difference is found to exist in the results of the different calculations ; and Playfair, in vol. v. of the Edinburgh Trans- actions, enters into the inquiry why, consi- dering the care with which observations have been made, they do not agree more hearly in the resulis?) Among the causes of this incon- sistency, besides unavoidable mistakes and the imperfections of instruments, the professor mentions local irregularities in the direction of gravity, particularly those occasioned by the attraction of mountains. ‘ There are,” says he, ‘‘no doubt, situations in which the measurement of a small arch might, from this cause, give the radius of curvature of the me- ridian infinite, or even negative.’ Another kind of local irregularity is that arising from the unequal density of the materials under the surface of the earth, and not far from it: errors thus produced might amount to 10” or 12”. And this cause of error is the more for- midable, not only because it may go to a great extent, but because there is not any visible mark by which its existence can always be distinguished. He also justly observes, that in order to avoid any material error in deter- mining the figure of the earth, the arches measured should be large, consisting each of several degrees, as an error would then be rendered inconsiderable, by being spread over a greater interval. We refer to Mr. Playfair’s paper for several ingenious formulz for ascer- taining the proportion of the earth’s axis, from different kinds of admeasurement. The rea- der may also consult the work of Clairaut mentioned above; Horsley’s “ Tracts ;” Ro- bertson’s “ Navigation ;” the papers of Wil- ‘ ams, Dalby, and Mudge, in the Phil. Trans. relative to the Trigonometrical Survey; Svan- EAR berg’s “Exposition des Operations faites en Lapponie,” &c.; Delambre, &c. “ Base du Systeme Métrique Décimal; Puissant, Geo- desic, 'Topographic,” Kc.: in several of which the principal geodesic operations and analytic formula are explained with considerable per- spicuity. Motion of the Earth—There are three prin- cipal motions of the earth. 1. A motion of rotation on its own axis; 2. A motion in an. orbit round the sun; 3. A motion of its axis about the poles of the ecliptic. These three motions are all mathematically derived from one single impulsion, supposed to be given to the earth at its creation, combined with the continued action of gravitation. The rotation of the earth on its axis, called its diurnal motion, is the most uniform we are) acquainted with. It is performed in 23"56'4"1, of mean solar time, or one sidereal day. The earth, or, more strictly speaking, the common centre of gravity of the earth and moon, describe an orbit round the sun, which orbit is of an elliptic form of small eccentri- city; the sun being placed in one of its foci. If we suppose the plane of this orbit extended to the fixed stars, it will trace in the heavens a circle called the ecliptic. The plane of the earth’s equator, which remains very nearly parallel to itself during the whole of this reyo- lution, is inclined to the ecliptic in an angle of 23° 28’. The points in the earth’s orbit, which are intersected by this plane, are called the equinoctial points. The motion of the earth in its orbit is very far from uniform; but it is so far regular, that, with the exception of some small inequalities caused by the action of the moon and planets, the radius vector or line joining the centres of the earth and sun describes equal areas or sectors of the ellipse, in equal times. See KEPLER’S Laws. The third motion of the earth, we have to notice, is that which produces the precession of the equinoxes. ‘The motion of rotation having produced a protuberant form in the equatorial regions of the earth, the continual action of the sun and moon on this surround- ing mass or annulus, produces a rotatory mo- tion in the axis of the earth round the axis of the ecliptic; the inclination of these axes re- maining the same. ‘This revolution is accom- plished in 25,063 years. [Tor the phenomena arising from these motions, see the articles DivRNAL, SEASON, PRECESSION of the Equi- noxes, &c. | The principal elements of the earth, as con nected with the sciences of astronomy and geography, according to the determination of Laplace, in the last edition of his “System (lu Monde,” are as follows: Equatorial diameter..,... 7924 Eng. miles Polar diameter............. 7908 Mean diameter.......:.... 7916 : Mean circumference... 24869 me Mean length of a degree 69°08 SUIIACE 1. cravacsouss 196862256, Solidity...........259726936416 sq. miles cubic miles EAR Density of the earth is 3°9326 times greater than the density of the sun, and about five times that of common water. Mass of the earth is +4752, of the mass of the sun. The weight of a body at the equator, is to the weight of the same body at the poles, as 1: 1°:00569. The length of a second’s pendulum at the equator is 39°027 inches, and at the poles 39°197 inches. The centrifugal force at the equator is about 52, of gravity. If the rotatory motion of the earth was seventeen times greater than it is, the cen- trifugal force would be equal to that of gra- vity; and therefore bodies at the equator would then have no weight. The mean handle or winch is fitted, and the cushion supported and insulated by a glass pillar; '» lower part of which is fitted into a wooden ‘ket, to which a regulating screw is adapted, ‘inerease or diminish the pressure of the shion against the cylinder. A piece of silk ‘nes from the under edge of the cushion, 1 lies on the cylinder, passing between it 1 the cushion, and proceeding till it nearly ets the collecting points of the conductor. '¢ more strongly this conductor is made to tere to the cylinder, the stronger is the de- ¢ of excitation. Before the cylinder, or osite to the cushion, is a metallic tube 1, supported by a glass pillar LM. This sometimes called the prime conductor, m only the conductor; and for the more (veniently trying experiments on the two (vers, and exhibiting the different states of cushion and conductor, there are two 2s to be fixed occasionally, the one to the ductor, the other to the cushion; on the er part of these are balls furnished with ing wires, that they may be set apart from 1 other at different distances; fig. 2 and 3 esent two views of the Electrical Plate | ELE Machine, the construction of which will be readily comprehended by the different letters which are referred to the same parts in both views. It will be necessary, before the electrical machine is put in motion, to examine those parts which are liable to wear either from the friction of one surface against another, or to be injured by the dirt that may insinuate itself between the rubbing surfaces. If any grating or disagreeable noise is heard, the place from whence it proceeds must be dis- covered, wiped clean, and rubbed over with a small quantity of tallow; a little of which should also be occasionally applied to the axis of the cylinder itself. The screws by which the frame is fixed should also be examined, and if they are loose, they should be tight- ened. Having examined the different parts of the machine, and put them in order, the glass cy- linder, and the pillars which support the cushion and conductor, should be well wiped with a dry silk handkerchief, to free them from the moisture which glass attracts from the air, being particularly attentive to leave no moisture on the ends of the cylinder, as any damp on these parts carries off the elec- tric fluid, and weakens the force of the ma- chine: in very damp weather it willbe proper to dry the whole machine, by placing it at some little distance from the fire. : Care should be taken that no dust, loose threads, or filaments, adhere to the cylinder, its frame, the conductors, or their insulating pillars ; because these will gradually dissipate the electric fluid, and prevent the machine from acting powerfully. When you are satis- fied of this, rab the glass cylinder first with a clean, coarse, dry, warm cloth, or a piece of wash leather, and then with a piece of dry, warm, soft silk; do the same to all the glass insulating pillars of the machine and appa- | ratus; these pillars must be rubbed more lightly than the cylinder, because, being var- nished, they may be damaged by too much friction. It may be proper in some cases to place a hot iron on the foot of the conductor, in order to evaporate the moisture which would other- wise injure the experiments. It may also be observed that, 1. To excite the machine, it is requisite to clean the cy- linder, and wipe the silk. 2. Grease the cy- linder, by turning it against a gréasy leather, till it is uniformly obscured. ‘The tallow of a candle will answer this purpose. 3. Turn the cylinder till the silk flap has wiped off so much of the grease as to render it semi-transparent. 4. Spread some amalgam on a piece of leather, and apply this against the turning cylinder. By this means the friction will immediately increase, and the leather must not be removed till it seems to have attained its maximum; then discontinue the application of it, and the effect of the machine will be found very powerful. The best kind of amaleam is made of zine S ELE and quicksilver. If a little of the latter be added to melted zine it renders it easily pul- verable, and more quicksilver must be added to the powder if very soft amalgam is re- quired. ELECTRICITY (from nrzxrpov, amber), is the name of an unknown natural power which produces a great variety of peculiar and sur- prising phenomena, the first of which are sup- posed to have been observed in the mineral substance called amber, whence they have been denominated electrical phenomena, and the laws, hypothesis, experiments, &e. by which they are explained and illustrated, form together the science of electricity, of which it is impossible to give more than a very slight historical sketch in the limits of our present article. ‘The electric property of amber was Known to Thales, who lived about 600 years before our wra, though Theophrastus, who flourished about 300 ygars after Thales, is the first author who makes any distinct remark on this subject. Some other of the ancients also speak slightly on this head, but still they confined its action to the two substances amber and jet, and knew nothing farther of its effect except its power of attracting light substances after being excited by friction. Electricity may therefore be considered as a modern science of no higher date than about the year 1600, in which year Gilbert, a native of Colchester, and a physician in Lon- don, published his treatise “* De Magnete,” in which are contained many considerable experiments and discoveries. We owe also to Boyle, Otto Guericke, Dr. Wall, and some others, a few ingenious experiments; but it was not before the year 1709 that any import- ant step had been made in this science. It was in this year that Mr. Hawksbee wrote on the subject of electricity, and noticed the great electrical power of glass, and the light proceeding from it; he also first heard the noise attending the excitation, and a variety of other phenomena relating to electrical at- traction and repulsion; at the same time he introduced a glass globe into the electrical apparatus ; and to this circumstance he was indebted for many of his most important dis- coveries. After his time there was an interval of near twenty years in the progress of this science, when Mr. Stephen Grey established a new wra in the history of electricity. To him we owe the discovery of communicating the power of native electrics to other bodies in which it cannot be excited, by supporting them on silken lines, hair lines, cakes of resin or glass; and a more accurate distinction than had hi- therto obtained between electrics and non- electrics: he also showed the effect of electri- city on water much more obviously than Gil- bert had done in the infancy of this science, The experiments of Mr. Grey were repeated by M. du Fay, member of the Academy of Sciences at Paris, to which he added many new observations and discoveries of his own. He observed, that electrical operatious are ELE obstructed by great heat, as well as by a moisi air; that all bodies, both solid and fluid, woul¢é receive electricity, when placed on warm 01 dry glass, or sealing-wax; that those bodies which are naturally the least electric, have the greatest degree of electricity communi cated to them by the approach of the excitec tube. He transmitted the electric virtue through a distance of 1256 feet; and first ob served the electric spark from a living body suspended on silken lines, and noted severa circumstances attending it. M. du Fay alse established a principle, first suggested bp Otto Guericke, that electric bodies attract al those that are not so, and repel them as soo) as they are become electric by the vicinity 6 contact of the electric body. He likewis, inferred from other experiments, that ther were two kinds of electricity; one of whie he called the vitreous, belonging to glass, roc crystal, kc. and the other resinous, as that ¢ amber, gum lac, &c. distinguished by the’ repelling those of the same kind, and attra ing each other. He observed, that commun: cated electricity had the same property as th excited; and that electric substances attra¢ the dew more than conductors. J Mr. Grey, resuming his experiments } 1734, suspended several pieces of metal o silken lines, and found that by electrifyir them they gave sparks, which was the orig) of metallic conductors; and, on this occasio} he discovered a cone or pencil of electr light, such as is now observed to issue fron metallic point. From other experiments } conclndes, that the electrical power seems | be of the same nature with that of thund and lightning. Desaguliers followed Mr. Grey in the pros cution of this science, and added considerab| information to the stock previously obtaine and several interesting experiments were n¢ made by various philosophers in Englap Germany, and France. But the most remar able discovery that had yet been made in th science, was in the end of the year 1745, aj beginning of 1746. This was the method giving the electric shock, or the accumulati of the power of electricity in a vial. Thish its name of the Leyden Vial, from Cuneeus native of Leyden, who exhibited it as he w repeating some experiments made by Mess) Muschenbrock and Allamand, professors the university of that city. It is said, he w not, however, the inventor. The merit oft} discovery (if any merit can be ascribed t discovery made by accident) belongs to % Van Kleist, dean of the cathedral at Cam Soon after this, however, a method of givi; the shock was discovered in Holland, | Cuneus; and the discovery of this power effect of the electric fluid immediately rai! the attention of all the philosophers in Euro! Many of them greatly exaggerated their counts; either from a natural timidity, or th love of the marvellous. Mr. Muschenbro} who tried the experiment with a very t! glass bowl, told Reaumur, in a letter writh ELE soon after the experiment, that he felt himself struck in his arms, shoulder, and breast, so that he lost his breath, and was two days be- fore he recovered from the effects of the blow and the terror. He added, that he would not take a second shock for the whole kingdom of France. Mr. Allamand, who made the expe- riment with a common beer-glass, said, that he lost his breath for some moments, and then felt such an intense pain all along his right arm, that he was apprehensive of bad conse- quences, but it soon after went off without any inconvenience, &c. Other philosophers, on the contrary, showed their heroism and mag- nanimity, by receiving a number of electric shocks as strong as they could possibly make them. After this discovery the science of electri- city became a general subject of couversation, numerous machines were constructed more and more powerful, and public experiments were made highly curious and interesting. In fact, from that period to the present day, searce a year has gone by without some addi- tional information on this important science, and various treatises have been written relat- ing both to the theory and practice of electri- city, the principal of which are by Adams, Cayello, Cavendish, Ferguson, Mahon, Mor- gan, Mairne, Van Marum, Van Swinden, &e. most of which may be advantageously consulted jor details of experiments and the principles of the science. Sce also Priesiley’s ‘ History of Electricity.” | ELECTROMETER (from »rexrpov, and iwetpw, I measure), an instrument contrived or measuring the quantity and determining he quality of electricity. There are various jorms given to this instrument, which will be jound explained in the works quoted in the preceding article. ‘The most common elcctro- Sbter is shown in Plate V. fig. 2. » ELECTROMICROMETER (derived as ‘beve, with the addition of j2sxpo;, small), an listrument, invented by Veau de Launay, to adicate and measure very small quantities of Hectricity. ) BLECTROPHORUS (from mnarxrpov, and \epw, Z bear), is a machine consisting of two lates, one of which is a resinous electric, and ne other metallic; and when the former is mee excited by a peculiar application of the tter, the machine will furnish electricity for ‘considerable time, from which circumstance derives its name, being thus denominated yits inventor Volta, of Como, in Italy, who dis- overed it about the year 1774, or rather earlier. -ELEMENTARY, any thing relating to se principles or elements of bodies. ) ELEMENTS of the Planets, in Astronomy. ne PLANET. ) ELements of Geometry, the title of a cele- ated and well-known work of Euclid. See fewern Y. Evements, in the Higher Geometry or Ana- is, denotes indefinitely small portions of ipo surfaces, and solids. See FLUXIONS. Zitements, in Physics, the first principles, ses oI or ingredients, of which bodies are composed. These are supposed to be few in number, un- changeable, and by their different eombina- tions producing that extensive variety of ob- jects which constitute the works of nature. ELEVATION (from elevare, to raise or lift up), denotes the height or altitude of any object. See ALTITUDE. Evevation of the Equator, in Astronomy, is an arch of the meridian, less than a quad- rant, intercepted between the equator and the horizon of the place. ELEVATION Of the Pole, is an arch of the meridian intercepted between the pole and the horizon of the place. ELEVATION of a Star, is an arch of a ver- tical circle contained between star and ho- rizon, and in the same manner is estimated the elevation of any other of the heavenly bodies. ELEVATION of a Piece of Ordnance, in the theory and practice of projectiles, is the angle the axis of the gun makes with the horizon. Angle of ELEVATION, the angle formed be- tween the line of direction and an horizontal line. ELEVATION, in Perspective, the representa- tion of the whole body or building, as. in architecture, elevation signifies the whole or principal face of a building. ELGEBAR, the name of the bright star in the foot of Orion, more commonly called Rigel. ELIMINATION, in Analysis, that opera- tion by means of which all the unknown quan- tities except one are exterminated out of an equation, whence the value of that one becomes determined, and hence by substitution the value of all the other quantities. This is a subject which in all its generality involves a variety of cases, and numerous and tedious investigations, which of course cannot be treated of in this place; we refer the reader, however, who is desirous of the most complete information on the theory of eliminations, to “ Théorie Général des Equations,” by Bezout, where the subject is treated of in all its gene- rality and detail. We shall give one or two of the most simple cases. Let there be proposed the two following equations of the first degree : ax+by+erzo dx +ey+f—ro Tn order to eliminate one of the unknown quantities, for example y, the lst equation is multiplied by e, and the 2d by 6, so that the co-efficient of y in both may be equal, then we have eax + bey + eco bdx + bey + bf =o Subtracting the second of these equations from the first, we have (ea—bd)x + ec—bf—o bf—ec ea—bd Having thus obtained the value of x, that of y may be found either by substituting this value: of x in either of the original equations, whence again, x = ELI or by multiplying the first of them by d, and the second by e, and then subtracting as above, in either way we find pte cd—af I= Ge—bd which solutions are general for all equations of two unknown quantities of the first degree, and may therefore be considered as formule of solution in such cases. By a similar proceeding, if there were given the three following equations of the first de- gree, viz. ax + by +ecz4+d=0 ex + fy + zth=o ixz+hy+lze+mro0 we should find Jha bhl—chk + dgk—ghm + cfm—dfl ~~ eek gedk + afl— bel + big —ife y — Sam—eem +chi—ahl +del —gid eet cekh —gdk + af l— bel + big —ife __ ahk—deh + dfi—afm+ bem—bih wees cek—gdkh+ afl—bel+ big—ife And by means of these general formule, we have the complete solution of all equa- tions of the first degree containing three un- known quantities. It is also obvious that by a similar operation we might eliminate any number of unknown quantities, but it soon becomes extremely la- ' borious, and a direct method of arriving at the result has, therefore, been considered a great desideratum; and several ingenious attempts have been accordingly made by mathema- ticians of the first eminence. Cramer has given a rule for this purpose in his “ Analysis of Curve Lines.” A paper on the same subject was published by Bezout, in the Memoires of the Acad. of Sciences for 1764, and again in his “ Théorie Général des Equations,” p. 171, of which we propose giving a slight sketch in the subsequent part of this article. Two other papers on this subject were printed in the Me- moires of the Academy for 1772, the one by Vandermonde and the other by La Place; Be- zout, however, justly considers the following as the simplest of any that has yet been published. New Method of Elimination in Equations of the first Degree, whether literal or numerical. Let u, x, y, z, &c. represent the unknown quantities, and let the number of them, as also the number of the equations, be n. Leta, b, c, d, &c. be the respective co-effi- cients in the first equation. a’, U, ¢, d', those of the second. a", b", ce’, ad’, those of the third, and so on. And let us conceive the known term in each equation to be effected by some unknown quantity which may be represented by ¢. Yorm the product uxyzt, of all the un- knowns, written in any order at pleasure ; only when this order is once determined on in any case, the same must be observed throughout. Change now successively each unknown for its co-efficient in the first equation, and observe to change the sign of each even term, . and this result is called the jist line. e line. ELL Change in this first ine each unknown for its co-efficient in the second equation, ob- serving, as above, to change the sign of each even term, and this result is called the second Change again in this second line, each un- known for its co-eflicient in the third equa- tion, observing still the same change of sign, and call this the third line. Continue thus the operation to the last equation inclusively, and the last line that you obtain will give you the value of each of the unknown quantities, as follows, viz. | Tach unknown quantity will be expressed by a fraction, of which the numerator is the’ co-efficient of the same unknown letter in the last or nth line, and the general denomi- nator will be the co-efficient corresponding with the unknown quantity ¢, which was in troduced in the beginning of the operation. — This will be understood from the following example: Let ax+by+emo avdx+by+e¢=0 Introducing ¢ these equations become, — ax+by+ctmo we+by+ctmo and form the product ryt. Now change z into a, y into 8, ¢ into e, anc change the sign of those terms which stan in an even place ; thus we have " ayt — bat + exy we Now again change x into a’, y into 0’, ¢ int c, changing signs as before, and we haye abit —acy—a'bt + be'x + a'ey—b'e: or (ab! — ba')t—(ac' —a'c) y + (bc'—b'e): be be’ — whence x = a seme ab’ — ba' f ‘ ae—a andine ace al BT Nak ab'—ba M Bezout, “Théorie Gén. des. Equat.” p. 171. ELLIPSE, is one of the conic sections formed by the intersection of a plane am cone, when the plane makes a less angle wit! the base than that formed by the base the side of the cone. The word is derived from éAAcnns, defectin and is thus denominated by Apollonius, b cause the square of the ordinate in this figur is always less than the rectangle of the para meters and abscisses. a There are three ways in which we may fine an ellipse; viz. 1. As being produced the intersection of a plane and cone, as W find it treated of by Apollonius and all th ancients. 2. According to its description 1 plano, as it is treated of by several of th moderns. And 3. As being generated by th motion of a variable line or ordinate, alon another line or directrix, whereby the prope ties of the curve are treated of, by mean the equation by which it is defined. 7 The several lines and points in and abot an ellipse, as the axis, diameters, paramet foci, &c. will be found defined and explain under the several articles in this Dictionat and we shall, therefore, in this place mere rr > ‘ ' ELL state some of the principal properties of the ‘figures, without, however, attempting their ‘demonstration, as that would extend this ar- 'ticle to too great a length. _ Properties of the Ellipse-—1. The squares ‘of the ordinates of the axis, are to each other ‘as the rectangle of the abscisses. H pe. On} : | '] T | Thatis, AF.FB:GF*::AD.DB:DE* | Or if one of the ordinates be taken at the centre as H-C, which thus become the semi- conjugate, then because AC.CB = AC?, ithe proportion becomes AC*: CH? :: AD.DB: DE? or by doubling | AB*: HI? ::AD.DB: DE? _ That is, As the square of the transverse axis Is to the square of the conjugate, So is the rectangle of the abscisses / _ To the square of their ordinate. | From which property is readily drawn the iquation of the ellipse; for, make AB=t, Hi=c, AD —2,and DE=y, then t 3 c% s w(t —ax): y* ( 2 Vhence y* = ~ (t2—2*) or is - Vv (tz —2?) vhich is the equation of the ellipse. Or taking DC =z, and tande to represent ue transverse and conjugate as before, then ese 3: Gt—a2) (Et +2): y* vhence y = 7 / (Gt — 2?) hich is the equation when the absciss com- ences at the centre. | 2 Since — = p, the parameter, by the defini- 'on of this line, we have also ; ips: 5t?— 27; y? 's ecerte y eral f /V(E ¢? — a ‘hich is the equation in terms of the para- veter and transverse axis, when the absciss 2gims at the centre, or i ' er f V/ (ti — x?) ‘hen it begins at the vertex. 2. The sum of two lines drawn from the To foci to any point in the curve is always {ual to the transverse axis, ‘Thatis, if F,f, be .€ two foci, and E, FE, two lines ‘awn from them any point E in A € curve, thenFE yas AB: From this pro- | B ELE perty is derived the common method of de- scribing the curve mechanically by points, or by a thread, thus: In the transverse axis take the foci I’, f,and any point I. Then will the radii A I, I B, describe, from the centres Ef, two ares intersect- ing each other in M, which will be a point in the curve. In like manner assume other points I, and thus de- termine other points m, m, &c. Then with a Steady hand a curve-line may be drawn through all the points of intersection, which will be the ellipse required. Or, otherwise, take a thread of the length A.B the transverse axis, and fix its two ends in the foci I’, f, (see the preceding fig.) Then carrying a pen or pencil P, round by the thread, keeping it always stretched, and its point will trace out the ellipse, as is evident from the property above stated. Other methods of describing the ellipse in plano are given in the subsequent part of this article. 3. Ifa tangent be drawn to any point in an ellipse, and. two lines drawn from the two foci to the point of contact, these two lines will form equal angles with the tangent, That is, if H T bea tangent to the ellipse at G, and the two lmes F G, fG from the two foci be drawn, then will the angle TG F.= the angle HGF. It is this property that gives the name foci to the two points F, f, for as opticians find that the angle of incidence is equal to the angle of reflection ; it follows from the above property, that rays of light issuing from the one focus, and meeting the curve in every point, will be re- flected back into the other focus, and ‘hence these points are denominated foci, or burning oints, This property may be otherwise illustrated by considering a billiard table in the form of an ellipse, then a ball being struck so as to pass through one focus in any direction, it will be reflected back again through the .other focus, then again through the first, and so on, as long as the ball continues in motion, 4. If there be any number of ellipses de- scribed on the same transverse axis, and any ordinate be drawn so as to meet all those ellipses, the tangents to the: several ellipses at: those points will all terminate in one common point in the prolongment of the transverse axis, Thatis, ifA HB, A H’B, be any two ellipses, having the same transverse axis ; and D HH’ be any common ordinate, then tke two ‘tan- ELE gents HT, H’F will be terminate in the common point T. Andas,this is necessarily true when A H'B becomes a circle, we have hence an easy me- thod of drawing a tangent to any point in an ellipse; which is as follows: Let H be the point to which a tangent is to be drawn; draw the right ordinate DH, and produce it indefinitely; on A B describe a semicircle meeting the ordinate produced in H’; join H'C, and draw H’T perpendicular to H’C, meeting the transverse produced in T, then TH will be the tangent required. 5. The ordinate H’T, in the circle, is to the ordinate HT of the ellipse, as the transverse axis of the ellipse is to its conjugate axis. And if a circle were described on the con- jugate axis, and an ordinate drawn as before, then the ordinate of the circle would be to the corresponding ordinate in the ellipse, as the conjugate axis of the ellipse is to its transverse. And hence it follows that the area of any ellipse is a mean proportional be- tween the area of the circles described on its two axes, 6. Every parallelogram circumscribed about an ellipse, at the extremities of any pair of its conjugate diameters, is equal to the rectangle of its two axes. 7. The sum of the squares of any pair of conjugate diameters is always equal to the sum of the squares of the two axes of the ellipse. The demonstration of the above properties, and numerous others, will be found in the -works of the several authors referred to under the article Conic Sectzons. Description of the Ellipse in plano.—Peside the two methods of describing an ellipse, given in proposition 2, there are several others, by instrumental operation, of which the fol- lowing are the most simple: 1. IftworulersY¥G, Gr fH, each equal in length to the trans- verse axis AB, have their extremities fix- ed in the foci so as to be moveable round those points; 4 |——~ Nh sal f--the teh Sim ti extremities of the rulers be connect- ed by a third ruler HG, which is equal in length to Ff, the dis- tance of the two foci so as to be moveable about the two points H and G, then if the ruler H G be moved round the centres F and J, the intersection of the rulers F G, fH, in E Wop : will describe the periphery of an ellipse, of which AB is the transverse axis, and F, f, the two foci. 2. Letthere be two rulers, A B, H1, set at right angles to each other, and let a third ruler, DEP, A be moved along, so thatthe points Dand FE. constantly touch the two rulers A B, 1 HI; so will the point P deseribe the peri-| phery of an ellipse. On this principle are con-. structed Eitiiptic Compasses, which see. 3. Ifone end C, of any two equal rulers CD, DP, which are moveable about the point D, like a car- penter’s joint rule, be fastented to the ruler A B, so as to be moveable about C; and if the end P, of the ruler D P, be drawn along the side of the ruler A B, then any point E, taken in the side of the ruler DP, will describe an) ellipse whose centre is C, conjugate axis = 2 EP, and transverse —2CD—2DE. | Area of an Ellipse-—Multiply the two axes) together, and that product again by “7854, which will be the area required. i) Area of an Elliptic Seyment.—Find the are of the corresponding circular segment, de-| scribed on the same axis to which the cutting. line or base of the segment is perpendicular. Then as this axis is to the other axis, so is the circular segment to that of the ellipse. ! Or find the tabular circular segment whose} versed sine is equal to the quotient, of the| height of the elliptic segment divide by its axis. ‘Then multiply together this segment,’ and the two axes of the ellipse, for the are : of the segment. it] Periphery of an Ellipse.—Put the transve 2 1 i axis = ¢, conjugate = c, andl — fh = dand t 3°14 16 = p, then + 81 2 3e __ 8%.5d | 3.5.70 POUT BT FR erg i pe &e. will be the periphery. La Or approximatively, Selpe Z) 1. ip x (¢ +c) Be tt 2 z 2 px4/ Ss ~ ial Bie | 6a. fete 3t+PQr ge 3. £p9 347 —_— — Eppa —S a In which last P is the perameter. Each’ 0 the above is a nearer approximation #hal that which precedes it. at Oe See the investigations of the above rules and several others relating to the rectification of the ellipse in Dr. Hutton’s “‘ Mensuration.” Sce also Edin. Trans. vol. iv. and v. Lon don. Trans. 1804; Landen’s‘ Memoires,”; Le gendre, Mem. de l’Acad. 1786; Legendre | | ELL Mem. sur les Transcendantes Elliptique ; and Euler’s Opuscula, Berlin, 1750. Infinite Eviiesk. See ELiiprorp. ELLIPSOID. See SPHEROID. ELLIPTIC, or ELuipricaL, something re- lating to an ellipse. Exvuietic Are, is any part of the periphery of an ellipse, the length of which is found as follows: Let ¢ represent the semi-transverse, and ¢ the semi-conjugate axis of any ellipse, z the distance of the ordinate from the centre, then the arc bounded by the ordinate and parallel semi-axis will be, aes Ath at hr igh E29: BETES g t Ellip. Are =z) 1+ G-2 Oe an at ok Btte?— 4tc* + 9g ig! 2 RT REO aT en Ong t? — ¢? make a 4 then t*—43qz 9 = 3 Bates ye ar ~ | —r§ foam ly Ra 3, Aro=3 9g pyre Sahcaes 42 ¢ 4 Ave — 19PO +09 C—21p)y Are TEC + gC 2 P)y | C being the whole axe where the arc begins, ind p, z, and y the corresponding parameter, absciss, and ordinate. | Exiiptic Compasses. See Compasses. Exuirtic Conoid, the same as spheroid. | Exuietic Dial, ah instrument usually made ‘o fold up for the convenience of the pocket. By this dial is found the meridian, hour of -he day, the rising and setting of the sun, &c. _ Exuiptic Motion of the Planets. See Ker- UER’s Laws. _ Exuiptic Spindle, is the solid generated by he revolution of any segment of an ellipse sbout its chord; the solidity of which may be ‘ound by the following formule : _ Put the perpendicular axe of the ellipse = 1, the parallel axe = 4; length of the spindle =I, distance of the centre of spindle and ‘lipse =e; and area of generating segment = A; then, a... a? [3 ? . Solidity = 157078 x aa —4cA ¢& b $ fhe Wevteseveee == ZX 7804 ; D*—4e 34D) ; vhere D is the greatest diameter of the pindle. » ELLIPTICITY of the Terrestrial Spheroid, ss the difference between the major and the ninor semiaxes; it is generally expressed in erms of the former, that is, of the radius of he equator. The quantity of the ellipticity ias been variously assigned by different ma- shematicians : Sir lsaac Newton, supposing the arth of uniform density, gave 34, for the el- ipticity ; Boscovich, from a mean of several dmeasurements, stated it at .},; Lalande, 4; Laplace, z4,; Sejour, =5,; Carouge, sos Krafft, .3,; and Playfair, from a theo- “em of Clairault applied to the heterogeneous : x x nearly. | EME spheroid, states it at+4,. Setting aside those Which are deduced from the hypothesis of uniform density, 35 Inay be admitted as the most probable value of the ellipticity. See EARTH. ELLIPTOIDES, an infinite ellipsis, i.e. an ellipsis defined by the equation ay™+" — ba” x (a—a”), wherein m 48’ 38”, the annual epact will be 104 21" 11’ 22"; that is,nearly 11 days. Conse- quently the epact of 2 years is 22 days; of EPH 3 years, 33 days; or rather 3, sinee 30 day: make an embolismic or intercalary month Thus the epact of 4 years is 14 days, and s¢ of the rest; and thus every 19th year the epact becomes 30 or 0; consequently the 20tH year the epact is 11 again; and so the cyele of epacts expires with the golden number, 0) lunar cycle of 19 years begins with the same | these are Julian epacts: the Gregorian de. pend upon the same principles, accounting only for the difference of the respective years, As the new moons are the same, that is, ay they fall on the same day every 19 years, s¢ the difference between the lunar and solaj years is the same every 19 years. And becam™ the said difference is always to be added te the lunar year in order to adjust or make ij equal to the solar year; therefore the said dif: ference respectively belonging to each | of the moon’s cycle is called the epact of the said year, that is, the number to be added te) the same year, to make it equal to the soll year. | Rule to find the Gregorian Epact. The difference between the Julian and Gre. gorian years being equal to the difference be- tween the solarand lunar year or 11 days; there fore the Gregorian Epact for any year is pied same with the Julian Epact for the preceding year; and hence the Gregorian Epact will be found by subtracting 1 from the golden number! multiplying the remainder by 11, and reject: ing the 30’s. This rule will serve till the yea 1900; but after that year the Gregorian Epaet will be found by this rule: divide the cen; turies of the given year by 4, multiply the re- mainder by 17; then to. this product add 46) times the quotient, and also the number 86, and divide the whole sum by 25, reserving) the quotient: next multiply the golden num- ber by 11, and from the product subtract the’ reserved quotient, so shall the remainder, after rejecting all the 30’s contained in it, be} the epact sought. nh The following table contains the golden numbers, with their corresponding epacts, till the year 1900. . TABLE OF GREGORIAN EPACTS. Golden Golden Epacts, Number. | EP3cts- |] Number. I O VIII} 17 ||IXvVv ai 11° TX 28 XVI IIT 22 9 XVII IV 3. I XI 20 || XVIII V 14 XII bea vI_ |} 25 [xu 12 lI Vil 6 || XIV {i 23 re “ ther - EPHEMERIDES, in Astronomy, tables calculated by astronomers, showing the pre: sent state of the heavens for every day at noon; that is, the places wherein all the pla nets are found at that time. It is from oe BP I ables that the eclipses, conjunctions, and \spects of the planets are determined; horo- ‘copes or celestial schemes constructed, &c. ‘he astronomers of most nations publish ephe- herides; of these the most celebrated are the Iphemerides of Bologna; the Nautical Al- janac, published in England; and the Con- ‘oissance des Temps, published in I'rance, by ‘I. J. Lalande, Delambre, and others. ‘ EPICUREANS, a sect of ancient philo- »phers who followed the doctrine of Epi- ‘arus. ' EPICURUS, a celebrated philosopher of targetium in Attica, who flourished about ')0 years before Christ. At the early age of ‘/2 years he gave astonishing proofs of genius, thich he afterwards improved by visiting 'thens, which was then crowded by the fol- iwers of Plato, the Cynics, Peripatetics, and te Stoics.. Here he established himself, and von attracted a number of followers by the veetness and gravity of his manners, and by 's social virtues. In his ethics he maintained at the supreme good of man consists m easure, and supreme evil in pain. Nature prself, says he, teach us this truth; and jompt us from our birth to procure whatever ives us pleasure, and avoid what gives us jin. This doctrine, which seemed to open ye way to sensuality and dissipation, pro- ered for Epicurus a great number of enemies, jrticularly amongst the Stoics; but he re- ited all the accusations of his adversaries by te purity of his morals. His followers were Imerous in every age and country ; but they ion degenerated from the comparatively pure atiments of their master, and placed their ppiness in gross and sensual pleasures. IEPICYCLE (from ex: upon, and xuxdros, a cle ona circle), in the Ancient Astronomy, Sa subordinate orbit or circle, which was oposed to move on the circumference of a ‘ger one, Called the different ; by means of sich one motion, apparently irregular, was olved into two that were circular and uni- ‘m. And when the observed motion. was “irregular and complicated as not to be re- ved with one epicycle, others were added a nearer approximation was obtained. This system owed its origin to a prejudice st seems to have been extremely ancient, favour of circular motion; and the problem \t principally engaged the attention of as- ‘nomers in those times, was to assign the per proportion of the different and epicycle ich should approximate nearest to absolute iervation. ZPICYCLOID, in Geometry, is a curve terated by a point in one circle, which re- ves about another circle, either on the wacity or convexity of its circumference, 1 thus differs from the common cycloid, ich is generated by the revolution of a circle aga right line; though the latter has some- es been assimilated with the former, by ‘sidering the right line as the circumference .circle whose diameter is infinite. he invention of epicycloids is ascribed to | Roemer, the celebrated Danish astronomer, } | EPI during his residence at Paris in 1674. These curves appeared to him to be the best form for the teeth of wheels, in order to diminish their mutual friction, and to render the action of the power more uniform ; hence he was led to consider them, and to this purpose they have been applied, though other forms are now generally preferred. However, De Lahire, in his “Traité des Epi- cycloides,” published in 1694, makes no men- tion of Roemer, and seems to claim for himself the merit both of the geometrical and mecha- nical part of the discovery. But Leibnitz, who resided in Paris in 1674, and the two fol- lowing years, says, that the invention of the epicycloids, and their application to mechanics, were the work of this Danish mathematician, and that he was esteemed the author of it. But Newton published an account of epicy- cloids before Roemer, in the first book of his “ Principia,” where he proposed a general and very simple method of rectifying these curves. After him J. Bernouilli, during his residence at Paris, showed how, by means of the integral and differential calculus, to deter- mine their area, rectification, &c.; and many of his * Lecons du Calcul integral,” are de- voted to this object. In 1694, M. de Lahire published his “ Traité des Epicycloides;” and in the ‘“‘ Memoires of the Academy” for 1706, he communicated to the public an extensive and elegant treatise on these curves; and Dr. Halley has also, in the Phil. Trans. No. 218, a very Curious and ingenious paper on the same subject. : EpicycLoips, are distinguished into exterior and interior. Exterior Epicycyorps, are those which are formed by the revolution of the generating circle, about the convex circumference of the quiescent circle, as GEHF in the following figure : Interior Epicyctoips, are these that are formed when the generating circle revolves on concave circumference, as E g G. The revolving or generating circle is called. the generant. EPI And the are of the quiescent circle passed over is called the base. yo these curves belong several curious pro- perties, of which we shall only mention a few of the most remarkable. ’ 1. If the generating and equiescent circle have to each other any commensurable ratio, then is the epicycloid both rectifiable and quadrable, although the area of the common cycloid, which is so much more simple in appearance, can never be completely obtained. 2 If the generating and quiescent circle are incommensurable with each other, then the area of the epicycloid cannot be found, but it is still in this case also rectifiable. 3. If in the interior epicycloid the diameter ef the generant is equal to the radius of the quiescent circle, the curve becomes a right line, equal and coincident with the diameter of the latter. ‘0 find the Length of any extertor Epicycloid. As the semi-diameter of the quiescent circle, Is to the sum of the diameters of the two circles ; So is double the versed sine of the are of the gencrant, which has passed over any part of the quiescent circle, Yo the length of epicycloidal are generated by the point which touched the quies- cent circle at the beginning of motion. When the whole arc is required the versed sine becomes the diameter of the generant. — The length of any are of an interior eycloid is found by the same proportion, only using the difference of the diameters, in the second, term of the proportion, instead of the sum. To find the Areas of Epicycloids. As the radius of the quiescent circle, ' {s to three times that radius, plus twice the radius of the generant ; So is the circular segment b H, To the epicycloidal sector 6 H¥; Or, so is the whole area of the generant, T’'o the whole area of the epicycloid E'G B. Which proportion holds good both for the exterior and interior epicycloids. yh Dr. Halley gives us a general proposition for the measurement of al] cycloids and epi- cycloids: thus, the area of a cycloid or epi- cycloid, either primary, or contracted, or pro- late, is to the area of its generating circle ; and also the areas of the parts generated in those curves, to the areas of analogous seg- ments of the circle ; as the sum of double the velocity of the centre and velocity of the cir- cular motion, to the velocity of the circular motion. The demonstration of which may be seen in Phil. Trans. No. 218. As to the tangents it has been known from the time of Des Cartes, that the line H 6 drawn from any point H, to that of the base which touches the circle, whilst this poimt is de- scribed, is perpendicular to the curve, and consequently to the tangent. Maupertuis, in discussing this subject, con- ceived one polygon torevolveupon another, the sides of which are respectively equal, so that EPO one of the angles described a curve, the peri phery of which is formed of ares, of circles and the area is composed of circular sector: and rectilineal triangles. He then determinec the proportion ofthe area, and of the periphery of this figure, to those of the generating po lygon; and finally supposed those polygons t become circles, the figure described to becomi an epicycloid; and the above-mentioned pro. portion, modified agreeably to this suppositio gave him the area and the periphery of thi epicycloid. Mem. de Acad. 1727. Spherical Epicycioips, are formed by { point of the revolving circle when its plan makes an invariable angle with the plane a the circle on which it revolves. Mossieanl Bernouilli, Maupertuis, Nicole, and Clairaut, have demonstrated several properties of thes! epicycloids in Hist. Acad. Sc. for 1732. Parabolic, Elliptic, &c. Eptcycioips. If | parabola be made to revolve upon anothe equal to it, its focus will describe a right lin perpendicular to the axis of the quiescen parabola; the vertex of the revolving parabol will also describe the cissoid of Diocles, an any other point of it will describe some o of Newton’s defective hyperbolas, having double point in the like point of the quiesce parabola. In like manner if an ellipse revoly upon another ellipse equal and similar to i its focus will describe a circle, whose centt is in the other focus, and consequently. th radius is equal to the axis of the ellipse; an any other point in the plane of the ellipse wi describe a line of the fourth order. ‘The sam also may be said of an hyperbola revolvin upon another equal and similar to it; for o of the foci will describe a circle, having 7 centre in the other focus, and the radius wi be the principal axis of the hyperbola; an any other point of the hyperbola will descrik; a line of the fourth order. Concerning thes lines, see Newton’s “ Principia,” hb.1.; E Lahire’s ‘“ Memoires de Mathematique,” & where he shows the nature of the epicyclo and its use in mechanics. See also Maclauril “ Geometria Organica.” a EPOCH, or Epocua, in Chronology, a ter. or fixed point of time, from whence the su ceeding years are numbered or reckone The word is from eroyn, from erexecy, to stl tain, stop; because epochs define or limi certain space or time. at Different nations make use of differe) epochs. ‘The Christians chiefly use the epot of the nativity or incarnation of Jesus Chris. the Mahometans, that of the Hegira; t] Jews, that of the creation of the world, or th of the deluge; the ancient Greeks, that of i Olympiads; the Romans, that of the buildi of their city; the ancient Persians and Ass rians, that of Nabonassar, &c. The doctrii. and use of epochs is of very great extent chronology. To reduce the years of one epo' to those of another, z.e. to find what year) one corresponds to a given year of another, period of years has been invented, which, cor mencing before all the known epochs, is, as: were, a common receptacle of them all, calli EQU ye Julian period. To this period all the pochs are reduced; 7.e. the year of this period vhen each epoch commences is determined. ‘o that, adding the given year of one epoch » the year of “the period corresponding with ‘s rise, and from the sum subtracting the year the same period corresponding to the other ‘poch, the remainder is the year of that other ‘poch. ' ‘The above epochs, as referred to the Julian, ‘re as follows ; viz. Creation of the world............... 706) _ TEMES Whe satugss Osc dda. ob RaOe | ‘Ss (Olympiads of the Gree ve began 3938 | == ‘Rome built, or Roman era...... 3961 > 3 ‘/Pra of Nabonassar. ..............4. 3967 = Si * Birth of Christ, or Christian era 4713 : ‘Hegira, or flight of Mohammed 5335 ) Epocn of Christ, is the common epoch wonghout Europe, commencing at the sup- ‘osed time of our Saviour’s nativity, Decem- ‘er 25; or rather, according to the usual ac- mnt, from his circumcision, or the Ist of ‘fanuary. The author of this epoch was an 'bbot of Rome, one Dionysius Exiguus, a ‘eythian, about the year 507 or 527. Diony- us began his account from the conception or iearnation, usually called the Annunciation, 'r Lady Day; which method obtained in the 'ominions of Great Britain till the year 1752, ‘efore which time the Dionysian was the same 5 the English epoch: but in that year the ‘regorian calendar having been admitted by ‘st of parliament, we now reckon from the st of Je anuary, as in the other parts of Europe, ccept i in the court of Rome, where the epoch * the incarnation still obtains for the date of leir bulls. 'EPROUVETTE, the name of an instru- ‘ent for ascertaining the strength of fired inpowder. EQUABLE Motion, is that whereby the ‘oveable body proceeds with the same con- ‘nued velocity, being neither accelerated nor itarded. / EQUABLY accelerated or retarded Motion, '¢. is when the motion or change is increased + decreased by equa! quantities or degrees, 1 equal times. > EQUAL, a term of relation between two more things of the same magnitude, quan- ity. or quality. | Wolfius defines equals to be those eit ‘hich may be substituted for each other with- ‘it any alteration of their quantity. ‘It is an axiom in geometry, that two things ‘hich are equal to the same thing, are also ual to cach other. And again, if to or from quals you add or subtraet equals, the sum or ‘mainder will be equal. *Eeuat Altitudes, in Practical Astronomy, ‘ae of the most practicable and certain me- hods of determining the time, and thus ascer- fining the error of a clock or chronometer, ' by observing equal altitudes of the sun or ?afixed star. Vor this purpose, all that is | * The true birth of Christ was four years earlier ‘an is reckoned in the common date, which cor- sponds to 4709 of the Julian epoch. EQU necessary is to observe the instant the sun or star is at any altitude towards the east, before the meridian passage; and the instant must likewise be marked when the same objeet at- tains exactly the same altitude towards the west, after the meridian passage; the mean between the above quantities will be the in- stant marked by the clock at the moment the sun or star was on the meridian. The pre- ceding operation, however, supposes that the decliuation of the object has not varied during the elapsed interval, but this with the sun sel- dom happens. The observation must, there- fore, be corrected by a table, or by a direct calculation. Equa. Angles, are those whose containing lines are inclined alike to cach other, or which are measured by similar ares of their circles, FQuar Arithmetical Ratios, are those where- in the difference of the two less terms is equat to the difference of the two greater. Equa Curvatures, are such as have the same or equal radii of curvature. EQuaL Figures, are those whose areas are equal, whether the figures be similar or not. JQUAL Geometrical Ratios, are those whose least terms are similar aliquot or aliquant parts of the greater. Equa. Solids, are those whose capacities are equal. EQUALITY, in Algebra, is a comparison of two quantities which are in effect equal, though differently expressed or represented. Equality is usually denoted by two parallel lines, as =; thus 4 + 48; that is, 4 added to 4is equal to 8. This character was introduced by Recorde, in the second part of “‘ Whetstone of Witte.” Des Cartes, and some after him, used the cha- racter &, and others again y, for expressing equality ; but the two lines are now generally adopted. In the solution of a nameral problem, which is to be rendered rational, if there be only one formula, to be equal to a square or other higher power, it is called a simple equality. Double Eauarity, in the Diophantine Ana- lysis, is when we have two formula contain- ing the same unknown quantity or quantities, each of which is to be made equal to a pertect power. Let there be proposed the following double equality; v2z. a+ bx = s* oes eher a Ee s and ¢t representing any squares whatever, merely indicating that both formule are to be perfect squares, By elimivating 2 in both equations, we have ad — be = ds* — bt? therefore ; y ds* = bt® + ad —,be d*s* = dbit? + ad* — bde and it now, therefore, only remains to find ¢* such, that dbt? + ad? —bde may be a square. Put this square = m’, then we have m* —-ad* + bde db that is, the double equality is reduced a sim~- Sodom EQU ple equality, which was the purpose of the operation. ; . Since then ¢? is determined, we have, from bs i i?—c the second of the original equations, z = ae m* — ad* a Dard For more on this subject see p. 374, Euler’s « Elements of Algebra.” Triple Equatity, is when there are three formulz of the same kind as above to be made periect powers. Ratio of Equa ity, is the ratio of two equal quantities. ; EQUANT, in the Ancient Astronomy, a cir- ele formerly imagined by astronomers in the plane of the deferent, for reguiating and ad- justing certain motions of the planets. EQUATED Bodies, on the Gunter Scale, is the name of two lines which relates to the comparison of the sphere and the regular bodies; they are, however, seldom given on modern scales. EQUATION, in Algebra, is any expression in which two quantities differently represent- ed are put equal to each other, by means of the sign = placed between them: thus 7ax +32—5, 5y? + 3y—a—0, 7x3 +3x’—11=0, &c. are equations; and these receive different names, according to the various circumstances of the powers, relations, and combinations of the unknown quantities which enter into them. A Literal EquaTION, is that in which all the quantities, both known and unknown, are ex- pressed by letters; as ax” + bx =e. A Numeral Equation, is that in which the co-efficients of the unknown quantity and abso- lute terms are given numbers; as 5x7 4-72 = 16, x3 — lax—1=—0, &c. A Simple Equation, is that in which the unknown quantity enters only in the first de- gree; as 7x35, ax + br=c, 3ax + 562 = 119, &c.; and these are always better expressed by putting all the co-efficients under a paren- thesis, with the unknown quantity outside; thus (a + b)x= ce, (8a + 56) x2 = 119, &c. See Simple Equation, A Quadratic Equation, is that in which the unknown quantity enters in the second de- gree; as ax* — b, ax* + bx =e, &c. Note. When only the second power enters, as ax* — 4, it is called a Simple Quadratic ; and when the second and first both occur in the same equation, it is called an Adfected Quadratic ; such are the equations aa* +b2 =e, x* + a— ba, &e. See QuavRratic Equation. A Cubic EQuaTION, is that in which the third power of the unknown quantity enters; as x3 + ax* + bx =e, 2°? + ax), ax3 =e, &e. The latter of which is called a Simple Cubic. See Cunic Equation and IRREDUCIBLE Case. A Biquadratic Eauation, is that in which the fourth power of the unknown quantity enters; as axt + bz? + cx* + dx =e; and when only the fourth power enters, it is called a Simple Biquadratic. See BrquapRaAtic Equa- sion. And generally an equation is said to be of EQU the 5th, 6th, &c. degree, according as % highest power of the unknown quantity is; any of these dimensions. ed Binomial Equations, are such as have ow two terms; as #5 =a, 2° = b5, a" =the x*—a—o,&e. See BINoMIAL Equations Determinate Equations, are those eq). tions in which only one unknown quantity « ters; or if more than one enter, there are alwig given as many independent equations as the are unknown quantities. See ELIMINATH of Equations. .. Indeterminate Equations, are those eqi- tions in which there are more unknown qui. tities than there are independent equatio; See INDETERMINATE Analysis. Reciprocal Equations, are those in whh the co-efficients of cach pair of terms, equey distant from the extremes, are equal to e¢ other; such as ax? +bxr® + ex5, &e. cx* +4 +a —0O; thus x3 + 327+ 32 + 1= 0,44 xt + ax? + bx* + ax + 1=0, are Recipe cal Equations, which see. Transcendental Equations. See TRANSCs DENTAL Equations. Symmetrical EQUATIONS. CAL Equations. ; Exponential Equations, are those in whh the exponent or index of the power is % known; as a* = 6, 2*¥=a,&c. See Exh NENTIAL. | Roots of an EeauaTIon, are those numb: or quantities which, when substituted for ¢ unknown quantity, will make the whole & pression become equal to 0, or zero. r Thus, in the equation 23 — 62? + 11255, or x3 — 6x? + lla — 6 — 0, the three res are + = 1, 2 = 2, + = 3; for each Olam quantities, substituted for x, will render whole expression = 0. The roots of an equation are divided io positive, negative, real, and imaginary or 1: possible. : Positive Roots, are those that have a pi tive sign prefixed or understood. ~~ Negative Roots, are those that have a nee tive sign prefixed to them. aa” Real Roots, are those that are expressedy any real or possible quantity. a , x =—1 + ./—3, and e =—1— the two latter of which are imaginary; tha’, they admit of no assignable value; yet tly are such, that when substituted for 2, @ whole equation becomes zero; or, which is ¢ same,(—1+ /—1)}=8,and(—1— ”—3}82 the same as 23, which is also equal to 8. Properties of . Equations —Every equatt has as many roots, real and imaginary, as tlre are units in the exponent of the highest po® of the unknown quantity. ee if one of the imaginary roots of an equat® be a + ./—b, another of its roots will¢ a— /— 6, Hence it follows, that the na See SYMMETI es EQU -rofimaginary roots in any equation is always »en; or, which is the same, they always en- r in pairs; therefore in an equation of odd di- vensions, that is, when the highest power of -e unknown quantity isan odd number, there ing an odd number of roots, one of them jast necessarily be real; whereas in an equa- ‘on of even dimensions, all its roots may be »aginary; such are the two equations x7 —6xz 14—0, 22*— 727? + llez*—32+11—0; .ither of which has any real root. Another curious property of equations is vis, that the sum of all the roots is equal to ‘2 co-efficient of the second term, with its sn changed from + to —, or from — to +. "ve sum of the product of every two of the ots, is equal to the co-efficient of the third im, without any change in its sign. The :m of the product of every three roots, is equal i the co-efficient of the fourth term, with its ¢n changed; and so on to the last or abso- le term, which is equal to the product of all i> roots, with the sign changed or not, ac- «ding as the number of terms in the equa- tn are odd or even. Or if we consider the signs of the roots f»mselves to be changed in the first instance, tn the several products aboye mentioned \ | require none of those conditions of chang- 1; signs, but they will always arise with the $n proper to them. Note. In this it is to be understood, that i, terms of the equation are arranged ac- c ding to the powers of the unknown quan- t; thus 2* + az*—! + ba*-? + ez™-3 + a'*—* &c. _Also the co-efficient of the first term is sup- ped to be unity, and the absolute quantity considered as standing on the left hand se; and in any case, where any power of {| unknown quantity is wanted, it must be uoduced with the co-efficient 0 prefixed ii t. _ “or example, in the equation x? — 6x? + 1- — 6 — 0, the three roots being 1, 2, and 3 ve have +2 +3 %=+4+ 6, the co-efficient 2d term with its sign changed -24+1.342.3— +11, the co-efficient ~ 3d term 2.3 =—+ 6, absolute term with its sign changed. Again, in the equation x+ — 25x? + 60z —36 = 0, the roots are 1, 2, 3, and—6; sup- p ng therefore the power x3, which is want- e; the equation becomes 2+ + 023— 23x? + 6 — 36 — 0. hen, according to what is stated above, +2 +3—6 0 ( sign changed 241.342.3—1.6—2.6-—3.6=— -2.3—1.2.6—2.3.6—1.3.6=—60 Sign changed 2,0 . Piha, 6 ae 77. 36 ‘eneration of EQuaTions, is a method in- EQU vented by Harriot for demonstrating the pro- pertics above mentioned: in order to which, he supposes successively wa 6 2 ee oe — 6 oe x—a—0, x—b—0, x—c=0, x—d=0, Ke. then multiplying these quantities continually together, he obtains x —a—O0 x —b—0 a2z— 3h +ab—0 x —e “Q ab xi— b sacl —abe—0 e§ be. x —d—0 ab 7) ac abe b ad acd pad d bd abd ed From which it is obvious, that the co-effi- cient of the second term is equal to the sum of all the roots, having their signs changed; the co-efficient of the third term is equal to the sum of the product of every two roots; the co-efficient of the fourth term is equal to the product of every three, with their signs chang- ed; and soon. But this method is objected to by modern authors, because it only proves that an equation may be generated, the co- efficients of which shall have the properties aboye-stated; and not that every equation whatever has necessarily these properties. It is, however, very ingenious, and throws con- siderable light upon the nature of equations. Lecroix and most other modern algebraists have pursued a different kind of proof, thus: having demonstrated first that every equation has one root, as for example, the equation g* axe—'! + 62°—* + &. p — 0 let this root, or value of x, be represented by k; then ke + ak*—' + bh"-? &e.p = 0 and consequently, (a* — k") + a(a*—1— h— ) + b (x*—2? — k"—9) + &e. = 0. Now, since the difference of any two equal powers is divisible by the difference of their roots, it follows that this whole equation is divisible by x —. And since the quantity which has been subtracted from the first equa- tion is equal to 0, it follows that this also is divisible by z —; which necessarily reduces it to an equation of the next inferior degree ; then this equation having also a root, which may be represented by 2’, it may be reduced by division to another still lower, and which will still have one root, and so on, till it be reduced to a simple equation. Every equa- tion haying therefore for its successive divi- sors, x —k, «x — kh’, x — kh’, &e. it follows, conversely, that every equation is made up of those factors, and consequently the above pro- perties have place in all cases, On this sub- EQU ject the reader may advantageously consult Bonnycastle’s “‘ Algebra,” vol. ii. Reduction of Equations, is of two kinds ; viz. first, the reduction of them from a higher to a lower dimension; and, second, the reduc- tion of them to some particular form, to pre- pare them for solution. The former of these cases is more commonly called the Depression of Equations, which see ; and the latter usually consists in exterminating the second term of the equation, this being the most eligible form for solution. See EXTERMINATION. Some authors also use the term Reduction of Equa- tions, for what is more usually and properly called the Solution of Equations. Solution of Eauations, is the method of finding their roots; which, however, can only be done in a direct manner, for the first four degrees, viz. in Simple, Quadratic, Cubic, and Biquadratic Equations; and the several me- thods of procedure, in each of these, is given under the respective articles. Equations that exceed the fourth degree cannot be solved by any direct rule (except in a few partial cases, in which there are certain relations either be- tween the roots or co-eflicients), although the subject has been investigated by many of the ablest analysts of Europe. We have, there- fore, no means of obtaining the roots in those cases, but by approximation. See APPRoxi- MATION. On the subject of equations the reader may consult, beside the works which treat gene- rally on algebra, Lagrange, “Sur la Reso- lution des Equations Numerique;” Bezout’s “Théorie Général des Equations ;” Waring’s “‘Meditationes Algebraic ;” and Lea “ On the Higher Equations.” Construction of Equations. See Construc- TION. : JQUATIONS in the Differential and Integral Calculus, are of different denominations; as Differential Equations, Equation of Finite Differences, Equations of Partial Differences, &e. Differential Equation, is that which con- tains in it certain differential quantities; and it is said to be of the first, second, third, &e. order, according as it involves the first, second, third, &c. differential. Thus the equation 3a*dx—2axdx + aydx—3y*dy +axdy=0 is a differential equation of the first order; and the solution of it requires that such func- tions of x any y may be found, which, when combined with their differentials, will render the whole expression equal to zero. Favation of Finite Differences, in the Theory of Finite Differences or Increments, is that into which the finite differences of the variable quantities of any function enter. Thus, taking Az to denote the finite incre- ment of any variable 2, and Ay that of any variable y; then Payt+ arcay + axvay + &c. =0 is an equation of finite differences. See INCRE- MENTS. EQU Eeuation of Partial Differences, is 1) which has place between the differential any function, and the differentials of the» riables on which it depends, combined w the variables themselves, and with or withe constant quantities. ‘Thus z being a funeti of a and y ganas ted pei dx dy is an equation of partial differences ; as is aj dz dz — +b— — amet ” da t dy *Y which are the simplest form of these equatio See PartiaL Differences. a Equation of a Curve, in Analysis, is” equation showing the nature of a curve, expressing the relation between any abse and its corresponding ordinate, or the relati of their fluxions, &c. Suppose we wished to deduce a gene equation to a circle; that, for example, 4 scribed from the centre C, and with a rad equal to the line CB. The rae ; Ay points in the circumference of this circle are distinguished A from all other points in the MC same plane, by being at a distance from the centre C equal to CB; and, consequently, where’ we assume the point P on that curve, } right lines CM and PM willbe sides of a rig angled triangle, whose hypothenuse CP equal to CB. Making, therefore, CM =! PM —y, CB—Tr, we have x? -— y* Sa whence we find y = ¥ (7? — 2’), an equat| which shows that when x or C Mis known | can find by computation, and without its | ing necessary to construct the figure y or P| or, at least, the relation of that line with - radius. ‘Taking, for instance, x = tr, ity be fl PO? SS Sr") ate ae ae It is easy to conceive that we may dedi from the same original expression the li PM, corresponding to all assumed points the line CB, comprised between C an The equation y = /(r? — 2°), proves also, well as the geometrical description of thes cle, that this curve cannot extend beyond for in order that the point M may be beyd h;, it is requisite that 2 should be greater fl] CB, or thany; in which case it would i come imaginary. ae authors referred to under that article. —- Equation of Payments, in Arithmetic, is’ finding the time to pay at once several des due at different times, and bearing no inter! till after the time of payment, so that no Ii shall be sustained by either party. aa The rule commonly given for this purposis as follows: Multiply each sum by the time! which it is due; then divide the sum of #¢ products by the sui of the payments, and @ quotient will be the time required. oy ce = EQU Thus, for example, A owes B £190. to be id as follows; viz. £50. at 6 months, €60. 7 months, and £80. at 10 months; what is e equated time at which the whole ought to paid, that no loss may arise cither to debtor creditor? By the rule /—~60 x 6 = 300 60x 7= 420 , 80 x 10 = 800 190 )1520(8 months equat. time. — 1520 i This rule, however, is founded on a suppo- ‘jon that the interest of the several debts nich are payable before the equated time, m their terms to that time, ought to be equal ) the sum of the interests of the debts pay- le after the equated time, from that time to ‘eir terms respectively; which, however, is t correct, as itis only the discount, and not 2 interest in the latter sums. In most cases, wever, that occur in business, the error is trifling that it will probably always be made e of, as being by far the most eligible and peditious method. ‘The true equated time fur two payments is pressed by the following formula: Let p and p’ be sums due, at the end of the ne n,n’, respectively; let also r represent the ie of interest, and x the required time of yment of the whole. — zx the true This formula is found on the principle of owing only simple interest. If compound serest be allowed it is more simple, being as lows: Let arepresent the sum of the several debts, d 6 the sum of their present values, com- ted as in the case of compound interest ; Jo r the interest on £1. for a year, and z the uated time required; then . __ log. a — log. b 2 log. (1 + 7) That is, from the logarithm of the sum of the veral debts, subtract the logarithm of the m of their present values, and divide the mainder by the logarithm of the amount of i. fora year, and the quotient will be the nated time. For an investigation of these formule, see tily’s *‘ Doctrine of Interest and Annuities;” )which this, and all other cases relating to isse important subjects, are treated of with 12 greatest perspicuity and ability: see also > bimini 8vo. * Arithmetic.” Equation, in Astronomy, is a term used to (press the correction or quantity to be add- { to, or subtracted from, the mean position 4a heavenly body, to obtain the true posi- EQU tion; it also, in a more general sense, implies the correction arising trom any erroneous sup- position whatever. Thus, for instance, the time of noon, as determined by taking equal altitudes of the sun, is first obtained by sup- posing the sun’s declination constant daring the whole interval, which false supposition is corrected by an appropriate equation. Equation to Corresponding Altitudes, is a correction which must be applied to the appa- rent time of noon (found by means of the time elapsed between the instants when the sun had equal altitudes, both before and after noon), in order to ascertain the true time. Tables of equations to corresponding alti- tudes, by the late Mr. Wales, are given in the Nautical Almanac for 1773; also in his traet for finding the longitude by time-keepers. EQuaTion of the Centre, called also Pros- thapheresis, or Total Prosthapheresis, is the dit- ference between the true and mean place of a planet, or the angles made by the true and mean place; or, which amounts to the same, between the mean and equated anomaly. The greatest equation of the sun’s centre may be obtained by finding the sun’s longi- tude, at the times when he is near his mean distances, for then the difference will give the true motion for that interval of time; next find the sun’s mean motion for the same interval of time; then half the difference between the true and mean motions will show the greatest equation of the centre. When the mean anomaly and eccentricity of an orbit are given, the equation of the cen tre may be readily obtained by the following admirable rule, given by Simpson, p. 47 of his ** Essays ;” viz. as radius to the cosine of the given anomaly, so is five-fourths of the eccen- tricity of the orbit to a fourth number; which number add to half the greater axis, if the anomaly be less than 90° or more than 270°, otherwise subtract from the same. 'Then say, as the sum or remainder is to double the eccentricity, so is the logarithmic sine of the given anomaly to the sine of a first arch, from three times which sine subtract double radius, the remainder will be the sine of a second arch, whose one-third. part, taken from the former, leaves the equation sought, Euler has particularly considered this sub- ject in Mem. de l’Acad. de Berlin, tom. ii. p. 225, seq. where he solves the following pro blems: 1. To find the true and mean anomaly cor responding to the planet’s mean distance from the sun; that is, when the planct is m the ex- tremity of the conjugate axis of its orbit. 2. The eccentricity of a planet being given, to find the eccentric anomaly corresponding to the greatest equation, Poy. 3. The eccentricity being given, to find the mean anomaly corresponding to the greatest equation. 4. From the same data to find the true anomaly corresponding to this equation. 5. From the same data to find the greatest equation, ‘ EQU 6. The greatest equation being given, to find the eccentricity. Euler observes, that this problem is very difficult, and that it can only be solved by approximation and ten- tatively in the manner he mentions; but, if the eccentricity be not great, it may be then found directly from the greatest equation. Thus, if the greatest equation = m, and the eccentricity = n; then 1K. 8 GOD: kr -ndbit st m= 2n + sh haga” +, ke. Whence, by reversion, may 587 5 _ moe ee es MN i 23 23.9 Where the greatest equation m must be ex- pressed in parts of the radius, which may be done by converting the angle m into seconds, and adding 4°6855749 to the logarithin of the resulting number, this will be the logarithm of the number m. The mean anomaly to which this greatest equation corresponds, will be ; ERE: ONE: BOE PNT ENA x = 90 i Bg Soa a bhi , &e. Whence, if to 90° we add 2 of the greatest equation, we shall sufficiently approximate to this mean anomaly. The same author gives us a table, in which may be found the greatest equations, the ex- centric and mean anomalies corresponding to these greatest equations for every hundredth part of unity, which he supposes equal to the greatest eccentricity, or when the distance of the foci and the transverse axis become infi- nite. The last column of his table also gives us the logarithm of that distance of the planet from the sun where its equation is greatest. By means of which, any eccentricity being given, the greatest corresponding equation may be found by interpolation. But the prin- cipal use of the table is to determine the eccen- tricity when the greatest equation is known: and without this help Euler thinks the pro- blem cannot be resolved. Equation of Time, in Astronomy, denotes the difference between mean and apparent time, or the reduction of the apparent unequal time, or motion of the sun or a planet, to equable and mean time, or motion. If the earth had only a diurnal motion, without an annual, any given meridian would revolve from the sun to the sun again, in the same space of time as from any star to the same star again; because the sun would never change his place with respect to the stars. But, as the earth advances almost a degree eastward in its orbit in the time that it turns eastward round its axis, whatever star passes over the meridian on any day with the sun, will pass over the same meridian on the next day when the sun is almost a degree short of it; that is, 3 minutes 56 seconds sooner. If the year contained only 360 days, as the eclip- tic does 360 degrees, the sun’s apparent place, so far as his motion is equable, would change a degree every day; and then the sidereal 2. - EQU , days would be four minutes shorter than { solar. 4 The equation of time is calculated by traci out the effects of three combined causes; t obliquity of the ecliptic, the sun’s unequ apparent motion therein, and the precessi of the equinoctial points. In consequence the first of these, in the first and third qu rants of the ecliptic from Aries; that is, t - tween Aries and Cancer, and between Li and Capricorn, the right ascension being It than the mean longitude, the point of rig ascension is to the west, and therefore t apparent noon precedes the mean noon; b in the second and fourth quadrants, name between Cancer and Libra, or Capricorn a Aries, the right ascension being greater th the longitude, or the mean motion taken the equator, the mean noon is westward, a therefore precedes the apparent noon. Bi even if the plane of the ecliptic coincided wy} that of the equator, there would be a correcti necessary ; for the apparent annual motion the sun being not quite uniform, a longer ¢ would be described in some days than othe that is, since the right ascension and lon tude would in this case be the same, the da) increments of right ascension would be t equal. And besides these, the third cat ought to be attended to, though it is toof quently disregarded. The equation of time in fact, equal to the difference of the sw true right ascension, and his mean longitu corrected by the equation of the equinoxes. right ascension. This was, we believe, fi shown by Dr. Maskelyne. . The mean and apparent solar days are ne} equal, except when the sun’s daily motion right ascension is 59’ 8”; this is nearly 1 case about April 15, June 15, September and December 24: on these days the eq tion is nothing, or nearly so; it is at the gre est about November 1, when it is 16/14”, _ Tables of the equation of time, as comput by Dr. Maskelyne’s rule for the noon of ea day, are given in the Nautical and o Almanacs. Sometimes they are computed. every degree of the sun’s place in the eclipt as in Gregory’s and Ewing’s “ Astron.” tables of this latter kind will not answer act rately for many years, because of the prec sion, the motion of the sun’s apogee, &e, that a frequent revision of the calcula will be necessary; and for which reason th are omitted in the present article. 4 EQUATOR, in Astronomy and Geograp. is a great circle of the sphere, equally dist from the two poles of the world. It is | the equator, because when the sun is in t circle the days and nights are equal in” parts of the world. Whence also it is call Equnoctial; and when drawn on maps a planispheres, it is called the Equinoctial Li or simply the Zine. ya Since every point of the equator is equa distant from the poles of the world, it follo that the equator divides the sphere into t equal hemispheres; the one of which towa' ; EQU the north pole is called the northern, and the other the southern hemisphere. As equal or mean time is estimated by the passage of ares of the equator over the meri- dian, it frequently becomes necessary to con- vert parts of the equator into time, and the converse; which is performed by the follow- ing analogy; viz. As 15°: 1 hour :: any are of the equator : the time it has been in passing. Or, conversely, 1 hour : 15° ::; any given time : the arc of the equator. From this circle are reckoned the latitude of places, both north and south, in degrees of the meridian. » EQUATORIAL, Universal, or Portable Ob- servatory, isan instrument intended to answer anumber of useful purposes in practical astro- nomy, independent of any particular observa- tory. li may be employed in any steady room or place, and it performs most of the useful problems in the science. See Plate LX. fig, 2. The principal parts of this instrument are, 1. The azimuth or horizontal circle, which represents the horizon of the place, and moves on a long axis called the vertical axis. 2. The equatorial, or hour-circle, representing the aquator, placed at right angles to the polar yxis, or the axis of the earth upon whieh it moves. 3. The semicircle of declination on which the telescope is placed, and moying on the axis of declination, or the axis of motion of the line of collimation. The peculiar uses of this equatorial are, l. ‘To find the meridian by one observation mily. Vor this purpose, elevate the equatorial sircle to the co-latitude of the place, and set he declination-semicircle to the sun’s decli- 1ation for the day and hour of the day requir- “d; then move the azimuth and hour-circles yoth at the same time, either in the same or sontrary direction, till you bring the centre of he cross hairs in the telescope exactly to ‘over the centre of the sun; when that is done, he index of the hour-circle will give the ap- »arent or solar time at the instant of observa- ion; and thus the time is gained, though the un be at a distance from the meridian; then urn the hour-circle till the index points pre- isely at twelve o’clock, and lower the tele- cope to the horizon, in order to observe some oint there in the centre of the glass, and that ‘oint is the meridian mark found by one ob- eryation only; the best time for this opera- ion is three hours before or three hours after welve at noon. 2, 'To point the telescope on a star, though jot on the meridian, in full day-light. Having levated the equatorial circle to the co-latitude f the place, and set the declination-semicircle 0 the star’s declination, moye the index of the ‘our-circle till it shall point to the precise ime at which the star is then distant from the aeridian, found in tables of the right ascen- ion of the stars, and the star will then appear a the glass. Besides these uses peculiar to his strument, it is also applicable to all the yoeppses to which the principal astronomical =e EQU instruments, viz. a transit, a quadrant, and an equal altitude instrument, are applied. For a full description of an equatorial in- verted by Mr. Short, see Phil. Trans. vol. xlvi.; and for one invented by Mr. Nairne, vol. lxi.; see also Hutton’s Math. Dict., Rees’s Cyclo- pedia, and Vince’s “ Practical Astron.” EQUIANGULAR (of equus, equal, and an- gulus, angle), in Geometry, is applied to ficures whose angies. are all equal; such are the square, and all regular figures. All equilateral triangles are also equiangular. An equilateral figure, inscribed in a circle, is always equiangular; but an equiangular tigure, inscribed in a circle, is not always equi- lateral, except when it has an odd number of sides. If the number of the sides be even, then they may be either all equal, or else half of them will always be equal to each other, and the other half to each otaer; the equals being placed alternately. See Hutton’s “Math. Mise.” p. 272. EQUIANGULAR, is also applied to any two figures of the same kind, when each angle of the one is equal to a corresponding angle in the other, whether each figure, separately con- sidered, be an equiangular figure or not; that is, having all its angles equal to each other. Thus two triangles are equiangular to each other, if one angle in each be of 30°, a second angle in each of 50°, and the third angle of each equal to 100°. Equiangulartriangles, according tothe above acceptation of the term, have not their like sides necessarily equal, but proportional to each other ; and such triangles are always similar lo each other. EQUICRURAL Triangle, is what we more usually call an isosceles triangle. The term is derived from eguus, equal, and crura legs, equal legs. EQUICULUS, EeuuLeus, or Eeuus Minor, in Astronomy, a constellation of the northern hemisphere. See EquuLeus. EQUIDIFFERENT, in Arithmetic, is when in a series of quantities there is the same dif- ference between the first and second, as be- tween the second and third, third and fourth, &e.; and they are then said to be continually equidifferent; but, if in a series of quantities there be only the same difference between the first and second, as between the third and fourth, fifth and sixth, &c.; then they are said to be discreetly equidifferent. .'Thus 3, 6, 7, and 10 are discreetly equidiiierent; and 3, 6, and 9 continually equidifferent. EQUIDISTANT, in, Geometry, a term of relation, between two things which are every - where at the same, or at equal distances from each other. EauipistaNnt Ordinates, Method of, is an approximation towards the area of a figure bounded'by a right line and curve. Dr. Hut- ton, in his “ Mensuration,” gives the follow- ing formula for finding the area of figures by this method, viz. Having measured any odd number of equi- distant ordinates, put the sum of the-first and . a + EQU last = A, the sum of the second, fourth, sixth, &e. = B, the sum of all the others = C,and the common distance of the ordinates = D; then A + 4B + 2C 3 And the same formula is applicable to the mensuration of solids, by taking the sections instead of the ordinates. See Dr. Hutton’s ‘* Mensuration,” p. 374. EQUILATERAL (from equus, and Tatus, side), is applied to those figures whose sides are all equal. ‘Thus, an equilateral triangle is that whose sides are all equal. ‘To find the area of an equilateral triangle, multiply the square of the side by £3. EQuivateraL Hyperbola. See HYPERBOLA. EQUILIBRIUM (from equus, and libro, to poise), Equipoised, in Mechanics, means an equality ef forces acting in opposite direc- tions, whereby the body acted upon remains at rest, or in equlibrio; in which state the jeast additional force being applied, on cither side, motion will ensue. A body in motion is also said to be in equi- librio when the power producing the motion, and the force whereby it is resisted, are so ad- justed that the motion may be uniform. HQuitLisrium of the Lever. See LEVER. EQUIMULTIPLEBS, the products arising from the multiplication of any two or more primitive quantities, by the same number or quantity. Thus 3a, 36 na, nb ma,mb Hquimultiples of any quantities have the same ratio as the quantities themselves; thus aro Rw na nb, a:b::ma:mb. EQUINOCTIAL, in Astronomy, a great circle ef the sphere, under which the equator noves in its diurnal motion. The equinoctial is conceived by supposing a semi-diameter of the sphere produced from a point of the equator, and there, by the rota- tion of the sphere about its axis, describing a circle on the immovable surface of the heavens. The poles of this circle are the poles of the world. The sphere is divided by it into two equal parts, the northern and southern. ¢ intersects the horizon of any place in the east and west points; and at the meridian, its cleyation above the horizon is equal to the co- latitude of the place. Whenever the sun, in his progress through the ecliptic, comes to this circle, it makes equal days and nights in all parts of the globe ; as he then rises due east, and sets due west, which he never does at any other time of the year. And hence the denomination from aquus and nox, night, quia acquat diem noctt. All the stars that are under this circle, or that have no declination, rise due east, and set due west. The equinoctial, then, is the circle which the sun describes, or appears to describe, at the time of the equinoxes: that is, when the length of the day is every where equal to that of night, which happens twice a year. From x D = area nearly. ' are equimultiples of a and b. ERA this circle is the declination in the heavens, or latitude of places on the earth, counted in’ degrees of the meridian. Upon this circle is reckoned the longitude 180° west, and 180° east; and in all 360°. Hence 1° of longitude’ answers to 4’ of time, 15’ to I’ of time, and 1” to 4” of time, &e. ; The shadows of those who live under this? circle are east to the southward for one half of the year, and to the northward during the” other half; and twice in a year, viz. at the, equinoxes, the sun at noon casts no shadow, being in their zenith. Eaquinocria Colure, is that passing through’ the equinoctial points. See CoLURE. + Eeauinoctiagt Dial, is one whose plane is- parallel to the equinoetial. The properties or principles of this dial are, * 1. The hour lines are all equally distant” from one another, quite round the circumfe= rence of a circle; and the style is a straight” pin, or wire, set up in the centre of the circle, perpendicular to the plane of the dial. : 2. The sun shines upon the upper part of this dial-plane from the 21st of March to the | 23d of September, and upon the under part of the plane the other half of the year. 4 EquinoctTiaL Points, are the two points wherein the equator and ecliptic intersect each other: the one, being in the first point of Aries, is called the vernal point; and the other, in the first point of Libra, the autumnal "point. t EQUINOX, in Astronomy, the time when - the sun enters one of the equinoctial points. | The equinoxes happen when the sun is in the” equinoctial circle; when of consequence the days are equal to the nights throughout the world, which is the case twice a year, viz. about the 2ist of March and the 23d of Sep- tember; the first of which is the vernal, and the second the autumnal equinox. For the’ time the sunis passing from one equinox te the other, see the article EarTu. Jt is found by observation, that the equinoc- tial points, and all the other points of the » ecliptic, are continually moving backward, or in antecedentia; that is, westward; which retro= grade motion of the equinoctial points, is what is called the precession of the equinoxes. See PRECESSION. ba EQUINUS Barbatus, a kind of comet. EQUULEUS, Eautcutus, and Equus Miz nor, the Horse’s Head, one of the northern” constellations. See CONSTELLATION. * ERA. See Aira. 7 ERATOSTHENES, a Greek of Cyrene, who was a librarian of Alexandria, under King Euergetes, son of Ptolemy Philadelphus. He died 194 years B.C. He was called the- cosmographer, because he was the first who discovered the method of measuring the bulk ! and circumference of the earth. What remains to us of his writings appeared from the Ox- — ford press, in 1672, 8yo. oy The measurement of the earth by Eratos- thenes, so much admired in its time, as a pro-— digy of human sagacity, has been transmitted — £ g ERR to us by Cleomedes: Cyel. theor. book i. chap. 10. Eratosthenes was informed, that at the time of the summer solstice the sun at noon was vertical to the city of Syene, situate on the borders of Ethiopia, under the tropic of Cancer. : EVO ‘of the curve PCR, and PD or HC the ab- 'sciss; whence it follows, that in order to de- termine the nature of the evolute we must ‘find PH and HC. Andsince PH =ME— ML, and HC= AD — AP, the following lines are to be found by the nature of the curve; viz. ME, AP, and AD, or its equal AL + LD, or AL + EC; and hence is derived the follow- ing rule: Having determined the value of CM, from cone of the expressions for the radius of curya- ture, make x, or y¥ = 0, which will determine AP the radius of curvature at the vertex, then CM — AP = CP, the length of the evolute. .Then find EL and CE from the ‘nature of the curve, and the equation of the evolute will be determined. ‘Phe fluxional expression for LE and CE, in terms of the fluxion of the absciss, of the ‘ordinate and curve, may be determined as ‘follows: Put absciss of the inyolute AL = 2, ordinate ML = y, and AM = 2, then from the nature of the radius of curvature, the chord baie Say 3 WE SS | —ry —y A therefore LE = ME—ML= — —Y¥. —1 Also by similar triangles MNS, CME MN:NS:: CM: CH, or ooh -CE— 7. Sey <2 YZ ips f, “oy ~ -~ ~ ) Or ife = 1; then CE = Sy ts In order now to illustrate this formula, Jet it be proposed to find the evolute of the com- “mon parabola. Let the preceding figure AMT represent a lad and AP = 2 az (a parabola, then CM = 5 (a being the parameter), therefore C P = | } (4 Gx + aye * = the length of the evolute. Q at 2 Mus we = 2242+", 422 — | ag. | Ax ak 3 i aes? consequently LE or PH = ’ Agt + ant (ME —ML === —y= by ar : ) 3 5 3 4x iz —— — at x = : , the ordinate of ar ar - the evolute. Again AD =AL°*+ LD =a2+ y 2 4x2 +a Ax . EC 3 to Land ar — = 3a +44, and consequently EVO HC —=AD—AP = 82 the absciss of the curve, Hence the square of the ordinate varies as the cube of the absciss, and therefore the evolute is the cubical parabola. Dealtry’s “ Tluxions.” : In like manner is found the evolute of any other curve. The evolute of the cycloid is another equal cycloid, which property was first discovered by Huygens, who by this means contrived to make a pendulum vibrate in an are of cycloid, by placing it between two cycloidal cheeks, and thus rendered the vibra- tions isochronous. See CycLoip and PEN- DULUM. The evolute of a spiral, or indeed of any other curve, may be described by finding the radii of curvature at several points in the inyo- lute; for then we shall have as many points in the evolute, through which if a curve line be drawn, it will be the evolute sought. Wolf, “ Elem. Math.” tom. i. p. 524, seq. or the “ Infinum. Petites” of M. le Marquis de VHopital; Simpson’s “ Fluxions,” vol. 1. p. 71, &c.; and Rowe’s “ Fluxions,” 3d edit. 1767, ch. vi. and vii. p. 103-132. Since the radius of an evolute is either equal to an are of an evolute, or exceeds it by some quantity, all the ares of evolutes may be rectified geo- metrically whose radii may be exhibited by scometrical constructions; whence we see why an are of a cycloid is double its chord ; the radius of the evolute being double the same, and the evolute of a cycloid being itself 2 cycloid, equal and similar to the involute, M. Varignon has applied the doctrine of the radius of the evolute to that of central forces ; so that having the radius of the evolute of any curve, one may find the value of the central force of a body ; which, moving in that curve, is found in the same point where the radius terminates; or reciprocally having the central force given, the radius of the evolute may be determined. Hist. de Acad. Roy. des Sci- ences, an. 1706. The variation of curvature of the line described by the evolution of a curve, is measured by the ratio of the radius of curvature of the evolute, to the radius of curvature of the line described by the evolu- tion. See Maclaurin’s “ Fluxions,” art. 402, prop. 36. Evotute, Imperfect. M. Reaumur has given a new kind of evolute, under this de- nomination. Hitherto mathematicians had only considered the perpendiculars let fall on the points of the convex side of the curve; if other lines not perpendicular were drawn upon the same points, provided they were alldrawn under the same angle, the effect would be the same; that is, the oblique lines would all in- tersect within the curve, and by their inter- sections form the infinitely small side of a new curve, whereof they would be so many tangents. This curve would be a sort of evolute, and would have its radii but an imperfect evo- lute, since the radii are not perpendicular to EXC the first curve. Hist. de l’Acad. &c. an. 1709. EvoLuTion, in Arithmetic and Algebra, is - the extraction of roots, being thus opposed to involution, which is the raising of powers. See EX FRACTION. EvoLution, in Geometry, the opening or unfolding of a curve, and making it describe an evolent. See EvoLute and INVOLUTE. EUTOCIUS, an eminent mathematician, who lived at the time of the decline of the sci- ences in Greece, was a native of Ascalon, in Palestine, and a disciple of Isodorus, one of the celebrated architects employed by the emperor Justinian. He probably flourished about the commencement of the sixth cen- tury, though we have no particulars respect- ing his life; but his works reflect much honour on his memory. He wrote elaborate and perspicuous “ Commentaries on the Books of Archimedes concerning the Sphere and Cy- linder ;”.and also on the first four books of the conics of “ Apollonius Pergeeus.” These commentaries have not only elucidated many difficult passages in those profound writers, but have tended to throw light on the history of mathematics. There have been many edi- tions of them, but the most magnificent was that in the edition of the works of Archi- medes, printed at Oxford in folio, in-the year 1792, which was prepared for the press by Torelli of Verona; and that in Dr. Halley’s edition of the eight books of Apollonius, pub- lished at Oxford in 1710. See Montucla’s Histoire des Math. tom. i. EXAGON. See Hexacon. EXCENTRIC. See Eccentric. EXCENTRICITY. See Eccentricity. EXCESS, Spherical. See SpHerican Ex- cess. EXCHANGE, in Arithmetic, is the reduc- tion of different coins or any denominations of money, whether there be real coins an- swering to them or not, from one to enother: or the metliod of finding how many of one species, or denomination, are equal in value ‘to a given number of another; in order to which it is necessary to know the value of the coins and moneys of account of different coun- tries, and their proportion to each other ac- cording to the settled rate of exchange. The several operations in this case are only dif ferent applications of the rule of three, which are explained in Bonnycastle’s, Hutton’s, and most books of arithmetic. Arbitration, or Comparison of Excuaner, determines the method of remitting to, or drawing upon, foreign places, in such a man- ner as shall be most advantageous to the merchant, Arbitration, is either simple or compound. Simple Arbitration, respects three places only. Here, by comparing the par of arbi- tration between a first and second place, and between the tirst and a third, the rate between the second and third is discovered; from whence a person can judge how to remit or ‘made in Algebra, it may be considered mere. EXH | draw to the most advantage, and to determin, what that advantage is. | Compound Arbitration, respects the cases tn which the exchanges among three, four, 0 more places are concerned. A person whi knows at what rate he can draw or remit di. rectly, and also has advice of the course 9 exchange in foreign parts, may trace ont ¢ path for circulating his money, through | or fewer of such places, and also in such order as to make a benefit of kis skill and credit | and in this lies the great art of such negocia, tions. See Hutton’s, Boniyeastle’s, or K eith’) “ Arithmetic.” | EXCLUSION, Method of, is a denomina tion given by Frenicle, to certain modes of in vestigation that he invented, which withon the assistance of algebra enabled him to answel with great readiness many problems relatin to the powers and property of numbers, whie were agitated in his time between Kermat Descartes, Wallis, and others. An explana: tion of it is given in the “ Memoirs of the Academy of Sciences” for the year 1693; bui since the great improvements that have bee, ly as a matter of curiosity, though it was thought of great importance by Fermat anc Descartes at that time. EXCURSION, in Astronomy. See Evow- GATION, . EXEGESIS, an ancient term for findin the roots of an equation. EXHALATION, the fume or steam is: suing from a body, and diffusing itself in the atmosphere, : EXHAUSTED Receiver, the vessel which is exhausted of its air in experiments witl the air-pump. B HE XHAUSTIONS, in Geometry. Method ol exhaustions is a way of proving the equality of two magnitudes, by a reductio ad absurdum, showing, that ifone be supposed either greater or Jess than the other, there will arise a con- tradiction. The method of exhaustions was of frequent use among the ancient mathema ticians, as Euclid, Archimedes, &e. It is founded on what the former says in his tenth book, viz. that those quantities whose differ ence is less than any assignable quantity, are equal; for if they were unequal, howeve small the difference might be, yet it might be: so multiplied, as to become greater than cither of them, and if not so, then it is really nothing, on This he assumes in the proof of prop. 1. book x. which imports, that if from the greater. of two quantities, you take-more than its half, and from the remainder more than its half, and so on continually, there will, at length, remain a quantity less than either of those! proposed. On this foundation it is demonstrated, that) if a regular polygon of infinite sides be in-| scribed in or circumscribed about a circle, the Space, which is the difference between the cirele and the polygon, will, by degrees, be’ | ; } j EXP tite exhausted, and the polygon become Oa to the circle. ‘Upon the method of exhaustions depends fe method of indivisibles introduced by Ca- Herius, which is but a shorter way of ex- ‘essing exhaustions; as also Wallis’s “ Arith- letic of Infinites,” which is a farther improve- ‘ent of the method of indivisibles; and hence, ‘so, the methods of increments, differentials, ‘ixions, and infinite series. (EXPANSION, in Physics, is the enlarge- ‘ent or increase in the bulk of bodies, in ‘usequence of a change in their temperature. ‘isis one of the most general effects of caloric ‘ing common to all bodies whatever, whe- ‘er solid or fluid. The expansion of solid hdies is shown by the PyroMeErTeER, and the (pansion of fluids by the ‘THERMOMETER. Je these articles. The expansion of fluids vary very consider- ly; but, in general, the denser the fluid the ‘ss the expansion: for instance, water ex- nds more than mercury, and spirits of wine bore than water; and commonly the greater ie degree of heat the greater is the expan- on; but this is not universal, for there are éses in which expansion is produced, not by increase, but by a diminution of tempera- ire. ‘Water furnishes us with the most remark- ‘le instance of this kind. Its maximum of (nsity corresponds with 42°.5 of Fahrenheit’s isrmometer, as has been lately ascertained | Mr. Dalton. If it is cooled down below 4°.5, it undergoes an expansion for every (gree of temperature which it loses; and at ¢° the expansion amounts, according to Mr. ulton, to -1, of the whole expansion which ‘iter undergoes when heated from 42°.5 to 2°. With this, more recent experiments incide very nearly; for by cooling 100,000 rts in bulk of water from 42°.5 to 32°, they ‘re converted to 100,031 parts. We are also lebted to the ingenuity of Mr. Dalton for the scovery of a very unexpected fact, that the ‘pansion of water is the same for any num- r of degrees above or below the maximum «density. Thus if we heat water ten de- fees above 42°.5, it occupies precisely the me bulk as it does when cooled down ten grees below 42°.5. Therefore the density ¢ water at 32° and at 53° is precisely the ime. Mr. Dalton succeeded in cooling jtter down to the temperature of 5° without fezing, or 37°.5 below the maximum point - density; and during the whole of that Inge, its bulk precisely corresponded with ‘2 bulk of water the same number of degrees ove 42°.5. ‘The prodigious force with which water ex- pes in the act of freezing has been long own to philosophers. Glass bottles filled )th water are commonly broken in pieces jen the water freezes. The Florentine aca- {micians burst a brass globe whose cavity ts an inch in diameter, by filling it with ap and freezing it. The force necessary ’ this effect was calculated by Muschen- Ex? brock at 27,720 Ibs. But the most complete set of experiments on the expansive force of freezing water, are those made by Major Williams at Quebec, and published in the second volume of the ‘ Edinburgh Transac- tions.” ‘This expansion has been explained, by supposing it the consequence of a tendency which water, in consolidating, is observed to have to arrange its particles in one determi- nate manner, so as to form prismatic erystals, crossing each other at angles of 60° and 120°. The force with which they arrange themselves in this manner ‘must be enormous, since it enables small quantities of water to overcome so great mechanical pressures. Various me~ thods have heen tried to ascertain the specific gravity of ice at 32°; that which succeeded best was to dilute spirits of wine with water till a mass of solid ice put into it remained in any part of the liquid without either sinking or rising. ‘The specific gravity of such a li- quid is 0.92, which of course is the specific gravity of ice, supposing the specific gravity of water at 60° to be 1. Thisis an expansion much greater than water experiences even when heated to 212°. Wesee from this, that water, when converted into ice, no longer ob- serves that equable expansion measured by Mr. Dalton, but undergoes a very rapid and considerable augmentation of bulk. EXPECTATION, in the Doctrine of Chances, is the value of any prospect of prize or property depending upon the happening of some uncertain event, the value of which in all cases is equal to the whole sum multiplied by the probability, that the event on which it depends may happen. See Demoivre’s Doc- trine of CHANCES. EXPECTATION, in the Doctrine of Life An- nuities, denotes that particular number of years which a life of a given age has an equal chance of enjoying, or the time which a person of a given age may justly expect to live. But Simpson has shown that this period does not coincide with that which writers on annuities call the expectation of life, except on a sup- position of a uniform decrease in the proba bilities of life; and Dr. Price adds, that even on this supposition it does not coincide with what is called the expectation of life, in any case of joint lives. See SURVIVORSHIP. The expectation of life coincides with the sums of-the present probabilities, that any single or joint life shall attain to the end of the first, second, third, &c. moments from this time to the end of their possible existence, or in case of survivorships, with the sum of the probabilities, that there shall be a survivor at the expiration of those periods. T'rom which principles Dr. Price has shown how Demoivre deduced his rules for determining the expectation of any given life. See Price’s “ Observations on Reversionary Payments.” Simpson’s table of the Expectation of Life, given in his “ Select Exercises,” p. 225, is as follows: Ex P TABLE Of the Expectations of Life in London. Age. | Expectation. Age. Expectation, 1 27°0 41 19:2 2 32°0 42 18'8 3 340 43 18'5 4 35'6 44 18:1 oO 36°0 45 17°8 6 56°0 46 17°4 7 35'8 47 17-0 8 35'6 48 16°7 9 35°2 49 16°3 10 34'8 50 16:0 ll 34:3 dl 15'6 12 33°7 52 15'2 13 33°1 53 14:9 14 32'5 54 14:5 16.5%) (4.319 55 14-2 16 313 56 13'8 17 30°7 57 13°4 18 30°1 58 13°1 19 29°5 59 12°7 20 28'9 60 12°4 21 28°3 61 12°0 22 27°79 62 11'6 23 27°2 63 11°2 24 26'6 64 10°8 25 26°1 65 10°5 26 25°6 66 10°1 27 25°1 67 9°8 28 24:6 68 9°4 29 24°] 69 9°] 30 23°6 70 8'8 31 23°1 71 8°4 32 22°7 72 81 33 22°3 73 7'8 34 21°9 74 vies) 35 21°5 75 72 36 211 76 6'8 37 20°7 77 6°4 38 20°3 78 6:0 39 19°9 79 o°5 40 . 19°6 80 a0 From this table the expectation of life, at any age, is found on inspection; thus, a person of 20 years of age has an expectation of 28°9 years, and in the same manner may be found the expectation at any other age. On this subject the reader may consult Demoivre’s “ Doctrine of Chances,” p. 288; Simpson’s “ Select Exercises,” p. 253; Dr. Price’s work above quoted, and Baily’s Treatise on “ Life Annuities and Assurances.” EXPERIMENT, in Philosophy, is a trial of the effect or result of certain applications and motions of natura] bodies, in order to dis- cover their natures, laws, relations, &c. EXPERIMENTAL Philosophy, that sys- tem of philosophy which is founded upon the results of various experiments; which thus furnish certain data, that are assumed as the unalterable laws of nature, and on which EXP finally rest every branch of modern philos| phical investigation. di; | EXPLOSION, a violent and sudden e plosion of an elastic fluid, by which it imm diately throws off any obstacle which has tendency to impede its progress; such is t explosion of fired gunpowder. f EXPONENT (from eapono, I express), Arithmetic and Algebra, denotes the numk, which expresses the degree of the power, root, of a quantity; thus 2, 3, », &c. aret exponents of 2d, 3d, nth, &c. powers; and 4, 3, &e. the exponent of the 2d, 3d, nth, &) root. Gi! Thus a?, a3, at, represent the square, cul, and mth power of a. And az, a3, a”, the square root, cube ro and nth root of the same quantities. The notation of powers and roots by meg of exponents, was one of the most import improvements ever made in the sciences arithmetic and algebra; asit is to this circu; stance we owe much of the facility with whi numerous operations are now performed. ‘1]} binomial theorem, the invention of logarith and various other advantages which the nh dern analysts enjoy, owe their origin to 1 mode of denoting powers and roots by ponents. For an account of the first int: duction of this notation, and its success} improvements, see the article ALGEBRA. also POWER. Exponent of a Logarithm. See Intl and LoGARITHM. Exponent of a Ratio, sometimes dence} the quotient arising from dividing the ante} dent by the consequent, thus the exponen} the ratio 1: 2, is 4; others however deli logarithms to be the exponents of ratios, i in this way they are considered by Kep' Halley, Cotes, &c. "Y EXPONENTIAL Calculus, is the met] of finding the fluxions and fluents of expon tial quantities. See FLuxions and FLUEN EXPONENTIAL Curve, is a curve which defined by means of an exponential equati EXPONENTIAL Equation, is any equal into which an exponential quantity enters) aise 8, x = a, OC. | The readiest method of solving exponen} equation is by means of a table of logarith thus: a 1. Let there be proposed the equation a? Taking the logarithms on both sides we hi log. b x log. a = log. b, therefore « = ion ry 2. Let a®” =, to find the value of x. | Assume &* = y, then a¥ = ¢, and bi y log. a = log.c; whence y eae on SD Now, therefore, b? = d, or x = Pes . g. 3. Let (a* —b*)* = a + b,tofinds. | Dividing by a + b, we have ’ | (a+ by (a— by =at+ b, or (a — b)x = (a + bb), EXT yonce a log. (a—b) = (1—2) log. (a +b); eof 2 __ log. (a + 6) je oh: Wr eeaces ~ log. (a — 6) rom which expression the value of x is dlily obtained. | iTence it appears, that if only the exponent ) inknown, the value of that exponent may ‘found by means of a table of logarithms ; jan approximation may be made towards it Shout logarithms, as follows: jet a, = b, assume x = m + 1 ‘ —, m being a {posed the nearest integral value of x, then ; vd 2 L Az eet se = a" x az' = b, or az = a | ck He at) — b', to the x’ power, so shall we have | 1 b'* =a; assume 2’ = m' + — x G. vad, sm 1 as before, b'” x b'2” =a, or b= Re /, prinat b's" — a’) ora’ = 0. roceeding again with this in the same iner, and finding the successive values of n’, m’, &c. we shall have ultimately 1 1 — =m +— m Lae m 1 a ——m + | x f i | ch values of m, m’, m", &c. being known, of x will be found by the formula given his purpose under the article, ConTINUED ctions. ;nother method of approximation is by the _of double position. See APPROXIMATION PosITION. nd for the method of converting such ations into series, see LOGARITHMS. yhen in the expontial expression both power and index are unknown, as in this, - a, then no direct method of solution can iven, at least no such rule has yet been dis- red, and we are therefore obliged to have vurse to the method of Trial and Error, or ible Position, by which means an approxi- e value may be obtained. See Position. 1 exponential equations of the form a* =a, be greater than unity there is only one , or value of x, which will answer the con- ms of the equation, but if @ be less than |y, then there are at least two values which functions of each other; that is, if x be root, and r be then found from the equa- r. ‘rr—i = x; we Shall have rx for another of the same equation. -XPONENTIAL Quantities, are divided into -rent orders; thus, x’ is called the first order, t Be testicved the second order; so on of others. -XPRESSION, in Algebra, is any alge- ‘cal quantity, simple or compound, as 3a, o+t7y, /(4a + b), &e. XTENSION, in Physics, is one of the q ral and essential properties of matter, the EXT extension of a body being the quantity of space which it occupies, the extremities of which limit, or circumscribe, the body. It is otherwise called the magnitude or size of the body. ‘The word extension, however, is com- monly used to denote the surface of a body only without regard to its thickness, . EXTERIOR Angle, that which is formed by producing the sides ofa figure. ‘The sum of all the exterior angles of any right lined plane figure are equal to four right angles. EXTERMINATION, is a term frequently employed by algebraists, to denote any opera- tion which has for its object the taking away any term or quantity, out of an equation or other algebraical expression. ‘Thus when there are given any number of unknown quantities, and as many independent equa- tions, then the values of those quantities are obtained by exterminating all of them except one, the value of which being then deter- mined, those of the other quantities will also become known. The method of performing these operations are very various, and much must necessarily be left to the skill of the analyst, yet the more common methods may be explained, and these will be found to apply in most cases; they are as follows: 1. Find the value of one of the unknown quantities in one of the equations, and substi- tute this value instead of that quantity in the others, which will reduce the number of equa- tions to one less. Then find the value of another unknown quantity in one of the new equations, and substitute its value in the other equations as before, which will again reduce the num- ber of equations to one less. Proceed in this manner till there be left but one equation and one unknown quantity, the value of which may then be found by the common rules, and thence the value of all those that have been exterminated. 2. Find the value of one of the unknown quantities in each of the equations, then by making these several values equal to each other, that quantity whose value was before found will be exterminated; then proceed in the same way with another of those quanti- ties, and so on till there remain only one equation, and one unknown quantity; with which proceed as in the rule ‘above given. 3. A third method of extermination is given under the article ELIMINATION. Let us illustrate the two preceding rules by the example given under that article. ax+by+c=—0 dz +ey+f—0O By the 1st Method. —c— by oe a substitute this in the second equation, and we have —de—bdy +ey + f—0, or acy —bdy = de —af; EXT _ de—af whence Y= f _ bf —eec andconvey «= erin? By the 2d Method. —ce—b Ist Equat. x= at y 24: Equat). i2= aia Whence © ra oes fh or cd + bdy = af + aey therefore (ae—bd)y — ed — af _ed—af i Y= Ge—bd ‘Ny Spo and ee us an 4 the same results as before. See other me- thods of extermination under the article Rr- SOLUTION of Equations. To exterminate any proposed Term out of an Equation. An equation being proposed, any term in that equation may be taken away; but ge- nerally this operation is confined to extermi- nating the second term, in order to prepare the equation for a numerical or literal solution, if it be of that kind which admits of such a solution. This is particularly necessary ii the solution of cubic equations, and some- times in biquadratics, though there are rules which renders it unnecessary in the latter case. The rule for this purpose is as follows: Let 2” = ax"—! + ba®-? + &c. = 0. 1 : ‘ Assume x = z = —4@, and substitute this v1) value instead of it in the equation proposed, so shall the resulting equation be of the form required, viz. the term involving the power, nm — 1 will be exterminated. Let it be pro- posed, for example, to exterminate the se- cond term out of the equation. x3 — 627 +42 +10—0. Assume x 2 + 2, then +a3 —2z3 + 627+122+4 8 —62—- —62?—UMz—A +427 = 4z+ 8 +10 = 10 whence z3 — 82+ 2=0 which is the form required. See Newton’s, Maclaurin’s, and Bonnycastle’s “‘ Algebras.” EXTERNAL Angles, are those formed without a figure, by producing its sides. Vhe external angle of a triangle is equal to the sum of the two internal angles; and in any figure whatever the sum of all the exter- nal angles is equal to four right angles. EX’TRA Constellary Stars, are those stars which are not ‘classed under any particular constellation, and are otherwise called un- formed stars. : EXTRACTION of Roots is the finding the requires. EXT . | roots of given numbers, or algebraical qua ties or equations. Extraction of the Roots of Numbers, is| find a number which, multiplied by itse] certain number of times, will produce given number. If only one multiplication required it is called the square root; if two, cube root; if three, the biquadratic or fourthn and so on, the number of multiplication be always one less than the number by wh the root is denominated. The rules for extracting the square et and biquadratic roots, are given under articles Square, Cube, and BIQuaDRATE, || we have therefore in the present place ¢ to. consider the general case, that is, of tracting any root whatever of a given num and for this purpose we shall employ the, lowing approximation. ‘ Let N be the given power or number wl} root is to be found, n the index of that poy y the assumed root (which is to be taken near the true root as possible), and A power. Then as(n+1)A +(n—I)N:(n+1)N +(00! A:: 7; the true root nearly. Or as (n+1) A + (n—1)N:2(AWN)i the correction, which must be added te subtracted from, the preceding assumed 1) according as it is less or greater than the required. | Then call this new root r, and use it a as a new assumed root, with which prog as before, and so on till you have obtaineq correct result as far as the nature of the) Note. Whatever be the number of fig assumed in the first root, that number} generally be doubled in the first operat and the same, at least, may be depended ¥ each time, viz. each operation will double number of figures assumed. Let it be proposed, for example, to fine! fifth root of 2..° . oi Assum.7r = 1, thenv5 = A=1,N=2,n) Whence the latter formula becomes — | (5 +1) 1 + (5—1)2:2(2 w1)::1: correc} Or 6+ 8:2::1: correction. . Or 14:2::1:-l4 correction. | Whence 1°14 is the new assumed | therefore 1:14 =r, and r5 = 1925 = A, { again fl (5+1) x 1:925 + (5— 1)2:2(2 #1°925):: L141 which in numbers is ts 19°550 : 150 :: 1°14 : 008699, therefore 1:148699 is the root required. | This rule is due to Dr. Hutton, having ¢ first given by him in his “‘'Tracts,” vol. 1.4 It will be found to be a particular case 0! general formula for extracting the roo equations given under the article APPROX! TION. e For the method of extracting the rot numbers by logarithms, see LoGarits and for the method of performing the +! by the Binom1aL Theorem, see that artic) t 4 EYE Root of simple Algebraical To extract the Quantities. Extract the proposed root of the co-efficient the numeral part, and divide the index the letter or letters by the index of the wer whose root is to be extracted for the ‘ral part, which connected will be the root juired, thus: YY + 6403 2° = + 442” Y¥— 32a yi = —2a’y &e. &e. extract the Root of any compound Algebraical Quantity. | ‘tind the root of the leading term, which ie in the quotient, and bring down the ‘ond term for a dividend. inyolve the root last found to the power xt below that whose root is to be found, 1 multiply it by the index of the power for ivisor. ‘hen divide the dividend by the isor for a new term in the root. Again olve this new root to the given power, and tract the result from the proposed quantity, 1 always divide the first term of the re- jinder by the divisor first found for a new (m; and thus proceed till the root is com- ytely obtained. Let it be proposed, for example, to find the se root of @® + 625 — 40273 + 96 x — 64. “ere by the rule the cube root of the first «mn is x*, whence tae + 6x2°—4023+4962— 64(2?4+2a—4 \a 1) 6x5 ‘Ean 6x54 1224+ 8a3= (a*+2zx)3 1) — 122+ j2° + 6a5—400?4+96—64—(a? +22%—4)3 jerefore x’ + 2a — 4 is the root required. \t will be observed, that in all such cases he above, the proposed formula must be janged according to the powers of the un- “own quantity or quantities. ‘or the Extraction of the Roots of Equa- wis, see APPROXIMATION. IXTRADOS, the outside of an arch of a )Ige, vault, &e. IXTREME and Mean Proportional. See PORTIONAL. IXTREMES, Conjunct and Disjunct, in eee! Trigonometry, are, the former, the %) circular parts that lie next the middle it, and the latter, the two that lie reinote in the middle part. These were the terms i lied by Lord Napier in his universal theo- 1, commonly called Napier’s Circular ‘ts. See CircucaR Parts. , YE, the organ of sight, consisting of eral parts, so adapted to each other, as to au wer the purpose of distinct vision when ved in a proper situation with regard to litt and shade. “he eye, though properly a subject of ana- liiy, is so connected with the doctrine of ; . / | EYE vision, that its structure must first be under- stood before any advances can be made in that theory, and as such it becomes a matter of philosophical inquiry, and must not there- fore be wholly omitted in the present work, although our limits will only admit of a brief illustration of its construction and principal mode of operation. The annexed figure represents a section of the human eye, made by a plane, which is perpendicular to the coats which contain its several humours, and also to the nose. Its form is nearly spherical, and would be exactly so, were not the fore-part a little more convex than the remainder ; the parts BF B, BAB, are, in reality, segments of a greater and less sphere. The humours of the eye are contained in a firm coat BF BA, called the sclerotica; the more convex or protuberant part of which, BAB, is transparent, and from its consistency, and horny appearance, it is called the cornea. This coat is represented by the space con- tained between the two exterior circles BF, BA. Contiguous to the sclerotica is a second coat ofa softer substance, called the choroeides; this coat is represented by the next white space, and extends along the back part of the sclerotica to the cornea. From the junction of the choroeides and cornea, arises the uvea, Ba, Ba, a flat opaque membrane, in the fore-part of which, and nearly in its centre, is a circular aperture called the pupil. The pupil is capable of being enlarged or contracted with great readiness; by which -means, a greater or less number of rays may be admitted into the eye, as the circumstances of vision require. In weak light, too few rays might render objects indistinct; and in a strong light, too many might injure the organ. Whilst the pupil is thus enlarged, or con- tracted, its figure remains unaltered. This remarkable effect is thought to be produced by means of small fibres which arise from the, outer circumference of the uvea, and tends towards its centre; this circumference is also supposed to be muscular, and by its equal action upon the fibres, on each side, the form of the pupil is preserved, whilst its diameter is enlarged or contracted. At the back part of the eye, a little nearer to the nose than the point which is opposite to the pupil, enters the optic nerve V, which spreads itself over the whole of the choroeides EYE like a fine net; and from this circumstance is called the retina. It is immersed in a dark mucus which adheres to the choroeides. These three coats, the sclerotica, the cho- roeides, and the retina, enter the socket of the eye at the same place. The sclerotica is a continuation of the dura mater, a thick membrane which lies immediately under the seull. ‘The choroeides is a continuation of the pia mater, a fine thin membrane which adheres closely to the brain. ‘The retina pro- ceeds from the brain. Within the eye, a little behind the pupil, is a soft transparent substance, ED E, nearly of the form of a double convex lens, the anterior surface of which is less curved than the pos- terior, and rounded off at the edges, EZ, E, as the figure represents. ‘This humour, which is nearly of the consistency of a hard jelly, decreasing gradually in density from the centre of the circumference, is called the crystalline humow’. Itis kept in its place by a muscle called the ligamentum ciliare, which takes its rise from the junction of the cho- roeides and cornea, and is a little convex towards the uvea, The cavity of the eye, between the cornea and crystalline humour, is filled with transparent fluid like water, called the aqueous humour. The cavity be- tween the crystalline humour and the back part of the eye, is also filled with a trans- parent fluid, rather more viscous than the former, called the vitreous humour. It is not easy to ascertain, with great ac- curacy, the refracting powers of the several humours; the refracting powers of the aqueous and vitreous humours are nearly equal to that of water; the refracting power of the crystalline humour is somewhat greater. The surfaces of the several humours of the eye are so situated as to have one line perpendicular to them all. This line, ADF, is called the axis of the eye, or optic axis. The focal centre of the eye, is that point in the axis at which the image upon the retina and the object subtend equal angles. This point is not far distant from the posterior surface of the crystalline lens, though its si- tuation is probably subject to a small change, as the figure of the eye or the distance of the object is changed. From the consideration of the structure of the eye, we may easily now understand how the notices of external objects are conveyed to the brain. Let PQR be an object, towards which the axis of the eye is directed; then the ray which diverge from any point Q, and | upon the convex surface of aqueous humo, have a degree of convergency given the they are then refracted by a double cony; lens, denser than the ambient medin. which increases the convergency; and if 1) extreme rays QH, QI, have a proper degy of divergency before incidence, the pen will be again collected upon the retina, al, and there form an image of Q. In the sa} manner, the rays which diverge from a other points, P, R, in the object, will be e! lected at the corresponding points, p, 7, the retina, and a complete image, p, q, 7, the object PQ R, will be formed there. ‘T impression, thus made, is conveyed to 4 brain by the optic nerve, which origina there, and is evidently calculated to ansy; this purpose. Wood’s “ Optics,” p. 123. | Here it will be observed, that since 4} axis of the several pencils cross each other O, the focal centre of the eye, the image up| the retina, is invested with respect to 4 object, and yet it furnishes the mind with1 idea of its being erect. This is a difficu} that has, produced considerable discussj amongst philosophers, and the most satisfi tory explanation which can probably be giy is, that experience alone teaches us what} on which they are painted, and thus give} the eye exactly the construction of a G@) gorian telescope. We have scena manuser treatise on “ Vision,” by Mr. Horner, in whi. this idea is adopted, and defended with gre ability; and it is to be hoped, that its genious author will be enabled to prese it to the public. ef Since writing the above, an abstract oft! above theory has been published, under t} title of an “ Essay on Vision.” x Eve Glass, in a telescope or other optic instrument, is the glass next the eye. | Bull’s Eye, the vulgar name of the st! ALDERARAN. a | t ih { L FAC ACTORS or Multipliers, in Arithmetic and -ebra, are those numbers or quantities, from multiplication of which other numbers or atities are produced, thus: 7 and 5 are the factors of 35; 3, 7, and 11, are the factors of 231. lsox+y and «—y, are the factors 2*— y’, a + ary + y’, and x — y, are the factors p__. 3, &e. he resolution of algebraical formule into - factors is very frequently of considerable ty, both in’ the Diophantine and Flux- | Analysis; and has therefore been much aded to by modern analysts, and particu- by Euler, both im his “ Algebra” and in * Analysis Infinitorum.” » give an idea of the advantage of this ution, we will assume one of the simplest S. ist it be proposed to find such values of a 'y, such that their difference x*— y*, may a integral square. irst, the factors of x?— y”, are x + y, and \y; that is, («x + y) («@—y) = 2*— y’. since this product is to be a square, we ? evidently have it so if we make each of tors a square, or the same multiple ofa re. ‘sume therefore x + y = mr x—y=ms* the product «* — y* = m’* 7” s*, which idently a square. But by addition and action these equations give us ) ORS (r* + 5?) 2 ate mM (7 we Bs “°) Gee ae 2 m,7, and s, may be assumed at pleasure, \take m = 2, then we have zr +s*,andy =1r’*— 8’; 1 expressions will obviously give integral 's of x and y, if r and s be taken integral ers. This is one of the most simple of the resolution of diophantine pro- by means of factors, but the same me- has been carried to a great length by *, Lagrange, Legendre, and other mo- nathematicians. Thus it has been shown, he numerical factors of a number, which ‘sum of two integral squares, prime to other, are also themselves the sums of iquares, or, Which is the same thing, no actors prime to each other but those ire the sums of two squares, can produce duct which is of that form. This and (al other curious properties of the nu- tal factors of certain formule may be } in Enler’s “ Algebra,” vol. ii.; his ilysis Infinitorum,” vol. i.; Legendre’s ai sur la Théorie des Nombres;” and in FEL F Barlow’s “ Theory of Numbers.” See also the articles Divisors and Propucts. FACTUM, is the product arising from the multiplication of several factors; thus 35 is the factum of 5 times 7. FACUL (the diminutive of fax, a torch), is aname given by some astronomers to cer- tain bright spots on the sun’s disc, which are said to be more vivid than the rest of his body; as macule denote those spots which are less bright. It seems, however, from modern observa- tions, that there is no appearance of such spots as those denominated facule; though the existence of the maculz, or dark specks, has been clearly ascertained. FAGNANO (JuLEs CHARLES), a distinguish- ed Italian mathematician, was born about the year 1690, and early in the eighteenth cen- tury published several memoirs, in the Italian journals, on the transcendental analysis; there are also many pieces of the same author in the Leipsic Acts, which were afterwards pub- lished by himself, with some other papers, under the title of ‘‘ Produzzioni Mathama- tiche ;” Pesaro, in 4to. 1750. Fagnano left,a son, Jean Francois, who fol- lowed the steps of his father, and published several interesting papers in the Leipsie Acts, for the years 1744, 1745, and 1746. FALCATED (from falz, a sickle), denotes one of the phases of the moon or planets, vul- garly called horned; such as the moon ap- pears in the third and first quarter. FALL, the descent or natural motion of bodies towards the earth. See ACCELERATION, DesceENT, and GRAVITY. FALSE Position. See Position. Farse Root, in Numbers and Equations, is a term used by Cardan to denote what we nov call their negative roots. ; FASCL®, in Astronomy, are the same as belts; as the fascize or belts of Jupiter. See BELTS. FATHOM, an English measure of six feet, or two yards. FATUUS, Jenes. See Ienes Fatwuis. FEBRUARY, the second month of the year, containing twenty-eight days for three successive years, and twenty-nine every fourth ear. ‘i In the first age of Rome, February was the last month, and preceded January, till the Decemviri made an order that February should be the second month, and follow January. FELLOWSHIP, the name of a rule in Arithmetic, useful in balancing accounts be- tween traders, merchants, &c.; as also in the: division of common land, prize money, and other cases of a similar kind. Fellowship is of two kinds, single and = FER double; or fellowship without time, and fel- lowship with time. Single ¥ELLOWSHIP, is when all the moneys have been employed for the same time, and therefore the shares are directly as the stock of each partner. The rule in this case is as follows: As the whole stock : the whole gain or loss :: each man’s particular stock : his particular share of the gain or loss. Exam. A bankrupt is indebted to A £1000., to B £2000., to C £3000.; whereas his whole effects sold but for £1200.: required each man’s share. Here the whole debt is £6000.; therefore 1000 : £200. A’s share As 6000 : 1200 ::-; 2000 : £400. B’s share 3000 : £600. C’s share Double FELLOWSHIP, is when equal or diffe- rent stocks are employed for different periods of time. The rule in this case is as follows: ’ Multiply each person’s stock by the time it has been engaged; then say, As the sum of the products : the whole gain or loss :: each particular product : the cor- responding share of the gain or loss. Exam. A had in trade £50. for 4 months, and B £60. for 5 months ; with which they gained £24.; required each person’s particular share. 50 x 4 = 200 60 x 5 = 300 Tom eh GO Te at esos Pan 900 22423) 300:£14. 8s. B’s gain. See Bonnycastle’s “ Arithmetic,” and most other authors on this subject. FERGUSON (JAMEs), an eminent experi- mental philosopher and mechanic, was born of very poor parents, in Bamffshire in Scot- jand, in 1710; in which humble situation he very early gave great proofs of an original and enterprising genius, which he afterwards dis- played much to his own credit and emolu- ment; having accumulated at his death, which happened in 1776, a sum of £6000. Mr. Fer- guson was fellow of the Royal Society, and author of the following works: 1. Select Mechanical Exercises, 8vo. 2. Introduction to Electricity, 8vo. 3. Intro- duction to Astronomy, 8yo. 4. Astronomy explained, on. Sir Isaac Newton’s Principles, -Syo. 5. Lectures on Mechanics, Hydrosta- tics, Hydraulics, Pneumatics, and Optics, 8vo. 6. A ‘Treatise on Perspective, Svo. A new edition of the Select Lectures on Mechanics, &e. has been lately published by Dr. David Brewster, with many valuable additions. FERMAT (PETER), a celebrated French mathematician, was born in 1590, and became, by his talents and acquirements, counsellor of the parliament of Toulouse. Fermat was intimately acquainted with all the first-rate mathematicians of his age, either personally or by correspondence. His works were collected and published at Toulouse in 1679, viz. sixteen years after his death, this eventhaving happened in1663. These BIG are entitled “ Opera varia Mathematica,’ 2 vols. folio; and contain, besides his edi’ and notes on Diophantus’s “‘ Algebra,” wh was first published separately in 1670, the lowing original pieces ; 1. A Method for the Quadrature of Sorts of Parabolas. 2, Another on Maxi Which serves not only for the determina of plane and solid problems, but for dray tangents to curve-lines, finding the centre gravity in solids, and the solution of quest concerning numbers; in short, a method + similar to the Fluxions of Newton. 3, Introduction to Geometric Loci, plane solid. 4. A Treatise on Spherical Tangene where he demonstrates in the solids the s things as Vieta demonstrated in pianes. , Restoration of Apollonius’s two books on P Loci. 6. A General Method for the Dit sion of Curve Lines. Besides a numbe other smailer pieces, and many letter, learned men. FERMENTATION (fermentatio, La an intestine motion, excited by the assisti of proper heat and fluidity between the tegrant and constituent parts of farinace saccharine, and a few other substances, which new combinations of their respec principles result. FIELD Book, in Surveying, a book | for setting down angles, distances, and marks, as they arise in the field practice; from which the map is constructed an contents computed. See SURVEYING. FIGURATE Numbers, are all those fall under the general form, n(n + 1) (n + 2) (n + 8)....(m $3 and they are said to be Ist, 2d, 3d, &c. 0 according as m — 1, 2,3, &c. thus: Nat. series, 1, 2,3, 4, &c. n Ist order, 1,3, 6,10, &c. ot) is 2d order, 1, 4, 10, 20, &e. niu + lat 3d order, 1, 5, 15, 35, &e. &e. ke. @ The most remarkable property of } numbers is, that if the nth term of any' be added to the n + 1 term of the preee order, the sum will be the » + 1 term 0 same order as the former. ‘This propel obvious, if we attend to the above ge forms; for ifn + 1, which is the x +1 n(n +1 n(n —— of the natural series, be added to nth term, Ist order, the sum » + 1 + 2. (mn +1) (m + 2) vt L.2 first order. Again: (n + 1) (nm + 2) | apes 1.2.3 @ (n + 1) (n + 2) (n + 8) 1: 4,2 and , isthen +1 term 0 | FIG shat is, the sum is the n + 1 term of the 2d rder; and in the same manner, a similar law ray be shown to have place with regard to ‘ny order m — 1 and m, Another property of figurate numbers, dis- covered by Fermat, is as follows: ' That if the nth term of the natural series be iultiplied by the x + 1 term of any order m, ae product will be equal to the m + 2 times ie nth term, of the order m + 1. Thus, in the preceding series, 4x15—=3 x 20 3 x 20—=4x 15 , And generally . ne piyee tat) | ] : | ee § are ae 1.2.3 This proposition, which is very obvious in le way in which we have defined figurate Baiyc is extremely difficult to demonstrate me difficulty as the binomial theorem; in- ed Fermat considered it as one of his most -eresting numerical theorems. See his Notes _ Diophantus, p. 16. It is obvious that these numbers are the co- icients of the terms of binomials, and it is s circumstance, in all probability, that in- ‘ced the early algebraist to pay so much ention to their formation, by which means arrive at a ready method of involving bino- al quantities; but since the general deve- (ment of the binomial theorem, in its pre- it form, no farther attention is paid to these inbers, except as matters of mere curiosity. ey were formerly distinguished into pyra- lal numbers, of different orders; for being | sums of series of polygonal numbers, they sht be conceived as forming pyramids, the he as a series of polygons, each less than other, but of the same number of sides, ald, when applied one upon another, form yramid. See Barlow’s ‘‘' Theory of Num- 3,” chap. ii.; see also POLYGONAL Numbers. ‘IGURE, in Arithmetic, is any one of the eral characters in present use, commonly ed digits; as 0, 1, 2,3, 4,5, 6, 7,8, 9. See TATION. IGURE of a Body, in Geometry and Men- tion, denotes generally its form or shape; ‘ah as all bodies are of some form or te, figurability is reckoned amongst the ntial properties of body or matter. A y without figure would be without limits, | must theretore be infinite. it URE, in the Conic Sections, according to Pe llonius, is the rectangle contained under latus rectum and tranyersum, in the hy- vola and ellipse. ' “GURE of the Earth. 4TH. ,GURE or Delineation of the Full Moon, is ‘presentation of this body, as it appears igh a good telescope. {parent FiGuRE, in Optics. See APPARENT. GURE of the Sines, Cosines, Versed Sines, See DEGREE and any other principles, involving exactly the - FIG Tangents, or Secants, &c. are figures made by conceiving the circumference of a circle ex- tended out in a right line, upon every point of which are erected perpendicular ‘ordinates equal to the sines, cosines, &c. of the corres- ponding arcs ; and then drawing the curve line through the extremity of all these ordinates 5 which is then the figure of the sines, cosines, &e. as in the following diagrams : Construction of the Figures of Sines, Cosines, §c. KE -C a Let ADB, &c. be the circle, AD an arc, DE its sine, CE its cosine, AE the versed sine, AF the tangent, GH the cotangent, CF the secant, and CH the cosecant. Draw a right line aa (see the figures below) equal to the whole circumference ADGBA of the cir- cle, upon which lay off also the lengths of several arcs, as the arcs at every 10°, from 0 at a, to 360° at the other end at a; upon these points raise perpendicular ordinates, upwards or downwards, according as the sine, cosine, &c. is affirmative or negative in that part of the circle; lastly, upon these ordinates set off the length of the sines, cosines, &c. corres- ponding to the arcs at those points of the line or circumference aa, drawing a curve-line through the extremities of all these ordinates; which will be the figure of the sines, cosines, versed sines, tangents, cotangents, secants, and cosecants. Where it may be observed, that all the curves are the same; viz. those of the sines and cosines, those of the tangents and cotangents, and those of the secants and cosecants; except that some of their parts are a little differently placed. It may be known when any of these lines, viz. the sines, cosines, &c. are affirmative or negative; that is, when they are to be set upwards or downwards, by observing the fol- lowing general rules for those lines in the Ist, 2d, 3d, and 4th quadrants of the circle. in the Ist and 2d are affirmative, in the 3d and 4th ... negative: in the Ist and 4th are affirmative, in the 2d and 3d ... ‘negative: in the Ist and 3d are affirmative, in the 2d and 4th .., negative: in the Ist and 3d are affirmative, in the 2d and 4th ... negative: in the Ist and 4th are affirmative, in the 2d and 3d ... negative: in the Ist and 2d are affirmative, in the 3d and 4th ... negative. Sines... } Secants..... ; Cosines..... Tangents... Cotangents Cosecants.. ; To find the Equation and Area, Se. to each of - these Curves. Draw any ordinate de; putting r — the ra- dius AC of the given circle, x = ad or AD any absciss or arc, and y = de its ordinate, U , FIG Which will be either the sine DE = s, cosine CE —c, versed-sine A E =v, tangent AI’ =?, cotangent GH =, secant CF =f, or €o- secant CH = g, accerding to the nature of the particular construction. Now, from the nature of the circle, are obtained these follow- ing general equations, expressing the relations between the fluxions of a circular are and its sine, or cosine, &c. gy rs is ae rv My —s Ce ~ 4/ Bre — 0 Brig age 3 —r¢ 5 Mea ca —ro PLP HP LL = fJfr—r ae oJ o—r And these me express the relation between the absciss and ordinate of the curves in ques- tion, each in the order in which it stands; where « is the common absciss to all of them, and the respective ordinates are s, ¢, v, t, 7, f, and ¢. And hence the area, &c. of each of these curves has been found as follows: Figure of the Sines. 1. In the fi igure of the sines, when the fluxion %} the area is found, its correct fluent gives + r/7*—s*=r?—re=r», the rectangle t a and versed sine, viz. — or +, as sis increasing or decreasing: which is a general expression for the area. When s = 9, as at a or b, this expression becomes 0, or 2r* j 80 that when ad becomes a quadrant, dul the area of afg. The length of the line, or figure of sines, is equal to half the perimeter of an ellipse, whose axes are to one another as /2 to 1, and whose conjugate axis is equal to the diameter of the circle, from which the line of smes is. conceiv- ed to be generated; as is ingeniously demon- strated by Mr. Wallace, R.M.C. at p. 187, No. viii. Leybourn’s Repository. Figure of the Cosines. @ aa Ff 2. In the figure of cosines, we have 7s, the rectangle of the radius and sine for the gene- ral area adec; and when af is ae to a qua- drant, the area becomes — 7”. rx AD— DE for the area. FIG . Figure of the Versed Sines. a, ec (27 nF 3. In the figure of the versed sines, we have When AD o: 31416 ae J , ad is a quadrant, this becomes = the area afg. And when AD or ad is¢ semicircle, it becomes 3°1416 r? = the are: abg in this figure. Figure of the Tangents. 4. In the figure of tangents, we have r* hyp. log. of 4 for the area ade; which when: becomes a quadrant, the area afg becom finite. Figure of the Cotangents. Np Wd t. ‘The same for the figure of cotangentay ginning at f instead of a. Figure of the Secants. 4 Cc 7. ¢ ; b b © a fF Zz k H. 5. For the figure of the secants, we ? y> x hyp. log. of f + a for the: ade; which expression is infinite, when! becomes the quadrant. Figure of the Cosecants, * 7 k a. . ‘ Gaf...6 re 4 The same might be remarked respe Ne figure. of cosecants, beginning at fi ins’ of a, | FIR - And hence the meridional parts in Mer- eator’s chart may be calculated for any lati- tude AD or ad. For the merid. parts are to the are of latitude AD:: as the sum of the secants to the sum of as many radii, or :: as the area ade; to ad x radius ac or AD x AC in the first figure. See Emerson’s “ Miscel- jlanies,” p. 232, &e. _ FILIPPO, acoinage of Milan, value 4s.83d. sterling. FIN AUS (Orontivs), in French, Finé, a professor of mathematics in the Royal Col- lege of Paris, was born in Dauphiné in 1494, and like most other mathematicians and astro- nomers of that period, was much addicted to astrology; he wrote, however, on various sub- jects, and an edition of his works, translated into the Italian language, was published in 4to. at Venice, in 1587 ; consisting of arithmetic, oractical; geometry, cosmography, astronomy, and dialling. Finzus died in the year 1555, in the sixty-first year of his age. FINITE, the property of any thing which ‘s bounded or limited, either in power, extent, ar duration. FINITOR, is sometimes used to denote the horizon. FIRE, is that subtile invisible cause by Which bodies are made hot to the touch, and 2xpanded or enlarged in bulk; by which fluids ‘are rarefied into vapour; solid bodies become luid, and are finally either dissipated and car- ‘ied off in vapour, or otherwise melted into glass. The opinion of the ancients respecting fire were various and fanciful. Ignorant of the eading facts which a theory is required to iccount for, and unassisted by experiments yw tools, they generally made use of words vhich convey no definite ideas. They called -t an active fermentation, an intestine motion, i repulsive agent, &c.; but no real attempt owards a rational investigation is to be found n their works. And though some of their assertions seem to coincide with the more vational modern theories, yet that apparent coincidence must be considered as being ac- sidental, as it is not founded upon systematic easoning. It must be acknowledged, how- ver, that almost all the opinions, either an- ‘ient or modern, respecting fire, may be divid- -d into two classes; for some of them assert- od that fire was nothing more than a violent gitation, in some unknown manner, of the varts of burning bodies; whilst others attri- ated it to something peculiar, and sud generis, vhich either existed in all combustible bodies, »r was communicated to them. The former, vhich is called the mechanical hypothesis, vas believed and maintained by the most ble philosophers, of much later and much aore enlightened times. Bacon, Boyle, and Yewton were of that opinion; and ihere seems o have existed a considerable struggle be- ween those distinguished persons and some elebrated chemists of their times, who main- ained that fire was a fluid of peculiar nature. ‘The former asserted, that the phenomena of FIR combustion could be accounted for on the supposition that fire consisted in nothing more than the violent agitation of the parts of the bodies concerned; but as no such motion could be produced without an adequate cause, they were considerably perplexed by it; and, in fact, their attempts toward an ex- planation are very confused. Boyle says, “that when a piece of iron becomes hot by hammering, there is nothing to make it so, except the forcible motion of the hammer im- pressing a vehement and variously determin- ed agitation on the small parts ofiron.” It is to be remarked, however, that the same phi- losopher, on observing the phenomena of the metallic bodies acquiring additional weight by their calcination, was induced to publish a treatise on the possibility of rendering fire and flame ponderable. Bacon defines heat (which he considers as meaning the same as fire) to be “an expansive undulatory motion in the minute particles of a body, whereby they tend, with some rapidity from a centre, towards a circumference, and at the same time a little upwards.” Newton did not at- tempt to assert any thing positive concerning it; but he conjectured that gross bodies and light might be convertible into one another ; and that great bodies of the size of our earth, when violently heated, might continue and increase their heat by the mutual action and re-action of their parts. ' It was not, however, till near the close of the last century, that we could be said to pos- sess any correct and rational theory of this principle; and for this, we are in a great measure indebted to the celebrated Lavoister, who introduced the notion of caloric, and by this means succeeded in explaining most of the phenomena of combustion: but our limits will not admit of relating the several experi- ments and conclusions which serve as the basis of his theory: indeed, these being sub- jects purely chemical, would be misplaced in the present work. The reader who is desirous of information on this subject, should consult the treatises on chemistry by Murray, Thomp- son, Davy, &c; in which he will find ample detail of every thing relating to this interest- ing inquiry. Fire Balls, in Meteorology. See METEOR, FIRKIN, an English measure containing the fourth part of a barrel; that is, 8 gallons ale méasure, or 9 gallons beer measure. A firkin of butter, soap, &c. is in Weight 56tb. FIRLOT, a Scotch measure containing about 21% pints of that country, or about 85 English pints. The barley firlot contains 31 pints, and the wheat firlot about 2211 cubic inches, being’ about 60 cubic inches more than the English bushel. FIRMAMENT (firmamentum, Latin). This word has been used with great latitude, as well by the sacred writers, as by poets and astronomers. Some old astronomers consider the orb of the fixed stars as the firmament; but in Scripture, and in common language, it is used for the middle regions, or the space ox U2 FLA expanse appearing like an immense concave hemisphere. Many astronomers, both ancient and modern, accounted the firmament a fluid matter; but those who gave it the name of firmament must have considered it as a solid. FIRMNESS, in Philosophy, denotes the consistence of a body, or that state wherein its sensible parts cohere, or are united together, so that a motion of one part induces a motion of the rest. In which sense firmness stands opposed to fluidity. See CoHESION. FIRST Mover, in the Ancient Astronomy. See Primum Mobili. FISHES, in Astronomy. See PISCEs. FISSURES, in the History of the Earth, are certain eruptions that horizontally or pa- rallelly divide the several strata of which the body of the terrestrial globe is composed. FIXED Bodies, generally denote those bodies which neither fire nor any corrosive has such an effect on, as to reduce or resolve them into their component elements, or abso- lutely to destroy them. Of fixed bodies, the principal are gold, platina, silver, precious stones, particularly the diamond, salts, &c. Fixep Ecliptic, a certain imaginary plane, which never changes its position in the hea- vens from the action of any of the parts of the solar system on each other; but, like a centre of inertia, remains immoveably fixed. The existence of such a plane is demonstrated by Laplace, who has shown the method of de- termining it from the situations, velocities, masses, kc. of the planets and other bodies, Fixep Signs of the Zodiac, an arbitrary denomination which some astronomers have given to the signs Taurus, Leo, Scorpio, and Aquarius. The particular corresponding sea- son being supposed most fixed when the sun is in those signs. Fixep Stars, are those which constantly maintain the same relative position with re- gard to each other, in contradistinction to the planets and comets which are constantly changing their relative positions. See Srar and CONSTELLATION. FIXITY, or Fixepngss, in Philosophy, the quality of a body which denominates and ren- ders it fixed; or a property which enables it to endure fire, and other violent agents, A body may be said to be fixed in two re- spects; viz. 1. When, on being exposed to the action of fire, or a corrosive menstruum, its particles are indeed separated, and the body rendered fluid, but without being resolved into its first elements: and, 2, When the body sustains the active force of the fire, or men- struums, whilst its integral parts are not car- ried off by fume. Each kind of fixity is the result of a strong or intimate cohesion be- tween the particles of the body. FLAME, the subtlest and brightest part of fuel, ascending above it in a pyramidal or co- nical figure, and has been conjectured by Newton to be a vapour red hot; which, how- ever, seems to be but an imperfect idea. We should rather say, that flame is an instance of combustion, whose colour will be determined FLA | by the degree of decomposition which takes, place. If it be very imperfect, the most re- frangible rays only will appear; if very per-_ fect, all the rays will appear, and the flame’ will be brilliant in proportion to this perfec-, tion. There are flames, however, which con- sist of burning particles, whose rays haye partly escaped before they ascended in the form of vapour; such would be the flame of a red-hot coal, ifexposed to such a heat as would gradually disperse it into vapour. When the fire is very low under the furnace of an iron- foundery, at the upper orifice of the chimney ared flame of this kind may be seen, which) is different from the flame that appears imme- diately after fresh coals have been thrown on the fire; for, in consequence of adding such a supply to the burning fuel, a vast column otf smoke ascends, and forms a medium so thick as to absorb most of the rays excepting the red. On this subject the reader may advan- tageously peruse the Rev. G. C. Morgan’s “Observations and Experiments on the Light of Bodies in a State of Combustion,” Phil. Trans. vol. lxxv. or New Abridgment, vol. xy. p- 668. FLAMSTEED, a celebrated English astro- nomer, and the first astronomer royal, was born at Derby in the year 1646, and very early discovered a great genius and taste for astro- nomical subjects; having, while yet a youth, computed both solar and lunar eclipses, and occultations of the moon and stars. ] Flamsteed was author of several small tracts | and of numerous papers in the Transaction: of the Royal Society, from vol. iv. to vol. xxix. | but his great work, and that which contain: the principal operations of his life, was thé ‘“ Historia Coelestis Britannica,” published ii! 1725, in three large folio volumes. The first 0 which contains the observations of Mr. Gas coigne, taken at Middleton in Yorkshire, an¢ those of his own, taken at Derby between thi years 1638 and 1643, with tables, &c, computes at the Royal Observatory between the year 1675 and 1689. 'The second contains his obser vations, made with an excellent telescope, 01 the zenith distances of the fixed stars, sun moon, planets, with their transits over the me ridian; also notes and observations on diameters of the sun and moon, with thei eclipses, and those of Jupiter’s satellites, be tween the years 1689 and 1719. The thin volume comprises a catalogue of the right as censions, polar distances, longitudes, and mag nitudes of nearly 3000 fixed stars. ‘The pre face to this volume contains an account of al the astronomical observations made before hi time, with a description of the instrument employed, and much other curious and highl important matter. The printing of this nobl work was not finished at the time of ou author’s death, and the care necessary to if completion devolved on Mr. James Hodgsor by whom it was published, in the year 1725. Flamsteed died at the conclusion of th year 1719, at the age of seventy-three years, Few men, says his biographer, possesse FLO more zeal and application in the pursuit of scientific knowledge than the first astronomer royal, and scarcely any man ever attained a higher respect amongst his contemporaries. In common life he was free and easy of access, and pleased with the company of those who, with scientific research, could unite their share in the convivial intercourse of life. PLEXIBLE ( flexibilis, Latin), is the pro- perty or quality of a body that may be bent. FLEXURE, or FLExton (flexio, Latin), is the bending or curving of a line or figure. Contrary Furxure. See INrLecrion and RETROGRESSION. _ FLOAT Boards, those boards which are fixed to the rim, or circumference, of under shot water-wheels, serving to receive the im- oulse of the stream, and by which means the nill is put into motion. | FLOATING Bodies, are those which swim on the surface of a fluid, the stability, equili- orium, and other circumstances of which form in interesting subject of mechanical and hy- lrostatical investigation, particularly as ap- tlied to the construction and management of ships and other vessels. _. The equilibrium of floating bodies is of two Ainds, viz. stable or absolute, and unstable or jottering. | In the one case, if the equilibrium be ever 0 little deranged, the bodies which compose ie system only oscillate about their primitive osition, and the equilibrium is then said to e firm, or stable. And this stability is abso- tte if it takes place, whatever be the nature f the oscillations; but it is relative if it only kes place in oscillations of a certain de- pription, vo } , In the other state of equilibrium, if the sys- -m be ever so little deranged, all bodies de- ate more and more, and the system, instead , any tendency to establish itself in its primi- \e position, is overset and assumes a new sition, entirely different from the former; jtd this is called a tottering, or unstable, equi- orium. “We may form (says a writer on this sub- ot) a just conception of these two states, by pposing an ellipse placed vertically on an nizontal plane; if the ellipse is in equili- um on its smaller axis, it is evident that, on a slight derangement, it will tend to sain its original position, after several small sillations, which will soon be terminated by > friction and resistance of the air; but if : ellipse be placed in equilibrium on its ‘ater axis, if once.it deviates from this posi- a, it will continue to deviate more and re, till it finally turns itself on its lesser s.’ In the above example, there is this | larkable circumstance ; the four positions equilibrium of the ellipse, on the extremi- ti of its two axes, are alternately stable and abe and this takes place in every case. ‘, Suppose two positions of stable equili- pam to take place in any body, and such tlt there does not exist between them any P ition of the same kind, if the body be placed , PLO in one of these positions, and is made to de- viate from it, and to approach the other, according as this deviation is greater or less, the body will either return to its original state, or will arrive at the other position. There will evidently, therefore, be some intermediate position, in which the body will neither tend towards one or the other of the former, but will remain at rest: but this state of equili- brium will be unstable, since, if the body be made to deviate ever so little towards one of the other positions, it will necessarily arrive at it. Hence, it appears, that if a body turn- ing round a fixed axis passes through several positions of equilibrium, they will be alter- nately stable and unstable. The stability of a floating body is the greater as its centre of gravity is lower than that of the displaced fluid, or as the distance between these centres is increased; it is for this reason that ballast is put in the lower part of vessels to prevent them from being overset. The nature of the equilibrium, as to stabi- lity, depends on the position of a certain point, called the meta centra, or centre of pressure ; which term was first adopted by Bouguer, in his “ 'Traite du Navire.” When the meta centre is above the centre of gravity, the equilibrium is stable; on the contrary, when the meta centre is lower than the centre of gravity, the equilibrium is tot- tering ; when the meta centre coincides with the centre of gravity, the body will remain at rest in any position it is placed in, without any tendency to oscillation. Laplace gives the following rule for deter-: mining whether the force which solicits the system, tends to restore it to the same state again when deranged from its primitive posi- tion, which is as follows: “If through the centre of gravity of the sec- tion of the surface of the water on which a body floats, we conceive a horizontal axis to pass, such that the sum of the products of every element of the section, multiplied by the square of its distance from this axis, be less than any other horizontal axis drawn through the same centre, the equilibrium will be stable in every direction; when this sum surpasses the product of the volume of the dis- placed fluid, by the height of the centre of gravity of the body above the centre of gra- vity in this volume.” This rule is principally useful in the con- struction of vessels which require suflicient stability to enable them to resist the effects of storms, which tend to submerge them. In a ship, the axis drawn from the stern to the prow, is that relative to which the sum above mentioned is a minimum; it is easy, therefore, to ascertain and measure its stability by the preceding rule. In order that a floating body may remain in equilibrium, it is also necessary that its cen- tre of gravity be in the same vertical line with the centre of gravity of the displaced fluid, otherwise the weight of the solid will not be FLU completely counteracted by the pressure of the displaced fluid. When the lower surface of a floating body is spherical or cylindrical, the meta centre must coincide with the centre of the figure, since the height of this point, as well as the form of the portion of the fluid displaced, must remain invariable in all circumstances. The place,of the meta centre is determined by the doctrine of forces combined with the elemen- tary principles of hydrostatics, by considering the form and extent of the surface of the dis- placed portion of the fluid, compared with its bulk and with the situation of its centre of gravity. According to Dr. Young, if a rec- tangular beam be floating on its surface, the height of the meta centre aboye the centre of gravity will be to the breadth of the beam, as the breadth is to twelve times the depth of the part immersed. Hence, if the beam be square it will float securely, when either the part immersed, or the part above the surface, is less than 24, of the whole; but when it is less unequally divided by the surface of the fluid, it will overset. If, however, the breadth be so increased as to be nearly one-fourth greater than the depth, it will possess a cer- tain degree of stability, whatever its density may be. Rees’s Cyclopedia. FLORIN, a German coinage, value from 2s. 4d. to 2s. 1d. sterling. Fiorin of Poland, is in value 6d. sterling. FLUENT, or Flowing Quantity, in the Flux- ional Analysis, is the variable quantity which is considered as increasing or decreasing ; or the fluent of a given fluxion, is that quantity whose fluxion being taken according to the rules previously laid down, will be the same with the given fluxion. See Inverse Method of FLUXIONS. FLuENTS by Continuation. TION. Contemporary FLUENTS, are such as flow together, or for the same time. And the same is to be understood of contemporary fluxions. When contemporary fluents are always equal, or in any constant ratio, then also are their fluxions respectively either equal, or in that same constant ratio; that is, if 2 = y, then is ge y;\orifa: ys:n21,thenis c: yi: n: 1; or ifa = ny, then is = ny. Correction of a FLUENT. See CORRECTION. FLuENTS by Series. See FLUXIons. Table of FLUENTS. See FLUXIONS. FLUID, or Fld Body, is that whose parts yield to the smallest force impressed upon them, and by yielding are easily moved amongst each other; in which sense it stands opposed to a solid, whose parts do not yield, but con- stantly maintain the same relative situation. The fluidity of bodies is accounted for by supposing them to be made up of a number of infinitely small particles, possessing, with re- gard to each other, an attractive and repulsive power, which being equal, places the whole system in equilibrium, whereby it obeys any external force impressed upon it. This hypothesis evidently places the several See CONTINUA- BLU* particles of a fluid at small distances front each other, and consequently supposes them to be porous, or to possess certain vacuities, the existence of which may be demonstrate from various experiments. Thus, water will” dissolve a certain quantity of salt; after which it will absorb a little sugar, and after that a little alum; and all this without increasing its | bulk: which evidently shows that the par-_ ticles of these solids are so far separated as- to become smaller than the vacuities, or inter-. stices, between the particles of the fluid. ) ] * Fluids are divided into Elastie and Non-. elastte. | Elastic Fiurps, are those which may be. compressed into a smaller compass, but which on removing the pressure resume again their, former dimensions; as air, and the yarioug. gasses. See AIR. , Non-elastic FLurps, are those which occupy. the same bulk under all pressures, or if they. be at all compressible it is in a very trifling | degree ; such as water and other liquids. Discharge of Fuurps. See DIscHARGE. | Pressure of Fiurps. See PRESSURE. 4 | Resistance of Ftuips. See RESISTANCE; see ' also the articles HypROSTATICS and HYyDRAU-, LICS. Aga FLUIDITY, that state or affection of b dies which renders them fluid. See FLUID. | FLUX and Rerivx of the sea. See TIDE, | FLUXION (fluxio, Latin), in the New. tonian Analysis, denotes the velocity with, which a flowing quantity increases by its, generative motion, by which it stands contra: distinguished from a fluent, or flowing quan, tity, which is constantly and indefinitely in- creasing, after the manner that a surface is described by the motion of a line, or a solid. by the motion of a surface. | Or, a fluxion may be otherwise defined, as the magnitude by which any flowing quantity would be uniformly increased in a given por- tion of time, with the generating celerity at any proposed position or instant, supposing it thence to continue invariable. See the fol- lowing article. a or Fluxional Analysis, is the algorithm, analysis of fluxions and flowing quantities distinguishable from the differential calculu both by its metaphysics and notation, but in all other respects the two methods are ide tical. See DIFFERENTIAL Calculus. by The invention of the fluxional analysis does more honour to the powers of the human mind, than perhaps any discovery of this or any preceding age; it opens to us a new world, and extends our knowledge as it were to infinity ; it carries us beyond those bounds which seems prescribed to our mental powers and Jeads to investigations and results whic must otherwise have ever remained in imp : netrable obscurity. * It is not, therefore, surprising to find that the honour of this brilliant discovery shoul have been contested with so much anxiety and warmth between two of the greatest ma: thematicians that probably any age ever pro = | FLU luced, or rather between the countries which rave birth to these celebrated opponents, for he authors themselves were perhaps the least nterested in the affair; they, without any upparent concern about the priority of the nvention, felt themselves happy in enjoying the rapid and extraordinary progress which t was continually making in their own hands and in those of the most eminent mathema- licians of that period. This harmony was not, however, of long duration, for in the year 1699, Fatio, in a treatise ‘‘ On the Line of swiftest Descent,” declared, that Sir I. Newton was the original author, insinuating at the same time that Leibnitz had borrowed his notions from New- ton, and that, after new modelling them, he wished to palm them upon the world as his own. This declaration gave rise to a long and angry controversy between Leibnitz, supported by the editors of the Leipsic Acts on the one side, and the English mathematicians on the other, though Newton himself never appeared in the dispute ; his honour, however, was become identified with that of his nation, and his ‘countrymen were earnestly interested in the business notwithstanding his apparent indif- ference as to the result. ' Previous to this decided assertion of M. Fatio no question as to priority seems to have been thought of, each was supposed. to have discovered his own peculiar method indepen- ‘dent of the other, and the difference in the notation and the circumstance, of neither of these authors ever mentioning the works of ithe other, seems to have given confidence to this supposition. | The subject, however, from this time began to be viewed in a totally different light, and various letters, papers, and pamphlets, were published to show to whom the original idea was due, a controversy which is of course too much extended to be introduced into an ar- ticle in a work of this kind; we must there- fore limit our remarks to the leading features of the evidence, which may be briefly stated | as follows: The first time any thing appeared pub- licly relating to this analysis was in 1684, when Leibnitz published his rules for certain eases in the Leipsic Acts, reserving to him- self their demonstrations, which were, how- ever, afterwards discovered by the brothers James and John Bernouilli, both of whom practised the differential calculus with great success. ; It should be observed, however, that though it was not till the year 1684 that Leibnitz published his rules, yet there is evidence that he had employed this method as early as 1677, at which time it very much resembled Dr. Barrow’s “ Method of Tangents,” which was published in 1670. Now Newton communicated his “ Method of Tangents” to Collins in 1672, which letter, together with another letter written in 1676, were sent to Leibnitz by M. Oldenburgh in FLU the same years, so that there is no doubt that Leibnitz had seen them previous to the ap- pearance of any thing of his own on this sub- ject, and it is certainly not improbable that he might avail himself of the hints they sug- gested. The above facts, which are well authenti- cated, would secure to Newton the priority of invention ; but it has been farther proved that he had actually employed this analysis so early as the year 1669, or perhaps some years before, so that no doubt can be entertained with regard to his title to the invention. Whether or not Leibnitz is equally entitled to be considered as the author of his own par- ticular calculus, shall be examined in the subsequent part of this article. The controversy of which we have spoken above was continued with considerable warmth and acrimony, till at length, in the year 1711, Leibnitz complained to the Royal Society, that Dr. Keill had accused him of publishing the “* Method of Fluxions,” invented by New- ton, under other forms and denominations, adding, that nobody knew better than Newton himself that he had borrowed nothing from him; and required that Dr. Keill should publicly disavow the ill construction which might be put on his words. The society, thus appealed to as a judge, appointed a committee to examine all the old letters, papers, and documents, that had passed among the several mathematicians, relating to the point in dispute; who after a strict examination of all the evidence that could be procured, gave in their report as follows: “That M. Leibnitz was in London in 1673, and kept a correspondence with Mr. Collins by means of M. Oldenburgh till September 1675, when he returned from Paris to Ha- nover, by way of London and Amsterdam : that it did not appear that M. Leibnitz knew any thing of the differential calculus before his letter of the 21st of June 1677, which was a year after a copy of a letter, written by Newton in the year 1672, had been sent to Paris to be communicated to him, and above four years after Mr. Collins began to com- municate that letter to his correspondents ; in which the method of fluxions was sufli- ciently explained, to let a man of his sagacity into the whole matter; and that Sir I. Newton had even invented his method before the year 1669, and consequently fifteen years before M. Leibnitz had given any thing on the sub- ject in the Leipsic Acts.” From which they concluded that Dr. Keill had not at all injured M. Leibnitz in what he had said. The society printed tis their determination, together with all the pieces and materials re- lating to it, under the tivle of “* Commercium Epistolicum de Analysi Promota,” 8vo. Lon. 1712. This book was carefully distributed through Europe, to vindicate the title of the English nation to the discovery ; for Newton himself, as already hinted, never appeared in the affair; whether it was that he trusted his honour with his compatriots, who were so zea- FLU lous in the cause, or whether he felt himself even superior to the glory of it. M. Leibnitz and his friends, however, could not show the same indifference; he was ac- cued of a theft, and the whole “ Commercium Epistolicum” either expresses it in terms or insinuates it. Soon after the publication, therefore, a loose sheet was printed at Paris, in behalf of M. Leibnitz, then at Vienna. It is written with great zeal and spirit; and it boldly maintains that the method of fluxiens had not preceded the method of diflerences, and even insinuates that it might have arisen from it. The detail of the proofs, however, on each side, would be too long, and could not be understood without a large comment, which. must enter into the deepest geometry. M. Leibniiz had begun to work upon a Commercium Epistolicum, in opposition to that of the Royal Society, but he died before it was completed. A second edition of the “ Commercium Fpistolicum” was printed at London in 1722: when Newton, in the preface and annota- tions which were added to that edition, par- ticularly answered all the objections which M. Leibnitz and Bernouilli were able to make since the ‘‘ Commercium” first appeared in 1712; and from the last edition of the “ Com- mercium,” with the various original papers contained in it, it evidently appears that New- ton had discovered his method of fluxions many years before the pretensions of Leibnitz. See Raphson’s “ History of Fluxions,” and the valuable account of the ‘“Commercium Epistolicum,” given in vol. xxix. of the Phil. Trans. or New Abridgment, vol. vi. p. 116, 153. There are, however, many persons, parti- cularly amongst foreigners, who support the cause of Leibnitz, maintaining that at least he was no plagiary, though there are few perhaps who are disposed to attribute to him the first invention; this glory is certainly due to Newton, and the only question is whether or not Leibnitz took any of his hints from what he had seen of Newton’s, or whether he fell upon the same thing independent of this as- sistance. This is a question which it is now difficult to determine with justice, though there are many circumstances which seems to bear hard against Leibnitz; at the same time it must be admitted, that he was an extraor- dinary genius, and that many advances had been made towards the fluxional or diffe- rential calculus, sometime before either New- ton or Leibnitz published «ny thing on the subject ; Des Cartes’s “ Method of Tangents,” Fermat's, Hudde’s, and Roberval’s ““ Methods de Maximis et Minimis;’ and Barrow’s “* Differential Triangle,” were so many ad- vances towards this elegant analysis, and it does not appear at all surprising, that with so much prepared to their hands, two such men as Newton and Leibnitz should be led to the same results though they took different means of arriving at them; this is at least the most FLU liberal idea, and it is such at the same tim: as is justified by several corroborating cir cumstances. . Having said thus much with regard to thi original discovery of the new calculus, it wil be proper to add a few remarks concernin its subsequent improvements. ; Newton himself brought it to considerable perfection, as may be seen in his Tract oy ‘“‘ Quadratures,” and in his “ Treatise oy Fluxions,” from which works it appears thay he not only laid down the whole theory of hi method both direct and inverse, but also ap- plied it in practice to the solution of many o the most useful and important problems in mathematics and philosophy. | But the elements of the docirine being de livered by its author in a very concise man ner, gave occasion to the pamphlet, entitled the “ Analyst,” published in 1734, by the! learned and ingenious Bishop Berkeley, in which he represents the doctrine of fluxions as being founded on inconceivable principles, and replete with false and specious reason- ings. He does not, he says, object to the! conclusions, but to the logic, and nical describes a fluxion as the ghost of a departed quantity. An answer to this attack was soon after, given under the signature of Philalethes Can- tabrigiensis, generally supposed to have been | Dr. Jurin; and various other letters and, pamphlets were published on the same side! of the question. But the most able answaal that were given to the “ Analyst,” were those by Robins and Maclaurin, the former in vol. ii, of his Tracts, and the latter in his elaborate’ "Treatise on Fluxions,” in which, besides, establishing the basis of the system upon a: secure foundation, he considerably extended it and showed its application to numerous phy- | sical problems. ib But before Maclaurin’s treatise, several. important improvements had been made in the fluxional or differential calculus, by John Bernouilli, who treated of the fluents-belong- ing to the fluxions of exponential expression Hi James Bernouilli, Craig, Cheyne, Cotes, Man- fredi, Taylor, Fagnanus, Clairaut, D’Alem- bert, Euler, Condorcet, Walmsley, Lagrange, Emerson, Simpson, Landen, Waring, Bezout, Bossut, Lacroix, &c. may also all be rank- ed amongst the promoters of this science. Beside which there are several elementary treatises by Hayes, Ditton, Holiday, L’Ho- pital, Hodgson, Lyons, Rowe, Vince, Deal- try, &e. Principles of the Fluxional Analysis. Within the narrow compass to which this article must necessarily be confined, the reader will not expect a minute detail of the prin- ciple of fluxions. It will be sufficient to ob- serve that all finite magnitudes are here con- ceived to be resolved into infinitely small ones, supposed to be generated by motion, as a line by the motion of a point, a superfices FLU a line, and asolid by a superfices; of which ty are the elements, moments, or dilffe- “Aces, The art of finding these infinitely small antities, or the velocities by which they are nerated, and of working on them, and. dis- vering other infinite quantities by their ans, makes whatis called the Direct Method | Fluxions. And the method of finding the ‘ents or flowing quantities, these fluxions ing given, is what constitutes the Inverse ethod. ‘What renders the knowledge of infinitely all quantities of such great use and extent \that they have relations to each other, which » finite magnitudes, whereof they are the initesimals, have not. Thus for example, in a curve of any kind jatever, the infinitely small differences of » ordinate and absciss have the ratio to sh other, not of the ordinate and absciss, 't of the ordinate and subtangent ; and, of sequence, the absciss and ordinate being own, will give the subtangent; or, which iounts to the same, the tangent itself. \Notation.—The method of notation in flux- is, introduced by the inventor, Sir I. Newton, is follows: The variable, or flowing quantity, to be iformly augmented, as suppose the absciss a curve, he denotes by the final letters c, y, 2; and their fluxions by the same let- s with dots placed over them, thus, v, x, 7, =. id the initial letters a, b, c, d; kc. are used express invariable quantities. ‘Again, if the fluxions themselves are also viable quantities, and are continually in- vasing or decreasing, he considers the velo- lies with which they mcrease or decrease, | the flaxions of the former fluxions or second xions, which are denoted by two dots over m, thus, ¥ x Z. After the same manner one may consider » increase and diminution of these, as their ixions also, and thus proceed to the third, rth, &e. fluxions, which will be denoted, 1s, y nz: y 2 z, &e. Lastly, if the flowing quantity be a surd, as r—y, he denotes its fluxion by (“Wx— y)’ ; - itis denoted by (<= ;) Sometimes, however, the fluxions of com- und quantities are expressed by placing > letter F, or f before them, thus, instead ( /(e — y))*, is written F. /(@ — y), or /(a—y). We shall, however, in the fol- ving article denote the fluxion by F. and > fluent by f.; so that, | we (9) Ydenote the v(z—y) a thee § fluxions of ; z+ ax* b+2 b+2 1 v(x pirat denotethe §* ie mn ax") _*" _ (fluents of : x + ax? xz + ax” Direct Method of Fluxions. It follows from 2 preceding definitions and notation that the L fraction i FLU fluxion of x being x; of y, y; of z, z, &e. that the fluxion of (a2 +y¥+2+4,c.)=% +y+z2, &e. Also the fluxion of x, being x, the fluxion of ax will be ax; ofax + by + ez, will be az + by + cz, and so on of other similar expressions, But when we have to find the fluxion of a rectangle of two variable quantities, as xy, the exact determination of it is by no means so obvious, indeed the whole obscurity which is said to belong to this doctrine, would be in a great degree obviated, if the fluxion of the above expression could be satifactorily obtained. We cannot of course enter into a discussion on this subject, and shall there- fore determine it in the way in which it is most commonly done, though we are aware at the same time that objections may be start- ed against it. Let two right A lines, DE and FG, ¢|....-------2---------neeneen= move parallel to Fp |: themselves from two other right lines, BA and BC, and gene- rate the rectangle DF. Let them always intersect each other in the line BHR, and let Dd (wx) and FE f(y) be the fluxions of the sides B D (x) and BF (y), and draw dm and fn parallel to DE and BD. The fluxion of the area BDH is Dmor yx, and that of the area B F H is Fx or xy, and therefore the fluxion of the whole rectangle BH = ry = BDH + BFGwillbe xy + xy. Hence it follows that the fluxion of a rectangle of two variable quantities, is equal to the sum of the fluxion of each, multiplied into the other quantity ; that is, flux.vy = xy + yx. Having thus determined the fluxion of a simple rectangle of two variable quantities, we may proceed from hence to determine the fluxions of every other quantity, both alge- braical and transcendental. B 1. To find the Fluxion of a Product consisting of any Number of variable Quantities. Multiply the fluxion of each quantity by the product of all the other quantities, and the sum will be the whole fluxion required. Thus, FEF. xyz .kyz i +yrz +2ay F.xyzwreyzwt+yrrw+eryw twrye and so on, for any number of quantities. For put xy = ¢, then xyz = $z. Now F. @z = 92 + 293 but @ = xy, and conseq. ? = xy + yZ%, sub- stitutes these values in the above expression, and we have, ; : BF. oz =F. xyz = xyz + yxz t+ zry. Again, in the second example, putryz =@, then F. cyzw =F. Qu= Ow +whs | but @=2xyz, and 0 =xryz+yrz+zxry substitute these values in the expression, EF. Qw = Ow + wQ; and we have, FB. cyzworyzwt+yxzw+zaeyw +wxyz and the samerule will obviously obtain what- FLU FLU ever may be the number of variables of which qd k Ld te h put gi the product is composed. ANC DAKO 24 ee VRENCO Earn aie pi) 2. To find the Fluxion of a Fraction. fore px?—' x =qz—-!z. But 22-1 = “i From the fluxion of the numerator multi- a? P ; BI plied into the denominator, subtract the fluxion — = 2#?—% =2x@~—Y¢ substitute this for z | of the denominator multiplied into the nume- 27 rator, and divide by the square of the deno- Pe i minator. we have pa?—! x = qx9592Z, Pi YY — yx ; hated 4 P S are | hati; bee oo oz =f" isk xd x ‘| y y 1 yQ-G 1 | For make 7 = = 9; thn F. -=¢@. Whence it appears, that the same rule ha y place both for integral and fractional indices Nowz=y 2, ae &=yO + OY; and consequently also for negative ones; tha is, _ Or, substituting ~ 5 - for d, we have a AEE Or Sie yx. - 1 1 nar—ly i saat Oy =z; or, Rona tics es and hs tik =e LY—YX nax-"-! y, | i x 5. To find the Fluxion of the Hyperbolic | Logarithm of a variable Quantity. yx + Ach =xy, whence? = that is, F. = — Hy¥— ye 4 Cor. If the Vers of the fraction be a Divide the fluxion of the quantity by th constant quantity, having its fluxion equal to staan Thee and it will be the fluxion re zero, the fluxion of such fraction is equal to dre 1us 1 minus the fluxion of the denominator drawn F. h. log. « = . into the numerator, and divided by the square of the denominator. For put 2 = 1 + y, then : eee h. log. (1+y) = y— 4 y* + Fy3 _—iyt +e) That is, F. sate a as is shown’ under the article logarithms therefore, i 3. To find the Fluxion of any Integral Power F. log. a eco eye y+ yf y—yys of a variable Quantity. y(l—yt+y* —y3 +e Multiply the given power by the index, but 1—y +y*—y3 + &e, | then reduce its index by unity, and multiply by the fluxion of the simple quantity. That is, Bsa® = na® 7.2. series; therefore, This rule follows immediately from what has been shown with regard to the fluxion of F.log. 1 +y)=y x —— a product; for it is obvious in the first place, that the number of terms in that result is ‘. always equal to the number of quantities; F. log.’ +y)= F. log. a= -. and that the number of variables in each am! term, is always one less than that number. If, therefore, we suppose them all equal to each other, so that the product becomes a power of any one of them, then it is evident’ that its fluxion will assume the form above equal to M x ~, where M is the modulus ( given; viz. is derived from the conversion of 2 inf 1+ y l+y : y Te +y | But since 1 + y= 2, y = 2; whence, The logarithm of one system is always coi vertible into any other system by means | a constant multiplier. Therefore i fluxion of any log. of a, | Fivvxr meee + eaux + xex—3a%x, the system. if it be the common, or Brigg: F. exee = F. xt 4235, logarithm, then M = 0°43429, &c. and for th and generally, . hyperbola logarithm, M= 1. me —1 ging ES ka 6. To find the Fluxion of exponential Quantitie 4. To find the Fluxion of a Quantity having that is, Quantities which have their a | a fractional Index. a flowing or variable Letter. ‘ This is performed by the same rule, viz. | These are of two kinds, viz. when the roc multiply the proposed quantity by the index, is a constant quantity, as e*; and when th reduce that index by unity, and multiply by root is variable, as y*. the fluxion of the simple quantity; that is, In the former case, put the proposed Cs] F. a2” = na”"—! x, whether n be integral or -ponential e* =z, a single variable quantity fractional. then také the logarithm: of each, so shall loy We have demonstrated this above, in the 2 =a x log. 63 take the fluxions of thes: case where nis an integer, and have therefore in this place only to attend to the case in which n i: P e—e"x x log. ey the fluxion of the propose Se aos oe ae gq exponential e*; and which, therefore, is equi so shall z = a x log.e; hence z =z X Jo; ms FLU the proposed quantity, drawn into the flux- n of the exponent, and also into the log, of e root, . Also in the second case, put the exponen- aly ==; then the logarithms give log. z = | . | x log. y, and the fiuxions vive= =a X log. z | ~ 2 . } : zx1 +2x Y ; hence 2 = za x log.y + citob _ : 4 i q . . . * oy substituting y® for 2) y* « x log. y +ay"—! , is the fluxion of the proposed exponential fe; which therefore consists of two terms, of ‘hich the one is the fluxion of the proposed ‘uantity considering the exponent only as constant, and the other is the fluxion of the ame quantity, considering the root as con- tant. 7. To find the Fluxions of Sines, Cosines, &e. Suppose we require the fluxion of sin. z, hat is, the sine of the angle or are denoted yz, we must suppose that by a motion of ‘me of the legs including the angle, it becomes » 4+ 2, then sin.(z + 2)— sin. z Is the fluxion fsin. z. But according to the formule for he sines of sums of arcs (see SINE and TRI- 4ONOMETRY), we have sin. (z +2) = sin. z sos. 2 + sin. 2 cos. 2 the radius being assumed qual to unity. But the sine of an arc inde- initely small does not differ sensibly from hat are itself, nor its cosine differ percep- -ibly from radius; hence we have sin. 21 2s and cos. 2 =1; and therefore sin. (z +2) = sin. z + 2 cos. z; whence sin. (2 + 2) — sin. *, or (sin z) * = 2 cos. z, viz. the fluxion of the sine of an are whose radius is unity, is equal to the product of the fluxion of the angle into the cosine of the same are. ~ Jn like manner the fluxion of cos. z, or cos. ‘z+2)— Cos. z= COS. Z COS. z—sin. 2 sin. 2 — cos. z, or since cos. (z + 2) = COS. z Cos. 2 —sin. z sin 2; therefore, because sin. 7 = 2, and cos. z —1, we have F. cos. z = cos. z — = sin. z — cos. z = — z sin. z, that is, the fluxion of the cosine of an are, radius being 1, is found by multiplying the fluxion of the are (taken with a contrary sine) by the sine of the Same arc. By means of these two formule, many other fluxional expressions may be found, wiz. _ F. cos. mz =— mz sin. mz. FE. sin. mz = + mz cos. mz. F. tan. Zz act —.-. Cos. * z s, z }. cotan.z,=— ———. sin. * z z sin. 2 iz, sec, v4 = TE. Wares cos.* z ¥ Zz COS. 2 FP. cosec. 2.5 aoe sin.* z — msin.™—! zz COS. Zz. —— mcos.™—! zz sin. z. 8. To find the second, third, §c. Fluxton of a ’ flowing Quantity. F, sin.” z F. cos. 2 These fluxions differ in nothing, except their order and notation, from first fluxions, os.) ; FLU being actually such to the quantities from which they are immediately derived; and therefore they may be found, in the same manner, by the general rules already de- livered, Thus, by the third rule, the first fluxion of x3 is 3a° x, and if 2 be supposed constant, or if the root « be generated with an equable celerity, the fluxion of 3277, or 3x x x’, will be 3a, X 2ax% = 6x27, which is the second fluxion of «3, and 6 23 will be its third fluxion; but if the celerity with which x is generated be variable, either increasing or decreasing then « being variable, will have its fluxion denoted by *, &c. In this case the fluxion of 3a” x a willbe, by the first and third rules, Gra X & + 3a” X & = 6x2” + 3272, the second fluxion of 23. And the third fluxion of x3 obtained in like manner from the last, will be 6% x x* + 6x X 2xH% + Gua xX & + 327% — 643 4+ 18h ¥%a2 + 3a?k%. Thus also, ify’ Ae then Yi a xe — Lee x” +nxax—'; and if z2*—zry, then 2zz = xy +yx, &e. Ifthe function proposed were ax”, we should find F. ax" = nax"—!a; the factors na and x being regarded as constant in the first fluxion nax"—! «, to obtain the second fluxion it will suffice 1o make x”! flow, and to multiply the result by naz; but F. a*—* =(n— 1) a*—?2; we have, therefore, 2nd F. aa®=n(n—l)aa"—? 2”. 38d FB. aa" =n(n—1)(n—2) ax"—3 x3, 4th F. ax” =n(n—1)(n—2)(n—3) aa”—4 at &e. ees &e. mth F. ax” = n(n—1)(n—2) .... (n—m +1) arm ym m being supposed not. to exceed x, for it is manifest that in the case of » being integral, the function a2” has only a limited number of fluxions of which the most elevated is in the nth, and which of course is expressed by the formula, nth F. ax” = n (n—1) (n—2)...3.2.1.02" in which state it admits no longer of being put into fluxions, as it contains no variable quantity, or in other words its fluxion is equal to zero. Let us illustrate some of the preceding rules by a few examples. EXAMPLES. Lia? ss One at (iii) 2. F.(a+2)° —6(a+ x5)x _ iii) 3. EF. (Qax+a°)y = 4(a—2) (2ax—x*) «x (iii) : once x Qwie (a? —a)— 3 F..(a* —x*)5.4 _ 8x ay eb (a* —a?)s 5. F. x3 y+ = 3x? xy* +403 y3 y A&iii) _ 2eary?—2yy x" z Sa SE LEN RENAN PE 6 RS gn Qyxx—2xy. ' By y (= 2a Fe BS SiG) __ Oxa"— nal a mI 1 ns an 7. ¥, x” —N 2 ‘ é _ (i and ii) gti PER'U , 2—y _syt@—Qri—ry g. F. x iy. See ty x _ ax—4r2 an Cr Cr 10 —r+1 _ & V (2x48) (On 4 23 — V (a? +2”) a vero ax — at(Ya* +2") jeie ciicSi yt 2h yy Vy +6*) ~— (y* + b°)3 Inverse Method of Fluxions—In the direct method of fluxions our object is to find the fluxion of any flowing or variable quantity, for which the proper rules have been laid down in the preceding part of this article, and the operation is therefore always direct and easily accomplished; but in the inverse me- thod, where it is required to find the fluents of given fluxions, the operation is much more difficult, as no rules can be laid down that will be sufficient for performing this in all cases; for though every flowing quantity has its peculiar fluxion, yet every fluxion has not its fluent, at least not without having recourse to infinite series, quadrature of curves, or other methods of approximation. There are, however, a few rules which may be useful in many cases, and as such it will be proper to enumerate them. We have before defined a fluent of a fluxion to be that variable quantity, whose fluxion being again taken shall produce the one ori- ginally proposed. And hence we draw the following rules or formula, being the inverse of those given under the direct method of luxions. 1. When any Power or Root of the variable Quantity 1s multiplied by the Fluxion of that Quantity. Substitute the variable quantity instead of its fluxion, which will increase the index of the power or root by unity; then divide the quantity by the index thus increased, and it will be the fluent required. ‘That is, fomera ‘Thus for example, f. 30° 7 x8 10 3 f. Sura 5 Bik em at aze fi Qi + 3y3j) =f. at + f.3yy = 322+ 5 ys 2. When the Root under the Vinceulum is a compound Quantity, and the Index of the Part or Factor without the Vinculum, increased by unity, is some Multiple of that under the Vinculum. Put a single variable quantity for the com- pound root, and substitute its powers and Daly sky fluxions instead of those of the quantity itse Find the fluent of this simple fluxion, a) then re-establish the value of the compou root, which will be the fluxion required, I’or example, let it be proposed to find fluent of (a* + au*)5 x32. Put a* + 2” =z, then «* = z—a’; and 2ra = 2, theref. 23% =1 (z—a’ (a* + x?)5 sir t 2h Zs Thiet L yg? oe > gro z Se 232 2 (7—@)z= ey o..k Now f. (£23 z—1a*z3z) = 3 3 5 T6273 —PD¢z2= 6 10 3 se (a +27)3— 2 @ (a? + x*)3 which may be farther reduced to 3 2 2\> 2 3 Z ig @ ay 24 MAP eee 3. When the Fluxion of a Quantity is divided by the Quantity itself. Then the fluent is equal to the hyperboli logarithm of that quantity; or, which is th same thing, the fluent is equal to 2:30258 &c. multiplied by the common log. of th same quantity. Thus the fluent | of = oF x~'x, is the hyp. log. of 2; 22. ; of ae is 2 x hyp. log. of x, or hyp. log. of x? i : vt is the hyp. log. of a+ 2; | 3 22 A pa , is the hyp. log. of a + x3; of of & pe of Aa hae is hyp. log. of (x + /a* +a’), H bs . “21 ye Ol 7 eee a is hyp. log. (a a+ 4/ x? + 2az) 2 Hs of 5s, hyp. log. of SEE, 2ax , eres 2 : of 2". is hyp. log. of 22” @ Ee arf a* +x a+ fat x 4. Many Fluents may be found by the direct Method of Fluxions, thus : ‘Take the fluxion again of the given fluxional expression, or the second fluxion of the fluent . . * } sought; into which substitute 7~ for and xz ‘ y . i | a for 7, &c. that is, make Xx, #, 2, as alspi Y, ¥; y, &e. in continual proportion. ‘Then divide the square of the given fluxional ex- pression by the second fluxion, just found, | and the quotient will be the fluent sought in. many cases. ; Or the same rule may be delivered thus: In the given fluxion # write x for «, y for y, &e. and call the result G, taking also the fluxton of this quantity G; then make Gi :: | ve FLU :F, so shall the fourth proportional F be .e fluent, as before. This is the rule of |. Paccassi. It may be proved whether this be the true ent, by taking the fluxion of it again, which it agree with the proposed fluxion, will show at the fluent is right; otherwise the method Is. Thus, if it be proposed to find the fluent of Here F = nx"-! x; write first x ix, and itis na"! x, orn x” —G; the fluxion | thisisG =n’? 2"-'a2; therefore G:F :: G:F imnes n° ®—' ae sna es 3s na* 32" = F, ‘e fluent sought. For a second example, suppose it be pro- jsed to find the fluent of ry + ay. Here _=a«y+eay; then writing x for a, and y jy, itis ry + xy or 2xy —G; the fluxion (which is 2xy + 2xy — 4G; thenG: Tf :: (: Fbhecomes 2xy +2ayi:arytauyii2zry :y — PF, the fluent sought A few other partial rules might be given, t as they are generally founded upon the ‘erse of the rules laid down under the direct thod, we shall merely repeat the formule, ‘re given, and leave their application to the ll of the reader ; thus, pm a LA @y tye) =2y ee Se : f y y : ir rm MC) Lf. max Sia ie mg pa Rafe 1% 22 2 aor x4 I : ptq f= = hyp. log. x weg. ex X log. e = ex fy x log.ytayy ay To find Fluents by Series. Nhen the proposed fluxion is of any form included in the preceding rules, nor given ‘he annexed table, there is no other method obtaining the fluent but by a series, at it it will generally be found most conve- at and direct, and will apply in all cases ; ‘rule is as follows: ixpand the radical or fraction into a series \the binomial theorem, and multiply the ral terms by the fluxion of the variable, a take the fluents of the several terms, ch will be the fluent required. should be observed, however, that the (ntities must be so arranged that the re- ing series may be a converging one, which ' always be effected by placing the greatest a3 foremost in the given fluxion. =. FLU Exam. Required the fluent of x (a*—27)3. 62° 2 ‘elit CAE TA FER ms 5a* 25a* 125a0 1 Ke. | é: (a? — 2*)$ = 2 . xx Qartx 6 xox tx fg—m 2eté Gate ha te 5a” = 2% at 125a° s Ke, f Now taking the fluents of each term, we have, 62:7 125at And the method is just the same in all si- milar cases. If the index of the proposed fluxion be fractional the series will be infinite; but when it is integral the series will terminate, and the upnt will be finite. Thus it will be found that, gm a aem— — aC ae ae EEE ach ALR 5 +, &c. +a™h. log. (x—a) eo et ag! I et+a m m—tl m being any integer number. For a variety of fluxions and fluents of this form, see Dealtry’s “* Fluxions.” The forms under which fluxions may arise are extremely various, and the method of de- ducing the fluents excessively laborious and tiresome; and as we have before observed there are many cases in which it is impos- sible to obtain the fluent in a finite form, and even when this may be done it is stil] very difficult in many cases to ascertain it, On which account several extensive tables have been constructed by Emerson, Cotes, Landen, &c. from which the fluent of a given fluxion is found merely by a reference to the forms in the table. From the above works and the tables of Vega we have selected the following fifty forms, which if they are not so general as. those given by some authors, they, are at least those which most commonly arise in practice, and therefore it is presumed may, frequently become useful in the practical ap- plication of this calculus. See an extensive chapter of fluents in Dealtry’s ‘“‘ Fluxions,” p. 214; see also Cotes’s ‘‘ Harm. Mens. ;” Demoivre’s “ Miscel. Anal. ;” Emerson’s “ Fluxions ;”” Walmsley’s ‘‘ Anal. des Me- sures,” &c.; and Landen’s work above quoted. To find Fluents by Continuation. See Con- TINUATION; and for the method by rationak fractions, see RATIONAL Fractions. Correction of a Fluent. See CORRECTION. +, &e. tah. log. (a +a) FLU FLU TABLE OF FLUENTS. NO. FLUXIONS. FLUENTS. ees Se CTT on I.| 2*—? se Eh rok ‘ Mt : 1 Ay Ul. |(aka® "1g" ye fe —— (a Ea") mir gmn—1y>, 1 mn Il. |———___, = + x —————— (a + "+! Muna (a + a” es | Ne (at =| an)—I nin l y ws a xm "| gamne 1 ~~ - _mna gn v. (“24 y <)> =e} gt yt j (me mY “i +e ke. oad ae R= anys ee, amy” zP, Ke. i" x ae ft +./ bx iu vil.; ————_—— — ee lor! | isk o spb ae ab rn wee / (ta = ba") , VIII # —+ A x are. tang bs ‘la + bx Toes eid a dee th) x 2 Ja ++ /bzx ‘ BIT erage dcciseat seal x log. ———_—___~ v2(+ az bz) /ab O° A(t a Ge 02) X. Ahead + x are. tang. ne fisted x2(a + bz) b , x PI § 2 ? 5 Se ene Se Fa o. 4% /bx* ba VERE o> ein x log y vox + S(bx a) Kieth ad => th his x are. tang. sal / (a — bx*) Vb a—bzx x 2 ? X111.| ———_"__ eho a laa em xk (ba + a) bigdhs , x 2 XIV. | ——____—__—__ + — x are. tan Pe ine at / (a — bx) vb 5° pee x bias 2 J/a+ JS/(at bx) LY) | a aay Le or, 2 a/(a + bx) Va x 10g J 2 x 2 ba —a ; ‘\a/ (bx — a) & a ‘SL eereprer eri al Ya x log. evict net Aare Sp XVIII av0e =e) a at ag Fy % are: tang. J bima ——— } ty aq e ae. 5 ivi 4% Ee =i2) , xx ae be =-+ b yan x arc, tang. ee & xXI| a 2 YS x Nog. EE ¢ XXII. be + lat aa Xx are. tang. Mild 4 ad TEES me Aes beet + sy X x log. , Sbeit Sa @) ; XXIV. Ss = — av GF) Th PE i a. es xxvlivGerta) [= + bevGart a +55 x log. | vba + v(be* + «| FLU FLU TABLE OF FLUENTS. NO. FLUXIONS. FLUENTS. XXVI. [a / (a — ba*) |= 4+ Lavy(a— ba’) ue 57h * ate: tang. Vf ae — bx’ x < « «22 wore v (bx + a) — a EE, fee Pe o. ae | Vibe ta) 5 b 4. P08 {vba + vet «)} eee th 3 / (a — bx) BXVIII. |j=—————- j= — — Ve i /(a — bx) 5 + Fp % are: tang Mie ae ee _ . at(Qbeta)y bx +a) a> XXIX. @22/(beta)i= + Oe ee OE xlog. | J ba+ v (beta)! I oe _ , «2(2ba—a)/(a— bx) a bx XXX. w2x/(a—bax)i= + I CEIGIEY tay EY Har W859: x are. tang. / = 24/ (0 - a) |. t a 6 an PEE RD SR inet Vv bx +v (beta)! \a/(a — bx) bx XXX.) ——____*_ |= eas > * arc, tang. xe = tatv(a be) rvb i N aa enh Se RE, = 2V(a+ bz) —2at x log. va tv @ bz) en 2 v (bx — a) — 2a% x are. ianeia/ua eee a a 2 2 — =uike Oe) =+ vV(a+ bx?)—at x gy ee teal Ae ss bat) x . a ag XXXVI. Be sim = + v (bu? — a) — a2 X are. tang. pills a - << Vil.}———_ - is % log. —~—— a (a x) na Gat i n—1, feexxvinr| ~*~ = : log. wlth nf a—x2" 2n/a J/ & — x” 1 Yd gil. x arc. tang. —— REX IX} = AY isthe oi a +a X arc. Cos) ———— Q2n/ a a + x?” els 1 ‘ ss i 1 = 2% MY (ae er) | pet aah 1? Ai Ys a+ 2) { n—1 4 2 XLI. pes aerse ad * x are. sin. devin,» had X arc. vers. ra Vv (a—a2™) | n Aas In a XLIl. ee ex + log. & + eee) eae) Nf a J/(ata")+Va PTE} ek ee ister ‘Aas | er aren eecyMeior Pad) dee a | a7 (" — a) N/ a J a N/a x XLIV. x/(dx— x)= 4 cir. seg. to diam. d. and vers. x SCY, FRCL ce ee wo Drier. \@ + ba + cx” Vv (4ac—b?) Say th J (4ac — b*) Kignpreneh ke ki talus Zi log. SARS “EMA” CASE E a — a+ bx— cx’ Vv (4ac +b?) Vv (a +-bx — ex”) ; x iad J (a+bx+cx7) b +2c XLVI. ;— —_——___ UN TF ee tan. : he ae yr a a x a/(4ac—b*) a th fi v (4ac— bt 7 / (a-ha — cx?) b V (4ac-+-b*) —(b—2ca GV ITS 5 fee eel Rae ohn ns TS Bs ge Tabata a: tis x ay Gato) S: v (a + bax — ex”) ET 9, h coe ena cate te 5X log. (a+ bx +cx”)——___—_. X afo.tang, oe a+ bx + cx e/ (4ac—6*) °" / (4ac—b* LX Hysol» Epes b V/ (4ac + b*)—(b—2cx) POD IT bree |— 5, x log.(atbe cx Maat Oy 8 ae eT Fists FOC FLY, in Mechanics, is a heavy weight ap- plied to some part of a machine, principally in order to render its operation uniform, though it is sometimes employed for the pur- pose of increasing the effect, as in the comimg engine. FLYING, a progressive motion through the regions of the air, like a bird or other winged animal. The art of flying has been attempted in all ages, and various schemes have been devised and attempted, both for rising in the air, and for passing through it. We have even some records by friar Bacon, and others, of the art having been really attained, yet they bear so much the appearance of fables that no credit can be given to them; and it may fairly be stated, that nothing of this kind was ever effected till the invention of balloons, a totally modern discovery, of which we have given an historical sketch, under the article AEROSTA- TION. We have seen a work lately published on the science of flying, in which the author, theoretically, has the motion of an eagle; but practically, it is probable, it would more resemble that of a stone. FOCAL Distance, in the Conic Sections, is the distance of the focus from some fixed point; viz. from the vertex of the parabola, and from the centre in the ellipse and hyper- bola. FOCUS, is that point in the transverse axis of a conic section, at which the double ordi- nate is equal to the parameter, or to a third proportional to the transverse and conjugate 2 axis; that is, if CFD = ok then the point F is the focus. These points are thus denominated from a remarkable property that obtains in all the three figures; viz. that if lines be drawn from the two foci f, F, to any point in the curve P, and a tangent RT be drawn at their point of meeting, then those two lines will form equal angles with the tangent, that is, cavities, these act against each other, and a vent the free motion that would ensue, ona supposition of the two bodies being perfootay polished planes. i Amontons, a member of the French acas) demy, scems to have been the first who made experiments connected with this subject. He: found that the resistance opposed to the mo- tion of a body upon an horizontal surface was, exactly proportional to its weight, and was) equal to one-third of it, or more generally to one-third of the force with which it was) pressed against the surface over which it was passed. He discovered also that this resist-/ ance did not increase with an increase of the’ rubbing surface, nor with the velocity with which it moved. Cah The experiments of M. Bulfinger authorized conclusions similar to those of Amontons, with} this difference only, that the resistance of frie- tion was equal only to one-fourth of the force with which the rubbing surfaces were presse together. “Hs This subject was also considered by Parent,) who supposed that friction is occasioned by! small spherical eminences in one surface be- ing dragged out of corresponding spherical cavities in the other, and proposed to deter-_ mine its quantity by finding the force which would move a sphere standing upon three equal spheres. This force was found to be to the weight of the sphere as 7 to 20, or nearly: one-third of the sphere’s weight. In inves-| tigating the phenomena of friction, M. Parent: placed the body upon an inclined plane, andy augmented or diminished the angle of inclina- tion till the body had a tendency to move, and the angle at which the motion commenced he: called the angle of equilibrium. The weight of the body, therefore, will be to its friction upon the inclined plane, as radius to the sme of the angle of equilibrium; and its weight | i git t FRI will be to the friction on a horizontal plane, as radius to the tangent of the equilibrium. Euler seems to have adopted the hypothesis of Bulfinger respecting the ratio of friction to the force of pression; and in two curious dis- sertations which he has published upon this subject, has suggested many important obser- vations, to which Mr. Vince seems afterwards to have attended. He observes, that when a body is in motion, the effect of friction will be only one half of what it is when the body has begun to move; and he shows that if the angle of an inclined plane be gradually increased till the body which is placed upon it begins to descend, the friction of the body at the very commencement of its motion will be to its weight or pressure upon the plane, as the sine of the plane’s elevation is to its cosine, or as the tangent of the same angle is to radius, or as the height of the plane is to its length. But when the body is in motion the friction is diminished, and may be found by the follow- m 15625 nn cos. w which » is the quantity of friction, the weight or pressure of the body being = 1, a is the angle of the plane’s inclination, mis the length -of the plane in 1000th parts of a rhinland foot, and x the time of the body’s descent. Re- specting the cause of friction, Euler is nearly of the same opinion with Parent; the only dif- ference is, that instead of regarding the emi- nences and corresponding depressions as sphe- irical, he supposes them to be angular, and imagines the friction to arise from the body’s ascending a perpetual succession of inclined planes. Mr. Ferguson found that the quantity of friction was always proportional to the weight of the rubbing body, and not to the quantity of surface; and that it increased with an in- crease of velocity, but was not proportional to the augmentation of celerity. He found also that the friction of smooth soft wood, moving upon smooth soft wood, was equal to one- third of the weight; of rough wood upon rough ood, one-half of the weight; of soft wood upon hard, or hard upon soft, one-fifth of the weight; of polished steel upon polished steel or pewter, one-quarter of the weight; of po- lished steel upon copper, one-fifth; and of polished steel upon brass, one-sixth of the weight. The Abbé Nollet and Bossut have distin- guished friction into different kinds, that which arises from one surface being dragged over an- other, and that which is occasioned by one body rolling upon another. The resistance which is generated by the first of these kinds of fric- tion, is always greater than that which is pro- duced by the second; and it appears evidently from the experiments of Muschenbroek, Schoe- ber, and Meister, that when a body is carried along with an uniformly accelerated motion, and retarded by the first kind of friction, the Spaces are still proportional to the squares of the times; but when the motion is affected by the second kind of friction, this proportionality ing equation « = tan. a in FRI between the spaces and the times of their de- scription does not obtain. The subject of friction has more lately oc- cupied the attention of Professor Vince of Cambridge. He found that the friction of hard bodies in motion is an uniformly re- tarding force; and that the quantity of fric- tion, considered as equivalent to a weight drawing the body backwards, is equal to M Mx W x : : x as where M is the moving force expressed by its weight, W the weight of the body upon the horizontal plane, S the space through which the moving force or weight de- scended in the time ¢ and 7 = 16°087 feet, the force of gravity. Mr. Vince also found that the quantity of friction increases in a less ratio than the quantity of matter or weight of the body, and that the friction of a body does not continue the same when it has different sur- faces applied to the plane on which it moyes, but that the smallest surfaces will have the least friction. Notwithstanding these various attempts to unfold the nature and effects of friction, it was reserved for the celebrated Coulomb to sur- mount the difficulties which are inseparable from such an investigation, and to give an ac- curate and satisfactory view of this complicated part of mechanical philosophy. By employ- ing large bodies and ponderous weights, and conducting his experiments on a large scale, he has corrected several errors which neces- sarily arose from the limited experiments of preceding writers: he has brought to light many new and striking phenomena, and con- firmed others which were hitherto but partially established. We cannot in a work of this kind follow M. Columb through his numerous and varied ex- periments ; all that can be expected will be a short abstract of the most interesting of his results; a few of which are as follow: 1. The friction of homogeneous bodies, or bodies of the same kind moving upon each other, is generally supposed to be greater than that of heterogeneous bodies; but Columb has shown that there are exceptions to this rule. 2. It was generally supposed that in the case of wood, the friction is greatest when the bo- dies are drawn contrary to the course of their fibres; but Columb has shown that the friction is in this case sometimes the smallest. 3. The longer the rubbing surfaces remain in contact, the greater is their friction. 4. Friction is in general proportional to the force with which the rubbing surfaces are pressed together, and is commonly equal to between one-half and one-quarter of that force. 5, Friction is not generally increased by augmenting the rub- bing surfaces. 6. Friction is not increased by an increase of velocity, at least it is not gene- rally so; and even in some cases the friction decreases with an increase of celerity. 7. The friction of cylinders rolling upon an horizontal plane is in the direct ratio of their weights, and in the inverse ratio of their diameters. From a variety of experiments on the fric- .] ie o. FRU tion of the axis of pulleys, Coulomb also ob- tained the following results: —When an iron axle moved in a brass bush or bed, the friction was 4 of the pression; but when the bush was besmeared with very clean tallow, the friction was only =; when swines-grease was inter- . 1 —; and when SF posed, the friction amounted to olive-oil was employed as an unguent, the fric- : ] ~ tion was never less than 4, or 7B When the axis was of green oak, and the bush of, guaia- cum-wood, the friction was 4 when tallow was interposed; but when the tallow was re- moved, so that a small quantity of grease only covered the surface, the friction was increased to s,, When the bush was made of elm, the friction was in similar circumstances = and oi, Which is the least of all. If the axis be made of box, and the bush of guaiacum-wood, the friction was 3% and {4 circumstances be- ing the same as before. Lf the axle be of box- wood, and the bush of elm, the friction will he =, and 44; and if the axle be of iron, and the bush of elm, the friction will be =4, of the force of pression. ~ On this subject the reader may consult Gregory’s “‘ Mechanics,” vol. ii. p. 24,44; Fer- guson’s “ Lectures” by Brewster, vol. ii; Desaguliers’s “ Exp. Phil.” vol. i.; see also the Phil, Trans. vols. i., xxxiv., XxxXvii., Jili., Ixxv., &e. FRIGID Zone. See Zone. FRIGORIPFIC Particles, is a term used by some early philosophers to denote what they considered the matter of cold; as modern phi- losophers make caloric the matter of heat. FROST, that state of the atmosphere which causes the congelation or freezing of water, or other fiuids, into ice. FRUSTUM, in Geometry, is the part of a solid next the base, left by cutting off the top or segment, by a plane paralicl to the base; as . the frustum of a pyramid, of a cone, of a co- noid, of a spheroid, or of a sphere, which is any part-comprised between two parallel cir- cular sections; and the middie frustum of a sphere is that whose ends are equal circles, having the centres of the sphere in the middle of it, and equally distant from both ends, Lo find the Solid Content of the Frustum of a Cone or Pyramid. Add into one sum the areas of the two ends and the mean proportional between them; then $ of that sum will be a mean area, or the area of an equal prism, of the same altitude with the frustum; and consequently that mean area being multiplied by the height of the frustum, the product will be the solid content of it. That is, if A denote the area of the greater end, a that of the less, and A the height; then (A +a+/Aa) x th is the solidity. . Or if S,s, be the sides of the bottom and “top of the frustum and ¢, the tabular number, for that figure, then (S? + sS +s?) x4 th= solidity of Frust. of Pyram. and (D* + dD+d?*) x thp = solidity of Frust. p "7854. ? FUL . of Cone, where D and dare the diameters, and Other rules for pyramidal or conic frustums may be seen in Hutton’s Mensuration, p. 189, 2d edit. 1788. q The Curve Surface of the Zone or Frustum of a Sphere, is had by multiplying the circumfe-_ rence of the sphere by the height of the frus-— tum. Hutton’s Mensuration, p. 197. ib And the Solidity of the same Frustum is found, by adding together the squares of the radii of — the two ends, and 4 of the square of the height of the frustum; then multiplying the sam by the said height and by the number 1°5708 5 — that is, (R* + 7? + $h*) x iph is the solid” content of the spheric frustum, whose height — is h, and the radii of its ends R and r, p being = 31416. Mensur. p. 209. 4 For the Frustums of Spheroids and Conovds, — either Parabolic and Hyperbolic, sec the seve-_ ral articles. 3 >: FULCRUM, in Mechanics, the prop or sup= port upon which a lever is sustained. b FULMINATING Powder (from fulmino, to- thunder), is a prepared powder which explodes upon the application of certain degrees of heat, with instantaneous combustion and prodigious — loud noise. . Simple fulminating powder without any me-~ tallic substance is thus prepared: Take three parts of nitre, two of purified péarl-ash, and— one of flowers of sulphur; mix the whole very accurately in an earthen mortar, and place if | on a tile or plate before the fire till it is per. fectly dry: then transfer it while hot into a’ ground stopper-bottle, and it may be kept. without injury for any length of time. Inv order to experience its efiects, pour from ten” to forty grains into an iron-ladle, and place it over a slow fire: in a short time the powder becomes brown and acquires a pasty consist+ ence; a blue lambent flame then appears on the surface, and in an instant after the whole — explodes with a stunning noise and a slight momentary flash. If the mass be removed. from the fire as soon as it is fused, and kept in a dry well-closed vial, it may at any time be exploded by a spark, in which case it burns, like gunpowder, but more rapidly and with greater detonation; but this effect cannot be produced on the unmelted powder, how accu= rately soever the ingredients of it are mixed. together. When fulmimating powder is in fusion, but not heated to the degree necessary to produce the blue flame, a particle of ignited charcoal thrown upon it will occasion imme+ .diately a remarkably loud explosion. 4 It appears that the ingredients of this pow= der do not acquire their fulminating property till combined by fusion; in other words, till the potash of sulphur form sulphuret of pot- ash: whence fulminating powder may also be made by mixing sulphuret of potash with nitre, instead of by adding the sulphur and alkali separate. t ie In all these the cause of the detonation or fulmination is not accurately understood, Ta FUN mple fulminating powder, there is a very ge portion of elastic gass evolved; in ful- inating gold or silver a much smaller; yet ie explosion in the latter case is infinitely reater than that in the former. FUNCTION. A quantity is said to be a imction of another quantity, when its value epends on that quantity and known, quanti- es only; and it is said to be a function of syeral quantities, wheu its value depends n those quantitics and known quantities aly. hus, if y = 2", .y <= (a + bx)", y = at, = log. x, &c.; or, in the equation "+-(a-+-by).2"—! +-(a+-dy-Ley’).x®—? + ,&ce.= 0 is said to be a function of 2. In the former ase y is called an explicit, and in the latter a implicit function of a. And if Sax” +b2", y= a+b", y =log.x+log.z, ke. rin the equation + (a+by)a"—',+, ke. © +(a + Wy) .2"-! 4+, &e. +atBytyy*...vy?).arze—P—™ 4+, &e, = 0 is a function of x and z. a, b, &c. m, n, Ke. eing constant quantities. ‘Such being the meaning of the term, algebra i general might be considered as the science f functions, since in it the various relations f the values of quantities are the objects of ivestigation; and in this manner it was cus- mary to regard functions, until Lagrange sserted (Memoirs of the Academy of Berlin, 772) that the theory of the expansion of func- ons, in series, contained the true principles * the differential or fluxional calculus; and y this theory demonstrated Taylor’s celebrat- 1 theorem, which he justly regards as the indamental principle of the calculus. He terwards collected and generalised his ideas a this subject, and published them in a work ititled “* Théorie des Fonctions Analytiques,” hich appeared in 1797; to this, in 1806, he Ided a commentary and supplement, under ie title of *‘ Lecons sur le Calcul des Fonc- ons.” Since this, the investigation of the pro- erties of the functions which result from 1e expansion of any function in a series, on ‘tributing an increase to one or more quan- ties in that function, has been considered as ie calculus of functions. The calculus of functions has the same ob- et as the differential calculus, taken im the ost extended sense; but it is not subject to 1e difficulties which are met with in the prin- ples and ordinary course of this calculus: it asides serves to connect the differential cal- ulus immediately with algebra, from which e may say it has till now made a distinct tience (Lagrange, Legon 1°®), Indeed the chief object of the calculus of inctions is to establish the principles of the ixional or differential calculus, without hav- lig recourse to velocities, or to vanishing or arto: small quantities. We will endea- ur to show that this has been accomplished: ad for this purpose we will give.as clear an count as our limits will permit, of some of | FUN the leading propositions in the calculus, and their mode of application. Let fx denote any function of x; then if a be increased by an indeterminate quantity 7, fx will become f(@ + 7); which, according to Lagrange, may be expanded in a series fx + pi + gi? +, &c. where p, g, &c. are func- tions of x, independent of 2*. To ascertain the manner in which the func- tions p, g, r, &c. are derived from fx, let us suppose that 2 is again increased by the inde- terminate quantity 0, which is independent of i; f (w +7) then becomes f (x + 0 + 7); and it is evident that the result must be the same, whether we suppose that it is x + 0 which is increased by 7, or that it is x which is increased byo +ziori+o; thatis, fi@totilasferito} Now since f(a‘+7) = fx + pi + qi? +r, Ke: and that p, q, 7, &c. are independent of 7, we have fy xt +0) = fe tp. +0) + qi +0) +r(2 +0)? +8(4 + 0)* + &e. =fetpit qv? + re + sit +&c. + po + 2qo0i + 3ro0i7+ 48013 + &e. + go* + &e. ; + ro? + &e. + sot + &e. and f(x + 0) = fx +po+qo*+ro3+sot + &e. As p,q, &c. are functions of x, we may sup- pose that when x becomes x + @, they will be- come p+pot+ &e,q+ dot+ kc; p’, 7, Ke. being likewise functions of,2, independent of o; hence fl@totil=feto+(ptpo t&eit (q+ qo+ &e.).27 + (r+r'o+ &e.) 8B +&e, = fxtpit qe + ri3 + si* +e. + po+ p'ot+q'o1* +7013 +&e. + qo + &e. +e. +e. 4+ ro3 + &e.+Ke. + so* + &e. And since this series must he identical with the former, the terms affected with the same powers of 7 and o must be the same, indepen- dent of any values of 2 and 0; consequently we shall have, 29- = DOr Os. 46 — 7; So. or q=> yal, sa, &e. Let f’x denote the co-efficient of 7, or that part of the second term in the expansion of f(«+%) which is independent of 2; and in like manner let fx, f’’x, fiw, &c. denote the co- efficients of 7 in the expansions of f” (x + 2), f' +0, fF" @ + 0, Ke. Thus we have p = fx; now p’ is the co- efficient of 7 in the expansion of p, when in p, x becomes x + 72; that is, p’ is the co-efficient of 2 in the expansion of f (x +1), or p= f"x and-.°, ¢ = is in the same manner, q' is the * For the objections which may be made to this, see Preface, p. xviii. of Woodhouse’s excellent work on the “ Principles of Analytical Calculation.” FUN co-efficient of 7 in the expansion o pf@t+h 2 2 Mos MIT or q' = 2 ; and ae “i? likewise s = 2.3 pat: 7 , &c. Substituting these expressions for Pp; 7; 7, &e. in the series fx + pi+qi?? +ri3 +, &c. we shall get this fundamental formula, f@torfut fruit: fe phe ae te) . +, &e. 2.3.4 where f'x, f’x, f''x, f'’ x, &c. are derived from SRS BHf'D Sf, Ke.; ; in the same manner, namely, by taking the co-efficient of 7 in the expansions of f (ew +2), f'(«@ +0, f’ («@ +d, ra (x + 2), &e. Hence, if for every function of a, we can find the co-efficient of 7 in the expansion of f(« +, we may find the co-efficients of all the other powers ‘of 7. f xis called the primitive function with re- gard to f’x, f’x, fx, &ce. which-are’ derived from it; and these are called derived func- tions, with regard to fa. fois called the first function , fx the second function, fx the third function, &c. of fx. if aletier 3 y be put for the function of a, and y', y’, y'", &e. denote the first, second, third, &e. functions of y; then when a becomes x oa i, ¥ will become yr yitd. a* 5 Yi +, &e, Ww e may here ae Ne faat Sx, f'n, &e.; y', y", &e. are the same, as the fluxional co- ce ys ,&e. aS eh , &e.; or as the é d* fx dpe D.f v5 oe D*f'x, &e.; yi Se, Dt y, &e.; and that the dive ‘ital agrees with Taylor’s theorem (“ Methodus Incrementorum,” p. 23), namely, that z flowing Gant an and becom- kg Te. oe OR 1.2.27 See ee efficients .——— differential co-efficients dy D.. ad x ing z + v, x becomes x + 2. dak | ys target &e- We will now illustrate what has been said by applying it to the expansions of the func- tions a”, a” and log. x, when 2 becomes «x + 7. First let sey ogg “then f(a +)=(e@ + "3 and it is easy to show, by the first operations of algebra, that the two first terms of the ex- pansion will be 2” 4+ ma™—17, whether m be an integer or a fraction, positive or negative ; consequently pi x= mx"—!; again, the co- efficient of a in (x + 2)"—-! expanded, is Mitt BP tarts IW AN hmmm 2 (m — 1). ” raid and in like manner fx =m. (m—1).(m—2) “Tdi &e. And since S(x@ +i) =fx + f'xi +e +, &e. + am + mam 1p pO Vom an 2 FUN. peat aa am—=3 43, Ke, which is the well-known binomial theorem. — Next let fa = az, then f(# +7) = atts - a® at; now a —(1t+a—l1=1 +i. (ga 2, t-(t—1).(i—2) sree sae rage Spek er + far DHL? + F(a—1) —, &es &e. omitting the following terms, since it is onl the co-efficient of i which is required ; or ce we sie A =(a—1)—}(a—1)*4+ F(a— 13 —, & at =1 + Ai +, &e. j . att+t = ax + Aa*.i 4+, &e. 4 consequently the co-efficient of 7 in at+i ey panded is Aa’, or fix = Aa’; hence f’x be ing the co- -efficient of iim Aarti expand 4 Aza’; and in the same manner, f’“x = Ase &e. Substituting aie % we shall have atti = at + Aa‘. peo Ai +, &e Zz meat (1 Ba r+. i +, &e Here, if we divide by a”, we nt : A3 . 2 3 Sc rcae .‘ "aa" +, &e, z— eS sa 3 &, or a sig, Wire ae yet AS ot +, Be which is the series for the expansion of a®,. Suppose now x = st a pe have a=1tA¢ D4 So Let e a the Tae of i Aga s thei 3 ts &e. a ae =1414! at * 55 + ke.=2.718281828459045 Again if x ee 3 1s 1 1 a —=1+1+=+ +,kce. =e “. «=e, or A is the logarithm of @ the base e. If Z. x denote the logarithm of 2 to the ba e that is the Naperean or hyperbolic logarith of x; and L.x denote the logarithm of # the base a; then « = e!-* and x = a"*, @ But a= — eA-L. ‘Mee Gh ahes.» (te celle By which equation, the logarithm of a nul ber being given, in any system of logarithi it may be converted into the Naperean log rithm of that number, and vice versa. If X =(a—1)—1}(a—1)* + at Ip +, s pene roe one 2 +a 1 | rads & Mf x 1 1 é ge ren pened So , om peat, As Aa WA Ni ibe oh a Py € : and 2 rite orz—a4 si l.x that is, i aie and I oye FUN xe (r—1)—1 (a—1* + F(a#— D3 —, Ke. { (@—1)—H@—1) + eI) 8c. i ; 1 Pee as 6 A » We will now proceed to the expansion of 4(x + 2). ; faeCt) —eficer (1 +=) = alt x al(i+-) e = ale th(1+3) sL@+)=betb(i ++)= x Bex i()—3G) ta G) 8 —%#(-)——(- —~{—) —, &e. tx i(3)—ale) tal) > Ki Sy Mean ae Sane Let + j}ti— 55 .i Tos » ket This expansion has been obtained without gain referring to the general expansion; we vill, however, show how the same series may e obtained from the expansion of f(a + 7) Ge Sat ps Ke. t =2? Saif 4 S(n—a)4it* ; +4 §@—D+if?— ke} (et Ba | ; —. ? —— . + §1—@—1)+(e—1)— | (x1) 44 &e.$ f4e ace! Neglecting all the terms containing the higher lowers of 7, since it is only the co-efficient of 2, hich is wanted. Now 1—(x—1) + (r—1)*—(a— 1) +, Kee. i 1 71 +(¢—)) | oL@t)=lep—.i rihag 7 x lonsequently, fax Px the second term of x hel 1 wile ‘ | acne — ‘ae Z d i Rewicx) or 7 (a + 2)~! expanded is : ee eed ee ay ent kes aa ae a. Aye et, ee 4 Hf —_ Ax’ in ‘ke manner f” =+ 2 fig sro Bee » Ax?’ Kat Substituting these in the general expansion, ye have “i BP Minted egy tt Brows et)=Let— (i P+, Ke ) s before, and ; Maes) 28 ate? don, a ee | x Qa:* Ean Gal Let us now proceed to find the derived ‘unction of the function of a quantity, which /uantity is itself a function of x, and likewise ae derived functions of the function of two or ‘nore functions of zx. | First, let y = Jp, p being a function of x; | FUN if p be increased by o, then fp will become Sp tf p.o Spee .o +, &e. But p being a function of « when 2 becomes x -+- 2, p be- comes p + p't +5 i, &e. oO pt + Ff 7 +, &e. and if this be substituted for o, Sp will become i 1 ; \ “i "gl 7) 4 fotpfp it+(PLPTPlP) #4, we, “y =Pf Pp Now let z be a function of p and q, which are functions of x. Suppose that this function is represented by f(p, q); and that f’ (p) de- notes the first function of f(p, q), considered as a function of p only, q being regarded as constant; and fq) denotes the first function of f(p,q), considered in like manner as a func- tion of q only. It is evident that the result must be the same, whether we substitute 2-+2z for x, in-p and q, at the same time, or first substitute it in p and then in q, in the result of the first substitution. ‘Therefore substitut- ing a +7 for x in p only, f(p, q) will become Sip Ot pf (p).i+, &c.; and substituting x7 for # in g only, in this expression, it will become FD DEPF OP? 5 +f Ds sahiiea et = PL Peed and by taking the first function of this we get 2", and so on. From this we likewise obtain y” in the fore- going case; for y" = (pip) = POPPY + PSP = PPPS PFPH=PLPLPSP . and taking the first function of this, we get y”, and so on. If y be an implicit function of x, y' may be determined, thus: Let the equation which determines the va- lue of y be denoted by F (x,y) = 0; then ¥ must be some function of x, which we will denote by fx. Substituting this for y we shall have F (x, fx), a function of x alone, which let be denoted by ¢2, and which is = 0, what- ever may be the value of a. Consequently it will be = 0 when x ++ 7 is substituted for x. Therefore we shall have Dae Ox + Qxi+t cei 4 &e. =o. And as this must be independent of any par- ticular value of i, each of the co-eflicients must be = 9, or Ou = 0, o'x =, Ox = 0, &e. Since @a = F(a, fx) = F (x,y), Qa will be the first function of F (2, y), y being considered as a function of 2. F@+y/FY~)=Fap=Urseo oe en fae and y = Fo) We shall get y"” by taking the first function of this, and so on. FUN The equations F" (x,y) = 0, F’(«,y) = 9, F’" (x, y) = 0, &e. are called first, second, third, &e. derived equations of the primitive I’ (2, y) =o. And any equation formed by a combi- nation of F‘(#, y) = 0 and F (x,y) =2, is call- ed an equation of the first order; of P"(a,y)=0, ¥’ (a, y) = 0, and F (a, y) = 0, an equation of the second order, and so on. The manner of finding the derived functions or equations, from the primitive functions or equations, may be called the direct analysis of functions; and the manner of arriving at the primitive functions or equations from the derived functions or equations, the inverse analysis of functions: thus corresponding to the direct and inverse methods of fluxions, or the differential and integral calculi. The difficulties attending the latter analysis, and the artifices necessary to be made use of in it, are the same as in these calculi, and for in- formation on this subject we must refer our readers to Lagrange, ‘“‘ Théorie,” p. 51; jus- qu’a, p. 115; “ Legons,” p. 111, et suiv., for what relates to the subject of primitive and derived equations; and to “ Théorie,” p. 117, et suiv. for the application to curves and to mechanics; see also Lacroix, “‘ Calculs. Diff. et. Integ.” Having shown how the several derived functions of ymay be found, when y=/f (p,q), p and q being each functions of the same quantity 2, and consequently how f/f, (p, q), may be expanded when in p and q, x becomes a +2; let us now consider how we shall find the expansion of a function of two quantities, which are independent of each other, when each is increased by an indeterminate quan- tity, for instance, of f, (x, y), When 2 becomes x + i, and y becomes y + 0. Here, as in the former case, we may suppose the changes in a and y to take place separately, that is, that y remains constant while 2 becomes 2 + 7; and then that x remains constant while y be- comes y + 0. We shall thus have different derived functions of f, (x, y); first, on the sup- position that f (x, y), is a function of x only, y being constant; and secondly, that it is a function of y only, x being constant. Not to confound these different derived functions, we will denote those derived on the supposi- tion of a being alone variable, thus f” (2, y), St” (@, y); those derived on the supposition of y being alone variable, thus f (a, y), S’" (@, y), &e.; and those derived on the sup- position partly of 2 variable, and partly of y variable, thus f” (x, y), f’’ (a, y), fo" (a, y), &e. . Thus we shall have in the first place Seti y=f ay +f? a pee OD i 4 Mt, R sD Yes teh substituting y + 0 for y, we shall have Si bay +0)=f (x, y+o) +f'.(2,y +0).0+ PU OYTO. 8,6 FUN + ‘ = f(x, y) nse ; +f (ay) 0+f" (ayy) do. 04.8 +E » % o* T e Y) . 2 07+ &e. ; ee Wl 7 +O) 08 + &e. bf in which the general form of the terms is | Yh aciaa (2x, y) A CRO I Sra ae 1 where m and n denote the order of the function, It is evident, that we must have obtained) the same result, if we had first supposed y to) become y + 0, and then x to become x + 7; consequently, to obtain the derived functida F(z, y), it will make no difference whether we first get f> (a, y), and then the first funes tion of this relative to y, f°’ (x,y); or, first get J’ (a, y), and then the derived function of this. relative to 2, f’’ (a, y); and the same of St" (2, y), fo" (2, y), &e. It therefore appears that the operations which are denoted by f% Jf’ &c. may be performed in any order without at all affecting the results. For example, let 2 oe x He ’ = 3 x* 7 FLY) af then f? (*, ¥) = ye and ‘1 6 1” . 2x43 fi @ N=—=S3 orf? (4, =— Sal ay 6x3 1 6x 4 f'@ya— rut also f” (x, y) = iss and ' 12x a FG x* p f"' @&yr=— re 5 of. Cesiy is —, and y y rk 18x ‘ ft (x, Y) _ ae na and the first functionay | j a tS (4, y), relative to x, is — a and relative x at , and so on. % y* “a place in order F (x, 9) +9 (yy) may =f"! (4, 9), that is, if F (a, y) =f” (, y), and @ (4,y) S J’ (ay); then must FY (x,y) = 9° («, y). 9 — Xx F a Spas may = Sy) So (See A cide ik and( —~. )= x — ye x fy? oe CaM 9h La x* fy? ~ (a* py? i We may here just observe, that f” (x, 9) LGD £9 Y); f" & WY), FG ye x, correspond to the partial differentials to y is And this condition must have that lor example, dx d(x, y) d(x, y) d(x y) @ (x, y) dye) Naa OMG eo tr ee f \; which therefore arise from the expansion of 2 function of two or more quantities, that are independent of each other, on attributing t each of those quantities an indeterminat increase. See PARTIAL Differential. a If we had an equation F (2, y, z) = 0, We should get the derived functions of z thus considering only x as variable, from what ha: FUN een given respecting F (a, y) = 0; we ave F', (a, y,z) + 2° F’ (z) = 0, and con- dering y alone as variable FY (*, y, 2) + %. EF’ (z) *(z) denoting the derived function of F (x, _z), considering z alone as variable. Hence aving 2’, and 2’ we may find 2%, 2”, 2’, &e. Having given the methods of expanding iferent functions, and of obtaining the de- ved functions, of functions of one or more aantities, whether those functions be ex- icit or implicit, and the quantities de- endant or independant of each other; we ill now show how this theory may be ap- ‘ied to those purposes to which the differen- al or fluxional calculus is usually applied. ‘ur limits will not, however, allow us to do is more than in a single instance, and we ink we cannot select one better calculated illustrate the application in general than the eory of contacts. But for this and other yplications to geometry, and likewise to echanics, it will be necessary to demonstrate e two following propositions: t. That y being a function, 7 may be taken so i a” aw aallin y + yi + oa a t3* + &c. that ty term shall exceed the sum of all the fol- wing terms by any quantity less than itself, ovided that the quotient of any term di- ded by the preceding is in no case infinite. swe do not think that our learned author s shown this, in a manner, by any means sar or satisfactory, either in his “ Théorie,” 12; or in his “ Lecons,” p.89; (since in the st the demonstration depends on the theory curves, and in the second, we doubt whether e property in question be not necessary to 3; preliminary proposition), we shall give a monstration on a similar principle to that ' Woodhouse, “ Prin. Anal. Cal.” p. 166. . é Pek tee yp In the series y+y't+ at as CR ypr? 2.3...p.(p+1)(p+2) y jm pF eR Se P+ +1 4 ipt2 , +, let the consecutive terms ym ; : im+l be such, that the 2...m.(m + 1) -eflicient of the latter, divided by that of the ym ; (m+1).y” icient of any other term whatever in the ‘ies, divided by that of the preceding. ‘mer, or is not less than the co- shi kya sen if ¢ be taken < mer (n any mber) the sum of all the terms after any m, as —”"— i? will be oe 2.3... aP yrptl (m+ 1). y™ i.3...p eee ny mi fF . ypt? (m + 1). uy fr (Pp +h). @t 2 myn FUN mag oe (ERY wir PV-0F9-0P) peers) pint an Wd aE DAG gett (pts 1) EET Oe, . Up+t’ Ta ya Wie Whats 5. (ptD.y? (p-++2). yt * CRSA: CREA ABE nyPt? PROG KAPPFS i di oii hw : (P+). (p 1). (p + 2) (p + 3) ny pth (p= 2). Prt ” (P+ 3) ypt2 Bs Ke, ? nyPt? n.yPrs * § (since Maule Ob EAL a i -y" is not > (p 7 yP lor (p+2)yrt! (p43). yPt? ¢ IT TSE rey &e.) ni 1 1 1 OR se eS Now, if x = 2, orz (m + 1). y”™ yp aa de ; < Soe OD att will exceed the sum of the following terms by a quantity>0; and it is evident by taking n greater, and Modes cinlin 1P is BURY ele oF made to exceed the sum of all the following yP 2.3...p ymth Ag! Hee Ga aay is in no case infinite. 2d. That y and « being functions of x, y= fx, u= Fx, « becoming « + i, and y /it consequently z less, that terms by any quantity < w, provided that and wu becoming y + yt + o. a? + 35 23. i“ “a . u . u . &e., and wu + wi tire a thats ® +. &c.; ifu — y, and w = y’, then? may be taken so small, that the value of another function of x, W le v= Oe, ory + vi + =? +553 + &e. cannot whenv = u = y fall between the above values of y and uw unless v = wv = y’. Let D =f (« + 1) — 9 (~-72) and AxzO(« + i) — F(*#—?) ‘i party ; yf —v" * yf" —v" * De ce 6 bch ea 43 paver ae ar 3+ &e, ze” hd — {5 +i? + —— 23 + &e. + 2 Re 2.3 ipa at aig gy — yl" oe » pas 3 + Ko, Am w—uv).t+ 2! 2" 3 & pin he Me we aide ceo oe cs —27it+ > a +535 + ift be put for y —v, and z forv—u. Now we have shown that in these 2 may be taken so small, that the first term shall exceed the FUN sum of all the following terms, and conse- quently that D shall have the same sign as y'—v’).t and A the same sign as (v'—w').2, or (v' — y') .2 (since wu = y’): that is, i may be taken so small that D and A shall have contrary signs; or, that if@(# +72)is < f(«+ 2), it shall likewise be < F (a + 7), and if 9 (« 4+ 7) is > f(x + 2), it shall likewise be > F(x +2); or, Which is the same thing, 2 may be taken so small, that the value of @ (« + 2) cannot fall between the values of f(« +7) and F (x +2), unless v= uw = yy’ In the same manner precisely it may be shown, thatifv =u= y,v mw =y', vo" = wu" =", &e. to vy =u? = y’; ‘and wptl = yPrl, then the value of @ (x +72) cannot fall be- tween the values of f (x + 72) and F (w 4+ 7) unless vPt! = upt! = yptl, To apply this to the contacts of curves, let x andy be the abscissa and ordinate of a curve whose equation is y =f, ¢ and u the abscissa and ordinate of another curve whose equa- tion is u = Ft, the abscissas of the curves being measured in the same line. Suppose that the curves have a common point, that is, that u = y; and for the sake of simplicity, let the absecissas in each be mea- sured from the same point, or t = 2, the equations will become y = fx, u = Fax. Let now another ordinate be drawn to each curve at the distance 7 from the former, that is, let x become «x + 2, y will become vit 7) a en y Bi Tae Be hos “i My weewet 3i + 5% a + &e.; then if w 73, &c.; and u, y the contact of the curves is such, that no other curve whose equation is v = $x can pass between them unless v uy, and vmuwmy,ifw —y' and u’ = 7" the contact is such, that no other curve whose equation is ’ = 9.x can pass between them unless » = u 9, Ot — of = oy" and in-géneral if w aig’, uw = ¢", Se. to upt! = ypt!, the contact is such, that the other curve cannot pass between them unless v =u = y, v =u ai en a yf. tO UP oo BP ye. this is evident from what has before been shown, for the value of the ordinate of the third curve cannot fall between the values of the ordinates of the other two, unless it have the above conditions. Hence the contacts of curves admit of different degrees, and calling one in which « —y' a contract of the first order, one in which uw’ = y', and wv’ = y", a contact of the second order, &c.; there are curves, which with a given curve admit enly of contacts of a certain order. For example, a right line can only have a contact of the first order, a circle can only have a contact of the second order, and a curve, whose equation is y = a@ + a,x + a,2”, &e. + ad,2%n can only have a contact of the order x with a given curve. These different degrees of contact require a certain number of constant quantities, which may be cailed elements of the contact. FUN We will now determine the conditions ¢ the contact of a right line with a curve. Le y = fx be the equation of the curve, and w= a + bt, the equation to a right line, ¢ bein measured from the same point and in the sa line asa. Ifu=y, and ¢t=2, the right lin and curve will then have a common point, an y=ur=a + ba; if likewise u' = y’, then bi tween this right line and the curve no othe right line can be drawn. For we shall ha the two equations y = u =a + ba, and 7: uw = 6, which determine the two quantitic a and 6,b=y',anda=y—y'x. To find th subtangent, we have, from the equation u = + bt, 6 = tangent of angle which this lir makes with the axis; consequently the sul tangent =; a ri which is the same asd ace dy” : To determine the conditions of the conta of the circle with a curve, let (¢—a)*+(u—b = c* be the equation to a circle, whose radit is c, a and 6 being the rectangular co-ordina to the centre. If y =u, when t =< the cire and curve will have a common point, at (a —a)* + (u— b)* = cc’, or ¥ L ; c—(a—ay} 23 af likewise y' = w = 1 ——_—_—_———_————,,, the circle and curve W Sct — (x — a)? t 2 have a contact of the first order; and o of the three quantities a, 6, c, will remain i determinate, since we have only the or panes | Hyplpee a t— a4 equations y — 6 + co (x — ay 4 : ! x—a bes =_— ——_—_——__———.. for their detern ; c* —(« —a)* . % nation; butifalso y’=u’= — ; c—(a—a)y we shall have three cquations for the det mination of a,b,c; consequently between tl circle and the curve no other circle can_ drawn. The first equation gives y—b= ; c?—(x—a)* ; and the second, ; c? — (x — a)? t 2 = x«—e U cy , and «—a= J — va +y%) 2 vz a e —(x«—a)*-*?=—— 2 U 3 ie ae Cl + ye (+): ‘ e y" — (1+ y2)2 Whence c = y U Re es: Be Samael v (i+y) y aT OFy) Va | a= xX | | GAG The circle thus determined, is called the cir- e of curvature, and the radius ¢ of this circle, ie radius of curvature. See CURVATURE. Having shown how this calculus may be yplied in one instance, we will conclude by iefly mentioning that Lagrange applies it ) maxima and minima, to the finding ofareas, jlidities, the lengths of curves, the contacts of irves of double curvature, and the most in- resting and difficult problems in the higher sometry, and afterwards to various funda- ental propositions in mechanics. ‘The reader /ho wishes for further information on this sub- Jet, we must refer to the works of Lagrange, ‘hich we have before mentioned; and if he ish for clear and accurate notions of the ibject, we would refer him to Woodhouse’s Prin. Anal. Calc.” where it is treated in the vlest manner, and where the errors into hich that most eminent mathematician La- ange, and many others have fallen, are wzaly pointed out. . The preceding article has been obligingly mmunicated to the author by S. H. Christie, sq. of the Royal Military Academy. FUNICULAR Machine (from funiculus, a ve), is a term used to denote an assemblage cords, by means of which two or many wers sustain one or many weights. ‘This iclassed by some authors amongst the simple chanical powers, and is the simplest of em all. In order to find the law of equilibrium in is machine, we must first reduce all the nwers which meet at one point to a single wer, by the method of Composition of Forces ; ttich single power must act in the direction ithe cord, this being evidently necessary for ablishing an equilibrium: following this me- TAGE, is a term employed to denote va- (us instruments used for measuring the state rarefaction in the air-pump, variations in barometer, &c. The Gace of the Air Pump, is of a variety ‘forms, as the Barometer Gage, the Syphon te, the Pear Gage, &c. Short Barometer GAGE is merely the lower it of a barometer, or a tube about eight or we inches in length, filled with mercury, and mersed with its aperture into a small quan- ; of the same fluid contained in a glass isel which forms the cistern. This gage is ther placed under the receiver upon the ‘neipal plate of the pump, or it is placed der a separate small receiver, on an auxi- Ly plate attached to some pumps for this [rpose. As this gage is not equal to a whole ometer, it will not begin to show the state (rarefaction till after three fourths of the air it GAG thod, we may reduce all the powers which act on different points of the cord, to a system of powers which act on the same point. Then reduce these powers, which act all on the same point, to one equivalent power, and we shall ultimately arrive at two powers only. which ought to be equal and acting in con- trary directions, in order to establish the per- fect equilibrium. See on this subject Gregory’s ** Mechanics,” art. 193, vol. i. FURLONG, an English measure of length, being $th of a mile; and therefore equal to 660 feet, or 220 yards, or 40 poles. FUSEE, in Clock-work, is a mechanical con- trivance for equalizing the power of the main- spring of a watch; for as the action of a spring varies with its distance from the quiescent po- sition, the power derived from the force of a spring requires to be modified according to circumstances, before it can become a proper substitute for a uniform weight, which is what it is intended to supply. In order, therefore, to correct this irregular action of the spring, the fusee on which the chain or catgut acts is made somewhat conical, so that its radius at every point may corres- pond with the strength of the spring, being greater and greater as the action of the spring becomes more and more weakened by un- bending. If the action of the spring diminish- ed equally, the fusee would then require a perfect conical figure; but the decrease of the power of the spring does not follow that uni- form law; and in consequence the figure of the fusee deviates from the form which it would otherwise take, and becomes a solid generated by the revolution of an equilateral hyperbola about its asymptote. See Martin’s “Math. Inst.” vol. ii. p. 264. G have been exhausted, that is, when the elasti- city of the remaining air is about one fourth of that of common air, the barometer itself being but about one-fourth of the usual height of the mercury; but after this the farther de- crease of elasticity is exhibited by the sinking of the mercury, and a graduated scale attached to the instrument. Thereis another gage, called the Long Baro- meter Gage, which works in a similar manner, and indicates the state of rarefaction on nearly the same principles. Syphon Gace. This differs from the short barometer gage, only in this: that imstead of terminating in a small cistern, in this gage the tube is bent, and rises upwards with its aperture, which by means of a brass tube is made to communicate with the inside of the pump, so that the ascending leg of the tube performs the oflice of a cistern. GAG Pear Gace. This is the invention of Mr. Smeaton, and-is thus denominated from its form, which resembles that of a pear. This gage does not indicate the rate of rarefaction as it proceeds, but shows the ultimate state to which it was carried after the re-admission of the air. The gage is suspended in the re- ceiver, and exhausted to the same degree ; but when this is carried on as far as is in- tended, the open orifice of the gage is let down into a vessel of mercury, which upon the re-admission of the air is forced into the pear, and thus the ultimate state ofrarefaction is determined. GAGE of the Barometer, is a contrivance for estimating the exact degree of the rise or fall of the mercury in the barometrical tube. It is obvious that when the mercury sinks in the tube it rises in the cistern, and vice versa; and as the distance between the. divisions graduated on the annexed scale, and the sur- face of the mercury in the cistern, is not truly shown by the numbers on the scale, errors must happen in determining the exact height of the mercury. To remedy this inconve- nience, a line is cut upon a round piece of ivory, which is fixed near the cistern: this line is accurately placed at a given distance from the scale, as for instance, twenty-seven inches; and a small float of cork, with a cylindric piece of ivory fixed to its upper sur- face, (on which a line must be cut at the dis- tance of two inches, exactly from the under surface of thé cork) is left to play freely on the quicksilver, and the cylinder works in a grove made in the other piece ; from this con- struction it appears, that if these marks are . made to coincide, by raising or lowering the screw which acts on the quicksilver, then the divisions on the scale will express the true measure of the distance from the surface. GAGE of the Condenser, is a glass tube ofa particular construction, adapted to the con- densing engine, and designed to show the exact density and quantity of the air contain- ed at any time in the condenser. Sea GAGE, is an instrument invented by Dr. Hales and Dr. Desaguliers, for finding the depth of the sea, the description of which is this. AB, (Plate VI. fig. 5), is the gage- bottle, in which is cemented the gage- “tube Ef, in the brass cap at G. The upper end of the tube I, is hermetically sealed, and the open lower end f, is immersed in mercury, marked C, on which swims a small quantity of treacle. On the top of the bottle is screwed a tube of brass HG, pierced with several holes, to admit the water into the bottle A B. The body K, is a weight hanging by its shank L,ina socket N, with a notch on one side at m, in which is fixed the catch J of the spring s, and passing through the hole L, in the shank of the ‘weight Ki prevents its falling out when once hung on. On the top, in the upper part of the brass tube at H, is fixed a large empty ball, or full-blown bladder I,which must not be so large, but that the weight K may be able to sink the whole under water. Gig ‘ ‘Fle instrument, thus constructed, is us in the following manner. The weight K b hung on, the gage is let fall into deep wate and sinks to the bottom; the socket Pe somewhat longer than the shank L, and the fore after the weight K comes to the bot the gage will continue to descend, till tl lower part of the socket strikes against 1 weight; this gives liberty to the catch to off the hole ‘whereby the weight K is di engaged ; when this is done, the ball or bladd, I, instantly buoys up the gage to the top the water. While the gage is below, th water, having free access to the treacle ¢ mercury in “the bottle, will by its pressw force it up into the tube; and E f, the height! which it has been forced by the greatest pre sure,v7z. that at the bottom, willbe shown byt mark in the tube which the treacle leaves bh hind it, and which is its only use. This sho into what space the whole air in the tube J is compressed, and consequently the height depth of the water, which by its weight pi duced that compression, which is the thi required. () If the gage-tube Ef were of glass, a se might be drawn on it with the point of diamond, showing by inspection what heig the water stands above the bottom. Butt length of 10 inches is not sufficient for fatho; ing depths at sea, because when all the aij such a length of tube is compressed into h an inch, the depth of water is not more th 634 feet, which is not half a quarter of a mi If to remedy this, we make use of a tu 50 inches long, which for strength may b musquet-barrel, and suppose the air compre ed into an hundredth »part of an inch; th! by saying, as] : 99 :: 400 : 39600 inches, 3300 feet ; even this is but little more fl halfa mile, or 2640. But since it is reas able to suppose the cavities of the sea b some proportion to the mountainous parts the land, some of which are more than th miles above the earth’s surface ; therefore explore such great depths, Dr. Hale contri a new form for his sea-gage, or rather for| gage-tube in it, which is as follows: BCI fic. 6,isa hollow metallic globe, communt ing on the top witha long tube A B, whi capacity is a ninth part of that globe. ‘On lower part at D, it has also a short tube DEI stand in the mercury and treacle. The aire tained in the compound gage-tube is compr ed by the water as before; but the degre) compression, or height to which the treé has been forced, cannot here be seen thro the tube; therefore, to answer that end, a sit der rod of metal or w ood, with a knob on) top of the tube A B, will receive the mar the treacle, and show it when taken out. . If the tube AB be 50 inches long, alll such a bore that every inch in length shal) a cubic inch of air, and the contents oft globe and tube together 600 cubic incl} then, when the air is compressed | within! hundredth part of the whole, it is evident! tveacle will not approach nearer than / 0 | GAL inches of the top of the tube, which will agree to the depth of 3300 feet of water as above. I'wice this depth will compress the air into half that space nearly, viz. 24 inches, corres- »onding to 6600, which is a mile and a quarter. Again, half that space, or 1} inch, will show Jouble the former depth, viz. 13,200 feet, or 24 miles, which is probably very nearly the rreatest depth of the sea. Bucket Sea Gace, is an instrument con- rived by Dr. Hales to find the diflerent de- irrees of coolness and saltness of the sea, at lifferent depths, consisting ofa common house- 10ld pail or bucket with two heads to it. (hese heads have each a round hole in the niddle, near four inches diameter, and covered vith valves opening upwards, and that they night both open and shut together, there is a ‘mallironrod fixed to the upper part of the lower a and at the other end to the under part f the upper valve; so that as the bucket de- cends with its sinking weight into the sea, hoth the valves open by the force of the water, vhich by that means has a free passage through ‘he bucket. But when the bucket is drawn ‘yp, then both the valves shut by the force of he water at the upper part of the bucket; vhereby it is brought up full of the lowest 'ea-water to which it had descended. } When the bucket is drawn up, the mer- ‘urial thermometer, fixed in it, is examined ; hut great care must be taken to observe the ‘egree at which the mercury stands, before ‘he lower part of the thermometer is taken ‘ut of the water in the bucket, as it would ‘therwise be altered by the different tempe- ere of the air. In order to keep the bucket in a right posi- ‘ion, there are four cords fixed to it, reaching bout four feet below it, to which the sinking eight is attached. Tide Gace. This is an instrument used for ‘etermining the height of the tides, by Mr. Bay- ty, in the course of a voyage towards the south ole, &c. in the Resolution and Adventure, in 1e years 1772, 1773, 1774, and 1775. ‘This istrument consists of a glass tube, whose fternal diameter was seven-tenths of an inch, ished fast to a ten-foot fir rod, divided into vet, inches, and parts; the rod being fastened » a strong post fixed firm and upright in the hater. At the lower end of the tube was an xceedingly small aperture, through which 1e water was admitted. In consequence of lis construction, the surface of the water in : tube was so little affected by the agitation "the sea, that its height was not altered the ‘nth part of an inch, when the swell of the sa was two feet, and Mr. Bayley was certain, hat with this instrument he could discern a ifference of the tenth of an inch in the height ‘i the tide. Wind Gace, is an instrument for mea- wing the force of the wind upon any given ‘uface. See ANEMOMETER, 'GAGING. See GAUGING. | GALAXY, Via Lactea, or Milky Way, in Stronomy, that long, luminous track or zone, GAL which encompasses the heavens, forming nearly a great circle of the celestial sphere. {t is inclined to the plane of the ecliptic at about an angle of 60°, and cuts it nearly at the two solstitial points. It traverses the constellations Cassiopeia, Perseus, Auriga, Orion, Gemini, Canis Major, and the Ship, wh¢re it appears most brilliant in southern latitudes; it then passes through the feet of the Centaur, the Cross, the southern Triangle, and returns towards the north by the Alter, the tail of the Scorpion, and the are of Sagit- tarius, where it divides into two branches, passing through Aquila, Sagitta, the Swan, Serpentarius, the head of Cepheus, and re- turns into Cassiopeia. This circle is described in the poem of Manilius, who seems to have been very for- tunate in his conjecture as to the cause of this light : ‘ “Anne magis densa stellarum turba corona Contexit flammas et crasso lumine candes Et fulgore nitet collato clarior orbis?’ Milton also speaks of it in the following beautiful and appropriate language : “A broad and ample road whose dust is gold, And pavement stars, as stars to thee appear, Seen in the Galaxy, that milky way, Which nightly, as a circling zone thou seest, Powdered with stars.” The ancients had many singular ideas as to the cause of this phenomenon; but modern astronomers have long attributed it to a great assemblage of stars, and Dr. Herschel has confirmed these conjectures, having discovered in a space of about 15° long, by 2° broad, no less than 50,000 stars. This, however, instead of satisfying the cu- riosity of astronomers, only gave rise to farther inquiries and hypotheses; amongst others, that of Dr. Herschel’s, which is very interest- ing: he supposes the sidereal universe to be distributed into nebulz and clusters of stars, and the milky way to be that particular cluster in which our sun is placed. See Phil. Trans. for 1784. See also NEBULE. GALILEO (GaLIiLeE!), a very celebrated mathematician and astronomer, was the son of a Florentine nobleman, and born in the year 1564. He had from his infancy a strong in- clination to philosophy and the mathematics, and made prodigious progress in these sciences. -In 1592 he was chosen professor of mathe- matics at Padua, and during his abode there - he is said to have invented the telescope ; or, according to others, improved that instrument, so as to render it fit for astronomical observa- tions. In 1611, Cosmo IL. Grand Duke of Tuscany, sent for him to Pisa, where he made him professor of mathematics, with a hand- some salary; and soon after, inviting him to Florence, gave him the office and title of ° principal philosopher and mathematician to his highness. He had been however but a few years at Vlorence before he was convinced by sad experience, that Aristotle’s doctrine, however GAL ill-grounded, was held too sacred to be called in question. Having observed some solar spots in 1612, he printed that discovery the following year at Rome; in which, and in some other pieces, he ventured to assert the truth of the Copernican system, and brought several new arguments to confirm it. For these he was cited before the Inquisition; and, after some months imprisonment, was released upon a simple promise, that he would renounce his heretical opinions, and not de- fend them by word or writing. But having afterwards, in 1632, published his “ Dialogues of the two Systems of the World, the Ptole- maic and Copernican;” he was again cited before ine Inquisition, and committed to the prison of that abominable court at Rome. In June, in the same year, the congregation con- vened, and in his presence pronounced sen- tence against him and his books, obliging him to abjure his errors in the most solemn man- ner; committed him to the prison of their office during pleasure; and enjoined him, as a saving penance, for three years to come, to repeat once a week the seven penitential psalms: reserving to themselves, however, the power of moderating, changing, or taking away altogether, or in part, the said punish- ment or penance. On this sentence, he was detained in prison till 1634; and his “ Dia- logues of the System of the World” were burnt at Rome. Thus did a few contemptible bigots exercise their tyrannical authority over this venerable philosopher at the age of seventy, making him renounce doctrines which he knew to be founded in truth and reason. Happily how- ever for the sciences, this infernal court has long been disarmed of its terrors, the truth has prevailed, the Copernican system has been established, and has now nothing to fear from the bigotry and intolerance of monks and priests. Galileo lived seven years after this, but the last three years of his life he was totally blind, which did not, however, prevent him from pursuing his scientific investigations, which he persevered in till his death, which happened in January 1642, being the 78th year of his age. From the time of Archimedes, nothing had been done in mechanical geometry, till Galileo, who being possessed of an excellent judg- ment, and great skill in the most abstruse points of geometry, first extended the bounda- rics of that science, and began to reduce the resistance of solid bodies to its laws. Besides applying geometry to the doctrine of motion, by which philosophy became established on a sure foundation, he made surprising disco- veries in the heavens by means of his tele- scope; and thus made the evidence of the Copernican system more sensible. He was the first who demonstrated that the spaces described by heavy falling bodies are as the aquares of their times of motion; and that a body projected inan oblique direction describes the curve a parabola. He invented the oycloid ; GAL a as well as the simple pendulum; and thought of applying it to clocks, but did not execute that design. He, together with his pupil Tor- ricelli, discovered that air had gravity, and endeavoured to compare it with water. Th short, he will ever be admired by true philo- sophers, as the greatest promoter of the’ sciences; he opened vast fields for the in- quiries of others, and ably assisted them by his inventions and discoveries. . Galileo wrote a number of treatises, many of which were published in his lifetime. Most of them were also collected after his death, and published by Mendessi in 2 vols, 4to., under the title of “ L’Opere di Galilee Galilei Lynceo,” in 1656. Some of these, with others of his pieces, were translated inte English and published by Thomas Salisbury, in his “ Mathematical Collections,” in 2 vols) folio. A volume also of his letters to several learned men, and solutions of several pro: blems, was printed at Bologna in 4to. _ Many of his pieces were, however, lost a his death, as some say, through the supersti tion of one of his nephews, or as others say through the artifice of his wife’s confessor at all events they were destroyed, in conse quence of their being supposed to contai doctrines which the Inquisition declared t be heretical. ; GALLON, an English measure of capacity, being equal to 4 quarts or 8 pints. —, Cub, Inches, The gallon, wine measure, contains 231 Ditto beer measure ............. 282 | Ditto dry measure..............4. 2684 GALYV ANI (Lewis), a modern philosophei who has had the honour of giving his nami to a newly discovered principle in nature, wa born at Bologna in 1737, where he practise: medicine, was publiclecturer at the university and reader in anatomy in the Institute of th same city. His reputation, as an anatomist and phy siologist, was established in the schools ¢ Italy, when accident gave birth to the dis covery which has immortalised his nam His wife, with whom he lived many yearsi the tenderest union, was at this time in declining state of health. As a restorativi she made use of a soup of frogs: and some ¢ these animals, skinned for the purpose, hay pened to lie upon a table in her husband jaboratory, upon which was placed an elet trical machine. One of the assistants in hi experiments happened accidentally to brin the point of a scalpel near the crural nerves | a frog, lying not far from the conductor. I stantly the muscles of the limb were agitate with strong convulsions. Madame Galvani, woman of quick understanding, and of a v4 entific turn, was present, and, struck witht phzenomenon, she immediately went to infon her husband of it. He came and repeat the experiment; and soon found that th convulsion only took place when a spark wi drawn from the conductor, at the time tl scalpel was in contact with the nerve. GAL From this simple experiment, or rather wecident, arose the new science of Galyanism, vhich has made, and is still making, such a ‘apid progress in the hands of modern che- nists and philosophers. Galvani died in 1798, a the 61st year of his age. GALVANISM, a modern and very inter- ssting branch of science, thus named after its elebrated discoverer, professor Galvani of 3ologna. (See the preceding article.) Galva- ism comprises all those electrical phenomena ‘ising from the chemical agency of certain netals with different fluids. _ We have noticed in the foregoing article Lhe humble origin of this science; and that by ecident it was discovered, that common elec- ‘tricity had the property of producing mus- ‘ular contractions in the limbs of animals ven a considerable time after death; and of ‘his Galvani more clearly convinced himself, y ascertaining that from whatever source the he same, was drawn, the effect of it was still : : | he same. ' But in one instance he found, that the mere ‘gency of a metallic substance, where he had 0 reason to suspect the presence of electri- lity, the limbs of a recently killed frog were ‘onvulsed; and after making several experi- nents, he ascertained that the convulsions nly took place when he employed dissimilar retals. Galvani’s experiments were repeated by aany eminent philosophers, both on the con- ‘nent and in this country. None of them, owever, added any thing new to what Gal- ani had himself discovered, excepting the tlebrated Volta, whose improvement was so fecided, that the science itself has nearly anged its name, taking that of Voltaism istead of Galvanism. When we view, indeed, the numerous facts hat have been added to the labours of Gal- ui, his discoveries form but a very small utOf the whole mass, whereas a great many ‘them are due to Volta. Yet when we collect again, that the investigation began ‘ith the former, and was in a great degree pro- oted by his own perseverance, we must ever ousider him as a principal-in this extensive vid of research; and cannot, without injus- 2e, deprive him of the honour which has en conferred upon him, of giving his own ‘ume to the science which he discovered and ‘omoted. Philosophy, however, is infinitely indebted Signior Volta, it being to him that we ive, in a great measure, the rapid progress lat has since been made in this interesting vanch of philosophy. ‘He repeated the experiments of Galvani, ‘id found that when two pieces of metal of flerent kinds were placed in different parts | an animal at the same time, that the metals ‘ere bronght in contact, or were connected : a metallic arc; as often as the contact was lade, convulsions were observed. He found jat the greatest was produced when the ] ’ ) ) GAL metals were zine and silyer. When several pairs of metals were employed, haying pieces of moist cloth between them, the effect ap- peared to increase as the number of pairs. This important discovery of accumulating the effects of this species of electricity was made by Volta in 1800, and’hence has been denominated the Voltaic pile. The apparatus first made by Volta consisted of a certain number of pairs of zine and silver plates separated from each other by pieces of wet cloth; the arrangement being as follows: zinc, silver, wet cloth; zine, silver, wet cloth, and so on. The silver plates were chiefly silver coins, the plates of zinc and the pieces of cloth being of the same size. He found this pile much more powerful when the pieces of cloth were moistened with a solution of common salt instead of pure water, and an apparatus consisting of forty pairs of plates he found to possess the power of giving a very smart shock, similar to that of a small electric jar; and that this effect took place as often as a communication was made between each end of the pile, and as long as the pieces of cloth remained moist. An account of this discovery was commu- nicated to the Royal Society, and published in the Philosophical 'Transactions. Since this time we have no account of any farther dis- coveries of Volta; but the science of Galva- nism has been since considerably extended by the researches and experiments of phi- losophers in France, England, and other countries. The first experiments made on the pile in this country were performed by Messrs... Ni- cholson and Carlisle, who observed, that on bringing the wires from each end of the column in contact with a drop of water, bub- bles of some elastic gas were disengaged, Which on closer examination they found to be hydrogen gas; this discovery gave rise to a great variety of experiments, and to many interesting results, but being chiefly chemical we cannot enter into any detail of them in the present article. The Galvanic energy evinced in the decom- position of bodies, which the experiments of Nicholson and Carlisle had first made known, was farther prosecuted by Mr. Cruickshank of Woolwich, to whom we are indebted for the invention of the Galvanic trough, which we have described under the article Galvanic BaTrery. This again led the way to other batteries of similar construction, but of a more powerful nature, by which it was found, that all the metals reduced into thin leaves were deflagrated with brilliant though dif- ferently coloured flames; and henceiorth Gal- vanism which had not before assumed an particular character, it being doubtful to what branch of science it properly belonged, was directed entirely to chemistry, and has since been the means of throwing great light on that interesting branch of human knowledge. Here, however, it ceases to belong to the 2 GAU subject of this work, and the reader, who is desirous of information on this head, is re- ferred to the several volumes of the Philoso- phical Magazine; the Philosophical Journal ; and the Transactions of the Royal Society since 1800; also to Davy’s, Murray’s, and Thompson’s “ Chemistry,” and the other works on this science. GAMING, See CHANcES and PRoBABI- LITIES. GAS, a generic name given by Van Hel- mont to elastic fluids, and now generally adopted. The term air was made generic by Priestley; but this seems to imply that elastic fluids are only modifications of common or atmospheric air, contrary to what is now known to be true; the term gas is therefore now properly preferred. Though the observations of Van Helmont, Boyle, Hales, Black, and Cavendish, seemed to leave no doubt that there existed three or four kinds of air, or fluids, agreeing in the property of elasticity, yet no very distinct notion was formed on the subject, till the discoveries of Priestley and those of Lavoisier. Since which time most chemical inquiries have been connected with the knowledge of gaseous bodies, and great addition has been made to their number, and to their distinctive properties. The number of compound gasses is very considerable, but this being a subject purely chemical, it would be improper to enter into their enumeration and properties im this work. The simple gasses are only three, viz. hydrogen, oxygen, and azoate or nitrogen. For the effect of different gasses on sound, see SOUND. GASSENDI (PETER), a celebrated French philosopher and astronomer, was born in a village in Provence in 1592. He very early discovered great talents, and at the age of 16 was made professor of rhetoric at Digne, and soon after professor of philosophy at Aix. Gassendi died in 1655, in the 63d year of his age, leaving his manuscripts to M. de Monmor, his friend and executor. Gassendi wrote against the metaphysical meditations of Des Cartes; and divided with that great man the philosophers of his time, talmost all.of whom were either Cartesians or Gassendists. To his knowledge in philosophy and mathematics, he joined profound erudi- tion“and deep skill in the languages. He wrote, 1. Three volumes on Epicurus’s Phi- losophy; and six others, which contain his own philosophy. 2. Astronomical Works. 3. The Lives of Nicholas de Pieresc, Epi- curus, Copernicus, Tycho Brahe, Purbach, and Regiomontanus. 4. Epistles, and other treatises. All his works were collected to- gether, and printed at Lyons in 1658, in six volumes folio. GAUGE-Line. See Gaucine-Rod. GAUGE Point of a solid, is used to denote the diameter of that circle, whose area is expressed by the same number, as is equal to GAU | the number of cubic inches in the solid. Thus 17°15 being the diameter of a cirel whose area is 231; this is called the gau point of the wine gallon, which contains 23) cubic inches. ix | GAUGING, the art or act of measuring the capacities of all kinds of vessels, an thence ascertaining the quantity of liquo which they contain. A Gauging of course forms a part of men suration, and is accordingly treated of by mos authors who have written on the latte subject; this, however, is generally with re ference to regular figures, as the frustums ¢ cones and conoids, of parabolic, hyperbolic and elliptic spindles, &c. But as casks ar seldom of any exact forms, these rules mus be regarded as merely theoretical, and ne applicable to the common cases that arisei practice. ' ‘To remedy the inconvenience arising fror such a number of rules, and at the same tim to abridge the labour attendant on several ¢ them, Dr. Hutton in his ‘ Mensuration” h given one general rule for all cases, and ex ceedingly simple in its application, which j as follows: a General Rule. Add into one sum, 39 times the square of the bung diameter, © 25 times the square of the head diameter, an 26 times the product of those diameters; —. multiply the sum by the length of the casl and divide the product by 114; then this la quotient divided by 231 will give the win gallons, and divided by 282 will give the a gallons. . Or, (39 BY + 25 H? + 26 BH) x oe il content in inches; which being divided t 231 for wine gallons, or by 282 for ale gallon will be the content. Exam. Let the length of a cask be 4 inches, the bung diameter 32, and the he diameter 24. } Here ne. 327 x 39 = 39936 ands iL eae: 247 x 25— 14400 aNG° te; 32 xX 24 x 26—= £19968 the sum........... ...- 74304 multiplied by....... 40 and divide by 114)2972160 PAW OR oy 4. sch decedent 26071 cub. i this divided by 231 gives 112 wine gallons, or divided by 282 gives 92 ale gallons. But the common practice of gauging performed mechanically, by means of t] gauging or diagonal rod, or the gaugi sliding rule, the description and use of w is as follow. é. GAUGING, or Diagonal Rod, is a rod or F adapted for determining the contents of cask by measuring the diagonal only, viz. the di gonal from the bung to the extremity of # apposite stave next the head. It is a squa rule, having four sides or faces, being usual four feet long, and folding together by mea of joints. GAU Upon one face of the rule is a scale of uches, for taking the measure of the diagonal ; o these are adapted the areas, in ale gallons, f circles to the corresponding diameters, like he lines on the under sides of the three slides ia the sliding rule, described below. And \pon the opposite face are two scales, of ale ‘nd wine gallons, expressing the contents of asks having the corresponding diagonals ; ind these are the lines which chiefly consti- lute the difference between this instrument md the sliding rule; for all the other lines pon it are the same with those in that in- trument, and are to be used in the same aanner. ' To use the Diagonal Rod. Unfold the rod ind put it in at the bung-hole of the,cask to le gauged, till its end arrive at the intersec- ‘on of the head and opposite stave, or to the urthest possible distance from the bung-hole, ind note the inches and parts cut by the mid- ile of the bung; then draw out the rod, and bok for the same inches and parts on the \pposite face of it, and annexed to them are sand the contents of the cask, both in ale nd wine gallons. Exam. Let it be required to find, by this od, the content of a cask whose diagonal aeasures 34°4 inches; which answers to the lask in the foregoing example, whose head ad bung diameters are 32 and 24, and length ‘0 inches; for if to the square of 20, half the sngth, be added the square of 28, half the ‘am of the diameters, the square root of the jam will be 34°4 nearly. | Now, to this diagonal 34:4 corresponds, pon the rule, the content 91 ale gallons, or ‘LL wine gallons; being but one less than ie content found by the former general rule 20ve given. Gavuoine Rule, or Sliding Rule, is a sliding ale particularly adapted to the purposes of nuging. It is a square rule, of four faces or des, three of which are furnished with slid- g pieces running in grooves. The lines j90n them are mostly logarithmic ones, or | stances which are proportional to the lo- arithms of the numbers placed at their ‘ids; which were fine lines placed upon ilers, by Gunter, for expeditiously perform- -g arithmetical operations, using a pair of compasses for taking off and applying the »veral logarithmic distances: but instead of .e compasses, sliding pieces were added, by ‘ir. Thomas Everard, as more certain and onvenient in practice, from whom this sliding ‘ile is often called Everard’s Rule. For the ore particular description and uses of which, -e Rue. See also Hutton’s “ Mensuration,” 564, 2d edition. The writers on gauging are, Beyer, Kepler, echales, Hunt, Everard, Dougherty, Shet- worth, Shirtcliffe, Leadbetter, Moss, Sy- ‘ons, &e. )GAUSS’S Theorem, is an expression used denote a theorem invented by Gauss, pro- |Ssor of mathematics at Strasburgh, for the Iution of certain binomial equations. ) GAU We have shown under the article BiInoMIAL Equations and PoLyGon, in what manner the solution of such equations are connected with the division of the circle, and how their roots may be found by means of tables of sines, cosines, &c. But Gauss’s theorem is the converse of this, and shows in what manner the sines and cosines of certain angles may be obtained by the numerical solution of ‘such equations. The artifice which he employs for this pur- pose, is too refined to admit of a clear eluci- dation in the narrow limits of this article; we can therefore only give the principles of his operations, but for the detail of them we must refer the reader to his work, entitled “ Dis- quisitiones Arithmetice,” or 10 the translation of the same under the title of ‘‘ Recherches Arithmetiques;” see also Legendre’s “ Essai sur la Théorie des Nombres,” or the last chapter of Barlow’s “Theory of Numbers.” Let «*— 1=0, be a binomial equation, then it is obvious that the real root of this equation is « = 1. And if now, according to the known theory of equations, we divide a —1 = 0, byx — 1 = 0, we have Freee | Sea Fa iene wen tel. O E—'T which latter equation necessarily contains all the imaginary roots of the original equation X,— 1 = 0; and our object is to give a slight view of the principles on which these roots are to be obtained. Let these roots be a, a’, a3, a+, &c. it being a property of these equations, that their roots are all powers of each other, as we have shown under the articles above referred fo. It is also obvious that their sum is equal to -— 1, that is, to the co-efficient of the second term with its sign changed from + to —; also the product of them = + 1. Now Gauss’s solution depends upon the subdivision of this period of roots into other less periods, so that the product, square, cube, &c. of these minor periods, shall pro- duce the sum or product of all the roots; but as we before observed, this part of it is too refined to admit of explanation in this place ; we shall therefore merely give the solution of one or two cases, which will be dound moré intelligible than any general observations on the division of his periods, which in fact-¢én- stitutes all the difficulty of the problem, Let it be proposed to find the imaginary roots of the equation 2° — 1 — 0, or of atta tar*¥tat+1—o. Let the four required roots be a, a’, a3, at; =a + at a’ +a? then pp —@B+a+a++ao—l and pt p =a 4+e°4+@0+4+at+=—1 _ Therefore the values of p and p’ are found from the quadratic equation. oe — nance. Sp also make Up! Pp’ + ail whence p =—$ +475 and op’ =~ E— 4 VG GEN which are the numerical values of the cosines of 72° and 144°, Let there now be proposed the equation x’ — 1 = 03 Or rm tad tattaitar>+t+ae+1—0 and let its roots be a, a*, a3, at, a5, a, p mata make $y =a + at PaO re Now multiply all these equations together, as also cach pair of them, we shall have P + P, tl + Pp" =i } ppt pp + pp 2 p = 1 therefore the values of p, p’, p” are contained in the cubic equation, ptp—2zpr=l the roots of which are p = 1:2469796 p = — 18019376 " = — 04450420 and these vaiues of p p’ p” furnish the three following quadraties, viz. x*—p2+1=20 x*>—px+1=0 x—p'x+1=—0 which contain among them all the imaginary roots of the proposed equation. When isa prime number of the form 4m +4 1, then the value of the root may be obtained from qua- dratics only; and the polygon of such a number of sides may be inscribed geometri- cally ina circle. See PoLYGon. GELLIBRAND (Henry), an English as- tronomer, having been professor of that science in Gresham College, was born in London 1597, and died in the same city of a fever in 1636, being only 39 years of age. Gellibrand was an intimate acquaintance of Briggs, and wrote the preface, and attended to the publication, of the ‘ ‘Trigonometria Britannica,” Briggs having died before the completion of that great undertaking. He Was a'so author of the following works; viz. 1. Trigonometria Britannica, or the Doc- trine of Triangles, being the second part of Briggs’s work above mentioned. 2. A small Tract concerning the Longitude. 3. A Dis- conrse relating to the Variation of the Mag- netic Needle, 4. Institution Trigonometri- cal, with its application to Astronomy and Navigation, 5. Epitome of Navigation, with the necessary ‘Tables; beside several manu- scripts which were never published. GEMINI, or Twins, one of the northern zodiacal constellations, denoted by the cha- racter 7. See CONSTELLATION. GENERANT, or Genitum, that which is generated, or supposed to be generated, by the motion of any point, line, or figure. The seneraitis always one dimension higher than the generating quantity, thus a line is ge- nerated by the motion of a point, a surface by a line, and a solid by a surface. It is generally a theorem in geometry, that the measure of any generant is always equal to the product of the generating quantity, Via GEO drawn into the path of the centre of gravity of the latter, whether its motion be rectilineal or rotatory. See Centro-Baryc. is GENERATED, is used by mathematicians to denote whatever is formed by the motion of a point, line, or surface; thus a line is said to be generated by the motion of a point, a surface by the motion of a line, and a solid by the motion of a surface. The same term is also sometimes used in a similar sense in arithmetic and algebra; thus 20 is said to be generated by the two factors 4 and 5, or 2 and 10; ab of the factors, a and b, &e. A GENERATING Line or Figure, in Geo- metry, is that line or figure, by the motion of which another figure or solid is supposed t be described or generated. In the fluxional analysis all kinds of quan- tities are supposed to be generated by the motion of other quantities, and the quantities thus generated are termed fluents. GENERATION, in Mathematics, denotes the formation or description of any geometri- cal figure or magnitude by the motion of another quantity or magnitude, of a dimensio one degree less. . GENESIS, in Mathematics, is nearly the same as generation, being the formation ofa line, surface, or solid, by the flowing of point, line, or surface. Here the moving line or figure is called the describent, and the line in which the motion is made the dirigent. GENITUM. See GENERANT. i GEOCEN'TRIC (from yn, earth, and xevrpoy, the centre), is said of a planet or its orbit, to denote its having the earth for its centre, The moon alone is properly geocentric. And yet the motions of all the planets may be considered in respect of the earth, or as they appear from the earth’s centre, and thenee called their geocentric motions. Hence alse the terms geocentric place, latitude, longitude, &e. being the place, latitude, longitude, &e of a planet as seen from the earth’s centre. GHODAISIA, (from yn, earth, and defo, J divide), is properly that part of geometry which relates to the division of lands, or othe surfaces; but it is now more commonly em ployed to denote those trigonometrical opera tions which have been executed for the pur pose of measuring the lengths of degrees ir different latitudes; or to the surveys of whole countries, such as that now carrying on it England under the direction and superintend ance of Col. Mudge of the Royal Artillery. — We have already given a brief sketch 0 these operations under Deerer, and for far ther details we must refer the reader to Colone Mudge's ‘‘ Account of the Trigonometrica Survey ;” Mechain and Delambre, “ Base dt systéme Metrique Decimal;” Swanberg, “ Ex position des Operation faites en Lapponie; and to the works of Puissant, entitled “ Geo desie,” and “'Traité de Topographie d’Ar pentage,” &ce. Geodesia, however, according to the original import of the term, denotet merely the division of land, under which sig nification of it we give the following problem GEO Pros. To divide a given triangle into two parts in any given ratio. Case l. When the Dividing Line is drawn from one of the Angles of the Figure. Let A BC be the tri- Cc angle, it is required to divide it into two parts jin any given ratio of m to n, by a line drawn from C to the opposite side A B. - D B Divide the base AB, in D, making AD: DB:: m:n; join CD, which will divide the triangle in the ratio required as is evident; because triangles of equal altitudes are to each other as their basis, | Calculation. AS AD: ABi:m:m + x therefore AD = AB x w™. m+n Case 2. When the Dividing Line is Parallel to one of the Sides of the Triangle, as AB. Let ABC be the pro- C, posed triangle. Make CE: EB:: m:n; erect D ED perpendicular to CB, and from C asa centre, and radius C D, describe the are DF; “ B draw FG parallel to AB, and it will divide the triangle as required. - For nN ie Rel : ined, But CE: EB:: min Or CE:CB::m:m 4+ nby const. Therefore CF? : CB*::m: m+n And since A CGF: A CAB:: CF? : CB? it follows that CGF: CAB::m:m + nas required. Calculation. Since C B*: CF7::m + n:m therefore (m + n) CF* = m. C B* whence CF /(m +2) =

RarrarGy FFs —,and 1 which the latter is the only possible valu x, the two former being imaginary or i sible. Sometimes the root of an equatio be represented by imaginary expr when it is in fact equal to a real quanti is the case in the solution of cubic eq j =o) a IM P the irreducible form, according to the me- od of Cardan. Albert Girard was the first thor who treated expressly on the imaginary ots of equations, and showed that every uation has as many roots, either real or inary, as is denoted by the highest power Phe index; see his “ Inventions Nouvelles ‘VAlgebra.” D’Alembert, in the Memoires Berlin for 1746, first demonstrated that ery imaginary expression may be reduced the form a “/—J, or b +a Y—1; and at the number of imaginary roots always ter in pairs, and consequently every equa- m of an odd dimension must have at least e real root, but an equation of an even gree may haye all its roots impossible. ‘aring also, in his “ Meditationes Alge- aice,”’ has treated largely on this head, see apters ii. and iii. of that work; in which Il be found many excellent observations on is subject, with rules for determining the ‘mber of imaginary or impossible roots in a ven equation of any dimension: the same s also been done by Newton in his “ Uni- ‘rsal Arithmetic.” by Maclaurin in his “ Al- ‘bra,” and by numerous other authors on 'e latter subject. IMBIBE, the same as ABSORB. IMMERSION, in Astronomy, is when a anet, comet, or other heavenly body ap- aches so near to conjunction with the sun, at it is enveloped in his rays, and lost to our servation. Immersion also denotes the beginning of an lipse, or of an occultation, when the body or y part of it first begins to disappear, either hind another body or in its shadow; as in eclipse of the sun, when the disc is first vered by the edge of the moon; or as in an lipse of the latter bod_*, when she first enters e terrestrial shadow. IMPACT, the single instantancous blow stroke communicated from one body in otion, to another either in motion or at rest. e PERCUSSION. IMPENETRABILITY, that quality of a idy which prevents it from being pierced or ‘netrated. IMPENETRABLE, that which cannot be netrated. IMPERFECT Number, that which is not ual to the sum of its aliquot parts. See erfect NUMBER. IMPERIAL Table, an old instrument used measuring land. IMPERVIOUS, that which cannot be en- red or penetrated. IMPETUS, in Mechanics. See Force, Mo- gNTUM, and MorTIon. IMPOSSIBLE, in Algebra, commonly sig- fies the same thing as Imaginary ; thus we y impossible, or imaginary roots, quantities, c. See IMAGINARY. ImpossisLe Forms of Equation, in the Inde- rminate Analysis, are those that will admit *no solution in rational numbers, neither in- gral nor fractional; such as 22* + 3y* =z, et 7y3 = 23, 32+ 7y* = z+, &e.; which 1 *« IMP are all impossible forms of equations, admit- ting of no rational solution. The investigation of these impossible forms does not seem to have engaged the attention of Diophantus; but his translators and com- mentators, as Bachet, Fermat, Father de Billy, &e. have all been led to the consideration of them, as the means of saving much unneces- sary labour, by being able to show in many cases, before any operation takes place, that the equation under consideration admits of no solution. ae The most general method of determining impossible forms is by means of the linear forms of squares, cubes, &c. Thus all square numbers are of one of the two forms, 42, or 4n +1; viz. all square num- bers are either exactly divisible by 4, or when divided by it as far as possible, they will leave a remainder 1; and therefore no numbers of the forms 4x + 2, or 4% + 3, can be squares; or 4x + 2 and 4x + 3 are impossible forms for square numbers. In the same manner, all square numbers are of one of the forms Sn, or 5n + 1; that is, square numbers, when divid- ed by 5, can only leave the three remainders 0, 1, and —1; or, which is the same, 0, 1, and 4; and therefore 5n + 2, and 5n + 3, are impossible forms for squares. Again, all square numbers are of one of the forms 72, 77 + Tf, 7n +4, or 7n + 2; and therefore 7x + 3, 7n +5, 7n +6, are all impossible forms. And thus is constructed the following table of the possible and impossible forms of squares to the prime moduli 2, 3, 5, 7, and 11. And the impossible forms of cubes are ob- tained in the same manner; thus we shall find that all cubes are of one of the forms 7, 7n + 1, or 7m + 6, and consequently all the other forms to modulus 7 are impossible forms for cubes; that is, no number of the form 7n+2, 7n+ 3, 7n + 4, 7n + 5, can be a cube number; in the same way it will be found that no number of the form 9x + 2, 9n + 3, On +4, 9n + 5,9n + 6, or 9n + 7, can bea cube number; that is, if a number, when di- vided by 7, leaves a remainder 2, 3, 4, or 5; or if, when divided by 9, it leaves a remainder 2, 3, 4, 5, 6, or 7; then in neither case is the number a complete cube. These are called simple or linear forms; we may by their combination arrive at other more compound cases, as in the following table: Impossible Forms of Squares. 923 bi Syt a Sn 6 y* =e" 62% + 3y* = = ba* + Gye ot x S27 + 3y? = 2 Ss? + 6y? = x3 Llaz? + 3y7= 2” bia? 6y” = 2” &e. &e. &e. =—-&e, (5p +2)x 4 Sqy* =z (5p +3) 5qy* =z (7p +322 Wqy=2 7p +5) Tqy* =z big eran 7qy =z (lip + 2) a? + 1lqy* = 2* (ilp + 6)a* + llgy*® =z BB INC ‘ (lip +. 7) a* + Ilqy* = 2* (lp + 8)2* + ligy* = 2 (I1l'p + 10) 2* + llgy? —'2* observing only, with regard to the last, that q must be taken prime to the modulus 11. Impossible Forms of Cubes. 22° t+ 7y? —'z3 2x oe Oy> = z5 p2e + 7? — 23 Ra EE ale 5, Axe + Tys = x3 Bat sees 5 2 EMS my Re pllfeomel 523 + Oy? = 23 Beste 7 ye 28 ke Bad + Os Ses Re ley AY leat 7 oo te Dae ee And these again may be further generalized, by writing them (7p + 228 t7qy? (9p + 2) 237+ OGy3 (7p + 3)23+7qy3 (9p + 8) x3 + 9qy3 (p+ Axe aTqy3 (9p + 4) a3 + 9gy¥5 (7p + 5x24 7qy3 (9p + 5) 23 + Oqy3 (7p + 9x t7qys (9p + 6) x3 + Ogy3 (7p + 10) &7qy? (Op + 7) x + 9gy¥5 No one of which equations can ever become equal to a cube, either in integers or fractions; provided that g be taken prime to the modulus with which it enters. A similar mode of investigation may be pur- sued with all the higher powers, the only dif- ficulty being in fixing upon a proper modulus; that is, such a number as shall have the most impossibleforms belonging toit, which requires a separate investigation (See Power.) But almost every power has some modulus that renders it expressible in three forms; thus, All 3d powers are of one ; : of the forms } “nor Tne] 4th powers. .......Jececcece 5n,or 5n +1 Sth powers... .......0. Lin, or ll n+ 1 6th powers..........00. 72, Or Tn +1 Sth powers........0...ce000 Wn, orl7n+1 Oth POW Ers NA, 0.0 19n,or19n+1 10th powers......... eae I1n,orlla +1 ke. &e. &e. &e. the 7th power are here omitted, not being reducible to a similar form. For the demon- stration of these, and a variety of other similar forms, see Barlow’s “ Theory of Numbers.” IMPROPER Fraction. See FRAcTION. IMPULSE, a momentary action, or force, such as that which arises from the sudden ex- plosion of fired gunpowder, or the momentum of a moving body. IMPULSIVE, relating to impulse. INACCESSIBLE, that which cannot be approached ; thus we say inaccessible height, distance, &c. See HEIGHT and Distance. INCEPTIVE, a word used by Dr. Wallis, to express what may be otherwise denomi- nated the principle of magnitude ; thus a point is inceptive of aline; a line ofa surface; and a surface of a solid. INCH, an English measure of length, being the twelfth part of a foot. . The French inch = 1:0657 Eng. inches, Scotch inch = 1:0054054 Ditto. INCIDENCE, in Mechanics and Optics, is used to denote the direction in which a body, or ray of light, strikes another body, and is & planes, towards each other, so that the line INC otherwise called inclination. In moving bodi their incidence is said to be Fines ot ao oblique, according as their lines of mo make a straight line, or an angle at the poi of contact. 4] Angle of INcIDENCcE, generally denotes angle formed by the line of incidence, a | line drawn from the point of contact perpé dicular to plane or surface on which the box or ray impinges. Thus ifa body Aim- | H pingesonthe planeDE, 4 at the point B, and © perpendicular BH be drawn, then the angle ABH is generally called the angle of incidence, H B A and ABD the angle of inclination, Dr. W lis, however, and some other old authors, ¢ the angle A BD the angle of incidence, ben that included between the line of incidene and the surfaces of the plane on which 1 body impinges, the complement of that an as ABH, being by these writers called 1 angle of inclination. Other distinctions a made by Wolfius in the denomination of the angles, but they are of very trifling impoh ance, | 1. It is demonstrated by writers on optic E) that the angle of incidence A BH, is ot equal to the angle of reflection H BC; or’ angle ABD = angle CBE. See REFLE TION. me! 2. It has also been demonstrated from e periment, that the angles of incidence ai refraction are to each other, accurately, | very nearly, in a given ratio. 3. That from air to glass the sine of angle of incidence is to the sine of the f fracted angle, as 300 to 193, or nearly as i to 9; and on the contrary from glass to al the angle of incidence to that of refraction, as 193 to 300, or nearly as 9 to 14. See R FRACTION. a Inc1DENcE of Eclipse. ‘Sec IMMERSION. — Axis of INcIDENCE, is the line BH in u preceding figure. 4 Line of Incipence, or Incipent Ray, | Catoptrics and Dioptries, is the line of dire tion in which a ray is propagated, as the lit AB in the above figure; this is also called tl INCIDENT Ray. : Point of INcIDENCE, is the point where tl incident ray meets the reflecting or refractit body, such is the point B. S| INCIDENT Ray. See Line of InctDENC INCLINATION, denotes the mutual aj proach or tendency of two bodies, lines, | i iy of their direction make at the point of co tact any angle of greater or less nagnitudll INCLINATION of a Right Line to a Plane, the acute angle which such a right line mz k with another right line, draw in the plan through.the point where the inclined line il tersects it, and through the point where it, also cut by a perpendicular drawn from “a point of the inclined lines. | INC INCLINATION of Meridians in Dialling, the gle that the hour live on the globe, which perpendicular to the dial plane, makes with = meridian. Inciinatron of an Jneident Ray, otherwise led the Angle of INCLINATION. See Angle TNcIDENCE. INCLINATION of a Reflected Ray, is the angle vich a ray after refraction makes with the is of inclination. INCLINATION of ‘the Axis ‘of the Earth, is 2 angle which it makes with the plane of > ecliptic; or the angle between the planes the equator and ecliptic. INCLINATION of the Magnetic Needle. See PPInG Needle. IncLINATION of a Planet, is an arc or angle mprehended between the ecliptic, and the ine of a planet in his orbit. See ELEMENTS the Planets. INCLINATION of a Plane, in Dialling, is the sof a vertical circle, perpendicular, both to : plane and to the horizon, and intercepted tween them. [INCLINATION of two Planes, is ‘the acute gle made by two lines drawn one in each me, through a common point of section, d perpendicular ‘to the same common sec- n. [NcLinaTIon, Angle of, in Optics, is the ne what is otherwise called the Angle of CIDENCE. INCLINED Plane, in Mechanics, as the me imports, is a plane which forms with an tizontal plane any angic whatever, forming e of the simple mechanical powers. ‘The ination of the plane is measured by the rle formed by two lines drawn from the ping and the horizontal plane, perpen- alar to their common intersection. Phe priacipal mechanical properties of the lined plane are as follows: ‘viz. |. When a body is sustained upon an in- aed plane, the sustaining power, or weight, i be to the weight. of the body, as the sine the plane’s inclination is to the sine of the gle, which the direction of the power makes ha perpendicular to the plane. ¥ Thus let A’B be ‘an inclined plane, W a ight sustained upon that plane by ‘the wer P, WE the line of the direction of the wer, and WC a perpendicular to the plane 3; then P:W :: sine < ABC: sine< FWC. hen WF coincides with WA, that is when 2 power ‘acts in‘a direction parallel to BA, m'the proportion becomes P ; W ;; sine of ane’s inclination : radius; diameter’ BC a semicircle be de- INC orP: W:: AC: BA; that is, the power: weight :: height of the plane : its length. 2. If two bodies keep each other in equili- brio, by a string passing over the vertex of two differently inclined planes, the weights of the bodies will be to cach other as the sines of the angles of inclinations of the opposite planes. A GB Cc That is, W: P :: sine < BCA: sine < ABC; or W:P:: BA: AC; because the sides of triangles are to each other as the sines of their opposite angies. 3. The velocity acquired by a body de- scending by the action of gravity down an inclined plane, is to the velocity of a body falling perpendicularly during the same time, as the height of the plane is to the length, 4. The force whereby a body descends down an inclined plane to the absolute force of gravity, as the height of the plane is to its jJength; which being a constant ratio for the same plane it follows that the force whereby the body descends is uniform, and conse- quently that it will produce a uniformly ac- celerated motion. Therefore all the laws laid down under the article ACCELERATION has equal place with regard to bodies on inclined planes, by merely substituting for gravity the product of gravity into the sine of the plane’s inclination. The same is also true with re- eard to the retardation in bodies projected up any given inclined planes. 5. Hence again the space descended down inclined planes is to the space perpendicularly described in the same time, as the height of the plane is to its length. 6. Consequently the velocities acquired, and the spaces descended by bodies down different inclined planes in the same time, are as the sines of the plane’s elevation. 7.1f AB be any in- clined plane, and DC be drawn perpendicular to AB; then while a body falls freely through the perpendicular BC, ano- ther body willinthe same A Cc time descend down the part of the plane BD. 8. Hence we deduce the following curious property of bodies descending down inclined planes. In any right-angled triangle having its hypothenuse BC perpendi- Bb cular to the horizon, a body will descend down any of its three sides BD, BC, DC, in the same jy, time. And therefore, if on the B D D scribed, the time of descending down any chords BD, BD, BD", &c. DC, D'C, DC, &e. BB2 INC will be all equal, and each equal to the time of falling freely through the diameter BC. 9. The time of descending down an in- clined plane, is to the time in falling through its perpendicular height, as the length of the plane is to its height. Consequently the times of descending down different planes of the same height are as the lengths of the planes. 10. A body acquires the same velocity in descending down an inclined plane, as in fall- ing perpendicularly through the height of the plane. Hence the velocities are alw ays the same in planes of the same altitude, whatever may be their degrees of inclination. 11, Ifa body descend down & any number of contiguous planes AB, BC, CD, &c. it will ultimately acquire the B same velocity, asa body which falls perpendicularly through the same height ED, sup- c posing no change in the ve-D locity in passing from one plane to another. 12. As this is true, whatever may be the number and magnitude of the planes, if we suppose each’ of them to be indefinitely small, they will form a curve line, which will thus be- come the path of the body. Whence it fol- lows, that a body acquires the same velocity in descending down any curve, as in falling perpendicularly, though the same height. See Gregory’s Mechanics, vol. i. INCLINERS, or Inciininc Dials, are such as are drawn on planes, that are not perpendicular to thé horizon. INCOMMENSURABLE Lines, are such as have no common measure, the diagonal and side of a square are incommensurable, being to each other as ./2 to 1; and conse- quently whatever number of parts the side of the square may be divided into, the hypothe- nuse will not be made up of any exact num- ber of such parts. INCOMMENSURABLE Numbers, or Numbers Prime to each other, ave those which have no integral common measure greater than unity. Tf numbers be incommensurable with each other, they are incommensurable also in power, that is no powers of such numbers can be commensurable with each other. INCOMPOSITE Numbers, the. same as Prime Numbers, which see. INCREMENT, in the Doctrine of Incre- ments, or Finite Differences, is the finite in- crease of a variable quantity; Dr. Brooke ‘Taylor, to whom we are indebted for this theory, denoted his increments by a dot under the variable quantity, thus the increment of x was denofed by * ; Emmerson also employs the same notation, ’ others have used a small accent, as 2’, or thus *. M. Nicole uses ano- A ther letter denoting the increment of x, or any variable, by x; but Euler, who seems to have given a permanent form to this branch of analysis, employs the character A; thus the increment of « = Az, increment of 4 hae Oy, &e.; see the following article. selence, to lend their kind assistance fort INC INCREMENTS, Method of, called by | French Calcul des Differences Finies, is an| teresting branch of analysis, invented by | celebrated Dr. Brooke Taylor, and publish by him in 1715, in a work entitled “‘ Metho Incrementorum,” in which both the dir} and inverse method are treated of in a yj learned and ingenious manner, as also { application of the doctrine to a variety of yi elegant and interesting problems; but {| novelty of the subject, and the concise m« of expression employed by its author, toget| with a very complicaied notation, rendei the work nearly unintelligible to any one skilled in analysis than the author hims} Fortunately however for the interest of | less able analysts, M. Nicole of the Academy of Sciences, undertook an illus tion of the subject, having g published his f paper on this doctrine in ‘the memoirs of t society for 1717, and afterwards two off papers in the same memoirs for the ye 1723 and 1724. And if his method is 1 general than that of Dr. Taylor, it has ‘ advantage of being extremely simple, a thus preparing the reader for more gene researches, by leading him on from step step with order and precision. Dr. Taylor himself also, in the Phil. Tra undertook an explanation of certain parts his work, and its farther application to so kind of series beyond those treated of in original: the same was also done by M. Mo mort, in the ‘ Transactions” for the ye 1719 and 1720; which latter gentleman, seems, had conceived some idea of the the before Dr. Taylor’s work appeared; and trifling altercation took place between the as to the originality of some of the notio which was claimed by both parties. In 1% Emerson published his ‘‘ Method of Ine ments;”” a work which, at least, does t author as much credit as any that he @ produced, but the notation resembles ii great measure that of Dr. Taylor, which m ders it now almost obsolete. Emerson ¢ pears to have been extremely anxious to bri the theory of increments to perfection; a earnestly urged those who were qualified the task, to pursue the paths he had open to them. ‘“ I cannot,” says he, “‘ promise tl I shall have time and leisure hereafter to p secute this subject any farther. And as have an earnest desire of advancing truth, a improving science, let me here entreat { friendly mathematicians, who are lovers advancement of this uncultivated branch. knowledge, yet in its infancy, or rather as} in the hands of Lucina, either according to yt model I have here laid before them, or sol better, if it can be found, so that by degre it may at length be brought to perfectio “* Hence,” says a writer on this subject, § appears that this author was even in 17 sensible of the neglect that had been paid the theory of iner ements ; and even now né€ half a century after this date, the subje Le INC snot been advanced, nay scarcely touched on by any English ‘mathematician, while eign authors are filling quarto volumes, ith the theory of differences finies, and its nost universal application to the most cu- us and important of mathematical inqui- s.”’ Euler, whose universal genius led him the investigation of every subject that was sful and interesting, did not leave the theory increments untouched, but has treated of in his usual masterly style, in a work enti- d, “ Institutiones Calcenli Differentialis,” {.in which he has given a new ferm, and ich extended the bounds of this important anch of analysis; and subsequent authors ve adopted his ideas, and rendered perma- nt the form he gave it.” Various other rks have since appeared, to illustrate and ider familiar the principles of this doctrine, y most complete of which is the “ Traité s Differences,” &c. par Lacroix. Bossut hs likewise a chapter on this subject, in his IDraité de Calcul Differentiel,” &c. where ) theory is treated in a very elementar y and i nprehensive manner; and the same is also ine by Cousin, in chap. iii. of the introduc- in to his “ Traité de Calcul Differentiel,” t. Rees’s Cyclopedia. We will now endeavour to illustrate this aject by a few simple problems, referring Is reader, poe is desirous of more minute |ormation, to the works above mentioned. Of the Direct Method of Increments. sl. The increment of a variable quantity, as ‘is the difference between that quantity in ) first state, and what it becomes after it has )2n increased by any certain finite quantity ; jich increase after Euler we shall denote by ic; hence Che increment of2 = Az UB ei diss Vecvesiae OF U7, (a A x)* — 27 | are aap of 23 — (a + Ax)} — x3 er aMleness tS hse of 2” = (x + Ax)™—2” ie) 22 Axe + Ax’ iz?) —=327 Ar +32 Ax’ + Ax ery — 0) A 2 BO Vim ant + ke. ‘hence it is obvious, that we may readily d the increment of any power of a variable antity, and as all expanded functions are mmonly made up of the several powers of : unknown quantity, it follows that this ne would extend to a very great variety of ses, but it sometimes happens, that it would -attended with considerable labour, when lifferent method of considering the subject uld lead us to the same result in a much lyre commodious manner; but as the limits ‘this article will only allow of a slight ab- act of the principles of increments, we can- {t enter into an explanation of the "methods Love alluded to, but must refer the reader ithe works mentioned in the preceding part | this article ; ; noticing only another very ference between 2 and (w+ Ax)” ing formule for these cases; viz. a(! res wre; aG). SE aa: Ax)(a+2Axn)... oo, SIN UG general rule relating to continued products of the form x, (x + Ax) («+2 Aa) (*#+3Azx)ke. Here we haye Ax TAY Ajx(e+Az)! yan } (ae Ax)(x +22) 3an(e+ Ax\x+2Azx) and generally the increment A jx (2+ Ax) (x+2Az) .... (n+ 1) (a+ Ax)(a+2Anz)... 2Ax(x-+ Az) (x -+-nAz) , iss (x +nAr) By means of these formule we may readily find the increment of.any function of a va- riable quantity, which consists only of integral powers. Suppose, for example, the increment of the function « — x3 + 32% + 4a + 10 were required. Here from the preceding formule we have Au A(x) +3 A(x) + 4A (2). Now A (x3) =3 a" Ax +32 A? Aa 3 A(a?) = 62 Art+3 Ax. 4A(z@)mt4Ax. Hence Au= 327 Ax+2(8A2* + 6Az) +A +3 Aa? +4Az. And we may proceed in the same manner in all similar cases. 2, Again the increment of i being the dif- 1 pita’ the Inere- z+ Ax ment of bs the difference between “5 and - a I &c, we readily deduce the follow- a(4 i= 2QrAx+t Ax (a + Ax) _ B8ar*Ax+38xr Ax + Ax a(4)=- BRT Te RAY. Fs ORES mx™—! Ax +m 5} xem? Ax? 4-&e, a(=)= Ty waa. + - Ax)” In a similar manner we find ae he 3 (a@ + Aa) Ax) —2A2x a(e+ Ax) (a+ 2Arx) 1 Ala caB= soe Cie Anka nae a —d3 Ax a(+ Ax) (@+2Ax)(#+3Az) And generally the increment 1 (a+u An —(n+1l)Azx x(a+ Ax) (e+ 2Ax)....@e+(n+ 1) Az) 3. But the most general formule for finding the increment of any function of a variable ONC guantity is that given by Dr. Taylor, the au- thor of this theory, which is as follows: Let Y represent any function whatever of the variable quantity x, then if x be increased by any finite difference A x, the value of A Y adopting the differential notation will be de- noted by the following formula, which is com- monly called TayLor’s Theorem, and is of the most extensive use in this theory, as well as in the differential calculus. That is, Y being any function of x, we shall have ay * Aady , Aad y Ady, . 57 a L &e. an = Sivplicn ie udibcaboan ti tee dadacks os where the law of continuation is obvious, and therefore requires no farther development ; it must however be remarked, that when A x is negatived, the terms of the above series must be taken plus and minus alternately. In order then to find the increment of any function of a variable quantity, we must take the successive orders of its fluxions, by means of which the fluxional parts in both the nu- merator and denominator will disappear; and we shall have the value of A Y, expressed in terms of x and Az; and this expression will always be finite, unless the function be tran- scendental. Let us propose the function y = ax 4+ ba” + 2°, to find the value of A Y, when z be- comesx +AnX, og y —adzx + 2badzx + 32°*dx ay 2bdx* + 6xdx’ 2 yx 6 dx dy = 0 Now these values being substituted in the general series, we shall have for the increment of y, Ay Ax(a+2br+327)+ Ax*(b4+32)4+Ax3 Again, given y =ax + ba* — ca + 2+, to find the increment of y. d y madzx+ 2brdx—8cex*dx+ 4x3dx Cais + 2hda* —6exrda* + 1227?dx* y= —6ceda3 + 242xdzx' d+ y = +24 dat* dy = 0 Whence making these substitutions in the ‘ general theorem, we have PY He Ax(a+2bx—Sex* +423) + Ax?(b—8ex my— + 627) + Ax (e+ 4x) + A xt. These examples will explain the method of applying the theorem to any case that may arise, it being as universal in its application in the theory of increments, or differences, as the binomial theorem is in the expansion of routs and powers: but like the latter, it was left by its author without demonstration; it has however received many since, from several able mathematicians, though there is no one of them perhaps that is quite so satisfactory as might be wished; Maclanrin’s rests upon the fluxional calculus, which it would be de- sirable not to introduce, if it could be obtained from more simple and obyious principles. ‘Another demonstration of this theorem, and which is esteemed the best, is given by L’Huilier in his work “ Incipioram Calculi INC e Diff. et Integ.” &e. 1796. 4. In all the preceding formule we considered the increment of the variable constant, because the limits of our arth would not allow of extending our remarks more than the most simple forms; it is obviai. however, that the differences by which ay riable is increased or decreased may be the! selves variable quantities, in which ease ‘ should have to find the second and third | crements, as in fluxions it is sometimes £ cessary to take the second and third fluxioy. these cases however but seldom arise, exer in the higher order of problems, which reader will not expect to find investigated a work of this description. | published at Geneva Of the Inverse Method of Increments. 5. In the Inverse Method of Incremen the question is to find the integral or funetic from its increment being given. We my therefore examine with attention the steps which we descend from a variable quanti to its increment; and then, by the rever operation, we may ascend to the integr when the increment is known. But this verse operation is attended with the very sat difficulties as the inverse method of fluxion for as in that, every fluent may be read put into fluxions; so may the increment any function be readily obtained; but it frequently difficult, and sometimes impossib to find the fluent of a given fluxion ; and so the method of increments, there are ma) cases that will not admit of integration; ¥ shall however give some of the most usual al obvious rules, and which will apply to t generality of examples, Let us first attend to the able quantity x. 6. Since A x= A (x); therefore recipr cally, // Ax=x. Andif we suppose A | constant (a supposition that has place in: that follows) we shall haye, | powers of a va f a J Axxl=z,or Axf.l=z, therefore /:1=% Since A ciprocally, S (22 ka + A x”) =x”, or, which is the sam J 2x hu + f. Ax* =x"; whence again, als Ax a> ra th . xe f hapa ped and hence 1} S a 2 2Ax . | (x?) = 2x2 Ax + Ax’, thereforer transposition, | P ~f. Ax a” At ! AAC fo = oo ——- < ae 4 2ZAx 2 Dax 2 f¥ a x | 2Ax 2 Again, since A (23) = 327 Ax + 3x Aa + 42%; therefore reciprocally, | So \ 3x" De + 3a Aa* + A x3)= 23; or, whie is the same, i. Si 8x* Dat f3x Ax +fAx—2z3; or, divic ing by 3 Aa, tS ae 3 x ———; whence x a - ‘ge — * Ar x— Se, 1; or, which ¥ 3Ax Sf Lh) ? 3 the same, 4 3 MY rAx “ — Gx 2 % i 7. We find in a similar manner by conti- uing to suppose A x, as constant, and sub- tituting always for the quantities contained under the particular sum their respective alues; the following results, for the integrals f the successive powers of x, in which we aye repeated the two preceding ones, for the ake of uniformity. x l ——-; af Page x” x . ey ve we ——}5 “a OPES 3 .z | eek x. LAX ee Ane 8.” OE EE BP "gt em) So ; [< 4Axn 2 A ‘ a at BAgr at Ax TA 2* = —-—— — ——-; LS SAa ” 2 3 a0 Bin xs ete ee Or eA 4 6A a ele. u 122 3 we &e. Where it is only necessary to observe, that ‘the proposed increment have any constant qultiplier, the integral above found must have ie same. 8. Hence we may find the integral of any metion made up of the powers of 2, effected r not with any constant co-efficients a, ), ¢, cc. For in order to find the integral of such nb increment, it is only necessary to find 10se of the different powers of x, and their am will be the integral required. Exam. 1. Required the integral of the’ in- rement a + bx + cx’. , coef iat OO Plo as Gy aff At ba* bx | a) == ‘ — — —— ie As $y 2 a3 caw” . ex Ax — eR 6 "And hence by a ba” bx cx} b . =~ Be en oa mee ack) = Sere eG 2 3Ax Cx” or As 2 6 Exam.2. Required the integral of ax+— 62”, onsidering A x as constant. Here we have et ax __ axt ax Ax ax Ax? SAx 2 3 30 ae __ —bzx3 ba® bx Ax ZAR 2 6 he sum of which expressions will be the vhole increment sought. 9. When it is required to find the integral INC of a quantity of any of the following forms, (A x being supposed constant), viz. _ («+ a) (x + a) (x + 2a) (x + a) (x + 2a) (x + 8a) (x + a) (w + 2a) (x + 3a) (a + 4a) We arrive at them by taking the actual product of those quantities, and “finding suc- ic a the increments of each of the terms: us xn. ar Kara fut fa= ce SPs ge S e+ a) e+ Qa) = f(x" +3ax+2 40 ae ee PE xcAx f* PPAR 2 TEX Sax” Sax mS HR ln Loeitcedh An BBi 2Azx 2 Qa*x py - = eo t x Ax ae Whence f.(w + a) (# + 2a) = —— ase 4 tAL , Sax? _ 3ax 2a’ x 6 2Ax 2 Az’ Again, fj(«@ +a) (@ +2 @+3QDE Sf (& + Bax* + lla®x+6a3)= fix LS rl aah a8 x” Ax ~~ 4Ar 2 4 6ax3 6ax* . 6GaxrAx , 64 SS ee B bebe iets J: sAx 2 + 6 hletnes: Vla*2* lla*x Bo eae 2 6a3 x b Grek SS J: Ax Whence by sadiBan, we have f. (x + a) | me a* Ax x Qa 3 oo (@ +20 (@4+3)= FP Aty 2ax3 3ax” axAx , lla*zx Gasix Ax 1 1 ZAX Ax llatx , Gabe . 2 Ax and so on of other similar quantities. And if it be required to find the integral of quantities of the form, x (x + ay x (a + a) (x 4% 4) Te taka (etta ttak &e. &e. We have in the same way, 5 Ev bah 2 —_ eee PBB KO me le BON be OR oe Fo eae xrAx ax? ax 6 + ZA “2 fx (e+a) (a@+2a=/. ay ae ar Sn 7 . oes x* Ah! OF ate + a Ax aia OS B ; oe Be __ Sax 3art Ax Qax* Qax bas = cee Le oe 2 The sum of which will be the increment re- quired. And in a similar manner we may find the integral to any other quantity of these forms. INC 9. Remark. Before we proceed any farther on this subject, it will be proper to attend to the correction of any integral; when from the nature of the problem under consideration such becomes necessary. As the increments of any variable quan- tities, x and x + a, are both expressed by A z, the constant part a, having no increment, so reciprocally, the integra! of the increment A 2, may be x, or x +a; therefore when we have found the integral of any increment, we must add to it a constant quantity, which will be zero, if the integral needs no correction; but areal quantity, positive or negative, in other cases which must be determined from the nature of the problem. ‘This remark will be of considerable importance in what follows. We might have here carried this subject to a much greater length, as no branch of ma- thematics offers a wider field for the exercise of analytical investigation than the inverse method of increments, except, indeed, the inverse method of fluxions, to which it is nearly allied ; but our limits will not admit of such development, and we must therefore proceed immediately to the application of this doctrine to the summation of series, to which it is very admirably adapted. Application of the Method of Increments. 10. The summation of series by the inverse method of increments is founded generally on this principle: ‘That if we have any series of quantities, as a, b, ce, d, e, &c. which are de- rived from each by some certain and invariable law, each term may be considered as the increment of the sum of all those which pre- cede it. Thus if at+b+c+d+te po A @+b+e+d+e+f=uz then it is obvious that 2’ — z = Az= f, and therefore conversely the integral of any one of those terms considered as an increment, will represent the sum of all the preceding part of the series. This being premised, we may proceed to the solution of the following examples. Exam. 1. Required the sum of » terms of the natural series. 14+24344465+4....n. Here by writing x instead of », the term next in order will be x +1, which being the increment of the series, we shall have f.(@+)) = the sum required. Now/ (@4+1)=f%+4//1; and by art. 7. Pract wth a Enna t jets a Oe es ee ig Tooele and since in this casé A x = 1, we have ao Mh wee cate nbn fOt=5 75 *q= Soo by writing again n jnstead of x, which is the sum of n terms of the proposed series, as is also evident from other considerations. This example offers an easy illustration of what has been observed at art. 9, of the cor- rection of an integral, which is necessary in many cases, the same as the correction of a ’ ae Thy a INC fluent is in the fluxional or integral calculy Suppose, for example, that instead of the pr ceding series beginning at unity, it had con menced from any other term as 7 ; the gener, law of formation would have been the sam and the increment would still have had th form « +1; and consequently the integra in the first instance, would be represented ¢ above ; vz. 1 x* +2 L(e+D= 5 But here a correction of the integral is n cessary, for from the nature of the serie when x = 7, the sum of the series is 7, th being the term at which the series com mences; whereas without a correction, w should have the sum — 28; we must theré fore write, — Te | __ xit+e : a | f(e+D)= = ‘+e; c being the correctig A and since when x = 7; cs ams +c—7:9 =+ ib +ce=7 we finde = — 21; after whic the sum of any number of terms of the pre posed series are readily obtained. Thus fe example, let » = 16, then the sum of th series beginning with the term 7, becomes | 167 + 16 2 « Exam. 2. Required the sum of the naturé series of cubes, 13 + 23 + 33 + 43 +, &c. ni, Here the general term is (x + 1)3; and th integral of this, that is £ (a + 1)3 = f. 23+ fp x* +3 fx + £1, is computed as follows; — 21 = 1165 as required. fac ens oe neh a= ad ste re stb ate x ; i addeaa ye re fod. =a And here since A x= 1, we have . 3 et a3 c= (2 iy tet) ara ag, oko Or by making a (% + 45 a7) Hence a very curious property with regar¢ to the sums of consecutive cubes ‘beginning at 1; vz. that this sum is equal to the square of the sum of all their roots. Thus 13 + 23 =(1 4+ 2)? | 13423433=(14243) | 13 +23 4334 43-(14243+4 4% &e. Ke. =n, the sum required i Exam. 3. Required the sum of. any pro posed number of terms in the series of tri- angular numbers. 5 1, 3, 6, 10, 15, &e. g In this case the general term is pS . ay And it will therefore be necessary to fing IND ee » integral of sl eld =i f(@+)) + 2) and this art. 8= } f(x +1) (a# + 2) de 1) (a4 +2) _ n(rw +1) (n+ 21, i 6AxX wit 6 y aking «—n; and Ar=1l. ‘The same may be otherwise found from 't. 7; thus, mee ti) Desa +t fsa rs [2 x x tA BS ee = Vv ee ___C as GAS. .°4 12 : 32° 32 fe =——_——— x —. — — 4 Ax ,And since A x = 1, this sum becomes ) . _ 3 a? ae _ r(e+1)(a-+2) f@+))(@4+2)=—+5+5= ‘d-the sums of any order of polygonal and furate numbers may be found in the same janner. iy 4. Required the sum of any number ‘terms of the scries, }E.24+2.343.444.5 +, &e. n(n +1) Here the general term of the series is x(a--1), \d it is therefore required to find the integral '+1)(«+2). First, by art. 8, we have |: 42 _x(a-Li)(r+2)_ n(n-+-1)(n+2) writing x—n, and Ax = 1. /'The same may also be found, as in the ex- ‘aple above, by article 7; but it is needless | repeat it, as it only differs from it in its vnstant factor. ‘We have in neither of the foregoing ex- oples used any correction, for it is obvious, a(x +1) (a+ 2) 6 at when x =o, the formula .o as it ought to be, and therefore the in- gral needs no correction. We had intended to have added here an ‘ample or two relating to the summation of finite series; but the article having already -ceeded its due limits, we have been obliged cancel them, and can therefore only refer e reader to the works mentioned in the eceding part of this article. Montucla, ist. des Math. Encyclopedie Methodique. ees’ Cyclop. vol. xix. INCURVATION of the Rays of Light. »e REFRACTION. INDEFINITE, that which is without any signed limits; thus we say an indefinite ie, meaning a line of any length. Some ithors use the word indefinite nearly in the me sense as we commonly attach to the rmibfinite. According to these, an indefinite 1e is that which is without termination; the ‘rmer, however, is the sense in which it is mimoniy employed by modern mathema- sians. INDETERMINATE, is nearly the same s Indefinite, the former being applied to num-~ ers, as the latter is to geometrical lines, jirfaces, &c. with this difference, however, hat the word indeterminate commonly implies | IND that number or quantity whose value cannot be assigned, and the former, that which may be of any magnitude. ; INDETERMINATE Analysis, is a very inte- resting branch of algebra, in which there are always given a greater number of unknown quantities than there are independent equa- tions, by which means the number of solu- tions is indefinite, though it commonly happens that certain restrictions are introduced, such as requiring integral or rational numbers, which frequently limit the number of solu- tions, and even in some cases render the pro- blem impossible. Diophantus, who is supposed to have flou- rished about the middle of the third century of the christian era, is the first writer on the indeterminate analysis, at least his Arithmetic or Algebra is the earliest treatise we know of on the subject. This work was commented on by the celebrated Hypatia; but her com- mentaries never reached our times, the first translation of the original being that published by Xilander in 1575, till which time little or nothing on this subject was known in Europe ; but it appears from a Hindoo Algebra, now publishing in this country, that these people were so early as the twelfth century particu- larly well acquainted with the indeterminate analysis, possessing some of those rules which have since been discovered by Euler and Lagrange, and which have been considered as very important advances in the Indeter- minate Analysis. It is but lately, however, that this work, entitled “ Bija Goneta,” has been known in this country, and what we know on the sub- ject of this branch of algebra is wholly inde- pendent of the knowledge of the Hindoos or Persians. ; After Xilander’s edition of Diophantus, | the subject remained unnoticed, till Bachet undertook another edition in 1621, and to this author it is we owe the g@neral solution of indeterminate problems of the first degree. Fermat made another edition of the same work, which was published after his death, in 1670; at which period this branch of ana- lysis was much cultivated, both in France and England; and problems were proposed by the mathematicians in one country as challenges to those of another. In this con- test Dr. Wallis, Lord Brounker, Fermat, Fre- nicle, Des Cartes, and Roberval, were the prominent disputants, and much light was thrown upon the subject during this rivalship. But it was not till towards the conclusion of the last century that the Indeterminate Analy-~ sis began to assume a permanent form, an im- provement that we owe, in a great measure, to Euler and Lagrange; though before this time several very reputable works on the sub- ject had been published by De Billy, Prestet, Kersey, Ozanam, Kirkby, &c. What Euler has done on the subject is con- tained in the Acta. Petro. and the second volume of his “ Elements of Algebra.” La- grange’s improvements are given in the IND Memoirs of Berlin, and in his Additions to the work of Euler above mentioned. To these we may add a paper in the Edinburgh Transactions, by Leslie; the ‘ Essai sur la Théorie des Nombres,” by Legendre; the “‘ Disquisitiones Arithmetice,” by Gauss; and Barlow's “ Elementary Investigation of the ‘Theory of Numbers,” the second part of which is wholly devoted to this subject. Solution of Indeterminate Problems. When amongst the several unknown quantities there are none that exceed the simple power, the problem is said to be of the first degree. If the second power enters, the problem is of the second degree. If the third enter, it is of the third degree, and so on, of the higher powers. Indeterminate problems of the first degree, are all included in the general form, as by shor ts, kc.’ = dj and they are all solved by means of the simple formula ap — bq = 1. That is, by finding the values of p and q in the equation ap—bq—=+1, we may thence immediately determine the values of x,y, and z, in the above more general form, our first investigation must therefore be di- rected to the solution of the latter equation, the general rule for which is as follows: Convert the fraction ; into a continued fraction, and then again to series of converg- ‘ing fractions, then the numerator and deno- minator of the fraction preceding © will be the values of p and gq, as is obvious from a property of these fractions mentioned under the article CONTINUED Fractions. Exam. 1. Let it be required to find the values of p and q, in the indeterminate equa- tion, l6p — 41g =1 Here the fractions converging towards are $, 4, 3, ie 48, therefore p = 18, and q= 7, which give 16.18 — 41.7 =1, as required. Exam. 2. Find the values of p and q, in the equation, 17p— l5q=—1. Here the converging series is, 16 41) bp therefore p — 7 and g =8, for17.7 — 15.8 = — I, as required. In the above solutions we found ap — bq =-+1, and ap — bg = —1, as the questions required; but we are frequently led to the solution ap — bq =-+ 1 when the question requires — 1, and toap— bq = —1 when the question requires + 1, which seems at first to destroy the generality of the rule; but these are easily converted from the one to the other, as follows: Let the proposed equation be ax — by = + 1, and suppose we have found p and q in the equation ap —bq =—1; we have then only to make « =bm—pand y = am—q, and it is obvious that we shall thus obtain a2 —by =a(bm— p)—bam—g= +1 x . ae ] IND as required. In this expression m is indet¢ mininate, and by means of which an inde number of solutions may be readily obtaine Again, if the proposed equation were az- by = +-1,and we had found ap—bg=+ we might still obtain an indefinite number solutions by making x = bm-+-p andy =a + q, for then it is obvious also that , a(bm +p) —b(am+q)=+1, and by means of the indeterminate m, an ind finite number of solutions may be obtained, | In order to illustrate this by an exampl let there be proposed the equation 1 I3a2—19y = 1. ] The converging series is 1, 3, 19, therefo| p = 3, and g = 2, which gives ap — bg =" or 13.3 — 19.2=—1. - Therefore the general values of x and y a x = 19m + p, and y = 13m + q, or x= 19m + 3, andy = 13m + 2. Assuming therefore m = 0, 1, 2,3, 4, & we have the following values of z andy: ne ao Ono 2 ee OF nis a = 3, 22, 41, 60, 79, 98, &c. y = 2, 15, 28, 41, 54, 67, Ke. which series may be continued at pleasure. If the proposed equation had been 132- 19y = —1; then having found p=3, a g = 2, as above, we must have made x = 19m — 3, and y = 13m — 2; and then by assuming m as before, we shou have m Oj By: 25) ys en, Oe De x — — 3, 16, 35, 54, 73, 92, &e. y = — 2, 11, 24, 37, 50, 68, &e. where it may be observed, that the successiy values of a and y, in both cases, form a serie of arithmeticals, and may therefore be cot tinued with great facility. “ —s a —— To find the general Values of x and y, in tl Equation ax — by me. In the first place, we must have either and 6 prime to each other, or if they have common measure c must have the same, fc otherwise the equation will be impossibl and in this latter case the whole equation ma be divided by that common measure, and thu reduced to one in which a and 6 are prime t each other: it will, therefore, only be nece sary to consider the quantities a and 6 a prime to each other. Also, after what 1 been taught in the foregoing proposition, w may always suppose that we know the cas ap—bq—+ 1; it will, therefore, only b necessary in this place to show how the genera values of a and y, in the equation az — by= + ec, may be deduced from the known ¢ ap—bg=+1}. 7 In the first place it is obvious, that since ap—bgr=+1, we shall have acp—beqr=+e; 4 but this furnishes only one solution; and i order to have the general values of x and IND re must substitute 2 = mb + ep, andy = ra + eg; which give a(mb + cp) —b(mat cq) = +0; he ambiguous sign +, in the two values of and y being +, when ap— bq has the same ign with e, but — when it has a contrary — ne. _ Exam, 1. Find the values of x and y, in the quation 9x — 13y — 10. First, in the equation 9p —13q— +1, we aye p=3, and g = 2, which gives 9p —18q =+1; and this being the same sign with 0, in the proposed equation, the general va- aes of x and y are ee oe § y= Im+ 2e, or tx = 13m + 30; ty= 9m + 20. _ And by assuming here m = — 2, — 1,0, 1,2, se, we have the following values of x and y. m — 2, — I, 0, 1, 2, 3, &c. x 4, 17,30, 43, 56, 69, &e. y 2, 11, 20, 29, 38, 47, &e. ach of which values has the required con- and itions for 9..446 — 13. 2— 10 9.17 — 13.11 — 10 9.30 — 13.20 — 10 9.43 — 13.29 = 10 &e. &e. &e. ' Exam. 2. Find the values of x and y, in the ‘quation 7x —12y ='19. _ First, in the equation 7p — 12q = —-1, we ave p — 5, and g =3; where — 1 has a dif- »rent sign from 19, in the proposed equation; ierefore the general values of x and y are i ae — 12m— 5.19 § y= 7m— 3.19, or i! ¢ = 12m — 95 ty=7m— 57; here, by taking m = 9, 10, 11, &c, in order hat x and y may be positive, we have fe x — 13. 25, 37; 49; Gly 73, 85, &c. y= 6, 13, 20, 27, 34, 41, 48, &c. nd in a similar manner may any possible quation, ax — by — + ¢, be resolved. a find the and general Values of x and y, in the : Equation ax + by = ce. | In the foregoing proposition, where the dif- “rence of two quantities was the subject of onsideration, we found that the number of slutions was infinite, providing a and b were rime to each other; but in considering ie sum of two quantities, asin the present ‘ase, the number of solutions is always limited, nd in many cases the equation is impossible ; _may, however, be demonstrated that the quation admits of at least one solution, if > ab — a — b, a and b being prime to each ther; and it is proposed in the present pro- osition to ascertain the exact number of ossible solutions, that any equation of this ‘ind admits of in integer number, and to oint out more accurately the limits of pos- bility. ' The solution of the equation az + by =e epends, like that in the foregoing proposition, pon the equation ap — by = + 1, though ‘s connection with it is not so readily per- i an : | IND Let ap — bq = 1; then we have also a.cp—~b.cq =e; and it is evident that we shall have the same result, if we make x —=ep—mb, andy = cg — ma; for this also gives a (ep — mb) — b (eq — ma) + ¢; assuming, therefore, for m such a value that cqg—ma may become negative, while cp—mb remains positive, we shall have a (cp — mb) + 6 (ma—eq) = ¢; and consequently « = cp—mb, and y=ma—cq; but if m cannot be so taken, that cg—ma may be negative, while cp — mb remains positi¥e, it is a proof that the proposed equation is im- possible in integer numbers. And onthe con- trary, the equation will always admit of as many integral solutions as there may be dif- ferent values given to m, such that the above conditions may obtain. And hence we are enabled to determine, a priori, the number of solutions that any proposed equation of the above form admits of: for since we must have cp > mb, and cq. < ma, the number of solu- tions will always be expressed by the greatest integer contained in the expression (Z Pee 4 a as is evident; because m must be less than the first of those fractions, and greater than the second; and therefore the difference be- tween the integral part of these fractions will express the number of different values of m, cy p b A en , b which is the same, we must consider Basa except when is a complete integer; or, ; ng ae a fraction and reject it, but not oA the reason for which is obvious, Exam. 1. Required the values of x and y, in. the equation 9x + I1By = 2000, and the number of possible solutions, in in- tegers. First, in the equation 9p — 13q — 1, we have at once p = 3 and g = 2; therefore the number of solutions will be expressed by PODS _ BOOKS = 461 — 444 = 17. And these are readily obtained from the for- mulze fx—=ep —mb ,y=ma—eq, or Ux = 6000 — 13m ty = 9m — 4000; in which, assuming m= 445, 446, &c. in order that 9m > 4000, we shall have the following solutions; each of which is deduced from the preceding one, by adding successively 9 for the values of y, and subtracting 13 from those of a; thus x — 215, 202, 189, 176, 163, 150, 137, &e, y= 5, 14, 23, 32, 41, 50, 59, &e. that is, 9.215 + 13.5 = 2000 9.202 + 13.14 = 2000 9.189 + 13.23 = 2000 9.176 + 13.32 = 2000 &e. &e. and IND Exam. 2. Given the equation lla +13y=190, to find the number of solutions, and the values of x and y. First, in the equation llp — 13q=—1, we have p = 6 and q — 5; therefore 190.6 __ 190.5 _ 9 _ gg — 1: 13 il whence it follows, that the equation admits of only one integral solution; and this is ob- tained from the seed ere —m yma —cq, or r= 100.6—13m 24 ‘foe 11m—190.5 where by taking m= 87, in order that ma—cq may be positive, we have x = 9, and y¥ = 7; which gives 11.9 + 13.7 = 190, as required. Our limits will not allow of a farther deve- lopment of these principles, in the present article ; but the reader who is desirous of far- ther information, may consult the second part of Barlow’s ‘“‘ Theory of Numbers,”’ where he ‘will find a complete investigation of this theory, as applicable to the solution of indeterminate problems of first, second, third, &c. degrees ; those of the second in particular being dedu- cible from the simple form p* — nq* = 1, as those of the first are from ap — bq = 1. The following table, or synopsis of indeter- minate formule, is also extracted from the same work; which the reader will find ex- tremely convenient in a great variety of cases, not only as connected with this subject, but in ascertaining the fluents of a numerous class of fluxions, falling under any of these radical forms. . SYNOPSIS OF INDETERMINATE FORMULEZ. FORM. I. Equation, az — by = +c. General value of x = mb + eq y—=matep In which expressions m is indeterminate, and the values of p and q result from the solu- tion of the equation ap — bg = + 1. FORM. II. Equation, ax + by =. ; General value of x = cq — mb ssdes.ovevcceesoncsesees = ma— cp Number of solutions = ot sis a The quantities p and q being ascertained as above, also m indeterminate. FORM. III. Equation, ax + by 4+ ez = d. § General value of x = (d — ez) q — mb y = ma — (d— cz) p The quantities p and q being found as above, eeeheeeseseretesetessoce : : 4 d also m indeterminate, and z any integer < — c FORM. IV. Equation, z* — ay* = 2?. ¢ General value of a = p* + aq” ? Ssh iy auditudieblga sige voce y a “pq A _ ERE we Z = p* — aq p> —Nq =+1, and m is indeterminate | IND i. In which expressions a is given, and p an gq are indeterminates that may be assumed | pleasure. FORM. V. | Equation a* + ay” = 2”. Fee ee eeeerereseresereses « a being given, and p and q indeterminates a above. FORM. VI. | Equation, ax” + bry + y* = z*. pone value of x = 2pq + bq Opa ap ban sopeeh Midas y=p —aq* pesvelvinests etesrsses Z = p* + bpg + aq?! where p and q are indeterminates, and a and. given quantities. | FORM. VII. Equation, ax* + bx = 2”. | § General value of x — bg" “ 2 j HP Mews diy! om where a and } are known quantities, and | and q indeterminates. " A i ~— eo — — FORM. VIII. Equation, m” x* + bx + ¢ = 2’. General value of x — ME Heim hi ) bq’ — 2mp “ ited Dee secsceseae 2 Mp* + meg” — bpg bq° —2mpq | where m, b, and ¢ are any given numbers, ant p and q indeterminates. | FORM. IX. Equation, ax? + ba + m? — 27. bq? —2mpq General value of x — > —_— a 2 1 mp” + amg’ — bpi p—a? where m, a, b, are known, and p and q inde| terminate. : eecetesas Beeeccetses eve v4 — FORM. X. | Equation, 2* — Ny? = + 1. : Gen. val. ofz' = (p-+qv N")+ (p—qv NP tqvN™) < (pg v Ny y= Dt dy N—(p—q NY 2VN : Where p and q arise from the equatior eeeeeoreeetsen except that it must be even or odd, as the case requires. See ConTINvVED Fractions. | | FORM. XI. Equation, x27 —Ny* = + A. ; General value of « = pm + Nqn y¥=pn Lqm | The values of mand x being found from the equation m*— Nu* = + A, ond p and q from & 4 : Sees artesesssosesereser the equation p* — Nq* = + 1, | | IND | FORM. XIl. | Equation, az? + bz? + ex + f? =2’. c* — 4bf? 4af* if Par. val. of x = (4bf? — oF vhere all the quantities a, b, c, and f are given. FORM. XM. Iquation, ax+ + bx? + cx* + dx + f* =2". } Par. val. of x = (8b f+ — 4edf* + d3)8f? vhere a, 6, c, &e. are all known quantities. | FORM. XIV. 4 Zquation, mat + ba? + cx? + dx + e= 2". { Par St ip 16¢7m+—64em®— 8cb?m? + b* | o""<"(8dmt— 4cebm* + 63) 8m* vhere also m, b, c, &e. are all known quan- ities. | FORM. XV. Zquation, m*x* + bx? + ex® + dx + f° 2". | d? + 8mf3 — 4cf* 4bf* = 4mdf 4m*d + 4mbf b> = 8m f — 4m*c In these expressions m, 6, c, &c. are known juantities; and with regard to the ambiguous ign, it must be observed that when it is taken 't in the numerator, it must be — in the de- iominator, and the contrary. ) Par. Var, Of. 2 = A as FORM. XVI. | Equation, ax? + ba* + ex A IW eee | (c? — 3b f3) OF? 27af* — c3 , b, c, &c. being known quantities. } Particular value of « — FORM. XVII. | Equation, mx? + bx? +ex+ d— 23. F 6 i) si se b3 — 27dm t Particular value of x = (Saint b) amt : : us 1, 6, e, &c. being given quantities as above. | FORM. XVIII. Equation, m?x? + b2* + cx + f3) = 2°. : _ (ce? — 3bf2) 9f3 Particular value of « = a7 mi fe — 3 Or a b3 — 27f3 m® 34 oF aero ee: WR et = Gem? — BY) Om? Homi _ 3mf* —e | Morice eeeeesee eeecee poerer HS — b— 3mif vhere also m, 6, c, &c. are known quantities. | FORM. XIX. i Equation, x* + axy + by? =z’. fe val. of « — t? — btu* — abu3 Le ibicessececnsse Y =a Bl7U-+ Satu’ +(a3 —b)u3 | Be iciscccee, Sm e® + atu +e vhere t and u may be assumed at pleasure. ; at age 4bf?c + c3) 8f? ~ 16¢*f*— 64af°—8cd*f* + d* IND FORM. Xx. Equation, x* + by? = 27, Gen. val. of x = t3 — btu? ; y = 3tu— bx3 git Oh ry t and u being indeterminates as above. SHO SSH Eee ese Pee ere eer ee sseree FORM, XXI. Equation, x” + by” = 2%, Gen. val. of « = t+ — 6bt?u* + but Vass ego wee Y = 4B — 4btu3 sein. ie ee Hy 098 t and u being indeterminates as above. FORM. XXII. Equation, «* + by’? = 2”, Gen. val. of « = t™ — Bi™—2 u2b + Jens ut? — &e. seestbieds soosee Yt u — yt™— 336 + gtr Ue Oe. es lost oe oe, where ¢ and u are ipdeterminates, and 1, «, 8, y, 0, &e. the co-efficients of (¢ -- u)”. FORM. XXIII. Equation, «3 + cy? = 2?, Gen. val. of « = 4t4 — 4ctu3 y = 8Hu +t cut Pee a EE 3 mah cus where ¢ and u are indeterminates. Peeesesesese On08 FORM. XXIV. _ Equation, 23 + ax*y + bay? + cy? = 22, Par. val. of t — — “bh 2euw+(@—b) wt i\2w Gen. val.of x = t? + 2cuw + acw? ° PACs y = 2tu — 2huw —(ab—c)w* where w and w are indeterminates, ‘on which also depends the value of ¢. Barlow’s “Theory of Numbers.” INDEX, in Arithmetic and Algebra, is the same as exponent. See EXPONENT. InpeEx of a Logarithm, called otherwise the Characteristic, is the integral part which pre- cedes the logarithm, and is always one less than the number of integral figures in the given number. Or otherwise, the index is always equal to the number of places that the units place of the proposed number is from the first effective figure, and is accounted positive when the first figure is to the left of the unit’s place, and negative when it is to the right. See LOGARITHMS. InDEX of a Globe, is a small hand fitted to the extremity of the north pole, which turning round with it points out the time upon the hour circle. INDICTION, or Roman INnDicTION (from indictio, order, or denunciation), in Chronology, is a term used for a sort of epocha amongst the Romans; the origin or commencement of which is not distinctly known. The Roman indiction consists of a cycle of fifteen years, which when expired begins anew, and goes on again in the same order without any de- IND pendence on the motions of the heavenly bodies. The popes since the time of Charlemagne have dated their acts by the year of the in- diction, which was fixed on the Ist of January, A.D. 313 At the time of the reformation of the ca- lendar, the year 1582 was reckoned the 10th year of the indiction. Now this date when divided by 15 leave a remainder 7, that is 3 less than the indiction, and the same must necessarily be the case in all subsequent dates; therefore to find the indiction for any year, divide the date by 15, and add 3 to the remainder, which will be the indiction, 1 thus in the present year 1812, we haye a = 120, and remainder 12; consequently 12 + 3 = 15, the indiction. INDIVISIBLES, in Geometry, are those small elements or principles into which any body or figure may be ultimately resolved. The method of indivisibles was first intro- duced into geometry by Cavalerius in 1635, in his work entitled ‘“‘ Geometria Indivisi- bilium.” Torricelli adopted it in some parts of his works, which appeared in 1664, and Ca- valerius himself made a new use of it in his treatise published in 1647. ~ According to the principles of this method, a line is said to consist of a number of conti- guous points, or surface of lines, and a solid of surfaces; and because each of these ‘ele- ments is supposed indivisible, if in any figure a line be drawn through the elements perpen- dicularly, the number of potmts in that line will be the same as the number of elements. Hence it follows, that a parrallelogram, prism, or cylinder, is resolvable into elements, or indivisibles, all equal to each other, paral- lel and like to the base, and a triangle, into lines parallel to the base, but decreasing from the base upwards in an arithmetical progres- sion, so also the circles which constitute the parabolic ‘conoid, and the several cocentric circumference which constitute a circle, and the successive circumferences which make up the surface of a right cone, all decrease ac- cording to the same law. This manner of considering magnitudes, is called the method of indivisibles, which is only the ‘method of exhaustions of the ancients, a little disguised and contracted; from which conclusions are drawn as principles without the trouble of demonstration; and it must be acknowledged also frequently without that ‘satisfaction. which ought to accompany ‘ma- thematical investigations. The extreme fa- cility, however, which it ‘affords im a‘ variety of problems, has induced many modern ‘ma- thematicians to adopt the method of Cava- lerius, in preference to the more ‘strict but tedious method of Euclid, and the other an- cient geometers. The great facility attending on the method of indivisibles is very obvious in the demon- stration of the celebrated theorem of Archi- INF medes, viz. “ Every sphere is two-thirds of jj circumscribing cylinder,” the principles ¢ which are as follows: Let ACB, DEF, and GHIK, represent hemisphere, inverted cone, and cylinder, hay ing equal bases and altitudes, and all standin on the common base line At K. Then iti easily demonstrated, that if any sections b made in these solids, by a plane passing parall to the common base, that the section ef of th cylinder is equal to the two sections ab, cd, ¢ the hemisphere and cone; and as this is th case in every parallel position of the cuttin plane, it follows that the sum of all the set tions of the cylinder, is equal to the sum ¢ all the sections of the hemisphere and cone and hence it is inferred, that the solidity ¢ the cylinder is equal to the solidity of the he misphere and cone. But the cone is a thin part of the cylinder, having the same base an altitude, whence it follows that the hemispher is equal to the remaining two-thirds. Se Bonnycastle’s Geometry. INDUCTION, is a term used by matheui ticians to denote those cases in which th generality of any law, or form, is inferred fror observing it to have obtained in several casé Such inductions, however, are very deceptive and ought to be admitted with the greates caution, as there are numerous cases in whie a law may obtain for.a considerable way, an ultimately fail when its uniformity is suf posed certain. A remarkable instance of th failure of induction appears in many of thos formule which have been given for prim numbers. ‘Thus the formula a7 + a2 4+ 41, b making successively « = 1, 2, 3, 4, ke. wi give a series, the first forty terms of whie are prime numbers, whence one might be ip duced to conclude the law to be universal ; fails however in the very next case, the for} first term being a composite number. Thi formula is mentioned by Eulerin the Memoir of Berlin for 1772, p.:36. INERTLE. See Vis Inertia. INFINITE, is a term applied to quantitie which:are oreater than any assignable quanti ties; also, “quantities that are less than assignable quantity, are said to be infini tag smalll. Infinite -also:is sometimes, though impre perly, used in the same ‘sense as indefinite, t denote a line or quantity to which no cer bounds or limits are prescribed. bi Infinite quantities are not necessarily equa but may have any ratio to each other; thus, line which is infinitely extended’ from @ cel tain point, in only one direction, is but” ‘that ‘which is Infinitely extended from same point in two directions. In the sam way we may conceive arectangular plane tob extended in one, two, three, or four differen EN-F >, ‘ections; and thus to form four different inite planes, being to each other as the imbers 1, 2, 3, and 4. In the same way a solid may be conceived ' be extended in six different directions ming infinite solids, which shall have to ‘ch other certain and determinate ratios. In the same way we may conceive an in- ite quantity, which is infinitely less than other infinite quantity; thus if two infinite cht lines be drawn paralle! to each other at y finite distance, in a plane infinitely ex- ided in all directions, the infinite space in- ided between them, wiil be infinitely less m the infinite plane in which they are un, for the breadth or distance between ase lines is infinitely less than that of the initely extended plane. Some anthors use the character oo to denote inity, or an infinite quantity, which symbol, en submitted to the usual rules of analysis, ound to possess several curious properties : : , : a isa being any finite quantity, we have a 0; 5 993 0x Coma. Thatis, ifa finite antity be divided by a quantity infinitely rat, the quotient is 0, or a quantity infinitely all. Jr if a finite quantity be divided by an in- tely small quantity or 0, the qnotient will a quantity infinitely great. If an infinitely vat quantity be multiplied by one infinitely all, the product will be a finite quantity. 1erson, in his “ Algebra,” has a chapter on properties of nothing and infinity. 4rithmetic of INFINITES, is a term applied Dr. Wallis, to a method invented by him the summation of infinite series, which plays great ingenuity; but the more ge- al method of fluxions renders it now of no ‘NFINITE Decimal, the same as circulating imal. See C1RcULATING Decimal and RE- ‘END. INFINITE Series. See SERIES. NFINITESIMAL, or Infinitely smallQuan- , is that which is so small, as to be in- aparable with any finite quantity whatever, t is that which is less than any assignable sntity. nthe Method of Infinitesimals, the element which a quantity increases or decreases is posed to be infinitely small, and is ge- ally expressed by two or more terms; 1e of which are infinitely less than the rest, ch being neglected, as of no importance, remaining terms form what is called the erence of the proposed quantity. The ns that are thus neglected are infinitely than the other terms of the element, and the same which arise in consequence of acceleration or retardation of the generat- ‘motion, during the infinitely small instant ime in which the element was generated; that the remaining terms express the ele- sat that would haye been formed in that } INF time, if the generating motion had continued uniform. These differences are, therefore, in exactly the same ratio to each other, as the generating motions or fluxions; and hence, though in- finitesimal parts of the elements are neglected the conclusions are accurately true, in con- sequence of what may be termed a compen- sation of errors, as is shown by Carnot ia his ‘“ Essay on the Infinitesimal Calculus.” See also Maclaurin’s “ Treatise of Fluxions;” In- Ac ae p. 389, 40, and Book I. art. 495 to 02. INFLAMMABILITY, that property of eam bodies by which they kindle or catch ire. INFLECTION (inflexio, Latin), called also diffraction, and ‘deflection, in Optics, is'a pro- perty of light, by reason of which, when it comes within a certain distance of any body, it will be either bent from it or towards it; which is’a kind of imperfect ‘reflection or refraction. This property was first taken notice of by Dr. Hooke, who shows that it differs both from reflection and refraction; and seems to depend on the unequal density of the consti- tuent parts of the ray, whereby the light is dispersed from ‘the place of condensation, which deflection is made in a perpendicular direction towards the surface of the opaque body. Some writers ascribe the first dis- covery of this property of light to Grimaldi, who published an account of it in his treatise “‘ De Lumine Coloribus et Iride,” printed in 1666 ; and Dr. Hooke, without any knowledge of what had been discovered by Grimaldi, communicated his observations on the same subject to the Royal Society in 1672, where he considers it as a newly discovered property of light. Newton also: ascertained the same from experiment, and La Hire found that the beams of the stars being observéd in a deep valley, to pass near the brow of the hill, are always more refracted in consequence of the proxi- mity of the hill. Although Sir I. Newton particularly ex- amined the phenomena relating to this subject under a considerable variety of circumstances, his observations were not correct, nor was the hypothesis, which he advanced to account for. it more fortunate; subsequent experiments and observations seem to reduce the phe- nomena of inflection to a single principle, viz. of the attraction of light towards bodies; which attraction becomes conspicuous when the rays of light pass within a certain distance of their surfaces. Besides their being bent, the rays of light are likewise separated into colours by the vicinity of bodies, and this pro- duces the singular phenomenon of the co- loured fringes that accompany the inflection. Various experiments have been made relative to the inflection of light by Miraldi, Grimaldi, Delisle, Mairan, Du Tour, Muschenbroek, and others, as well as Newton; an account of which experiments the reader may ‘see in INF Priestly’s “ History of Vision, Light, and Co- lours,” Part VI. sect. 6. Point of INFLECTION of a Curve in Geometry, is a point where the curve begins to bend or turn the contrary way. K R K. A. B A BA B If the curve-line A RK be partly convex, and partly concave towards a right line AB, or towards any fixed point, then the point R which divides the convex part of the curves from the concave is called the point of inflec- tion, if the curve continued beyond R pre- serves the same curvature as in fig. 1. But if when continued beyond this point it returns back again, towards that part or side whence it took its origin, it is then called the point of retrogression, as in fig. 2 and 3. And if after the retrogression the concavity of the two parts lie the same way, it is calleda ramphotd, as in fig. 2, but if they lie the contrary way, having their convexities towards each other, it is called a ceratoid; the former being de- rived from aos, beak, and ssdos, like resem- bling a beak; and the latter from xepas, horns, from its resemblance to horns. There are various methods of finding the points of inflection or retrogression of curves, of which the following is perhaps the most simple. From the nature of curvature it is evident, that while a curve is concave towards its axis, the fluxion of the ordinate decreases, or is in a decreasing ratio with regard to the fluxion of the absciss; and on the contrary, this fluxion increases, or is in an increasing ratio to the fluxion of the absciss, when the curve is convex towards the axis; and hence it fol- lows that these fluxions are in a constant ratio at the point of inflection or retrogression, where the curve is neither concave nor con- 2 x q - . vex; that is, —, or 4 isa constant quantity, 4 x the fluxion of which is therefore at this point — 0. Whence we draw this general rule. Put the given equation of the curve into fluxions, from which equations of the fluxions find =, or z. and take again the fluxion of this x . fraction, and make it equal to 0; and from this last expression find also = or 4 , and by equating these two, together with the general equation of the curve, x and y will be de- termined, being the absciss and ordinate an- swering to the point of inflection. Let it be proposed, for example, to find the point of inflection in the curve, whose equa- tion is ; 2 Pee x The fluxion of aa* =a*y + 27y, is2axxz= ax* ary + 27° y,ory = INF . Pe a? a dy + Qxyh 4 x%y; whence = = y+ @ayxr +n et Sana And now making again the fluxion of 1, quantity = 0, we have Qua (ax — xy) = (a> + x”) (ax— yx —K 2 2 = ts Wi x wag eee | a~ — x* And now, equating these two expressio we have az + x x a? + x? 1 ners ee a —x a—y 2x a— Ye which being reduced, gives 2x* = a* — 3x? =a’, and x =a V4; and this valuc x substituted in the original equation of e x whence ~ oe ax” 4 a3 curve, becomes y = ———; = J =} a +a Fa the ordinate at the point of inflection. Ex. 2. Again, let it be proposed to find point of inflection in a curve whose equat is b*y = ax” — x3, The fiuxion of this expression, gives by = axe —3x* x; or 4 aa Taking again the fluxion of this expressi and making it equal to 0, we have | Qab? x — 6b? xx = 0, or 2ab* — Gb’ x;) because y does not enter; whence * = which is the absciss answering to the pom) inflection sought. | Ex. 3. Let it be proposed to find the pi of inflection, or retrogression, in the ev parabola, with the equation ; yrmat ¥ (a — 2a x + ax’). By taking the fluxion, we have (—2a* + 2ax)z Yi & (q3 —2Qa*x + ax*)3 Taking again the fluxion of this express} and making it equal to 0, according to} above rule, we have ax (a3 —2a*x 4 ax* i ad 4 z Q(—2a*x + 2axx) x (2ax— 2a ) becal ~3(a3—2ax + ax?) 3 y is not found in the second fluxion, wl} expression being reduced, gives x = a; | this value substituted for x, gives also y =i hence the point required is that, answell to these conditions of the absciss and ordin| which is a point of retrogression; as willl pear from considering the nature of the cul Ex. 4. Let it be proposed to investi the point of contrary flexture, in the et) commonly called the witch; of which} equation is y7* = a’ —a’x. J Taking the fluxion of this expression, _—(a* + ZY: Now again making the fluxion of this pression = 0, we have 4 —4ytyx+ (Qxry +2yz\ (ae +y*) al 2x (a? + y*) y—A4y? xy + 2yx (a + y’) whence : —2y (a> + y*) Qn (a* + y*) — 4y* x have 2y yx + yx = —a*z;or4 x i 0 INT And now equating this value of a, with the regoiny, there is obtained | | 2y(a + y) _ at Hy? Zx(a* +y")—4y*x Oya |, 4y*x = 2x (a® + y*) — 4y* x, or4y”? = + y*, whence y = a 4; which being jbstituted in the original equation, gives .= 3a; which are therefore the values of x .d y, answering to the required point of flection. _We have here only considered those curves at have parallel ordinates, and which are erred to one common axis; but it is ne- (ssary to employ a different method when 2 curves are referred to a common focus; {s subject however our limits will not allow (investigating, and we must therefore refer i> reader for information on this subject to f» several writers mentioned under the ar- ile Curves; see also Donna Agnesi’s “Ana- ical Institutions.” INFORMES Stelle, Unformed Stars, are fase which are not reduced into constella- ns, and are otherwise called sporades. The jzients left many unformed stars, but these ih been mostly formed into new constella- us by the moderns. INGRESS, in Astronomy, is the entrance jthe sun into any of the signs of the zodiac, y ticularly Aries. »;NORDINATE Proportion, is wheré the } er of the terms is not regular. NSCRIBED Figure, in Geometry, is one ‘ich has all its angles in the sides or peri- sry of another figure, in which the former ‘aid to be inscribed. lo inscribe a circle in a ingle ABC.—Bisect any of the angles A and B, . from the point of inter- tion O, let fall the per- dicular OD; so will OD ~f\ une radius, and O the A Bo, 'tre of the circle required. 'o inscribe a circle in a regular polygon, |. polygon in a circle. See PoLycon. on Hyperbola, that which lies wholly wi lin the angles of its asymptotes; as does ‘common or conical hyperbola. INSTANT, an infinitely. small and indi- dle portion of duration, being with regard ime what a point is in respect to a line. NSULATED Body, in Electricity, denotes idy that is supported by electrics, or non- luctors, whereby its electrical communi- on with the earth is interrupted. VNTACT A, the same as ASYMPTOTE. NTEGER, or InteGraL Number, is an {, or an assemblage of units. NTEGRAL Calculus, is the reverse of the | rential calculus, and corresponds with our irse method of fluxions; the finding of an gral to a given differential being the same ‘inding the fluent of a given fluxion, and ened by the same rules. See FLux- , and DIFFERENTIAL Calculus. qi | } | | INT INTEGRANT Parts of a Body, in Philo- sophy, are those parts which are of the same nature with the whole. INTENSITY, or Zntension, in Physics, de- notes the power or energy of any quality, or action of a body; thus we say, intensity of heat or cold, intensity of a blow, &c. INTERCALARY Day, denotes the odd day which is inserted in the calendar every fourth year. See BIssexTILeE. INTERCEPTED Axis, in the Conic Sec- tions, is the same as ABSCISS. INTEREST, is the allowance made for the loan or forbearance of a sum of money, which is lent for, or becomes due, at a certain time ; this allowance being generally estimated at so much per cent. per annum; that is, so much for the use of 100 pounds for a year, and this rate is by law fixed not to exceed 5 pounds, or in other words the greatest rate of interest must not exceed 5 per cent. per annum. Interest is either Simple or Compound. Simple interest, is that which is allowed upon the principal, only for the whole time of the loan, or forbearance. The money lent or forborne is called the principal. The sum paid for the use of it the interest. The interest of 100 pounds for one year, is called the rate per cent. And the sum of any principal, and its interest together, the amount. To find the Interest of any Sum for any Period, and at any given Rate per cent. Say, as £100. is to the rate per cent.; so is any principal to its interest for a year. And then, as one year is to this interest, so is any given time to the interest for that time. — Suppose it were required to find the inte- rest of £519. 13s. 4d., for 34 years, at 5 per cent. per annum. First, 100: 5:: £519. 18s. 4d. 5 £25.98 6 8 "20 s. 19.66. 12 d. 8.00 Again, 1; 33 ::::25 19 8 2 2 7 7 fist 178 £90 18 10 2 That is, the interest of the proposed sum for 1 year, is £25. 19s. 8d.; and for 33 years, it is £90. 18s. 10d., as required. Again, to find the interest of £250. or 24 ars at 5 per cent. ay ‘As £100. > £5. :: £250. : £12. 10s. As 1 year: £12. 10s. :: 25: £31. 5s. The calculation may however be made more simple and general, as follows: if y = the interest of £1. per annum, CC INT p = any principal or sum lent, == the time of the loan or forbearance, i — the interest of the given principal for the time ¢, a = the amount of the principal and its interest for the given time. Then it is obvious that 7 = pir, and a =p + ptr=p(i + tr); from which theorem the following formule are readily deduced ; viz. Twn. @p(l+rt) =—p+ptr 2 wot ET ae a es eer SS SS Sam pt 4 tots ae pr RS, Ba By means of which theorems all circum- stances relating to the simple interest of money are readily computed. 1. Thus repeating the former example, let it be required to find the interest and amount of £250. for 24 years, at 5 per cent. Here r = ‘05, t = 2°5, p = 250; therefore the interest i= ptr = 250 x 2'5 x 05 = £31°5 theamounta = px ptr = 250 x 315 = £2615 TABLE Of the Interest of £1. for any Number of Days at different Rates of Interest. INT | | 2. Again, suppose it were required; ¥ principal put ont at 5 per cent. for 24 y would amount to £281. 10s. | Here a = 281'5, t = 2°5, and r = ‘06; th fore ~ ange ee SRT Oo. eet P= {prt 1425 x 05 1125 3. Let it now be required to find at rate of interest £250. must be put out, amount to £280. 10s. in 23 years. | Cae 1) 0 5 05, wi: pt 250 X 23 - | corresponds to 5 per cent. 4. In what time will £250. amount £281. 10s. at 5 percent per annum. Here t 72 = 3! = 2-5, or 2 To prot) 260% 05S) a As the computations of interest are of g importance in all mercantile transaction) has been found convenient to have exten tables of interest computed, whereby interest of any sum, for any time, and a rates of interests is found by inspection: these works the most approved are thos| Baily, Morgan, and Smart. The following tables will much facili) the computation of simple interest. J Here r = ee a fa, No. of Days: 3 per cent. 3 & ahalf percent. 1 .0000821 -0000958 2 .0001641 .0001916 3 .0002465 -0002876 4 .0003287 .0003835 5 .0004109 .0004794 6 .0004931 .0005753 7 .0005753 -0006712 8 .0006575 .0007671 9 .0007397 .0008630 19 .0008219 .0009589 20 .0016438 .0019178 30 .0024657 .0028767 40 .0032876 .0038356 50 .0041095 .0047945 60 .0049315 0057534 70 .0057534 .0067123 80 .0065753 .0076712 90 .0073972 .0086301 100 .0082191 .0095890 200 .0164382 .0191780 300 .0246573 -0287670 4 per cent. 4&a half per cent. 5 per cent. .0001095 .0001232 .0001369 .0002191 .0002465 .00027389 .0003287 .0003698 .0004109 .0004383 .0004931 .0005479 .0005479 .0006164 .0006849 -0006575 .0007397 -0008219 .0007671 .0008630 .0009589 .0008767 .0009863 .0010958 .0009863 .0011095 .0012328 .0010958 .0012328 .0013698 .0021917 .0024657 .0027397 .0032876 .0036986 | .0041095 .0043835 .0049315 .0054794 0054794 .0061643 .0068493 .0065753 .0073972 .0082191 .0076712 -0086301 .0095890 -0087671 .0098630 .0109589 .0098630 0110958 .0123287 .0109589 .0125287 .0136986 0219178 .0246574 .0273972 .0328767 0369861 .0410958 This table it is obvious will furnish, by the addition of two or three of its numbers, the interest for any number of days, and the fol- lowing will in the same way find it f or number of years. INT IN T TABLE ‘og the Interest of £1. for any Nwnber of Years not exceeding 25, at different Rates of Interest. No-of Years- 3 per cent- 3 &ahalfper cent. 4 per cent. 4 &ahalf percent. 5 per cent. i ) 1 .0300000 .0350000 -0400000 .0450000 -0500000 2 0600000 .0700000 6800000 .0900000 -1000000 3 0900000 .1050000 . 1200000 -1350000 - 1500000 4 .1200000 -1400000 - 1600000 -1800000 -2000000 5 - 1500000 . 1750000 .2000000 .2250000 .2500000 6 - 1860000 .2100000 -2400000 -2700000 -3000000 7 .2100000 .2450000 .2800000 .3 150000 .3900000 8 .2400000 | .2800000 3200000 .3600000 -4000000 fi 9 .2700000 .8 150000 .3600000 -4050000 -4500000 38000060 .38900000 -4000000 -4500000 .0000000 f .3300000 3850000 4400000 -4950000 .5300000 | 8600000 .4200000 -4800000 .2400000 -6000000 } 3900060 .4550000 0200000 .0850000 .6500000 | -4200000 4900000 2600000 .6300000 “7000000 } .4500000 8250000 .6000000 6750000 £7500000 i 4800000 _. 6600000 -6400000 -7200000 -8000000 } 2100000 .0950000 6800000 | .7650000 .8500000 2400000 .6300000 -7200000 _ | .8100000 -9000000 i 9700000 -6650000 -7600000 .8550000 .9500000 ) 6000000 -7000000 .8000000 .9000000 1.0000000 .6300000 .7390000 .8400000 -9450000 1.0500000 6600000 .7700000 .8800000 .9900000 1.1000000 .6900060 .8050000 9200000 1.0350069 1.1800000 .7200000 .8400000 .9600000 1.0800000 1.2000000 .7500000 .8750000 1:0000000 1.1250000 1,.2500600 ‘To find the Interest of any Sum, for a given : Time, by the preceding Tables. Add together interests for the several pe- »ds corresponding with the proposed rate of ircent. and that sum multiplied by the prin- yal will be the interest required. -Exampce. Required the interest of £400. * 4 years 123 days, at 43 per cent. Tabular interest for 4 years ‘1800000 Miebeths 142<9-}0<.40-- ss9 LOOrdays . ‘0123287 20 days 0012328 3 days 0003698 1939313 400 0725200 20 45040 12 9245V bon 3} 69920 estas £17. 11s, 4id. is the interest re- ired. ;COMPOUND Interest, is that which arises m any sum or principal in a given time, increasing the principal, each payment by 2 amount of that payment, and hence ob- ning interest upon both interest and prin- al, ‘Hence it follows that except for the sake { brevity compound interest would require ~ ay | s. 11 d, 4 no other rule than what is given for simple interest, for it is only necessary to find the interest upon the given principal; then add this interest to the principal, and find the interest upon this, and so on, through the whole number of payments. But this would be attended with immense labour if the time was at all considerable, and therefore other methods of computation be- come necessary. Now, in the first place, it is obvious that if we know the amount of £1. for any period, the amount of any other principal p, will be equal to p times the former; we may there- fore limit our investigation to the simple case of finding the compound interest and amount of €1. for any given number of years at any given rate per cent. Let then r be the amount of £1. for one payment, whether yearly, half yearly, or quar- terly. ‘Then, as the amount is always pro- portional to the principal, we shall have, MeL SP fOr See OM, KC. that is if the amount of £1.for 1 payment be r the amount of £1. for 2 payments =7* the amount of £1. for 3 payments =r? the amount of £1. for x payments = 7” and consequently the amount (a) of any prin- cipal (p) for (n) payments = pr’, which being for the conveniency of calculation put into the logarithmic form, becomes log. of a = log. p-t+-n log. r. Suppose, for example, the amount of £60. CC2 INT for 15 years, at 5 per cent. were required, the payments being due yearly, Since the interest of £100. is £5.; the interest of £1. is °05, and the amount of £1. im one payment 1°05. Hence by the above formula, log. 1:05 = 0°021189 mult. by 15 15 O'317835 log. 50 = 1698970 IN:T The natural number to this logarithm ,; 103°946 or £103. 18s. 10d. the amount; fire which, if the principal be subtracted, it y leave the compound interest itself. ) By this means the amount and compow interest of any sum, for any period and ra: per cent. may be readily computed. Butt same may be done still more expeditiou; by means of the following table, which ¢ hibits the amount of £1. for any number years not exceeding 25, Hence the follo » Ing rule: TABLE TI. Showing the Amount of £1. in a given’ Number of Years not exceeding 25, any Rate of Compa Interest, from 3 to 6 per cent. Years. 3 per cent, 3 &a half per cent. 4 per cent. i 6 per cent. 1 4 &a half percent, 5 per cent. _ J 1.03000 1.035000 1.040000 1.045000 1.050000 1.060000 2 1.060900 1.071225 1.081600 1.092025 1.102500 1.123600 | 3 1.092727 1.108718 1.124864 1.141166 1.157625 1.191016 — 4 1.125509 1.147523 1.169859 1.192519 1.215506 1.262477 5 1.159274 1.187686 1.216653 1.246182 1.276282 1.838226 © 6 1.194052 1.229255 . 1.265319 1.302260 1.340096 1.418519 ) 7 1.229874 1.272279 1.315932 1.860862 1.407100 1.503630 | 8 1.266770 1.316809 1.368569 1.422101 1.477455 1.593848 — 9 1.804773 1.362899 1.423312 1.486095 1.551328 1.689479 — 10 1.343916 1.410599 1.480244 1.552969 1.628895 1.790848 — 11 1.384234 1.459970 1.539454. 1.622853 1.710339 1.898299 | 12 1.425761 1.511069 1.601032 1.695881 1.795856 2.012196 13 1.468534 1.563956 1.665074. 1.772196 1.885649 2.132928 14 1.512590 1.618695 1.731676 1.851945 1.979932 2.260904 | 15 1.557967 1.675349 1.800944 1.935282 2.078928 2.396558! 16 1.604706 1.733986 1.872981 2.022370 2.182875 2.540352 | 17 1.652848 1.794676 1.947900 2.113377 2.292018 2.692773 18 1.702433 1.857489 2.025817 2.208479 2.406619 2.854339 | 19 1.753506 1.922501 2.106849 2.307860 2.526950 5.025599 20 1.806111 1.989789 2.191123 2.411714 2.653298 5.207135 | 21 1.860295 2.059431 2.278768 2.520241 2.785963 5.399564 22 1.916103 2.131512 2.369919 2.633652 2.925261 5.603537 © | 23 1.973587 2.206114 2.464716 2.752166 | 3.071524 5.819750 | 24 2.032794 2.283328 2.563304 2.876014 3.225100 4.048935 - 25 2.093778 2.363245 2.665836 3.005434 3.386355 4.291871 Find the amount in the table for the given number of years and rate of interest, and multiply this amount by the principal for the amount required. 'Thus in the preceding ex- ample, Amount by the table =2-67893 Principal 50 , 103°94650 = £103.18s.10d. as before. This table is only computed for yearly pay- ments, and therefore when the payments are different from these, or when the number of years or payments exceed 25, recourse must be had to the preceding rule. The accumulation of money, when placed at compound interest, after a certain number of years, goes on exceedingly rapid, and in some instances appears truly astonishing. Thus Mr. Baily has shown in his. valuable “ Treatise on Interest,” that 1 penny put out at 5 per cent. compound interest, at the bir of Christ, would, in 1810, have amounted a sum exceeding in value 357 millions solid globes of standard gold, each in mag? tude as large as this earth! The exact nw ber of globes, according to this genilemal computation, is 357474600. INTERIOR Angle of a Polygon, that whit is formed internally by the meeting of any the sides of the figure. ; INTERNAL Angle. See ANGLE. ~ - INTERPOLATION, in Analysis, is t method of finding any intermediate term in series, its place or distance from the first el being given. This is commonly effected } means of the successive differences of # given terms, and is therefore sometimes ez ll the differential method, though this latter e pression is more commonly employed to ¢ note the method of summing of series INT ; t ans of their differences; but in the present éicle we shall only consider the interpolation « the ferms, and for the rest must refer the y.der to.article SERIES. The method of interpolation was first invent- eby Briggs, and applied by him to the calcu- fon of logarithms. His principles were fowed by Wallis, Reginald, and Mouton. S Isaac Newton, in lemma 5. lib. iii. Phil. Encip. Mathem. gave a most elegant solu- {1 of the problem for drawing a curve line tough the extremities of any number of gen ordinates; and in the subsequent pro- » sition applied the solution of this problem hat of finding from certain observed places 9s comet, the place of it at any given inter- ndiate time. Waring has also resolved the sie problem, and rendered it more general, phout having recourse to finding the suc- sive differences. Nicole in the Memoirs ythe Academy of Sciences, and Stirling in 1 “ Tractatus et Summationes, et Interpo- gone Serierum Infiniitorum,’ &c. as also \wton, in his “ Methedus Differentialis,” 1 ¢ made a-happy application of these prin- es to various purposes. vhis theory is of very extensive use, not 'y in pure analysis and geometry, but in »ous ether subjects of mathematical in- /y and-computation, and particularly in sonomy; we shall therefore endeavour to lain the principles upon which it is found- « and show its application in a few cases to )ctical operations. jirst, then, let a, 5, c, d, e, f, &c. represent } series of similar quantities, and let the isrence between the first and second, the emd and third, the third and fourth, &c. 21s be taken; and these several remainders » form what is called the first order of dif- sauce; then again let the differences of ve differences be taken in the same way; t the differences of these last again the te, and so on, which will give the follow- |result, observing, that for the convenience _xhibiting the operation, we have only re- ‘ed the first remainders in each successive } raction. ries a, b, c, d, e, f, &e. sdifLa— 6 Hifa—2b+ e 7 lif.a—3b+ 3ce— d onf.a—46+ 6ce— 4d+e Nliff.a—5b +10c—10d+ 5e—f iff. a —6b + 15e— 20d + l5e—6f +g '&e. Ke. ow the co-efficients of these terms are tectively the same as those of the co-effi- ts of the binomial, and the order of their ration evidently follows the same law, » therefore we may conclude with equal »inty, that the nth difference of any series vaantities, will be expressed by the for- ty 3 n(n— 1) wet ‘n(n — 1)(n — 2) (n— 3) 1.2.3.4 n (n—1)(n—2) Maku x as € — ke, INT _ Now it is obvious that if the given quan- tities be such that any order of their diffe- rences become equal to 0, any one of those quantities may be accurately expressed in functions of the others; thus for example, suppose the fourth difference to become zero} that is, i a—4b+6e—4dt+e—0 —a+4b+4d—e 6 and it is obvious that any other of these quan- tities might be expressed in a similar manner: and therefore if all those quantities but one be known, that one may be ascertained. Thus, by way of illustration, sappose we had the three squares 107 — 100, 87 =64, and 7* = 49, and the square of 9 was required; since the third differences of squares equal 0, we should have 9» — 100 + 3.64— 49 = 81, and the same is obviously true of any terms of which the differences vanish. i But if the differences do not vanish, then any intermediate term found by this method only approximates towards the true result, which is however sufficiently correct in a number of cases; thus in finding any loga- rithm of which those consecutive to it are given the above formula may be successfully employed, for though, in facet, the differences of logarithm never become zero, yet their fourth differences are so small, that by con- sidering them as 0, the error will not affect the truth of the result to 8 or 9 places of decimals. Exam. Given the logarithms of 101, 102; 104, 105, to find the logarithm of 103. Here, calling the log. of 101 =a, of 102 = b, 103 = c¢, 104=d, and 105 =e; and consi- dering the fourth differences of these logs. = 0, we shall have from the formula a—4b+ be—4d+e=—0 alte 4(6+d)—(a+e) 6 Hence the following computation: log. 101 = 20043214 =a log. 102 = 2:0086002 = d log. 104 = 2°0170333 = d log. 105 — 20211893 = e 40256335 =b+d 4 16°1025340 — 4 (b + d) 40255107 =a+e |. 6)12-0770233 log. 103 = 2°0128372 as required. then will ¢ = subt. This method of finding the intermediate logarithms between others that are known, though of little importance in the present state of the sciences, was of very essential service to the original computors; and to whom the invention of it is due, or more pro- perly to Briggs alone, who seems to have been the first that entered upon the investi- gation of this theory. This doctrine is applied with great success in various astronomical operations, and is the BNOTE means of saving, in many cases, immense la- borious calculations. ‘Thus, for example, in finding the places of some of the planets, whose motion is not very rapid, it will be sufficiently accurate to find their places by calculation for every fourth or fifth day, and then by means of the methods above described, their places for all the intermediate days may be found by interpolating between the known terms, which method will give a result much nearer the truth, than by proportional parts, because this supposes a uniformity both in motion and time, which is not correct. Again, in computing the moon’s place for any particular hour, supposing its place for every day at noon to be given, the method of interpolations may be applied with great suc- cess, the result having scarcely any sensible difference from those that arise from actual computation, and we may thus frequently avoid one of the most laborious of astrono- mical calculations. By this means also the place of a comet, at any particular time, may be ascertained, from observations made on it prior to, and subsequent to, that precise period, as also the times of the equinoxes and solstices, which are determined much more accurately by this means than can be done by proportional parts, for in this we are obliged to suppose that the sun’s declination increase and decrease pro- portionally to the distance of this body from the equinoxial point; which is evidently a false hypothesis. In fact astronomy has deriv- ed more assistance from this theory than any other of the mathematical sciences, although it has been applied to other purposes with very great success; but in order to render its application thus general a much more ex- tended investigation of the theory becomes necessary than can be allotted to this ar ticle ; the reader, therefore, who is desirous of more minute information should consult Newton’s “* Methodus Differentialis ;” Stirling’s “ T'rac- tatus et Summationes et Interpolatione Serie- rum Infinitarum ;” A paper, by Mayer, in the Acta. Petro. tom. ii. p. 198; a Memoir, by Lalande, in the Acad. of Sciences for 1761, and the article INTERPOLATION, Encyclopedié Methodique ; see also Rees’s Cyclopedia. INTERSCENDENT Quantities, in Algebra, are those that haye radical indices; as aV re av 3, &c. INTERSECTION, the cutting of one line or plane by another; the point or line in which they meet being called the point or line of intersection. When two Jines or two planes intersect each other, the vertical or opposite angles are equal. INTERSTELLAR, is used by some authors to denote those parts of space which are be- yond the regions of the solar system. INTESTINE Motion of the Parts ofa Body, is that which takes place amongst the com- onent parts, INTRADOS, the internal curve of the arch of a bridge. JON INVERSE Proportion, Rule of Three, Rat} &e. See the respectiye terms. a InveRSE Method of Fluxions. Method of PLuxtons and FLUENT. | InveRSE Method of Tangents, is the meth of finding a curve belonging to a given t; gent; being thus distinguished from the dire method, or that of finding the tangent fic the properties of the curve. } INVERSION, InvERTENDO, is the inve; ing the terms of a proportion; EER 11; 0,'s.. 257. eieaaane a:b then by inversion... b:a@:: Euclid, def. 14, book v. INVESTIGATION, in Mathematics, ¢ notes the tracing out the solution of any pi posed. problem, or an examination of its sé | ral properties, &c. INVOLUTE Cwe, in the Higher Geomet is that which is traced out by “the extrem} of a thread, as it is folded or wrapped abe another figure or curve. See EVOLUTE. — INVOL UTION, in Algebra and Arithme is the raising a given number or quantity any proposed power. In arithmetical and simple algebraical quantities, this consists oI in multiplying ihe quantity as many times: itself as is necessary to produce ‘the giv power; the number of operations being @ less than the index of the power to be P duced; thus See Inve axa =a’ ......2d power axXaxa Sau 3d power @XaX a Xo Foe. Ath power; Ke. So also 2x2 re ARRAS 2d power of 2 | 2x2x-2 pense, Ae ree HI 3d power of 2” 2x2x2XV=16...... 4th power of 2, And in the same way we may proceed W compound algebraical quantities; but for bit mial ones, it is best to make use of the bil mial theorem, the general form of which as follows: viz. (a + 6)" =a® + na®—1b + it A Se 5 J) a"! n (n — 1) (n — 2) n—3)3 & 1 1.2.3 : a Whence (a + b)* =a’ +2ab +0’ (a+ b’=a+3¢°b4+3ab* +03 . (a + b)* = a+ + 4036 + 6a’*b* + 4a} ny &e. ke. See BINOMIAL JOANESSE, a coin of Portugal, va £1. 15s. 11d. JOINT Lives, in the Doctrine of Annuit are such as continue the same time, or dur the existence of both parties. Sce LirEa nuities. % Universal Joint, in Mechanics, is an inv tion of Dr. Hooke, adapted to all Kinds oft tion. See his “‘ Cutlerian Lectures,” prift in 1678. ry JONES (WILLIAM), an eminent mathe tician, was born in North Wales in 1675,% died in 1749, being at that time one of Vice Presidents of the Royal Society. published the two following works, hes | IRR jome papers in the Phil. Trans, vol. xliv. 1xi. di. _ 1. A new Compendium of the whole Art of lavigation, 8vo. London, 1702. 2. Synopsis Palmariorum Matheseos, or a ew Introduction to the Mathematics, 8vo. London, 1706. He also published, by permission of Sir I. lewton, an elegant edition of several of his japers, entitled 8. Analysis par quantitatum Series, Flux- ones, &c. cum Enumeratione Linearum Tertii Irdinis, 4to. 1711. _ JOURNAL, in Navigation, is aregister kept \¢ sea by the master and other officers, in rhich is noticed every incident of the voyage ; s the rate and course of sailing, the state of je weather, tides, currents, astronomical ob- ervations, latitude, longitude, &c. &e. ‘In all sea journals the day.is divided into rice twelve hours, those before noon being harked A.M., for ante meridian, and those fter noon by P.M., post meridian. IRIS, the name of the rainbow. Ir1s Marina, the Sea Rainbow. This ele- ‘ant appearance is generally seen after a vio- ant storm, in which the sea water has been wich agitated. The celestial rainbow has reat advantage over the marine one, in the rightness and variety of the colours, and in eir distinctness one from the other; for in he sea rainbow there are scarcely any other olours than a dusky yellow on the part to- yards the sun, and a pale green on the oppo- ite side; the other colours are not so bright r distinct as to be well determined: but the ea rainbows are more frequent and more umerous than the others. It is not uncom- ‘ion to see twenty or thirty of them at a time ‘tnoon day. Observ. sur l’Asiz, p. 292. IRRATIONAL Number, the same as SuRD. IRREDUCIBLE Case, in Algebra, is an xpression arising from the solution of certain quations of the third degree, which always ‘ppears under an imaginary form, notwith- tanding it is in fact a real quantity ; but the eduction of it to a rational or irrational finite ‘xpression, has at present resisted the united fforts of many of the most celebrated mathe- aaticians of Europe. Every cubic equation aay be reduced to the form x? + ax = 6; and hen, according to the common rule, : a/b as ee es oe BSE oy AY GaN i HV ot Et 5—v Gt 4° 27 ee Cusic Equations. a3 27 3 5 ive; and therefore when < a the quan- | | Now when ais negative, — is also nega- as 27 i imaginary, because we cannot extract the ‘quare root of a negative quantity ; and this is vhat constitutes that which is generally called he Irreducible Case. This difficulty soon pre- jented itself to Cardan, after Tartalea had : | \ity under the inferior radical, viz. / : — IRR communicated to him his method for the solu- tien of cubie equations; which rule is now commonly, though very improperly, attributed to the former. Cardan informs Tartalea, in a letter dated August 4, 1539, that he under- stood the solution of the equation x3 +4 axz=b, & a3 and also of x3 — ax — b, when =< > Sy? but 2 3 when = < os his attempts always failed ; and he therefore begged of Tartalea to clear up his difficulty, by sending him the solution of the equation x3—92—= 10. Tartalea was himself perfectly aware of this difficulty, but he was by no means satisfied with Cardan’s conduct, whom he at that time suspected to be about publishing, as his own, the rules he had taught him; and therefore instead of giv- ing him an explicit answer, he writes to him in the following terms:—‘ M. Hieronime, I have received your letter, in which you write, that you understand the rule for the case % 3 x3 — ax — b, when “ is greater than es but 3 jr when exceeds GZ you cannot resolve the equation, and therefore you request me to send you the solution of the equation «3 —9x = 10. To which Lreply, that you have not used a good method in that case, and that your whole process -is entirely false. As to resolving you the equation you have sent, I must say that I am very sorry that I have already given you so much as I have done; for I have been informed, by a credible per- son, that you are about to publish another algebraical work, and that you have been boasting through Milan, of having discovered some new rules in algebra. But take notice, that if you break your faith with me, I shall certainly keep my word with you; nay, I even assure you, to do more than I promised.” Dr. Hutton’s Math. Dict. article ALGEBRA. Tar- talea, however, notwithstanding what he says in this letter, was himself well aware of the difficulty in question, as appears from some of his private memoranda; and from that time to the present, which is near 300 years, the same impediment remains, notwithstanding the re- peated attempts of many very distinguished mathematicians; in fact, there is great reason to suppose, independently of the failure of so many ingenious men, that the formula is inex- pressible in any other finite form than that un- der which it naturally arises by the solution. But though no direct analytical solution can be feund in the irreducible case, other methods of determining the roots have been invented; such as by means of series, a table of sines and tangents, the usual approxima- tion, &c. The two first methods exhibit the solution to the eye in as neat a manner as can be desired, but they are both commonly attended with great labour in the practical operation. Another method of solution is by a table computed for the purpose, the general principle of which is as follows: IRR Let «3 — az = b be any cubic in the isre- ducible case; this may be reduced to a depen- dent oe y3 — y = d; and since in the 1 d* first — > a, so in this last 5- > —, or ee 27 4 d*< > or dcannot exceed ‘38491. Whence again it follows, that in equations falling under the irreducible case, when they are reduced to the form y? — y = d, the value of y cannot exceed 11549, nor be less than 1. On this principle, therefore, I have constructed a table of all the values of y to five places between these limits, and those of d to eight places, whereby the solution of such equations is obtained by one or two simple operations, true to eight or nine places of figures. See Rees’s Cyclopedia, article IRREDUCIBLE Case; also Leybourn’s Math. Repository, No, 11; and for the solution by sines and tangents, see Cusic Equations. The reason why Cardan’s rule fails in this particular case is, that we make two supposi- tions in order to arrive at the solution, which are inconsistent with each other. Let, for example, 2? — ax = b, be an equation falling 3 under the irreducible case, having os - Now the suppositions are m +n? — b, mn = — and itis to be shown that these two condi- tions cannot have place at the same time, under the conditions of the problem, at least only when m3 = n3, or when the equation has two equal roots, for since m* + n3 = b, the greatest product m3 n3 is when m3 = n3, or when they are each equal to 46, and con- sequently when pte product = 152, Now 43 we suppose m3 n? — re and since is greater Zz than = by hypothesis, it is obvious that we make an impossible supposition, and conse- “quently the result of it must a dot be 2 ‘imaginary. But if m3 = 73 then =e = -—, and 27 the solution may be obtained. It may be fur- ther remarked, that when an equation falls under the irreducible case, it will have three real roots; and conversely, that every equa- tion having three real roots, unless two of them are equal, must according to Cardan’s rule fall under the irreducible case, as may be shown thus. Let «3 — ax = 6, be an equation having three real roots p, qg, 7, of which let 7 be the greatest; then from the known theory of equa- tions p +tqtr= pr +aqr bb pqr tn Whence lst, p + gq = —r 2d, r(p+_ ERI Gi ORGY —pqr~= 3d, (p Cry te ‘ Iso Now in the case of equal roots we sho have p = q; but this is not the case we ar considering, having shown above thatthe go lution will obtain on that supposition. Sing then the roots are unequal, let p be greate than g, and make p = q + d, and we shal have by substituting ‘this value of Pp in th second and third equations Qq+dp—Gtd)= a Qqtd) x@tdg= bd 3q° + 3qd + d* — dt, 2q3 + Sard + adh x wo 6 and it remains to be shown, that with these values of a and 6, =4a3 is necessarily prema than £5’. Or 3} 39? + 3dq + antsy £5} 293 Bq°d + d*g ( 3 or} G+dg+ yal > }G+igdtidegt which is obvious by simply expanding those quantities, the former being po + opr + 4ptr + 3p3 rs 4+ 4 pred spr? + ap; the latter p® + + 3p5y +34 ptr? + 3p3r3 + prt Therefore when the three ee of a cubit + equation 23 — az = b are real, 5 is greater Zz i than ae and consequently the solution of i falls under the irreducible case. Hence then it appears, that when a cubic equation has only one real root, that root will be found by the formula of Cardan; but when all thr roots are real, no one of them can be obtained by that method ; which is the only analytics solution that has yet been discovered, ane probably the only one the equation admits of. See Phil. Trans. vol. Ixviii. part 1, art. 42; vol. Ixx. p. 387. : IRREGULAR, that which deviates fide the usual form or rules; thus, in geometry, a polygon which has not all its sides and angle equal, is called an irregular polygon. ISAGONE, is used by some old authors to denote a fizure having equal angles ; the word being derived from soos, equal, and Yywvic, ale gle. ISOCHRONAL, or IsocHRONOUS (from sos, equal, yore, tune) i is applied to such Cae tions of a pendulum as are performed i in equ times. Of which kind are all the cycloidal vibe tions or swings of the same pendulum, whether the ares it describes be longer or shorter; for when it describes a shor ter arc, it moves so much the slower; and when a long one, pro- portionally faster. See PENDULUM. id ISOMERIA, is a term used by Vieta to denote the operation of freeing an equatidl from fractions, ISOPERIMETRICAL Figures, in Geome> try, are those which have equal perimeters. € ISOPERIMETRY (from saos equal, Tepsy meTpov, perimeter), in Mathematics, isa branch of the higher geometry, which treats LSO ‘the properties of isoperimetrical figures’ . of surfaces contained under equal perime- s, of solids under equal surfaces, of curves equal lengths, &c. Of the foregoing heads » two first may be considered as containing » elements of the science, which relates incipaliy to the maxima et minima of diffe- it surfaces and solids, when bounded by ires of equal perimeters, but of a greater less number of sides, and posited in a dif- ant order. The other head, which relates the maxima et minima of curves, are pro- ms of another kind and of the most difficult ure, Which have engaged the attention and sreised the talents of many of the greatest thematicians of modern times; as Newton, ibnitz, the Bernoullis, Euler, Lagrange, &c. | gave rise to many warm and angry dis- es, particularly between the brothers, John | James Bernoulli, of which an account is en under the article BRACHYSTOCHRONE. “he problems in which it is required to find, ong figures of the same or different kinds, ee which within equal pcrimeters shall wprehend the greatest surfaces, and those ds which under equal surfaces shall con- | the greatest volumn, had long engaged attention of mathematicians before the ention of fluxions; and different methods been devised for the solution of them by scartes, ermat, Sluze, Hudde, and others ; ch were all supplanted by the simplicity ‘ generality of the new analysis: after which the elements of the science seems to e been lost sight of by mathematicians, » were all engaged in the solution of the 1er order of isoperimetrical problems. ‘dmpson was the first who condescended reat of the more elementary parts of this Ines by giving in his ‘‘ Geometry” a very peeing chapter on the maxima et minima seometrical quantities, and some of the dlest problems concerning isoperimeters. ; next who treated the subject in an ele- tary manner, was Simon L’ Huillier of Ge- t, who in 1782 published his treatise ‘“‘ De atione mutua Capacitatis et T'erminorum rarum,” &c. His principal object in the position of that work, was to supply the siency in this respect, which he found in t of the Elementary Courses, and to deter- 2 with regard to both the most usual sur- s and solids, those which possess the mi- am of contour with the same capacity; reciprocally, the maximum of capacity the same boundary. Legendre has also idered the same subject in a manner some- t different from either Simpson or L’ Huil- inhis “ Elemens de Géométrie ;”’ and Dr. sley has a paper on the same subject in Phil. Trans. vol. Ixxv., for 1775. These, ever, principally relate to the elementary Ositions of isoperimetry ; but what relates e higher geometry, forms the highest order reblems, which as we before observed called into action the talents, and excited »assions of some of the ablest mathema- ns of Europe; having led to a dispute ISo which, for want of competent and impartial judges, remained undecided for many years, and has since been termed the “ war of pro- blems,” on account of the great interest it ex- cited, and the determined and able manner in which each party supported its opinions and contested those of its opponents: a brief sketch of which will be found under the arti- cle BRACHYSTOCHRONE, as well as the solu- tion of that celebrated problem. As to the rest, the reader who is desirous of farther information, may consult the papers mention- ed in that article; as also the essays of several able mathematicians who have introduced the subject into their respective works. Simpson, in his “Tracts,” has a chapter en- titled, “‘ An Investigation of a General Rule for the Resolution of Isoperimetrical Problems of all Orders.” He has also given the sojution of several isoperimetrical problems, in his “ Doctrine of 'luxions.” Maclaurin has like- wise a chapter on the same subject, in his “ Treatise of Fluxions.”” To these may also be added, Emerson, Le Seur, Bossut, and La- croix; each of whom has introduced this doc- trine into their respective works: but the two writers who have most contributed in bringing to perfection the theory of isoperi- metry, are Euler and Lagrange, the former having, beside several memoirs in the Acta Petro. a tract on this subject, entitled “ Me- thedus inveniendi Lineas Curvas Proprietate Maximi Minimive gaudentes;” which, with a very few execcptions, is what it was intend- ed to be, a complete treatise, containing essen- tially all the requisite methods of solutions, with great variety and abundance of examples and illustrations: there were still, however, some defects in this work, for want of a better algorithm, or more compendious process of establishing the theorems, and certain supple- mental formulae; which defects have been finally removed by Lagrange, in his admir- able and refined “ Calculus of Wariations ;” and a small treatise on the same subject has lately been published by Mr. Woodhouse, in which are combined the history and progress of the science, with such observations and remarks as seem most calculated to render it instructive and familiar to the English student; see also Rees’s Cyclopedia, article Isoprrt- METRY. ISOSCELES Triangle (from ios, equal, and oxshoc, leg), is a triangle of two equal legs or sides; such is the triangle ABC. The angles at the base of an isosceles tri- angle are equal, and if the sides be produced, the angles under the base are also equal. (Euclid, v. 1). JUN If the line BD be drawn perpendicular to the base, it will bisect the base and the ver- tical angle; or if.it be drawn to bisect the base, it will be perpendicular to it. JULIAN Calendar, Epoch, Period, Year, &c. See the respective articles. JUNO, the name of one of the new planets, situated between the orbits of Mars and Ju- piter, was discovered by Mr. Harding, at the observatory of Lilienthal, near Bremen, on the evening of the Ist of September, 1804. While this astronomer was forming an atlas of all the stars, as far as the eighth magnitude, which are near the orbits of Ceres and Pallas, he observed, in the constellation Pisces, a small star of the eighth magnitude, which was not mentioned in the “ Histoire Celeste” of La Lande; and being ignorant of its longi- tude and Jatitude, he put it down in his chart as nearly as he could estimate with his eye. Two days afterwards the star disappeared, but he perceived another which he had not seen before, resembling the first in size and colour, and situated a little to the south-west of its place. He observed it again on the Sth of September, and finding that it had moved a little farther to the south-west, he con- cluded that this star belonged to the planetary system. The-planet Juno is of a reddish colour, and is free from that nebulosity which surrounds Pallas. Its diameter, and its mean distance, are less than those of the other new planets. It is distinguished from all other planets by the great eccentricity of iis orbit; and the effect of this is so extremely sensible, that it passes over that half of its orbit which is bisected by its perihelion, in half the time that it employs in describing the other half, which is farther from the sun. From the same cause, its greatest distance from the sun is double the least distance, the difference between the two distances being about 127 millions of miles. Though there is no nebulous appearance around the planet Juno, yet it appears, from the observations of Schroeter, that it must have an atmosphere more dense than that of any of the old planets of the system. A very remarkable variation in the brilliancy of this planet has been observed by this astronomer. He attributes it chiefly to changes that are going on in its atmosphere, though he thinks it not improbable that these changes may arise from a diurnal rotation performed in 27 hours, The following elements were calculated by Buckhardt. Annual revolution ........... wee 4yrs, 128 days Mean longitude, 31st Dec. 1804, noon........... Eiiwee a 4932217! 23" Place of ascending node....... 5§21° 6’ 0” Place of perihelion in 1805... 18 29° 49’ 23” PSCCOMUTICIG I s.... 050500 cagp cedan . 0.25096 Inclination of orbit. ......... cay eee. 0! Dittosarag. 2a. 3n task peek: 5s 13° 4 Mean distance from the sun in English miles............... 253,000,000 Mean distance.....ccccccccsesseves 20007 . FUP Diameter in English miles, according to Schroeter...... 1425 | Apparent mean diameter, as | seen from the earth, ac- cording to Schroeter ......... 3”.057 JUPITER, in Astronomy, is the large: planet in our system, and the fifth in orde from the sun with regard to the old planets but the ninth, including the four new planet lately discovered between him and Mar, Jupiter is attended by four satellites, whic were first discovered by the venerable Galile on the 8th of January, 1610. See SATEr LIfEs. He is denoted by the character 2. This planet is farther remarkable for: th faint appearance of stripes, or belts, wi which he is surrounded. These differ mue at different times, and even at the same tim in telescopes of different powers. ‘They aj pear usually of an uniform tint, but whe viewed to the greatest advantage they seei to consist of a number of curved lines; an are now generally supposed to have som connection with the atmosphere of the plane They were first discovered by Zuppi am Bartoli, two jesuits, before the year 1660, | which year they were observed by Campat with a refracting telescope of his own col struction. See Betts. And for the sever elements of the orbit of Jupiter, see ELEMEN§ of the Planets. : Jupiter is the largest, and after Venus t) most brilliant of all the planets; and eve Venus he sometimes surpasses in brightnes He performs his sidereal revolution in 433% 14" 18’ 41”, 0; or in 11862 Julian years; bj this period is subject to some inequality. F performs his synodical revolution in abo 399 days. is His mean distance from the sun is 5:2 that of the earth being considered as unit which makes his mean distance above 4} millions of miles. ‘The eccentricity of h orbit is 0482; half the major axis being tak« as unity. His mean longitude at the cor mencement of the present century was , 3° 22° 36”, 1; and the longitude of his pe. helion was, at the same time, in 0° 11° 8’ 38% but the line of the apsides has an appare motion according to the order of the signs, 56”, 7 in a year, or of 1° 34 33”, 8 in century. | His orbit is inclined to the plane of t ecliptic in an angle of 1° 18’ 51”, 5; which’ observed to decrease nearly 22’, 6 in a ce tury. His orbit at the commencement of ti present century, crossed the ecliptic, in 3° 25’ 34", 2; but the place of the nodes has: apparent motion in longitude, according. the order of the signs of 343 in a year, 57’ 12"4 in a century. k, The rotation on his axis, is performed 9 55’ 49"7; and his axis forms an angle 86° 541 with the plane of the ecliptic. B diameter is equal to 91,522 miles; and con quently he is above 1331 times larger thi the earth; his polar diameter is to his eqt torial as 9287 to 1, or nearly as 13 to 1 + KEP he mass of Jupiter, compared with that of e sun as unity, is +067-00" and its density to at of the sun is *909501 to 1. The light id heat received by this planet from the n, supposing it to be inversely as the square ‘the distance, is 037 of that received by the rth. ‘A body which weighs one pound at the wth’s surface, would, if removed to equa- rial regions of Jupiter, weigh 2°281 pounds. As seen from the earth, the motion of Ju- ter appears sometimes to be retrograde ; e mean are which he deseribes in this case about 9° 54’; and its mean duration is about " days. This retrogradation commences finishes when the planet is not more dis- nt than 115° 12’ from the sun. His mean ' ’ ] KEP apparent equatorial diameter is 882; being greatest when in opposition, at which time it is equal to 476. Laplace’s “ Systeme du Monde.” JURIN (James), a celebrated physician, mathematician, and philosopher, in the early part of the last century. He was author of several ingenious compositions, particularly “An Essay on Distinct and Indistinct Vi- sion,” printed at the end of vol. ii. of Dr. Smith’s *‘ System of Optics;” beside several controversial papers against Robins, Michel- lotti, &c. and many papers on different sub- jects in the Phil. Trans. from vol. xxx. to vol. Ixvi. At the time of his death, which hap- pened in 1750, Dr. Jurin was secretary to the Royal Society, but his age when this event happened is not stated. | K ss Ly x EILL (Dr. Joun), an eminent mathema- sian and philosopher, was born at Edinburgh 1671; and studied in the university of that ty. Keill was author of several works, but as more distinguished in his time for the con- ‘icuous part he took in the dispute between bibnitz and Newton concerning the inven- m of Fluxions. bn was approved of both by Newton and the her members of the Royal Society, and veral of his letters are given by Collins in e “ Commercium Epistolicum.” ‘The prin- pal works of this author (besides what are ntained in vols. xxvi. Xxvil. XXxvili. and ‘ix, of the Phil. 'Trans.) are as follows, viz. 1. Introductio ad Veram Physicam, 1701; cond edition, 1705; a third edition in Eng- h, 1736. { 2. Introductio ad Veram Astronomiam, ‘18; which was afterwards translated by the ithor himself into English, and published ider the title of 3. An Introduction to True Astronomy in (21. Besides these he published some other orks of less importance, relating to Burnet’s id Whiston’s “ ‘Theories of the Earth,” &c. KEPLER (Joun), a very eminent astro- ymer and mathematician, was born in the uchy of Wirtemberg, in 1571. He very urly discovered a great taste for mathematical id astronomical inquiries, and at the age of } years had acquired so great a reputation at he was appointed professor of mathema- bs at Gratz in Stiria. He afterwards, on the 'vitation of Tycho Brahe, settled in Bohemia, ‘id on the death of that eminent astronomer eceeded to his title of “mathematician to \s imperial majesty,” a distinction which he His conduct on this occa-~ enjoyed the remainder of his life, though in consequence of some irregularities in the pay- ment of the salary or pension attached to the appolutment, he is said to have been very much distressed in his pecuniary affairs, and died at Ratisbon in 1630, in his 59th year, whither he had repaired to solicit his legal arrears. Maclaurin in drawing a sketch of the writings of this great man observes, that to him we owe the discovery of the true figure of the orbits, and the proportion of the motions of the solar system. 'This astronomer had a particular passion for finding analogies and harmonies in nature, after the manner of the Pythagoreans and Platonists. Three things, he tells us, he anxiously songht after from his early youth ;—Why the planets were six in number? Why the dimensions of their orbits were such as Copernicus had deseribed from observation? And what was the analogy or law of their revolutions? He songht for the reasons of the first two of these in the pro- perties of numbers and plane figures without success. But at length reflecting that while the plane regular figures may be infinite in number, the regular solids are only five, he imagined that certain mysteries in nature might correspond with this limitation inherent in the essence of things: he therefore en- deayoured to find some relation between the dimensions of those solids and the intervals of the planetary spheres; and imagining that a cube inscribed in the sphere of Saturn would touch by its six planes the sphere of Jupiter, and the other four regular solids in like manner fitted the intervals that are be- tween the spheres of the other planets; he became persuaded that this was the true reason why the primary planets were precisely KEP six In number, and that the author of the world had determined their distances from the sun, the centre of the system, from a regard to this analogy. Being thus, as he imagined, possessed of the grand secret of the Pythagoreans, and pleased with the discovery, he published it in 1596, under the title of “ Mysterium Cosmographicum.” He sent a copy of this book to Tycho Brahe, who did not approve of the speculations contained in it, but wrote to Kepler, urging him first to lay a solid foundation in observation, and then by ascending from them, to strive to come at the causes of things: and to this advice we are indebted for the more solid discoveries of Kepler... See KEpLer’s Laws. The great sagacity of this astronomer in the planetary motions, suggested to him some views of the true principles from which these motions flow. He speaks of gravity as of a power that was mutual between bodies; and says, that the earth and moon tend towards each other, and would meet in a point so many times nearer to the earth than to the moon, as the earth is greater than the moon, if their motions did not hinder it. He adds, also, that the tide arises from the gravity of the waters towards the moon. But not having notions sufficiently just of the laws of motion, he was unable to make the best use of these ideas; nor did he steadily adhere to them, for in his “ Epitome of Astronomy,” published many years after, he proposes a physical ac- count of the planctary motions derived from different principles. Besides his astronomical discoveries we are indebted to Kepler for some important labours relating to Logarithms and other branches of mathematical science: his principal works are, Cosmographical Mystery, 1596; Optical Astronomy, 1604; Celestial Physics; New Ephemerides; Copernican System, in six books, the first three in 1618, and the others in 1622; Logarithms, 1624; astronomical tables, called, “'The Rodolphine Tables ;’ Epitome of Astronomy, 1635. Besides several other works of less importance. KEPLER’S Laws, a term used by astronomers to denote certain analogies between the dis- tances of the planetary bodies and their times of periodic revolution, as also between the rate of motion of any revolving body, whether pri- mary or secondary, and its distance from the central body about which it revolves; to which may also be added, the figure of the planetary orbits, and the position of the central body ; these in order in which they were discovered stand as follows: | 1. Equal areas are described in equal times. That is, if a line be supposed to join the central and revolving body, this line passes over or describes equal areas in equal times, whether the planet be in its aphelion, peri- helion, or in any other part of its orbit. K IR 2. The planets all revolve in elliptic orbits situated in planes passing through the centr of the sun; the latter body occupying one ¢ the foci of the ellipse. 3. The squares of the times of revolutioy of the several planetary bodies, are as th cubes of their respective distances from th sun. It is impossible in this place to enter int a history of these discoveries, and the diffi culties and prejudices which Kepler had 4 encounter in bringing them to perfection; bu the reader may see this subject fully illustrate: in Dr. Small’s “ Account of the Astronomiea Discoveries of Kepler.” i) KepLer’s Problem, is the determining th true from the mean anomaly of a planet, 6 the determining its place in its elliptic orbi answering to any given time; and so name from the celebrated astronomer Kepler, why first proposed it. See ANOMALY. i The general state of the problem is thi to find the position of a right line, whi | passing through one of the foci of an ellips shall cut off an area, which shall be in am given proportion to the whole area of th ellipse ; which results from this property, tha such line sweeps areas that are proportiona to the times. 3 t Many solutions have been given of thi problem, some direct and geometrical, othen not; viz. by Kepler, Ward, Newton, Keil Machin, &c. See Newton’s Principia, lib. i prop. 31; Keil’s Astron., Lect.23; Phil. Trans Abr. vol. viii. p. 73, &e. ye KERSEY (Joun), an able English mathe matician, who flourished towards the close 0 the seventeenth and beginning of the eigh, teenth century, and is chiefly known in th scientific world by his “‘ Elements of Algebra, in two vols. folio, whichis an ample and com: plete work, containing a full explanation 6 the problems of Diophantus; he was autho likewise of Dictionarum Anglo-Britanicum, or General English Dictionary. ii KEYSTONE of an Arch, the middle vous} soir, or that immediately over the centre ol the arch. a KILDERKIN, a liquor measure containing 18 gallons beer measure, and 16 gallons ale measure. “| KIRCHER (Atuanasivs), a celebrated ma thematician, was born at Fulde in 1601. He wrote an immense number of books, amount ing altogether to twenty-two volumes fo io, eleven quarto, and three octavo. It m st, however, be allowed, that their utility bears) no proportion to their magnitude, and 1 would therefore be useless to give a catalogue of them in this work, as there is no one of them but has been superseded by more mo dern authors. Kircher died in 1680, in the 80th year of his age. aa ‘ ‘ABEL, a long thin brass ruler used with angent line on the edge of a circumferentor take angles of altitude, distances, &c. LACERTA Lizard, a new northern con- Hlation. See CoNnsTELLATION. LAGNY (Tuomas Fantet De), an emi- at French mathematitéian, was born at ons in the year 1660. He is said to have eived a bias for mathematical studies by ying accidentally met with Fournier’s clid, and a book on Algebra, from which ie he gave himself up wholly to these ences. He came to Paris in 1686, and was m after appointed tutor to the Duke de ailles;s He became a member of the ademy of Sciences, and was appointed by uis XIV. royal hydrographer at Rochfort; ‘sixteen years afterwards he was recalled Paris, and made librarian to the king, with onsiderable pension. He died in the year 44, and in his last moments, when he no ger knew the persons who surrounded bed, one of them, through a foolish cu- sity, asked him, ‘ What is the square of ” to which he replied, as it were mecha- ally,144. His works are, 1. New Methods the Extraction and Approximation of ots. 2. Elements of Arithmetic and Al- ra. 3. On the Cubature of the Sphere. A General Analysis, Method of resolving »blems; and 5. Several papers in the Me- irs of the Academy. Lagny excelled in hmetic, algebra, and geometry, in which made many important discoveries. He nd the measures of angles in a new mce, called “ Goniometry;” in which he ad the value of angles to great accuracy ‘Means of compasses, without scales or les of any kind. He paid great attention cyclometry, or the method of measuring circle ; and calculated, by means of infinite es, the ratio of the circumference of a le to its diameter, to 120 places of figures. rens. sALANDE (JosepH Jerome Le FRAn- 3), a celebrated French astronomer, was a at Bourg, in the department of L’ Ain, in y, 1732. He was first intended for the _ but his genius having been very early ted to astronomical subjects his first in- jon was given up; and he followed his onomical pursuits under the celebrated nonier, with the greatest success. At time when Lacaille was preparing for his age to the Cape of Good Hope, for the LAM L purpose of determining the parallax of the moon, aiid his distance from the earth, La- lande, though then only 18 years of age, was, through the influence of Lemonier, appointed to make the corresponding observations | at Berlin, a task which he executed to the en- tire satisfaction of the Academy of Sciences, and with the greatest credit to himself, After this period he enriched the memoirs of this learned body, by numerous papers on astro- nomical and various other subjects. Lalande was also an associate of the principal learned societies in Europe, and was for many years the centre of communication amongst the most celebrated of their members. After a: long and useful life in the pursuit of science, he died on the 4th of April, 1807, being then in the 75th year of his age. Beside the many papers which he published in the Memoirs of the Academy, he was author of “ A Traité Astronomique,” the first edition of which was published in 1764. He also republished the “ Histoire des Mathema- tiques” of Montucla, enlarging it to 4 vols, 4to. principally from the papers of Montucla. In the midst of his other labours he likewise drew up his “ Voyage d’Italy;” “'Traité des Canaux ;’ “ Bibliographié Astronomique ;” ‘“‘ Abregé de Navigation Historique, Theo- rique, et Practique;” he also edited the French edition of Halley’s “Tables ;’ and superin- tended the publication of the “ Connoissance des 'Tems,” from 1760 to his death, beside many other pieces of less importance. - LAMBERT (Joun Henry), an eminent mathematician and astronomer, was born at Muhlhausen in Sundgaw, belonging to Switz- erland, Aug. 29, 1728. His parents being poor he had the greatest difficulty to contend with in the pursuit of his studies, which he nevertheless pursued with unbounded success; and was, after a time, assisted in them by a party of gentlemen to whom his talents were made known. In 1757 he was nominated a corresponding member of the Scientific So- ciety of Gottingen; and the year following, having travelled to Paris, he became inti- mately acquainted with D’ Alembert, and other eminent scientific men, and was ultimately elected a member of several learned societies. He published a “ Treatise on Perspective,” another on ““Photometry,’’ beside many smaller works, as his “ Letters on the Construc- tion of the Universe ;”’ which were afterwards digested, and published in English, under the LAN fitle of “'The System of the World.” Most of his mathematical pieces were published in a collected form by himself in three volumes, in which almost every branch of mathematical science has been enriched with his improve- ments and additions. Lambert died in 1777, in the 50th year of his age. LAMIN, in Physics, are extremely thin plates, of which solid bodies are supposed to be made up. These are indeed rather ideal than real; but such a conformation is fre- quently supposed for the sake of simplifying the solution in a great variety of physical problems. LAMP EDIAS, a term sometimes applied to denote a bearded comet. LANDEN (JoHN), an eminent English mathematician, was born at Peakirk, near Peterborough, in January, 1719. He became an early proficient in the mathematical sci- ences, and was a contributor to the Ladies’ Diary in 1744, and continued his labours in that useful little performance nearly till the time of his death. Mr. Landen published, in the Phil. Trans. for 1754, “ An Investigation of some Theorems,” which suggested several remarkable properties of the circle, &c.; and in the following year he published a volume entitled “‘ Mathematical Lucubrations.” This title was intended to inform his friends and the public, that the study of mathematics was, at that time, rather the pursuit of his leisure hours, than his principal employment. ‘They contain a variety of tracts relating to the recti- fication of curve lines, the summation of series, the finding of fluents, and many other points in the higher branches of mathematics. From this time, to the year 1766, he gave the world several valuable works; and on the 16th of January of this year, he was elected a fellow of the Royal Society: soon after which he in- serted in the Phil. 'lrans. “ A Specimen of a new Method of comparing curvilineal Areas ;” by means of which many areas are compared which did not appear to admit of comparison by any other method, a circumstance of con- siderable importance in that part of natural philosophy which relates to the doctrine of motion. ‘These are but a small part of the works which he produced, and which have given celebrity to his name. In the year 1731, 1782, 1783, he published three small tracts, «On the Summation of converging Series;” in which he explained and showed the extent of some theorems, which had been given for that purpose by De Moivre, Stirling, and Thomas Simpson, in answer to what he conceived to have been written in disparagement of those excellent mathematicians. Mr. Landen was author of a work published in two volumes, and at difierent times, entitled, “‘ Memoirs.” "Lhe second volume contains his last labours on the solution of the general problem con- cerning rotatory motion. It comprises, also, a resolution of the problem relating to the motion of a top; with an investigation of the motion of the equinoxes, in which Mr. Landen . LAT has, first of any one, pointed out the cause Sir Isaac Newton’s mistake in his solution this celebrated problem. He lived to see t volume completed, and received a copy oI the day before his death, which happen January 1l5th, 1790, at Milton, in the 7 year of his age. LARBOARD, the left hand side of a s} when a person stands on board with his towards the head of the vessel. LATERAL Equation, is a term used some old authors for what is now more co monly called a SimpLE Equation. LATION, is sometimes used for trans tion, or change of place. | LATITUDE, in Geography or Navigati is the distance of a place from the equat reckoned on an arch of the meridian, int cepted between its zenith and the equator. North LAtiItuDE, is that which falls in northern hemisphere; viz. between the equ tor and the north pole. . South LATITUDE, is that which falls in t southern hemisphere, or between the equa and south pole. \ ed Parallels of LATITUDE, are small circles the sphere supposed parallel to the equa and are thus called because they show 1 latitude of places by their intersection W the meridians, all places falling under 1 same circle being in the same latitude. | The quadrant, or meridian, intercepted tween the equator and either pole, is divid into ninety degrees, and numbered both wi from the equator to the poles; and the latit of any place is equal to the measure of 1 arch intercepted between the equator and tf place, and which is said to be north or sou according as it is situated towards the no or south pole. . The latitude of a place, and the elevati of the pole above the horizon of that pla are terms frequently used indifferently the ¢ for the other, being in fact equal to each oth This will appear from the following figu, where the circle ; ZUQP is a cir- cle of the terrestrial sphere, Z any place upon it, HO the horizon, EQ the equator, P the pole, and N the corres- ee ponding pole of the heavens, which may considered as infinitely distant from P, | situated in the continuation of the terrest axis PP. Now the elevation of the pole. measured by an observer at Z, will be angle WZN; ZW being drawn parallel the true horizon HO; and because N is definitely distant from P, a line drawn fro1 to N may be considered as parallel to P and consequently the angle WZN = an OC P the elevation of the pole; and beca E ZAP, and ZPO, are both quadrants, if th be taken from each the arch ZP, there | remain OP = EZ; but OP is the eleva LAT ‘the pole, and EZ is the latitude of the ace; therefore these quantities are equal, id may be used indifferently the one for the her. The knowledge of the latitude of places is the greatest importance in geography, na- gation, and astronomy; it may therefore be oper to state some of the best methods of ‘termining it both by sea and land; one these is by finding the elevation of the pole, aich is always equal to the latitude of the hse, as shown above, and this is done by servations on the pole-star or any other Riri otar stars, thus: Either draw a true meridian line, or find e time when the star is on the meridian th above and below the pole, then at these nes with a quadrant or other instrument ether with Des Cartes, and of gravity Wi Newton; though he has not reconciled the principles, nor shown how gravity arose fro the impulse of this ether, nor how to accou for the planetary revolutions in their re pe tive orbits. His system is also defective, | it does not reconcile the circulation of # ether with the free motions of ‘the comets all directions, or with the obliquity of | planes of the planetary orbits, nor does hey solve other objections to which the bypothes of the vortices and plenum is liable. , | zt 4 ef Soon after the period just mentioned, dispute commenced concerning the inve ti of the method of fluxions, which led ¥ Leibnitz to take a very decided part in op sition to the philosophy of Newton. From1 ! goodness and wisdom of the Deity, and] principle of a sufficient reason, he conclude that the universe was a perfect work, or ft best that could possibly have been made; @ that other things, which are evil or incomm dious, were permitted as necessary con quences of what was best: that the mater) system, considered as a perfect machine, @ never fall into disorder or require to be § right; and to suppose that God interposes it, is to lessen the skill of the author, andi perfection of his work. He expressly charg an impious tendency on the philosophy | Newton, because he asserts that the fabrie the universe and course of nature could 1 continue for ever in its present state, but) process of time would require to be re-es! blished or renewed by the hand of its ii framer. The perfection of the universe, consequence of which it is capable of 0 tinuing for ever, by mechanical laws, in_ present state, led Mr. Leibnitz to distingu between the quantity of motion and the for of bodies; and whilst he owns, in oppositi to Des Cartes, that the former varies, to mai tain that the quantity of force is for evert same in the uniyerse; and to measure t LEM es ties. gd eibnitz proposes two principles as the :dation of all our knowledge ; the first, that : impossible for a thing to be, and not to at the same time, which he says is the adation of speculative truth; and secondly, nothing is without a sufficient reason why principle, he says, we make a transition 4 abstracted truths to natural philoso- . Hence he concludes that the mind aturally determined, in its volitions and tions, by the greatest apparent good, and -it is impossible to make choice between gs perfectly like, which he calls indis- tibles; from whence he infers, that two gs perfectly like could not have been pro- ed eyen by the Deity himself: and one on why he rejects a vacuum, is because parts of it must be supposed perfectly like vach other. Fer the same reason too, he cts atoms, and all similar parts of matter ; ach of which, though divisible ad infinitum, scribes a monad (Act Lipsic 1698, p. 435) ctive kind of principle, endued with per- jon. The essence of substance he places ction or activity, or, as he expresses it, omething that is betwecn acting and the Ity of acting. He affirms that absolute is impossible, and holds that motion, sort of nisus, is essential to all material stances. Each monad he describes as esentative of the whole universe from point of sight; and yet he tells us in of his letters, that matter is not a sub- ce, but a substantiatum, or phenomené bien é See Maclaurin’s “ View of Newton’s osophical Discoveries,” book i. ch. 4. EMMA, (from aeuSaw, L assume), in Ma- atics, denotes a previous proposition, laid n in order to clear the way for some fol- ng demonstration, and prefixed either to rems in order to render their demonstra~ Jess perplexed and intricate, or to pro- is to make their resolution more easy and t. Thus, to prove a pyramid one third of ism, or parallelopiped of the same base height with it, the demonstration of which ie ordinary way is difficult and trouble- e, this lemma may be premised, which is ed in the rules of progression, viz. that the of a series of square numbers in arith- cal progression, beginning from 0, as 1, +16, 25, 36, &c. is always subtriple of the of as many terms, each equal to the test; or is always one-third of the great- erm multiplied by the number of terms, IMNISCATE,in the Higher netry, the name of a curve ie form of a figure of 8. If ~ be represented by 2, PQ 1, and the constant line AB C by a, the equations ay = Y—x*, or a? y* = a* a* — *xpressing a lineof the fourth r, will denote a lemniscate, ng a double point in A. re may be other LEMNIs- | of bodies by the squares of their ve- ould be so, rather than otherwise; and by . LEW CATES, as the CASSINOID or CasstneAD Ellipse ; but the one above defined is the simplest. This curve is manifestly quadrable. For since ay = 2x,/a* — x’; the fluxion of the ENGL R AS ie / a* —2x*; of which the 1 fluent is } a” ee (a* —x*) 3, the general area of the curve. This when x = a becomes simply ¥a* = AQB. A right line may cut this curve in 2 points, as NQ; or in 4, as mupq; even the right line BA Cis conceived to cut the curve in 4 points, the double point A reckoning as two, For solutions of the problem to assign equal ares in the lemniscate, see Leybourn’s Ma- thematical Repository, N. 8. vol. i. p. 204-209. LENS, a piece of glass or other transparent substance, having its two surfaces so formed that the rays of light in passing through it have their direction changed, and made to converge or diverge from their original paral- lelism, or to become parallel after converging or diverging. Lenses receive particular de- nominations according to their form; as convex, concave, plano convex, plano concave lenses, and meniscuses. Convex LENS, is one which is thickest in the middle. If only one side is convex and the other plane, it is called a plano convez lens, such is AF, in the following figure; but if i¢ be convex on both sides, it is called a convexe convex, or double convex lens, as BG, Concave LENS, is that which is thinest in the middle; but it is also divided in plano concave and concavo concave, as in the former case; such are two lens CH, DI. And when the lens is concave on one side and convex on the other, itis called a meniscus, as EK, in the above figure. See MENIscus. In every lens the right line perpendicular to the two surfaces, is called the axis of the lens; the points where the axis cuts the sur- face, are called the vertices of the lens; also the middle point between them is called the centre, and the distance between them the diameter. Some confine the term lenses to such as do not exceed half an inch diameter, those that exceed this being termed lenticular glasses. Lenses are either blown or ground. Blown Lenses, are small globules of glass, melted in the flame of a lamp by a blow-pipe, or otherwise. Ground LENSES, are such as are ground to the required form, by means of machinery for this purpose. See some ingenious contrivances for grinding lenses explained in Brewster’s edition of Ferguson’s Lectures; see alse Phil. rans. vol. x}i, p. 556. DD LEN Optical Properties of Lenses.—Parallel rays refracted at a convex spherical surface of a denser, or a concave of a rarer medium, into which they pass, are made to converge. But if refracted at a concave spherical surface of a denser, or convex of a rarer medium, they are made to diverge. For let. DA, HH GC, be two rays N of a parallel pen- p cil, passing out of @ = ae a rarer medium into a denser, and M incident upon the convex spherical surface A C B, whose centre is E. Let GCE pass through the centre of the surface, and it suffers no refraction. Join EA, and produce it to H; also produce DA to K; and let DA be refracted in the direc- rection A F'; then D A H is the angle of inci- dence, and EA F the angle of refraction of this ray, and since it passes out of a rarer medium into a denser, the angle EA F is less than the angle HA D, and therefore less than the angle K AK; add to each of these the angle A EF, and the two angles F A E, A EF are together less than the two KAE, AEF; and there- fore they are less than two right angles, con- sequently AF and CE if produced will meet. Again, when the ray passes out of a denser medium into a rarer, and the surface of the medium into which they are refracted is spherically concave, the rays still converge as before. The construc- N tion remaining | D the same, since theray DA passes out of a denser medium into a rarer, the angle of incidence D AE= AEC, is less than the angle of refraction HAF, and therefore, as above, it follows that AF and C E produced will meet. Whence it follows, that if rays are made to traverse a double convex lens, they must con- verge more rapidly then through a plano con- vex one, for they are first refracted at the con- vex surface of a denser medium on entering the lens, and are again refracted at the con- cave surface of the rarer medium in passing again into the air, and must therefore have a double degree of convergency. Thus the two rays DA, DA, which would have had their point of concourse in T, in a medium of the same density as the lens BG, will have it in E, in consequence of the second reiractiou in passing from the lens again into the rarer medium of the air. By a chain of reasoning similar to that em- LEN ployed in the preceding articles, it ma shown, that rays in passing out of a rarer dium into a denser, the surface. of that which they are refracted being spheri concave, or in passing from a denser mec into a rarer, the surface of that into w they are refracted being spherically con will in both cases be made to diverge. Whence again it follows, that if rays made tg traverse a double concave lens, must diverge more rapidly than in pas through a plano concave lens, for in this | they are first refracted at the concave sm of a denser medium on entering the lens, | are again refracted at the convex surface rarer medium, in passing again into the air, must therefore have a double degree of vergency. Thus the rays DA,. DA, whose direct arising from the first refraction would be AT, are changed in consequence of the see refraction into BE, BE, by passing from denser medium of the lens into the rarer: dium of the air. . Rules for finding the Foci of Lenses.— focus of a convex spherical lens, is dis from the vertex a diameter and a half of conyexity nearly. In a plano convex ]} the distance of the focus from the verte equal to a diameter of the convexity, if segment do not exceed 30°. Or the gen rule is, 4s 107; 193 :: the radius of conve : the refracted ray, taken to its concoi with the axis. In double convex glasse the same sphere, the focus is distant from| vertex about the diameter, if the segmen’ not exceed 30°. But when the two conv ties are unequal; that is, when they are } ments of different spheres, then the rule i the sum of the radii of both convexiti the radius of either convexity alone :: the dius of the other convexity : the distane the focus from the vertex. ny It must be observed, however, that the which fall near the axis of any lens, are united so near the vertex as those which farther off; nor will the distance be so gi in a plano convex lens, when the convex! is towards the object, as when the plane } is towards it. And hence it follows, that viewing any object with a plano convex k the convex side should always be turned ¢ wards; and the same remark has place burning with such a glass. To find the Foci geometrically.—Dr. Ha has given a general method for finding foci of spherical glasses of all kinds, both « cave and convex, exposed to any kind of r either parallel, converging, or diverging, follows: to find the focus of any parcel of r LEV » iverging from, or converging to, a given point 1 the axis of aspherical lens, and making the ame angle with it, the ratio of the sines of ~ ofraction being given. Suppose GLa lens, P a point in its surface, “its pole, C the centre of the spherical seg- ent, O the object or point in the axis to or om which the rays proceed, and OP a given 1y; and suppose the ratio of refraction to 2asrtos. Then making CR to CO, ass 'r, for the immersion of a ray, or as r tos r the emersion (7.e. as the sines of the angles . the medium which the ray enters, to the wrresponding sines in the medium out of hich it comes; and laying CR from C to- yards O, the point R will be the same for all ne rays of the point O. Lastly, drawing the dius PC, continued if necessary; with the tre R, and distance OP, describe an arc itersecting PC in Q. The line QR being yawn, shall be parallel to the reflecting ray ; hd PF being parallel to it, shall intersect the iS in the point F, the focus sought. Or ake as CQ: CP:: CR: CF, which will be ie distance of the focus from the centre of e sphere. And from this general construc- ym, he adverts to a number of particular sim- € cases. . Dr. Halley gave also an universal algebraical eorem, to find the focus of all sorts of optic asses or lenses. See the Phil. Trans. No. 5, or Abr. vol. i. p. 191. LEO, the Lion, one of the old northern nstellations. See CONSTELLATION. (Leo Minor, a new northern constellation. 1e CONSTELLATION. . Cor LIONIS, the Lion’s Heart, a fixed star the constellation Leo; and is otherwise illed ReGuus, which see. LEPUS, the Hare, a southern constella- m. See CONSTELLATION. 'LEUCIPPUS, an ancient Greek philoso- ier, who flourished about 420 years before srist. LEVEL, an instrument employed in ascer- ining a horizontal line, of which there are ‘rious sorts; as the ‘Air Levet, which shows the line of level | means of a bubble of air inclosed with me fluid in a glass tube of an indeterminate agth and thickness, and having its two ends ‘rmetically sealed: an invention, it is said, -M. Thevenot. When the bubble fixes itself a certain mark, made exactly in the middle the tube, the case or ruler in which it is LEV fixed is then level. When it is not level the bubble will rise to one end. This glass tube anay be set in another of brass, having an aper- ture in the middle, where the bubble of air may be observed. The liquor with which the tube is filled is oil of tartar, or aqua secunda; those not being liable to freeze as common water, nor to rarefaction and condensation as spirit of wine is. Plumb LeEvet, that which shows the hori- zontal line by means of another line perpen- dicular to that described by a plummet or pendulum. his instrument consists of twe legs or branches, joined together at right angles, whereof that which carries the thread and plummet is about'a foot and a half long ; the thread is hung towards the top of the branch. The middle of the branch where the thread passes is hollow, so that it may hang free every where: but towards the bottom, where there is a little blade of silver, whereon is drawn a line perpendicular to the telescope, the said cavity is covered by two pieces of brass, making as it were a kind of ease, lest the wind should agitate the thread ; for which reason the silver blade is covered with a glass to the end, that it may be seen when the thread and plummet play upon the perpen- dicular. The telescope is fastened to the other branch of the instrument, and is about two feet long; having a hair placed horizon- tally across the focus of the object-glass, which determines the point of the level. The tele- scope must be fitted at right angles to the perpendicular. It has a ball and socket, by which it is fastened to the foot. Water LEVEL, that which shows the hori- zontal line by means of a surface of water or other fluid; founded on this principle, that water always places itself level or horizontal, The most simple kind is made of a long wooden trough or canal; which being equally filled with water, its surface shows the line of level. And this is the chorobates of the an- cients, described by Vitruvius, lib. viii. eap. 6. ‘The water-level is also made with two cups fitied to the two ends of a straight pipe, about an inch diameter, and three or four feet long, by means of which the water communicates from the one cup to the other; and this pipe being moveable on its stand by means of a ball and socket, when the two cups show equally full of water, their two surfaces mark the line of level. This instrument, instead of cups, may also be made with two short cylinders of glass, three or four inches long, fastened to each ex- tremity of the pipe with wax or mastic. The pipe is filled with common or coloured water, which shows itself through the. cylinders, by means of which the line of level is determined; the height of the water, with respect to the centre of the earth; being always the same in both cylinders. This level, though very sim- ple, is yet very commodious for levelling small distances. See the method of preparing and using a water-level, and a mercurial level, an- nexed to apgihe quadrant, for the same pur- ID VEV pose, by Mr. Leigh, in Phil. Trans. vol. x1. 417, or Abr. viii. 362. Where works of moderate extent are car- yied on, and where the perfect level of each stratum of materials is not an object of im- portance, the common bricklayer’s level, made in the form of an inverted T; thus, L, having a plumb suspended from the top, and received in an opening at the junction of the perpen- dicular with the horizontal piece, will answer well enough. The principle on which this acts is, that as all weights have a tendency to gravitate towards the centre of the earth, so as the plumb-line is a true perpendicular, any line cutting that at right angles must be a horizontal line at the point of intersection. But the most complete level that has ever yet been invented, is the spirit level of the late Mr. Ramsden, of which we haye given a representation in Plate XI. fig. 1. The level gr is surmounted by a telescope OMI, and the whole fixed on a stand resembling that of the improved theodolite, the adjustment of the instrument being effected by means of the screws shown in the figure. But the reader who is desirous of a complete description of the adjustment and use of this instrument, must consult Gregory’s Dictionary of Arts and Sciences, where they are both given at considerable length. And a variety of other levels, though none so perfect as this, may be seen in De Lahire’s and Picard’s treatises on Levelling, and in Biron’s description of Ma- thematical Instruments. LEVELLING, the finding a line parallel to the horizon at ene or more stations, to de- termine the height or depth of one place with respect to another, for laying out grounds even, regulating descents, draining morasses, conducting water, &c. Two or more places are ona true level when they are equally distant from the centre of the earth, Also one place is higher than another, or out of level with it, when it is farther from the centre of the earth; and a line equally dis- tant from that centre in all its points, is called the line of true level. Hence, because the earth is round, that line must beacurve, and make a part of the earth’s circum- ference, or at least be parallel to it, or concentrical withit ; as the line BC FG, which has all its points equally dis- tantfrom A, the cen- tre of the earth, considering it as a perfect globe. But the line of sight BDE, &c. given by the operations of levels, is a tangent, or a right line perpendicular to the semi-diameter of the earth at the point of contact B, rising always higher above the true line of level, the farther the distance is, is called the apparent line of level. Thus, CD is the height of the appa- rent leyel above the true level, at the distance LEV BC or BD; also EF is the excess of height at F, and GH at G, &c. The difference, it s evident, is always equal to the excess of the secant of the arch of distance above the radius of the earth. | The common methods of levelling are suffi- cient for laying pavements of walks, or for, conveying water to small distances, &c.; but in more extensive operations, as in levelling’ the bottoms of canals, which are to convey | water to the distance of many miles, and such’ like, the difference between the true and the} apparent level must be taken into the account, Now the difference CD between the true and apparent level, at any distance BC or BD, may be found thus: by a well-known property of the circle (2AC + CD): BD 3) BD:CD; or because the diameter of the. earth is so great with respect to the line C D,) at all distances to which an operation of leve } ling commonly extends, that 2AC may be, | ‘| | safely taken for 2AC + CD in that propor tion without any sensible error, it will b 2AC:BD:: BD: CD, which therefore is = BD? , BC” nearly: that is, the diff bal Ba Tie nearly: iat is, the difference be-) 4 | tween the true and apparent level is equa | to the square of the distance between thé places, divided by the diameter of the earth and consequently it is always proportional the square of the distance. , Now the diameter of the earth being nearly) 7958 miles, if we first take BC = 1 mile, then) BC? ‘ a ae becomes 7, of a mile, whi ch is 7.962 inches, or almost eight inches, for the height of the apparent above the true level a the distance of one mile. Hence, proportions ing the excesses in altitude according to squares of the distances, the following table is obtained, showing the height of the apparent above the true level for every 100 yards of dis- tance on the one hand, and for every mile ol the other. | sia the excess | Dif. of Level, | or CD. ' Distance, or | Dif. of Level, || Distance, Be, or CD. BC Inches. Feet. Inches. I 0.026 0 oz 0.103 0 2 0.231 0 4 0.411 0 8 0.643 2 8 0.925 6 0 1.260 10 7 ¥ 1.645 16 7 2.081 23 11 4 2.570 32 6 3.110 42 6 3.701 53 9 FF 4.344 66 4 4 5.038 80 3 F 5.784 95 7 = 6.580 112 2.9 7.425 130 1 LEV By means of this table of reductions, we an now level to almost any distance at one peration, which the ancients could not do ut by a great multitude; for, being unac- nainted with the correction answering to any istance, they only levelled from one twenty ards to another, when they had occasion to ontinue the work to some considerable ex- ont. This table will answer several useful pur- oses. Thus, first, to find the height of the pparent level above the true, at any distance. f the given distance is in the table, the cor- sction of level is found on the same line with .t thus at the distance of 1000 yards, the cor- action is 2°57, or two inches and a half nearly; nd at the distance of 10 miles, it is 66 feet 4 iches. But if the exact distance is not found 1 the table, then multiply the square of the istance in yards by 2°57, and divide by 000,000, or cut off six places on the right for ecimals ; the rest are inches: or multiply the juare of the distance in miles by 66 feet 4 iches, and divide by 100. 2dly, To find the extent of the visible hori- yn, or how far can be seen from any given sight, on a horizontal plane, as at sea, &c. appose the eye of an observer, on the top of ship’s mast at sea, is at the height of 130 feet xove the water, he will then see about 14 iles all around. Or from the top of a cliff by € sea-side, the height of which is 66 feet, a arson may see to the distance of near 10 miles 1 the surface of the sea, Also, when the top a hill, or the light in a light-house, or such ke, whose height is 130 feet, first comes into ‘e view of an eye on board a ship, the table iows that the distance of the ship from it is t miles, if the eye is at the surface of the ater; but if the height of the eye in the ship 80 feet, then the distance will be increased y near 11 miles, making in all about 25 miles . distance. ‘3dly, Suppose a spring to be on one side of hill, and a house on an opposite hill, with a ley between them, and that the spring seen om the house appears by a levelling instru- ent to be on a level with the foundation of ie house, which suppose is at a mile distance om it; then is the spring eight inches above ie true level of the house; and this difference ould be barely sufficient for the water to be ‘ought in pipes from the spring to the muse, the pipes being laid all the way in the ‘ound. 4thly, If the height or distance exceed the nits of the table, then, first, if the distance » given, divide it by 2, or by 3, or by 4, &c. ll the quotient come within the distances in e table; then take out the height answering the quotient, and multiply it by the square ’ the divisor, that is, by 4, or 9, or 16, &c. r the height required: so if the top of a hill just seen at the distance of 40 miles, then 40 vided by 4 gives 10, to which in the table iswer 664 feet, which being multiplied by }, the square of 4, gives 10614 feet for the sight of the hill. But when the height is Lm Vv given, divide it by one of these square num- bers, 4, 9, 16, 25, &c. till the quotient come within the limits of the table, and multiply the quotient by the square root of the divisor; that is, by 2, or 3, or 4, or 5, &c. for the dis- tance sought: so when the top of the peak of Teneriffe, said to be about 3 miles, or 15,840 feet high, just comes into view at sea, divide 15,840 by 225, or the square of 15, and the quotient is 70 nearly ; to which, in the table, answers by proportion nearly 103 miles; then multiplying 103 by 15, gives lod niles and 2, for the distance of the hill. All that has been previously stated has been said without any regard to the effect of refrac- tion in elevating the apparent places of ob- jects. But as the operation of refraction in incurvating the rays of light proceeding from objects near the horizon is very considerable, it can by no means be neglected, when the difference between the true and apparent level is estimated at considerable distances. It is now ascertained (see REFRACTION) that for horizontal refractions the radius of curva- ture of the curve of refraction is about seven times the radius of the earth; in consequence of which the distance at which an object can be seen by refraction, is to the distance at which it could be seen without refraction, nearly as 14 to 13, the refraction augmenting the distance at which an object can be seen by about a thirteenth of itself. By reason of this refraction, too, it happens, that it is ne- cessary to diminish by + of itself the height of the apparent above the true level, as given in the preceding table of reductions. ‘Thus, at 1000 yards, the true difference of level, when allowance is made for the effect of refraction, will be 2°570 — ‘367 = 2203 inches. At two miles it would be 32 — 44 = 273 inches; and sO on. Prony has given, at the end of his “ Archi- tecture Hydraulique,” a table computed on this principle, extending from 50 to 6000 French toises; and showing, in three distinct columns, the difference between the true and apparent levels; first without regarding refrac- tion, next considering it, and then a column showing the difference ef the results. To find the height H of a mountain, its an- gle of apparent elevation E, the are A of a great circle of the earth included between the foot of the mountain and the place of the ob- server, and the apparent angle C made at the top of the mountain between the plumb-line and the apparent first place of the observer on the earth’s surface, M. Lambert gave this theorem, R being the radius of the earth: _ R sin. (90 + E— 4,A), EO 50) gin, (C= 2 A whence H is immediately found. Other for- mule are deduced by M. Laplace for the same purpose; but they are too complex to be in- serted here. LEVELLING, is either simple or compound ; the former is when the level points are deter- mined from one station, whether the level be fixed at one of the points or between them; LEV and the latter, or compound levelling, is no- thing more than a repetition of several such simple operations. Vor the practical operations in levelling, the reader is referred to the treatises on this sub- ject by De Lahire, Picard, and Lefebure. LEVELLING Staves, instruments used in levelling, serving to carry the marks to be ob- served, and at the same time to measure the height of those marks from the ground. They usually consist of two mahogany staves, ten feet long, in two parts, that slide upon one another to about 5,4, feet, for the greater con- venience of carriage. They are divided in 1000 equal parts, and numbered at every tenth division by 10, 20, 30, &c. to 1000; and on one side the feet and inches are also some- times marked. A vane slides up and down upon each set of these staves, which by brass springs will stand at any part. These vanes are about ten inches long and four inches broad; the breadth is first divided into three equal parts, the two extremes are painted white, the middle space divided again into three equal parts, which are less; the middle one of them is also painted white, and the two other parts black; and thus they are suited to all the common distances. These vanes have each a brass wire across a small square hole in the centre, which serve to point out the height correctly, by coinciding with the hori- zontal wire of the telescope of the level. LEVER, a straight bar of iron, wood, &c. supposed to be inflexible, supported on a ful- crum or prop by a’single point, about which all.the parts are moveable, and is generally considered as the first of the mechanical powers, being the simplest of them all. Its principal use is in raising great weights to small heights, or in moving heavy blocks of stone, &c. to short distances, &e. Levers are divided into four kinds, accord- ing to the position of the fulcrum, weight, and power, aud are thus distinguished: A Lever of the first kind, has the fulcrum or prop between the weight and the power: thus, if AB bea rod or bar, W a weight attached to the end A, Pa power acting at the other end B, and C the fulcrum or prop that sup- ports AB; then C is the centre of motion, and AC and BC are the arms of the lever AB. Of this kind are balances, scales, pincers, scis- sars, &c. A Lever of the second hind, has the weight W between the power P and the fulerum C; as oars, rudders, bellows, cutting-knives fixed at one end, &e. ‘to be moved. Op A Lever of the third kind, has the power | between the weight W and the fulcrum € such as tongs, sheep-shears, a man raising | ladder, the bones and muscles of animals, & A bended Lever is the fourth kind; asa clay hammer drawing a nail. This, however, i only a species of the first kind, beeause th fulcrum is between the power and the bod Let the weights W and P be attached t the ends of the inflexible line or lever A aud suppose C is the fulcrum or prop suppor ing the weights; then if they are equilibri the distances AC and CB will be reciprocall as the weights; that iss AC: CB:: BP: W or AC x W=CB xP. ; 4 Let 8 be the centre of the éarth; then be cause the weights W and P gravitate toward the centre S, the three forces in equilibrio ac in the directions AS, BS, and SC. 7 Draw CR, CD, perpendicular to AS, BS respectively, and CH, CG, parallel to B§ AS; and the three forces will be as the side of the triangle SCG, or CHS; hence CH HS (or CG):: P: W. - Ss lg i ; * _ But in the parallelogram C HSG, the oppo site angles at H and G are equal, consequenth the angles CHR, CGD, are also equal, anc therefore the triangles CHR, CGD, are si milar. S. whence........ Rt RR “i GH d CaGks ae bats ..2.te% ae ree CH: CGP.) .: Va therefore, by equality, CR:CD::P :W OF hass saben seadacts the eb CA: CBP... Wa For CA and CB, and the respeetive per pendiculars CR and CD, are not sensibly di ferent either in length or position. Pe Cor. 1. Hence if the weights W, P, or th | LEV veight W and power P in equilibrio, move on 1c fulerum or centre of motion C, the arcs or paces Ww, Pp, described in the same time, rill be as the radius CW, CP; therefore the veight W x Ww = power (or weight) P x ‘p, and since the velocities will be as the res Ww, Pp, the momenta of W and P are qual. i %. ®e - all - - . . “aL = ®». me. ™. tay *. Meu. Cor. 2. And if WCR be a bended lever, nd the power act perpendicularly to WC D, ne weight W and power P will be in equi- ‘brio when WC, DC, the perpendicular dis- ‘inces from the centre of motion C, are reci- rocally as the weight and power; that is, we: DO: Pe W. WV Cc , Cor.3. Therefore when the power P acts ob- quely against the end of the lever W B, the reight W and power P are reciprocally as VC and the perpendicular C D, the two dis- imees of the directions of the forces from the Ww Cc B ontre of motion C; thatis, WC: DC::P:W. ‘ence if WCD be a bended lever, and the eight W and the power P act perpendicu- ly to the arms CW, CD; then WC x W :C€D x P, as in the straight lever. ‘Cor. 4. When several weights W, S, D, P, sting on a straight lever WP, are in equi- orio, the sum of the products of the weights, ultiplied by their respective distances from seir support C on one side, will be equal to ve sum of the products on the other; that is, CxS8S+WCxW=DCxD+PC xP. For the effort of each weight to turn the ver, is as the weight multiplied by its dis- nee from the fulcrum C, and therefore the im of the efforts on one side must be equal ) those on the other, in case ofan equilibrium. Cor. 5. Hence the place of the fulcrum is vadily determined, when the length of the ver WP, and the weights W, P, are given fee fig, to Cor.1). For W:P::CP:CW; | ) LEV and by composition, W +P: W, P (the length) 2: P: CW; thatis, the length must be divided into two parts, having the proportion of the weights. A lever of the second or third kind may be reduced to the first: thus, Conceive the lever CB to be equal to CD, then it is manifest that if the power P were removed to p, but acting in a contrary diree- tion, the equilibrium would still remain, and we should have DC x p= CW x W;; that is, DC x P= CW x W. Hence, in the lever of either kind, if the weight and the power are multiplied by their respective distances from the fulcrum, the products will be equal when there is an equi- librium. ; Hence it follows, that the beam of a pair of scales being a lever of the first kind, its arms ought to be exactly of the same length; for should there-be any difference, equal weights when placed in the scales will not rest in equi- librio. The obvious method of trial, however, is to weigh any lody very accurately in one scale; then if the weight and body change places, and either end preponderates, the scales are imperfect or false ; but when we know what the body weighs in each scale, its true weight may be found thus: let W and w denote the weights of the body 6, in the two scales whose distance from the fulcrum are d, 0; then db = 3w, and 36 = dw whence 6* = Ww, orb = / Ww; that is, its true weight is a geometrical mean between the least and greatest weights found by the false scales. See BALANCE. For the application of the principles of the lever to the Roman STEELYARD, see that article. : Let the compound lever SD be composed of three levers of the first kind, DA, AB, BS, acting upon one another; the fulcrums being at C, O, R. Then P:W::CA.OB.RS:CD.OA.RB; when the power P, and weight W, are in equilibrio. D * * S Cc 0 BR sistas P CD.P cA the force at A; eC lu Fin Gd GA st OB: OA: ——: force at B; ae. pp... CD-OA:P , CD.OA.RB.P said ‘CA. OB-RS- For CA:CD::P: CA.OB” Me CA.OB = W, the force at S. Li.B Therefore | | CD: OA-RB.P=CA.OB.RS. W. And a similar conclusion is derived in the other kinds of levers, by making use of the respective distances from the props or ful- crums. On this subject see Gregory’s ‘“ Me- chanics,”’ vol. i. LEVITY, in Philosophy, the opposite to gravity, or that supposed quality of certain bodies which gives them a power of ascent, being thus opposed to gravity, by which they have always a tendency to descend. The ancients supposed several different bodies to be possessed of levity, but the error has long since been detected, and the principle itself excluded from every system of philosophy. LEUWENHOEK (AnrTony), a celebrated Dutch philosopher, was born at Delft in 1632, and acquired a great reputation throughout all Europe, by his experiments and discove- ries in Natural History, by means of the mi- croscope. He particularly excelled in mak- ing glasses for microscopes and spectacles ; and he was a member of most of the literary societies of Europe, to whom he sent many memoirs. Those in the Phil. Trans. and in the Paris Memoirs, extend through many volumes; the former were extracted and pub- lished at Leyden in 1722. He died in 1728, at ninety-one years of age. LEYDEN Phial, in Electricity, is a glass phial or jar, coated both within and without with tin-foil, or some other conducting sub- stance, which may be charged and employed in a variety of useful and entertaining experi- ments, Oreven flat glass, or any other shape, so coated and used, has also received the same denomination. Also a vacuum produced in such a jar, &c. has been named the Leyden vacuum. The Leyden phial has been so called, be- cause it is said that M. Cunzeus, a native of Leyden, first contrived, about the close of the year 1745, to accumulate the electrical power in glass, and use it in this way. But Dr. Priestley asserts, that this discovery was first made by Von Kleist, Dean of the Cathedral in Camin; who on the fourth of November, 1745, sent an account of it to Dr. Lieberkuhn, at Berlin. LIBRA, the Balance, one of the zodiacal constellations, denoted by the character =. See CONSTELLATION. Lrpra, in Mechanics. See BALANCE. Lisra also denotes the ancient Roman pound, which was equal to about 5040 of our Troy grains. It was likewise the name of one of their gold coins, equal in value to 20 de- narii. See Phil. Trans. vol. 1xi. p. 462. LIBRATION of the Moon, in Astronomy, is more particularly applied todenote an apparent irregular libratory motion of that body about her own axis, whereby we see a little more than one-half of the lunar disc; or rather, it is in con- sequence of our seeing more than one-half of it, that the moon appears to have such a mo- tion; for although the term libration, from the Latin, libratio, to balance, agrees perfectly LIB well with the appearances observed, still must not be understood in a positive sens the appearance itself arising from a total different cause from that which the word seén to indicate. In order to illustrate this, let y conceive a visual ray drawn from the cer tre of the earth to the centre of the moon: th plane drawn through the latter centre perpe} dicularly to this ray, will cut the lunar gloh according to the circumference of a cire| which is, with respect to us, the apparent dis; If the moon had no real rotatory motion, i) motion of revolution solely would discover us all the points of its surface in succession the visual ray would therefore meet that su, face successively in different points, which 1 us would appear to pass, the one after th other, to the apparent centre of the lunar dis¢ The real rotatory motion counteracts t effects of this apparent rotation; and bring back constantly towards us the same face | the lunar globe. Suppose, now, that the rotation of the moo is sensibly uniform ; that is to say, that it doe not partake of any periodical inequalities (th supposition is at least the most natural whie¢ we can make, and it is conformable to obse) vations); then one of the causes which pr duce the libration will become evident; for th motion of revolution partaking of the period! cal inequalities, is sometimes slower, som times more rapid: the apparent rotation whie it occasions cannot, therefore, always exacfl counterbalance the actual rotation, which mains constantly the same; and these tw effect will surpass each other by turns. T points of the lunar globe ought, therefore, t appear turning sometimes in one directio} sometimes in another, about its centre; an the resulting appearance is the same as if th moon had a little vibratory balancing from on side to the other of the radius vector draw from its centre to the earth. Itis this whic is named the libration in longitude. Several accessory but sensible causés mt dify this first result. The spots of the moo do not always retain the same elevation abo the plane of its orbit; indeed some of then by the effect of the rotation, pass from on side of this plane to the opposite side. Thes circumstances indicate an axis of rotatio which is not exactly perpendicular to th plane of the lunar orbit; but according as t axis presents to us its greater or its smalle obliquity, it must discover to us successivel the two poles of rotation of the lunar spheroi¢ hence it is we perceive, at certain times, som of the points situated towards these poles, an lose the sight of them afterwards, when the arrive nearer the apparent edge: this is calle the libration in latitude. Itis but inconside rable, and therefore indicates that the equato of the moon differs very little in position fron the plane of its orbit. ‘te Finally, a third illusion arises from the ob server being placed at the surface of the eartl and not at its centre. Towards this centre i is that the moon always turns the same face LIF { the visual ray drawn thence to the centre rthe moon would always meet its surface at i: same point, abstracting from the preceding qualities. It is not the same with respect ) the visual ray drawn from the surface of » earth; for this ray makes a sensible angle ith the former, by reason of the proximity of y moon; this angle is at the horizon, equal ‘the horizontal parallax: in consequence of s difference the apparent contour of the aar spheroid is not the same for the centre \ the earth, and to an observer placed at its srface. This, when the moon rises, causes ne points to be discovered towards its upper wge, which could not have been perceived im the centre of the earth: as the moon es above the horizon, these points continue | approach the upper edge of the disc, and length disappear, while others become visi- )) to its lower edge: the same effect is con- fued during all the time that the moon is ‘ible; and, as the part of its disc which ap- pars highest at its rising is found lowest at setting, these are the two instants when » difference is most perceptible. Thus the aar globe, in its diurnal motion, appears to willate about the radius vector drawn from ; centre to the centre of the earth. This fenomenon is designated by the name of uirnal libration. For a popular illustration _ this phenomenon, see Laplace’s “Systeme ) Monde ;” and for a physical explanation, te Mem. of Berlin for 1764 and 1780, by grange. {LIBRATION of the Earth, is a term applied ‘some astronomers to that motion, whereby 3 earth is so retained within iis orbit, as that axis continues constantly parallel to the is of the world. See PARALLELISM. This Copernicus calls the motion of libra- n, and may be illustrated thus: suppose a ybe, with its axis parallel to that of the earth, inted on the flag of a mast, moveable on its is, and constantly driven by an east wind, tile it sails round an island; it is evident 2 painted globe will be so librated as that L axis will be parallel to that of the world, in ery situation of the ship. LIFE Annuities, are such periodical pay- ints as depend on the continuance of some rticular life or lives; and they may be dis- guished into lives, to commence imme- mately ; and annuities, to commence at some ture period, called reversionary annuities. The limits of this article will not admit of br entering upon the theoretical investigation this important subject; we must, therefore, ntent ourselves with merely a popular illus- ition of the principles upon which the com- ‘tations are conducted, and for more minute formaton the reader is referred to the works annuities, mentioned in the subsequent irt of this article; but, first, it will be proper give a sketch of the progress of this impor- sat branch of political economy. ‘The first writer who attempted the develop-, ent of a rational theory of life annuities was an Hudden, or Hudde, which was farther LIF extended by De Witt, the celebrated Dutch pensionary, in a tract printed at the Hague in 1671, under the title of “De vardye van de Lif-renten,” &c. In 1692, Dr. Halley gave in the Phil. Trans. “ An Estimate of the Degrees of Mortality of Mankind, drawn from curious Tables of the Births and Funerals at the City of Breslaw,”’ &c.; in which paper the author presented a very perspicuous view of the principles of an accurate theory, and from this the first correct table was computed. De Moivre took up the subject where it was left by Halley, and in 1724 published the first edition of his “ Annuities on Lives,” founded on the principle of an equal decrement of life; an hypothesis which, though it very much simplified the computation, and furnished an elegant theory, could not be admitted into practice without injury either to the insurer or the person insured. In 1742 T. Simpson published his tract “On the Doctrine of Annuities and Reversions ;” in which the tables were computed from the absolute observations on the bills of mortality ; and the same subject was farther prosecuted by the same author, in his ““Select Exercises,” published in 1752; Dodson, in his ‘ Mathe- matical Repository,” published in 1753 and 1755, also treated on the same subject, but adopted the hypothesis of De Moivre. The science remained in this state, without much improvement, till the publication of Dr. Price’s treatise, in 1769, entitled ‘“ Observa- tions on Reversionary Payments,” &c. This work was written with a view to oppose and destroy the injurious effects and evil inten- tions of a class of men (unfortunately to be found in every stage of society), who, under pretence of establishing societies for the be- nefit of old age and of widows, were only forming schemes to allure and to defeat the hopes of the ignorant and the distressed. His efforts were eventually crowned with success: and those bubble societies have long since met with the fate which he so truly predicted. In this laudable pursuit, Dr. Price saw the necessity of more accurate observations on the mortality of human life; in order to determine with more correctness the value of annuities, and to show more forcibly the futility and ex- travagance of the schemes that were issued by those societies. By the assistance of some public-spirited individuals, he obtained cor- rect registers of the rate of mortality at North- ampton, Norwich, Chester, and other places in England. But still, the computation of the values of annuities, according to these obser- vations, wasa work so tedious and unpleasant, that little hopes were entertained of profiting by those researches: and Dr. Price suffered three several editions of his treatise to pass over without affording any additional informa- tion on this subject. At length the fourth edition appeared (1783) enriched with several valuable tables of annuities on single and joint lives, at different rates of interest, de- LIF duced not only from the probabilities of living as observed at Northampton, but also from the probabilities of living, as observed in the kingdom of Sweden at large. The great addition which Dr. Price hasmade to our means of information respecting tais science, and the assiduity with which he thus promoted some of the best interests of man- kind, deserves the highest commendation: and his labours on this subject entitle him to our warmest praise. The primary object which he had in view has been fully answered; and his treatise was admirably adapted to that end. lo every other respect, however, it is far from being complete: and the reader will look in vain for the most common eases that occur in practice. Indeed, those subjects which are to be met with, do not readily pre- sent themselves, owing to the loose and irre- gular manner in which they are treated. Dr. Price’s object was not so much to insert what was new, as to illustrate (by some striking examples) a few of the leading problems, with a view to oppose the pernicious schemes that disgraced the age in which he lived. But, those schemes having long since vanished, his observations may now be considered rather as a beacon to posterity. The next treatise on this subject was by Mr. Morgan, published in 1779, under the title of “The Doctrine of Annuities and As- surances ;” and in 1783, Mr. Baron Maseres published his “Principles of the Doctrine of Life Annuities,” &c. Some foreigners, as Bernoulli, Gregory, Fon- tana, Kerseboom, Duvillard, Deparcieux, Crome, Sprengell, &c. have attended to some of the insulated branches of this subject. But there were no works which deserved the name of treatises, except the above, till Mr. Francis Baily published his very complete and valu- able work, entitled “ The Doctrine of Life Annuities and Assurances analytically inves- tigated and explained.” In this performance the arrangement of the author is logical; his demonstrations as perspicuous as the nature of the subject will allow; the new notation by which he simplifies his theoretical processes is ingenious; and the numerous practical rules which he has deduced from his investigations are plain and free from ambiguity. He has eareiully guarded against a loose and unscien- tific use of terms; and. has so contrived the subdivisions of his work, as to keep himself tolerably free from needless repetitions. The tables, which are in number fifty-nine, and occupy one hundred pages, are neatly and (as far as we may venture to speak from a pretty cautious inspection) accurately printed. We therefore think it right to recommend Mr. Baily’s treatise to such of our readers as wish to investigate this interesting subject Ere | minutely: others may be satisfied with | following tables and remarks: +i The value or present worth of an annu for any proposed life or lives, it is evide depends on two circumstances; the inter of money, and the chance or expectati of the continuance of life. Upon the furn only depends the value or present worth of annuity certain, or that whichis not subjeet the continuance of a life or other contingent but the expectation of life being a thing x certain, but only possessing a certain chan it is evident that the value of the certain; nuity, as stated above, must be diminished proportion as the expectancy is below ¢ tainty: thus, if the present value of an annu certain be any sum, as suppose £100. and{ value and expectancy of the life be one-h; then the value of the life anunity will be o half of the former, or £50; and if the val of the life be only one-third, the value of 1 life annuity will be but oue-third of £1 that is, £25. 6s. 8d.; and so on. it The measure of the value or expectancy life, depends on the proportion of the nu of persons that die, out of a given number, the time proposed. thus, if fifty persons | out of one hundred, in any proposed ti then, half the number only remaining alive,a one person has an equal chance to live ort in that time, or the value of his life for time is one-half; but if two-thirds of the nu ber die in the time proposed, or only one-th remain alive, then the value of any life one-third ; and if three-quarters of the num die, or only one-quarter remain alive, th the value of any life is but one quarter; @ soon. In these proportions, then, must 1 value of the annuity certain be diminished, give the value of the like iife annuity. It is plain, therefore, that in this business is necessary to know the value of life at all 4 different ages, from some table of observatio on the mortality of mankind, which may sh the proportion of the persons living, out ol given number, at the end of any propos time; or from some certain hypothesis, of sumed principle. Now various tables a hypotheses of this sort were given by 14 writers on this subject: but the same table probabilities of life will not suit all places; long experience has shown that all places 4 not equally healthy, or that the proportion the number of persons that die annually, different for different places. It may BD therefore, be improper to insert here a ¢0) parative view of two of the principal tabl that have been given of this kind, as belo where the first column shows the age, and t other columns the number of persons oe that age, out of 1000 born, or of the age in the first line of each column. 1 \ oe : LiF | LIF TABLE UL. ibiting the Decrease of Life, at all Ages from 1 Year to 90, as deduced from the Bills of Mortality in ) London and Northampton. 4 Ages. | London. oe | Ares.} London. | amotm, || Ages.|_ London. fee bo eo Pe * | ampton. U } 100 1000 23 : 174 279 1 | 680 738 24} 3 167 265 | 2) 545 625 25 159 261 3} 492 385 26 153 254 | 452 262 27 147 247 5} 426 544 || 28 141 | 239 6] 410 530 29 5 135 232 7} 397 ols 30 : 130 225 8 | 338s 510 31 125 | 218 9} 33 o04 32 211 10 373 498 33 36 96 16 204 il 367 493 34 197 12) 361 488 35 190 13 356 484 36 . ‘ 185 ) 14 351 480 37 r( 176 15 | 347 475 38 I 169 16 343 470 39 162 ‘17 338 465 40 Je 155 18 | 334 459 41 . 148 19} 329 453 42 141 20| 325 | 447 |] 43 134 21 S21 440 44 127 2 316 433 45 120 TABLE IL. } ° joing the Value of an Annuity on One Life, or Number of Years Annuity in the Value, supposing Money | to bear Interest at the several Rates of 3,4, and 5 per Cent. Years value at | Years value at | Years value at A Years value at | Years value at | Years value at 3 per cent. 4 per cent. 5 per cent. ge. 3 per cent, 4 per cent. 5 per cent. 18's 16°2 14:1 41 13°0 Lisa | 10°2 189 16:3 14:2 42 12°8 11°2 10°1 19°0 16°4 14:3 43 12°6 111 10°0 19°0 16°4 14°3 44 12°5 11°0 9:9 190 16°4 14°3 45 12:3 10°8 9°8 19°0 16°4 14:3 46 121 10°7 : 18:9 163 14°2 47 11:9 18°7 16:2 141 48 118 18°5 16°0 14:0 49 116 18°3 15'8 13°9 50 11-4 181 156 13°7 51 11:2 17:9 15°4 13°53 52 11:0 17°6 15-2 13°4 a3 10°7 17°4 15°0 13°2 54 10°5 H 17:2 14'8 13°0 55 10°3 f 17°0 14:7 12:9 56 10°1 i 16°8 14:5 12°7 a7 9:9 89 16°5 14°35 12°6 58 9°6 8°7 ié3 141 12°4 59 9°4 8°6 16°! 14:0 12:3 60 9:2 8°4 159 13 8 12°1 Gl 8’9 $2 15°6 13°6 12:0 62 8°7 81 15°4 13°4 11°8 - 63 8°5 79 15'2 13°2 117 64 83 77 15°0 13°1 116 65 8:0 75 14°8 12:9 11°4 66 7°38 73 146 12°7 11°3 67 76 . 14°4 12°6 11-2 68 7-4 14:2 12°4 110 69 yi | 14°1 12:3 10°9 70 69 13°9 12°] 10°8 71 67 13°7 11°9 10°6 rie ee 65 13°5 11°8 10°35 73 62 13°3 116 10°4 74 59 132 15 103 | 7% 5°6 LIF | LIF *, 1 TABLE fIl. For the Value of an Annuity upon the Longer of Two given Lives. aan ™ Age of Age of Value at Value at Value at Age of Age of Value at Value at Value at Younger.| Elder. 3 per cent. 4 per cent. 5 per cent. |} Younger.| Elder. 3 percent. | 4 percent. | 5 per cent, 10 23°4 19°9 17°1 55 17°4 151 15 229 19°5 16°8 60 17°0 148 20 22°5 19°1 16°6 30 65 16°6 14°5 25 22°2 18°8 16°4 70 16:1 141 30 21:9 18°6 16'2 79 15°6 13°7 35 21°6 18°4 16°1 — —_—_—|—__—— 10 40 21°4 18°3 16°0 35 18°3 15'8 45 21°2 18°2 15°9 40 17°8 15°4 50 20°9 18°0 15°8 45 17°4 15°1 55 20°7 17°8 157 50 17°1 148 60 20°4 17°6 15°5 30 55 16°7 14°5 65 20°1 17°4 15°3 60 16°3 14:2 70 19°8 172 15°1 65 15'8 13°8 79 19°5 169 14'8 70 15°3 13°4 | | 75 14'8 13°0 15 22'8 19°3 16°7 + ——_ |—_—— 20 22'3 18°9 16°4 40 17°3 15°0 25 21°9 18°6 16:2 45 16°8 146 30 21°6 18°3 16°0 50 16°3 14°2 35 21°3 181 15:9 40 55 15°9 13°9 40 211 17°9 15°7 60 15°4 13°5 15 45 20°9 178 15°6 65 14:9 13°1 50 20°7 17°6 15°4 70 14:5 12°7 55 20°4 17°4 15°3 73 14:0 12:3 60 20°1 17°2 15:2 ——_——_ a 65 19°8 16°9 150 45 16:2 14°2 70 19°4 16°6 14:7 50 15°7 13°8 7 189 16°3 14°4 55 15°2 13°4 | -_-_ —__——| 45 60 14°7 12°9 20 21°6 18'°3 15'8 65 141 12°5 25 21°1 17°9 15°5 70 13°6 12:0 30 20°7 17°6 15°'3 79 13:1 11°6 35 20°4 17°4 151 |j-——— a 4 20°1 17°2 15°0 50 150 13°3 20 45 19°9 17°0 14:9 55 14°5 12°9 30 19°6 16°8 14°7 50 60 139 12°4 55 19°4 16°6 14:5 65 13°3 120 60 19°] 16°3 14:3 70 12°8 11°5 65 18°7 16:0 141 73 12:3 11°0 70 18°2 15°7 13:8 || —— ——_—_——_—_|—————_- 73 17°7 15°3 135 55 13°6 12°4 _——_——$——$.,§ $$ | | ———____|-___ 60 13°0 11:9 . 25 20°3 17°4 151 55 65 12°4 113 30 19°8 17°0 14'9 70 11°8 10°8 35 19°4 16°7 14°7 75 113 10°3 40 19°2 16°5 145 ||-——— —_—— | — 45 18°9 16:3 14°3 60 12:2 11:2 25 50 18°7 16°1 14:2 60 65 11°5 10°6 55 18°4 15°9 14:0 70 10°9 10°1 60 18°0 156 13°8 75 10°3 9°5 65 17°6 15'3 13°6 | > 70 17°2 15°0 13°3 65 10°7 10°0 94 7 75 16°7 14°6 12:9 65 70 10°0 9°4 89 4 , 75 9°3 87 83 35 | 188 16:2 14:2 70 9-2 8°6 $27 30 | 40 | 184 15°9 140 75 8-4 79 76 7 50 17°8 15°4 13°6 75 75 76 72 69 LIF he uses of these tables may be exemplified 1e following problems: , yp. 1. To find the Probability or Proportion € Chance, that a Person of a given Age con- nues in being a proposed number of Years. | ‘hus, suppose the age be 40, and the num- of years proposed 15; then, to calculate he table of the probabilities for London, able 1, against 40 years stands 214; and inst 55 years, the age to which the person st arrive, stands 120; which shows that, of persons who attain to the age of 40, only of them reach the age of 55, and conse- ntly 94 die between the ages of 40 and 55, ; evident therefore that the odds for attain- the proposed age of 55, are as 120 to 94, s 9 to 7 nearly. yp. 11. To find the Value of an Annuity for a proposed Life. ‘his problem is resolved from table 2, by cing against the given age, and under the yosed rate of interest; then the corres- ding quantity shows the number of years chase required. For example, if the given be 36, the rate of interest 4 per cent., and proposed annuity £250. Then in the table yppears that the value is 12:1 years pur- se, or 12°1 times £250.; that is, £3025'1. \fter the same manner, the answer will be ad in any other case falling within the limits he table. But as there may sometimes be asion to know the values of lives computed aigher rates of interest than those in the le, the two following practical rules are joined; by which the problem is resolved ependent of tables. 2ule 1. When the given age is not less than years, nor greater than 85, subtract it from /then multiply the remainder by the per- ity, and divide the product by the said ainder added to 2: times the perpetuity ; hall the quotient be the number of years chase required. Where note, that by the petuity is meant the number of years pur- se of the fee-simple; found by dividing by the rate per cent. at which interest is koned. txam. Let the given age be 50 years, and rate of interest 10 per cent. Then sub- sting 50 from 92, there remains 42; which Itiplied by 10, the perpetuity, gives 420; _ this divided by 67, the remainder in- ased by 2} times 10 the perpetuity, gives nearly, for the number of years purchase. srefore, supposing the annuity to be £100., value in present money will be £630. 2ule 2. When the age is between 10 and years, take eight-tenths of what it wants of which divide by the rate per cent. increased 1:2; then if the quotient be added to the ae of a life of 45 years, found by the pre- jing rule, there will be obtained the num- . of years purchase in this-case. For ex- ple, let the proposed age be 20 years, and rate of interest 5 per cent, Here taking LIG 20 from 45, there remains 26; eight-tenths of which is 20; which divided by 6:2, quotes 3°2; and this added to 9°8, the value of a life of 46, found by the former rule, gives 13 for the number of years purchase that a life of 20 ought to be valued at. And the conclusions derived by these rules, are said by Simpson to be so near the true values,j computed from real observations, as seldom to differ from them by more than one- tenth or two-tenths of one year’s purchase. The observations here alluded to, are those which are founded on the London bills of mor- tality. And a similar method of solution, ac- commodated to the Breslaw observations, will be as follows: viz. Multiply the difference be- tween the given age and 85 years by the per- _petuity, and divide the product by eight-tenths of the said difference, increased by double the perpetuity, for the answer. Which, from 8 to 80 years of age, will commonly come within less than one-eighth of a year’s purchase of the truth. Pros. 11. Vo find the Value of an Annuity for the longest of two Lives; that is, for as long as either of them continue in being. In table 4, find the age of the youngest life, or the nearest to it, in col.1, and the age of the elder in col. 2: then against this last is the answer in the proper column of interest. Exam. So, if the two ages be 15 and 40, then the value of the annuity upon the longest of two such lives, is 21'1 years purchase, at 3 per cent. OF TED caceas vain gaan’ at 4 oo... Shoes OL, LOF te wacsees onsen 0a ductins Ce Gh anteas Bitake Note. In the last two problems, if the younger age, or the rate of interest, be not exactly found in the tables, the nearest to them may be taken; and then, by proportion, the value for the true numbers will be nearly found. Rules and tables for the values of three lives, &c. may also be seen in Simpson, and in Baron Maseres’s “ Annuities,” &c. All these calculations have bcen made from tables of the real mortuary registers, differing un- equally at the several ages. But rules have also been given upon other principles, as by De Moivre, upon the supposition that the de- crements of life are equal at all ages; an as- sumption not much differing from the truth, from 7 to 70 years of age: and for the rea} probabilities of life as deduced from observa- tions, by Mr. Morgan and by Mr. Baily, see the last-mentioned author’s book on this sub- _ ject, quoted and commended above. LIGHT, that principle or substance which renders objects perceptible to our sense of seeing. ‘This is, perhaps, one of the most in- teresting subjects that falls under the con- templation of the philosopher; at the same time it must be acknowledged to be one that is as little understood, and upon which opi- nions are as much divided, as any of the most abstruse subjects. of philosophical inquiry. Some consider light as a fluid, per se; while LIG others consider it merely as a principle, and attribute it to a sort of pression, or vibra- tion propagated from the luminous body through a subtle ethereal medium. But not- withstanding the imperfection of our know- ledge, with regard to the nature and cause of light, repeated experiments and observations have made us acquainted with several of its properties; such as its INFLECTION, REFLEC- TION, REFRACTION, &c.: for which see the re- spective articles. Of the Motion of Light.—The ancients con- sidered light as propagated from the sun and other luminous bodies instantaneously; but the observations of the moderns have shown that this was an erroneous hypothesis, and that light, like any other projectile, employs a certain time in passing from one part of space to another, though the velocity of its motion is truly astonishing, as has been mani- fested in various ways; and, first, from the eclipses of Jupiter’s satellites. It was ob- served by Roemer, that the eclipses of those satellites happen sometimes sooner and some- times later, than the times given by the tables of them; and that the observation was before or after the computed times, according as the earth was nearer to, or farther from J upiter, than the mean distance. Hence Roemer and Cassini both concluded that this circumstance depended on the distance of Jupiter from the earth; and that, to account for it, they must suppose that the light was about fourteen mi- ~ nutes in crossing the earth’s orbit. This con- clusion however was afterward abandoned and attacked by Cassini himself. But Roe- mer’s opinion found an able advocate in Dr. Halley; who remoyed Cassini’s difficulty, and left Roemer’s conclusion in its full force. Yet, in a memoir presented to the academy in 1707, M. Maraldi endeavoured to strengthen Cas- sini’s arguments, when Roemer’s doctrine found a new defender in Mr. Pound; see Phil. Trans. No. 136. It has since been found, by repeated observations, that when the earth is exactly between Jupiter and the sun, his satellites are seen eclipsed about eight minutes and a quarter sooner than they could be ac- cording to the tables; but when the earth is nearly in the opposite point of its orbit, these eclipses happen about eight minutes and a quarter later than the tables predict them. Hence then it is certain that the motion of light is not instantaneous, but that it takes up about sixteen minutes and a half of time to pass over a space equal to the diameter of the earth’s orbit, which is near 190 millions of miles in length, or at the rate of near 200,000 miles per second; a conclusion which is placed beyond every possibility of doubt, by the aberration of the stars discovered by the ce- lebrated Dr. Bradley. See ABERRATION. Of the Momentum of Light.—We have he- fore observed that much diversity of opinion existed with regard to the materiality or im- materiality of light, viz. whether it is a fluid per se, or whether it be merely a principle Sonsisting in pulsations or vibrations; and LIG thus rendered sensible to our optic net sound is to our organs of hearing. +84 The ingenious Dr. Franklin, in a dated April 23, 1752, expresses his dissj faction with the doctrine, that light con; of particles of matter continually driver, from the sun’s surface, with so enorme swiftness. ‘‘ Must not,” says he, “ the sn est portion conceivable, have, with suc motion, a force exceeding that of a 24 pou discharged from a cannon? Must not the; diminish exceedingly by such a waste of ter; and the planets, instead of drawing ne to him, as some have feared, recede to ere distances through the lessened attract Yet these particles, with this amazing mo! will not drive before them, or remove, Jeast and slightest dust they meet with; } the sun appears to continue of his ancien] mensions, and ‘his attendants move in 4 ancient orbits.” He therefore conjectt that all the phenomena of light may be nh properly solved, by supposing all space fh with a subtle elastic fluid, which is not vis} when at rest, but which, by its vibrati\ affects that fine sense in the eye, as thost the air affect the grosser organs of the and even that different degrees of the vi tion of this medium may canse the app ances of different colours. Franklin’s “ per. and OGbserv.” 1769, p. 264. And the celebrated Euler has maintai the same hypothesis in his “'Theoria Luci Colorum ;” where he urges some’ farther} jections to the materiality of light, bey those of Dr. Franklin, as above stated. Th objections, however, Dr. Horsley took ¢ siderable pains to obviate, though it musi allowed that some of them still remain in| force. g Others, on the contrary, have attemptec prove the materiality of light, by determin the momentum of its component particles by showing that they have a force so as, their impulse, to give motion to light bod M. Homberg, Ac. Par. 1708, Hist. p. imagined that he could not only dispel pieces of amianthus, and other light s} stances, by the impulse of the solar rays, | also that by throwing them upon the end ( kind of lever, connected with the spring d watch, he could make it move sensibly quich from which, and other experiments, he in’ red the weight of the particles of light. “A Flartsoeker made pretensions of the sat nature. But M. Du Fay and M: Mai made other experiments of a more accur kind, without the effects which the formerl! imagined, and which even-proved that ’ effects mentioned by them were owing to rents of heated air produced by the burn glasses used in their experiments, or s0! other causes which they had overlooked. However, Dr. Priestley informs us, that Michell endeavoured to ascertain the mom! tum of light with still greater accuracy, @) that his endeavours were not altogether Wi? out success. Having found that the inst eo §, LIM ent he used acquired, from the impulse of he rays of light, a velocity of an inch in a jecond of time, he inferred that the quantity f matter contained in the rays falling upon he instrument in that time, amounted to no nore than the 12 hundred millionth part of a yrain. In the experiment, the light was col- ected from a surface of about three square pet; and as this surface reflected only about je half of what fell upon it, the quantity of natter contained in the solar rays, incident ‘pon a square foot and a half of surface, in a 2cond of time, ought to be no more than the 2 hundred millionth part ofa grain, or upon ne square foot only, the 18 hundred millionth art of a grain. But as the density of the vys of light at the surface of the sun, is 45,000 mes greater than at the earth, there ought ) issue from a square foot of the sun’s sur- ce, in one second of time, the 40 thousandth ut of a grain of matter; that is, alittle more an two grains a day, or about 4,752,000 ‘ains, which is about 670 pounds avoirdupois, 6000 years, the time since the creation; a tantity which would have shortened the n’s semidiameter by no more than about 10 et, if it be supposed of no greater density an water only. | But after all; these experiments and com- ttations must be considered as very vague a unsatisfactory ; and it may be added, that je material hypothesis is almost wholly re- jeted by the most celebrated chemists and ilosophers of the present day. LIGHTNING. See THUNDER. LIKE Quantities, in Algebra, those which nsist of the same letter and power. See efinitions, article ALGEBRA. LIMB, the outermost border, or graduated (ge, of a quadrant, astrolabe, or such like ithematical instrument. ‘The word is also ed for the arch of the primitive circle, in y projection of the sphere in plano. imb also signifies the outermost border or ge of the sun and moon; as the upper limb (edge; the lower limb; the preceding limb, side; the following limb. Astronomers serve the upper or lower limb of the sun or von, to find their true height, or that of the atre, which differs from the others by the nidiameter of the disc. LIMIT, is a term used by mathematicians some determinate quantity, to which a ‘iable one continually approaches, and may ne nearer to it than by any given difference, t can never go beyond it; in which sense ‘irele may be said to be the limit of all its {cribed and circumscribed polygons; be- ise these, by increasing the number of their ten can be made to be nearer equal to the ae than by any space that can be proposed, | vever small it may be. uiMiITs of the Roots of an Equation —By ling the roots. of an equation, is to be un- stood the finding two such numbers that » shall be greater, and one less than the trequired; by which means an approxima- 1 is evidently made towards the true root, : ! ) ‘LIN and the nearer these limits approach towards each other, so much the more accurate will be the approximation. Lagrange, in his “'Traité de la Resolution des Equations Numeriques,” has carried the method of limits to its utmost possible perfection, by showing in all equa- tions, how the limits of each of its roots may be ascertained, and has shown that the method of approximation employed by Newion, and in fact every method except that of his own, is defective in this respect; viz. that between the limits ascertained in their operation there may be one, two, or more roots, and conse- quently, that they are not necessarily the limits of one root, but merely the limits be- tween which one at Icast of the real roots of the equation must lie. The nature of this ar- ticle will not admit of our-enterimg into an explanation of the process of this celebrated analyst ; we can therefore only refer the reader to the work itself. See also the article Ap- PROXIMATION, Encyclopedié Methodique; La Croix’s and Bonnycastle’s Algebra; as also the Meditationes Algebraic of Waring. LIMITED Problem, that which admits of but one, or of a certain number of solutions, in contradistinetion to an unlimited or inde- terminate problem, which may admit of an infinite number of solutions. LINE, in Geometry, is, according to Euclid’s definition, that which has length without thick- ness. Lines are either right or curved: A Right or Straight Line, is that which lies all in the same direction between its extremes or ends. : ; A Curve Line, is that which continually changes its direction. Curve LinEs, are again divided into alge- braical, geometrical, and mechanical, or trans- cendental. An Algebraical or Geometrical Lint, is that which may be expressed, that is, the relation between its absciss and ordinate by an alge- braical equation. And such lines are divided into orders, according to the dimensions of the equations by which they are represented. Sec CuRVE. Mechanical and T'ranscendental Lines, are those which cannot be expressed by finite algebraical equations, See TRANSCENDENTAL. Besides the above distinctions, lines receive other denominations according to their ab- solute or relative positions, as parallel, perpen- dicular, eblique, tangential, &c.; for which see the respective terms. Lines have again other distinguishing ap- pellations, as they are introduced into the dif- ferent sciences of astronomy, geography, dial- ling, perspective, kc. ; as Line of the Apsides ; of the Nodes; Horizontal, Hour, Equinoctial, &e. Lines; each of which will be found illus- trated under the respective articles. Linz also denotes a French measure of length, being the 12th part of an inch. LINEAR Equation, the same as SIMPLE Equation. Linear Number, is that which relates to length only, or to one dimension, as the num- POC ber which expresses the side or perimeter of a geometrical figure. ’ LineAR Problem, that which can be solved by means of right lines only. LIQUID, that which besides having the property of fluidity has also a peculiar quality of wetting other bodies immersed in it. LITERAL Equation, in Algebra, is that which is expressed in general terms by letters ; being thus distinguished from a numerical equation, in which all the coefficients are given numbers. LIZARD. See Lacerta. LOADSTONE. See MAGNET. LOCAL Problem, is used by some writers to denote a problem that admits of an indefi- nite number of solutions. See Locus. LOCI, the plural of Locus. LOCO Motion, the power of changing place. LOCUS Geometricus, denotes a line, by which a local problem is solved. ‘Thus let it be required to find a point P such, that if two lines be drawn from it to the two given points A and B, the included angle APB shall be equal to a given angle. P P A B Here it is obvious, thatifon A B asegment of a circle be described capable of containing the given angle, that any point P in the periphery of that segment will answer the required condition; and therefore in this case that segment is the locus geometricus. If the locus which the equation or problem requires be a right line, it is called a /ocus ad rectum; if a circle, locus ad cireulum; if a pa- rabola, locus ad parabolum; and the same for the ellipse, the hyperbola, &c. The first of these are also called plane loci, and the latter solid loci. Apollonius of Perge wrote two books on plane loci, in which the object was, to find the conditions under which a point, varying in its position, is yet limited to trace a right line, or 2 circle given in position. These books are Jost, but attempts have been made at restora- tions by Schooten, Fermat, and R. Simson; the treatise ‘‘ De Locis Planis,” of the latter geometer, published at Glasgow, 1749, is a very excellent performance, in all respects worthy ofits celebrated author. Besides the above-mentioned writers, the doctrine of loci has been treated of by various other mathe- maticians, as Craig, Maclaurin, Des Cartes, De lVHopital, &c.; the latter of whom has two chapters on this subject in his Conic Sec- tions. Leslie, in his Geometry, has also a chapter on plane loci, which contains several of the most simple propositions of this kind. We can only in this place give a single ex- ample, in order to illustrate the nature of the LOC : problems classed under the general term— plane loci; to which we shall add a few oft] most remarkable cases, leaving their inves) gation to the reader’s ingenuity. ‘; 1. Let ED, DF, be & two right lines, perpen- dicular to each other, and let the right line AB be supposed to slide between g¢ these lines, always touch- ing them both at its ex- tremities; then the locus of its centre C is the circumference of a given circle. D | Analysis. Join DC; then because the ba! of the triangle ADB is bisected in C, a cire! described from C as a centre, and with tl radius A C or C B, will pass through the poi D; for the angle ADB being a right angle, necessarily falls in the circumference of tl) semi-circle ADE; consequently AC, €/ and CD, are all equal to each other. B) AC being half of AB is given, therefore D is also given, whence the locus of the point, bisection C, is a circle described from D wi the radius DC. f Composition. From D, with a distance equ to half’ the given line, describe a circle; tl is the locus required. t For, draw the radius D C, make AC =D) and produce AC to B. Because AC =D) the angle ADC=DAC; but the ang} DAC and DBC, are together equal to a rig angle, and therefore equal to ADC and BDt whence the angle D B C is equal to the ang) BDC, and consequently the side D C is equ to BC. The segments AC, BC, are th) each of them equal to DC; and hence AB itself double DC, or is equal to the giv straight line. 2. If from two points there be drawn ty, straight lines, of whose squares the differen is given, the locus of their point of concour will be a right line given in position. © which is the same, if the base of a triang) and the difference of the squares of thet sides be given, the vertex of the triangle Ww) fall in a right line given in position. . 3. If the base and vertical angle ofa triang be given, the locus of its vertex will be 1) circumference of a given circle. Ra 4. If the difference of the sides, and t) radius of the inscribed circle of a triangle given, the locus of its vertex is a right li given in position. . 5. If two given unequal perpendiculars drawn toa right line given in position, a their opposite extremities be joined, the loc of the point of intersection will be a right li given in position. } 6. If in any triangle the base be given, a! the sum of the squares of the other two sidi the locus of the vertex is a given circum) rence. tS 7. If from given points there be dra\ straight lines, whose squares aré together eq to a given space, their point of concow LOC ‘ill terminate in the circumferenee of a given rele. 8. If triangles be inscribed in a given seg- ‘ent of a circle, and from the vertex on either de (produced if necessary) there be taken ther way, a right line always in a constant tio to either of the sides, or to their sum or fference, the loci of the points su described ill be circles. — ; With regard to the higher order of loci, it only a particular case of what we have con- ered more generally under the article curve; atis, you have here either the generation the curve given to find its equation and ‘ace its properties, or the equation being ven you are required thence to determine i reneration. 'We shall also give one problem under this ad, and a few of the most simple cases of e higher order of loci. ‘If on any given right line A B, there be cen any variable distance A L and from L, the same direction, any given invariable stance L.M; and if with the centres L and and radii LA, B M arcs be described, it is juired to determine the nature of the curve nich is the locus of P, the point ef inter- i “n eer eee ee A CL HMO B K eC AB—a LM=—b; BM = BP= 9, i having drawn PO perpendicular to AB, /-BO=«2z. ThenBL=9+6; LO=o@ b— 2x; LP=AL=a—b—9Q; and be- /ise LP? — LO* = BP*— BO’, we have in fabols (e —b— 0’ —(9+-b—2)' = 9 — 22; ence a*—2(a—2x)b = 97? 4+2(a—a2)O; i adding — b* + b* + (a—2)? to one side /lits equal (a —x)* to the other side, there (alts a — 6’ + fa—b—2xYu(a—x +9). Now take AC=LM=6, draw CD per- | dicular to A B, and make AD = AB =a; bn CD* =a — tb’; CO* = (a— b — x)’, i (OA+BPY= (a—x + )* whence we i¢ DO AO + PB; or PB= DO— AO. lence it will be easy to derive an alge- j.cal equation for the rectangular co-ordi- (2 of the curve; for we have only to put —y, to substitute “(x + y”) for @, and lear the equation of radicals. The equa- ( thus found will show the curve to be of fourth order; but the curve and its prin- i properties may be more readily deduced (1 the property above investigated; viz. »=DO—AO. The curve will consist of equal and similar parts, lying on different 's of A B it will be a sort of oval inclos- ithe point B on every side. ‘he following are some of the simplest cases 1¢ higher order of loci: | ; | | LOG 1. The base and sum of the sides of a plane triangle being given, the locus of its vertex is an ellipse, 2. 'I'he base and difference of the sides of a plane triangle being given, the locus is an hyperbola. 3. The locus of that point, from which, if perpendiculars be drawn to three right lines given in position, and such that the sum of their squares shall be equal to a given space, is an ellipse. And the same is true whatever be the num- ber of lines given in position. 4. Ifa triangle, given in species, have two of its angles upon a straight line given by position, and the side adjacenit to those angles passing through a given point, the locus of the angle opposite that side is an hyperbola. 5. Let A, B, be two given points in the right line A B given in position; and let C, D, be two given points without that line; also let CV, DV, be drawn meeting AB in F and G, so that the rectangle A F x BG isgiven: the locus of the point will in all cases be a conic section. 6. Let AB be a given straight line, and P a given point without it, let CP D be drawn meeting A Bin C; and let CP be to PD, as AC to CB; the locus of the point D is a given hyperbola. 7. When the base of a triangle is given, and one of the angles at the base double the other, the locus of the vertex is an hyperbola. 8. The locus of the angles of a parallelogram, formed by drawing tangents at the vertices of any two conjugate diameters of an ellipse, is also an ellipse concentric with the former. The above cases, and several other curious properties of this kind, the reader will find investigated in Leybourn’s “ Mathematical Repository :” the subject is also ably treated by Craig in his “Treatise on the Quadrature of Curves;” in De l’Hopital’s “Conic Sections ;” Maclaurin’s ‘“ Algebra,” and most of the other writers, quoted under the article Curve. LOG, an instrument used on shipboard for determining the rate of the vessel of which there are different constructions ; see Robert- son’s-and other Treatises on Navigation. LOGARITHMIC Curve, or Locistic Curve, is a curve having its abscisses in arithmetical progression, and its corresponding ordinates in geometrical progression, so that the ab- scisses may be considered as the logarithms of the respective ordinates, from which pro- perty the curve has derived its name. Np — D_ Let P, Q, R, S, &c. represent the logarith- mic curve, then if any absciss AB = x, its ordinate BQ —y, AP. = 1, dnda =a certain constant quantity, or the-mfodulus ef the lg- EE LOG garithms, then the equation of the curve is x =a X log. y = log. y*, the fluxion of which r - _ ay . . e gives x =-——; whence y: x :: ya, but.in y ’ the curve y: x:: 4: subtangent A'T, there- fore the subtangent of this curve is always equal to the same constant quantity a the modulus of the logarithms. To find the Area contained between any two ordinates of the Logarithmic Curve. ay The fluxion of the area A or yx, is y X = ay, which corrected gives A = a (AP—y) =4(AP— BQ) = 2.x:PV—AT x BY; that is, the area between any two ordinates is equal to the rectangle of the constant sub- tangent, and the difference of the ordinates ; and hence when the absciss is infinite, and consequently the last ordinate equal to zero, then the infinitely long area AP Z is equal to AT x AP, or double the triangle A PT. To find the Content of the Solid formed by the Revolution of the Curve about rts Axis AZ: The fluxion of the solid § = py? « =py’x 2 pe = payy, where p = 3'14159, &c.; and the correct fluentisS = ipa x (AP*—y’*) =ip x AT x (AP?— BQ”, which is half the difference between two cylinders of the common altitude a, or AT, and the radii of their basis AP, BQ. And hence, supposing the axis infinite towards Z, and consequently the ordinate at its extremity zero, the content of the infinitely long solid, will be equal to ipa xX AP? = ip x AT x AP’, or half the cylinder on the same base, and its altitude AT. This curve greatly facilitates the conception of logarithms, and affords a very obvious proof “of the very important property of their flux- ions, or very smail increments; namely, that the finxion of a number, is to the fluxion of its logarithm, as the number is to the sub- tangent. The logarithmic curve has been treated of by a number of very able mathematicians, as Huygens, who first proposed it, Le Seur, Keil, the Bernoullis, Halley, Emerson, &c.; see the latter author’s Treatise on Curve Lines, p. 19. Locaritumic or Logistic Spiral, is a curve having similar properties to the above, but dif- ferently constructed ; thus, divide the quadrant of a circle into any number of equal parts in the points A, B, D, &e. ; and from the radii CA, CB, CD, &c. cut off CA, Cb; Cd, &c. con- tinually proportional, then the curve passing through the points A, 6, d, &e. will be the logarithmic spiral. Hence the several areas are as the loga- rithms of the ordinates; and hence the deno- mination of the curve. pod. E LOGARITHMS, (from Acyop, ratio, and cesOuop, number), the ratio of numbers, are the. indices of the ratio of numbers to one another, or they are a series of numbers in arithmetical | progression, answering to another series of, numbers in geometrical progression; or, which conveys a still more simple and unembarrassed idea of these numbers, they are indices of the powers of a certain radix, which, when involved to the power denoted by the index, is equal to the given number; thus, if 7* =a, r¥ —b, r7 =e, then is x the logarithm of a, y the logarithm of 6, z the logarithm of e, Ke, where r is called the radix of the system, and may be assumed any number at pleasure, unity only excepted. These numbers are of the greatest possible use in almost all arithmetical and trigonome:) trical operations, because by help of them multiplication is performed by addition ; dig sion by subtraction; involution by multipl cation; and evolution by division, as wil appear immediately from the latter of the pre ceding definitions. For since 7 = a, 74 =a a and y being the logarithms of a and 6, we have immediately from the first principles © algebra, ‘es MX 9! Sage ry = ab multiplication ; —— Pee: SS PO ae ah 1 4 &} xt = an evolution. 4 Vrom which it is obvious that the logarithr, of the product ab, is equal to the sum of th logarithms of a and b; the logarithm of th quotient a divided by 5, is equal to the diffe ence of the logarithms of a and 6; the logs rithm of the nth power of a, is equal to times the logarithm of a; and the logarith of the uth root of a is equal to the nth part « the logarithm of a. - | Therefore universally to multiply two nu bers together, we must take the sum of the logarithms; to divide one number by any ther, we must subtract the logarithm of tl divisor from the logarithm of the dividen To involve a number to any power we m muluply the jogarithm of the number by tl index of the power. And to extract the ro of a number we divide the logarithm of number by the index of the power who root is to be extracted; but each of the rules require a little farther illustration, whi, may be seen in any table of logarithm; b before we proceed any farther let us attend the history of this brilliant discovery. hs These properties of the indices of numbe were taken notice of by Stifelius, and e by Archimedes in his work on the number of the sands; but it is to Baron Napier, Merchiston, in Scotland, that we are indeb for the happy idea of applying such numbe tu the purposes of arithmetical and trigot metrical calculation, which first appeared his. “ Mirisici Logarithmorum Canonis L scriptio,” published at Edinburgh in 16 This work was translated by Mr. Edw a «ee 3 division =r" a involution =rn a LOG Wright, and published by his son in 1616. ‘he method of constructing the table was eserved by the ingenious author till the sense if the learned, upon his invention, should be ‘nown ; nevertheless Kepler, in his “ Chilias uogarithmorum,” &c. published in 1624; ' Speidell, in his “ New Logarithms,” pub- ished in 1619; Ursinius, in his “Table of uogarithms,” 1625; and many other mathe- ‘gaticians constructed small tables conform- ‘bly to the plan of Lord Napier. But of all ‘hose who assisted in the construction of lo- ‘arithmic tables, Briggs is the most conspi- ‘uous; it was he who first suggested our pre- ent system, the advantages of which are iwalculably greater than those first construct- 'd by Napier, at the same time that he la- oured more than any one in the construc- on of them. _In the present state of analysis many com- aratively short methods may be employed ‘ this purpose, that were unknown to the uly writers, and for want of which the labour ‘tending the first computation was exceed- igly great, notwithstanding they had certain jeans of abridging the operation in particular ses; a minute and interesting account of Jhich, with an explanation of their several ‘odifications, is given by Dr. Hutton in the ‘troduction to his Mathematical Tables, to hich work the reader is referred for every formation on this subject. The publications relating to logarithms are » numerous, that we can only find room to ‘ention a small portion of them; but as it is seful to know which are reputed the best Athors, and particularly the best editions of ie same authors, we shall subjoin the follow- ‘g list, which may be considered as contain- g the most respectable and accurate works ‘this kind. 1. The first Canon of Logarithms for na- ‘ral Numbers, from 1 to 20000, and from 00 to 101000, was constructed and pub- shed in 1624, by Briggs, with the approbation the inventor, Lord N apier. (2. Briggs’ Logarithms, with their difference ‘10 places of figures; as also the logarithmic es, tangents, &c. by George Miller, London, 13). 18, “Trigonometria,” by Richard Norwood, ‘531, containing a table of logarithm, from 1 | 10000, besides sines, tangents, &c. '4.“Directorium, Generali Uranometricum,” Francis Bonaventure Cavalerius, Bologna, 32. This work, beside the usual table of Jarithms, contains several new and useful ‘bles of sines, versed Sines, &c. and some ‘her original matter. 5. In 1643 appeared the “Trigonometria,” ‘the same author, which may also be con- ‘lered an interesting work. 6. “Tabule Logarithmic,” by Nathaniel ‘we, London, 1633. In this work the lo- rithms are given to eight places of figures, ' every number from 1 to 100000, and lo- ‘uithmic sines, tangents, &c. to every 100th it of degrees to ten places. Loc 7. “ Trigonometria Britannica,” by John Newton, London, 1658. Here the logarithmic tables are put in the most convenient form, being nearly the same as is now adopted by authors of the present period. 8. Adrian Vlacq, also published different editions of Logarithmic Tables, which have been since republished; these are generally considered very accurate and useful tables, particularly the edition of 1631. 9. Sherwin’s Mathematical Tables, pub- lished in 8vo. London, 1704, form the most complete collection of any we have yet no- ticed ; containing, besides the logarithms of all numbers from 1 to 101000, the sines, tan- gents, secants, &c. versed sines, both natural and logarithmic, to every minute of the quad- rant. The first edition was printed in 1706, but the third, published in 1742, as revised by Gardiner, is considered as superior to any other. The fifth and last edition, published in 1717, is so incorrect that no. dependence can be placed upon it. The third edition above mentioned, which is called Gardiner’s Tables, was republished at Avignon in France, in 1770; but this is not considered so accurate as the original one by Gardiner himself. -10. An “ Antilogarithmic Canon,” for rea- dily finding the number corresponding to any logarithm, was begun by the algebraist Harriot, and completed by Warner, the editor of the former’s works; but it was never pub- lished for want of proper encouragement. But a complete canon of this kind was pub- lished by James Dodson, 1742, in which the numbers answering to each logarithm from 1 to 100000 are computed to 11 places of figures. 11. In 1783 was published, by M. Callet at Paris, a very neat and useful collection of Logarithmic Tables ; and in 1795 an enlarged edition of the same work, under the title of “Tables Portative de Logarithms.” This is an. elegant work, beautifully printed and sterio- typed at the celebrated Didot’s press; it is more correct than the former edition, though it contains a few errors not noticed in the list of errata. 12. Dr. Hutton’s Mathematical Tables, containing the common hyperbolic and lo- gistic logarithms ; also sines, tangents, secants, and versed sines, both natural and logarithmic ; together with several other tables useful in mathematical calculations. ._To which is pre- fixed, a history of the discoveries and writings of the most celebrated authors on this subject. This work was first published in 1785, since which time it has passed through five editions, and is much esteemed for its accuracy. 13. Taylor’s Tables of Logarithmic Sines and Tangents, to every second of the quad- rant; to which is prefixed, a Table of Loga- rithms from 1 to 100000. This isa very valuable work, and has a useful introduction, composed by the late astronomer. royal, Dr. Maskelyne. | 14. Vega’s Tables, published in Latin and EE2 LOG German, is also a very excellent performance, particularly the second edition. 15. Another very accurate and extensive collection of tables, computed for the decimal division of the circle, by Borda, and revised and augmented by Delambre, was published in Paris. This work is held in great esteem by the French; but it is of little use to Eng- lish mathematicians, on account of the par- ticular division of the circle; it is, however, preceded by a very perspicuous and scientific investigation of the most useful logarithmic series, and trigonometrical formule, and may therefore be read with interest by the general mathematician. Besides the authors above mentioned many others have treated on the subject of loga- rithms, among the principal of whom are Hal- ley, Leibnitz, Mercator, Cotes, Brook Taylor, Euler, Maclaurin, Wolfius, Keill, and Simp- son. Having given this sketch of the history and progress of the discovery of logarithms, let us before we quit this subject endeavour to ex- plain the way in which the more modern mathematicians have investigated the proper- ties of logarithms, and the series that have been invented for the more ready computa- tion of them. : We have before defined a logarithm to be the index of a certain number, called the radix, which being raised to the power de- noted by that index or logarithm, will produce the given number. If therefore +z — N, then z is the logarithm of N, and 7 is the radix of the system. Now, first in order to find an analytical expression for N in terms of z and 7; 7” must be con- verted into a series, for which purpose it may be put under the forma rod+er—})yr214+2¢40¢—))+ (r—1)* + &e. 142 ; (r—N—Er—-1)? + (r—)3 —Ke. t ’ x. tee Nh. wha. Seba KS ¢ ,2 aoe +250 1 —(r—1) Pet = TtAgw + A’x? + A"x3 + Ke. that is, by writing | A =(r—1)— i (r—1¥ +4. — D3 —& ee. A’ = (r—1)— (r—1)3 + &e. Al ec. where A, A’, A”, &c. are constant but un- known; but of which the law by which they are connected with each other may be thus obtained : . Let x be increased by any mdeterminate quantity, then r*+7=1+A(a+z)+A'(a+z)? + &e, A®—-D (x +2)"; that is, by simply writing 2 +z for 2 in the preceding series, or expanding the powers of « + z, and stopping at the first two terms, we have a (a—1) 1.2 ere = PAs eee) +A’ (x? +2xr z+&c.) oe A”) (23 43272 + &e.) + ACD (an 4+ na"—z + kc.) . + AM (art) 4 (n4+-1) arz + &e. LOG Aga, rtcr*t rei (l+Aa+ A’a? + A’2x? + &e.) x (L+Az+A’2*+A"%z3 + &e.) “| the actual multiplication of which gives yt? =—14+A(e+z)+ A’a* +A%x3 + come) Ate z+ A Aa?2 he Whence, by comparing the correspondin| terms in the two expansions, we have 2 2N'= Mor AU = A333 A'— ha A3 < us ” Raat 3 therefore A’ o- A 1.2.3 in the same wayA” = A# , 1.2.3.4 and generally A@-D = A 1.2.3&e.n A Ash And consequently 2 r7—N—1 $Act Soot 1.2.3 which is the analytical expression for aj number in terms. of the radix r, and log) rithm «2; but the reverse of this, by whiy the logarithm is expressed in terms of | number and radix is the formula, which inquiry. . This may be found as follows: In the preceding article we found r?s=N=—1+Arct Ar x + At ie eer Ale bby yf where A =(r—1)—3(7r—1) +4 1) and if now we make B=(N—1)—7(N—1)° + 7(N—D)? we shall have on the same principles 2 3 Ne=1+ Bet c+ z+ & But ; Nz rv? = 1 + Are pit ales e1,.g a3 23 + &e. . whence by comparing the coefficients of both series we have 2 eZ 2 3 »3 Min Tg ee, a , Ale ass B* g 10827 (23 STO ES ae each of which gives the same result, | eA Ax = B; whence we obtain immediately eB (N11) — TNS a a AW (r—)N)—i(r— 1) $3 (r — 1} which is the analytical expression, for | logarithm of any number N, in terms of itl and the radix of the system; that is, writ a instead of N, m4 @—D—} (a1 + Hai (F—1I)—4 71° +47 1) 1 log. Or, log. Laas 4= ta—idtio— pose =) 1+ 0D This, however, must only be considere# a simple algebraical method of expressil logarithm; for it does not always answer! LOG yurposes of calculation; thus if @ be any vumber greater than unity, it is obvious that he series in the numerator will either con- verge very slowly, or otherwise will diverge, nd the same with regard to the denominator, upposing r to be equal to 10, as it is in the ommon system; in fact, the terms of the eries are larger, the more remote they are rom the beginning, and consequently no aumber of them can exhibit either exactly, or early, the true sum. Let us therefore inves- igate the method of submitting these to cal- ulation; in order to which we will repeat ‘gain our last series, viz. og.-lta= ta—ia+iad—tat st &e. F—l)—f0—1)* + 40—1) — &e. nd here since the denominator is always a onstant quantity, when the radix of the ystem is given, we may make (M=(r—D— 4 (r—1)7 +4.r— D3 — &e. vhich renders the above expression still more imple, as in that case it becomes barely | | 1 M* | Whence again by subtraction ig. paces ; a—ta* + 4a3—tat + Ke. ; Or, taking a negative 2 ” 5 g¢.1—a— ) af — 403 hat — Be. REN? FW rosa: Ba ke, ¢ rg. Naa patie +30 +74 4+. on a—t1 Now a = baer “He if therefore we sub- pong + i i : . al. jitute in the foregoing expresssion — $1? in ‘ead of a, it becomes | ae fifeezaah (2) 4 es ua WN a tal see | Ebi » CS) tse$ hich series must necessarily converge be- ‘ase the denominator of each of the fractions ' greater than its denominator ; still, how- ver, when a is a number of any considerable jagnitude, the decrease in the terms will be Jrslow as to render the formula useless for th purposes of calculation. . The limits of this article will not, however, Amit of our entering into an investigation of ‘ie series suited to the several cases that may ise, but for the sake of reference it will be \seful to give some of the most useful formula \r this purpose, which are as follows; vz. I | §(@— 1-4-1 +4 5 etl > M. (a— 1)? — ke. } \ Ih 1 é a— 1 (¢ 4 4 - 1 42 Loge a = x }( a +8 F 3 | aa LOG 3. Log. a —2 x ; ()+ (SS) + \ (2-22) 40} san t(est)—eety as 5. Log. 3 (‘ 2) — &e.} a dts? (ar) (a7 bv? ee Pita. Mae a+b rh eyes a arb at (<< FAI Ge ( 1 61 1 7. Log. a = log. (a— os is! gee: iach hil Mm La bis 1 1 senna os 3 o nw VO peered as § U a 8. Log. a = log. (a YN+7* jens: ] J 2 (ack lie Pea ee 9. Log. a = log. (a — 2) + Gol x eam + r, M a—l bei 1 oy sap Hee sye eS To the above may be added the following, which will be found useful on many occa- sions, 1 ; 10. Log.a =a x J(a—« '\— 1(@ —a-?) +4 (a ~ a7) — &e. ; H. Log. (@ + 2) = log. a + = x ee: a Pirogngizt ay tyia* ? re Soren & iat Par aah 1 2 12. Log. (a-—z) = log. a — + x }2 +d 3 ~ thea ek ae er , + &e.} 2 ( 2 o. pace a4 — « — Ss 13. Log. (a + log. a M ae Os =) Se asaks taker) + &e$ 14, Log. a=, x Cy a— 1)—1 (7/ a—1)” + i(ya—1— ke. } These formule, which we have selected principally from Bonnycastle’s Trigonometry, might have been extended to a much greater length, but those that are given will be found to embrace the generality of cases; and will be found useful on various occasions. — i _ At present we have assumed the series which constitutes the denominator m our first expression or known quantity, which we have LOG represented by M.; it will however be proper, before we conclude this article, to offer a few remarks upon the absolute value of this series, according to any given radix. First then, since a—ta*+ $a3—iat*4+ &e. (r—1)—4 (r—1)* +4 (r— 1 8— Ke. . the denominator and numerator of this frac- tion are totally independent of each other, and therefore r may be assumed at pleasure, and the value of the whole denominator com- puted for any particular magnitude assigned to this letter. Or otherwise, the whole deno- minator may be taken equal to any quantity, and the value of r itself determined by com- putation, The latter method, at first sight, log. l+a— appears the most eligible, for by assuming the - whole denominator equal to unity it disap- pears entirely, and the expression becomes fog. 1 +a ma—ta®?+ 3e—tat+k&c.; there are however inconveniences attending this system, that do not appear upon a slight view of the subject, but which are, notwith- standing, very evident upon a farther investi- gation. Jn the case in which the whole de- nominator is assumed equal to unity, the value of x, the radix of this particular system is found to be 2°7182818284, &c., and the fraction = becomes — 1. These constitute what are called hyperbolic logarithms, and which are treated of under that article in the present work; we shall therefore enter no farther upon the subject in this place, than is necessary to show the defect of this system for general purposes, when compared with that now in common use, a defect which is by no means compensated by the trifling ad- vantage attending their computation. In the common system, the radix 7 is assumed equal to 10, the same as the radix of our scale of notation ; and hence arises a most important advantage, which is, that the logarithm of all numbers expressed by the same digits, whe- ther integers, decimals, or mixed of the two, have the same decimal part; the only altera- tion being in the index or characteristic of the logarithm. For the radix being 10; 0, 1, 2, 3, &c. will be logarithms of 1, 10, 107, &e. that is, 10° = 1; 107 = 10,:10* = 100, &c.; and therefore to multiply or divide a number by any power of 10, we have only to add or subtract the number expressing that power from the integral part of the logarithm, and the decimal part will still remain the same, by which means the tables of logarithms are much more contracted than they could be with any other radix, for in the hyperbolic system, or in any other which has not its radix the same as that of the scale of nota- tion, every particular number requires a par- - ticular logarithm, and this circumstance would either swell the tables to an unmanageable size, or if they were kept within the present limits, frequent computations would become necessary; so that in either way it is clear that the advantages of the present logarithms, LON much more than counterbalance the ext trouble in computing them. This in fact onl consists in multiplying the hyperbolic loga: rithm by a constant factor, viz. the reciproca of the foregoing constant denominator repre sented above by ar” ‘the value of which wher 1 2:30258509" Hence it is obvious that different systems o logarithms are connected together by constan multipliers, and by means of which a logarithn may always be converted from one scale t another. Thus the hyperbolic logarithm of ! number is converted into a common logarith by multiplying the former by 4342944, any the latter is transformed into the former b multiplying it by 2°30258509. On the analytical investigation of the pre perties of logarithms, see the introduction t Callet’s ‘“‘ Tables, Portative des Logarithms; Cagnoli’s “ Trigonometry ;” Bonnycastle’s an W oodhouse’s and other treatises on the sam subject. Brigg’s LOGARITHMS, are the logarithms i present use, being more frequently call common logarithms. Hyperbolic LoGarttumMs. See HyPersotl Logarithms. Imaginary LoGARITHM, is used to denol the logarithm of an imaginary or negatiy quantity; the introduction of which int analysis is frequently of considerable use, pa ticularly in finding the fluents of certai| fluxions, which sometimes come out the ly garithm of imaginary quantities, whence the| may be transformed into circular arcs, se tors, &c. See Bernoulli, Oper, tom. i.; Mi claurin’s Fluxions, art. 762; Cotes, Harmon| Mens. p.45; Walmsley, Anal. des Mes. p. 6 Euleyr’s Analysis Infinitorum, vol. i. p. 7 and particularly M. A. 8S. Missery’s Théor des Quantités Imaginaires, Paris, 8vo. 1801, Logistic LOGARITHMS, are certain logarithn of sexagesimal numbers or fractions used astronomical calculations. The logistic logarithm of any number | seconds is the difference between the cor mon logarithm of that number and the log! rithm of 3600, the number of seconds in) degree ; the use of these numbers are in col puting a proportional part in minutes al/ seconds, or hours and minutes, or other se| agesimal divisions. See Hutton’s Introdul tion to his Mathematical Tables, p. 144. LOGISTICA, or Logistical Arithmetic, | sometimes used to denote the operations pt formed on sexagesimal fractions. LONG (RoGeEr), an English astronomic professor, was born in 1679, and received }) college education at Cambridge; he becar master of Pembroke Hall, and Lownde professor of astronomy. He is chiefly knov as an author, by a treatise on astronomy, } two volumes ; the first of which was publish! in 1742, and the second in 1764. He wi : 3) also the inyentor of a curious astronomi! + — 10 is &e. = *43429448, &e LON machine, erected in Pembroke Hall; which we have already described under the article Giose. Dr: Long died in 1770, in the 81st year of his age. Lonc Measure. See MEASURE. LONGIMETRY, the measuring of lengths or distances, both accessible and inaccessible. Accessible distances, are measured by the application of some measure a certain number of times, as a foot, chain; &c. And inacces- sible distances are measured by taking angles, &c. by means of proper instruments, as the circumferentor, quadrant, theodolite, &c. "This embraces a great number of cases according to the situation of the object and observer, a variety of which are given in Hutton’s Men- »suration, and Bonnyeastle’s Trigonometry. ‘See also some cases solved under the article ALTITUDE. LONGITUDE, in Asironomy, is the an- gular distance of any star or celestial body from the vernal equinoctial point, that is, ifa great circle pass through a star perpendicular to the ecliptic, the arc of the ecliptic inter- cepted between the intersection of it with this circle, and the vernal equinoctial point will be the longitude of the star. : _ Loneirupe, in Geography and Navigation, | is the measure of the angle included between ‘the meridian of any place, the longitude of ‘which is required, and a certain fixed meri- dian from which the longitude is reckoned ; or it is the number of degrees, minutes, &c. intercepted between a certain fixed point of the equator, and the intersection of the me- ridian of the place with the same circle. N Thus let ENQS represent the terrestrial sphere, EQ the equator, and NS the poles, Nm, Nm’, Nm’, &c. meridians passing by the zeniths of the places G, H,I, K. Then if Gm be considered as the fixed meridian, the longitudes of the places H, I, K, will be measured by the arcs mm’, mm’, mm’, &C., that is, by the number of degrees of the equa- tor intercepted between the point m, and the intersections of the several meridians with | that circle. Hence it appears that all places situated i under the same meridian have the same lon- ’ gitude, as for example, all places falling under * the meridian Nm’; and therefore, when only } the longitude of a place is given, it may be ) ‘situated any where within that circle. Or if ’ only the latitude be given, the place may be | ‘situated any where in that parallel of latitude ; LON thus for example, all places falling on the circle pp will have the same degree of lati- tude, so that with neither of these separately can the situation of the place be determined ; but if both the latitude and longitude be given, then the exact situation becomes known. If, for instance, a place is known to be situated in the parallel of latitude pp, and at the same time, that it is also under the meridian Nm’, then it is obvious that the — exact position of that place is known, as it must necessarily be situated in the intersec- tion of these two circles, as at H. Now navigators can always find the lati- tude of the ship by observations on the sun, circumpolar stars, &c.; and it is obvious, therefore, how highly important it is for them to possess a similar easy method of finding their longitude, which would fix their position on the elobe to the greatest possible degree of accuracy. But unfortunately this is not the case, and the problem of finding the lon- gitude, though now much simplified, is still very embarrassing, and the result cannot ulti- mately be depended upon to the same degree of accuracy as in ascertaining the latitude. We cannot, in this article, enter into a de- tailed account of the several methods that have been proposed for this purpose; it will be sufficient to observe, that the only correct method is by ascertaining the difference in the time, between the place from which the longitude is reckoned, and that whose longi- tude is to be found; which difference of time converted into degrees, &c. will give the lon- gitude required. Since the earth performs one diurnal revo- lution in 24 hours, or, which is the same, since the sun makes one apparent revolution in that time, it is obvious that he passes over 15° every hour, and therefore, if one place is situated 15° to the westward of another, the sun will pass over the meridian of that place one hour later than at the former; and if it be 18° to the eastward, it will pass over it one hour sooner, and so on in proportion for any other difference in time, whence it is obvious that if the difference in time be given, the difference in longitude becomes known; or if the difference in longitude be given, the dif- ference in the time is also readily computed. Most nations assume for a fixed meridian that which passes over their metropolis, or some other remarkable place Thus the English reckon their longitude from the me- ridian of the observatory at Greenwich ; the French from the meridian of Paris; the Spaniards from that of Madrid, &c.; and as we before observed, the great object is to ascertain the difference in time between these fixed places and that whose longitude is sought, that is, the time which elapses be- tween the sun’s passing the meridians of the two places. Now if a time-piece could be so accurately constructed as neither to. gain nor lose, or if it either gained or lost, provided the rate of its deviation was constant, this would be ail LON. that is required for the complete solution of this problem. Yor supposing the watch to be set to the true Greenwich time, then when- ever it is 12 by the watch, the sun will be upon the meridian of Greenwich, therefore if the sun passes the meridian of any place when it is 11 o’clock by the watch, it shows the place to be 15° to the east, or that the place was in 15° east longitude; and if it is one o’clock by the watch, the place would then be in 15° west longitude, and so on for any other time. ‘This circumstance, and the re- wards that have been offered by different covernments, have induced several ingenious mechanics to direct their attention to the accurate construction of chronometers; and if they have not succeeded to the full extent of their wishes, they have carried the work- manship to an astonishing degree of perfec- tion, and have claimed and received. the rewards due ito their ingenuity and _ per- severance, Another method of finding the longitude is by astronomical observation, that is, by mea- suring the distance between the moon and certain fixed stars, which in consequence of the rapid motion of the moon, varies every instant; and therefore this distance being ac- curately ascertained by proper instruments, the exact corresponding time at Greenwich also becomes known by means of tables com- puted for that purpose, and published every year in the Nautical Almanack. Hence havy- ing the time at Greenwich, and the time at the ship, the difference in time between the two places, and consequently the difference in longitude becomes known. The great importance of this problem in navigation, has induced the government of several nations to offer very noble rewards to those who should by any means facilitate the necessary computations, or add to the ac- curacy of the methods at present made use of for this purpose. in the year 1598, the government of Spain offered a reward of 1000 crowns for the solution of this problem; and soon after the states of Holland offered ten thousand florins for the same. Encouraged by such offers, in 1635, M. John Morin, professor of mathema- tics at Paris, proposed to cardinal Richlieu a method of resolving it; and though the com- missioners who were appointed to examine this method, on account of the imperfect state of the lunar tables, judged it insufficient, Cardinal Mazarin, in 1645, procured for the author a pension of 2000 livres. In 1714, an act was passed in the British parliament, allowing £2000. towards making experiments; and also offermg a reward to the person who should discover the longitude at sea, proportioned to the degree of accuracy that might be attained by such discovery ; viz. a reward of £10,000. if it determines the longitude to one degree of a great circle, or sixty geographical miles; £15,000. if it de- termines the same to two-thirds of that dis- tance; with other regulations and encourage- LON : ments, 12 Ann. cap. xv.. See also Stat. 14,) Geo. IL. cap. xxxix.; and 26 Geo. II. cap, xxv, But by Stat. Geo. I11. all former acts. concerning the longitude at sea are repealed, except so much of them as relates to the appointment and authority of the commis- sioners, and such clauses as relate to the | publishing of nautical almanacks, and other useful tables; and it enacts, that any person who shall discover a method for finding the longitude by means of a time-keeper, the principles of which have not hitherto been made public, shall be entitled to the reward of £5000. if it shall enable a ship to keep her longitude, during a voyage of six months within sixty geographical miles, or one de- gree of a great circle; or to a reward of £10,000. if within thirty geographical miles, or half a degree of a great circle. But if the method shail be by means of improved solar | and lunar tables, the author of them shall be entitled to a reward of £5000. if they show the distance of the moon from the sun and stars within 15” of a degree, answering to about 7’ of longitude, after making an aliow- ance of half a degree for the errors of observa- tion; and after comparison with astronomical | observations for a period of eighteen and a half years, or during the period of the irregu- larities of the lunar motions. Or that in case any other method shall be proposed for finding the longitude at sea, besides these before mentioned, the author shall be entitled to £5000. if it shall determine the longitude within one degree of a great circle, or sixty geographical miles; to £7500. if within two- thirds of that distance; and to £10,000. if within half the said distance. John Werner of Nuremberg, appears to be the first who proposed the method of finding the longitude, by observing the distance be-. tween the moon and a star, in his annotations on the first book of Ptolemy’s Geography, : printed in 1514. This method was also re- commended by Gronce Finé of Briancon, in his -work -“ De Invenienda Longitudini,”’ printed in 1524, and by several other writers. However, before the time of our illustrious countryman Sir I. Newton, the lunar theory was very defective, and consequently but little progress could be made in navigation by means of lunar observations; indeed, the first great advance towards the perfection of this method, may be dated from the year 17865, when professor Mayer of Gottingen sent a manuscript copy of his Lunar Tables to the British admiralty, claiming at the same time some one of the rewards, promised by par- liament, which he might be thought to merit. | In these tables the arguments are investigated on the Newtonian principle of universal gra- vitation, and the maxima of the equations are deduced from his own observations, and those of Dr. Bradley and others. The above’ tables were delivered to Dr. Bradley to be examined; who having compared them with a great number of his own observations, was | fully convinced of their excellency; and in| LON sitters which he wrote to Mr. Cleveland, ecretary to the admiralty, he spoke in the aost positive terms of their accuracy and eneral utility. He observes in his second »tter, that from a comparison of above cleven undred observations, the difference did no shere amount to more than one minute; and ierefore, says the doctor, it may reasonably e concluded, that so far as it depends upon he lunar tables, the true longitude of a ship t sea may be found in all cases within halfa egree, and generally much nearer. _ Profes- ow Mayer died in the beginning of the year 762, and left a still more complete set of iar tables, which were soon after transmit- xd to the Board of Longitude by his widow, w which she received, in the first instance, ‘3000. Euler also received £300. in con- deration of Mayer having availed himself of ie lunar theory of this mathematician. These ibles, together with a set of Mayer’s solar ibles, were printed at London in 1770, nder the inspection of Dr. Maskelyne, from hich all the articles in the Nautical Alma- ack relating to the sun and moon were com- ited. An edition of the lunar tables, im- coved by Mason, from Dr. Bradley’s series ‘Junar observations, published in the Nau- cal Almanack in 1774, was printed at Lon- on in 1787, and reprinted at Paris, in the Connoissance des Temps,” for 1790. “These ew tables,” says Dr. Maskelyne, “ when mmpared with the above-mentioned series of servations, seems to give always the moon’s ngitude in the heavens correctly, within half degree.” Dr. Maskelyne very much de- rves the thanks of every navigator, and of e public, for his great industry and exer- ms, in bringing this method of ascertaining e longitude at sea into general practice. this “ British Mariner’s Guide,” printed at ondon in 1763, he very strongly recommends and gives new precepts for making the ne- ssary observations and computations. The les for computing the effects of refraction d parallax, which he had formerly com- unicated to the Royal Society, and which ere published in the “ British Mariner’s aide,’ were again communicated to that. ciety in 1764, and published in vol. liv. of \e¢ Phil. Trans. In 1765, he presented a ‘emorial to the commissioners of longitude, which he shows, that in his voyage to St. selena, and return thence, he made frequent |Paesation of the distance of the moon from )e€ sun and fixed stars, with Hadley’s qua- jant, from which, by the help of Mayer’s jaar tables, he computed the longitude of 'e ship from time to time, and from the near jreement of the observations, especially at jaking the land, when the ship’s common ickoning was very erroneous, he inferred, ‘a the error in the longitude, thus deter- imed, would never exceed onc degree; and | concludes his memorial by saying, “ I am {2refore authorised to conclude, that nothing i wanted to make this method. generally jacticable at sea, but a nautical ephemeris, LON an assistance which is much to be hoped from this board.” And after several examinations of captains and mates of East Indiamen, who had made use of the same method, the Board of Longitude came to the following resolu- tions: “That. it is the opinion of this board, upon evidence given of the utility of the late professor Mayer’s lunar tables, that it is pro- per the said tables should be printed, and that application should be made to parliament, for power to give asum, not exceeding £5000. to the widow of the said professor, as a reward for the said tables, part of which has been communicated to her since her husband’s decease; and also a power to give a reward to persons to compile a nautical ephemeris, and for authority to print the same when compiled, in order to make the said lunar tables of general utility.” These tables of Mayer were long held in the highest estimation, and are indeed still a very valuable collection; but the best set of lunar tables, at present, are those of Burg, published by Delambre, and since reprinted by professor Vince in the last volume of his Astronomy. ; . The above resolution of the Board of Lon- gitude was immediately put in execution, and accordingly, since the year 1766, a work en- titled the “ Nautical Almanack and Astrono- mical Ephemeris,” has been published an- nually, by order of the commissioners of longitude, and which was conducted under the inspection of Dr. Maskelyne, the late astronomer royal, till bis death, and is now placed under that of his successor I. Pond, Kisq. This is generally allowed to be the most accurate publication of the kind that has hitherto appeared. See Nautical ALmaA- NACK. At the same time with the first Nautical Almanack, a set of tables to facilitate the ecal- culations, was published by Dr. Maskelyne ; in which are given two. excellent methods, with their demonstrations, for clearing the moon’s distance from the effects of refraction and parallax ; the first by Mr. Lyons, and the other by Mr. Dunthorne; also two examples for ascertaining the longitude from observa- tion. About twelve years after, this edition being sold off, it became necessary to reprint _ it; previous to which it was new inodelled by Dr. Maskelyne, and the second edition ap- peared in 1781; with which, and the Nautical Almanack, every person ought to be provided, who wishes to ascertain a ship’s place from celestial observations. After this time, publications on the longi-. tude by observations, became very numerous, not only in England, but in all parts of the continent; almost every book on navigation and astronomy published in Britain, since the commencement of the Nautical Almanack, contains a few rules for this purpose. Be- side the works above mentioned we have many other writers on this subject, as Ro- bertson, Adams, Bishop, Dunn, Emerson, Heath, Margett, Nicholson, Vince, Kelly, LON Bonnycastle in his Trigonometry, &c.; but the most complete work on this subject, is Mackay’s “ Theory and Practice of finding the Longitude,” in which all the necessary rules are laid down with perspicuity, and a set of very useful tables, for this purpose, forms a second volume; but with regard to tables, those of Joseph De Mendoza Rios are by far the most extensive and accurate, particularly the second edition, published in 1809. This latter author has also a paper on this subject, in the Phil. Trans, for 1797. At present we have confined our remarks prin- cipally to English writers, but our neighbours on the continent have taken an equal interest in bringing to perfection this important nan- tical problem, and various ingenious tracts have been published on the subject in Hol- land, France, and Germany ; and other papers have been given from time to time, in the memoirs of the different foreign academies. Monnier, in a work entitled, ‘‘ Abregé des Pilotage,” published in. Paris in 1766, men- tions this method of finding the longitude. In the same year Pere Pezenas published his “¢ Astronomie des Marins et Avignon,” which, besides examples to the problems in the “ Astronomie Nautique,” of the celebrated Maupertuis, and other articles, contains va- rious methods of finding the longitude from lunar observations. In the year 1766, M. de Charnieres present- ed tothe Royal Academy of Sciences at Paris, his ““Memoire sur les Longitudes en Mer,” which he published the following year; and his treatise entitled ‘“‘Experimences sur les Longitudes,’’&c. appeared in 1768. In this tract he has shown the utility of his instru- ment, called the Megameter, in observing the distance between the moon and a star. This instrument is constructed on the same prin- ciple as the object glass micrometer, and is only applicable to the measuring of distances when they are less than 10 degrees. In 1772, the same author published his ‘Theorie et Practique des Longitudes in Mer ;” which also contains a particular detail of the above mentioned instrument, with plates; the me- thod of reducing the observations for the lati- tude, longitude, &c. The Abbé de Lacaille recommends the lunar method as the only practical one at sea. He was however sensible, from his own expe- rience, of the errors to which it was then liable. In his edition of Bouguer’s “ Navigation,” printed at Paris in 1781, he says, “La grande incertitude a laquelle nous avons dit qu’ etoit sujette la méthode d’employer les observa- tions de la Inne faites sur mer, ne doit pas decourager le marin, ni la Juirendre suspecte, puisque dans les voyages de long cours ou l’on a essayé beaucoup de vents contraires, et de long coups des vents, il arrive souvent, qu’ aux atterrages, on se trouve en erreur de sept ou huit degrees sur la longitude, estimee selon les regles du pilotage” This subject is also treated in a masterly manner in the sixth LON volume of Bezout’s “Cours des Mathemati ques,” Paris, 1781; and in Catlet’s edition o Gardiner’s “ Logarithmic Tables,” Paris 1783. , | | An excellent tract was also published a Paris in 1787, entitled “ Description ef Usagi du Circle de Reflexion,” par le Chevalier Borda; in which that instrument is. very par ticularly described, its use shown in nautica observations, and the manner of calculatin; these observations. | Lalande likewise treats very perspicuous) on this subject, both in his “ Exposition d Calcul,” and in his “ Astronomie.” In th former of these works he says, p. 168, ‘‘San une methode pour trouver les longitudes, navigation est toujours incertaine ; l’estimé laquelle on a recours, peut etre fautive de : or 500 lieues, apres quelques mois de navig¢ tion, et jeter ceux qui navigaent dans les plu grand dangers.” He then recommends th lunar method of finding the longitude at sea and shows the method of constructing an using Lacaille’s ‘“‘Chasses de Deduction,” fe clearing the apparent distance from the effec) of refraction and parallax, and the method ¢ deducing the ship’s longitude from observi tion. In his “Astronomy” he enumerates, i a particular manner, the advantages attendin| the lunar method; viz. that it does not nece sarily depend upon a single observatio| neither does it require an extreme precisi€) in taking the altitude of the moon, nor # height of the pole; it depends very little ¢ the moon’s declination, and requires only | clear horizon; and after mentioning the exae| ness which Dr. Maskelyne and Lacaille hi found it susceptible of, and the preferen| they had given to it, he concludes by sayin} “Tl ne nous reste donc, qu’a inviter les nay gateurs aen étudies les calculs, a en acqu rir Phabitude, et a rendre cette practiqi| aussi générale, qu’elle est utile pour la nay gation.” f This method of finding the longitude | sea, being then allowed on all hands to be f} preferable to any other at present known, aij improvement ta it, either by means of né tables, or in instruments for taking the di) tances, will always be considered of the 1 most importance ; and though much has Ma already done towards bringing it to perfecti¢ yet much more remains to be done; al which, it is hoped, may yet be effected perseverance, for which the honours and ¥ wards held out by the government will | doubt act as a powerful stimulus. z Degrees of Loneitupe. It is obvious, fre the figure in the preceding article, that t} length of a degree of longitude varies with t parallels of latitude, being every where as t) cosine of the latitude. The following table shows the length of, degree of longitude, corresponding to eve’ degree of latitude from the equator to t) pole, as expressed in English and geograph cal miles. - —_ LON LonGITUDE Stars, is a term frequently used 0 denote those fixed stars which have been ‘elected for the purpose of finding the longi- aude by lunar observations; these are as fol- OWS: 1. aw Arietes, a small star without the zo- liac, about 22° to the right hand of the Pleiades. 2. Aldebaran, in the Bull’s-eye, a large con- spicuous star, lying about half way between the Pleiades and the star which forms the western shoulder of Orion. 3. « Pegasi, a star about 44° to the right of x Arietes, being nearly in a line with this lat- ter star and the Pleiades. 4. Pollux, a little to the northward of Alde- baran, being the Jeft-hand one of two bright stars in the constellation Gemini. 5. Regulus, about 38° SE of Pollux, being the southernmost of four bright stars to the N.E. of Aidebaran. 6. Spica Virginis, a white sparkling star, about 54° S. BE. of Regulus. | 7%. Antares, lying to the right hand of Re- gulus, and about 45° from Spica Virginis. , 8. Formahault, lying about 45° to the south of a Pegasi. _ 9. « Aguila, a star about 47° to the west- ward of « Pegasi. LONGOMONTANUS (Curistian), a fearned astronomer, born in Denmark in 1562, LUC eihase. |ceot Miles. | Eo’ ies.’ ll Lathase, | Geog, Miles. | Engr Mites. |{ Lartade.| Geos, Mites. | Exe’ mitce. 0 | 6000 | 6920 || 30 | 51:96 | 59°93 || 60 | 30-00 | 24-60 1 | 5999 | 6919 | 31 | 51-43 | 59-32 || 61 | 29°90 | 33:55 2 | 6996 | 6916 || 32 | 50°88 | 5869 || 62. | 2817 | 32-49 3 | 5992 | 6911 || 33 | 6032 | 58:04 || 63 | 2724 | 31:42 4 | 59:85 | 69:03 || 34 | 4974 | 57:37 || 64 | 2630 | 30:34 5 | 5977 | 6894 || 35 | 49:15 | 5669 || 65 | 2536 | 29°25 6 | 59°67 | 6882 || 36 | 4854 | 55°98 || 66 | 24:40 | 2815 7 | 6955 | 6869 || 37 | 47:92 | 5527 || 67 | 23-45 | 27°04 8 | 5942 | 6853 || 38 | 4728 | 5453 || 68 | 2248 | 25°92 9 | 5926 | 6835 || 39 | 4663 | 53°78 || 69 | 21°51 | 24°80 10 | 59:09 | 6815 || 40 | 4596 | 53:10 | 70 | 2052 | 23-67 11 | 5889 | 6793 || 41 | 4528 | 5223 || 71 | 1953 | 2253 12 | 5869°| 6769 || 42 | 4459 | 51-43 || 72 | 1854 | 21:38 13 | 5846 | 67-43 || 43 | 43:88 | 5061 || 73 | 17:54 | 20-23 14 | 5822 | 6714 || 44 | 4316 | 4979 || 74 | 1654 | 19°07 15 | 5795 | 6684 || 45 | 4243 | 4893 || 75 | 1553 | 17°91 16 | 5767 | 6652 || 46 | 4168 | 4807 || 76 | 14:52 | 1674 17 | 5738 | 6618 || 47 | 4092 | 47:19 | 77 | 13:50 | 1557 18 | 5706 | 6581 || 48 | 4015 | 4630 | 78 | 12:48 | 1439 19 | 5673 | 6543 || 49 | 39:36 | 45:40 || 79 | 11:45 | 1320 2 | 5638 | 6527 || 50 | 3857 | 4448 || 80 | 10-42 | 12°02 21 | 5601 | 6460 |) 51 | 3776 | 4355 || 81 9:38 | 10:83 22 | 55:63 | 6406 || 52 | 3694 | 4260 |} 82 8:35 9°63 23 | 65:23 | 6370 || 53 | 3611 | 41-65 || 83 731 843 24 | 5481 | 63:22 || 54 | 3527 | 40-68 || 84 6:27 7:23 2 | 5438 | 62:72 || 55 | 3441 | 39°69 || 85 5:22 6:03 96 | 53:93 | 6220 || 56 | 33:55 | 38-70 || 86 4:18 4:83 27 | 53:46 | 61:66 || 57 | 3268 | 3769 || 87 3:14 3°62 98 | 5297 | 6110 || 58 | 31°79 | 3667 || 88 2:09 2°42 99 | 52:48 | 6052 | 59 | 30:90 | 35°64 || 89 1:05 1-24 at the village of Longomontum, whence he took his name, He was the son of a poor la- bouring man, and had innumerable difficulties to encounter in obtaining that knowledge, for which he is now so justly remembered and admired. He was assistant to Tycho Brahe at his Observatory, and afterwards Professor of Mathematics at Copenhagen, where he died in 1647, at eighty-five years of age. The principal work of this author, is his ‘ Astro- nomica Danica,’ first printed in 4to. 1621, and afterwards with additions and improvements, in folio, in 1640. LOUIS d’Or, a gold coin, value 19s. 11id., old coinage; and 18s. 94d. of the new. Louis @’Or, Double, of Malta, is in value 1l. 19s. 5d. LOXODROMIC Curve, or Spiral (from Aokoc, oblique, and dIgouos, cowrse), the path of a ship when her course is directed constantly towards the same point of the compass, there- by cutting all the meridians at the same angle. See Ruums Line. LOZENGE, a term used by mechanics and some old authors for rhombus and rhom- bord. LUCIDA Bright, an appellation used by way of distinction to several stars; as Luerda, Corona, Hydre, Lyre, Kc. LUCAS (pE BurGo). See Pacceo.t. LUCIFER, a name given to the planct a] , LUN Venus, when she appears in the morning be- fore sun-rise. LUMINARIES, a.term employed by way of eminence to denote the sun and moon. LUNA, the Moon. See Moon. LUNAR, any thing relating to the moon; thus we say, Lunar Cycle, Lunar Month, Lunar Year, &c. See Cycie, Montu, YEAR, &e. Lunar Distance, in Navigation, is a popular term used to indicate the problem of finding the distance of the moon from the sun or some fixed star, for the purpose of ascertaining the longitude. We have already, under the article Loner- TUDE, given a sketch of the history and pro- gress ‘of this interesting problem; and we pro- pose in the present place to point out in a general manner the nature of the operation, and to show in what way the longitude of a place is connected with the lunar distance. Now, in the first place, the moon, of all the celestial bodies, is that whose motion in the heavens is the most rapid; for as it makes its revolution in about a month, it necessarily passes through an arch of twelve degrees in twenty-four hours, or half a degree in an hour; and consequently its angular distance with any of the other celestial bodies is sensibly different at every instant, and the object of the present inquiry is to ascertain by obser- vation this distance, and thence to infer the apparent time of the observation at Green- wich: then finding also the apparent time at the ship, the difference will be the longitude in time between Greenwich and that place, Now the Nautical Almanac contains the true distance of the moon from the sun, and from certain fixed stars for every three ‘hours throughout the year; and: consequently the distance at any intermediate hour, minute, and second, may be determined by the com- mon method of proportional parts; and con- versely, the time may be found answering to any given distance. ‘ Wecannot in this place enter into all the details of this problem; we shall, therefore, limit our remarks merely to that part which relates to what is called Clearing the Lunar Distance; that is, clearing the observed dis- tance from the effects of refraction and paral- jax, and thence deducing the true distance, which is necessary in order to find the true time on which the final result depends. In this problem the apparent distance of the moon from the sun or a star, as found by ob- servation, is given, and also their apparent altitudes or zenith distances, to find their true distance as seen from the earth’s centre. Since the observed altitude of any celestial object is affected by refraction and parallax, the effects of which are always produced in a vertical direction, it is obvious that the ob- served distance of any two bodies will also be affected by the same causes. Withregard to the fixed stars the parallax vanishes, so that their places are changed by refraction only ; but in observations of the moon particularly, the effect of parallax is very sensible, on ac- LUN count of her proximity to the earth; for whid reason the distance of the moon and any cele tial object is, for the most part, considerabh different from the observed distance. It may also be remarked, that since th refraction of the sun, at the same altitude, T always greater than his parallax, his true plac will be lower than his apparent place; an because the moon’s parallax, at any giver altitude, is always greater than the refractioy at that altitude, her true place will be highe than her apparent place. Z s M This being premised, let ZM be the ob served zenith distance of the moon, and Za her true zenith distance ; Mm being the dif ference between the moon’s refraciion and he parallax in altitude. Also let ZS be the ob served zenith distance of the sun or star, an¢ Zs its true zenith distance : Ss being the dif ference between the sun’s refraction and Pe rallax, or the refraction of a star. Then since in the triangle ZS M, the thre sides ZS, ZM, SM, are given, the vertica angle SZ M may be found by the known casi of oblique-angled triangles. And because the triangle Zsm, the two sides Zs, Zm, and the included angle sZm, are also known, th true distance sm may also be found in a simi’ lar manner from the common rules of sphert- cal trigonometry. But this method, though direct and obvious requires three separate statings, or analogies for obtaining the true distance; it may, how, ever, be rendered more commodious in praé tice, by incorporating the analytical formula for finding the angle Z and the side sm, intt a single expression; which when converte into logarithms gives the following rule, neelll the altitudes instead of the zenith distances Rule. Take the difference of the appar altitudes of the moon and star, or moon ani sun, and half the difference of their altitudes also take half the sum and half the difference of the apparent distance and difference of the apparent altitudes. Then to the log. sines @ this half sum and half difference, add the log cosines of the true altitudes (as corrected ft semi-diameters, refraction, parallax, and | dip. by means of the tables calculated for these purposes), and the complements of the log. cosines of the apparent altitudes, and take half the sum. he From this half sum take the log. sines half the difference of the true altitudes, find the remainder amongst the log. tangent: which being found, take out the correspondit log. cosine ‘without taking out the arch, whi is unnecessary. Lastly, ‘subtract this log. CO- sine from the log. sine of half the cilerol of the true altitudes, increased by 10 in LUN Jex ; and the remainder will be the log. sine half the true difference. Thus, for example, let there be proposed ) following data, to find the true distances; Apparent dist. ) and ©........ 51° 28! 35” Apparent alt. })’s centre...... 12 30 Apparent alt. ©’s centre...... 24 48 True alt....... )’s centre...... 13 20 42 True alerts @’s centre.. ... 24 45 57 Apparent altitude of ©...... 24 48 Apparent altitude of )...... 12 30 Different apparent altitudes 12 18 True altitude of ©...... 53: 24 45 57 True altitude of ).......... 13 20 42 2)11 25 15 1 difference true altitude.... 5 42 377 | Apparent distance..........0++ 51 28 35 | Different apparent altitude.. 12 18 | 2)63 46 35 Pe staid... 2... OT) es SEIS 31 53 172 | 2)39 10 35. UE difference... eee 19 35 174 Then by the foregoing rule we have the Jowing computation: \Log. sine ....... 31°53’ 172” ... 9°7228488 . ‘Log. sine ....... 19 35 172 ... 9°5253759 Co. log. cos..... 12 30 ... 0'0104185 Log. COS.......+. 13 20 42. ... 9:9881119 Co. log. cos..... 24 48 ... 0°0420206 Log. COS... ...00 24 45 57 ... 99580990 2)39°2468743 | 19°6234371 ‘Log. sine........ 5 42 87 ... 89978159 Log. tan. of an arch,......0... 10°6256212 ‘Corresponding log. oosine..... 9°3625337 Log. sine........ 25 84 54% ... 9°6352822 51.9, 49 True distance f This is the direct method of determining 6 true distance, independently of any other bles than those of common logarithms, and gat are found in the Nautical Almanack ; it as this is the most laborious operation nnected with the longitude problem, various ver rules have been devised, which by the of certain tables render the operation ‘ach more simple and expeditious; but ina ork of this kind, we cannot properly enter ton the problem under that point of view, “consequence of our not having the neces- Ty tables to refer to. The most approved of 'ese rules may be seen by consulting Mac- omer Longitude; see also Maskelyne’s fariner’s Guide,” the Requisite Tables, and other works referred to in the preceding of this article, by Borda, Caille, Delambre, + Obertson, &e. &c. Ps LUN LUNATION, the time between one new moon and another; consisting of 29 days, 12 hours, 44minutes, 354 seconds. See Moon. LUNE, LunuLa, in Geometry, is the space included between the arcs of two unequal-cir- cles, forming a sort of crescent, or half-moon; the area. of which may in many cases be as accurately determined as that of any recti- linear figure. The lune was the first curvili- near space of which the quadrature was as- certained, and this is said to have been first effected by Hippocrates of Chios; though others say it was discovered by CGinopides of Chios. However this may be, the former geo- meter has generally had the honour of the discovery attributed to him, and the figure still bears his name, being commonly deno- minated the lune of Hippocrates; the con- struction of which is as follows: On the diameter © of a semicircle de- scribe a right-an- gled triangle, of which the angular point will necessa- rily fallin the cir- A Cc B cumference. Then on each of the sides AD, DB, describe a semicircle, and the two figures AGFD, DHEB, will be lunes; and the area of them will be equal to the area of the right- angled triangle ADB. For circles, and con- sequently semicircles, being to each other as the squares of their diameters; and since AB? = AD* + DB’, therefore the semicircle ADB=AGD + DHEB; from these equal spaces, taking away the common segments AFD and DEB, there remain the two lunes equal to the triangle ADB; and therefore if the two sides AD, DB, become equal, as in the following figure, the two lunes are each equal to half that triangle, and consequently the quadrature of them is determined, being each equal toa given rectilinear figure. This is what is properly called the lune of Hippo- crates, and it was the only one of which he could determine the area; for though he, in all cases, had the measure of the space of both together, yet it was only in the case of equa- lity that he could find the area of the single lune, though he could always determine a lune that should be equal to any given rectilinear space. D This, as we have observed above, was the first instance of the quadrature of a curvili- near space; that is, of its being shown equal to a rectilinear figure; for properly speaking, it is not absolutely a quadrature, as was that of Archimedes, when he demonstrated that every parabola was two-thirds of its circum- scribing rectangle. Hippocrates arriving at his result only step by step, by subtracting MAC equal quantities from equal spaces, and hence, finally, as by chance, coming to a case in which a curvilinear area is equal to a recti- linear one. _ This discovery of Hippocrates, it seems, in- spired him with great confidence of being able to find the measure of the circle itself; and the reasoning which has been attributed to him on this subject, though very erroneous, is still extremely plausible. Hippocrates sup- posed a semicircle A DEF, in which he drew the three chords, or radii, AD, DE, EB; and on each of these chords he described a semi- circle, and a fourth, as F, equal to them. Then the four semicircles AGD, DEH, EIB, and F, being each equal to a quarter of the semicircle A DEB, they are therefore toge- ther equal to it; and taking away from each the small segments AGD, DHE, EIB, we shall have on the one side the rectilinear figure A DEB, equal to the three lunes, together with the semicircle F. If therefore the area of the lunes be taken away from the rectilinear ADEB, there will remain the area of the semicircle F, equal to a given rectilinear space. This reasoning, however, though ingenious, is still very defective, in consequence of the M MMlacuin E, in a general sense, signifies any thing that is used to augment or to regu- late moving forces or powers; or it is any instrument employed to produce motion so as to save either time or force. The word is derived from unxavn, machine, invention, art ; and is therefore properly applied to any agent in which these are combined, whatever may be the strength or solidity of the materials of which it is composed. The term machine, however, is by common usage generally re- stricted to a certain class of agents, which seem to hold a middie place between the most simple organa, commonly called tools or in- struments, and the more complicated and powerful termed engines; this distinction, however, has not place in a scientific point of view ; all such compound agents being ge- nerally classed under the term machines, the simple parts of which they are compounded, being termed MECHANICAL Powers. MAC lunes employed in this case being differer from those of which Hippocrates had foun the quadrature; for that, as we have seen, - bounded by a quadrant of one circle and th half of another, whereas those in the aboy figure are bounded by a semicircie and th sixth part of another circle, which is very di ferent from the former; and the quadratu! of it, as difficult to obtain as that of the cire itself. Hor other curious properties of lune) see Ozanam’s ‘‘ Math. Recreations,” aad th papers of David Gregory, Caswell, and Walli in Phil. Trans. No.259; and for the soliditic of the solids generated from them, see BD Moivre’s paper, Phil. Trans. No. 265. LUPUS, the Wolf, a southern constellatio) See CONSTELLATION. ; LYNX, anorthern constellation. See Cor STELLATION. LYONS (Israkt), a reputable mathem: tician, and an excellent botanist, was born « Jewish parents in Cambridge, in 1739. H was author of several works, but the only on necessary to mention here is his “‘ Treatise 6 Fluxions,” published in 1758. He was appoin ed Astronomical Observer, in Captain Phipps voyage towards the North Pole in 1773; th duties of which office he discharged much t the satisfaction of the Board of Longitud by whom he was appointed. His death hay pened about two years after his return fro} this voyage. LYRA, the Harp, a northern constellatio; See CONSTELLATION. “ Pi Machines are again classed under differer denominations, according to the agents b| which they ave put in motion, the purpos¢ they are intended to effect, or the art in whie they are employed; as Electric, Hydrauli Pneumatic, Military, Architectural, &e. Ms CHINES. ee, | Montucla, at the conclusion of the 1 ie volume of his “ Histoire des Mathematiques, has given an extensive catalogue of differe: works, which have been written for the pu. pose of describing and exhibiting the mo! important and curious machines, both ancier and modern, of which we haye selected ‘th following, for the information of those wh may not possess that useful history. my 1. The first, and most interesting moder work of this description, is entitled “ Led Verse et artificiose machine del capitano Ago} tino Ramelli dal ponte della Tresia, &c. & composte in lingua Italiana et Francese; MAC igi, 1588, in folio, (in Germany) in 1620. s is a very scarce work, seldom to be met 1 but in choice libraries. . * Machine nove Fausti Verantii cum laratione, Latina, Italica, Hispanica, Gal- , et Germanica.” Venetiis, 1591, 1625, olio, with figures. . Recueil de plusieurs Machines militaires, pour la Guerre et récréations, par Frangois mpourel et Jean Appus, 1620, 4to. . Heinrich Zeizings, theatrum machina- 1, Leipsig, 1621. . . A century of Inventions, &c. by Edward rerset, Marquis of Worcester. London, 3, in 12mo, , Les dix livres d’architecture de Vitruve. translated into French by Claude Per- ‘It; 1673, folio. . Veterum mathematicorum, Athenzi, lodori, &c. 1693, folio. This learned and ous edition of the ancient Greek machi- ans, was begun by Thevenot, and finished La Hire; but it relates principally to mi- 'y engines. “ Theatrum machinarum universale,” by Jacob Leupold, Leipsic, seven volumes, ), 1724, 1727, 1774. This is the greatest . most complete work of this kind that r was published. The first volume is little than an introduction to the work; the mad and third volumes contain a descrip- _of hydraulic machines; the next two mes relate to machines for raising weigits, ‘theory of levelling, and other subjects ; | the sixth treats principaliy on machines nected with the construction of bridges ; seventh volume is entitled, ‘ Theatre ametico geometrique,” where the author ts of all instruments employed in these sciences. This work would have been sh more considerable if its author had lived ompilete the immense task he had under- in. . Ashort Account of the Methods made of in laying of the Foundation of the ‘ss of Westminster Bridge, by Charles ielye, 1739. t ). The Advancement of Arts, Manufac- as, and Commerce; or, a Description of idon, 1778, 1779, folio. hesides the above-mentioned works, many ‘ul particulars may be gathered from Strada, ‘son, Beroaldus, Bockles, Beyer, Lunpergh, 1 Zyl, Belidor’s ‘“ Architecture Hydrau- (e;” Desagulier’s ‘‘ Course of Experi- Philosophy ;” Emerson’s *‘ Mechanics.” |) Royal Academy of Sciences at Paris ealso given a collection of machines and tions approved of by them. This work, lished by M. Gallon, consists of 6 volumes uarto; contain engraved representations {he machines, with their description an- i ; e might have carried the enumeration of wks of this kind to a much greater length, | the above are the most interesting, and ‘ul Machines and Models, by A. M. Baily," MAC the reader who wishes for farther information on this subject may consult the work of Montuela above mentioned. But we ought not to omit to mention, in this place, the second volume of the “ Architecture Hy- draulique,” of Prony, and the second volume of Gregory’s “‘ Mechanics ;” the first of these relates principally to steam-engines; but the latter contains a description of the most useful modern machines for various purposes. Maximum Effect of Macuines, is the greatest effect which can be produced by them. In all machines that work with a uni- form motion, there is a certain velocity, and a certain load of resistance, that yields the greatest eifect, and which are therefore more advantageous than any other. It is obvious that a machine may be so heavily charged, that the motion resulting from the application of any given power will be but just sufficient to overcome it, and if any motion ensue it will be very trifling, and therefore the whole effect very small. And again, if the machine is very lightly loaded, it may give great velocity to the load; but from the smallness of its quantity the effect may still be very inconsiderable, conse- quently between these two loads there must be some intermediate one that will render the effect the greatest possible. And this is equally true in the application of animal strength as in machines, and both have been submitted to strict mathematical investigation, the for- mer being founded on numerous experiments and observations on the best method of apply- ing animal strength, and the measure of it when applied in different directions. We caunot enter into this investigation, as it would carry us beyond the due limits of this article, we must content ourselves therefore with merely stating the results, and must refer the reader who wishes to see the inves- tigations to Gregory’s ‘* Mechanics,” vol. i. ; Prony’s “ Architecture Hydraulique,” from art. 487 to 507; the last edition of Ferguson’s “* Mechanics,” by Brewster; and to Maclau- rin’s “* View of Newton’s Discoveries.” 1. The maximum effect of a machine is produced when the weight or resistance to be overcome is just 4 of that which the power when fully exerted is just able to balance, or of that resistance which is necessary to reduce the machine to rest; and the velocity of the part of the machine to which the power is applied should be one third of the greatest velocity of the power. 2. 'The moving power P and the resistance R being both given; if the machine be so constructed that the velocity of the point to which the power is applied be to the velocity of the point to which the resistance is applied, as9R : 4P, the machine will work to the greatest possible advantage. 3. This is equally true when applied to the strength of animals; that is, a man, horse, or other animal will do the greatest quantity of work, by continued labour, when his strength is opposed to a resistance equal to 4 of his MAC natural strength, and his velocity equal to 4 of his greatest velocity when not impeded. Now, according to the best observations, the force of a man at rest is on an average about 70lbs.; and his greatest velocity, when not impeded, is about 6 feet per second, taken at amedium. Hence the greatest effect will be produced when the resistance is equal to about 31ilbs. and his uniform motion 2 feet per second. ) The strength of a horse at a dead pull is generally estimated at about 420Ibs. and his greatest rate of walking 10 feet per second ; therefore the greatest effect is produced when the load = 1862ibs. and the velocity 49, or 33 feet per second. . 4, A machine driven by the impuise of a stream, produces the greatest effect when the wheel moves with one third of the velocity of the water. For more on this subject, besides the works mentioned in the preceding part of the article, the reader may consult Bossut’s Hydrod. tom. i. art. 390; Smeaton’s “ Experimental Inquiry,” &c.; Phil. ‘Trans. vol. li. p. 100, &e. and Euler, Acta Petro. tom viii. p. 230. MACLAURIN (Co Lin), a very celebrated mathematician, was born in Argyleshire, in February 1698. In his 12th year he is said to have made himself master of the first six books of Euclid in a few days without any assistance, having met with the book by ac- _ cident and studied as it were for his amuse- ment. In 1717 he was appointed professor of ma- thematics in the Marischall college, Aberdeen, and was soon after admitted a fellow of the Royal Society, of which he was a very useful member. ‘The principal works of this author are as follows, viz. 1. Geometria Organica, London, 172). 2. A Treatise on Fluxions, in 2 vols. 4to. Edinburgh, 1724. 3. A Treatise on Algebra; and an Account of Sir I. Newton’s Philoso- phical Discoveries. The two last works, how- ever, were not published till after our author’s death, which happened in June 1746, being then in his 49th year. Besides these works he published several papers in the Phil. Trans. He gained the prize of the Royal Academy of Sciences for 1724; and shared that for 1740 with the celebrated D. Bernoulli and Euler. MACULA, in Astronomy, dark spots ap- pearing on the luminous faces of the sun, moon, and even some of the planets. In which sense macule stand contradistinguished from faculze, which are luminous spots. The solar macule are dark spots of an irre~ cular changeable figure observed in the face of the sun, first taken notice of by Galileo, in 1610, soon after he had finished his telescope, and about the same time by Hanriot, and afterwards accurately observed by Scheiner, Hevelius, Mr. Flamstead, Cassini, Kirch, &c. Phil. Trans. vol. i. p. 274; vol. Ixiv. p. 1, 194. Many of these macule appear to consist of heterogeneous parts; of which the darker and more dense are called by Hevelius nuclei, MAG and are encompassed, as it were, with a pheres somewhat rarer, and less obscure; the figure both of the nuclei and entire x cule is variable. In 1644 Hevelius obser a small thin macule, which in two days 4 extended to ten times its bulk; appear withal much darker and with a larger nue and such sudden mutations are freq The nucleus, he observed, began to fail sibly before the spot disappeared; and | before it quite vanished it broke into fo which in two days again re-united. So macule have lasted two, three, ten, fifte twenty, thirty, but seldom forty days; thot Kirch observed one in 1681, which remait from April 26th to the 17th of July. spots move over the sun’s disc with a mot somewhat slower, nearer the limb than centre; that observed by Kirch was twe days visible on the sun’s disc, for fifteen di more it lay behind it, it being usual to rett to the limb whence they departed in twen seven, and sometimes in twenty-eight days Various hypotheses have been invented account for these spots, some consider them as dark clouds floating in the solar) mosphere, others as real excavations, wh have not the property of propagating light the other parts of this luminary, the depth some of them having been estimated at m than 4000 miles, and their orifices much) ceeding this in diameter; some of them | deed, from the angle they subtend, must 4 ceed in size the whole globe of the earth.” Dr. Herschel has offered some conjectu on this subject in the Phil. Trans. for If and 1801; in which he conjectures that) variations in the state of the weather, in ferent years, may arise from the greater: less number of macule of the solar dise. { other papers on this subject, Phil. Trans; vols. Ixiv. part i. and ii. and vol. Lxxii. part see also the articie Sun. oe MASTLIN (MicHAcL), a noted Gem astronomer of the 16th century, a great s porter of the Copernican system, and w. it is said, first converted Galileo to the sa opinion; he numbered amongst his py the celebrated Kepler, and was author several astronomical and philosophical wor Meestlin died in 1590. / i MaGeELtanic Clouds, the name given three permanent whitish appearances, rese bling the milky way, near the south pe being distant from it about 11 degrees. MAGIC Circle of Circles, is an invent} of Dr. Franklin’s, founded on the same pr ciples and possessing similar properties to1 magic square of squares, by the same auth This consists of eight concentric circles 2 eight radii, in the circumferences of va the natural numbers from 12 to 75 are posited, that the sum of the number in circumference, together with the central nu ber 12, is equal to 360; and the numbers each radius, including always the central n ber, also is equal to 360. Besides the a these circles possess several other curid bend MAG »roperties, which the reader will find ex- slained in Dr. Franklin’s Exp. and Obser. ito. 1769, as also in Ferguson’s Tables and fracts, 1771, p. 318. _ The position of the numbers in each radius s as below, beginning with the outward num- yer and proceeding thence towards the centre, ] . . vhich is always 12, and common to each vadius. st rad. 2d rad. 3d rad. 4th rad. sth rad. 6th rad. 7th rad, eth rad, 62 73 14 % 30 41 46 ~=° ‘57 aa 15 «©°720«6068°:«C56 C47 401 Tl #64 2 16 #39 32 55 48 17 22 «65 «(670 «649 «(54 «68338 (66 Sb. 18 37 .34 568 50 19 20 67 68 51 52 35 36 160 75 12 27 «+28 «+43 «#344 = «459 26 13 #74 Gl 58 45 42 29 ml2..12 12 12 12 12 2 2 _ Here the successive horizon lines will re- resent the several circumferences, and the ertical ones the radii, the sums in each being 160. | Macic Lantern, a well-known optical in- trument, by means of which small painted igures, on the wall of a dark room, are mag- nified to any size at pleasure. This machine is represented in the above igure; A BC Dis a common lantern, to which s added the tube bkle, bhe is the lens that hrows the light of the candle or lamp a, or he object de, and 1 is the lens which mag- ifies the image fg on the white wall. Now, tis obvious that if the tube bnklme be con- racted, and the glass kl be brought nearer he object de, the image fg will be enlarged ; md on the contrary, if the tube be protracted, image of the object will be diminished. juler, who left no part of science untouched, as proposed an improvement in the magic antern. See Noy. Com. Petro. vol. iii. p. 363. Maaic Square, is a square divided into ells, in which the natural numbers from 1 to he proposed square are so posited that the um of each row, whether taken horizontally, ertically, or diagonally, is equal to a certain iven number; thus in the annexed figure, vhich contains 9 cells, the sum of the numbers n each row is equal to 15. This is a subject which must be allowed 0 be more curions than useful, and probably MAG owes its origin and its name to the supersti- tion of early times. Pythagoras, and some of his followers, attributed great virtue to par- ticular numbers, and paid considerable atten- tion to these kind of problems; and some latter writers have pursued the same subject, not, however, always for the sake of the magic of it, but for the curious properties that are connected with it. In the latter class may be mentioned Mos- chopulus, a modern Greek, and Cornelius Agrippa; though these had each probably some view to the magical properties of these squares; but Bachet, Frenicle, Rallier, De Lahire, and Dr. Franklin, and some others that might be enumerated, investigated the same subject.with very different views, and produced many very curious results. De Lahire gives the following rule for filling up the cells in any square consisting of an odd number of units, viz. Place the least term 1, in the cell imme- — diately under the middle or central one, and the rest of the terms, in their natural order, in a descending diagonal direction, till they run off either at the bottom or on the side. When the number runs off at the bottom, carry it to the uppermost cell, which is not occu- pied, of the same column that it would have fallen in below, and then proceed, descending diagonally again as far as you can, or till the numbers either run off at bottom or side, or are interrupted by coming at a cell already filled. When any number runs off at the right hand side, then bring it to the farthest cell on the left hand of the same row or line it would have fallen in towards the right hand; and when the progress. diagonally is interrupted, by meciing with a cell already occupied by some other number, then descend diagonally to the left from this cell Hill an empty one is met with, where enter; and thence proceed as before. This rule, with reference to the above square, will be readily understood by the ingenious reader without farther explanation. Those, however, who are desirous of farther information may con- sult the Memoirs of the Acad. Scien. for 1706 and 1710; Saunderson’s “ Algebra,” vol. i. and Ozanam’s “ Recreations.” Maeic Square of Squares, is an extension given to oe square, by Dr. Franklin. Here F MAG a great square of 256 little squares, in which all the numbers from 1 to 16% or 256, are placed in 16 columns, which, taken either hori- zontally or vertically, possess several curious properties, but which we are obliged to omit detailing, for the introduction of more im- portant matter; the reader however will find an ample account of this square of squares in Franklin’s Exp. and Qbser. p. 350, edit. 4to. 1769; and in Ferguson’s Tables and Tracts, 1771, p. 318. : ‘MAGINI (JoHn ANTHONY), a noted as- trologer and astronomer of the 16th century, was born at Padua in the year 1536. He was the author of a number of different works, which, considering the state of the sciences at that period, were very honourable testi- monies of his talents; they are, however, now of little or no interest or utility, and we there- fore omit the detail of them in this place; they may be seen in Dr. Hutton’s Math. and Phil. Dict. ; article MAGINI.. MAGNET, Loadstone, or Leading Stone, is an iron ore, or ferruginous stone, found com- monly in iron mines, of various forms, sizes, and colours. It is endowed with the property of attracting iron, of pointing itself in a cer- tain direction, and of communicating the same property to iron and stcel bars. It is known by the works of Plato and Aristotle, that the ancients were acquainted with the atiractive and repulsive powers of the magnet; but it does not appear that they knew of its directive property, which is of all others the most useful and interesting. The powers of the magnet have excited the wonder of the ancients, and the admiration of the moderns, and within the last two cen- turies many singular and remarkable magne- tical properties have been discovered, some: of which will be found under the articles Compass, DippinG Needle, VARIATION, Ke. a few others of a more general description are mentioned in the subsequent part of this article. It has been observed above that a magnet has the power of communicating its pro- perties to an iron or steel bar, and such a bar then becomes a magnet itself, and such is generally to be understood by the term magnet in the observations that follow. The Poles of a Magnet, are those points which seem to possess the greatest power, or in which all the virtue appears to be concen- trated. The Magnetical Meridian, isa vertical circle in the heavens, which intersects the horizon, in the points to which the magnetical needle when at rest directs itself. The Axis of a Magnet, is a right line which passes from one pole to another, _ Lhe Equator of a Magnet, is a line perpen- dicular to the axis, and exactly between the poles. The distinguishing and characteristic pro- perties of a magnet, are as follows: 1. Its atiractive and repulsive power. 2. Its directive power. * M AG 3. Its dip or inclination to a or below the horizon. 4. Its power of communicating its owr properties to certain other bodies. aa Phenomena of the Magnet.—These are ve numerous, but some of the principal ones are as follows: % 1..A magnet, when freely supported eithe by a thread or in a light vessel on water, will place itself in a direction nearly coinciding with the poles of the earth. See Compass. 2. This direction of the needle is not the same in all parts of the world, nor in the samé place at different times. See VARIATION. 3. A needle, which is not magnetised, being exactly balanced, will, if touched by a magne so as to communicate that property to it have its equilibrium destroyed, one of its ex tremities dipping considerably below the ho rizontal plane. See Dippine Needle. | 4. The centres of action of a magnet are a a very small distance from its extremitie and the law of attraction from these centre; are reciprocally as the squares of the dis tances. re 5. When a magnetic needle is put out & its natural line of direction, the force wit! which it tends to regain that position varie as the sine of the angle subtended by it natural direction, and that in which it 7 placed. 6. In every magnet there are two poles of which the one points northwards, the othe southwards; and if the magnet be divider into any number of pieces, the two poles wil be found in each piece. The poles of a magne! may be found by holding a very fine shor needle over it, for where the poles are thi needle will stand upright, but no where else.| By the following method also, the situatio! of the poles, and the direction of the (supposed megnetic effluvia in passing out of the ston may be exhibited to the sight: Let there be strewed about the poles q the stone, on every side, some iron or sted filings on a sheet of white paper; these smah particles will be affected by the effluvia of th! stone, and so disposed as to show the cours) and direction of the magnetic particles i every part. ‘Thus, in the middle of each pole it appears to go nearly straight on; toward the sides it proceeds in lines more and mor, curved, till at last the curve-lines, from bot poles exactly mecting and coinciding, fort numberless curves on each side, nearly ¢ a circular figure. ) A small artificial magnet may be used i this experiment instead of the real magne! with a similar effect. If the table on whie| the paper rests receives a few gentle knock so as to shake the filings a little, otherwis the action of the magnet will not be sufficier to dispose properly those particles which li at a considerable distance. 7. These poles, in different parts of th globe, are differently inclined towards a poi under the horizon. ‘ 8. These poles, though contrary to eae! point above MAG rer, help mutually towards the magnet’s raction, and suspension of iron. 9. If two magnets be spherical, one will en or conform itself to the other, as either them would do to the earth; and after they ye so conformed or turned themselves, they ve a tendency to approach or join each aer; but if placed in a contrary position, ey repulse each other. 10. If a magnet be cut through the axis, e segments or parts of the stone, which be- “e were joined, will now repel each other. 11. If the magnet be cut perpendicular to axis, the two points which were before con- ned, will become contrary poles; one in 2 one, and one in the other segment. 12. [ron receives virtue from the magnet application to it, or barely from an ap- oach towards it, though it do not touch it; d the iron receives its virtue variously, cording to the parts of the stone it is made touch, or even approach to. 13. If an oblong piece of iron be any way plicd to the stone, it receives virtue from mly lengthways. 14. A needle touched by a magnet will nits end the same way, towards the poles the world, as the magnet itself does. 15. Neither the magnet, nor needles touched fit, conform their poles exactly to those of |; world, but have usually some variation hm them; and this variation is different in ferent places, and at divers times in the ine places. a6. A magnet will take up much more iron, wen armed or capped, than it can alone. 17, A strong magnet at the least distance fm a smaller or a weaker, cannot draw to it piece of iron adhering actually to such a saller or weaker stone; but if it touch it xan draw it from the other; but a weaker gnet, or even a small piece of iron, can ‘ww away or separate a piece of iron con- uous to a larger or stronger magnet. \{8. In north latitudes the south pole of a }enet will raise wp more iron than its north ! 1é, 19. A plate of iron only, but no other body erposed, can impede the operation of the net, either as to its attractive or directive ality. 20. The power, or virtue of a magnet, may ‘impaired by lying long in a wrong position, also by rust, wet, &c. and may be quite stroyed by fire, lightning, &c. 21. A wire being touched from end to end ih one pole of a magnet, the end at which ‘1 begin will always turn contrary to the c that touched it: and if it be again touch- ‘the same way with the other pole of the gnet, it will then be turned the contrary way. 22. If a piece of wire be touched in the iddle with only the pole of the magnet, with- ‘ moving it backwards or forwards; in that ce will be the pole of the wire, and the two tis will be the other pole. ; (3B. If a magnet be heated red hot, and fain cooled, either with its south pole to- : | MAG wards the north, in a horizontal position, or with its south pole downwards in a perpen- dicular position, its poles will be changed. 24. Hard iron tools well tempered, when heated by a brisk attrition, as filing, turn- ing, &e. will attract thin filings, or chips of iron, steel, &c. and hence we observe that files, punches, augres, &c. have a small degree of magnetic virtue. 25. The iron bars of windows, &c. which have stood a long time in an erect position, grow permanently magnetical; the lower ends of such bars being the north pole, and the upper end the south pole. 26. Tongs and fireforks, by being often heated, and set to cool again, in a posture nearly erect, have gained this magnetic pro- perty. Sometimes iron bars, by long standing in a perpendicular position, have acquired the magnetic virtue in a surprising degree. A bar about ten feet long and three inches thick, supporting the summer beam of a room, was able to turn the needle at eight or ten feet distance, and exceeded a loadstone of three pounds and a half weight; from the middle point upwards it was a north pole, and down- wards a south pole. 27. A needle well touched, it is known, will point nearly north and south; but if it have one contrary touch of the same stone, it will be deprived of its faculty; and by another such touch it will have its poles in- terchanged. 28. A magnet acts with equal force in vacuo as in the open air. The smallest magnets have usually the greatest power in proportion to their bulk. the Phil. Trans. for 1751, which is as follows: — Procure a dozen bars, 6 of soft steel and 6 of hard; the former to be each 3 inches EE? . MAG long, 4 of aninch broad, and the 20th of an inch thick, with two pieces of iron, each half the length of one of the bars, but of the same breadth and thickness, and the six hard bars to be each 53 inches long, | an inch broad, and ¢sths of an inch thick, with two pieces of iron half the length, but the whole breadth and thickness of one of the hard bars; and let all the bars be marked with a line quite around them at one end. Then take up an iron poker and tongs, or two bars of iron, the larger they are, and the longer they have been used, the better; and fixing the poker upright betw een the knees, hold to it, near the top, one of the soft bars, having its marked end downwards by a piece of sewing silk, which must be pulled tight by the left hand that the bar may not slide; then grasp- ing the tongs with the right hand a little below the middie, and holding them nearly in a vertical position, let the bar be stroked by the lower end from the bottom to the top, about ten times on each side, which will give it a magnetic power sufficient to lift a small key at the marked end, which end, if the bar were suspended on a point, would turn to- wards the north, and is therefore called the north pole; and the unmarked end is for the same reason called the south pole: four of the soft bars being impregnated after this manner, lay two of them parallel to each other, at £ of an inch distance, between the two pieces of iron holding them, a north and south pole against each piece of iron: then take two of the four bars already made magnetical and place them together so as to make a double bar in thickness, the north pole of one even with the south pole of the other; and the re- maining two being put to these, one on each side, so as to have two north and two south poles together, separate the north from the south poles at one end by a large pin, and place them perpendicularly with that end downwards on the middle of one of the pa- rallel bars, the two north poles towards its south end, and the two south poles towards its north end; slide them three or four times backward and forward the whole length of the bar; then removing them from the middle of this bar, place them on the middle of the other bar as before directed, and go over thatin the same manner; then turn both the bars the other side upwards, and repeat the former operation; this being done, take the two from between the pieces of iron, and placing the two outermost of the touching bars in their stead, let the other two be the outermost of the four to touch these with; and this process being repeated till each pair of bars have been touched ‘three or four times over, which will give them a considerable magnetic power. Put the half dozen together after the manner of the four, and touch them with two pair of the hard bars placed between their irons, at the distance of about half an inch from each other; then lay the soft bars aside, and with the four hard ones let the other two be im- pregnated ; holding the touching bars apart MAG at the lower end near two-tenths of an inef to which distance let them be sepatfated aft they are set on the parallel bar, and. brougl together again before they are taken off: th being observed, proceed according to the m thod described ‘above, till each pair has Dee touched two or three times over; but as’ th vertical way of touching a bar will not sive quite so much of the ‘magnetic virtue as will receive, let each pair be now touch once or twice ove r, in their parallel positio between the sh a with two of sen bars le same time the ebrti cad of one from middle over the south end, and the south the other from the middle over the north ef of a parallel bar; then bringing them to fi middle again without touching the parall bar, give “three or four of these horizont strokes to each side. The horizontal tout after the vertical, will make the bars as stro as they possibly can be made, as appears] their not receiving any additional stren when the vertical touch is given by a number of bars, and the horizontal by th of a superior magnetic power. e This whole process may be gone through about half an hour; and each of the lar bars, if well hardened, may be made to] twenty-eight Troy ounces, and someti ; more. And when these bars _ are thus 1 ir same size ‘its fall rina in le SS ihe ten n nutes ; and therefore will answer all the p poses of magnetism in navigation and € perimental philosophy, much better thant Joadstone, ‘which has not a power sufficie to impregnate hard bars. The half ‘a being put into a case in such a manner that no two poles of the same name may together, and their irons with them as bar, they will retain the virtues they recei { but if their power should, by making expe . ments, be ever so far impaired, it may be! stored. without any foreign assistance i few minutes. And if perchance a mu larger set of bars should be required, t will communicate to them a suflicient pow to proceed with; and they may, in as time, by the same method, be brought to “ full strength. im Other methods of making artificial magné may be seen in the Phil. Trans. No. 414; 4 8, for 1744; art. 3, for 1745; and art. 2, f 1747; as also in vol. xv. and Ixix. See a Cavallo’s Nat. Phil. vol. iv.; Haiiy’s Ni Phil.; and Michell’s Treatise on Artifiel Magnets. e | MAGNE TISM, that quality or constitut of a body and its pores whereby it is render magnetical. RS MAGNETISM of the Earth, is that proper of the terrestrial globe, from which the me netism of the ordinar y magnets, the directi of the magnetic needle, and other pheno a are derived; and upon which they necessar depend. This is obvious, since almost” ihe phenomena, which may be exhibited wi fe) MAG Usual magnet, may be also exhibited with ie earth, as far as it can be tried. The phenomena of the compass and the pping needle in different parts of the world, ad the magnetism naturally acquired by soft on, when properly situated, are exactly imi- ted with a common magnet, or a terreila or obular magnet; the only deviation being iat which relates to the magnetic attraction ‘iron, which if it exists, is not perceptible in ve earth. If the earth attracted iron in con- ‘quence of its magnetism, the action of gra- ty still having its independent effect, iron onld be more accelerated in its fall than 1y other substance, and the weight of it ould be greater as it approached towards ie magnetic poles; but neither of these cir- mstances has yet been shown to have place, id we are therefore ignorant at present whe- ter or not the earth differs from the common agnet, even in this particular, for if such jomally did attain with regard to iron bodies, is reasonable to suppose that if must exist | a very small degree, which it would be ex- emely difficult to detect by any experiment iat it is within our power to perform. )Laws of MAGNETISM, are certain results, hich, from numerous and accurate experi- ents, are found to obtain in all natural and tificial magnets ; the principal of which are efly stated under Phenomena of the Mac- ET. MAGNIFYING, is used to denote the ap- arent enlargement of an object, by means of convex lens, or some other optical instru- vent, particularly the microscope. See LENs, Ticroscore, TELESCOPE, &c. )Maeniryinc Glass, is a popular term for ay convex glass or lens which has the pro- arty of magnifying | MAGNITUDE, is used to denote the ex- msion of any thing, whether it be in one direc- jon, as a line; in two directions, as a surtace; in three directions, which constitute a body + solid. | Geometrical MAGNITUDES, may be conceived »be generated by motion, as a line by the iotion of a point, a surface by the motion of ine, and a solid by the motion of a surface. Apparent MaGNitUDE of a body, is that hich is measured by the angle which that ody subtends at the eye; at least this is that is always to be understood by this ex- ression in the science of optics, though in tality the apparent magnitude depends not aly on the visual angle, but also upon the sup- feed distance of the ebject we are viewing. The mind jadges of the magnitude of dis- ant objects on two principles ; viz. with re- ence to the optic angle, and the distance f the object from the eye, the latter arising ut of our experience, which shows that dis- ice diminishes that angle; and therefore, ‘fithout being aware of the deduction, we ways make a certain compensation agree- bly to the supposed distance. If we seea ian, or any other known object, at a distance, conveys to our mind the idea of a certain lagnitude, which we attribute to it entirely —_*- M AI independent of the angle which it subtends; but if the object is unknown, then both the distance and the angle are considered in form- ing an idea of its magnitude; it is thus that we sometimes deceive ourselves with regard to the size of an object if we are mistaken as to its distance ; thus a small bird on the branch of a tree may appear to be a larger bird at a greater distance, or the contrary; thus alsoa fly imperfectly seen in the comer of a window, may have the appearance of a crow flying in the open air; these are illusions which com- monly happen, and which most persons will recollect to have experienced at one time or other. It is on this principle that some writers have accounted for the different apparent magnitudes of the sun and moon, and the apparent distance of two or more known stars, when seen near the horizon; and when they have a greater angle of elevation. When we see the moon, for example, at a considerable elevation, there being no intervening objects between that luminary and the eye wherewith to make a comparison of its distance, we in- tuitively suppose it nearer than when we ob- - serve it in the horizon; because there are then numerous objects, many of them at great distances, and the moon being evidently be- yond them all, we thence suppose its distance greater than in the former case, and the optic angle being still nearly the same, we attribute to it a greater magnitude, in the horizon than in the zenith, because we suppose its distance then to be the greatest: hence also the apparent figure of the heavens, which instead of having the form of a perfect concave hemisphere, the eye being in the centre, it has always the appearance of being considerably flattened in the upper part, or being a segment consi- derably less than a hemisphere. This how- ever is a subject that has much engaged the attention of astronomers and philosophers, from the earliest'period to the present time ; and various hypotheses have been advanced to account for it, none of which are perhaps perfectly satisfactory. See Smith’s Optics, vol. i. p. 63; James Gregory, Geom. Par. Univs. p. 141; Biot, Astronomié Physique, vol. 1; Reobin’s Tracts, vol. it.; and Phil. Trans. vol. 1. and ti. old abridgment. MAIGNAN (EMANUEL), a celebrated phi- losopher, was born at Thoulouse in 1601, and very early discovered a superior genius for mathematical and philosophical subjects, and could solve with ease and accuracy a variety of geometrical problems, before he had ever seen Euclid’s or any other work on that sci- ence. In 1648 Maignan published at Rome his “‘ De Perspectiva Horaria;” and in 1652, he published his Course of Philosophy, in 4 vols. 4to. When this work appeared, the adherents to the Aristotelian system insisted that it was impossible to reconcile the author’s opinion with the truths of religion; which objection Maignan undertook to refute, in a work entitled ‘ Philosophia Sacra,” the Ist volume of which appeared in 1662, and the 2d in 1672; which was followed by a “ Dis. MAN sertation De usu licito Pecunix ;” Maignan died in 1676, in the 75th year of his age. MALLEABILITY, the property ofa solid that is hard and ductile, and which may there- fore be beaten, forged, and extended under the hammer without breaking; as is the case with all metals not excepting quicksilver, but of these gold possesses this property in the highest degree. See Ductitiry. MANFREDI (Eustacuio), a celebrated astronomer and mathematician, born at Bo- logna in 1674, His genius was always above his age. He was a tolerable poet, and wrote ingenious verses while he was but a child. And while very young he formed in his father’s house an academy of youth of his own age, who became the Academy of Sciences, or the Institute, of that city. He became professor of mathematics at Bologna in 1698, and su- perintendant of the waters there in 1704. The same year he was placed at the head of the college of Montalte, founded at Bologna for young men intended for the church. In 1711, he obtamed the office of astronomer to the Institute of Bologna. He became mem- ber of the Academy of Sciences of Paris in 1726, and of the Royal Society of London in 1729; and died the 15th Feb. 1739. His works are: 1. Ephemerides Motuum Ceoles- tium ab anno 1715, ad annum 1750; 4 vols. 4to. ‘The first volume is an excellent intro- duction to astronomy; and the other three contain numerous calculations. 2. De Tran- situ Mercurii per Solem, anno 1723; Bologna, 1724, 4to. 3. De Annuis Inerrantium Stel- Jarum Aberrationibus; Bologna, 1729, 4to. Besides a number of papers in the Memoirs of the Academy of Sciences, and in other places. (Hutton’s Dictionary.) MANLIUS (Macus), a Latin astronomical poet who lived in the reign of Augustus Cesar, and wrote an ingenious poem on the stars, entitled “ Astronomicum,” the best edi- tions of which are those of Joseph Scaliger, 4to. 1600; Bentley’s, 4to. 1788, and that of Burton, 8vo. 1783. MANOMETER (from paves, rare, and Mitpoy, measure), an instrument, intended to measure the rarefaction and condensation of elastic fluids in confined circumstances, whe- ther occasioned by variation of temperature or by actual destruction, or generation of por- tions of elastic fluid. It is sometimes called manoscope. Mr. Boyle’s statical barometer was an instru- ment of this kind, it consisted of a bubble of thin glass, hermetically sealed, about the size of an orange, which being counterpoised when the air was in a mean state of density, by means of a nice pair of scales, sunk when the atmosphere became Jighter, and rose as it grew heavier. This instrument would evidently indicate the changes of density of the atmo- sphere, but it leaves us uncertain as to the cause, whether it is from the change of its own weight, or of its temperature, or of both. The manometer constructed by Mr. Rams- den, and used by Captain Phipps, in his voy- MAP age to the north pole, was composed of a tab of a small bore, with a ball at the end; ¥ barometer being at 29.7, a small quantity 0 quicksilver was put into the tube, to take of the communication between the external and that confined in the ball, and the parte the tube below this quicksilver. A seale ij placed on the side of the tube, which mar the degrees of dilatation arising from the im crease of heat in this state of the weight 0 the air, and has the same graduation as tha’ of Fahrenheit’s thermometer, the point o freezing being marked 32. In this state therefore, it will show the degrees of heat i the same manner as a thermometer. Buti the air becomes lighter, the bubble encloser in the ball, being less compressed, will dila itself, and take up a space as much larger aj the compressing force is less; therefore the changes arising from the increase of heat wil be proportionably larger, and the instrumen will show the differences in the density of th air arising from the changes in its weight an¢ heat. Mr. Ramsden found, that a heat equa to that of boiling water, increased the magni tude of the air from what it was at the freez ing point 4°4 of the whole. Hence it follow that the ball and part of the tube below th beginning of the scale, is of a magnitude equa to almost 414 degrees of the scale. If th height of both the manometer and therm meter be given, the height of the baromete may be thence deduced by this rule; as the height of the manometer increased by 414, i to the height of the thermometer increased b 414, so is 29.7 to the height of the barometer MAP, a plane figure representing the sur face of the earth, or a part of it, according te the laws of perspective. Or a map is a pro jection of the globe, or of any part of it, on¢ plane surface, representing the forms and di mensions of the several countries and rivers with the situation of mountains, cities, an¢ other remarkable places. 18 Universal Maps, are those which exhibit the whole surface of the earth, or the two hemi pheres. r Particular Maps, are those which exhibi some particular region, or country. ; Each of these kinds of maps are calle¢ geographical or land maps, in contradistine tion to hydrographical or sea maps, represent: ing only the sea or sea coasts, and properly called CHARTs, which see. In the construction of maps particular at tention must be paid to the following circum: stances: viz. 1. That all the places have theil just situation with regard to the principa’ circles of the earth. 2. That the magnitude: and forms of the several countries have the same proportion as on the surface of the earth 3. That the several places have the same dis tance and situation with regard to each othe! as on the earth itself. | The exact attainment of these objects de pends upon a strict adherence to the laws 0’ PERSPECTIVE and PROJECTION, for the par ticulars of which see those two articles; an for an abstract of La Croix’s paper on th » ‘major being 1. MAR tojection of maps, see the Introduction to Pinkerton’s Geography. he first maps are said to have been con- structed by Anaximander about 400 years before Christ, but the maps of Ptolomy are the first in which meridians are drawn, as also parallels for the more exact determination of places, which was a ereat improvement; all maps before his time being merely laid down according to their measured distances, with- out reference either to latitude or longitude. Little improvement was made in the construc- tion of maps from the time of Ptolomy to that of Mercator in the sixteenth century, since which time numerous other authors have treated on the subject, and the greater part of the globe is now represented by maps of very accurate construction. MARALDI (James Puitip), a learned astronomer, and nephew of the celebrated Cassini, was born in the county of Nice in 1665: he was a most indefatigable observer of the heavens, and had in contemplation the construction of a catalogue of the fixed stars, but he died before it was completed in 1729, in the 64th year of his age. Maraldi pub- lished no distinct work; but the Memoirs of the Academy of Sciences, of which he was a smember, bear ample testimony of his talents and exertion, as almost every volume, from ‘the year 1699 to 1729, contain various in- teresting papers relating to astronomy and other brauches of the mathematics. MARCH, the third month of the year ‘amongst the modern, but the first of the / Roman year; it consists of 31 days. MARINE Barometer. See BAROMETER. MARINER’S Compass. See COMPASS. MARIOTTE (Epme), an eminent French mathematician and philosopher, was born at “Dijon early in the seventeenth century, and died in 1684. He investigated a number of curious philosophical subjects, such as the collision of bodies; the pressure and motion of fluids; the nature of vision; the properties of atmospheric air, &c.; on which subject he _had several ingenious papers in the Memoirs ‘of the French Academy of Sciences, from yol. i. to vol. x. His works were all collected and printed at Leyden in 1717, in2 vols. 4to. MARK, an old English coin, value 13s. 4d. sterling. MARS, in Astronomy, is one of the plancts in our system, the fourth in order from the sun, and consequently the next above our earth. ‘The character by which it is repre- sented is g , a rude representation of a man holding a sphere. This planet, which is known in the heavens by his red and fiery appearance, performs his ‘revolution in his orbit in 6864 23" 30' 39”, or in 1'881 Julian years. His mean distance from the sun is 1°524, the distance of the earth being taken as unity, which makes his —_ mean distance 142 millions of miles. The eccentricity of his orbit is 093, the semiaxis His mean longitude for 1800 was 2° 4° 7’ 2-3; the longitude of his perihe- tion being then 11* 2° 24 239; but the line MAR of his apsides has an apparent motion accord~ ing to the order of the signs of 1/5"9 ina year, or 1°51’ 35 in a century. The orbit of Mars at the same time crossed the ecliptic in 1° 18° 1’ 18”, but the place of the nodes has an apparent motion in longitude, according to the order of the signs, of 268 in a year, or 44’ 415 in a century. The rotation on his axis is performed in 1¢ 0" 39’ 218 ; and his axis is inclined to the ecliptic in an angle of 59° 41' 49-2. His mean diameter is equal to 4398 miles; consequently he is rather more than ith the size of our earth. His mass, compared with that of the sun considered as unity, is g-gégzq- ‘The propor- tion of light and heat, received by him from the sun, is 43; that received by the earth being considered as unity. He has a very dense but moderate atmos- phere; and he is not accompanied by any satellite. As viewed from the earth, the motion of Mars appears sometimes retrogade. The mean are which he describes in this case is 16° 12’; and its mean duration is about 73 days. This retrogradation commences, or finishes, when the planet is not more than 136° 48’ from the sun. Mars changes his phases somewhat in the same manner as the moon does from her first to her third quarter, according to his various positions with respect to the earth and the sun; but he never becomes cornicular, as the moon does when near her conjunctions, His mean apparent diameter is 97; which augments in proportion as the planet ap- proaches its opposition, when it is equal to 29-2, His parallax is nearly double that of the sun. At the poles of this planet there has been observed bright spots, which are, however, variable in their size and figure; and from the observations of Dr. Herschel, it seems proba- ble that they proceed from snows accumulated in his polar regions during their long winter, these having been observed to decrease and increase, according as they are directed to or from the sun. MARTIN (BENJAMIN), a very eminent English artist and mathematician, was born in 1704, and died in 1782, in the 78th year of his age. He was author of a great number of ingenious treatises on scientific subjects, the principal of which are as follows: viz. The Philosophic Grammar; being a view of the present state of Experimental Phy- siology or Natural Philosophy, 1735, 8vo. A new, complete, and universal System, or Body of Decimal Arithmetic, 1735, 8vo. The Young Student’s Memorial Book, or Patent Library, 1735, 8vyo. Description and Use of both the Globes; the Armillary Sphere and Orrery, Trigonometry, 1736, 2 vols. 8vo. System of Newtonian Philosophy, 1759, 3 vols. New Elements of Optics, 1759. Mathematical In- stitutions, viz. Arithmetic, Algebra, Geometry, and Fluxions, 1759. Natural History of Eng- land, with a Map of each Country, 1759, 2 vols. Philology, and Philosophical Geogra- MAT phy, 1759. Mathematical Institutions, 1764, 2 vols. Lives of Philosophers, their inven- tions, &c. 1764. Introduction to the Newto- nian Philosophy, 1765. . Institutions of Astro- nomical Calculations, 2 parts, 1765. Descrip- tion and Use of the Air Pump, 1766. De- scription of the Torricellian Barometer, 1766. Appendix to the Description and Use of the Globes, 1766. Philosophia Britannica, 1778, 3 vols. Gentleman’s and Lady’s Philosophy, 3 vols. Miscellaneous Correspondence, 4vols. System of Philclogy. Philosophical Geogra- phy. Magazine complete, 14 vols. Principles of Pump Work. ‘Theory of the Hydrometer. Doctrine of Logarithms, MASKELYNE (NevIL), an eminent as- tronomer and mathematician, who filled the important situation ef astronomer royal of England for 46 years, with the greatest credit to himself, and much honour and advantage to his country. Dr. Maskelyne was born in London in 1732, and very early made con- siderable progress, both in his classical and scientific pursuits. In 1758 he was elected a fellow of the Royal Society, and soon after was appointed to go to the island of St. Helena, to observe the transit of Venus over the sun’s disc on June 6th, 1761; which was to deter- mine the important point relating to the sun’s paraliax, and consequently its distance and magnitude, and hence that of all the other planets; it unfortunately happened, however, to be cloudy, which prevented the necessary observation from being made; but other as- tronomers were more successful, and there- fore nothing was lost, except the confirmation that it would have added to the other results. It was in this voyage that he practised the Junar method of finding the longitude at sea, which he afterwards promoted to the utmost of his power by the publication ef the Nau- tical Almanack, which owes its origin to his recommendation and advice. He also pub- lished the British Mariner’s Guide; Requisite Tables to be used with the Nautical Alma- nack. It would far exceed our limits to enu- merate the many improvements and correc- tions which were by him introduced into astronomy, but many of them may be seen in Vince’s treatise on that subject. Dr. Mas- kelyne died on the 9th of F ebruary, 1811, in the 79th year of his age. MASS, the quantity of matter in any body, which is always proportional to, and may be truly estimated by its weight, whatever be its figure or magnitude. MATERIAL, relating to matter. MATHEMATICAL, relating to mathema- tics. MATHEMATICAL Seci, is one of the two leading philosophical sects which arose about the beginning of the seventeenth Century ; who founded their doctrines upon experiments and mathematical principles. MATHEMATICS, has been defined by ‘Some writers the science of quantity, but it is more properly the science of ratios; for it is not quantities themselves which are the sub- jects of mathematical investigation, but the M A T ratio that different quantities of the same kind have to each other. ee The term mathematics is derived from polnoss, mathesis, discipline, science, represent= ing with justness and precision, the high idea that we ought to form of this branch of human knowledge. In fact mathematics are a me- thodical concatenation of principles, reason- ings, and conclusions, always accompanied by certainty, as the truth is always evident; an advantage that particularly characterizes accurate Knowledge and the true sciences, with which we must be careful not to associate metaphysical notions, conjectures, or even the strongest probabilities. The subjects of mathematics are the com- parisons of magnitudes, as numbers, distances, velocity, &c. Thus, geometry considers the relative magnitude of bodies; astronomy, the relative distances and velocities of the planets; mechanics, the relative powers or forces of different machines, &c.; some determined quantity being first fixed upon in all cases as” a standard of measure. | Mathematics are naturally divided into two classes ; the one comprehending what we call pure and abstract ; and the other, the compound or mixed. “i Pure Matuemarics relate to magnitudes. generally, simply, and abstractedly; and are. therefore founded on the elementary ideas of quantity. Under this class is included arith- metic, or the art of computation; geometry, or the science of mensuration, and compari-— sons of all kinds of extensions; analysis, or: the calculations and comparisons of magni- tudes in general; and lastly, geometrical ana- lysis, or the combination of elementary geo- metry and analysis. Mixed MATHEMATICS, are certain parts of physics, which are by their nature susceptible of being submitted to mathematical investi- gation; we here borrow from incontestible experiments, or suppose in bodies, some prin- cipal and necessary quality, and then by a methodical and demonstrative chain of rea- soning deduce, from the principles established, conclusions as evident and certain, as those which pure mathematics draw immediately from axioms and definitions. To this class belong mechanics, or the science of equili- brium, and motion of solid bodies. Hydro- dynamics, in which the equilibrium and motion of fluids are considered. Astronomy, which relates to the motion of the celestial bodies. Optics, or the theory of the effects of light ; and lastly, acoustics, or the theory of sound. Mathematics are otherwise divided into practical and speculative. | Practical MatHEMATICS, is the application » of these principles to the various uses of society. ft Speculative MATHEMATICS, is that by which we merely vontemplate the properties of things, without at all considering their practical ap- plication. . As to the origin of these sciences, Josephus dates it before the flood, and makes the sons of Seth observers of the courses of the heavenly MAT dies, adding, that in order to perpetuate sir discoveries, and to secure them from the uries either of a deluge or a conflagration, xy had them engraven on two pillars, one of me, the other of brick ; the former of which, says, Was yet standing in Syria in his time. ch authority as this, however, has but little sight with philosophers of the present day, can only be considered as fiction, and as ch is deserving of little attention. It is in- ed impossible to fix the origin of mathema- s with any thing like chronological preci- m, as tliere is every reason to suppose that must be dated back to the remotest ages. hen mankind began to relinquish their wan- ring and savage life, and general laws, or uventions, were established for its main- yance; when by common consent it was reed that every individual should provide his own subsistence, without seizing what is in the possession of another; necessity d self-preservation, the two great springs to ysical exertion, first gave rise to the most aful arts. Huts were built; iron was forged; > lands were divided; and the courses of » stars were observed. With regard to the t, it would seem at first sight to have been ort of amusement, and not in any degree nnected with the economy of society; but on a nearer and more accurate view it wil] pear no less useful than either of the former. ‘was seen that the earth yielded sponta- ously many of the most common of its pro- ctions, and with which man in his savage te was enabled, or rather under the neces- y of subsisting ; but it was soon discovered it there were others of far greater utility, lich required labour and cultivation to bring perfection, and that it was necessary to sulate this by the proper seasons of the ar: thus the ground was sown, the harvest is reaped, and the fruits of the year were sserved, till the bounty of nature furnished arther supply. All these observations and performances, ‘first extremely rude and unskilful, were anected with mathematics by a secret tie, ough for a long time they had no other fide than experience and blind custom. The siduous labour required in hunting or fish- x, and the other business of the field, did tallow man to ascend to general and ab- ‘use ideas, but confined the limits of their mughis and actions to those of their physical nts. At length perhaps arose a Newton, 10 collecting together the traditionary know- lge, observations, and remarks of his pre- cessors, formed them, from the pure efiorts his own genius, into a rude system, sufli- “nt at least to show his fellows, that man S$ a being formed to hold a much more wated rank in the scale of creation than d been before imagined. It was then, says Bossut, that man beheld ‘th new eyes the magnificent spectacle ich nature exhibited on all sides to his fases and imagination; he learned to ex- | | M A T amine things, and to compare them with each other; ideas acquired from physical objects were transported as it were into an intel- lectual world ; the phenomena of nature were studied with discriminating attention, and the mind was impressed with a desire to know the causes by which they were produced. Geo- metry, confined at first to the art of measur- ing the fields, was extended to other purposes, and gave rise to loftier and more difficult pro- blems. Astronomy was enriched by regular observations, and by several instruments adapted to increase their number, and to give them the requisite degree of accuracy and connection. Machines were invented, in which a skilful combination of the several powers of wheels and levers was employed for transporting the heaviest loads: in a word all the parts of mathematics successively ad- vanced; and their progress would in all pro- bability have been more rapid and conside- rable had not fanaticism and ambition too frequently obscured the flame of genius for a long series of ages; but as a fire concealed beneath the embers, it resumed its lustre in happier times, and at length burst forth with that flood of light which now illuminates so great a portion of the earth. We cannot here enter into the history and progress of the mathematical sciences, as it would occupy more room than can be allotted to this article; we shall therefore merely state that they seem first to have assuméd a per- manent form amongst the magii or priests of Egypt, whence they were transplanted into Greece, and here it is that we first begin to possess any authentic record of their progress, and of the names of those that cultivated them with the greatest success ; of which an interesting account is given in Bossut’s “ His- tory of Mathematies;” the first edition of which, consisting of one 8vo. volume, has been translated into English by Mr Bonny- castle. This brings down the history to the jatter end of the last century ; but a second vo- lume has been recently published at Paris by the above author, in continuation of the pre- ceding, detailing all the modern improvements up to the present time. But for a luminous and full account of the rise and progress of mathematics, the reader must consult Mon- tucla’s ‘“‘ Histoire des Mathematiques,” in 4 vols. 4to. The following table or biographical chart exhibits, under one point of view, a concise history of these sciences, viz. the dates, names, and discoveries, of all the most eminent ma- thematicians from the earliest authentic re- cords to the present time. The most cele- brated are printed in small Roman capitals, and the discoveries by which they are most distinguished placed opposite their name, or otherwise that particular branch in which they most excelled. This table we have copied from Rees’s Cyclopedia, being more perfect than that given at the end of Bossut’s History, English edition. M AT MAT .” CHRONOLOGICAL TABLE ~ _ Of the most eminent Mathematicians, from the earliest Period to the present Time. Cen- turies. Beginning. Middle. B.C. 600 | Conrucrus, 722 B. C. I Eee Ce A Tuaces, Gr. predict an eclipse Anaximander, Gr. celestial globes Era of Nabonassar, 747 B. C. |Carron the Centaur, 960 B. CG, Anaxagoras, Gr. philosophy Pytraacoras, Gr. 47th Euchi 500 |Cleostratus, Gr. astronomy Anaximenes, Gr. sun-dial System of astronomy 400|PLavo, Gr. geometry and philos. Luctemon, Gr. astronomy Meton, Gr. metonic cycle Hippocrates, Gr. quadrature of|G2nopides, Gr. geometry lunes Zonodorus, Gr. geometry Eupoxus, Gr, geometry and as-| Pyruras, Gaul, navig.and astro tronomy. Archytas, Gr. math. and philos, ; Aristeus, Gr. conic sections Denostratus, Gr. quadratix Menechmus, Gr. geometry 300} Aristotle, Gr. philosophy ~ Calippus, Gr. astronomy Dinocrates, Gr. architecture Theophrastus, Gr. hist. and math. Xenocrates, Gr. philos. and math. 200| Apollonius, Gr. geom. and conic| ARcHIMEDES Kucuip, Gr. elements of g sections Aristarchus. Gr, astronomy and optics Eratosthenes, Gr, measure a de-|Aratus, Gr. poet _and astronomy gree Aristillus, Gr. philos. and astron *. Nicomedes, Gr. conchoid pea ne a eR eS ee) 100 HIPPARcHuUs, Ge length of year, number the stars Ctesibius, Gr. water-pumps fiero, Gr, Hero’s fountain clep- sydra 0 |Cleomedes, Roman, astronomer |Czsar, Jul. reforms the calendar Possidomius, Rom.mech.and mat Geminus, Rhodes, geom. & astron.|Sosigenes, Egyptian, astronomy |Theodosius, Rom. spherics Era.|Manilius, Rom. poet. and astron. A.D.|Manlius, Roman, astronomer Menelaus, Rom. spherical trigo-\Jamblicus, Syria, philosophy — Q |Vitruvius, Roman, architecture 4 nometry. 100|Nicomachus, Gr. mathematics |Protomy,Claud, Egypt, almagest Frontinus (Sextus) Rom. Engin.|Hypsicles, Gr. mathematician Diophantus, Gr. Diephan. analys. Jamblicus, also of Syria, philos. Pappus, Gr. eehe frical loli oc Theon, Gr. philosophy ‘ 400} typatia (daugh. of Theon.) com.|Proclus, Gr, comment. on Euclid] Diocles, Gr. cissoid on Diophan. Serenus, Gr. geometry Arithemius, Rom. architec. domes Eutocius, Gr. geometry {sodorus, Rom. architecture £00| Marinus, Naples, Geometry 600) Alexandrian library destroyed 642 A. D. Beda (the venerable) Eng. Mot 700 Almansor (the victorious ) astron.| Hero (the younger ) Gr. geomet §00|Almaimon (Arab. Prin.) astron.|Alfragan, Arab. astronomy Albategni, Arab. astronomy ms Alrashed, Persia, astronomy Thebit Ibn Chora. Arab. astro 900 (Gebert.) Silvester 1. Spain, math. 1000 |Ibn Ionis, Arab, astronomy Geber Ben Alpha. 4rab. com. on almagest 1100} Athazen, Arab. optics and astron. MAT MAT CHRONOLOGICAL TABLE Of the most Eminent Mathematicians, from the earliest Period to the Present Time. Cen- cures. — A.D. tz0U|Leonarp de Pisa, first Euro-| Alphonso (K. of Castile), Alphon.| Bacon, Eng. philosopher pean algebraist table Campanas, theory of planets Nassir Eddin, Persian, astronomy|Halifax, or Sacrobosco, Eng.| Vitellia and Pecam, optics mathematician Jordanus Nemorarius, mathema. Middle, End, ne * Beginning, 1300] Albano, Jtaly, physic and math, |John of Saxony, astronomy Ascoli, Jéaly, mathematician ReEGIoMONTANDS, or Muller, Vi-|Lucas de Burgo, or Pacceoli, enna, astronomy Germ, algebra usa, Cardinal, astronomy Bernard, of Granolachi, astron. tlenry, (D. of Visco ), sea charts |Novera, Dominic, Italy, astron. Ulug. bieg (Zartar Pr.), astron. [490| Bianchini, Italy, astronomy Moschopulus (mod. Greek) magic squares Purbach, Vienna, astronomy 11500)Corernicus, Germ. system of|Viera, France,angular sections |Braue, Tycho, Dan. astronomer astronomy Ferrari Rothman Bacon, Lord Ff. Eng. philosopher Apian Maurolycus |Memmius Stiffelius GattLeo, italy, law of falling Buteo Nonius Mercator Ubaldi Guido bodies Cardan Sturmius Ramus Venatorius Bombelili Digges Commandine ‘Tartaglia Recorde Zemberti Byrgius Ghetaldus Durer Albert Werner Reinhold Castelli Meestlin Ferreus Clavius Rheticus . 11600] Bricas, Eng. present system of CAvALERIus, Milan, indivisibles | Bernouttt, J. Swiss, mathemat. ' logarithms BROUNKER, freland, contin. fract.| Barrow. Eng, mathematics Des Cartes, Fran. equation of|Fermat, France, max, et min. Hooker, Eng. oe and mech, : curve lines theory of numbers Huveens, Hol. evolute of curves Kepier, Germ. laws of celestial] PAscat, France, doctrine of pro-|LEtsnrrz, Germ. differential cal- motions babilities culus Neper, Scot. logarithms Watts, Eng. arith. of infinites |Hoprrar, 1’, France, mathematics forriceuui, Italy, gravity of the| Bartholin Riccioli RoemMeR, Dan. progressive mo- atmosphere Borelli Roberval tion of light Bayer Horrox Bullialdus Slusiis Amontons Lieutard Beaugrand Kircher Deschales Snellius Auzout Maraldi De Beaume Lucas Valerius |Frenicle Tacquet Bachet Molyneaux Ceulen Metius Girard, Albert Techirnhausen |Fagnani Olcenburgh De Dominis Otho Gregory,J.&D. Vincent, St. Gr.| Flamsteed Ozanam Gassendi Oughtred Henrion Viviani Grimaldi Peli : Gellibrand Pitiscus Hevelius Viacq Guido Grandi — Picard Guldin Planudes Horrebow Ward, Seth. ae Ee oe Porta Baptista \Mersennus Witt, Jn. de ersey Schooten Longomontanus Howiaiis : Kinghuysen Wren Harriot Ursinus Lagney 1700| Newton CrarrauT, Fr. mathematician ALEMBERT, D’, Fr. partial diff, | BERNOULLI, Jn. Swiss, mathema-|Mactaurin, Scot. mathematician| EULER, Germany, mathematies Moivre, DE, Eng. mathematician| LANDeN, Lng. residual analysis tician l Brapwey, Eng. aberration of the|Smeson, Eng. mathematician stars Cores, Eng. mathematician Cavior, Eng, increments Billy, de Brackenridge Cassini, D. & J. Craig Gravesande Keill Lahire, Laloubere Lansber . Manfridi Marchetti Vierbomius Bellidor Omerique, d’, |Bouguer Hugo Bougainville Pemberton Caille, V Prestet Collins Saunderson Courtivron Saurin Cramer Stirling Dodson Ulloa Dollond Varignon Fatio Verbiest Fountain W olfius Goldbach Guisnée Bernoulli, N. Bernoulli, D. LALANDE, fr. astronomy Herman MASKELYNE, Eng. astrono my Jacquier Warne, Eng. mathematics Koenig Agnesi, donna Horsley Long Atwood Kestner Mairan Bailly Montucla Marriotte Berkeley Pingre Maupertuis Bezout Price Mayer Borda Robinson Montmort Boscovich Steward Nicole Carnot Vandermond Riccati Condorcet Vega Robins Diderot W argentin Seur, le iumerson Simson Walmsley MAT MATTER, in Philosophy, whatever is ex- tended and capable of making resistance ; hence, because all bodies, whether solid or fluid, are extended, and do resist, we conclude that they are material or made up of matter. That matter is one and the same thing in all bodies, and that all the variety we observe arises from the various forms and shapes it puts on, seems very probable, and may be concluded from a general observation of the procedure of nature in the generation and destruction of bodies: thus, for instance, water rarified by heat becomes vapour; a great collection of which forms clouds; these con- densed descend in the form of hail or rain; part of this collected on the earth constitutes rivers; another part mixing with the earth enters into the roots of plants, and supplies matter to, and expands itself into various species of vegetables. In each vegetable it appears in one shape in the root, another in the stalk, another in the flowers, &c. Hence various bodies proceed ; from the oak, houses, ships, &c. from hemp and flax we have thread ; thence our various kinds of linen; these de- generate into rags, which receive from the mill the various forms of paper, &c. According to Sir Isaac Newton, it seems highly probable that God in the beginning formed matter into solid, massy, impenetrable, moveable particles, or atoms, of such sizes and figures, and with such other properties, and in such proportion to space, as most con- duced to the end for which he formed them, and that these primitive particles being solids, are incomparably harder than any porous bodies compounded of them, even so hard as never to wear or break in pieces; no ordi- nary power being able to divide what God himself made one in the first creation. While these particles continue entire, they may compose bodies of one and the same nature and texture in all ages ; but shouldithey wear away, or break in pieces, the nature of things depending on them may be changed. Water and earth composed of old worn particles and fragments of particles, would not be of the same nature and texture now, with water and earth composed of entire particles in the be- ginning; and therefore, that nature may be lasting, the changes of corporeal things are to be placed only in the various separations and new associations of motions of these per- manent particles, compound bodies being apt to break, not in the midst of solid particles, but where those particles are laid together and only touch In a few points. Dr. Berkeley argues against the existence of matter itself, and endeavours to prove that it is a mere ens rationis, and has no existence out of the mind. Some late philosophers have advanced a new hypothesis concerning the nature and essential properties of mat- ter. The first of these who suggested, or at least Sli an account of this hypothesis was fl. Boscovich, in his “ Theoria Philosophize Naturalis;” he supposes that matter is not MAT impenetrable, but that it consists of physica points only, endued with powers of attraction and repulsion, taking place at different dis. tances; that it is surrounded with varion spheres of attraction and repulsion, in the same manner as solid matter is generally supposed: be. Provided therefore that any body moye with a sufficient degree of velocity, or have ¢ sufficient momentum to overcome any powel ofrepulsion that it may meet with, it will find no difficulty in making its way through any body whatever. If the velocity of such body in motion be sufficiently great, Boscovich con tends, that the particles of any body throug which it passes, will not even be moved on) of their place by it. With a degree of velocity something les: than this, they will be considerably agitated and ignition might perhaps be the conse, quence, though the progress of the body in motion would not be sensibly interrupted and with a still less momentum it might not pass at all. Mr. Mitchell, Dr. Priestley, ar some others of our own country, are of t same opinion. See Priestley’s ‘‘ History of Discoveries relating to Light,” p. 390. In conformity to this hypothesis, this autho maintains that matter is not that inert sub. stance that it has been supposed to be; tha powers of attraction or repulsion are neces sary to its very being, and that no part of ii appears to be impenetrable to other parts Accordingly he defines matter to be a sub stance, possessed of the property of extem sion and of powers of attraction or repulsion which are not distinct from matter, and foreigr to it, as it has been generally imagined, bu absolutely essential to its very nature ame being; so that when bodies are divested 0 these powers they become nothing at all. In another place Dr. Priestley has given a some- what different account of matter; according to which it is only a number of centres 6 attraction and repulsion; or, more properly of centres not divisible, to which divine agency is directed; and as sensation ane thought are not incompatible with these powers, solidity or impenetrability, and con quently a vis inertig only having been thought repugnant to them, he maintains that we haye no reason to suppose that there are in mar two substances absolutely distinct from each other. ; But Dr. Price, in a correspondence with Dr. Priestley, published under the title of “A Free Discussion of the Doctrines of |] terialism and Philosophical Necessity,” 1778) has suggested a variety of unanswerable ob jections against this hypothesis of the pene trability of matter, and against the conclu sions that are drawn from it. The vis inertia of matter, he says, is the foundation of all that is demonstrated by natural philosophy concerning the laws of the collision of bodies’ This, in particular, is the foundation of New: ton’s philosophy, and especially of his three laws of motion. Solid matter has the powel of acting on other matter by impulse; but ‘eae MAU ar es matter cannot act at all by impulse ; nd this is the only way in which it is capable f acting, by any action that is properly its wn. | If it be said that one particle of matter can .et upon another without contact and impulse, or that matter can, by its own proper agency, ‘ttract or repel other matter which is at a dis- ‘ance from it, then a maxim hitherto univer- lally received must be false, that “ nothing ‘an act where it is not.” Newton, in his etters to Bentley, calls the notion, that matter yossesses an innate power of attraction, or hat it can act upon matter at a distance, and \ttract and repel by its own agency, an ab- urdity into which he thought no one could vossibly fall. And in another place he expressly disclaims he notion of innate gravity, and has taken yains to show that he did not take it to be an ‘ssential property of bodies. By the same kind if reasoning pursued, it must appear that patter has not the power of attracting and epelling; that this power is the power of ome foreign cause, acting upon matter ac- sording to stated laws; and consequently hat attraction and repulsion not being actions nuch less inherent qualities of matter; as wach, it ought not to be defined by them. And if matter has no other property, as Dr. viestley asserts, than the power of attracting and repelling, it must be a nonentity; because this is a property that cannot belong to it. esides, all power is the power of something ; nd yet if matter is nothing, the very idea of t is a contradiction. If matter be not solid sxtension, what can it be more than mere Farther, matter that is not solid, is the same ith pore; and therefore it cannot possess what philosophers mean by the momentum or force of bodies, which is always in proportion to the quantity of matter in bodies, void of sore. ’ MAUPERTUIS (PETER Louis Morceau E), a celebrated French mathematician and hilosopher, was born at Malo, in 1698, and was there privately educated till he attained his 16th year, when he was placed under the celebrated professor of philosophy, M. Le Bland, in the college of La Marche, at Paris ; while M. Guisnee, of the Academy of Sci- ences, was his instructor in mathematics. For this science he soon discovered a strong inclination, and particularly for geometry. ‘He likewise practised instrumental music in his early years with great success ; but fixed on no profession till he was 20, when he en- tered into the army; in which he remained about five years, during which time he pursued his mathematical studies with great vigour. After that time he devoted himself entirely to science, and in 1723 became a member of ‘the French Academy, and about five years | after was chosen a fellow of the Royal Society of London. If 1736 he was sent with other academicians to the North, to determine the ‘figure of the earth, which service they per- MAX formed with reputation. At the invitation of the Prince of Prussia, afterwards Frederic the Great, he went to Berlin, and was appointed president and director of the Academy there. He was of an irritable temper, and had a dispute with Koenig, professor of philosophy at Francker, and another with Voltaire, who exerted his satirical talents against him. He died at Basil, on a visit to the Bernoullis, in 1759. The works which he published were col- lected in 4 vols. 8vo. published at Lyons, in 1759; where also a new and elegant edition was printed in 1768. These contain the fol- lowing works. 1. Essay on Cosmology. 2. Discourse on the different Figures of the Stars. 3. Essay on Moral Philosophy. 4. Phi- losophical Reflections upon the Origin of Languages, and the Signification of Words. 5. Animal Physics concerning Generation, &c. 6. System of Nature, or the Formation of Bodies. 7. Letters on various Subjects. 8. On the Progress of the Sciences. 9. Elements of Geography. 10. Account of the Expe- dition to the Polar Circle, for determining the Figure of the Earth; or the Measure of the Earth at the Polar Circle. 11. Ac- count of a Journey into the Heart of Lipland to search for an ancient Monument. i2. On the Comet of 1742. 13. Various Acadezaical Discourses, pronounced in the French and Prussian Academies. 14. Dissertation upon Languages. 15. Agreement of the different Laws of Nature which have hitherto appeared incompatible. 16. Upon the Laws of Motion. 17. Upon the Laws of Rest. 18. Nautical Astronomy. 19. On the Parallax ofthe Moon. 20. Operations for determining the Figure of the Earth, and the Variations of Gravity. 21. Measure of a Degree of the Meridian at the Polar Circle. Besides these works nume- rous papers, by the same author, were pub- lished in the Memoires of the Academies of Paris and Berlin, from the year 1724 to 1756 inclusive. MAXIMA et MINIMA, in Analysis and Geometry, are the greatest and least value of a variable quantity, and the method of finding these greatest and least values, is called the Methodus de Maximis et Minimis; which forms one of the most interesting inquiries in the moderna analysis. This subject was considered. geometrically by some of the most ancient mathematicians, particularly by Apollonius, in the 5th book of his Conics;* and there are still a few problems of this kind, which suc- ceed better by the geometrical than by the analytical method; their number, however, are very limited compared with those which may be elegantly performed by analysis. ‘To the latter therefore we shall principally direct our attention, only showing in a few cases how the same may be accomplished by means of the pure elements of geometry. The method of maxima et minima, accord- ing to the analytical doctrine, first arose at the beginning of the seventeenth century, aiter the invention of Des Cartes for express- M A X ing the properties of curve lines by means of algebraical equations, and classing them into different orders, according to the degree of the equation which expressed the relation be- tween the absciss and ordinate. Besides the method of Des Cartes, we have also those of Fermat, Hudde, Huygens, Sluze, and some others, which are now all supplanted by the general and elegant method of fluxions: yet as these several methods may be considered as so many steps towards the discovery of the latter, it will be interesting to have a brief abstract of them, in order to show how slow and progressive are the steps to knowledge, and by what imperceptible degrees we arrive towards perfection. Fermat’s Method of Maxima et Minima.— The principle upon which Fermat formed his operation consisted in this; that when the ordinate of a curve was the greatest possible, if we augmented the variable quantity x (which represents the absciss by an indefi- nitely small quantity e, the ordinate corres- ponding to this absciss will be equal to the former, or will approach towards. equality in- definitely near; or, which is the same, the increase or decrease of an ordinate when it approaches indefinitely near its maximum or minimum is nothing; and therefore these two ordinates may be considered as equal, whence an equation is obtained, from which cancel- ling the like quantities and all those powers of e beyond the first, because they are inde- finitely small with regard to the others, and dividing the other terms by e, the value of x will be obtained; that renders the function a maximum or a minimum. For example; let it be proposed to find that value of x inthe equation y* = 2ax — x’, which renders y a maximum. Increase the variable quantity by e; then y? = 2a (x + e) — (x +e)’, or Qax— a =2ax— 2? + 2Qae—2ex + e’, or 0 = 2ae—2ex by rejecting e*, which is indefinitely small; whence again, Dae = 2ex, or x — a. Again; required the value of « in the equa- tion y3 = ax*—x’, which renders the whole function a maximum. Making as before x = x + e, we have x? — «3 = ax*—x3 + 2axe + ae’ — 3ex*— 3e*a—e?; or, O= + 2axre— 3ex? by suppressing those powers of e above the first; whence Sex” = 2axe-or x — 2a, These examples will be sufficient to show the spirit of Fermat’s rule, which is in prin- ciple much the same as the fluxional method, only that it wants that generality and ele- gance which constitute the distinguishing characteristic of the latter. Des Cartes's Method.—This consisted in mak- ing two of the roots of the equation equal to each other, in which case two of the ordinates of the curve became equal, and thus indicated the maximum or minimum state. This, how- MAX ever, being much less eligible than the pre ceding we will not enter into farther expla- nation, but proceed to Hudde’s method, whieh is in principle the same as Des Cartes, bu more elegant and concise. Hudde’s Method of Maxima et Minima— This, as we have observed, consisted like Deg Cartes, in making two of the roots of the pro- posed equation equal to each other, and for which he gave the following rule ; viz. Mul tiply each term of the equation, arranged according to the powers of a, by the terms of an arithmetical progression ; viz. the first by the first, the: second by the second, &c. an the equation thus obtained will indicate the maximum or minimum required. Let us take for example the equation above, ax* — a3 = 3, Arranging this equation according to the powers of x, and supplying the deficient term, we have . x3 —ax? + Ox —y3=—0 Arith. prog. 3, 2, 1, 0 ' 323 —2ax* —0 ° 3x3 = 2ax*, or x = Xa, as above, - Again; let a*— ax + y* — 2by + b* be proposed, — 4 Writing this, x*—ax + (y>—2by + 0?) =0 Arith. prog. 2, 1, 0 2x*— ax —0, orx=ia. This rule, though not so general as could be wished, is still extremely simple and inge- nious; and considering the state of analysis at the period it was discovered, it is highly creditable to its author; to whom we are also indebted for several other analytical and geo- metrical improvements, Huygens’s Method of Maxima et Minima.— As the rule of Hudde, described above, was a simplification of that of Des Cartes, so the following one is founded on the principle of Fermat, and ean only be considered as a sim- plication of his. Instead of substituting 2 + e for 2, and then cancelling the like terms, suppressing those in which e rises to a higher power than the first, and finally dividing by e; Huygens, as also Sluse, arrive at the final equation at once by the following simple rule. Multiply each of the terms in which x is found, by its expo- nent, rejecting all those into which it does not enter, divide the result by x, and make the whole equal to zero; and the equation thus arising will give the value of x required, | For example; required the value of « in the equation a 3ax”* —3 = y3, Multiplying each of those terms by the ex- ponent of x in them, we have Gax*— 323; then dividing by x H 6ax* —3 z2* =0, or 323 —6ax=0, or x*—2ax+a* =a*,orx mata; that is, x =0; or 2a. This rule differs in no respect from our fluxional operation, except that we divide by « instead of 2; yet the generality of the MAX ‘er is such, that the methods above de- ibed have long been forgotten, and are y given here as an interesting historical w of the methods employed by our prede- sors; and in this respect they are entitled particular notice ; for in them is evidently itained the germ of the modern analysis. rther advances were made in these kind of prations in the method of tangents; of these differential triangle of Barrow is parti- arly interesting, but they are foreign to / presentinquiry. See TANGENTS. ‘the Method of Maxima et Minima according ito the Fluxional or Differential Calculus. .. The fluxion of a quantity when it isa ximum or a minimum, is equal to zero, i), This is obvious from the definition of a ‘ion, for this being the measures, or rates ncrease or decrease, of a variable quantity ; en this quantity becomes a maximum or a 1imum, its fluxion must be = 0, because that point it admits of no farther increase lecrease. ). Ifa quantity be a maximum or minimum, ‘power or root of that quantity must then dently be a maximum or minimum. For power or root of a quantity will increase \decrease as long as the quantity itself in- jases or decreases, and no longer. . Any constant multiple, or ,part of a intity, which is a maximum ora minimum, st also be a maximum or a minimum. For | multiple or part of a quantity will in- ase or decrease as long as the quantity If increases or decreases, and no longer; refore when its fluxion is made equal to 0, the constant multiplier may be neg- ited. . The fluxion of a constant quantity — 0. ’ this admitting of no increase or decrease no fluxion, or its fluxion = 0. . To divide a given number (a) into two h parts #, and that 2” y= may be a maxi- ince x + ¥ =a, and 2” y” = a maximum, ‘flaxion of each — 0, the former because it ‘constant, and the latter because it is a ‘ximum; whence z+y—0 feeray ce se tne yy = 0; | n the first we have —z = + y; and sub- futing this in the second, gives myn x”) —na™ y"—) x = 0, oF ee eee yt Or my — nx; whence ; : ¥=— ow eer n Jonsequently x + — 2 =a, or | 7 | _ ma na | ~~ m+n? men 'm =n, then the two parts are equal. lence to divide a quantity (a) into three parts, i z, 80 that xyz may be a maximum; the 2€ parts must be all equal amongst them- Ves. For whatever one of ihe parts may and y = / MAX be, if it be constant, the product of the other two will be the greatest when they are equal to each other; and in the same manner if we consider any one of the parts as constant, the rectangle of the other two will be the greatest when they are equal to each other; whence it is obvious that the product will be the greatest when the three parts are equal to each other. And in the same manner, if the given quantity be divided into any num- ber of parts, the product of them, or the pro- duct of any equal power of them, will be the greatest when the several parts are all equal amongst themselves. 2. To divide a given number (a) into two such parts, « and y; that the sum, of their al- ternate quotients may be a maximum. Here we must have x + y = 4, Boing anna and — + Y — q@ maximum. q x Now since the first is constant, and the lat- ter a maximum, we have x = — y, and ay — ye , ye—ary_g y* at mage ag’ Or substituting for x, its equal — y, this becomes _ ; _ yy tay Pie + yy z z = 0, or y <7 ey tyywyy tery x” a y? 7 1 1 Whence we have = Pics Tis Ok x= y; that is, each of the required quantities is equal to ja. We might have given here numerous other problems relating to the maxima et minima of quantities, as no branch of mathematics offers a greater variety of pleasing and inter- esting questions ; but our limits will not admit of farther detail, as it is necessary to offer a few other remarks more immediately connect- ed with the principles of the doctrine. In our definition we stated a maximum of minimum to be the greatest or least state of a variable function, which indeed is the real meaning of the expression, and in this sense it must be still understood when applied to physical problems. But, analytically, we must understand this term in a more general sense, signifying that state of a variable function, which, if the variable quantity upon which it depends, be either increased or diminished, the whole function will decrease or increase according as itis in its maximum or minimum state ; so that a function may admit of several maxima and minima, and the object of the fol- lowing remarks is to point out the method of ascertaining the number of each, and which value gives the maxima and which the mi- nima, our operation being the same In both cases. In the two preceding problems the result- ing equations are only of the first degree, and consequently admit of only one solution ; but it may happen that the, resulting equation is of higher dimensious, and therefore contain > MAX several roots, and it then becomes necessary to have some means of distinguishing which of those roots give the greatest, and which the least result; and for this Lacroix gives the following general rule. Rule. Let y represent any function of 2; find the value of x in the equation ¢=0, which value substitute for x in the equation oe z x maximum; if positive, a minimum; and if zero, neither a maximum nor a minimum, = 0; then if the result be negative, y is a unless also 5 = 0, and then it will depend : .y . upon the sign of we and so on; and the same H i process being observed with regard to each of the roots of the fluxional equation, the num- ber of maxima and minima will be obtained. Let us illustrate the preceding rule by an example. 1. Find y = 2+ — 82° + 2247— 242 + 10, a@ max. et min. Here ‘ — 4x3 —242* + 442 24-0 Where x = 1, 2, and 3. And it is required to find which of these roots answer to the maxima, and which to the minima. | Now ee — 1227 — 482 + 44. x And here making x = 1, 2, 3; the results are respectively +, —, +; therefore the root 2 answers to the maximum, and the other two to the minima. 2. Let there now be proposed the function y = x —7x* +1943 — 2527 +162+10 Here 4 = 5x+—28a3 + 57x?—50xr+16—=0 x ’ And the roots of this equation are 1, 1,2, 1 3. Now 4 = 2023 — 8427 + 114x— 50 x which = 0, when x =1; therefore the root 1 gives neither a maximum nor a minimum, unless A = 0; which upon trial does not ob- x tain. But by assaming x= 2, in this equation, the result is—4; and consequently this value of x answers to a maximum. And by submitting the other root 13, to the same test, a similar result will be obtained. We will add another example, with which we must conclude this article. 3. To find when the function y = x3 — 1827 4- 962 — 20 becomes a maximum or a minimum. Here 2 =3 x*— 362 +96—0, xv im which equation the roots are z = 4,2 = 8. Now ¥ — 6x — 36, x MEA Here the root 8 gives % positive. And the root 4 gives #, negative. x Therefore the former answers to the mj mum and the latter to the maximum. If the fluxional equation has no real then it follows that the proposed funet admits of neither a maximum nor a minimu but increases or decreases ad infinitum. For more on this subject see Craswe “ Treatise on Maxima et Minima,” la published, and most authors on the doctr of fluxions. See also an ingenious chapter the maxima et minima of geometrical qu} tities in Simpson’s “ Geometry.” MAXIMUM Effect of Machines. MAcHINES. : MAY, the fifth month of the year, com ing of 31 days. | MAYER (Toptss), a German astronor} and mechanician, was born at Manpach, Wirtemberg, in the year 1723; and in 1751) was nominated mathematical professor at university of Gottingen, and soon after was mitted a member of the Royal Society in1 town. From this time every year of his’ was distinguished by discoveries in geoni and astronomy. He invented many us instruments: he applied himself to st) the theory of the moon: he extended ‘his} servations to the planet Mars, and the fi stars, determining the places of the lat and ascertaining that they possess a cer! degree of motion relative to their respee systems. ‘Towards the close of his short ihe magnetic needle engaged his attent! to which he assigned more certain laws t those before received. 'To all his pursuits applied with such indefatigable assiduity, he died literally worn out with labour, in 1% at the age of thirty-nine. The principal we which he gave to the public were, “A and General Method of resolving all Geof trical Problems, by means of Geometi Lines ;” “ A Mathematical Atlas,” in wh all the mathematical sciences are comp in sixty tables; “ A Description of a Li Globe,” constructed by the Cosmograp Society of Nuremberg, from new obset tions ; “‘ Maps ;” and several valuable pa’ in the Memoirs of the Royal Society of , tingen. His table of refractions, dedu) from astronomical observations, agrees °t that of Dr. Bradley; and his theory ofl moon and astronomical tables and preett were so well received that they were rewaie by the English Board of Longitude with premium of three thousand pounds, wf sum was paid to his widow after his deecé These tables and precepts were publishel 1770. | MEAN, is a middle state between twa tremes; thus we say, Mean Distance, Mo) Time, &c. ; Arithmetical, Geometrical, and it monical Mean. | Arithmetical Mean, is half the sum oft MEA o quantities: thus, sit! is arithmetical »an between a and b. Geometrical MEAN, is the square root of the oduct of any two quantities; that is, / ab is 2 geometrical mean between a and b. Harmonical MEAN, is double a fourth pro- rtional to the sum of two quantities and e quantities themselves; thus, @ + bia A is the fourth proportional, and Qba a+b therefore the harmonical mean between a id b. These three means may be found geometri- My, as follows: ‘Let DB and BC be any two lines between hich the means are required to be found. ‘n DC, equal to the two, as a diameter, de- sribe the circle DAGCH, and at the point _erect the perpendicular BC; take BA equal » the radius OC, and produce it to cut the ‘rele again at H: so shall BA be the arithme- cal mean, BG the geometrical mean, and ‘HH the harmonical mean, between the given nes DB, BC. This is evident, for OB — OC = DE hee + a arithmet. mean BG = v(DB x BC), geometrical mean BH — BB x BC __2DB=BC ere DBA’ | harmonical mean. It is obvious, from the above construction, yat the arithmetical mean is greater than the armonical, and the harmonical greater than ae geometrical. p if ] equired between two quantities, they may e found analytically as follows: viz. a and 6 ing the two quantities, we shall have mean Jab, _ means ?/ a’b, ¥ ab’, means 4/ a3b, 4/ ab”, 7 ab}, means %¥/ a*b, ¥/ a3b*, ¥ ab’, 7/ ab+, means Y a%b, °/ atb*, Y 026°, 6, arb*, S/ ab>, | &e. Ke. &e. more than two mean proportionals are MEA Pappus, in his “Collections,” treats of these means, and mentions several properties relat- , ing to them; and amongst others, the follow- ing, viz. a, 6, and ec, being three continued terms, either arithmetical, geometrical, or har- monical; then in the Arithmeticals...... a: a3: Geometricals ......@:¢::a@—e:ce—b Harmonicals.......@:b::a—e:ce—b For more on this subject, see PROGRESSION and PROPORTION. Mean and Extreme Proportion. See Pro- PORTION. Mean Anomaly, Axis, Diameter, Distance, Motion, Time, &c. See the respective articles, MEASURE, in Geometry, denotes any cer- tain quantity assumed as unity, with which other homogeneous quantities are compared. Measure of an Angle, is the number of de- grees, minutes, &c. contained in the arc of a circle comprised between the two lines form- ing that angle, its angular point being the centre. Common MEASURE, in Fractions. See Com- MON Measure. Measure of a Line, is its length compared ls some determinate line; as a foot, yard, C. Measure of a Surface, is the number of square units contained in it, whether that unit be a foot, a yard, mile, or other quantity. MEasvreE of a Solid, is the number of cubic units contained in it; as inches, feet, miles, &e. Measure of a Number, is such a number as will divide another number without a re- mainder. Measuvre of a Ratio, is its logarithm in any system of logarithm; thus, the measure of the ratio 2:3, or 3, is the logarithm of 3, or the log. of 2 minus the log. of 3. Measure, in Mechanics, is used in various ways; as the measure of the mass, momentum, velocity, &c.; for which see the respective articles. Measures, in Commerce, are of various de- nominations, according to the nature of the things of which the quantity is to be determined. They are also different for the same things, in different countries. Of these we shall give the most important; but our limits will not admit of entering at much length on this sub- ject. a—c:ce—b GG MEA MEA eal A Long Rood 1339°2 _ A Hogshead = 13235°7, or 16 Gallons 4 ® TABLE ; t Of the several Standard Measures. y; g a eee “ ENGLISH. fn LONG MEASURE. WINE MEASURE. Barleycorns Pints. 3= 1 Inch 2= 1 Quart 36—=° 12= °§1 Foot 8- 4= 1=—1Gallon—231Cub. Inches 108= 36> 3= 1 Yard 336 168 42—1 Tierce 994=" 198='164 =" 53= ‘1 Pole 504= 252= 63=11=1 Hogshead 23760= 7920— 660= 220= 40=1Furlong] ¢72— 336— 84—2 —1i=1 Puncheon 190080—63360—5280—1760— —320=8—1 Mile 1008= 504=126=3 =2 —11=1 Pipe Also 2016=1008=—252—6 =4 =—3 —2—1 Tun 4 Inches ae Hand 231 Cubic Inches = 1 Gallon * 6 Feet 1 Fathom x, ee hes ‘ 10 Gallons = 1 Anker 3 Miles = | League a Fi. OT 18 Ankers = 1 Runlet 60 Geographical Miles — 1 Degree 912 Gallons Bae Fp APR 694 English Miles cre | Degr ee nearly Path , By 360 Degrees, or 25000 Miles, is equal to the Circumference of the Earth nearly aie Ani neat Weaecaee 4 } Pints. i aft CLOTH MEASURE. 2— 1 Quart { 90— | Nail S= 4= 1 Gallon & BoM 16 4S ty Ay" tia 144—= 72= 18= 2=1 Kilderkin Pz 27 =- 12 = 3 = 1.Ett Elemish 288=144—= 36= 4=2=1 Barrel at 45 = 20—5 —1 Ell Enslish 432—216= 54= 6=3=14=1 Hogshead © 546 = 24 —6'—= LEN French 576288 —372— B42 “=14=1Pancheon 864—432—108—12—6—3 —2 $e wey Butt Nae SQUARE MEASURE. The Ale Gallon contains 282 Cubic Inches, 144— 1 Foot 1296 Dy sal a4; sWiard 39204 2721= 361= 1 Pole tomWiucod Grade Mae Mant 156816010890 =1210 = 40=1 Rood tgalhy Gallon Ms rf ers. war oes) emt anaes — SHEPARD A560 —4840 —160—4=—1 Acre ieee Upipank Bi Also, 5 Yards = 1 Pole 64= 8= 4= 1 Bushel % 40 Poles = 1 Rood 256— 32— 16—= 4= 1 Goomb | 4 Roods = 1 Acre 512—= 64= 32= 8= 2= 1 Quarter 2560—320—160=40—10= 5=1 Wey ‘ ‘Ap CUBIC MEASURE. 5120—640—320—80—20—10—2—1 a nehes, | 1728 = 1 Foot | 2684 Cubic Inches = 1 Gallon i 46656 = 27 = 1 Yard 36° Bushels. = 1 Chaldron of Coals” 5 SCOTCH. LONG MEASURE. MEASURE OF CAPACITY. Eng. inches. Eng. Cub. Inch. An Elk creat Oy ce A Gill = 6:462 A Fall <—i2eos A Mutchkin = 25°85 A Furlong =) eo A Choppin = _ 51:7 A Mile = 71424 A Pint = ,103°4 A Link = 8928 A Quart ei a Te ; A Chain, or Short Rood = 89°28 A Gallon =. 82725 MEA METRE SYSTEM. SUPERFICIAL MEASURE. Eng. Yards. Are, a Square Decametre........ 119-6046 DICCATE yo. 0- cass steed os ele cgi antie 1196:046 EAECCADALC «. }. The apparent diameter of the moon varies according to her distance from the earth; when nearest to us itis 33/311; but at her greatest distance it is 29’ 21'"9; so that her mean apparent diameter is 31’ 265, and her mean horizontal parallax is equal to 57' 342. The phases of the moon are caused by the reflection of the sun’s light, and depend on the relative positions of the sun, the earth, and the moon. An eclipse of the moon can take place only at the time of her opposition to the sun; and is caused by her passing through the shadow of the earth; which shadow is three and a half times longer than the distance between the moon and the earth; and its breadth, where it is traversed by the moon, is about two and two-thirds times greater than the diameter of the moon: the breadth of the earth’s shadow, where it is traversed by the moon, is equaj to the difference between the semi-diameter of the sun, and the sum of the horizontal parallaxes of the sun and moon. The moon cannot be eclipsed, however, if her distance from the place of her node, at the time of her opposition, exceeds 13° 21’; be an eclipse. The duration of the eclip will depend on the apparent diameter of 4 moon, and on the breadth of the shadow the point where she traverses it. ; The sun cannot be eclipsed unless the nade be in conjunction; and then only when ft) centres of the sun and moon are in a line, | nearly in a line with the eye of the spectat on the earth. In the first case, if the a parent diameter of the moon be greater thi that of the sun, the eclipse will be total; b if it be less, it will be annular. In other cas, the eclipse will be partial. | The sun cannot be totally obscured for longer period of time than four minutes ; b} the moon may be obscured for a much long period. “| Eclipses, of which there cannot be less th two in a year, nor more than seven, general in the same order and magnitude at the @ of 223 lunations; for in that number of SYI dical revolutions there are 6585¢ 74 42™ 31 and 6585° 18" 41™ 456 there are ninete mean synodical revolutions of the moo nodes; therefore at the end of 6585° 7" 24™3] the moon’s mean longitude will be only 28/@ behind the mean place of her nodes. The atmosphere of the moon, if it has a must be extremely attenuated, more rare th we can produce with our best air-pumps. T’ light of the moon is only one-300000th part that of the sun; and the refraction of the ra} of light, at the surface of our earth, must } at least 1000 times ‘greater than at the st: face of the moon: this, at least, is the opini of many of our most celebrated astronome| and it would perhaps be thought presumpti} in us to doubt such authority. 4 When we look at the surface of the mo} with a good telescope, we find its surfa} surprisingly diversified. Besides the lar dark spots that are visible to the naked e we perceive extensive valleys, and long rid of high elevated mountains projecting tht shadows on the plains below. Single mot tains here and there rise to a great heig, while hollows, which from accurate measu} ment are found to be three miles deep, al almost perfectly circular, are excavated in ft plains. The margin of these circular cavit is frequently elevated a little above the gel: ral level, and a lofty eminence rises in #} centre of the cavity. 4 When the moon approaches to her op sition with the sun, the elevations and i pressions on her surface in a great meas} disappear, while her disc is marked with} number of brilliant points and permané radiations. . ‘These various appearances haye been ac rately represented in maps of the moon’s si face. This was first attempted, but in a vd rude manner, by Riccioli. Hevelius, in 3 “Selenographia,” afterwards gave more jit (lelineations of the lunar disc, during the whe of her progress round the earth. A mapf the full moon was drawn by Cassini, and bi but if it is within 7° 47', there will a i MOO | a ® MOO een copied, though extremely incorrect, into nost of our modern treatises of astronomy. ixcellent drawings of the moon were also nade by Tobias, Mayer, and by Mr. Russel ; vut the most accurate and complete are those )f the celebrated Schroeter, who has given _ighly magnificent views of several parts of the aoon’s surface. The most favourable time for jlewing the lunar disc, is when she is about ve days old; the irregularities in her surface eing then the most conspicuous. It is difficult to say, with any degree of pro- ability, what the immense cavities, of which ve have been speaking, are, or may have een; but we cannot help thinking that our arth would assume nearly the same appear- nd who can say but that this may have been wie case in the lunar regions? Astronomers jave formerly supposed that the dark part of ve moon’s surface were large lakes and seas; ut it is obvious, on an attentive observation ith a good telescope, that there are ridges id unevennesses in those parts, which plainly dicate that they are not fluid but solid, like ie other parts of her surface; whence it ap- ars, that there is very little of any fluid mat- \r in this luminary, and hence probably is ye reason that her atmosphere is so different jom our own; if, indeed, this can be justly yferred from the circumstances usually ad- slced as arguments in favour of this hypothe- 8, which we confess appears very doubtful. ) We have observed, that mountains may al- ays be seen on the surface of the lunar Sc; and even volcanoes have been frequently jserved, from which some philosophers have pposed the eroliths, that at times fall to the tth, to have been projected: which it ap- ars, from computation, would only require velocity of about 8200 feet per second, to use them to pass from this body to the rth. Moseleretion of the Moon. See ACCELERA- ON. VAge of the Moon, is the number of days jice the new moon, which is found by the \lowing rule: To the epact add the number and day of month, which will be the age required, if is than thirty ; and if it exceed thirty, sub- ict this number from it, and the remainder ‘Il be the age. See Epact. inpea Moon, is a remarkable phenome- nrelating to the rising of this luminary in > harvest season. During the time she is the full, and for a few days before and after, ial about a week, there is less difference in >time of her rising between any two suc- ssive nights at this than at any other fe of the year. By this means she affords i immediate supply of light after sun-set, ‘ich is very beneficial in gathering in the its of the earth; and hence it is, that \S lunation has been termed the harvest on, th order to conceive this phenomenon, it ty first be considered, that the moon is uce if all the lakes and seas were removed ; MOT always opposite to the sun when she is full ; that she is full in the signs Pisces and Aries in our harvest months, these being the signs opposite to Virgo and Libra, the signs occu- pied by the sun about the same season; and because those parts of the ecliptic rise in a shorter space of time than others, as may easily be shown and illustrated by the celes- tial globe, consequently when the moon is about her full in harvest, she rises with less difference of time, or more immediately after sun-set, than when she is full at other seasons of the year. Horizontal Moon. See Apparent Macni- TUDE. | Moon Dial, is a dial which shows the hours of the night by the light of the moon. MOORE (Sir Jonas), a respectable ma- thematician, Fellow of the Royal Society, and Surveyor-General to the Ordnance, was born at Whitby in Yorkshire, in 1620. He was author of several works, but is more deserv- edly remembered for the support and protec- tion which he rendered to the science, than for any immediate improvement or discovery of his own; the important offices that he filled engaging too much of his time to allow him to follow the subject so far as he’might other- wise have done. To him wé owe the first establishment of the Royal Observatory at Greenwich, and the appointment of Mr. Flam- stead to that important office; as also for the institution of the mathematical school belong- ing to Christ Hospital; for the use of which, he composed a part of a mathematical course, but died before it was completed, about the year 1681. The list of his works, as given by Dr. Hut- ton, are as follows: 1. A new System of Ma- thematics, being that above mentioned, 2 vols. 4to. 1681. 2. Arithmetic, in two books; viz. Arithmetic and Algebra: to which is added, some other pieces on the Ellipse and the other Conic Sections, 8vo. 1660. 3. A Mathema- tical Compendium, or useful practical Pro- blems in Arithmetic, Geometry, Astronomy, Navigation, &c. 12mo. fourth edition, 1705. 4. A general Treatise on Artillery, from the Italian of Tomaso Moretii, 8vo. 1683. MORTALITY. Bills of mortality are ac- counts or registers specifying the number of births and burials which happen in any town, city, or parish, and on which is founded the doctrine of Life Annuities and Assurances. See LIFE Annuities. MOTION, or Local Motion, in Mechanics, is acontinued and successive change of place, or it is that affection of matter by which it passes from one point of space to another, Motion is of various kinds, as follows: Absolute Morton, is the absolute change of places in a moving body independent of any other motion whatever; in which general sense, however, it never falls under our ob- servation. All those motions which we con- sider as absolute, are in fact only relative ; being referred to the earth, which is itself in eee absolute motion, therefore, we HH2 ; MOT must only understand that which is so with regard to some fixed point upon the earth; this being the sense in which it is delivered by writers on this subject. Accelerated Motion, is that which is con- tinually receiving constant accessions of velo- city. See ACCELERATED Motion. Angular Motion, is the motion of a body as referred to a centre, about which it re- volves. Compound Motion, is that which is pro- duced by two or more powers acting in dif- ferent directions. See PARALLELOGRAM of Forces. Equable Motion, or Uniform Motion, is when the body moves continually with the same velocity, passing over equal spaces in equal times. _ Natural Motion, is that which is natural to bodies, or that which arises from the action of gravity. Relative Motion, is the change of relative place in one or more moving bodies; thus two vessels at sea are in absolute motion (ac- cording to the qualified signification of this term) to a spectator standing on the shore, but they are only in relative motion with regard to each other. Retarded MovioN, is that which suffers con- tinual diminution of velocity, the laws of which are the reverse of those for accelerated motion. See ACCELERATION and RETARDA- TION. Projectile Motion, is that which is not na- tural, but impressed by some external cause; as when a ball is projected from a piece of ordnance, &c. See PROJECTILE. Rectilinear Motion, that which is performed in right lines. Rotatory Motion. See Rotarion. Laws of Motion, as delivered by Newton in his “ Principia,” and on which he has sup- ported the whole system of his philosophy, are the three following: 1. Every body perseveres in its state of rest or uniform motion in a right line, until a change is effected by the agency of some ex- ternal force. 2. Any change effected in the quiescence or motion of a body is in the direction of the force impressed, and is proportional to it in quantity. 3. Action and reaction are equal and in contrary directions. When speaking of these axioms, or laws of motion, it ought always to be recollected that they are not the efficient operative causes of any thing. A law presupposes an agent; for it is only the mode, according to which an agent proceeds: it implies a power, for it is -the order according to which that power acts. Abstracted from this agent, this power, the law does nothing, is nothing: so that a law of nature or of motion can never be assigned as the adequate cause of phenomena, exclusive of power and agency. The Newtonian axioms are, in reality, in- termediate propositions between geometry MUL and philosophy; through which mechanics becomes a mathematical branch of physies, and its conclusions possessed of such herence and consistency among themselves, and with matter of fact, as are rarely to be found in other branches, which admit not of so intimate a union with the science of quantity. . The evidences from which our assent to these axioms is derived, are of various kinds, 1. From the constant observation of our senses, which tend to suggest the truth of them in the ordinary motion of bodies, as far as the experience of mankind extends. 2. From experiments, properly so called. 3, From ar guments @ posteriori. One or other of these kinds of evidence generally forms a parto our most valuable treatises on mechanics an¢ physics: but there is a fourth way in whiel the truth of these axioms may be deduced which is that in which they are shown to bi laws of human thought, intuitive consequence of the relations of those ideas which we hav of motion, and of the causes of its productio; and changes. on Quantity of Motion. See MOMENTUM. , Motion, in Astronomy, is still farther di vided into diurnal, annual, horary, sidereal, &e for which see the respective terms. | Motia Perpetual Motion. See PERPETUAL and OrrFryReus’s Wheel. ; MOTIVE Force. See Motive Force. | MOTRIX, that which has the power ¢ faculty of moving. MOVEABLE, that which is susceptible | motion. ; MoveaBLe Feasts, in the calendar, are tho: which do not fall always upon the same di of the month, such as Easter, and all tho depending upon it. MOVEMENT, is sometimes used in tl same sense as motion, but more commonly implies the internal parts of machines, par. cularly of clocks and watches. MUI.LER (Joun), commonly called ] giomontanus, from Mons Regius, or Koning berg, where he was born in 1436. Mull was one of the greatest astronomers of t age in which he lived; he was the pu and companion of the celebrated Purb and a participator in many of his m useful labours. It was Purbach who fi reduced the trigonometrical tables of six from the old sexagesimal division of 1 radius to the decimal scale. He suppos the radius to be divided into 600,000 eq parts, and computed the sines of the ares every ten minutes, in such equal parts oft radius, by the decimal notation. This proj of Purbach was perfected by Regiomontan who not only extended the sines to eve minute, the radius being 600,000, as design by Purbach, but afterwards disliking tl scheme, as evidently imperfect, he compu them likewise to the radius 100,000, for evi minute of the quadrant; Regiomontanus a introduced the tangents into trigonomel the canon of which he called foecundus, | MUL cause of the many great advantages arising from them. Beside these things he enriched trigonometry with many theorems and pre- cepts. Indeed, excepting for the use of lo- garithms, the trigonometry of Regiomontanus is but little inferior to that of our time. His “Treatise on both Plane and Spherical Trigo- nometry,” is in five books; it was written about the year 1464, and printed in folio at Noremberg in 1533. In the fifth book are various problems concerning rectilinear tri- angles, some of which are resolved by means of algebra; a proof that this science was not wholly unknown in Europe before the treatise of Lucas de Burgo. Regiomontanus was also author of some other works, as “The New Theories,” of Pur- bach, which were left incomplete by the lat- ter. “The Astronomicon,” of Manlius and Ephemerides, showing the motions of the heavenly bodies for thirty years to come, be- sides several translations from the Greek and Arabian authors; he died at the age of 40, at Rome, to which city he had been called by Pope Sixtus the 1Vth, to assist in the re- ormation of the calendar. MULTANGULAR Figure (from multus and angulus), is one that has many angles, and consequently many sides. See PoLYGON. MULTILATERAL Figure, (from multus nd /atus), a figure of many sides. MULTINOMIAL (from multi, many, and 2omen, name), an algebraical quantity con- sisting of more than three terms, though this s sometimes called a multinomial, but more sommonly a trinomial. See 'TRINOMIAL. MULTINOMIAL Zheorem, is a general ex- oression or formula, for determining any power wr root of a given quantity consisting of any 1umber of terms. This theorem was first siven by De Moivre, No. 230, Phil. Trans. (697, who also pointed out the law of its erms; but it was afterwards simplified by Euler in his “ Calcul Differential,” c. viii. yart 2; and the same has been done by (rbogast in his “ Calcul des Derivations,” i. 12. _ The general form of this theorem, as given n Bonnycastle’s Algebra, vol. i. p. 206, is as ollows: A + Ax + Aa? +Az3.... Aan)™ = A™ + 0 1 ee 3 ” 0 x + 3mAB BAB. — + 2mAB nA 30§ 23 21 +(m—1) AB 0 (m—2)AB Yo 12 ~ 4mAB 4 +(3m—1) AB] 2 3 1 eS. + (2m—2) AB( 4A 22 0 + ( Sy 1 3 Where B = A”; and B, B, B, &c. are the " 0 1 2 3 co-efficients of the terms immediately pre- + &e. 2mAB +(m—DAB 20 1,8 MU L ceding those in which they first appear. And the manner of applying this theorem to any particular case, is by substituting the numbers or letters in the given example, for A, A, A, 2 &c.; and m, as in the binomial theorem, which see. Thus for example, let it be proposed to cube the series 1+ av fa? + x? + o* + &e, , Here A=1,A—1, A =1, &c. and m=3. 1 2 Whendd epee Learns ove’ Te 0 0 mAB=3x1x1=3=—B by) .¢ 1 —1)AB 2mAB + (m ) AS ae DA 3 Birsut 0 3mAB + (2m— 1) AB + (m— 2) AB 3 0 74 "dS glee 3A on 0 ato = B, ke, be, Therefore ‘ 1 + 3a + 627 + 1023 + 152+ 4+ &e. is the cube required. Again, required the cube root of the series 1 + ta + 42° + 223 + &e. Here A=1,A =1,A = }, &c., and m = 4. 2 0 1 Whence 0 mAB=4xix =iG—yv)=12=8 2A 0 3m AB + (m—1) AB 4+ (m—2) AB 3.0 2 1 bee De ee 3A iii 0 gis = B, &e. &e. Therefore 1 + tx + 72° + Jy x? + &e. is cube root of the series proposed. See Bon- nycastle’s Algebra, as above referred to. MULTIPLE, in Arithmetic, is a number which contains another number a certain number of times, thus 18 is a multiple of 6, or of 3, or of 9, &c. Common MULTIPLE of two or more numbers, is that which contains those numbers a certain number of times; thus 36 is a common mul- tiple 4 and 9, being equal to 9 times the first and 4 times the second. To find the least common Multiple of several Numbers. Reduce them all to their prime factors, then the product of the greatest powers of those prime factors is the least common multiple required. Let it be proposed to find the least common multiple of 12, 25, and 35, or the least number that will divide by each of them without a remainder, MUL Here 12=3 x 2’; therefore 3 Xx 2? x 5* x 7= 210, the least common multiple required. MULTIPLE Ratio. See Ratio. MULTIPLICAND, in Arithmetic, is one of the factors in multiplication, being that which is multiplied by the other, which is called the multiplier. MULTIPLICATION, is one of the prin- cipal rules in arithmetic and algebra; and consists in finding the amount of a given number or quantity, called the multiplicand, when repeated a certain number of times ex- pressed by the multiplier; and this amount is generally termed the product; also the mul- tiplier and multiplicand are commonly called factors. Multiplication is either simple or compound. Simple MULTIPLICATION, is when the pro- posed quantities are integral numbers, Rule. Place the multiplier under the mul- tiplicand, so that units may fall under units ; tens under tens; and so on. ‘Then begin‘at the right hand, and multiply every figure in the multiplicand, by each of the figures in the multiplier. Find how many tens there are in the product of every two simple figures, and set down the remainder directly under the figure you are multiplying by, or if there be nothing over, a cipher. Carry as many units as there were tens to the product of the next figure, and proceed in like manner till the whole is finished. ‘Then add all the separate products together for the answer. Proof of MULTIPLICATION. 1. Invert the operation, by making the multiplier and mul- tiplicand change places, and if you thus ob- tain the same result, it is highly probable the work is right. 2. Cast out all the 9’s from the multiplier, multiplicand, and product; and multiply the overplus of the two former together, and cast the 9’s out of this product; then if this re- mainder be the same as that arising from the total product, the operation is probably right, but if not it is certainly wrong. This proof depends upon a singular property of the number 9; wiz. that any number di- vided by 9, will leave the same remainder, as the sum of its digits when divided by the same number. This method of proof is generally attributed to Dr. Wallis ; but it is of a much earlier date, being given by Lucas de Burgo in his “ Summa de Arithmetica,” printed in folio at V enice in 1494, 3. Another proof for multiplication is drawn from a particular property of the number 11, which is this; that the sum of the digits in the odd places, that is, the Ist, 3d, Sth, &c. being taken from the sum of the digits in the 2d, 4th, &c. places; the remainder when di- vided by 11, will leave the same overplus, as the whole number when divided by 11. If the former sum be greater than the latter, as many times 11 must be added to it, as will render the latter swm the greater of the two. MUL | 25 = 5*,and35=5 x7; This being observed, the proof by this number will be the same as in the former case. EXAMPLE. | 45684 multiplicand : __ 4374 multiplier | 182736 | 319788 | 137052 | 182736 | 199821816 product proof by 9 proof by 11 oO 4 O £ Ie Dy The other proof by inverting the operation depends upon this; that the ‘product of two numbers is the same whichever of the two is the multiplier; or generally that @ times J, is the same as 6 times a; which though generally considered as an axiom, is in fact a propo- sition, and one that is not very easily demon- strated. Compound MULTIPLICATION, is the method of finding the product arising from a com- pound and simple quantity. Rule. Place the multiplier under the lowest denomination of the multiplicand; and mul- tiply this denomination by the multiplier, Find how many units of the next higher de- nomination are contained in the product; set down the remainder, and carry the units to the next product, with which proceed as be fore, and so on through all the denominations to the last; and the result will be the answer required. Note. If the multiplier exceed 12, the opera: tion will be much simplified as follows: 1. If the given multiplier be a composite number, multiply successively by each of it factors, instead of the whole number at once 2. If the given multiplier be not a compo: site number, take that which is nearest to it and multiply by its factors as before ; then ade or subtract as many times the first line, as the number so taken is less or greater than the multiplier. Also if there be any fractional part belong ing to the multiplier, take such part of the multiplicand, as this fraction is of a unit, ane add it to the result before found. EXAMPLES. Multiply £7. 13s. 44d. by 24. Ge ae 713 41 4 30 13 6 6 24—-4x6 answer £184 1 O ee ee MUL Multiply £3. 15s. 63d. by 173. 7ix4x44144 4x4— 60 8 8 P= 815 “6k ; Sea ee answer 173 — £66 1 11% Cross MULTIPLICATION. See DUODECIMALS. Muttip.ication of Fractions, is performed y the following Rule. Reduced all mixed numbers to im- oper fractions; and then multiply the nu- nerators together for a new numerator, and he denominators together for a denominator, vhich will be the answer required. Note. All factors that are common both to he numerators and denominators, may be ‘ancelled or omitted, by which means the ‘esult obtained will be in its lowest terms. EXAMPLES. i $x $39 0X $X BoE XEXG= HE XIEL e. 31x 4i= x Bosxsa?. 41°364 7328 multiply by product 303-115 392 ; . | , multiply by 14 1468 product *0205 52 Contracted MULTIPLICATION. See CONTRAC- TION. Mu ttiPxication of Circulating Decimals, is performed by converting the circulates into their proper fractions, and multiplying them together by the rule for multiplication of frac- tions. See Circulating DECIMALS. Mu tTiPxication in Algebra, is the method of finding the product arising from the multi- plication of any two or more indeterminate quantities; which may be divided into two cases. Case 1. When the multiplier and multipli- cand are both simple quantities. MU L Rule. Multiply the co-efficients of the twe quantities together, as in arithmetic; and an- nex to the result all the letters in both factors. But if there be two or more letters that are alike, as aa, bbb, xxxx, &c. enter into the product, these are to be represented by a’, b3 7, C0. ‘ And with regard to the sign to be prefixed the following rule must be observed. ; 1. If the factors are both affected with the sign +, or both with the sign —; the sign to be prefixed to the product must be +. But if one of the factors be +, and the other —, the sign of the product must be —. This is generally expressed by saying, like signs pro- duce plus, and unlike signs minus. EXAMPLES. l 4abx 6ab = 24aabb = 24a* b> 2. 62yXx . 12* ge eeoe yy" 3 Sayx—4ay” =—122*y3 4.—4abx —3cd =+12abed Case 2. When the multiplier and multipli- cand are both compound quantities. Rule. Place the multiplier under the multi- plicand. Then beginning at the left-hand, multiply each term in the multiplicand, by each term in the multiplier, by the last rule; and the sum of the several products will be the answer. Note. It is best to remove the leading term of yal product, one place more to the right and, EXAMPLES. a+ 5b 7a*+ 5a b* a + 6 5 Ye Sa | a~+ ab Q2lat + 15a} b* ab+b” | + 7a3b* + 5a*bt a + 2ab +b? | 2lat + 22a3b* + 5a*b* 7y +42 Be Yo 22" y — z 32 yt, te 22% 7y? + 4yz 9a? y* — 6xy2” — Tyz— 42° + 6xy2z*—42” 7y* — 3yz—4z7|9x*y * — 42” MUuttTIPLicaTiIon of Algebraic Fractions, is performed by the following Rule. Reduce all mixed expressions to im- proper fractions; then multiply all the nu- merators together for a new numerator; and the denominators together for a new deno- minator, and the result will be the product required. . Note. All factors that are common to the. numerators and denominators, may be can- celled or omitted. EXAMPLES. y Bale baer bdo _ Ahe ; wed u 3a*b aRee Liye ee Sa) ms Oe sdk fave eS Cee ah How ed eS , ra ae O30) Oana NAP MULTIPLICATION of Surds, is the method of finding the product of two or more izational factors. ‘ Rule. Reduce the given surds to their sim- plest form; and the radical parts thus arising to like radicals; then multiply the co-efficients together for a new co-efficient, and the surds together for a new surd; which being an- nexed to the preceding product, wil! be the answer required. Note. 1f the radicals be not the same, but the quantities under them be equal, the mul- tiplication will be performed by the addition of the indices representing those radicals, EXAMPLES. 1. 7X SOM B.S 105 2. V3x3y X JI8etmeyY3ry X3 72x = 32 V6x°y¥ = 32? V6y Y 7abx2/ 4a? c= 28a' ch—ai/ Whe axamzatim—ad — Yas n p no+mp 3 4 5. .a"™ xX aq=a 6 mq “faa tmp 1 1 n m ae 2. a" X bm mann x bun — i ebm MULTIPLICATION by Logarithms. See Lo- GARITHMS. MULTIPLICATION Table, a small table con- taining the products of all the simple digits, at least this is its natural limits, though it is most commonly carried as far as 12 times 12. It is otherwise called Abacus Pythagoricus, from Pythagoras, who is commonly said to be the inventor of it, though it appears highly probable that it must have had an earlier origin, as we can scarcely conceive any arith- metical operations to be carried on without it. The Abacus Pythagoricus was most likely a table of this kind carried to a much greater extent, which would have been a great ac- : j P| NAP % commodation to the Greeks on account o their embarrassed and complicated notation See ARITHMETIC of the Greeks. , MULTIPLIER, or Multiplicator, the num. ber by which another is multiplied. MURAL Arch (from murus, a wall), a wall, or arched wall, placed exactly in the plane o| the meridian, for fixing a large quadrant, sex- tant, or other instrument, to observe the me- ridian altitude, &c. of the heavenly bodies. Tycho Brahe was the first who used a mural arch in his observations; after him Hevelius, Flamstead, De Lahire, &c. used the same means. See a description of the mural arch at Greenwich observatory, con. structed by Mr. Graham, in Smith’s Optics, book iii. ch. 7, with the improvements of Mr, Bird, in the “ Method of constructing Mural Quadrants,” published by the commissioners of longitude in 1768. MUSICAL or Harmonial Proportion. See PROPORTION. MUSSHENBROEK (Peter), a very emi- nent mathematician and philosopher, was born at Utrecht about the year 1700. He was pro- fessor of mathematics first in that city, and afterwards at Leyden, where he died in 1761. His works, which are very reputable to his memory, are as follows: 1. Elements of Physico-Mathematics, in 1726. . 2. Elements of Physics, 1736. 3. In- stitutions of Physics, 1748. 4. Introduction to Natural Philosophy, 1762; and again in French by Sigaud de la Fond, 1769, in 3 vols. 4to. under the title of a Course of Experi- mental and Mathematical Physics. Besides these works Musshenbroek had several papers in the Paris Memoirs, from the year 1734 to 1760. - MYRIAD, the number ten thousand. N NABONASSAR, first king of Chaldea, me- morable for the Jewish era which bears his uame, and which corresponds to Feb. 26, 747, before Christ. See Epocn. NADIR, that point of the heavens which is diametrically under our feet, or opposite to the zenith which is directly over our heads. The zenith and nadir are the poles of the horizon. The Sun’s Napir is sometimes used to de- note the vertex, the earth’s shadow arising from the interception of the solar rays. NAPIER, or Neper (JOHN), baron of Mer- chiston in Scotiand, the celebrated inventor , of logarithms, was born in 1550. Napier had very early discovered great mathematical talents, and was particularly engaged in as- tronomical observations and computations, and consequently experienced all the diffi- culties which then attended this subject in consequence of the tedious operations of mul- tiplication, division, &c. which enter into them. He therefore was extremely anxious to abridge this labour, and his rods or bones were probably his first attempt of this kind. See Napier’s Rods. But his admirable in- vention of logarithms rendered useless every other contrivance directed to this end, and will Pee r aye . (ira Abdr ao at \“ iy, 2 5, cy), > a Pe erie. ; RPry aL A eS rete ON, ~ NAVIGATI Bes Bo J TH SNE | ae; LJ | os L PLL OY) VO , d LES ee YO PORK 3 f Ss X= S SESE 5 -i8 a ra a re iit Fj F re Engraved by Sam Lacey. Pee? - London Fablished July 11813. by G&S Robinson & the rest of the Proprietors, ee. * : , , 3} . j it ; ‘ ¥ , R “e . - » oats ae » posit 2 hl * oe NAP ver perpetuate his name, and render his me- tory dear to every astronomer and mathe- iatician to the latest posterity. See LoGa- ITHMS. It is not, however, on this invention only iat his fame is established; for he also made mnsiderable improvements in spherical tri- onometry, &c. particularly by his catholic or niversal rule, being a general theorem by hich he resolves all the cases of right-angled yherical triangles in a manner very simple id easy to be remembered, namely, by what > calls the five circular parts. See CIRCULAR ‘arts. His construction of logarithms too, »side the labour of them, manifests the great- it ingenuity. Kepler dedicated his Ephe- erides to Napier, which were published in ie year 1617; and it appears from many pas- ges in his letter about this time, that he counted Napier to be the greatest of his ‘e in the particular department to which he »plied his abilities. The last literary exertion of this eminent “rson was the publication of his ‘“‘ Rabdology id Promptuary,” in the year 1617 ; soon after ‘hich he died at Marchiston, the 3d of April _ the same year, in the 68th year of his ve. The list of his works are as follows: ‘Logarithmorum Canonis Descriptio,” 1614; ‘Mirisci Logarithmorum Canonis Construc- »,” &e.; Una cum Annotationibus aliquot vetissimi D. Henrici Briggii in eas memo- itum appendicum.” Published by the au- ‘or’s son in 1619. ‘“ Rabdologia, seu Neu- jzrationis per Virgulas libri duo,” 1617. This intains the description and use of the rods ( bones; with several other short and in- nious modes of calculation. His letter to Anthony Bacon, entitled, ‘Secret Inventions profitable and necessary | these Days for the Defence of this Island, id withstanding Strangers, Enemies to God’s ‘uth and Religion;” dated June 2, 1596. 'Napier’s Rods or Bones, a method con- ived by Lord Napier, for the more easy per- ming of the arithmetical operations of mul- ilication, division, &c. These rods are five inumber, made of bone, ivory, horn, wood, ( pasteboard,. &c. ‘Their faces are divided /.0 nine little squares, each of which is parted io two triangles by diagonals. In these (‘le squares are written the numbers of the tiltiplication table ; in such manner as. that (2 units, or right-hand figures, are found in (> right-hand triangle: and the tens, or ts left-hand figures, in the left-hand trian- f. ‘To multiply numbers by Napier’s bones, (pose the rods in such manner, as that ty top figures may exhibit the multipli- cad, and to those on the left hand, join the iL of units: in which seek the right-hand fare of the multiplier; and the numbers cor- ponding to it, in the squares of the other tls, write out, by adding the several num- i's occurring in the same rhomb together éd their sums. After the same manner write ‘t the numbers corresponding to the other ‘ures of the multiplier, disposing them under NAV one another as in the common multiplication ; and lastly, add the several numbers into one sum. For example, suppose the mul- 5978 tiplicand 5978, and the multiplier 937 937. From the outermost triangle 41846 on the right-hand figure, which 47934 corresponds to the right-hand fi- 53899 gure of the multiplier 7, write out Ze07356 _ the figure 6, placing it under the se line. In the next rhomb towards the left, add 9 and 5; their sum being 14, write the right- hand figure 4 against 6; carrying the left-hand figure 1 to 4 and 3, which are in the next rhomb; join the sum 8 to 46 already set down. After the same manner, in the last rhomb, add 6 and 5, and the latter figure of the sum 11, set down as before, and carry 1 to the 3 found in the left-hand triangle; the sum 4 join as before on the left hand, 1846. Thus you will have 41846 for the product of 5978 by 7. And in the same manner are to be found the products for the other figures of the multiplier; after which the whole is added together as usual. NATURAL Day, Year, &c. YEAR, &c. NATuRAL Philosophy. See PHILosopny. NAVIGATION, the art of sailing, or of conducting a vessel on the ocean, and is usually divided into navigation common, and navigation proper; the former relating to what is otherwise called coasting, or sich voyages as are conducted along the coast or shore of a country, and in which the navigator seldom or never loses sight of land. And the other to those voyages made from one country to another, through the trackless paths of the largest seas and oceans. - The origin of navigation, like that of all the other arts and sciences of ancient daie, is hidden in obscurity, some attributing it to one nation, and some to another. The Phoe- nicians, however, particularly those of Tyre, are now more generally considered as the first people who made any great advances in this important art. ‘These were afterwards fol- lowed up by the Carthagenians, who dis- covered the Fortunate Islands, or the Canaries; and even, according to some authors, America was visited by this enterprising people, but of this there is not sufficient proof. From Carthage and Tyre commerce and navigation were transferred principally to Alexandria, which latter city, when under the Romans, was only inferior to Rome itself, the latter being supplied with its merchandize wholly from the magazines of the former. Constantinople became afterwards the cen- tre of commerce, and navigation was for a long time pursued with great ardour by the merchants of that city; after this time it began to spread itself, though slowly, amongst the several European cities and nations, Genoa and Venice are particularly distinguished for the active part they took in promoting this important branch of human knowledge. The crusades, however, contributed in a See Day, NAV great measure to the revival, or at least to the more rapid progress of commerce and na- vigation; for the Genoese, the Pisans, Vene- tians, &c. farnished the transports which carried those vast armies, composed of all the nations of Europe, into Asia upon this wild enterprise, and likewise supplied them with provisions and military stores. Navigation was not, however, confined only to these religious fanatics; other travellers, beside those whom a mistaken religious zeal sent forth to Asia, ventured into remote coun- tries, from the prospect of commercial ad- vantage, or from motives of mere curiosity. Of these the most eminent was Marco Polo, a Venetian of a noble family, about the year 1269; and he was succeeded about half a century after by Sir John Mandeville, an Englishman. But the present art of navigation owed its rise to the invention of the mariner’s compass in the beginning of the fourteenth century ; and made considerable progress during the voyages that were began in the year 1418 by Henry Duke of Visco. This learned prince was particularly skilled in cosmography, and employed a person from the island of Majorca to teach navigation, and to make instruments and charts for the sea. These voyages being greatly extended after the discovery of Porto Santo and Madeira, the art was improved under the succeeding monarchs of Portugal; so much that Roderic and Joseph, physicians to John II. together with one Martin de Bohemia, a Portuguese native of the island of Fayal, a pupil of Re- siomontanus, about the year 1485, calculated, for the use of navigators, tables of the sun’s declination, and recommended the astrolabe for taking observations at sea. About this time Columbus, a native of Genoa, having thoroughly acquainted himself with the dis- coveries and observations of the Portuguese, and having also considerable experience in the art of navigation, made a proposal to John II. of exploring a passage to India by sailing directly towards the west across the Atlantic Ocean. John rejected this proposal in a very dis- honourable manner; and Columbus instantly quitted the kingdom, and landing in Spain towards the close of the year 1484, he resolved to propose his plan to Ferdinand and Isabella, with whom, after repeated ap- plication and long delay, he ultimately suc- ceeded. In August, 1492, Columbus, fur- nished with a small armament of three ships, set sail, and steered directly for the Canary Islands; from thence, holding his course due west, he stretched away into unfre- quented and unknown seas. After many dif- ficulties, he at length arrived at Guanahani, one of the large cluster of islands called the Lucaya, or Bahama Isles. He also discovered Cuba, Hispaniola, and several other small islands; and having left a small colony in a fort at Hispaniola, returned to Spain in March, 1493. In September following he set out on NAV his second voyage, and sailed by the Leewar Islands to Hispaniola. In a third voyage, ur dertaken in the year 1498, he discovered th continent of America; and in the same yea Vasco de Gama returned to Lisbon, from voyage to the East Indies by the Cape « Good Hope. Columbus, it is said, before h attempted the discovery of America, consulte Martin de Bohemia, as well as others, an during the course of his voyage instructed th Spaniards in navigation ; for the improvemet of which art, the emperor Charles V. afte wards founded a lecture at Seville. This a was considerably improved by the discover of the variation of the compass, and by tl use of the cross-staff. At length there wer published, in Spanish, two treatises containir a system of the art, which were in gre esteem, the first by Pedro de Medina, at Va ladolid, in 1545, called “‘ Arte de Navegar the other at Seville in 1556, by Martin Corte under the title of “‘ Breve Compendio de Sphera y de la Arte de Navegar, con new Instrumentos y Reglas.”’ This is said to hay been composed at Cadiz in 1545; and a tran lation of it was published in London, whi passed through several impressions. Besid the improvement which the art of navigati received from the proposals of Werner, aj Gemma Frisius, for finding the longitude, was much promoted by Pedro Nunez, » Nonius, who composed a treatise on this su ject so early as 1537, in the Portuguese la guage, which thirty years after was printed Basil in Latin, with additions, under the ti of “‘ De Arte et Ratione Navigandi.” In tl work the errors of the plane chart are expose and the problem of determining the latitu from two observations of the sun’s altitut and the intermediate azimuth, is resolv In 1577, Bourne published his treatise, ¢ titled “A Regiment for the Sea,” and intend as a supplement to that of Cortes. The mistakes of Medina were well expos by Michael Coignet, a native of Antwe who, in 1581, published a small treatise, titled “ Instruction Nouvelle des Points p excellens et necessaires touchant l|’Arte Navigar.” In the same year Robert Nj man published his discovery of the dippi needle, in a pamphlet called the “ Ni Attractive,” to which is always subjon William Burrough’s ‘‘ Discourse of the * riation of the Compass.” In 1594, capt John Davis published a small treatise, enti the ‘“ Seamens’ Secrets,” which was m esteemed in its time. The writers of © period complained much of the errors of plane chart, which continued still in though they were unable to discover a pre remedy; till Gerrard Mercator contrived universal map, which he published in k without clearly understanding the prineij of its construction; these were first discove by Mr. Edward Wright, who sent an ace of the true method of dividing the meric from Cambridge, where he was a fellow Mr. Blundeville, with a short table for # NAV pose, and a specimen of a chart so divided. »se were published by Blundeville, in 1594, ong his exercises; to the later editions of ich was added his “ Discourse of Universal ps,” first printed in 1589. However, in 9, Wright printed his ‘“ Correction of tain Errors in Navigation,” in which work shows the reason of this division, the man- of constructing his table, and its uses in ease A second edition of this treatise, h farther improvements, was printed in 0; and a third edition, by Moxon, in :7. The method of approximation, by what alled the middle latitude, now used by our va occurs in Gunter’s works, first printed 1623. About this time logarithms began ¢ introduced, which were applied to navi- ion in a variety of ways by Gunter; though first author who applied the logarithmic les to the cases of sailing was Thomas lison, in his “ Arithmetical Navigation,” ited in 1625. In 1635, Gellibrand printed Discourse Mathematical on the Variation he Magnetic Needle,” containing his dis- ery of the changes to which the variation is ject. In 1631 Richard Norwood published excellent “ Treatise of Trigonometry,” iy to the invention of logarithms, parti- uly in applying Neper’s generals canons ; for the farther improvement of navigation, ndertook the laborious work of measuring yegree of the meridian for examining the ange of the log-line. Of this affair he has ona full and clear account in his ‘ Sea- i’s Practice,” first published in 1637; where jalso describes his own excellent method etting down and perfecting a sea-reckon- ) &e. This treatise, and that of trigo- 1etry, were continually reprinted, as the cipal books for learning scientifically the of navigation. What he had delivered, cially in the latter of them, concerning subject, was contracted, as a manual for ors, in a very small piece, called his “ Epi- e,”’ which has gone through innumerable ions. About the year 1645, Mr. Bond ai in Norwood’s ‘“ Epitome,” a very it improvement in Wright’s method, by a erty in his meridian line, whereby his jsions are more scientifically assigned than eae himself was able to effect; which deduced from this theorem, that these sions are analogous to the excesses of logarithmic tangents of half the respec- latitudes, augmented by forty-five degrees ve the logarithm of the radius; this he rwards explained more fully in the third ion of Gunter’s works, printed in 1653; ithe demonstration of the general theorem { supplied by James Gregory of Aber- 1, in his “ Exercitationes Geometrice,” tted at London in 1668; and afterwards yJr. Halley, in Phil. Trans. No. 219, and yCotes, Phil. Trans. No. 388. In 1700, ' Bond, who imagined that he had dis- sxred the longitude, by having found out wtrue theory of the magnetic variation, ‘lished a general map, on which were de- NEB lineated curve lines, expressing the paths where the magnetic needle had the same variation: the positions of these curves will indeed continually suffer alterations; but they should be corrected from time to time, as they have already been for the years 1744 and 1756, by William Mountaine and James Dobson. The allowances proper to be made for leeway are very particularly set down by John Buckler, and published in a small tract first printed in 1702, entitled ‘‘ A New Com- pendium of the whole Art of Navigation,” written by William Jones. As it is now generally agreed that the earth is a spheriod, whose diameter at the poles is shorter than the other, Dr. Murdoch pub- lished a tract in 1741, in which he accom- modated Wright’s sailing to such a figure ; and Maclaurin, in the same year, gave a rule in the Phil. Trans. No. 461, for deter- mining the meridional parts of a spheriod, which speculation he has farther prosecuted in his “ Fluxions,” printed at Edinburgh in 1742. The principal foreign writers on navigation are Bartolomew Crescenti, of Rome, in 1607; father George Fournier, at Paris, in 1633; John Baptist Riccioli, at Bologna, in 1661 ; father Millet Dechales, in 1674 and 1677; the sieur Blondel St. Austin, in 1671 and 1673; M. Dassier, in 1683; M. Sauveur, in 1692; M. Bouguer, in 1698; father Pazenas, in 1733 and 1741; and M. Peter Bouguer, who, in 1753, published a very elaborate treatise on this subject, entitled ‘“ Nouveau Traité de Navigation,” in which he gives a variation compass of his own invention, and attempts to reform the log, as he had done in the Memoirs of the Academy of Sciences for 1747. He is also very particu- lar in determining the lunations more accu- rately than by the common methods, and in describing the correction of the dead reckon- ings. This book was abridged and im- proved by M. de Lacaille, in 1760; to these we may add Don George Juan, of Spain, in 1757. For an account of the several steps that have been pursued for the discovery of the longitude, see LonerrupE. For an account of Hadley’s quadrant, and its use in nautical observations, see QUADRANT. For the me- thod of making artificial magnets, and their use, see MaGnet, Compass, and NEEDLE. Those who are desirous of perusing a fuller account of the progress of navigation, may consult Dr. James Wilson’s preface to Dr. Robertson’s ‘“ Elements of Navigation,” in two volumes, octavo, in 1772; a work de- servedly held in the highest estimation. NAUTICAL Almanac, Chart, Compass, Planisphere, &c. see the several substantives. NEAP, or NEEpP-Tides. See TIDES. NEBULA, in Astronomy, faint luminous spots in the heavens, some of which consist of clusters of telescopic stars, others appear as luminous spots of different forms. The most considerable is one in the midway between NEB the two stars on the blade of Orion’s sword, marked §, by Bayer, discovered in the year 1656 by Huygens; it contains only seven stars, and the other part isa bright spot upon a dark ground, and appears like an opening into brighter regions beyond. Dr. Halley and others have discovered nebule in different parts of the heavens. In the Connoissance des Temps, for 1783 and 1784, there is a catalogue of 103 nebulz, ob- served by Messier and Mechain. But to Dr. Herschel we are indebted for catalogues of 2000 nebulz, and clusters of stars, which he himself has discovered. Some of them form a round compact system, others are more irregular, of various forms, and some are long and narrow. The elobular systems of stars appear thicker in the middle than they would do if the stars were all at equal distances from each other; they are. therefore condensed towards the centre. That stars should be thus accidentally disposed is too improbable a supposition to be admitted; he supposes, therefore, that they are brought together by their mutual attractions, and that the gradual condensation towards the centre is a proof of a central power of such akind. He observes also, that there are some additional circum- stances in the appearance of extended clusters and nebule, that very much favour the idea of a power lodged in the brightest part; for although the form of them be not globular, it is plain that there is a tendency to sphericity. As the stars in the same nebule must be nearly all at the same relative distances from us, and they appear nearly of the same size, their real magnitude must be nearly equal. Granting, therefore, that these nebule and clusters of stars are formed by mutual attrac- tion, Dr. Herschel concludes that we may judge of their relative age by the disposition of their component parts, those being the oldest which are most compressed. He sup- poses, and indeed offers powerful arguments to prove, that the milky way is the nebule of which our sun is one of its component parts. See GALAXY. Dr. Herschel has also discovered other phe- nomena in the heavens, which he calls nebu- lous stars; that is, stars surrounded with a faint Juminous atmosphere of large extent. Those which have been thus styled by other astrenomers, he says, ought not to have been so called, for on examination they have proved to be either clusters of stars, plainly to be distinguished by his large telescopes, or such nebulous appearances as might be occasioned by a multitude of stars at a vast distance. The milky way consists entirely of stars ; and he says, “‘ I have been led on by degrees from the most evident congeries of stars to other groups in which the lucid points were smaller, but still very plainly to be seen; and from them to such wherein they could but barely be suspected, until I arrived at last to spots in which no trace of star was to be discerned. But then the gradation of these jatter were by such connected steps as left ~ which is involved in a shining fluid, 0 NEG no room for doubt, but that all these phe mena were equally occasioned by stars y ously dispersed in the immense expanse the universe.’ In the same paper is given an account some nebulous stars, one of which is t described: “‘ Noy. 13, 1790. A most singt phenomenon! A star of the eighth magnitu with a faint luminous atmosphere of a cirer form, and of about three minutes in diame The star is perfectly in the centre, and atmosphere is so diluted faint, and eq throughout, that there can be no surmise 0, consisting of stars, nor can there be any do of the evident connection between the a | phere and the star. Another star, not m less in brightness, and in the same fiel¢ view with the above, was perfectly free fi any such appearance.” Hence, Dr. Hersc draws the following consequences: grant the connection between the star and the; rounding nebulosity, if it consist of stars y remote, ‘which gives the nebulous appearal the central star, which is visible, must: immensely greater than the rest; or if, central star be no bigger than common, h extremely small and compressed must. those other luminous points which oce the nebulosity! As by the former supposit the luminous central point must far exe the standard of what we call a star; so in latter the shining matter about the cel will be too small to come under the sa denomination; we therefore either hay central body which is not a star, or a} nature totally unknown to us. ‘This last ¢ nion Dr. Herschel adopts. He has, howe been since led to suspect that some of th which he has given as nebulze were comet great distances, and it is not improbablet this was one of this kind. Light reflec from the star could not be seen at this ¢ tance; besides, the outward parts are nei as bright as those near the star. Moreové cluster of stars will not so completely acco for the mildness or soft tint of the light these nebule as of a self-luminous flu * What a field of novelty,” says Dr. Herse “‘is here opened to our conceptions! as ing fluid of a brightness sufficient to reac from the regions of a star of the 8th, 9th, 1 11th, 12th “magnitude, and of an extent considerable as to take up 3, 4, 5, or 6) nutes in diameter.” He conjectures that 1 shining fluid may he composed of the perpetually emitted from millions. of sti See Phil. Trans. vol. 1xxxi. p. 1. NEEDLE Compass, Magnetic, Dipping,| see the several adjectives. NEEDLE, Variation of. See VARIATION. NEGATIVE, in Algebra and Arithme the reverse of positive, as negative index, ponent, quantity, sign, &e. ° NEGATIVE Index, of a logarithm, are which are affected with a negative sign; St are the indices of all numbers less than un See LoGARITHMS. NEG NeGATIVE Exponent or Power, is that which affected with a negative sign, being the m in which the reciprocal of any power is mmonly put to prepare it for expansion o a series, and is equally convenient for ny other algebraical operations. Thus gh wedge Be po) Sree, se 8 = — Ox ®, —_ x we (a + x)” —x) (a+2)—", &.; — 2, —3, and—n ing negative indices. That these expressions are equivalent to sh other, may be shown as follows : [he known rule for the division of powers, algebra, is to subtract their indices. 1 a? I tee ne CAS, OF x x3 I {n the same way = OE sshelnt bac eee ae 1 the same for any other similar expres- ns. NEGATIVE Quantities, are those quantities ich are preceded or affected with the ne- ‘ive sign. See the following article. NEGATIVE Sign, in Algebra, is that cha- ter, or symbol, which denotes subtraction, ng a short line preceding the quantity to subtracted, and is read minus; thus - b denotes that the quantity 6 is to be en ieee the quantity a, and is read, a wus 6, Che introduction of this character into al- »ra has given rise to various controversies, h regard to the legality or illegality of tain conclusions depending upon it; some intaining, that as a negative quantity is in If totally imaginary, it ought not to be roduced into a science, the excellency of ich depends upon the rigour and certainty ts conclusions; while others, running into opposite extreme, have endeavoured to strate what will not admit of illustration ; 1 thus, like other zealots, have been the atest enemies of the cause they were so volumes 8vo. printed in several editions nis works, in different nations, particularly edition with a large Commentary, by the » learned Jesuits, Le Seur and Jacquier, ae volumes, 4to. in 1739, 1740, and 1.2, '. A Paper concerning the Longitude; wn up by order of the House of Com- ns, ibid. ». Abregé de Chronologie, &c. 1726, under direction of the Abbe Conti, together with 1e observations upon it. . Remarks upon the Observations made ‘n a Chronological Index of Sir I. Newton, Phil. Trans. vol. xxxiil. ’. The Chronology of Ancient Kingdoms onded, &c. 1728, 4to. ‘. Arithmetica Universalis, &c. ; under the vection of Mr. Whiston, Cantab. 1707, There are also English editions of the ie, particularly one by Wilder, with a ‘mentary, in 1769. A Latin edition by tilian, 2 vols. 4to. Amsterdam, &c. '0. Analysis per Quantitatum Series, Flux- 3s, et Differentias, cam Enumeratione Li- rum Tertii Ordinis, 1711, 4to.; under the ection of W,. Jones, Esq. F.R.s. ‘1. Several Letters relating to his Dispute 1 Leibnitz, upon his Right to the Inven- _ of Fluxions ; printed in the Commercium ‘stolicum D. Johannis, Collins, &c. aliorum ‘Analysi Promota Jussu Societatis Regiz fram, 1712, 8vo. | 2. Postscript and Letter of M. Leibnitz the Abbé Conti, with Remarks, and a ter of his own to that Abbé, 1717, 8vo. which is added, Raphson’s History of xions, as a Supplement. 3. The Method of Fluxions, and Analysis Infinite Series, translated into English n the original Latin; to which is added, ‘erpetual Commentary, by the translator, . John Colson, 1736, 4to. ‘4, Observations on the Prophecies of Daniel the Apocalypse of St. John, 1733, 4to. ‘5. Newtoni Elementa Perspective Uni- ¥isalis, 1746, 8vo. 6. Tables for purchasing College Leases, 2, 12mo. ''7. Corollaries, by Whiston. 8. A Collection of several pieces of our a/hors, under the following title; Newtoni NIC Is Opuscula Mathematica Philos, &c. Philo} collegit J. Castilioneus; Laus. 1744, 4to. 8 tomes. 19. Two Treatises of the Quadrature of Curves, and Analysis, by Equations of an Infinite Number of Terms explained ; trans- lated by John Sewart, with a large Commen- tary, 1745, 4to. 20. Description of an Instrument for ob- serving the Moon’s distance from the fixed Stars. Phil. Trans. vol. xlii. 21. Newton also published Barrows’s Op- tical Lectures, in 1699, 4to. and Bern. Varenii Geographia, &c. 1681, 8vo. 22. The whole Works of Newton, were published by Dr. Horsley, 1779, 4to. in five volumes. NEWTONIAN Philosophy, the doctrine of the universe, and particularly of heavenly bodies, their laws, affections, &c. as delivered by Sir Isaac Newton. The term Newtonian philosophy is applied very differently; some authors under this phi- losophy, include all the corpuscular philosophy, considered as it now stands, corrected and reformed by the discoveries and improvements made in several parts thereof by Sir Isaac Newton. In which senseit isthaiGravy ide calls his elements of physics, “ Introductio Philosophiam Newtonianam.” And in the same sense the Newtonian is the same with the new philosophy, and thus stands con- tradistinguished from the Cartesian, the Pe- ripatetic; and the ancient corpuscular phi- losophy. Others, by Newtonian philosophy, mean the method or order which Sir Isaac Newton observes in philosophising, viz. the reasoning and drawing conclusions directly from phe- nomena exclusive of all previous hypotheses: the beginning from simple principles, deduc- ing the first powers and laws of nature from select phenomena, and then applying those laws, &c. to account for other things. And in this sense the Newtonian philosophy is the same with experimental philosophy, and stands opposed to the ancient corpuscular. Some, by Newtonian philosophy, mean that wherein physical bodies are considered mathematically, and where geometry and mechanics are applied to the solution of the appearances of nature. In which sense the Newtonian is the same with the mechanical and mathematical philosophy. Others again, by Newtonian philosophy, understand that part of physical knowledge which Sir Isaac Newton has handled, im- proved, and demonstrated, in his “ Principia.” While others mean the new principles which Sir Isaac Newton has brought into philosophy ; the new system founded thereon ; and the new solutions of phenomena thence deduced; being that which characterises and distinguishes his philosophy from all others. NICOLE (Francis), a celebrated French mathematician, was born at Paris in 1683, and died in 1758, in his 75th year. He was author of many papers on different mathe- NOL tical subjects, the whole of which are inserted in the Memoirs of the Academy of Sciences, from the year 1707 to 1747 inclusive, an ac- count of which may be seen in Dr. Hutton’s Math. and Phil. Dictionary. NIEUWENTYT (BERNARD), an eminent Dutch mathematician and philosopher, was born at Westgraafdyk, in North Holland, in August, 1654, and died in May, 1730, at 76 years of age. He was author of several works in the Dutch, French, and Latin languages, the principal of which are as follows: 1. The Religious Philosopher, proving the Existence of a God. 2. A Refutation of Spi- noza. 38. Analysis Infinitorum, 1695, 4to. 4. Considerationes secunde circa Calculi Differentialis Principia, 8vo. 1696. 5. A 'Trea- tise on the New Use of Sines and Tangents ; and a 6th work on the subject of Meteors. NIGHT, that part of the natural day during which the sun is below the horizon; though the twilight, both in the morning and evening, is sometimes considered as forming part of the day. See Day. NOBLE, an old English coin, value 6s.8d. NOCTILUCA, a species of phosphorus. See PHOSPHORUS. NOCTURNAL (from the Latin, nox, night), an instrument for taking the altitude of the stars, &c. formerly used at sea, but long since discontinued. NocTuRNAL Arch, the arch described by any of the celestial bodies during the night. NODATED Hyperbola. See “Hyrernoa. NODE Nodus, in the Doctrine of Curves, is a small oval fizure, made by the intersection of one branch of a‘curve with another. See CURVE. Nope, in Dialing, denotes a small hole in the gnomon of a dial, which indicates the hour by its light, as the gnomon itself does by its shadow. NopEs, in Astronomy, are the opposite points where the orbit of a planet crosses the ecliptic. Ascending Novk, is that where the planet ascends from the south to the north side of the ecliptic, which is denoted by the character &, and denominated dragon’s head. Descending NobDE, is that where the planet descends from the north to the south side of the ecliptic, which is denoted by the cha- racter 8, and is called the dragon’s tail. The right line joining these two points, is called the line of the nodes. It appears by observation that in all the planets, the liue of the nodes continually changes its place, its motion being in antece- dentia, or contrary to the order of the signs, the particular quantity for which, in each planet, will be found under their several names. NOLLET (the Aspe JoHN ANTHONY), a reputable French philosopher, was born in the district of Noyon, the 19th of November, 1700, and died in 1770. He was member of most of the learned societies in Europe; and was author of several works, viz. 1. Receuils des Lettres sur V'Electricité, 3 vols. 12mo. 1753. NOT . Essai sur l’Electricité des Corps, 1 4 ieee 3. Recherches sur les Causes pai culiers des Phenomenes Electriques, 1 y 12mo. } Besides which, he had numerous papers the Memoirs of the Academy of Sciene from the year 1740 to 1770 inclusive. / NONAGESIMAL Degree, called also M Heaven, is the highest point or 90th degree the ecliptic, reckoned from its interseeti with the horizon at any time. NONAGON (from yove, nine, and yw angle), a figure of nine angles and nine sid The angle ; at the centre ofa nonagon is 4 the angle subtended by its sides "140°, a its area when the side is 1 = 2 nat. tang. 4 = 6°1818242. NONES, were certain days in the Rom calendar, viz. the 5th day of the months Jar ary, l’ebruary, April, June, August, Septemb November, and December. - NONIUS, or NUNEZ (PETER), an emin Portuguese mathematician and physician, born at Alcazar, in Portugal, in 1497, a died in 1577, at the age of 80 years. Fn this author is derived the name of the inst ment called the nonius, from his having 4 scribed it in one of his works, ‘‘ De Crepi culis,” although he does not himself claim 1 invention of it; but says it was probablyi unknown to Ptolomy. The invention of it however, more commonly attributed to Vi NIER, by which name itis also sometimes call and under which it is described in this wor) The list of the works of Nonius are as} lows: 1. De Arte Navigandi libri duo, 1530. 2. Crepusculis, 1542; Annotationes in Aris telem. 4. Problema Mechanicum de Me Navigii ex Remis. ‘5. Annotationes in P nitarum Theorias Georgii Purbachii. 6. Lil de Algebra in Arithmetica y Geometra, 156 Nonius. See VERNIER. NORMAL, is the same as perpendien) See PERPENDICULAR and SUBNORMAL. NORTH, one of the four cardinal points NorTH Star. See Po.e Star. NORTHERN Signs, are those that are the north side of the equator, viz. Aries, tT rus, Gemini, Cancer, Leo, and Virgo. — NORT HING, in ‘Navigation, is the dif ence of latitude which a ship makes in si ing towards the north. NOTATION, in Arithmetic, is the meth of expressing, by means of certain symbols characters, any proposed number or quanti In the modern analysis, notation implies method of representing any quantity or 0} ration, and the judicious selection of proj symbols for this purpose, is an important ¢ sideration, which every author who und takes to write on this subject should p cularly attend to. In the common scale of notation evi number is expressed by means of the ten ¢ racters, 0, 1, 2, 3, 4, 5, 6, 7, 8,9, by giving each digit a local as well as its proper or ' tural value, the discovery of which was p <= NOT ps one of the most important steps that has er been made in mathematics, and does much honour to its inventor as any other the history of this science. To whom we are indebted for this improve- nt is not known, nor even the nation whence lates its origin. Some authors have attri- ted the honour of it to the Arabs, others to » Greeks, and others again to the Indians ; ich latter people have certainly a priority of im to both the other nations ; but whether "y were the original inventors, is at present, 1 probably ever will be, undetermined. e simplicity of this method of expressing antities, and the universal application of it every species of calculation, render it so viliar to all our ideas of numbers, that we e sight of the ingenuity of the invention, that which ought most to distinguish it; tis, the obvious and accurate principles on ich it is founded; and instead of consider- it as a most ingenious discovery, we are ' to treat it as a necessary consequence owing immediately from the nature of num- ' itself. That this is however a mistaken ‘ion, is evident from the notation of the seks and Romans, to whom this method is unknown; in fact, it does not appear to ‘e been introduced into Europe till the er end of the tenth century, when it was t taught by the Arabs to the Spaniards, h whom they had a communication; and ice it soon after became generally known ill the European nations, and seems to have n brought into England as early as the r 1150. “he Arabs, it is evident from several of ir manuscripts, derived their knowledge of sither directly, or through some other na- i, from the Indians, it being entitled, in 1e of their works, The Indian Arithmetic ; ,as we before observed, whether this latter ion originally made the discovery is un- wn. With regard to the characters or its, by which numbers are at this date versally expressed, they seem to be ‘the ie, With a very slight alteration, as those t were originally employed for that -pur- ¢; but their forms are not such as to indi- 2 their origin, though some authors have overed more ingenuity than judgment, endeavouring to trace them to the Greek iabet, and hence inferring, contrary to ry evidence, a Grecian origin to our pre- it system of notation. See the different hmetical characters, (Plate X.) nthe common, or denary scale of notation, value of every digit increases in a tenfold Iortion from the right towards the left; 3, 1111 = 1000 + 100 + 10+ 1= 103 + +10+1; also 3464 — 3.10% + 4.107 + +4, and so on of others; the distance any figure from the right indicating the rer of 10, and the digit itself the number of se powers intended to be expressed, on ch obvious principle itis evident, that any aber whatever may be represented with *2 and simplicity; but since also a number NOT may be expressed by assuming any other radix instead of 10, the curious reader will inquire how it happened that the number 10 came to be adopted as the radix of our nota- tion in preference to any other. The fact is, that long before the invention of our present notation, all nations, at least with a very tri- fling exception, divided their numbers into periods of 10, and this singular coincidence of different people, unknown and unconnected with each other, has been the subject of phi- losophical investigation and inquiry from the time of Aristotle to the present day, though it is now generally allowed to have had its origin in the formation of man, that is, to his having 10 fingers, by means of which it is highly probable that computation, or at least numbering, was first effected; as we always see children, in making their first efforts in calcu- lation, have recourse to this means for assist- ing their memory; and hence we may infer that the present division of numbers into periods of 10, had its origin as soon as num- bering was attempted, that is, as soon as man began to associate with each other. See some ingenious remarks on this head in Montucla’s ** Histoire des Mathematiques,” tom. 1. This, therefore, being the case, with regard to the division of numbers, the choice of the radix 10 for the present system was the most natural of any number that could have been selected, though it was not the best; for there is no doubt that 12 is much better adapted to this purpose; the advantages of it, however, are net such as to lead us to expect, or even to wish, that it should ever be sub- stituted for that, which long established cus- tom has rendered so familiar to all our ideas of number. But notwithstanding the present scale is as convenient as can be desired, an investigation of the general principles on which it is founded, and their application to other numbers as radii, have not escaped the speculative mathematician ; and our rea~ ders will therefore expeet some informatien on the subject. Of the different Scalés of Notation. In general any number N may be repre- sented by the formula N=a.r"+br"—!+ cr”—? + &e. pr* +qr+u, where r the radix, may be any number what- ever, and a, 6, c, &c. integer numbers less than r. And according as r is assumed = 2, 3,4, &c. the different scales or systems re- ceive the following denominations. If7* = 2 itis termed the Binary scale TULA O° Lies tevetase teva cee Ternary scale PF RSUae Yoke Wi rteady aa.98 Quaternary scale Poe TGR. cnskss eras Quinary scale FiO tasvesen take i iettd. 2 Senary scale TH 1O ol. seteceee tasters Denary scale Fake m ld oP Tee orraduae-' »» Duodenary scale And since the co-efficients a, b, c, &c. are always less than r, therefore it follows that for any scale we must have as many characters, including the cipher, as is equal to the num- il NOT Her expressing the radix of the system; thus the characters are for the Binary scale....... 0,1 Ternary scale..... 0, 1,2 Quaternary scale 0, 1, 2,3 Quinary scale..... 0, 1, 2, 3, 4 Senary scale....... 0, 1, 2,3, 4,5 Denary scale...... 0,1, 2,3, 4, 5,6, 7, 8, 9 ’ And hence it follows, that for the duode- nary scale two other characters are neces- sary for expressing 10 and 11; assuming, therefore, 10 =, and 11 = 7; the digits of this system are . 0, 1, 2, 3,'4, 5, 6, 7, 8, 9, 9, 7 To trunsform a Number from the Denary or Common Seale, to any other Scale. Rule. Divide the number successively by the radix of the given scale, and the several remainders will be the digits of the number required. Exam. Transform 1810 into the binary and ternary scales. 2)1810 3)1810 2) 905 rem. = 0 3) 603 rem. = 1 2) 452 1 3) 201 0 2) 226—_ 0 3) 67 —— 0 ek aa 3) 22—— 1 2),b6i42-1, 34 By Fee Cok 2) 98 =44 (0 Beep Q) 14!) 0 TO AW Nap see Se oF cs | OV eres 3 ~O i Therefore 1810 = 11100010010 in the bmary scale. And ry scale. 2. Transform the two numbers 844371 and 215855, from the common to the duodenary scale. 1810 — 2111001 in the terna- 12)844371 12)215855 12)70364rem. =3 12)17987rem. > ll=r 12)5863 8 12)1498 =ll=qc 12)488——_ 7 12)i24—- = 10 =@ 1240 8 12)l0 — = 4=4 123—— 4 “0——=10=9 o— 3 Hence 844371 = Ania in the duode- And (215855 = 0497 § nary scale. And thus a number is readily transformed from the denary to any other system of which the radix is given, and hence we find 1000 is expressed in the following manner according to the value of the radix r. Ifr = 2, 1000 = 1111101000 ria (38,1000 = 1101001 r— 4, 1000 — 33220 7 = 5,,1000 — 13000 5,5. LOUD = 4344 eee ta AO oe 2626 TOSS 5, AS ee 1750 y= 9,1000=— 1331 r = 10, 1000 = 1000. r = 11, 1000 = 820 | ry = 12,1000 = 674 | With regard to the transformation of nunber from any other system to the comm scale, this is readily effected; for from w) has been before observed, the distance of 4) digit from the right indicates the power of radix, and the digit itself the number of th powers intended to be expressed; thus, 348873, and $4¢r7, in the duodenary se; become (3.125 4+ 4.12+ + 8.423 4- 8.12*4+ 7 § + 32844371 l 10. 12+ + 4.123 +10.12*+ 11 + 11 = 215855 and in a similar manner the transforma’ may be effected with any other radix. — more on this subject in Barlow’s ‘“ Elen tary Investigations,” &c.; see also Du NARY Arithmetic. . | NoraTIOn of the Greeks. These people three distinct notations; the most simpl which was, by making the letters of t alphabet the representatives of numbers, 8, 2; y,3; and soon. Another method by means of six capital letters, thus, I [s pow | 1; [were] 5; A [Sexe | 10; H [Hex 100; X [xr] 1000; M [pve] 10000: | when the letter II enclosed any of these, cept I, it indicated the the enclosed let be 5 times its proper value, as stated_ab thus, |a| represented 50; Ju{| 500; |x] and so on. This kind of notation was used to represent dates and similar cé but for arithmetical purposes they had aj organised system, in which thirty-six” racters were employed, and by meat which any number not exceeding 10000 might be expressed; though in the firs stance it appears that 10000, or a myria the extent of their arithmetic. Our digits 1,°2,''3, 4, Op They express- ed by the a B, Y> ) &, Sy ‘¢ letters..c..6c8 ’ 10, 20, 30, 40, 50, 60, 70,8 For ourtens, as i, X, Ay By Vy zy 0, 4 ra » % 7,4 They employ- ed the letters The. hundreds eee aah 0 oT, uO, x, t,o And the thou- sands by..... i pe; B, % 3, fy. £6 ¢ | That is, they had recourse again to th racters of the simple units, which were tinguished by a small iota or dash p below them; and with these charact number under 10000 was readily expré and this, as we have observed above, wi some time, the limit of their arithmetic, terwards 10000, or a myriad, was repres by M; and any number of myriads by M} under the number of them. Thus, a Y x M, M, M, represented 10000, 20000, 130000, &c | NOT ter this character followed the number of hits: thus Ay.6449 represented 339999 ; but ) M’ ‘is character for myriads being inconvenient, ‘as afterwards changed for Mv set after the jumber of myriads ; so that the above number ‘as sometimes written AyMv 0449, and this jaracter was latterly entirely omitted, and the iyriads were distinguished from the simple nits only by a point placed between the two. By these means the notation of the Greeks is extended to 100000000 ; and it was after- i farther improved by Archimedes, Apol- aius, &c. The following numbers, when com- pred with what has been stated above, will ‘rve as an illustration of this method. | 91 = ala 9917382 = Aha. ComB 9999 = 4240 171000 = sda 17382 = ern 647499 = £0.2040 8036 — as 1110000 = dios 16420 — = sux 3000001 = ra 4001 — Ya 99999999 = 9749.01 ‘See some farther illustration of the Greek thmetic under the article ARITHMETIC. ‘Notation of the Hebrews, resembled, in a ‘eat measure, that of the Greeks above de- ‘bed ; thus, ta, 2, 3, 4, -5,.6, 7, -8,.9, 1e Hebrews ‘used their ‘letters wrourtens,as 10,20,30,40,50,60,70, 80,90, employ- led Fis shes . Lad 2, 2 D, 3,.D, 0, DB, ¥- ts, She oats | Dea Cly Wise, Ihertly kee ‘dreds they i», oA fuk 4p aes NB ark ah | And for representing thousands, they had ain recourse to. their simple units; distin- ishing them only by two dots, or acute a i “ cents ; thus x, or rs, expressed 1000 ; 13,2000; 10000; andsoon. Sometimes they express bir 1000 by the character 8 with two dots irked above it, which is an abbreviation of i> word mbes, or oDdx, signifying one thou- ad, or thousands; to which other letters 're prefixed, expressing the number of »usands at pleasure; thus x for 1000; Na, £90; 7, 4000; s, 10000; ND, 100000, &e. d from these complete or round numbers, they may be so called, they formed com- und numbers, as in the following table: x 1 tt AIO ‘o>. 12 ‘op! Ltd. ime. 3 13 mow 3848 im) 14 ap 625 ti et ,25 me: 1001 aes a> 36 N¥YN 1017 aa 47 Int 4004 nm) 68 ayyp 100074 / xp 101 vare—«1727 .. . NOT At is to be observed, that the number 15 was never expressed by 71, » signifying 10, and n 4; because they considered it to be profane to use one of the names of the Deity for a uum- ber; they therefore denoted 15 by yp, } being 6, and 1» 9; the sum of which is 15. For the same reason, }0, that is, 9 and 7, is used in- stead of », 10 and 6, to express 16. The twenty-two Hebrew letters express numbers as far as 400; and the five remain- ing hundreds, to complete the 1000, are ex- pressed by different forms of five of the letters, which seem to have been invented expressly for this purpose. See Rees’s Cyclopedia, ar- ticle CHARACTER. NoraTIon of the Romans. This is still em- ployed by us for dates and other similar pur- poses; and is too well known to require a very minute description. ‘The Roman nume- ral characters are seven in number; viz. L. one; V. five; X. ten; L. fifty; C. a hundred; D. or I9. five hundred; M. one thousand: this Jast number is also sometimes expressed by Dq. or by CIg. And by the various combi- nations of these characters, any number what- ever might be expressed, as in the following table: mn As often as any character oa ll is repeated, so many times — “Cis its value repeated. A less character before a 4 — I1V.< greater, diminishes its value Uby the less quantity, Posen V A less character after a 6. VIL) greater, increases its yalue by the less quantity. yf Nance i FM . 5 VILL. Oo = LX: Ips 3X. Lives Al 40 = XL. DO! se. 60..—> Lax. 100 = C. For every < added, 500 = D. or Lo.) this becomes ten umes as much. ( For every vO fe one at each end, this 1000 = M. or CI5. becomes ten times as much. we ( A lime over any 5000 = I99. or V.< figure increases it . 2 1000 fold. 6000 — VI. 10000 = X. or CCI9g. 60000 = I9992. 60000 = LX. 100000 = M. or CCCI999.- 1000000 = MM. or CCCCI9Q999- For the various notations made use of in the several branches of Analysis, Astronomy, Geometry, kc. see the respective articles; see also CHARACTER and SIGN. I12 NUM NOVEMBER, the eleventh month i in the Julian year, but the ninth in the year of Ro- mulus, beginning with March, whence its name. In this month, which contains thirty days, the sun enters the sign }, usually about the twenty-first day of the month. NUCLEUS, the Kernel, is used by Heve- lius and some ‘other astronomers for the body of a comet, which others call its head, as dis- tinguished from the tail or beard. Nucleus is also used by some writers for the central parts of the earth and other planets, which they suppose firmer, and separated from the other parts, as the kernel of a nut is from the shell. NUMBER, in its most extended significa- tion, has a reference to every abstract quantity that can be made the subject of arithmetical computation; but, in a more limited sense, it signifies only several things of the same kind, and is defined by Euclid to be a multitude of units. Numbers of this latter kind are termed integral, to distinguish them from the other kind of numbers, which are of various deno- minations ; as exponential, fractional, loga- rithmic, surd, &c. See EXPONENTIAL, FRAc- TION, SURD. Integral NuMBERS are distinguished into va- rious classes, as follows: Absolute, Abstract, Abundant, Amicable, Cardinal, Circular, Com- posite, Coner ‘ete, Figurate, Homogenial, Irra- tional, Or dinal, Per rfect, Polygonal, Prime, Rational, &e.; for which see the respective adjectives, Properties of NUMBERS, are certain theo- rems relating to the Divisors, Forms, Powers, Prodicts, &c. of numbers; many of which are given under the respective articles, and some others of a more general nature are enumerat- ed below. 1. The product of two numbers is the same, whichever of the two is the multiplier. 2. If any number p, be prime to two other numbers, « and J, it is also prime to their pro- duct ab. 3. Every integer number, without excep- tion, is composed of, and may be resolved mto, different terms of the geometrical series 1, 2, 4, 8, 16, &e. 4. Every integer number, without excep- tion, may be made up by “the addition and subtraction of different terms in the series 1, 3, 9, 27, 81, &c.; and therefore with sucha series of weights, any number of pounds may be ascertained. 5. The dfitctenes between any cube and its root, is divisible by 6; between any fifth power and its root, is divisible by 10; and, generally, if n be a prime number, the difference 2” — “7 is divisible by 2n. 6. Every number whatever is the sum of three, ora less number of triangular numbers; of four, or a less number of squares; of five, or a less number of pentagonals; and so on for hexagonal, heptagonals, &c. This theorem was invented by Fermat, but it has never yet been demonstrated generally ; an attempt at NUM it was made in vol. xxi. of Nicholson’s Phil sophical Journal, but it is by no means sati factory. The only cases that have received complete demonstration, are the two first, f triangular and square numbers. For other properties of numbers, see Div sors, Factors, Forms, Powers, PRIME ‘a bers, ‘Propucts, &e. Theory of NuMBERS, is the investigation an demonstration of certain theorems Yelating the properties of numbers. This is a subje with which the ancients were little acquait ed; in fact, their method of notation was ; insuperable impediment to any investigatio of this kind. Diophantus, however, may said to have treated on the properties of nw bers; but it was his commentators, Bael and Fermat, who first laid the foundation the present theory: the latter, in particul discovered many fine theorems, which s§' retain the name of their inv entor; but th were mostly given without demonstration, his notes on Diophantus, in one of which distinctly mentions a work that he was Pp paring on the subject of numbers; but was unfortunately lost at his death. WI Fermat thus left incomplete, was taken up Euler and Lagrange, who demonstrated se ral. of his theorems, the former in the Aj Petro. and the latter in the Memoirs of Berl and, latterly, a very ingenious work, entit ve Essai sur la Théorie des Nombres,” in 4 was published at Paris by Legendre; and; other about the same time by M. Gauss, Strasburg, entitled “ Disquisitiones Arithn tice,” and which has since been transla} into French by M. Poullet de Lisle. 7 two works contain, beside the demonstrat of the greater part of Fermat’s propositie several ‘interesting and original theorems | latter of which authors has, by means of ¢ tain numerical properties, been able to SO any equation of the form 2— 1 = 0, proy ing n be a prime number; and ifn be also | form 2” 4~1, he obtains this solution by q dratic equations only, and is thus enabled inscribe geometrically, in a circle, a polygol seventeen sides; which was far from b considered possible, till the publication of} celebrated work. See an investigation explanation of this theorem, and many ot Of a curious and interesting nature, in low’s “Elementary Investigation of the The of Numbers.” F Golden NUMBER. See GOLDEN Number CYCLE. NuMBER of Direction, in Chronology, s one of the 35 numbers between the Ea limits, or between the earliest and latest d on which it can fall; viz. between March and April 25, which is 35 days, and is so cal because it serves as a direction for find Easter for any year; being, indeed, the nt ber that expresses how many days after ae of March Easter-day falls. Thus, Eas’ day, falling as in the first line below, the nt ber of direction will be as on the lower lim NUM »* ; March. ! nn aster-day, 22,23, 24, 25, 26, 27, 28, 29, 30, 31 ‘o.of Direct. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 April. jorerear Chaat le aster-day, 1, 2) 3, 4, 5, 6,.7," 8, &e- o. of Direct. 11, 12,13, 14, 15, 16,17, 18, &e. ‘dso on, till the number of direction on the wer line be 35, which will answer to April , being the latest that Easter can happen. rerefore add 21 to the number of direction, id the sum will be so many days in March + the Easter-day ; if the sum exceed 31, the ess will be the day in April. To find the Number of Direction.—Enter the lowing table with the Dominical Letter on > top, and the Golden Number on the right nd; then where the columns meet is the mber of direction required for that year. GiN.| A} Bj} Ci} D|E];F,G mel 1,29") 27,|. 8 | 19 | 30 | 24 | 25 2 19 | 13] 14] 15] 16] 17 | 18 3 TRE ite as Se al a ot aS ae a | 26 17°27), 214).22. | 23 | 24 [25 mem | 12) .13 | 141.15 | 16) 10). 11 6 |.33 | 34 | 35 | 29 | 30 | 31 | 32 7 19 | 20 | 21 | 22] 23 | 24 | 18 8 Poti thet Ort eh Oe LL | 9 | 26 | 27 | 28 | 29} 30] 31 | 32 MIO. | 19.4 2001 21 |-15 1 16 |.47°), 18 ll Sle Gais e+ | oe | 10 bed 112 | 26 | 27 | 28 | 29 | 23 | 2 | 2 Y 13 12/13 | 14] 15] 16 | 17 | 18 14 DZi, Geta, bf} 2,8 bee P15 | 26 | 20] 21 | 22 | 23 | 24 | 24" —~=«16 12°) 182)14 PISS 10 Et 17 | 83 | 34 | 28 | 29 | 30} 31 z 18 19 | 20 | 21 | 22 | 23) 17] 18 mo 112) Gi ay si 2110; It Thus, for the year 1790, the Dominical Let- being C, and the Golden Number 5; in » column C and in the line 5 stands 14, the mber of direction; to this add 21, which es 35, subtract 31 and there remains 4; ich is the day of April on which Easter-day lin the year 1790. NUMERAL Figures, or Digits, are those ares by which all numbers are expressed arithmetic. See NoTATION. NuMERAL Letters, are the seven Roman ca- als I. V. X. L. C. D. M. See Notation the Romans. NUMERATION, in Arithmetic, is the art reading, or estimating, the value of any mber expressed by the ten characters 1, 2,3, . any how combined or repeated; and is, srefore, the reverse of notation, which is ex- »ssing by those characters any number pro- e . In order that the nine significant figures 0 express not only units, but tens, hundreds, | NUT thousands, &c. each character has a local value given to it, as stated under the article Notation; so that though when alone they ‘express only units, yet in the second place they denote tens; in the third, hundreds; in the fourth, thousands: thus, 5555 represents five thousands, five hundreds, fifty and five. Hence then, to express any written number, or to assign the proper value to each charac- ter, beginning at the right hand, divide the proposed number into periods of three places each, and consider two of these as forming one period of six places: then each of these greater periods has aname common to all the firures of which it is composed ; the first six being units; the next six, millions ; the third, millions of millions, or billions; the fourth, trillions; and so on: also every half period of three places is read separately, or by itself, so many hundreds, tens, and units; only after the left-hand half of each period, the word thousands is repeated; and at the end of each complete period, its common name is added. Thus, 4,591, is four thousand, five hun- dred and ninety-one. 346,718, is three hundred and forty- six thousand, seven hun- dred and eighteen. 47,671,600, is forty-seven million, six hundred and seventy-one thousand, six hundred. seven billion, four hun- dred and eighteen thou- sand, four hundred and sixty-seven million, six hundred thousand. Tt should be observed, however, that the French enumerate their numbers in a different manner, calling the first three, hundreds and units; the next three, thousands; the next three, millions; the next three, billions; the next three, trillions, &c. See BiLion. NuMERATOR of a Fraction, is that number which stands above the line, and shows how many parts the fraction consists of; as the de- nominator represents the number of parts into which the unit is supposed to be divided. _ NUMERICAL, or NuMERAL, something that relates to number. NUMERAL Algebra, are those cases in which numbers are employed, in contradis- tinction to Literal Algebra, or that in which the letters of the alphabet are made use of. NUTATION, in Astronomy, a kind of tre- pidation, or tremulous motion of the axis of the earth, whereby its inclination to the plane 7,418,467,600,000, is of the ecliptic is not always the same, but va- . ries backwards and forwards some seconds ; and the period of these variations is nine years. This nutation was discovered by Dr. Bradley, who published an account of his dis- covery in the year 1737. This is an obvious result of the Newtonian system of attraction, the first principle of which is known to be, that all bodies mutually attract each other in the direct ratio of their masses, and in the inverse ratio of the squares of their * OBJ distances. If the terrestrial orbit was a circle, and the globe itself a perfect sphere, the at- traction of the sun would have no other effect than to keep the body in its orbit, and world cause no irregularity in the position of its axis; but neither is the earth’s orbit a circle, nor its bodya sphere ; for the earth is sensibly protuberant towards the equator, and its orbit is an ellipsis, which has the sun in its focus. Now when the position of the earth is such, that the plane of its equator passes through the centre of the sun, the attractive power of the sun acts only so as to draw the earth to- wards it, still parallel to itself and without changing the position of its axis, and this happens at the equinoxes. In proportion as the earthrecedes from those points, the sun also goes out of the plane of the equator, and ap- proaches that of one or other of the tropics; the semi-diameter of the earth, which is then exposed to the sun, being no longer equal, the equator is more powerfully attracted than the rest of the giobe, which causes some altera- tion in its position and its inclination upon the plane of the ecliptic: and as that part of the orbit which is comprised between the an- tumial and vernal equinox, is less than that which is comprised between the vernal and autumnal, it follows that the irregularity caused by the sun, during his passage through the northern signs, is not entirely compensated by that which he causes during his passage through the northern signs; and that the pa- rallelism of the terrestrial axis, and its ineli- nation with the ecliptic, will be a little changed. The same effect also which the sun produces upon the earth by its attraction is also pro- duced by the moon, which acts with greater force in proportion as it is more distant from O - y Ops ECT, any thing presented to the mind, either by sensation or imagination: but, in Optics, it is more particularly employed to denote that thing which we intended to mag- nify or to represent. OxJect Glass, in Optical Instruments, is that which is placed towards the object, the other extreme lens being called the eye-glass, being that to which the eye is directed. For the method of finding the true centres of object-glasses, see Smith’s ‘“ Optics,” book 3, chap. iii.; also Phil. Trans. vel. xIviii. p. 177. OBJECTIVE Line, in Perspective, is any line drawn on the geometrical plane, the re- presentation of which is sought in the draught or picture. OBJECTIVE Plane, is any plane situated in more influenced than at any other time; § OBL the equator: now at the time when the node concur with the equinoctial points, its greates latitude is added to the greatest obliquity ¢ the ecliptic. At this time, therefore, the powe which causes the irregularity in the positio of the terrestrial axis acts with the greates force; and the revolution of the nodes of th moon being performed in eighteen years, itj clear, that in eighteen years the nodes wi twice concur with the equinoctial points and, consequently, twice in that period, ¢ once every nine years, the earth’s axis will b that it will havea kind of balancing backwar and forward, the period of which will be nin. years, as Dr. Bradley had observed; and th balancing he calls the nutation of the terrestru axis. See Phil. Trans. Nos. 406 and 485. | Several effects necessarily arise from th nutation of the terrestrial axis; of which th principal, and that which is most readily pa ceived, is the change in the obliquity of th ecliptic, the quantity of which ought to var from that cause, eighteen seconds in aboy nine years. The nutation changes also th) declination, right ascension, aud longitud of the fixed stars; but their latitude is n¢ affected by it, because the ecliptic is immo able, as connected with this theory. For the mathematical investigation of th eflects of the system of universal attractio in producing the nutation, &c. see d’Alen bert’s “* Recherches sur la Precession dé Equinoxes;” Silvabelle’s Treatise on the sam subject ; Walmsley’s treatise ““De Prece: sione Equinoctiorum,” &c. in the Phil. Trani 1756; and more particularly the “ Mechaniqu Celeste” of Laplace; Biot’s “ Traité Elémer taire d’Astronomie Physique,” and other work on physical astronomy. if the horizontal plané, whose perspective repre sentation is required. ; OBLATE, flattened or shortened. , OBLATE Sphere. See SPHEROID. 7 OsLATE Figure of the Earth. See DEGREL OBLIQUE, aslant, indirect, or deviatin either from perpendicularity or parallelism. OBLIQUE Angle, Ascension, Descension, D rection, Force, Line, Plane, Projection, Sai ing’, Sphere, &c. See the several substantive! OBLIQUITY of the Eeliptic. See Ect £9 (od es OBLONG, is properly a right-angled para lelogram, of which the length and breadth ar unequal, ‘The same word, however, is cor monly employed to denote any figure whic is longer than it is broad, or even a solid OBS us a prolate sphere is sometimes called an Jong spheroid. OBSERVATION, in Astronomy and Navi- tion, denotes the measuring, with some in- rument proper for the purpose, the angular stance, altitude, &c. of the sun, moon, or her celestial body. ¥ OBSERVATORY, OBSERVATORIUM, @ ace destined for observing the heavenly bo- es; ora building usually in form of a tower, jsed on some eminence, and ‘covered with terrace for making astronomical observa- ons. The more celebrated observatories e, ‘1. The Greenwich Observatory, or Royal bservatory of England, was built in 1676, y order of King Charles If. at the solicita- on of Sir Jonas Moore and Sir Christopher Vren; and furnished with the most accurate istruments by the same, particularly a noble xtant of seven feet radius, with telescopic ghts. The person to whom the province of bserving was first committed, was Mr. J. ‘amstead; a man who, as Dr. Halley ex- resses it, seemed born for the employment. ‘or the space of fourteen years, with un- rearied pains, he watched the motions of the lanets, and particularly thuse of the moon, as vas given him in charge; that a new theory f that planet being found, exhibiting all her regularities, the longitude might thence be ‘etermined. In the year 1690, having pro- ‘ided himself with a mural arch of seven feet iameter, well fixed in the plane of the meri- jan, he began to verify his catalogue of the ixed stars, (which hitherto had depended al- ogether on the distances measured with the extant) after anew and very different manner ; iz. by taking the meridian altitudes, and the novements of culmination, or the right ascen- ion and declination. This instrument he was so pleased with, hat he laid the use of the sextant almost vholly aside. Royal employed for thirty years; in the course f which time nothing had appeared in public vorthy so much expense and preparation: so hat the observer seemed rather to have been smployed for his own sake, and that of a few riends, than for the public; though it was x0torious the observations that had been made were very numerous, and the papers swelled 0 a great bulk. : This occasioned Prince George of Denmark, n the year 1704, to appoint certain members of the Royal Society, viz. the Honourable Fr. Robarts, Sir C. Wren, Sir Isaac Newton, Dr. and Dr. Arbuthnot, to inspect Mr. Flamstead’s papers, and choose out of them purposing to print them at his own expense : but the patron of the work dying before the impression was half finished, it lay still for some time; till at length it was resumed by order of Queen Anne, and the care of the press com- mitted to Dr. Arbuthnot; and that of correct- ing and supplying the copy to Dr. Halley. uch was the rise and progress of the “ His- Thus was the Astronomer — such as they should think fit for the press; p OCC toria Coelestis ;” the principal part whereof is the catalogue of fixed stars, called also the Greenwich Catalogue. Flamstead was succeeded by Dr. Halley; and Dr. Halley by Dr. Bradley, in 1742, so deservedly celebrated for his discovery of the aberration of the stars (see ABERRATION), and the nutation of the earth’s axis; after Dr. Brad- ley, the appointment was conferred upon Mr, Bliss, in 1762; who was succeeded, in 1765, by Dr. Maskelyne, the late worthy astronomer royal; upon whose demise, in 1811, this impor- tant office was conferred upon Mr. Pond, by whom it is.at present filled. The Greenwich Observatory is found, by very accurate observation, to lie in 51° 28° 30” north latitude. 2. The Paris Observatory, built by Louis XTV. in the Fauxbourg St. Jaques. Itis a very singular, but withal, a very magnificent building; the design of M. Perrault. It is eizhty feet high, and at top is a terrace. It is here M. de Lahire, M. Cassini, &c. have been employed. This observatory was begun in 1664, and finished in 1672. The difference in longitude between this and Greenwich Ob- servatory, is 2° 20” 15’ east. In the Paris Observatory is a cave, or cellar, of one hundred and seventy feet descent, for making various experiments, particularly such as relate to congelations, refrigerations, indu- rations, conservations, &c. In this cave there is a thermometer of M. de Lahire, which is always at the same height, which indicates the temperature of the place to be always the same. 3. Tycho Brahe’s Observatory was in the little island Ween, or the Scarlet Island, be- tween the coasts of Schonen and Zealand in the Baltic. It was erected, and furnished with instruments, at his own expense ; and was called by him Uraniburgh. In this place he spent twenty years in observing the stars. We might enumerate here some other ob- servatories, as that of Pekin, erected by a late emperor of China in his capital upon the recommendation of the jesuit missionaries, and that of the Bramins at Banares, in the Fast Indices, of which an interesting account is given by Sir R. Barker, in the Phil. Trans, b. vii. p. 598, with a plate representing and illustrating several particulars, as enumerated in the memoir; beside many other observa- fories in different countries, which our limits will not admit of detailing. OBTUSE, literally implies or dull, in contradistinction to acute, or pointed. : Ostuse Angle, Angular Section, Cone, Hy- erbola, &c. See the several substantives. OCCIDENT, in Astronomy and Geography, is the same as westward, or point of the horizon where the sun sets. A planet is said to be occident when it sets after the sun. OccipENT Egquinoctial, that point of the horizon where the sun sets, when he crosses ihe equinoctial, or enters the sign Aries or Libra. any thing biunt sharp, OCT ~OccivENT Estival, that point of the horizon where the sun sets at his entrance into the sign Cancer. OcciDENT Horizon. See Horizon, | OcciDENT Hybernal, that point where the sun sets when he enters the «sign Capri- corn. . OCCULT Line, in Geometry, a dry or ob- scure line, which is drawn as a necessary part of the construction of a figure or problem, but which is not intended to appear after the plan is finished. « OCCULTATION, the obscuration of a planet or star by the interposition of the moon, or other planet, between it and our eye. Circle of Perpetual OCCULTATION. CIRCLE. OCEAN, that immense collection of water which forms the greater portion of the terres- trial globe, being commonly estimated at double the space occupied by the dry land. See EarTH and SEA. OCTAGON (from oxrw, exght, and yanc, angle), in Geometry, is a figure of eight sides and angles, which when the sides and angles are all equal is called a regular octagon, and when they are not both equal, an irregular octagon. The angle at the centre of an octagon is 45 degrees, and the angle of its sides 135 degrees. The area of a regular octagon whose side is 1—2(1 + V2) — 48284271; and therefore when the side is s, the area = 48284271 s’, and the radius of its circumscribing circle = See $ v(2— 72) On a given Line AB to construct a regular Octagon. On the extremities of oF the given line A B, erect the indefinite perpendi- G culars A F, B E, and pro- duce AB both ways to m and ». Bisect the angles mAT', nBE, by the lines H\.---} AH and BC, and take “Qi if” AH and BC both equal ” A>=—"B to AB. Draw GH and DC parallel to AF or BE, and each equal to AB; then from G and D as centres with radius AB, describe arcs cutting the perpendiculars in F and E. Join FG, FE, and ED, soisABCDEFGH, the octagon required. To inscribe an Octagon in a given Circle. Inscribe a square in the given circle; then bisect each of the four equal arcs intercepted by the sides of the square, which will be the arcs subtended by the sides of the octagon. See PoLyGon. j . . OCTAHEDRON, or OcTAEDRON (from oxtw, eight, aud ida, seat), one of the five re- gular bodies, contained under eight equal and equilateral triangles. Or an octahedron may be conceived to be made up of eight equal triangular pyramids, whose vertices unite in OLD one common point, which is the centre of th solid, and of its circumscribed spheres. To find the Surface and Solidity of an Octa hedron, the Side of one of its equal Faces being given. 1, Let s represent the given side, then - surface = 287.3 = 3°4641016 s? solidity = 48372 = '4714045 s3 The Radius of the cireumseribing and inscribec Sphere being given to find its Side or Linea Edge, its Surface and Solidity. Let R and r represent the radii of the cir cwmscribing and inscribed sphere, then side = Rv2=rv6 te - surface — 4 R? V3 = 12773 } solidity = 4 R3 = re Or writing s for the side, we have radius of circum. sphere £s./2 radius of inscrib. sphere £5 ./6 See Hutton’s Mensuration, p. 251, 2d edition, OCTANT, the eighth part ofa circle. OcTANT, or Octile, is also an ancient term in Astronomy, to denote one of the aspects, viz. when two planets are distant from cach other 45°, ‘i OCTAVE. See Cuorp. b OCTILE. See Ocranrt. 4 OCTOBER (from oxtw, eight), being the eighth month of the ancient Roman calendar, but the tenth according to the Julian year. Thi: month contains 31 days, on about the 22d 0 which the sun enters the sign Scorpio. Wy OCTOGON. See OcraGon. i ODD Number, that which cannot be divided into two equal integral parts, or which, when: divided by 2, leaves a remainder 1. i ODDLY Odd Number, that when divided by 4 leaves 3 for a remainder, or that which is of the form 4n + 3. i ODOMETER, is an instrument for mea- suring the distance travelled over by a post chaise or other carriage; it is attached to the wheel, and shows, by means of an index and dial-plate, the distance gone over. . OFFSETS, in Surveying, are those short perpendiculars that are measured on the sides of irregular figures for the more accurate de- termination of the area. See SURVEYING. Orrset Staff,.a staff or rod used in survey- ing for measuring offsets. It is commonly made of light wood ten links in length, divided: and numbered from one end to the other. _ OLBERS, a name sometimes given to the planet Pallas, discovered by Dr. Olbers, March: 25th, 1802; but since the discovery of a second planet by the same astronomer, viz. Vester, March 29th, 1807, the name Olbers is usually changed for that of Pallas, to prevent con- fusion between the two. See PaLLas. OLDENBURG (Henry), was born in the! duchy of Bremen, in the Lower Saxony, about the year 1626, but spent most of his time im England, and was one. of the first members. of the Royal Society; in the transactions of which he published several papers, and. as secretary to the society he superintended the OPT iblication of the early part of that work, vz. ym No, 1, in 1664, to No. 136, 1677; the lowing year, 1678, Mr. Oldenburg died at yout the age of 52. OLYMPIAD, in Chronology, a period of ur years, by which the Greeks reckoned eir time; being thus called from the olympic mes held every fourth year, during five days ‘ar the summer solstice, on the banks of the rer Alpheus, near Olympia, a city of Elis. ‘The first olympiad began the year 3938 of e Julian period, corresponding to 776 years fore Christ. The computation by these games ended th the 404th, being the 440th of the present ristian era. See Epocu. OMBROMETER, a name given by Mr. ckering to the pluviameter or rain-guage. OMPHALOPTER, a name sometimes ven to a convex lens. OPACITY, that quality of bodies which anders them opaque, in contradistinction to msparency. See TRANSPARENCY. OPAKE, or Opaque, not translucent nor msparent, or that which prevents the free mission of the reys of light. OPERA Glass, in Optics, is so called from use in playhouses, though it is sometimes led a diagonal perspective, from its construc- nm. It consists of a tube about four inches if; in each side of which there is a hole, i exactly against the middle a plane mirror, ich reflects the rays falling upon it to a ivex glass; through which they are refracted the concave eye-glass, whence they emerge rallel to the eye at the end of the tube. e peculiar artifice of this glass is to view a son at a small distance, so that no one i know who is observed, for the instrument ints to a different object from that which is wed. and as there is a hole on each side, it impossible to know on which hand the fect is situated, which you are viewing. ¢ position of the object will be erect through | concave eye-glass. JPHIUCUS, a northern constellation. See INSTELLATION. IPPOSITE Angles, Cones, Sections, &c. See GLE, CONE, Conic Section, HYPERBOLA, &e. IPPOSITION, in Astronomy, is that as- 't of any two heavenly bodies, when they | diametrically opposite each other. See iPECT. IPTIC, or Optical, any thing relating to science of optics. As Optic Angle, Axis, For which see the several sub- See also LENS. Yetic Inequality, in Astronomy, is an ap- entirregularity in the motions of the planets | other celestial bodies, being thus called ause it does not arise from any real in- ‘ality of the moving body, but from the ation of the eye of the observer. See PARENT Motion. etic Nerves, the second pair of nerves inging from the crura of the medullar oblon- ‘a, and passing thence to the eye. See Eye. e's Pencil, See Prncit of Rays, . | | FF ‘Optic Place of a by the eye. _ Optic Pyramid, is a pyramid formed by the visual rays proceeding from the eye, and pass- ing through the extremities of any picture, when these rays are continued to terminate in a plane perpendicular to the observer. Optic Rays. See Ray. OPTICS, the science of Vision, including catoptrics and dioptrics, and by some authors perspective is also included under the general term optics; as also chromatics, or the theory of colours, and all the phenomena of visible objects. See the several articles Catoptrics, CuHRomATICcs, Dioptrics, Lens, Mirror, Vi- SION, &e. History of Optics. The reflection of the rays of light, from a polished surface, is a cir- cumstance so common and obvious, that it could not long remain unnoticed by the most superficial observer ; even the equality of the angle of incidence and reflection appears to have been known from earliest periods; but this was not the case with regard to the angle of incidence and refraction. It was, indeed, known at an carly period that an oar, or other straight piece of wood, partially immersed in water, no longer appeared straight, yet ages after this elapsed before any attempts were made to ascertain the relation between the angles of incidence and refraction. Em- pedocles was the first on record that wiote systematically on light; and Euclid composed a treatise on the ancient optics and catoptrics ; dioptrics being less known to the ancients, though it was not quite unnoticed by them; for among the phenomena at the beginning of that work, Euclid remarks the effect of bring- ing an object into view by refraction, in the bottom of a vessel, by pouring water into it, which could not be seen over the edge of the vessel before the water was poured in; and other authors speak of the effects of glass globes, &c. both as burning glasses, and as to bodies seen through them. Euclid’s work, the genuineness of which has been doubted, is chiefly on catoptrics, or reflected rays; in which he shows the chief properties of them in plane, convex, and concave surfaces, in his usual geometrical manner, beginning with that concerning the equality of the angles of incidence and reflection, which he demon- strates; and in the last proposition, shows the effect of a concave speculum, as a burning glass, when exposed to the rays’ of the sun. The effects of burning glasses, both by refrac- tion and reflection, are noticed by several others of the ancients, and it has been thought that the Romans had a method of lighting their sacred fire by some such means. Aris- tophanes, in one of his comedies, introduces a person as making use of a transparent stone to cancel a bond that was against him, by thus melting the wax on which such instru- ments were then executed. If we give credit to what some ancient historians have written concerning the exploits of Archimedes, we shall be induced to think that he constructed planet, is its place as seen GPT some very powerful burning mirrors. It is even allowed that this eminent geometrician wrote a treatise on the subject of them, though it be not now extant; as also concern- ing the appearance of a ring or cirele under water, and therefore could not have been ignorant of the common phenomena of refrac- tion. We find, indeed, many questions con- cerning optical appearances in the works of Aristotle. This author was sensible that it is the reflection of light from the atmosphere which prevents total darkness after the sun sets, and in those places where he does not shine in the day-time. He was also of opinion, that rainbows, halos, and mock suns, were all occasioned by the reflection of the sun-beams in different circumstances, by which an im- perfect image of his body was produced, the colour only being exhibited, and not his pro- per figure. The ancients were not only ac- quainted with the more ordinary appearances of refraction, but knew also the production of colours by refracted light. Seneca says, that when the light of the sun shines through an angular piece of glass, it shows all the colours of the rainbow. These colours, however, he says, are false, such as are seen in a pigeon’s neck when it changes its position; and of the same nature, he says, is a speculum, which, without having any colour of its own, assumes that of any other body. It appears also that the ancients were not un- acquainted with the magnifying power of glass rlobes filled with water, though they probably knew nothing of the reason of this power; and it is supposed that the ancient engravers made use of a glass globe filled with water to magnify their figures, that they might work to more advantage. Ptolemy, about the mid- die of the second century, wrote a considerable treatise on optics. This work was supposed, till very lately, to have been lost, but from some late Paris journals it appears to have been recently discovered in a library of that city, and Delambre read an interesting account of it to the academy, from which it appears that Ptolemy was not only acquainted with the refraction of light, but had determined the ratio of the angles of refraction as compared with those of incidence; and it is singular that his results differ very little from those of Newton, not more than might have arisen from his using water and glass of a little different specific gravities to those employed by Newton; he was not, however, aware of the ratio of the sines of those angles. He there also treats of the effects of refractions in astronomical ob- servations, and shows that it decreases from the horizon to the zenith, where it is zero, &e. &e. The more early astronomers were not aware that the intervals between the stars appear less when near the horizon than in the meridian, and on this account they must have been much embarrassed in their observations; but it is evident that Ptolemy was aware of this defect by the caution which he gives to allow something for it, whenever recourse is had to ancient observa- OPT tions. This philosopher also advances a ver ingenious hypothesis to account for the re markably great apparent size of the sun anc moon when seen near the horizon. The mind he says, judges. of the size of objects by mean; of a preconceived idea of their distance fron us; and this distance is fancied to be greate when a number of objects are interposed be tween the eye and the body we are viewing which is the case when we see the heavenl bodies near the horizon. In his Almagesi however, he ascribes this appearance to a re fraction of the rays by vapours, which actuall enlarges the angle under which the luminarie appear, just as the angle is enlarged by whie an object is seen from under water. af Alhazen,an Arabian writer, was the nexta thor of any importance, who wrote on this sul ject about the year 1100. Alhazen made man experiments on refraction, at the suriace be tween air and water, air and glass, and watt and glass; and hefice he deduced several pri perties of atmospherical refraction, such a that it increases the altitude of all objects i the heavens; and he first advanced that th stars are sometimes seen above the horize by means of refraction, when they are reall below it; which observation was confirm by Vitellio, Walther, and especially by t observations of Tycho Brahe. Alhazen 0 served, that refraction contracts the diamete and distances of the heavenly bodies, and th it is the cause of the twinkling of the sta This refractive power he ascribed not tot vapours contained in the air, but to itsd ferent degrees of transparency. And it w his opinion, that so far from being the cau of the heavenly bodies appearing larger ne the horizon, that it would make them appe less; observing that two stars appear neat together, in the horizon, than near the me dian. This phenomena he ranks among tical deceptions. We judge of distance, says, by comparing the angle under whi objects appear, with their supposed distan¢ so that if these angles be nearly equal, a the distance of one object be conceiy greater than that of the other, this will, imagined to be larger. And he farther serves, that the sky near the horizon is alwe imagined to be farther from us than any otf part of the concave hemisphere. " In the writings of Alhazen, we likew find the first distinct account of the mae ing power of glasses, and it is not improba that his writings on this head gave rise tot useful invention of spectacles; for he sa that if an object be applied close to the hi of the larger segment of a sphere of gla will appear magnified. He also treats of” appearance of an object through a globe, a says, that he was the first who observed | refraction of rays in it. K 9 In 1270, Vitellio, a native of Poland, pI lished a treatise on optics, containing allt was valuable in Alhazen, and digested i better manner. He observes that light always lost by refraction, which makes obje Om year less Iuminous. He gave a table of the ults of his experiments on the refracting wer of air, water, aud glass, corresponding different angles of incidence. He ascribes ' twinkling of the stars to the motion of the in which the light is refracted ; and he il- trates this hypothesis by observing, that y twinkle still more when viewed in water (in motion. He also shows, that refraction 1ecessary, as well as reflection, to form the abow; because the body which the rays upon is a transparent substance, at the face of which one part of the light is always ected, and another refracted. And he kes some ingénious attempts fo explain re- stion, or to ascertain the law of it. He 9 considers the foci of glass spheres, and apparent size of objects seen through m, though with but little accuracy. See FRACTION. Jontemporary with Vitellio was Roger Ba- 1, a man of very extensive genius, who ote upon almost every branch of science ; ugh it is thought his improvements in op- s were not carried far beyond those of Al- ren and Vitellio: to him, however, has mn attributed the invention of the magic tern. One of the next who distinguished iself in this way was Maurolycus, teacher of thematics at Messina. In a treatise ‘De mine et Umbra,” published in 1575, he nonstrates, that the crystalline humour of eye is a lens which collects the rays of light ling from the objects, and throws them ym the retina, where the focus of each pencil ormed. From this principle he discovered | reason why some people are short sighted, l others long sighted; also why the former relieved by concave glasses, and others convex ones. Yontemporary with Maurolycus was John ptista Porta of Naples. He discovered the aera obscura, which throws considerable it on the nature of vision. His house was ‘constant resort of all the ingenious persons \Naples, whom he formed into what he led an academy of secrets, each member ng obliged to contribute something that snot generally known, and might be useful. this means he was furnished with materials his “ Magia Naturalis,” which contains his fount of the camera obscura, and the first tion of which was published, as he informs when he was not quite fifteen years old. also gave the first published hint of the gic lantern, which Kircher afterwards fol- ved and improved. - His experiments with camera obscura conviticed him, that vision verformed by the intermission of something y the eye, and not by visual rays proceeding n it, as had been formerly imagined; and was the first who fully satisfied himself and ers upon this subject. He justly considered eye as a camera obscura, and the pupil } hole in the window shutter; but he was | : staken in supposing that the crystalline ‘nour corresponds to the wall which re- Yes the images; nor was it discovered till | ORB the year 1604, that this office is performed by the retina. He made a variety of just remarks concerning vision, and particularly explained several cases in which we imagine things to be without the eye, when the appearances are occasioned by some affection of the eye itseif, or by some motion within the eye. He re- marked also, that, in certain circumstances, vision will be assisted by convex or concaye glasses; and he seems even to have made some small advances towards the discovery of telescopes. Other treatises on optics, with various and gradual improvements, were af- terwards successively published by several authors, particularly by Newton, whose genius nowhere displays itself to greater advantages than in the beautiful theory which he has esta- blished, and of which abstracts will be found under the articles CoLour, PRism, Rerrac- TION, TELESCOPE, &c. These were published in the Phil. Trans. and afterwards in his New Theory of Light and Colours. Since this work we have had many treatises on optics, but the limits of this article will not admit of an enumeration of them, particularly as we have already mentioned all the principal ones under the articles CaToprrics and Dioprtrics; to give only brief accounts of the several discoveries that have been made in this science, would occupy a larger space than can here be allowed. We must, however, mention the excellent work on optics, by Dr. Smith, 2 vols. 4to.; an abridgment of which was made by Dr. Kipling, for the use of the students at the universities, entitled ‘““ Ele- mentary Parts of Dr. Smith’s Optics,” &e. 1778; and an elaborate “ History of the pre- sent State of Discoveries relating to Vision, Light, and Colours,” by Dr. Priestley, 4to. 1772; a work highly instructive and enter- taining to persons who have a taste for phy- sics. See also the several articles Catop- TRIics, Dioprrics, Mirrors, &c. as men- tioned at the beginning of this article. ORB, a spherical shell, or hollow sphere, whence the denomination of orbs to the seve- ral supposed moveable spheres, of which the ancients conceived the heavens to consist; and by the introduction of which they en- deavoured to account for all the celestial motions and phenomena. ORBIS Magnus, the great orb in which the sun was supposed to revolve. ORBIT, the path of a planet, comet, or other celestial body. The orbits of the several planets were, even after the restoration of the Pythagorean system of astronomy by Coper- nicus, supposed to be circular, having the sun in their common centre, which, indeed, was such a rational and simple hypothesis, that it is not at all singular both Copernicus and Kepler, as well as the other astronomers of that day, were so unwilling to give up this idea. However, Kepler, after an immense number of observations upon the planet Mars, found that it was impossible to reconcile his observations with that theory, and he there- fore abandoned it; but now, though he ORF changed the figure from a circle to an ellipse, he still, by supposing the sun in the centre, found nearly the same difliculty in accounting for some of the observed phenomena. At length, however, it happily occurred to him to place the sun in one of the foci, with which position every observed irregularity perfectly agreed; and by that perseverance, for which he is so eminently distinguished, he came finally to those fundamental laws, which still bear his name, viz. ; 1. The planets all revolve in eliptic orbits situated in planes passing through the centre of the sun, the latter body occupying one of the foci of the ellipse. 2. Equal areas are described in equal times. That is, if aline be supposed to join the central and revolying body, that line passes over or describes equal areas in equal times. 3. The squares of the times of revolutien in planetary bodies, are as the cubes nf phen distances from the sun. See KEepLer’s“Laws. For the inclination of the several orbits to the plane of the equator, their eccentricity, &e. see Eccentricity, and Elements of the PLANETS. Orsits of the Comets. See Comets. ORDER of a Line, in the Theory of Curves, ~ js denoted by the dimension of the equation by which the line is defined or expressed ; it being said to be of Ist, 2d, 3d, &c. order, ac- cording as the equation is of the Ist, 2d, 3d, &e. degree or dimension. See Curves and LINES. ORDINATE, in the Theory of Curves, is any right line drawn from a point in the ab- sciss to terminate in the curve; and if it be drawn perpendicular to the absciss, it is called a right ordinate. It is a general property of the ordinates of a curve, that when they are perpendicular to fhe axis, or principal diameter of a line of the 2d order, that is, in the circle and conic sec- tions, the ordinates are all bisected by that axis, making the sum of all the ordinates on one side equal to that of those on the other; and in lines of the 3d order, where a line may cut the curve in three places, the ordinate on the one side is always equal to the sum of the ordinates on the other; and the same for curves of any dimension, viz. the sum of the ordinates on one side of the axis, is always equal to the sum of those on the other. For the use of these lines in defining the nature of a curve, and determining its equa- tion, see Axssciss, Curves, and EquaATIoN of Curves. ORFFYREUS’s Wheel, in Mechanics, is a machine so called from its inventor, which he imagined to be a perpetual motion. ‘This ma- chine, according to the account given of it by M. Gravesande, in his “ Oeuvres Philoso- phiques,” published by Allamand, Amst. 1774, consisted of a large circular wheel, or rather drum, twelve feet in diameter, and fourteen inches in depth, and very light; as it was formed of an assemblage of deals, the inter- vals between which were covered with waxed ORR * cloth, in order to conceal the interior parts, it. The two extremities of an iron axis, ¢ which it turned, rested on two supports. Q giving the wheel a slight impulse in eith direction, its motion was gradually accelera ed; so that after two or three revolutions. required so great a velocity as to make twent; five or twenty-six turns in a minute. Th rapid motion it actually preserved during th space of two months, in the chamber of # Landgrave of Hesse, the door of which wi kept locked, and sealed with the Landgraye own seal. At the end of that time it wi stopped to prevent the wear of the material The professor, who had been an eye-witne to these circumstances, examined all the & ternal parts of it, and was convinced that the could-‘not be any communication between _and any neighbouring room. Orffyreus, hey ever, was so incensed, that he broke the m chine in pieces, and wrote on the wall, that was the impertinent curiosity of professi S. Gravesande which made him take this ste The prince of Hesse, who had seen the terior parts of this wheel, being asked } S. Gravesande, whether, after it had been} motion some time, there was any change 0 servable in it, and whether it contained at pieces that indicated fraud or. deception, a swered both questions in the negative, declared, that the machine was of a vé simple construction. 4 ORGANICAL Description of CuRVES, | the method of describing them on a plane } means of instruments. Of these, the cor passes and ruler are the most simple, and a the only instrument admitted into plane | elementary geometry; but other instrumen have been invented for the description of ¥ rious curves; such are those described - the articles Elliptic, Compasses, Conchoid, rs lipse, Hyperbola, Parabola, &c. See also ( this subject, Schooten’s “De Organica Con Sect. in Plano;’’ Newton’s ‘“ Universal Arit metic ;” lV’ Hopital’s “ Conic Sections ;” Ma laurin’s “‘Geometrica Organica,” &e. } ORIENT, the east or eastern point of f horizon. ; ORIENT Equinoctial, that point of the horiz where the sun rises when in the equinoctia ORIENT Estival, is that point of the hor where the sun rises in the middle of summ Orient Hybernal, that point of the horiz where the sun rises in the middle of winter. ORIENTAL, any thing or place situa to the eastward of an observer. ., Fi ORION, one of the grandest of the ancie constellations, situated in the southern hem phere. See CONSTELLATION. | Orion’s River, another name for the e¢ stellation Eridanus. i ORRERY, an astronomical instrument exhibiting the several motions of the heave bodies. The first machine of this kind W constructed by Mr. Graham, but its name derived from one made by Mr. Rowley | the Earl of Orrery, which was supposed Sir R. Steel to be the first ever construete Ose d he therefore gave it the above name in nour of the Earl, and attributes the inven- nto Mr. Rowley; which name it has ever ice retained, though the error on which it ss adopted has been long corrected. ORTHODROMICS, in Navigation, is the ne as Great Circle Sailing. 'The word is rived from opbos, right or straight, and dpoos, tance ; viz. the straight or shortest distance, ich can only be in tlie are of a great circle. ORTHOGONTAL, in Geometry, is the same rectangular or right-angled. ORTHOGRAPHIC Projection of the here, is that projection which is made upon plane passing through the middle of the 1ere, by an eye placed vertically at an in- ite distance. ORTHOGRAPHY (from op605, right, and “On, description), in Geometry, is the art of wwing or delineating the front of an object, d exhibiting the heights or elevations of the eral parts. The same is also to be under- od by an orthographic plan in architecture d fortification. ORTIVE, or Eastern Amplitude, in Astro- ny, is an arch of the horizon intercepted tween the place where a star rises, and the st point of the horizon. OSCILLATION, in Mechanics, vibration, the reciprocal ascent and descent of a pen- lum. Axis of OscILLATION, is a right line passing ‘ough the point of suspension parallel to the rizon. Centre of OsciLLATION, is that point in a rating body into which, if all the matter of + body were collected, the vibrations would performed in the same time. See CENTRE Oscillation. uygens, in his work “De Horologia Os- latorio,” first gave a general solution of the yblem of oscillatory bodies, by reducing all mpound bodies or pendulums to simple ies; or, which is the same, by finding their mmon centre of oscillation. This solution founded on this principle, that the common atre of gravity of several bodies, connected ether, must return precisely to the same ight whence it fell; whether those weights urn conjointly, or whether after their fall +y return separately, each with the velocity ad then required. This hypothesis gave rise to various discus- ns, some adopting it and others objecting it; and others again, who were disposed to nk it correct, were not willing to adopt it thout demonstration. After various disputes, James Bernoulli pub- ied, in the Memoirs of the Academy of iences for 1703, a perfectly satisfactory geo- ‘trical demonstration of Huygens’s hypothe- , by referring all the different weights to a er; and after his death, which happened 1705, his brother John gave another demon- ation, much more simple than that of the mer, which he published in the same Me- irs for 1714; and at the same time, Dr. 00k Taylor published his demonstration in OVA. his “ Methodus Incrementorum,” which gave rise to some dispute between these two authors, each accusing the other of plagiarism; the particulars of which may be seen in the Leipsic Acts for 1716, as also in the works of John Bernoulli, published in 1742. For the method of investigation, see CENTRE of Oscil- lation and PENDULUM. OSCULATION (from the Latin, osculatio, hissing), in the Theory of Curves, denotes the contact between any curve and its osculatory circle; or that circle which has the same cur- vature as the curve at the given point of oscu- lation. ; If AC be the E evolute of the involute curve AEF, and the tangent CE the 43 radius of cur- vature at the point E, with which and the centre C, if the “XC circle BEG be described, this circle is said to osculate, or kiss the curve AEF, in the point E; which point E is called the point of osculation; CE the osculatory radius, or radius of curvature; the circle BEG the oseu- latory circle; and the evolute AC the locus of all the centres of the osculatory circles. See Ranius of Curvature. Point of osculation is also used to denote the concourse of two branches of a curve that touch each other; which differs from a cusp, or point of retrocession, in this, that in the latter case, although it forms a point of con- course, the curve ter- minates there; where- as in a point of oscula- tion, the curve does not terminate but is con- tinued on both sides. Thus, in the annexed figure, A is a point of retrocession, and B a point of osculation. OSCULATORY Cirele, or Kissing Cirele, is the same as the circle of curvature; such is the circle BEG in the preceding figure, article OscuLATION. See several properties of these .circles in Maclaurin’s ‘ Algebra,’’ “Appendix de Linearum Geometricarum,” &c. theo. ii, sect. 15, &e.; see also Rapius of Curvature. OSTENSIVE Demonstration, is a direct geometrical demonstration, in contradistine- tion to an apogogical one, or that which de- pends upon areductio ad_absurdum. OTACOUSTIC, a name sometimes given ¥ G to a hearing trumpet, and other instruments | for improving the sense of hearing, OVAL, an oblong curvilinear figure, having two unequal diameters, and bounded by a curve line returning into itself. Under this general definition of an oval is inclfded the ellipse, which is a regular oyal; and all other OUG figures which resemble the ellipse, though without possessing its properties, are classed under the same general denomination. We have shown, under the article ELLIPSE, different mechanical methods of describing ellipses; and we shall therefore only give in this place a practical method of describing an oval, which to the eye very much resemble the conical ellipse; this is as follows: Set the given length re and breadth AB, CD, to bisect each other per- pendicularly at E; with the centre C and radius AE describe an arc to cross AB, in F and G; v -then with the centres F and G, and radii AF and BG, describe two little arcs, and HI and KL for the smaller ends of the oval; and then with centres C and D, and radius CD, de- scribe the arcs HK and IL, for the flatter or larger ares of the figure. Oval is also employed in the theory of curves, to denote certain figures which in any way approach towards the form of an oval, as above described. ‘These curves are commonly expressed by some equation of even dimensions above the se- cond; as a*y?=—2*+a2x3, which denotes the oval B, < in shape of the section of a B pear through the middle; which is easily described by points. For if a circle be described whose diameter is AC =a, and AD be perpendicular and equal to AC; then taking any point P 4, in AC, joining DP and cA drawing PN _ perpendi- cular to CA, and NO parallel to the same line; and then taking PM — NO, the point M willbea point in the oval sought. ni =) C Stone’s Math. Dictionary. OUGHTRED (WILLIAM), an eminent Eng- lish mathematician, was born at Eaton in Buckinghamshire, in 1573; and died in 1660, at eighty-six years of age. He was author of several works, the principal of which are as follows: viz. ? 1. Arithmetice in Numero et Speciabus Institutio, 8vo. 1631; reprinted, with addi- tions, and published under the title of a “Key to Mathematics,” in 1648; and a third edition of the same, in 1652. 2. The Circles of Proportion, and a Hori- zontal Instrument, 4to. 1633. 3. Description and Use of the Double Ho- rizon Dial, 8vo. 1636. 4, Trigonometria, 4to. 1657; and another edition of the same work in English, with Tables of Sines, Tangents, &c. WN. Oughtred, besides these works, left several - manuscripts.at his death, of which some were published at Oxford in 1676, under the title of ger Dutch Mathematica, hactenus inedita,” yo. OZA The characters introduced into algebra b this author, in his work above mentioned were as follows: x to denote multiplication :: for proportion ++ for continued proportion 3 for great — for less. Dr. Hutton’s Math. Dict. OUNCE, an English weight, being the six teenth part ofa pound Avoirdupoise, and th twelfth part of a pound Troy. ‘The former; equal to 16 drams, and the latter to 20 ee weights, or 480 grains. : OX- -Kye. See Scroptic. 4 Ox-Gang, or Ox-Gate of Land, an om \ measure of land, commonly estimated at abo fifteen acres. OXYGONE Tirangle, is the same as } otherwise called an acute-angled triangle. OXYGONAL, acute angular. OZANHAM (JAmMEs), an eminent Frene mathematician, was born at Boligneux i Bressia, in 1640; and very early discover great mathematical talents, having written work on this subject when only fifteen yea of age. He afterwards published several y luable treatises, of which Dr. Hutton hj given the following catalogue : 1. A Treatise on practical _Geometr 12mo. 1684. 2. Tables of Sines, Tangents, and Secant with a Treatise on ‘Trigonometry, 8vo. 1686 3. A Treatise of Lines of the first. Crd of the Construction of Equations, and of oe metric Lines, &c. 4to. 1687. 4. 'The Use of the Compasses of Prop tion, &c. with a Treatise on the Division Lands, 8yo. 1688. 5. An universal Instrument for readily 1 solving Geometrical Problems without Cale lation, 12mo. 1688. 6. A Mathematical Dictionary, 4to. 169 7 A general Method for Drawing Dia Ke. 12mo. 1693. 8. A Course of Mathematics, in 5 ve 8vo. 1693. 9. A Treatise on Fortification, ancient a modern, 4to. 1693. 10.. Mathematical and Philosophical F creations, 2 vols. 8vo. 1694; and again, w. additions, in 4 vols..1724. 11. New Treatise on Trigonometry, 120 1699, 12. Surveying and measuring all Sorts Artificer’s Works, 12mo. 1699. 13. New. Elements of Algebra, 2 vols. 8 1702. 14, Theory and Practice of Perspecti 8yo. 1711. 15. ‘Treatise of Cosmography and Geog. phy, 8vo.. 1711. 16. Euclid’s, Elements, by De Chales, ¢ rected and enlarged, 12mo. 1709. 17. Boulanger’ S practical Geometry, | larged, &c. 12mo. 1691. 18, Boulanger’s Treatise on the Sphe corrected and enlarged, 12mo, PAL PAL P PAccroLt (Lucas), commonly known by he name of Lucas de Burgo, was one of the arly writers on arithmetic, algebra, and as- ronomy, soon after the revival of the sciences a Europe. He translated Euclid into Latin, x rather he revived that of Campanus, which e corrected and augmented with notes; but - did not appear till 1509, in folio. He also ublished in Italian, “Summa de Arithmetica feometria Proportioni € Proportionalita,” 494, folio; another work on the “ Rules of ‘erspective,” and some others of less im- ortance. Paccioli died early in the sixteenth entury. PAGAN (BLAISE FRANCOIS CoMTE DE), a ery eminent French mathematician and en- ineer, was born at Avignon in Provence, in $04, and died in 1665, in the sixty-first year fhis age. Pagan was most distinguished in ismilitary character, and for the works which wrote on the theory of fortification; but s other performances are also very honour- le testimonies of his scientific acquirements. hese are as follows: viz. 1. Treatise on For- fication, published in 1645. 2. Geometrical heorems, 1651. 3. Theory of the Planets, 357. 4. Astronomical Tables, 1658. PALILICUM, another name for the star Idebaran. : ‘PALM, an ancient long measure taken om the extent of the hand. The Roman palm was of two kinds; the veat palm, taken from the length of the hand, iswered to our span, and contained twelve gits, or fingers breadths, or nine Roman ches, equal to about eight and a half Eng- thinches. The small palm, from the breadth the hand, contained four digits or fingers, jual to about three English inches. ‘The Greek palm, or doron, was also of two nds; the small contained four fingers, equal little more than threé inches; the great Im contained five fingers. The Greek double m, called dichas, contained also in pro- irtion. The modern palm is different in different aces, where it is used. It contains vw Inc. Lines. MPRA lan — aby’; asin fig. 3; where x = AP the absciss, y = PQ the or- dinate, b= A B, c= AC, and a = a certain given quantity. Sometimes, however, biqua- dratic parabolas are expressed by other equa- tions, but we cannot enter into a detail of them in this place. Cartesian PARABOLA, is a curve of the second order, expressed by the equation ry = az3 + ba? + cx + d, containing four infinite legs ; viz. two hyperbolic and two parabolic, as in the annexed figure. Cubic PARABOLA, is also a curve of the se- cond order, haying two infinite legs tending contrary ways. If the absciss touch the curve in a certain point, the relation between the absciss and ordinate is expressed by the equation . . PAR ymax? +ba? +exr+d ead when J, c, and d are each Zero, the equa- tion then becomes y = az’, as in the annexed figure. The area of the cubic parabola is equal to three quarters of its circumscribing cylinder ; but it cannot be rectified even by means of the conic sections. Diverging PARABOLA, is a name given by Newton to a species of five different lines of the third order, expressed by the equation y? = ax + ba* + cx + d. The Ist is a bell-formed parabola, having an oval at its head, as in fig. 1; which is the case when the equation 0 = ax? + b2* + ex + d has three real and unequal roots. Fig. 1. Fig. 2. The 2d is also bell-formed, with a punctum conjugatum, fig. 1; which happens when the above equation has its two less roots equal. The 3d is when the two greater roots are equal, as in fig. 2. The 4th is when the same equation has two imaginary roots, as in fig. 3 ig. 3. ies The 5th is when the roots of the proposed equation are all equal to each other, as in fig. 4; the most simple equation being in this case py” — x3; which is the semi-cubical, or Neilian parabola. If a solid, generated by the rotation of a semi-cuhical parabola about its axis, be cut by a plane, each of these five parabolas will be exhibited by its sections, according to ihe positions of the cutting plane. Infinite PaRapowas, or PARABOLOIDES, are parabolas of the higher orders, defined by the general equation a”™—" 2” = y”; inall of which the proportion of the area to that of the cir- cumscribing parallelogram, is as m to m +n. Semi-Cubical PaRAsBota, that whose equa- tion is py” = x3. See Diverging PARABOLA. PARABOLIC Asymptote. See ASYMPTOTE. PARABOLIC Ares, are the ares of the curve of a parabola, the length of which may be found by the following formula: Let p = parameter, x any absciss from the vertex, and y the corresponding right ordi- nate; also make ba = q and /(1 + q*) = sg then 1. Parabolic are Fig. 4. = tp ; qs + hyp.log.(q+s) ; PAR fh 2. Parabolic arc =2y}1 io = 39° 3.59! T$2.6.9 | 0-4 oee ke ce q ,_1:3¢ Be severeraee ware =2y } 1+ LA re | —_— 3 .5q” + 5" C &e. i Where A, B,C represent the preceding term To which may be added, the following a proximations: 4. Parabolic are =2 isd + 4°) nearly. Bin, Seto are =2$ v (YY? +42") Ts moe PARABOLIC Frustrum, or Zone, is the spa included between any two parallel double q ¢ dinates. y Let Dd represent the two ends, and af perpendicular distance between them ; the D3 — ds area of zone = 3a x D? da When d =0 the area of the whole Pe bola becomes area of parabola = ZaD. ‘| PaRABoLic Pyramidoid, is a solid gen er ed by supposing ali the squares of the o nates applicates to the parabola, so placed th the axis shall pass through all their centi at right angles; in which case the aggrega of the planes wil! form the solid called Parabolic Pyramidoid; the solidity of whi is equal to the product of the base, and h the altitude. Parapo.ic Spiral, or Helicoid. See Spiri ParapoLic Spindle, is the solid generat by the rotation of a parabola about any dow ordinate; the solidity of which is found” follows : % Let m denote the middle diameter, the length of the spindle; then 1. Solidity = -418879 Im? And the solidity of a middle frustrum of su solid, is Solidity — 05236 1(8m* + 3d? + 4dm), where d denotes the diameter of the end the spindle. See Hutton’s Mensuration. PARABOLOID, or Parasotric Conoil the solid generated by the rotation of # a bola about its axis, which remains fixed ; surface and solidity of which may be fous the following formule: i Let y represent the radius of the 2 base, x the altitude of the solid, and 3°1416 then 1. Surface = apt LS (ye + 4x°)2 -- | 2. Solidity = ay? z, % Frustrum of a PARABOLOID, is the lower | lid formed by a plane passing parallel to base of a paraboloid. Let D denote the greater diameter, @ t less, P the parameter, and / the height 7 frustrum 3°1416 = p; then tA (Pe +p ya +a} P —— i. Surface 2, Solidity = *3927h (D* + d’*) or when d = 0. X oP ; Solidity = -3927A D* for the whole para- . boloid. These formule only obtain when the base of the frustrum is perpendicular to the axis of _ the solid. - PARACENTRIC Motion (from mapa, near and xevrpov, centre), is the motion of a planet to- wards the centre of attraction or thesun. The orbits of the planets being elliptical, they are sometimes nearer and sometimes more remote from the sun; and the difference in this dis- tance is what is called the paracentric motion. PARALLACTIC Angle, is the angle sub- tended by two lines drawn from the centre of a planet, the one to the centre of the earth, and the other to some point on its surface. See PARALLAX. | PARALLAX (Ilepaaara§ss), in Astronomy, an arc of the. heavens intercepted between ithe true and apparent place of any of the heavenly bodies ; that is, between its place as viewed from the centre of the earth, and from some point on its surface. : Thus, let ABC be the earth, C its centre, vv'v" three different planets, or three ditferent posi- tions of the same planet; then its true places in _ these positions, as seen from the centre C, and | as referred te the hea- Yens, will be VV'V’; but their apparent places will be w, w’, w", and the arcs wV, w’ V', and Vw", will be the measures of the parallaxes, or of the parallas- | tic angles V vw, v'v'w’, &e. or of Due, Dv'C, &e to which they are equal. The Horizontal ParaLuax, or that which has place when the star is in the horizon, is the greatest; the angles and arcs both diminish ‘from the horizon to the zenith, where it be- _comes nothing, as is obvious from the above ‘figure. Whence it is obvious, that the parallax diminishes the apparent altitude of a body; but that this diminution is less and less, as the altitude becomes greater and greater: and it has, therefore, a contrary effect to refraction, -which always increases the apparent height of any of the celestial objects. _ Parallax receives particular denominations, according to the circle upon which it is com- puted. PARALLAX of Altitude, is the difference be- tween the true and apparent altitude of a pla- net, as above described. PaRaLLax of Right Ascension and Descen- sion, is an arch of the equinoctial, by which the parallax of altitude increases the ascen- sion and diminishes the descension. ~Paratiax of Declination, is an arch of a circle of declination, by which the parallax of ew ae ee altitude increases or diminishes the declina- tion of a star. PARALLAX of Latitude, is an arch of a circle of latitude, by which the parallax of altitude increases or diminishes the latitude. Menstrual PARALLAX of the Sun, is an an- gle formed by two right lines; one drawn from * the earth to the sun, and another drawn from pe sun to the moon, at either of their quadra- ures. _ PaRALiax of the Annual Orbit of the Earth, is the difference between the heliocentrie and geocentric place of a planet, or the angle at any planet subtended by the distance between the earth and sun. PARALLAX of the Planets, The exact de- termination of the parallax of the planets is of the greatest importance in astronomy, as it is from that, or indeed from the parallax of any one of them, we estimate their several distances. For if the parallax, or the angle which is subtended by the terrestrial radius, at any planet be known, its distance from the earth is also known; whence its distance from the sun, as also the distances of all the other planets, will be known also, from the third law of Kepler, viz. that the squares of the pe- riodic times are as the cubes of the distances. _ To illustrate this it may be observed that in very small angles, as are those of the pa- rallaxes, and while the side subtending those angles remains constant, the angles wilt be reciprocally as the distances; that is, if DC be very small with regard to CY, then ae the angle DY C: angle DMC ::CV:CM nearly. For in this case we may consider DV —CV and DM = CM, also DC will not differ sensibly from the are of the circles which subtends the angles at V and M, and DC tting 3°1416 = —_— therefore putting 16 = p we have Sy T Gl for the measure of the angles D V C, and DC 2px CM which are therefore reciprocally as ©V and CM. It is obvious, therefore, that the parallax of any oue of the planets being found, that of all the others may be determined because their proportional distances from the sun are accu- rately known from Kepler’s Law, above men- tioned. There are many different methods of deter- mining the parallax of the planets; but being an extremely delicate operation, on account of the smaliness of the angles, we had several different results; till the parallax of Vengs was determined by means of her transits over the sun’s dise, in 1761 and 1769; and as it is on this that a great part of our,knowledge depends respecting the absclute distances of the planets from the sux. it will be proper to KK 2 for the measure of the angle DMC, PAR give a sketch of the principles on which it resis, without, however, attempting to enter into the minutize of the observations and cal- culations. . Let V.V’ (fig. 1. Plate FX.) represent the orbit of Venus, and E E’ that of the earth; V and E being the positions of these two bodies at the true time of Venus entering upon the disc of the sun, that is, to an ob- server situated at the earth’s centre, by whom her place in the heavens would be referred to v, which would be also the place of the eastern limb of the sun. But to an observer situated at A, Venus would be referred in the heavens to w, and the eastern limb of the son tos; and consequently, to such an ob- server, Venus instead of appearing on the sun’s disc, would be distant. from him by the arc S W, that is, by an are equivalent to the difference of the parallax of Venus and of the suv. Andinthesame manner, supposing the earth, during the transit, to have moved in its orbit from E to E’, and Venus from V to V’; it is obvious, from whatis said above, that at the time when this planet was just quitting the sun’s western limb, to an: ob- server situated at the centre E, would be referred by an observer at A tov’, and the western limb of the sun ats’; so that, in the first case, sne would not enter the sun’s disc till after the true time, and in the latter, she will appear to leave the suun’s dise before the true time; and as in some parts of the world, in the transit above alluded to, Venus was seen both to enter and emerge from the sun’s limb by the same observer, the whole difference between the true and apparent time of the transit became known, and half this difference was the time Venus was in passing over the ere ws; whence the measure of that arc be- came known, or, wuich is the same, the differ- ence between the parallax of Venus and that of the sun. But to those observers who could not see both the immersion and emersion, the difference between the true and apparent time of cither, when converted into seconds of a degree, was what determined the differ- ence of the parallax of the two bodies. Now this difference being known, as also the proportional distances of the Earth, Venus, and the Sun, their absolute parallaxes were de- termined, this, as we have shown above, being reciprocally as the distances. Let m: n re- present the ratio of the distance Ev to ES, also P and p the parallaxes of Venus and the Sun, which are required, and P—p = g, the difference of their parallaxes as found above; then fm .29Re3 Pong whence n—mim::P—pmq: —L-— Rembtgon ar it’ the parallax of the sun. In this manner, aided by subsequent obser- vations, &c. Laplace has detérmined the pa- rallax of the sun to be 8”2, whence its distance is found to be about 94. million miles. See Distance of the Planets. ; PAR a And hence the parallaxes of the severa planets are determined as below: Sun....... 8".7 Mercury 14”. ns Greatest horizontal paral- Mee lel of the..... ee Shs Jupiter. oan Saturn... 1°02) Uranus. 0* 1 Parauax of the Moon. This is much mor considerable than in any other of the heavenl bedies on account of its proximity to the earth and is much easier determined than any of others; one of the most simple and correc methods being as follows: “te Observe the moon’s meridian altitude wit the greatest accuracy, and mark the momer of observation: this time being equated, com pute her true longitude and latitude, and fror these find her declination, and from her dec nation, and the elevation of the equator, fin her true meridian altitude ; if the observed a titude be not meridian, reduce it to the tm for the time of observation ; take the refractio from the observed altitude, and subtract the re mainder from the true altitude; and the rm mainder is the moon’s parallax ; by this meat the parallax of the moon has been found follows, viz. | Greatest parallax......... 61’ 32" Least parallax............. 54 4 Mean parallax..........0.. o7 = 48 But Laplace makes her} ai mean: parallax ..........secea+es 5 ord fy * PARALLAX of the Fixed Stars. As the di tances of the heavenly bodies are determine by means of their parailaxes, every possib’ method has been attempted to ascertain th parallax of some of the fixed stars, but the distance is so immense that not only hay they no parallax, as compared with the te restrial radius, but even the whole diamet of the earth’s orbit, though near 200 millic miles, is not sufficient to render any differenc in their apparent places at all evident, whic if it only amounted to one second, would } discoverable by modern instruments. This di tance, therefore, great as it is, is not mo than a mere point compared with the distance of the nearest of the fixed stars. PARALLEL, in Geometry, is applied to lin figures, and bodies, which are every whe equidistant from each other, or which, if evi so far produced, would never meet. PARALLEL Right Lines, are those whic though infinitely produced, would never me which is Euclid’s definition. To draw a line CD parallel to a given line AB throngh a given point P. PromP draw any line & | oe meeting AB in some point Q; then mal the angle QP D = the angle AQP, so sh: DPC be the parallel line required. This : | PAR ractice, however, is better done by a parallel | ‘uler. ‘ f : PaRALLEL Ruler, an instrument consisting »f two wooden, brass, or steel rulers, A B and i DF H C } : : I i A E GB 4 D, equally broad throughout, and so joined ogether by the cross blades EF and GH, as 0 open to different intervals, and accede and | ecede, yet still retain their parallelism. 'The Ase of this instrument is obvious; for one of ‘the rulers being applied to a given line, an- ‘Mher drawn along the extreme edge of the Mher will be parallel to it; and thus, having : “iven only one line, and erected a perpendicu- ‘ar upon it, we may draw any number of paral- el lines or perpendiculars to them; by only \bserving to set off the exact distance of every ine with the point of the compasses, '! But the best parallel rulers are those whose ‘Mars cross each other, and turn on a joint at ‘heir intersection; one of each bar moving on ' eentre, and the other ends sliding in grooves 8 the rulers recede. PARALLEL Circles, Planes, Rays, Sailing, iphere, &c. See the several articles. PARALLELS of Altitude, Declination, Latitude, ce. See the articles ALTITUDE, Dectina- ton, Latirupe, &ce. PARALLELEPIPED, or PaRAtietoprt- ep, or PARALLELOPIPEDON, is a solid figure ‘ontained under six parrallelograms, the oppo- Nite of which are equal and parallel; or it '; a prism whose base is a parallelogram, ‘ee Prism. i PARALLELISM, the quality of being pa- allel. t) PaRALLectam of the Earth’s Axis,is used to " enote that invariable position of the terres- “ial axis, by which it always points to the same “oint in the heavens, abstracting from the ihifiing effect of nutation, &c. See NuTaTion. PARALLELOGRAM, in Geometry, is a Mnadilateral right-lined figure whose opposite des are parallel. | PARALLELOGRAMS receive particular deno- shinations according to the equality or inequa- “ty of its sides and angles. Thus a rectangle, rhombus, rhomboid, and fare, are each a particular species of paral- itograms, for the properties of which see the »veral articles. _ Other properties common to every paralle- J.gram may be enumerated as follows: / 41. A parallelogram has its opposite sides ‘ad angles equal to each other, and the dia- onal divides it into two equal triangles, its: DD c } jbo i os PAR 2. The two adjacent angles of any paralle- logram are together equal to two right an- gles. 3. Parallelograms having equal bases and altitude are equal; on equal basis they are to each other as their altitudes; and with equal altitudes they are to each other as their bases; and generally parallelograms are t@ each other in the compound ratio of their bases and altitudes. 4. The sum of the squares of the diagonals of any parallelogram is equal to the sum of the squares of the four sides. 5. The complements abont the diagonals of any parallelogram are equal to each other, See COMPLEMENT. PARALLELOGRAM of Forces, is a term used to denote the composition. of forces, or the finding a single force that shall be equi- valent to two or more given forces when acting in given directions, the principles of Which may be thus illustrated. The simultaneous action. of two impulsive forces P, P’ on a body A, which would ime press upon it separately the velocities V, V’, in the directions AC, AC’, will cause that body to move uniformly over the diagonal of the parallelogram whose sides are in the di- rections of those forces. Imagine that the body A is placed on a plane ACC’, which moves uniformly in the direction AC’ with such a velocity as in each unit of time will carry it over a space equal to the line AC’; itis certain that this body, considered with relation to the plane on which it is placed, has no motion; yet if a spectator, fixed timmoveably out of that plane, observes the body A, he will attribute to it a motion equal and parallel to that of the plane. Now, if we conceive that any impulsive force what- ever, P acts upon the body A in the direction PAC, and impresses upon it such a velocity that in a unit of time it would pass over a space equal to AC, there can be no doubt that if the body were acted upon by this force only, it would be found at the point C, at the termination of the unit of time. ' But since, in consequence of the motion of the plane, the line AC advances towards C’B by a mo- tion uniform and parailel, so that it would really coincide with C’B at the end of a unit of time, it is obvious that the point. C will then coincide with the point B, and that, of consequence, the body A, which-partakes of the motion of the plane, ought to be found in B at the end of the first un itoftime. We may prove, in like manner, that at the end of any part or multiple whatever, 'T, of this unit, the PAR body A, moving with the same velocity AC, ought to run over a proportional space Ac=T x AC, while the common motion constrains the line A c to pass parallel to itself over a distance Ace’ =T x AC’. This line coincides there- fore with c’b, and consequently bis the place of the body A at the end of the time T. And it is manifest that all the points 3, 6, that may be determined by the same reasoning, are found on the same diagonal A B, since Ac: eb::AC:CB. The body A, therefore, ac- tually describes the diagonal A B. But be- sides this its motion along this line must be uniform; for Ab: AB:: Ac: AC :: T x AC:AC::T:1; that is to say, Ab is to A Bas the time employed in passing through Ab to that occupied in passing over AB. Consequently the motion of the body A, along the diagonal A B, is uniform. Since a body at rest on a moveable plane has the same motion as the plane, it is clear that if the plane were at rest, but that the body moved uniformly according to the right line P’AC’, with the velocity AC’ equal to that which would be impressed upon it by the force P’, and received at the point A from the force P a velocity A C in the direction P AC, it would describe uniformly the diagonal A B of a pa- rallelogram formed upen the sides AC, AC’, which represent the velocities of the body in those respective directions, while the diagonal A B represents its new velocity. We may likewise show that if a body be acted on by two similar variable forces (for the same time) whose directions and magnitudes are expressed by the adjacent sides of a pa- rallelogram ‘concurring in the body, it will describe the diagonal of the parallelogram. Let the forces act by impulses, at the be- ginning of equal particles of time, and let Ac’, e'C'C'c and Ac, cC, Ce, be the relative magnitudes of corresponding impulses. Then by the action of the two first impulses the body will, by the preceding article, describe the diagonal Ab; and by the next two the diagonal 5B, of the parallelogram dd’, whose sides bd’, bd, are equal and parallel to the representatives of those new impulses; but the forcés are similar, therefore the paralle- lograms ce’ dd’, are similar; and having pa- rallel sides and a common point d, they exist about the same diagonal AB. The same may be-shown for a third pair of impulses, and so on, ad libitum. Let now the particles of time be evanescent and the forces inces- sant, and the same demonstration will obtain. if the forces by which the body is urged in the directions AC, AC’, be not similar, it will move in some curve line, whose nature will depend on the relation of the forces. ' Hence it appears that in order to find a force that shall be equivalent to two forces, whose quantity and diréction is given, we have only to find: the diagonal of the paral- lelogram, by the sides of which the given forces are represented, and this will expréss the quantity and direction of the force which is equivalent to them both. BD BehA PAR Thus if AB, AC A B represent the quan- “ tity and direction of two forces act- ing upon a body at ° A, we first com- plete the paralle- ie | logram ABCD, ani the diagonal AD will represent the equivalent force required. | And in the same manner may be found the equivalent to three or more forces, by first reducing two of them to one single and equi- valent force, then this and one of the other given forces, and so on till they are all re- duced to one equivalent force. 4 D Suppose, for example, it were required t compound the three forces AB, AC, AD; a1 to find the quantity and direction of one single force, which shall be equivalent to the three given forces. # First reduce the two AC, AD toone AE, by completing the parallelogram A D EG; then reduce the two AE, AF, to one EAE by the parallelogram AE} B. So shall the single force A F be the direction and quan tity, or as the quantity of the force, whic shall of itself produce the same effect as if a the three, AB, AC, AD, acted together. And hence again conversely, any single direct foree may be resolved into two or moreé oblique forces; which is done by merely de: scribing any parallelogram, such that th line representing the given single force maj be its diagonal; and as these may be an indefi nite number of parallelograms having the same diagonal, so may any single force be re solved in an indefinite number of ways int two or more oblique forces, that shall produce the same effect as the single given force. On this subject the reader may consul Gregory’s “‘ Mechanics,” and the article Dy: NAMICS, Supplement to the Encyclop. Brit. PARALLELOGRAM of the Hyperbola, is used to denate the modulus, or any of the equa parallelograms inscribed between the curve and its asymptote. See HyPERBOLA. % Newtonian PARALLELOGRAM, is an invention of Sir Isaac Newton, for finding the first ter of an infinite converging series. See Newton’! “ Method of Fluxions and Infinite Series,’ p. 9. Colson’s Comment on that treatise p. 192; Maclaurin’s “‘ Algebra,” p. 251; an Cramer’s “ Analysis des Lignes Courbes,’ p. 148. ” PARALLELOGRAM Protractor, a mathema tical instrument, consisting of a semicircle 0) brass, with four rulers in form of a parall gram, made to moye to any angle: one 0 PAR these rulers is an index, which shows on the semicircle the quantity of any inward or out- ward angle. PARALLELOGRAM, is also the name of an instrument for copying plans, otherwise called a PentaGRATH, which see. | PARALLELOPIPED. See PARALLELE- (PIPED. ) PARAMETER, a certain and constant right line in each of the three conic sections, ‘and otherwise called the Latus Rectum. The word is derived from the Greek wapa, and perpoy, equal measurer, because it measures tthe conjugate axis by the same ratio, which rhas place between the axes themselves, being ‘alway a third proportional to them, wz. a \third proportional to the transverse and con- jugate axes in the ellipse and hyperbola, and which is the same thing, a third proportional to any absciss and its corresponding ordinate in the parabola. See Conic Sections. PARDIES (Ienatius GAsTON), an inge- nious French mathematician, was born in the province of Gascony, in 1636, and died in 1673, when only 37 years of age. Pardies was author of several works, of which the following are the principal. 1. Horologium Thaumaticum duplex, 4to. 11642. 2. Dissertatio de Motu et Natura ,Cometarum, 8vo. 1672. 3. Discourse du /Movement Local, 12mo. 1670, 4. Klemens ‘de Geometrie, 12mo. 1670: this has been ' franslated into several languages; in English iby Dr. Harris, in 1711. 5. Discours de la ) Connoissance des Betes, 12mo. 1672. 6. Lettre ,d’un Philosophe a un Cartesien de ses Amis, i2mo. 1672. 7. La Statique ou la Science des Forces Mouvantes, 12mo. 1673. 8. De- scription et Explication de deux Machines |propres a faire des Cadrans avec une grande /facilite, 12mo. 1673. Remarques du Mouve- ment de la Lumiere. 10. Globi Coelestis in ‘tabula plana reducti Descriptio, folio, 1675. /Part of his works were printed together at the Hague 1691, in 12mo.; and again at Lyons, 1725. Pardies had a dispute with | Sir Isaac Newton, about his “ New Theory of | Light and Colours,” in 1672, His letters are inserted in the Phil. Trans. for that year. PARENT (AnTHONY), a reputable French mathematician, was born in Paris in 1666, and died in 1716, having at his death nume- /rous manuscripts on various mathematical » and philosophical subjects ; besides which he ) published the following complete works, viz. , 1. Elemens de Mechanique et de Physique, | i2mo. 1700. 2. Recherches de Mathematiques et de Phy- ; sique, 3 vols. 4to. 1714. | 3. Arithmetique Theorico-Pratique, 8vo. 1714. _ Besides a great number of papers in the » volumes of the Memoirs of the Academy of . Sciences, from the year 1700 to 1714, having ) several papers in almost every volume upon a ) variety of branches in the mathematics. PARHELION, or PaRHELIUM, (from wapa, near, and nAsos, sun), denotes a mock sun or | | } PAR meteor, appearing as a very bright light near the sun, being formed by the reflection of his * tei in a cloud properly situated to receive rem. PART, Aliquant, Aliquot, Circular, &e. . See the several articles. PARTIAL Differences, Theory of, is one of the higher branches of the modern analysis for the development of which we are prinei- pally indebted to d’Alembert, though the sub- ject had been previously investigated by other mathematicians. He was led to this discovery in attempting the solution of certain problems relating to the vibration of musical chords ; and afterwards employed it with great success in his Theory of the Causes generale des Vents; the resistance of fluids, and other very delicate investigations. But, as we have cbserved, some little progress had been made before d’Alembert commenced his investi- gations. Thus Fontaine had already prepared and adopted the notation, and Nicholas Bernoulli in his paper on Trajectories Acta. Eruditorum, 1720, found the relation which has place be- tween the partial differences of a function of two variables. Clairaut also discovered the saine theorem, and employed it in two of his papers ou the Integral Calculus, in the Me- moirs of the Academy of Paris for 1739 and 1740; in the second of which he deduces certain results from the consideration of curve surfaces; and from some remarks first suggest- ed in this paper, Lagrange was:induced to un- dertake, and finally completed, his theory of particular integrals. See Memoirs of Berlin for 1774... But of all those who preceded d’Alembert, in the doctrine of partial differences, Euler has perhaps the greatest claim to priority of invention, he having completely integrated an equation of this description in 1734, in vol. vii. of the Acta. Petro.; but afterwards forgot or neglected his new calculus, not perhaps being aware of its extensive application to physical problems. This application we owe entirely to d’Alembert, and the great ingenuity and sagacity which he has displayed in the integra- tion of his equations cannot fail of placing him amongst the first of modern mathematicians. Integration of Equations of PartiaL Dif- ferences. Let wbeany function of x, y, and z, and 3, d, D, the characteristics of their diffe- rentiation relatively to each of the variables, then an equation compounded in any manner, with any of the quantities, i Y; zy, 1&¢. dw du Du &e dx’ dy’ dz’ : ddu ddu Ddu & dx dady’ dxdz is called an equation of partial differences, and which may be compounded or not, with constant or known quantities. ‘The degree of the equation is estimated by the highest power of any of the differentials that enters into it. du, du e! , Thus 7 + cea a ts is a partial differential equation of the first degree, the solution or integration of which depends upon finding such relation between the variables « and y and the function w; that the latter being first differenced with x as variable and y constant, and divided by dz; plus the same differenced with y as variable, and x constant, and divided by dy, the sum of the two partial differences a cc “ may be equal to zy, and the hethiod ‘of performing this’is called the integral calculus of partial di iff erences. rome AA iter yt 0 is also an equation of partial differences, the integral of which is u = 2? y + y*x For differencing this first relatively to x va- riable, and dividing by dx, we have rstoe Ss 2ay Fg?) du And the same with reference to y gives . du ——= 2 + 2ry; dy By Tf consequently ae + ~ ty? +4ary. ‘This integral is ouikes merely by the re- ' verse operation by which the function was thrown into partial differences, and is therefore very obvious; but when the equation arises in the solution of a problem or otherwise, pro- viding its origin is not known, its integr ation is frequently extremely difficult, and requires the most profound and accurate analytical judgment. Let there be meat the equation ; ddu _ a> odu dy* "ae § du=pdx + qdy; dp =rdx-+sdy; Suppose 4 dg=sdx+tdy, then t =a?r dq= sdx +a’ rdy, and therefore } 44 — —=ardx ose. . Whence adp + dq =(s + ar) (dx + ady) which is one complete differential, and there- fore its equal (s + ar) (dx + ad ¥) must be one likewise; and consequently s+ar is some function of x + ay, and ap + q another func- tion also of x + ay. Let therefore ap + gq =f (« + ay) ‘and ap—q—f (@ — ay) the latter being obtained in the same manner as the former. _ § (dx +ady). f (« +ay)+ Whence 2adu= ; (dx —ady). f(a—ay) and integrating, 2an = O(#+ay)+0(a—ay) an integral equation, whatever may the func- tions @ ‘and 9, which functions are to be de- termined from the nature of the problem. See Mem. of Berlin for 1747, 1748, 1750, and 1763; vols. i. and ii. of the Memoires of Turin; and the Opuscules of d’Alembert, and the other papers referred to in the preceding part of this article. PARTICLE, the minute part of a body, PAR or an assemblage of several atoms of which | natural bodies are composed. This term i frequently used as synonymous with atom, corpuscule, and molacule, but sometimes it i distinguished from them. 4 PARTICULAR Integral, in the Integra Calculus, is that which arises ip the integra- | tion of any differential equation, by giving ay particular value to the arbitrary quantity, or quantities that enter into the general integral, ‘Thus for example, in the equation * “2xrdy — ydy* f pda 5 cea . the general integral is i y = 4m (x — m) and by making m =1, this becomes y= 4 (a — 1) which is a par ticular integral. We callalso particular integrals those whic are obtained by means of an equation o inferior dimensions ; thus in the above, if we make x — y, we shall have .. _ 2ydy yd y" Sa. ive is which obviously answer the conditions of the equation. , So also in the equation te xdx+ydymdy v(x" ag Meee 23 bi it is easy to see, that 2? + y* =a* answers_ the condition of the equation, and is therefore called by some a particular integral. But neither this, nor that found above, wat + Ye are deducible from the general integration of the equation whence they are derived, aud they thus differ from the particular integral above defined; so that this term ought not t be employed to denote the two distinct Cases; and accordingly some authors distinguish th integrals arising from such equations into three kinds, viz. complete integral, which is tha drawn from the general integration; an i= complete integral, which is that found as above, by means of an equation of inferior dimen; sions ; and lastly a particular integral, which | is that arising from giving to the general in= determinate “quantities in the complete in- tegral some particular value. Others again, as “Lagrange in his ‘‘ Lecons des Fonctions,”. instead of incomplete integrals calls then valeures singulaire. ’ 'Phese incomplete or singular integrals were first noticed by Clairaut, and were treated by Euler as insolvable paradoxes; but Lagrang in the Memoirs of Berlin, has explained this” apparent anomaly, by showing that the inte gration of a differential equation of the eh degree, necessarily produces one constant arbitrary quantity, and the integration of o equation of the second degree gives rise to tw such quantities, and gener ally a differenti equation of x dimensions give place to n arbi- trary quantities, which may be -assumed at pleasure. ° If therefore in the case of two variables, w ve consider these as the rectangular co-ordinates: ofa curve, itis obvious that we may imagin as many different curves to be described, there TR be partiqular values Biv ems to ‘the a) PED veneral indeterminate quantity; and if it hap- ens that these curves have one constant juantity common to them all, then the same angent touching all the curves, must involve ja its equation the clement of one of those jurves, yet this tangent is not one of the curves, ind therefore its equation is not obtainable yom the complete integration. See Memoirs f Berlin for 1774 ;) the ‘“ Theorie des Fonc- sons Analytiques;” and the ‘“‘ Lecons des /onetions,” by Lagrange; see also Montucla’s | Hist. des Math.” vol. ili. p. 180. PASCAL (BLAtse),a very excellent French yathematician and philosopher, was born at Htermont, in Auvergne; in 1623. He disco- ered very early an extraordinary genius for 1athematical and philosophical inquiries, nd is said to have formed a system of geo- at the early age of 12 years, without aving seen any work on the subject, these aving been carefully kept from him by his lither lest they should divert him from his jterary pursuits. | After this he was permitted to follow his iathematical studies, and at 16 years of age je published a treatise on the conic sections, jad three years after invented what is com- only called Pascal’s arithmetical machine, articularly useful in a variety of computa- ons. ‘I'o Pascal we are also indebted for the mous experiment of the Puy de Dome, which as the first confirmation of the pressure of ‘mosphere, and this gave rise to his work on lis subject, which he afterwards divided into yo treatises, the one entitled “ A Disser- |tion on the Equilibrium of Fluids ;” and the ther on the “ Weight of the Atmosphere,” hich were published after his death. He so published, under the name of A. d’Etton- lle, the celebrated problem of the Roullette ‘Cycloid. Pascal is also equally renowned r his elegant composition, as for his mathe- atical and philosophical genius, particularly tr his celebrated “ Lettres Provinciales,” jblished in 1656, under the name of Louis 1s Montalte. | These and some other pieces were collected to five volumes, 8vo. and published at the jague, and at Paris in 1779, by the Abbé ‘ossu, which may be considered as the first mplete collection of the works of this cele- ated author. | Pascal was always of a very delicate con- litution, which he finally sunk under in 1662, ‘tag at the time of his death only in his 39th sar, PAVO, Peacock, a southern constellation, le CONSTELLATION, PECK, an English measure, the 4th part a bushel. PEDOMETER, or Popometer (from ass, ot, and petpov, measure), a mechanical in- jument in form of a watch; consisting of rious wheels, with teeth catching in one an- her, all disposed in the same plane ; which by fans of a chain or string, fastened to a man’s ot, or to the whcel of a chariot, adyance a PEN notch each step, or each revolution of the wheel; so that the number being marked on the edge of each wheel, one may number the paces, Or measure exactly the distance from one place to another. PEGASUS, the Horse, a northern constel- lation. See CONSTELLATION. PELECOID, or PELECOIDEs (from qeAcxus, hatchet, and «do:, form), hatchet form, in Geo- metry, contained under the two inverted qua- drantal arcs A B and A D, and the semicirele BCD. C B ae rh E ef A 'The area of the pelecoid is demonstrated to be equal to the square AC; and that again to the rectangle EB. Itis equal to the square AC, because it wants of the square on the left hand the two segments A B and A D, which are equal to the two segments BC and CD, by which it exceeds it on the right hand. PELL (Dr. JoHN), an eminent English mathematician, was born at Southwick, in the county of Sussex, in 1610, and died in Lon- don in 1685, at the age of 74.. He was author * of numerous papers and treatises on mathe- matical and astronomical subjects, but none of them are now of sufficient importance to be enumerated in this place; but a complete catalogue of them may be seenin Dr. Hutton’s Math. Dict. To Dr. Pell we owe the first solution of the general indeterminate equation x* — ay” = 1, which was proposed by Mermat as a Challenge to all the English mathema- ticians. See Euler’s “ Algebra,” vol. ii. p. 78, 2d English edition; to the same author we are also indebted for the method of ranging and numbering the several steps in the solu- tion of algebraical problems, as followed by _ Ward in his “ Young Mathematician’s Guide,” and by l’enning in his “‘ Algebra.” PENCIL of Rays, in Optics, denotes a number of rays diverging from some luminous. point, which, after falling upon and passing through a lens, converge again on entering the eye. PENDULUM, in Mechanics, denotes any heavy body so suspended that it may vibrate, or swing backwards and forwards, about some fixed point by the action of gravity. The vibrations of a pendulum are called its oseid- lations, the time of each being counted from the time of its descent from the highest point on one side, till it attains the highest point on the opposite side. ; Picks A pendulum, therefore, is any heavy body B, suspended upon, and moving about som fixed point at A-as a centre. “ Here the point A is called the centre of motion, or of suspenston, and the line PQ parallel to the horizon, the axis of oscillation, and that point in the body B, into which, if all the matter of the body was condensed, so as still to perform its oscillations in the same time, is called the centre of oscillation; and the distance of this point from the point of suspension, is accounted the length of the pendulum. See Centre of Oscillation. Pendulums receive particular denomina- tions, according to the materials of which they are composed, or the purposes they are intended to answer; for an account of which see the subsequent part of this article; we shail here confine our remarks to the) theory of the simple pendulum. Simple PenpuLuM, theoretically considered, is a single weight attached to a string sup- posed to be devoid of gravity, and oscillating freely without resistance, either from friction or the air. Properties of a Simple Pendulum thus consti- tuted, and vibrating in small circular Ares. 1. The vibrations of the same pendulum, or of different pendulums of the same length, and in the same place, in very small arcs of circles, are always performed in the same time. 2. The velocity of the bob, in the lowest point of the arc, will be very nearly as the length of the chord of the arc which it de- scribes in its descent. 3. The times of vibration of different pen- dulums are, to each other, as the square roots of their lengths. Or their lengths are as the squares of the number of vibrations performed in the same time. . 4, On the supposition above made, viz. that there is neither friction nor resistance op- posed to the motion of the pendulum, that motion would be perpetual, that is, the force which it acquires in its descent would carry it up to the same height on the opposite side, which would be again repeated in its descent, and thus it would continue to oscillate con- stantly through the same arc, and conse- quently its vibrations uniform and perpetual. But as it is impossible to divest it of these two retarding forces, it is obvious, as well from theory as practice, that without the application of some external force, the vi- brations will be shortened in every ascent, and the motion of the pendulum ultimately cease. 5. The time of vibration in a circular arc PEN : vof any sensible magnitude, is expressed by the following formule : eee d § la. 1.07 : — =p / 4g rep sigs 2 d + 2° .4*.d? | 1*.37.5" a3 q Peed * Ko. { Where d = 2 AB, or twice the length ; the pendulum, a = CB the versed sine 0 half the are of vibration, g = 16, half the force of gravity, and p = 3'1416 the circum ference of a circle whose diameter is 1. 6. When a is very small, as we have sup posed in the preceding cases, then writing 2) instead of d, and omitting all the terms in th above series beyond the second as being in considerable, we shall have Meal ha a a\_p l , time =p \/ a x0 +a)=EV (82 + a). 7. Or ifinstead of using the versed sine the half arc, we introduce the degrees in tha arc, the expression becomes : l d* time =p\/ —x (Qa += ). 2g 5234 Properties of Pendulums vibrating in cyclord Ares. , 1. It is a known property of the cycloi that its evolute is a cycloid, similar and equé to the former. If therefore a pendulum b E EB Cc suspended from C, the point of concurren¢ of two equal inverted semicycloidal cheek and be made to vibrate between them, tl bob will itself describe the cycloidal a HAK. The properties of a pendulum th suspended are as follow: 2. The time of oscillation in all arcs, wh ther great or small, are performed in the sat time. a 3. The time of vibration in ‘any arc, is) the time in which a heavy body would fall the force of gravity through half the length the pendulum; as the circumference oi circle is to its diameter. Now by the laws falling bodies 4/ & = the time of a falling through £ J, or CA; therefore puttit p = 31416, the circumference of a circ whose diameter is 1, we have : asWi pi / pV x vibration of a pendulum whose length is 2 4. It is obvious in this formula, as well in those given above, that the time of vibr tion depends not only on the length of # the time | PEN endulum, but also on the value of g, which ; different for different latitudes; and con- lequently the lengths of the pendulum de- igned to vibrate any proposed number of imes in a second, in any place, must be re- ulated by the value of g, or by the force of jrayity at that place. ) 5. The value of g in the latitude of London § 16,4, feet, or 193 inches, which being sub- tituted for g in the expression p he = ra 108 tives 7 = 39°11, or 394 inches for the length tf the second’s pendulum, which is extremely rear the true length as determined from ac- farate experiments, which show it to be about 92 inches. _ 6. From the above formula may also be ound the force of gravity in any particular face, the length of the second’s pendulum eing previously found from experiment; for, prl {uaring the above, we have —— = 1, org = pl, the descent of a heavy body in one ‘cond, in terms of the length of the second’s sndulum; and this is a much more accurate ethod of finding the force of gravity, than ay other which depends simply on experi- jents. | 7. In speaking of the simple pendulum as brating in very small arcs, we found that the me of one vibration was very nearly = p | vA l a s ’ 20 x ( + <), where a is the versed ne of half the are of vibration, whereas in a veloid the time is simply p = V so and erefore the time lost each vibration in the rmer case is —? x ai a or estimating by Sl 2g aes l e degrees of the arc it is ——— x 5 Orie 52524 ie” id hence we may compute how much the mdulum must be shortened in the former se to make its vibrations correspond with te vibrating as in the latter case. 8. We haye seen above that the length of e pendulum varies in different latitudes in nsequence of the variation in the force of avity, and on this principle is computed the lowing table, which shows the length of the ‘cond’s pendulum for every 5 degrees of lati- de from the equator to the pole. Deg. of Length of Deg. of Length of Latitude. Pendulum. Latitude. Pendulum, 39°027 50 39°126 39°029 39°142 39°032 39°158 39°036 39-168 39°044 39°177 39°057 39°187 39070 39'191 39°084 39°195 39°097 39°197 39°111 PE N° The foregoing laws, &e. of the motion of pendulums, cannot strictly hold good, unless: the thread that sustains the ball be void of weight, and the gravity of the whole ball be collected into a point. In practice, therefore, a very fine thread, and a small ball, but of a very heavy matter, are to be used. But a thick thread and a bulky ball disturb the mo- tion very much; for in that case, the simple pendulum becomes a compound one; it being much the same thing, as if several weights were applied to the same inflexible rod in several places. M. Kraft in the new Petersburg Memoirs, vols. vi. and vii, has given the result of many experiments upon pendulums, made in dif- ferent parts of Russia, with deductions from them, from thence he derives this theorem: if x be the length of a pendulum that swings seconds in any given latitude /, and in a tem- perature of 10 degrees of Reaumur’s thermo- meter, then will the length of that pendulum for that latitude be thus expressed, viz. x = (439.178 + 2.321 x sin. 21) lines of a French foot. And this expression agrees very nearly, not only with all the experiments made on the pendulum in Russia, but also with those of Mr. Graham's, and those of Mr. Lyons, in 79° 50' north latitude, where he found its length to be 441.38 lines. What is stated above of the uniform laws relating to the vibration of pendulums shows very obviously their great utility in regu- Jating the motion of clocks, or for other equal division and measurement of time. Galileo was the first who made use of a heavy body annexed to a thread, and sus- pended by it for measuring time, in his ex- periments and observations: but according to Sturmius, the first person who observed the isochronism of pendulums, and made use of it in measuring time, was Ricciolus; after him Tycho, Langrene, Wendeline, Mer- senne, Kircher, and others, observed the same thing; though without any intimation of what Ricciolus had done. But Huygens was the first who demonstrated the properties of the pendulum, and probably the first who applied them to clocks. He demonstrated, that if the point to which a pendulum is sus- pended was perfectly fixed or immoveable, and all friction and resistance removed, that a pendulum once set in motion, would for ever continue to vibrate without diminution of motion, and that all its vibrations, (sub- ject to these conditions) would be perfectly isochronal. Hence the pendulum has uni- versally been considered the best chronome- ter, as all pendulums of the same length perform their vibrations in the same time, without regard to their different weights, and it has been suggested, by means of them, to establish an universal standard of measure for all countries. On this principle, M. Mouton, canon of Lyons, has a treatise “ De Mensura posteris transmittenda.” But it has been suggested that if a long PEN | pendulum were made to swing in the plane of the meridian, and another of equal length in a plane perpendicular to the meridian, some difference might be found in their vibrations from the centrifugal force arising from the earth’s rotation about its axis. See Phil. Trans. No. 468, sect. 1. The method of making a pendulum vi- brate in the are of a cycloid, and the pro- perties resulting from such construction, we owe, as stated above, to Huygens, who has treated at length on this subject both theore- tically and practically, in his ‘‘ Horologium Oscillatorum sive Demonstrationes de Motu Pendulorum ;” and every thing relating to the motion of pendulums has since been de- monstrated in different ways, and particu- larly by Newton in his ‘‘ Principia,” where he has extended the same theory to epicy- cloids. Fiaving said thus much with regard to sim- ple pendulums, it remains only to offer a few remarks on the nature of compound pendu- lums, as applied to.clocks and other machines where uniform motion is required. We have seen above that pendulums in different latitudes require to be of different lengths, in order that they may perform their vibrations in the same time; but besides this there is another irregularity ia the motion of a pendulum in the same place, arising from the different degrees of temperature. Heat expanding, and cold contracting the rod of the pendulum, a certain small variation must necessarily follow in the time of its vibration, to remedy which defect various methods have been invented for constructing what are com- monly called compensation pendulums, or such as shall always preserve the same distance between the centre of oscillation and the point of suspension ; but of these we have only room to describe one or two of the most approved construction. Compound or Compensation PENDULUMS, have received different denomination from their form and materials, as the gridiron penduhun, mercurial pendulum, lever pendulum, &e. Gridiron PENDULUM. This consists of five rods of steel, and four of brass, placed in an al- ternate order, the middle rod being of stcel, by which the pendulum bail is suspended ; these rods of brass. and steel, are placed in an alter- nate order, and so connected with each other at their ends, that while the expansion of the steel rods has a tendency to lengthen the pen- dulum, the expansion of the brass rods acting upwards tends to shortenit. And thus, when the lengths of the brass and steel rods are: duly proportioned, their expansions and con- tractions. will exactly balance and correct each other, and so preserve the pendulum invariably of the same length. The Mercurial PENDULUM was the invention of the ingenious Mr. Graham, in consequence of several experiments relating to the ma- terials of which a pendulum might be formed, in 1715. Its rod is made .of brass, and branched towards its lower end, so as to.em- PEN ; brace a cylindrical glass vessel 13 or 14 inches | long, and about 2 inches in diameter; which being filled about 12 inches deep with mer. cury, forms the weight or ball of the pendus_ lum. If upon trial the expansion of the rod. | be found too. great for that of the mercury, more mercury must be poured into the vessel} if the expansion of the mercury exceeds that of the rod, so as to occasion the clock to. gain with an increase of temperature, some mercury must be taken out of the vessel, inv order to shorten the column. And thus may the expansion and contraction of the quick- silver in the glass be made exactly to balance the expansion and contraction of the pendulum rod, and thus preserve the distance between the centre of oscillation and the point of Sus : pension invariably the same. a Reid’s Compensation PENDULUM, 18 a ole a recent invention of Mr. Adam Reid a of Woolwich, the construction of which is as follows: AN is a rod of wire, and ZZ a hoilow tube of zinc, which slips on the wire, being stop- ped from falling off by a nut N, on which it rests; and on the upper part of this cylinder of zinc rests the heavy ball B; now the length of the tube ZZ being so adjusted to the length of the rod AN, that the expansions of the two bodies shall be equal with equal degrees of temperature; that is by making the length of the zinc tube an to that of the wire, as the expansion NW ie of wire is to that of zine, it is obvious vi that ball B will in all eases preserve the same distance from A; for just so much as it would descend. by the expansion of the wire downs wards, so much will it ascend by the expan-_ sion of the zinc upwards, and consequently — its vibrations will in all temperatures be- performed in equal times. S Bailistic PenpuLuM. See Bauistic Pen- dulum. - PENETRABILITY, the capability of be- ing penetrated. . PENETRATION, is used principally t denote the forcible entry of one solid bod within another by means of a projectile mo-_ tion, communicated to the former, which enables it to displace those parts of the latter with which it comes in contact. Or the pe netration may be otherwise produced by the action of some percussive force acting upon one of the bodies when in contact with the other; these two cases, however, differ rather in circumstances than in principle, and there-— fore in the slight sketch we shall give of this” subject, we shall consider the penetra body to be projected with a certain velocity, — and impinging upon the fixed body in a di-— rection perpendicular to its surface. = This is a subject of considerable importance in military and naval gunnery, and has been accordingly treated of by different writers or these subjects; Dr. Hutton in particular h made several experiments on different sub- stances, and with different charges of powder, a = , *s oe ere . PN and different weight of shot, in order to pro- ‘cure data from which the penetration in other cases may be determined, The mean results of the most accurate of his experiments, as iven in vol. iii, of his Tracts, are stated elow. ° Diam. of iron shot | Penetration, Velocity in Substance. feet. in inches, 1600 Elm 1:96 20inch. 1200 Ditto 1:96 15 1500 Ditto 2°78 30 1060 Ditto 2°78 16 -*’ 1200 Oak 504 34 ™ 1800 Earth 5°55 15 feet ifgs In these experiments the gun was placed 0 near the object, that the initial velocity of the ball was not changed, and the penetration was made lengthways of the timber, which in the first four cases was sound elm, cut off near the root of the tree; the other two experi- ments were made by Robins, and given by ‘that author, with some others, in his Tract on gunnery. The first two experiments with he velocity 1600 and 1200 feet, give 20 and inches for the penetration being precisely a the ratio of the two velocities; but this atio does not obtain in the next two experi- nents; in fact, it is.pretty obvious that this iannot be the case, unless the resisting force f the wood was uniformly the same through- mt, which cannot be; its density about the renetrating body increasing every instant, in onsequence of the parts forced in by the ball, vhich accounts for the results as deduced by Jr. Hutton on this hypothesis not agreeing vith each other. ' Supposing the resisting force of the wood » be uniform, and representing it by f, the elocity by v, and the space described by s, Iso g = 16 J, feet, the space described by body by gravity in one second, we shall have 2 a the principle of forces f = =~ and ap- lying this formula to each of the above cases w the same substance, each of them will be und to give different results, which if the sistance was uniform, ought all to agree. ith each other, or nearly so; and the same djection must necessarily have place with ‘gard to the times of penetration. It is tobable, however, that this hypothesis, of niform resistance, may not much affect the sults in drawing a comparison between the netration of bodies of different densities, agnitudes, &c. because here we do not re- ire any absolute quantities but their ratios; regard to each other, ander this point of ew, ifd is made to represent the diameter the ball, n its density, that of water being 1e, f the height of a column of water equal ) the resisting force of the wood or body metrated, a — *7854, g — 16 fect, then it ay readily be shown from the doctrine of } AEP... 2 a rees that f — woe , and's = cide 6sg 6 hich formule, and the preceding experi- ; from PEN ments, all cases of penetration on the sub- stances there given may be determined. From these formule it appears, that the depth penetrated, is as the density and dia- meter of the ball, and the square of the ve- locity, divided by the strength, or resisting force, of the matter or obstacle; so that if equal balis be discharged against the ob- stacle, the depths will be as the squares of the velocities. See Hutton’s Tracts, vol. iii. p- 283. PENNY, an English copper coin, the 12th part of a shilling. PENNYWEIGHT, the 20th part of an ounce Troy. This weight derived its name from being exactly the weight of an ancient English silver penny. PENTAGON (from ware, jive, and yuu, angle), in Geometry, is a figure of five angles, > and consequently also of five sides, and when these are both equal it is called a regular pentagon, but otherwise it is irregular. The angle at the centre of a pentagon is 72°, and the angle of its sides 144°. The area of a pentagon, whose side is one, is 1°7204774; consequently, when the side is s, the area = s* x 17204774. To inscribe a Pentagon in a given Circle. Draw the diameters, Ap, nm at right angles to each other, and bisect the radius oninr. From the point r, with the dis- tance rA, describe the arc As, and from. the point A, with the dis- ; tance As, describe the arc sB. Join the points A, B, and the line AB being carried five times round the circle, will form the pen- tagon required, To. describe a Pentagon on a given Line. Make Bm perpendi- cular to AB, and equal to one half of it. Draw Am, and produce it till the part mn is equal to Bm. From A and Bas centres with the radius Se Bn, describe arcs cut- = A ting each other in 0. And from the point 0, with the same radius, or with o A, or 0B, de- scribe the circle ABCDE, Apply the. line AB five times round the circumference of this circle, and it will form the pentagon re- quired. Note. If tangents be drawn through the angular points ABCDE, a pentagon. circum- scribing the circle will be formed; and if the arcs be bisected, the circle will be divided into ten parts, which answers to a decagon, which is thus readily constructed. PENTAGRAPH, an instrument with which designs of any kind may be copied in any proportion at pleasure, without being skilled in drawine. ene San, . oe" « a PEN The pentagraph is made of brass, and con- sists of four levers ABD E, the two longest of which, Aand B, are jointed together at their ends, the other two, D, E, are also jointed to- gether at one of their ends, and to the levers at the others. In this manner the instrament always forms a parallelogram, aAa=e ie, and aBe = aDe; f, g, and h, are three tubes upon the levers, two of which, f, g, slide along upon their respective levers, and can be fixed at any point by screws; any of these tubes is adapted to receive either a fulcrum or fixed centre, round which the whole instrument turns a blunt point or tracer, to pass over the original design, which is to be copied, or a crayon to draw the figure or copy of the original design; these three points must be always in one right line, and by the construc- tion of the levers, if they are once set ina line they will continue in it through any of its motions. The proportion in which the pentegraph will reduce any figure will be easily calculated from the same principles as the lever ; viz. that the magnitude of the figures described by either of the points, will be in the same proportion to each other, as the distances of these points from the fulerum; thus if the point f be the fulcrum, andif the distance from f to ¢ be half the distance from f to h, the size of the figure described by the point 2, will be half the size of the figure described at the same time by the point h. The fulcrum, as we have said be- fore, can be changed, as also the pencil and the tracer, andany of the three can be applied to either of the tubes upon the levers, if the tracer is placed in the tube A, the pencil in g, and the fulcrum at f, any figure described by the tracer A will be exactly copied one half the size by the pencil at g ; and if, on the contrary, the pencil is placed at A, and the tracer at g, the figure drawn by the pencil will be twice the size of the original traced at g. PENUMBRA, in Astronomy, a faint or partial shade observed between the perfect shadow and the fall of light in an eclipse. This arises from the magnitude of the sun, for were he only a luminous point, the shadow would be everywhere perfect; but in conse- quence of his diameter it happens, that a place, which though not illuminated by the whole body of the sun, may notwithstanding receive a part of his rays. ‘This may be illustrated as follows: let S re- present the sun and M the moon, then it is ob- vious since luminous rays proceed from every part of the sun’s disc, there will be no part of the shadow in which the light will be totally intercepted, except that included within th rays proceeding from the extreme edges ¢ the sun and moon ABC, and EDC, the othe part of the shadow, vz. from C to H, and experiencing only a partial interception, an consequently in those parts a faint light wi be observed, proceeding from the darke shades at C, diminishing both ways to H an I, where it is lost in perfect light. Penumbras must be constant attendants: all eclipses whether of the sun, moon, or pl: nets, primary or secondary; but with us the are most obvious in eclipses of the sun, whic is the case above alluded to. PERAMBULATOR, an instrument f measuring distances, called also a pedomete waywiser, and surveying wheel. | This wheel is so constructed as to measu out a pole, or 164 feet in two revolutions, ar consequently its circumference is 83 feet, at its diameter 2°626 feet nearly. 2 To the wheel is attached a sort of donb pole where the machinery is contained, whit works by means of the revolution of t wheel; and at the other end is a dial whi exhibits the number of miles, furlongs, & passed over; the whole being driven forwa by a person on foot, or drawn along by a cz riage to which it is sometimes attached. This machine is extremely convenient a expeditious, but it is not always quite ace rate, in consequence of the unevenness of t road. q PERCH, in Land Measure, the 40th part a square rood, containing 304 square yards PERCH, is also sometimes used as a meast, of length, being equal to 53 yards, or 164 fe and is otherwise called a rod or pole. PERCUSSION, in Mechanics, the striki of one body against another, or the sh arising from the collision of two bodies. T is either direct or oblique. by Direct PERCUSSION, is when the impt takes place in a line perpendicular to— plane of impact. Oblique PERCUSSION, is that which t place in any direction not perpendicular the plane of impact. . The Theory of PERCUSSION, is a subj which has much engaged the attention of losophers, particularly with regard to the 6 parison of percussion and pressure, one pa maintaining a perfect congruity between th two forces, while others assert their total comparability, observing that the least qu tity of percussion is greater than any press’ however great; for, say they, the moment of a body is measured by its mass into velocity, if therefore the body A moves’ a velocity », while the body B is at rest, | PER yas no velocity; the momentum of the former s A x v, and of the latter B x 0, and conse- uently the former is infinitely greater than e latter. . But however plausible this reasoning may \ppear at first sight, it is evidently erroneous is to the fact. Daily experience will con- vince us, that though the advantages gained vy bodies, moving swiftly, are very great over hose which oppose merely a resistance of oressure, yet that they are by no means in- inite. Numerous circumstances will suggest themselves to the mind, which prove, that, dhysically speaking, we may balance any vercussive force by an equivalent one of mere oressure, or even we may make the latter greater so as to overcome the former. _ The pile engine offers a remarkable con- Jirmation of this equality, or even preponder- ace on the side of pressure. It has, for in- stance, been found, that in driving piles in a iniform sandy soil of the same density to 47 eet, the piles could not be driven more than 16 feet by any percussive blow that could be ay by the engine; that is, the riction and resistance of the soil which may xe considered as a pressive force, was greater han any percussive force that could be em- jloyed by the pile engine, although the ram- ners made use of were extremely great. And hence when we are computing the fect of a pile engine, it will be necessary to Penatc first the quantity of percussion that ‘s equivalent to the resistance and friction opposed to the pile; as no momentum short of this, or even just equal to it, will produce my effect, and when the momentum is greater ‘han this, it is only the difference between the two that is effective in producing motion inthe pile. And to this circumstance must de attributed the many erroneous solutions hat appeared a few years back to the ques- ion, “ What must be the height of a pile mgine to produce the greatest effect in a riven time?” This question, at first sight, ap- years to be the same, with asking how high nust the pile engine be to produce the great- st momentum in a given time; but by using this principle the solution always gave the leight = 0; that is, the greatest effect will ye produced when the rammer is left at rest m the top of the pile. But if instead of proceeding thus, we first ‘stimate, or find from experiment, the height o which the rammer must be drawn, in order hat its momentum may be equivalent to the vesistance of the pile, and then considering he difference between this and any greater ‘nomentum to be only the effective part, a rery rational solution will be obtained. But before entering upon the solution of this wroblem, it will be proper to offer a few far- her remarks with regard to the comparability f percussion and pressure, because the solu- jion ultimately depends upon a proper com- arison of those quantities, and a want of due | apation to which, seems to have been the | PER cause of the erroneous results generally de- duced in the solution of this problem. Without, indeed, entering into a discussion concerning the congruity or incongruity of these forces, it is obvious, that they may be so employed as to produce the same or equal results. A nail, for example, may be driven to a certain depth into a block of wood by the blow of a hammer, or it may be sunk to the same depth by the pressure of a heavy body ; whence, and from numerous other instances, it is obvious, that pressure and percussion, whether congruous or incongruous in their na- ture, are at least comparable in their effects. With regard to the above problem, the re- sistance and friction of the soil against the pile may, as above observed, be considered as a pressure, and the object of our inquiry is to establish a comparison between this resist- ance or pressure of the soil, and the momen- tum of the ram, or what part of the whole generated momentum of the latter is em- ployed in overcoming the resistance of the former, in order to determine the effective part of the stroke, which ought alone to be considered in estimating the maximum effect; because any single momentum, less than that which is equivalent to the resistance, would produce no effect whatever. Now it being admitted that pressure and momentum are at least comparable in their effects, it must also be granted that there is some determinate momentum of the ram, that is equivalent to the resistance of the pile, and the height necessary for producing this momentum must be the first object of our research, which it is obvious, from the various circumstances that may arise in the applica- tion of the pile engine, can only be deter- mined by experiment. Suppose, for example, that it is found by experiment, that a momentum m drives the pile a inches, and a momentum M drives it 6 inches, and let it be required thence to deter- mine the momentum y, which is equal to the resistance of the pile. The first effective momentum is m — y, and the second M — y; and now admitting each effect proportional to the cause which pro- duces it, we have m—y: M—y:: 4: 6 or bm — by = Ma— ay __Ma—bm aie Tipe « which is the momentum equal to the resist- ance of the pile. Find the height due to this momentum, and call it A; then the solution of the problem will stand as follows: Let x be the required height to produce the max, a the velocity per second in drawing up, t the time in falling, g = 167; feet. Then supposing the momentum to vary as the velocity, and calling the weight of the ram w, we haye 2ot —v, vw =m, the generated momentum, = time in drawing up, » “ = time in falling =, rVYeta/e ‘aV/gZ 2/gu =v, or 2w/gz = the whole time, m, by substituting oe%. +5 stead of é. / o 5 Also h being the height due to the momen- tum above determined, this will be expressed by 2w / gh,and consequently 2 /gx—2wv gh is the eifective momentum, which divided by the time necessary for the production of it must be a maximum; that is, 2w /gx—Wwy/gh U/Z+a/xX Ww a JVh I or ———____—___ = a max. B/E + a/zr Fits thrown into fluxions, gives i}s-@vg +ava2)— /ge tv gh— 5 x a/gZ — rh Nc kd Q/a eee vi)! —0 which reduced gives $/ eu =a J * + /gh, or u/s a ana +2/ghx, or JS hh t—2 /hx = a-— Vg : Whence vim vhs (ht tat) ea} vhe/ (b+ $ayy which is the height required. In the above solution we have only deter- mined the height when any single momentum, divided by the time necessary for the produc- tion of it, is a maximum; whereas the pro- blem requires, that the sum of allthe momenta divided by the whole time should be a maxi- mum. In order to determine this we must assume a certain law of resistance, to which, applying the preceding principle of solution, a general result will be obtained. But as this assumed law of resistance must be totally sup- posititious, and probably very different from that which arises in practice, the preceding determination (supposing the value of h to have been found by experiment towards the conclusion of the operation of the engine) is probably more accurate than that which would result from the above supposition of a variable law of resistance. For it must be observed that 2, as found above, is always the same function ef h; that hincreases as the blows are repeated, and con- sequently x also increases each blow ; but the distance between the top of the pile and the top of the engine also practically increases each stroke, by the depth the pile has pene- trated. or > % PER found as above for any single stroke, is ver nearly, if not accurately, that which ansy for the whole sum. , ¥ Let us now return to our original subjee and endeavour to trace the causes of the ¢ ference between the effects of impact and ¢ of simple pressure. course of its motion strikes another body als perfectly hard, the variation in motion ough) to be produced in an indivisible instant, an locity, and the velocity after the shock, the shall not be any intermediate velocity. Br if the motion of the body were modified by | pressure, or by a constant force, such as tha of gravity, then it would change by sensib degrecs, and have undergone a determing variation at the end of'a certain time. It is therefore the law of continuity whie distinguishes the effects of pression from tho of percussion, when the hardness is infinit but as such hardness no where exists, si matter always possesses a certain degree ¢ elasticity, and a limited cohesion of particle which may be surmounted, we may inquire percussion, considered physically, conform to the law of continuity. Now when a bod strikes another, two effects have place in ea First, The parts in contact yield to the ac ib of the stroke and become compressed, so thé the figure of the bodies is altered by a flatter ing or impression, which obtains in the pa in contact, and in the neighbourhood of thos parts. Secondly, When the flattening or i pression has arrived at the greatest degree ¢ which the bodies are susceptible, then t | inherent elasticity tends to destroy the ir pression, and effaces it either wholly ori part; this produces a mutual action and‘ action of the bodies, which is continued they are no longer in contact. Thus, as soo as the bodies come into sensible contact, com pression must begin; for we may suppose th bodies to be two balls, which will therefor touch only in one point. The mutual pr sure which is necessary in order to produ the retardation of A and the acceleration ¢ B, is exerted only on the foremost particle ¢ A and the hindmost particle of B; but n atom of matier can be put in motion, or ha¥ its motion any way changed, unless it be act ed on by an adequate force. The force urging any individual parti must be precisely competent to the producti of the very change of motion which obtains that particle. Except the two particles wh come into contact in the collision, all the othe particles are immediately actuated by th forces which connect them with each other and the foree acting on any one is genera compounded of many forces which conne that particle with those adjoining. Therefor when A overtakes B, the foremost particle® A is immediately retarded ; the particles be hind it would move forward, if their mutua connection were dissolved in that instant ; bu PER this remaining, they only approach nearer to the foremost striking particles, and thus make a compression, which gives occasion for the inherent elasticity to exert itself, and by its re-action retard the following particles. Thus each stratum (so to conceive it), continuing ‘in motion, makes a compression, which occa- sions the elasticity to re-act, and by re-acting to retard the stratum immediately behind it. ‘This happens in succession: the compression and elastic re-action begin in the anterior ‘stratum, and take place in succession back- _ward, and the whole body gets into a state of ,compression. All this is done in an instant (as we commonly, but inaccurately, speak); that is, in a very small and insensible moment of time; butin this moment there is the same gradual compression, increase of mutual ac- ‘tion, greatest compression, common velocity, subsequent restitution, and final separation ; as in the case of bodies with a slender spring interposed, or even in the case of mutual re- pelling magnets. In all the cases the changes of motion are produced by the elasticity, or the repulsion, and not by the transfusion of the force of motion. The changing force is, in- deed, inherent to the bodies, but not because ithey are in motion; the use of the motion is fo give occasion, by continued compression, for the continued operation of the inherent elasticity. | Hence it appears, that the law of continuity \has actually place in the impact of bodies, and that no alteration in their motion takes place without their previously partaking of all the in- termediate alterations. Itis true this alteration Salways produced during an extremely short hiterval of time, and this occasions the great lisproportion which is observed between the fects of impact and those of pressure; but it § notwithstanding inconceivable, as Mr. At- wood remarks, “that any really existing body ihould pass from quiescence into finite motion, w from one degree of finite motion to an- ther, without having possessed all the inter- nediate degrees of velocity;’ and hence it ollows, that the phenomena of collision may ve considered of a kindred class to those which we occasioned by accelerating or retarding orces, and act by insensible degrees, in order 0 produce a finite effect. _ For farther information on the theory of vercussion, see the work of George Juan, a jpanish author, entitled ‘Examen Maritimo;” *rony’s ‘* Mech. Hydraulique ;” Gregory’s _Mechanics;” and the article IMPULSION, Sup- lement to the Encyclop. Britan.; see also ur article CoLLIsion. | Centre of Percussion. See CenTRE. , PERFECT Number, is one that is equal to he sum of all its divisors, or aliquot parts uclid, book vii. def. 22); thus 6 6 6 Sa ae 28 28 28 28 28 Ma ta tiget fa og vhich are therefore perfect numbers. PER To find a Perfect Number. Find 2"— 1 a prime number, then will 2"-1 & (2" — 1) be a perfect number. For 2" —1—=—1 +242? 4 23 + 2+, &e. QW! and it is demonstrated by Euclid, in the last prop, of book ix. that if the above series be continued till its sum be a prime, then that sum multiplied by the last term of the series will be a perfect number; which, therefore, is identically the same as the above formula, and the truth of the theorem, under its analytical form, may be demonstrated as follows: Since 2” — 1 is. a prime, put it = 6, and 2*-' = 2” = a”; then the above reduces to a”b. Now Ry, at+ri_. ] the sum of the divisors of a™6 — —_—__- x —l biti _ ] b—1 the number itself is considered as a divisur, which is excluded in perfect number; there- fore the sum of the divisors, exclusive of the number itself, is . See Drvisor. But in this formula, amtt__qy #l4+1—] Waa ob ee and it only remains to be demonstrated, that this is equal to a”6; or that a™tl__] §*— 1 a—l 6—1 and this will be evident, by re-establishing again the value of a and 4, as above; 2-1) eV) 2—1 ~ Qn—bD—1 (22 — 1 + 1) = 2" (2 — 1) and 2a”b—=2,2"—!(2"— 1) = 2” (2*— 1), which are identical. And consequently every number of the form 2”~1 (2n —1), the latter factor being a prime, is a perfect number, the sum of fts divisors being equal to the number itself. If nm— 2then2 (2* —1)=6 m= 3......27 (2% —1)=28 pac Do li then —(2" = 3) n= 5...... 2+ (2° —1)—496 MEST Ut 2° (27 —1)=-8128 TN ct 277(2?? — 1385503836 ENT cnt 216217 — 1) 8589869056 MIS i sdh 238(279 — 1) — 137438691328 pam 5 ae 23°(23' — 1) 2305843008139952128 which are all perfect numbers. The difficulty, therefore, of finding perfect numbers, arises from that of finding prime numbers of the form 2n — 1; which is very laborious. Euler ascertained, that 23! -—1— 2147483647 isa prime number; and this is the greatest at present known to be such, and consequently the last of the above perfect numbers, which depends upon this, is the greatest perfect number known at present, and probably the greatest that ever will be discovered; for as they are merely curious, without being useful], it is not likely that any person will attempt to find one beyond it. PERIGEE, or Peric£um (from sreps, about, and yn, earth), in the Ancient Astronomy, sig- nified the nearest approach of the sun, or any of the planets, to the earth; or rather that point of their orbit when at their least dis- LL PER - tance: a term which the moderns have chang- ed to perihelion, because it is the earth that is in motion and not the sun, as was supposed by the ancients. ‘The term perigee is, how- ever, still properly used, as applied to the moon, comets, &c. to denote their nearest ap- proach to our planet. See PERIHELION. PERIHELION, or PerRineELiIuM (from rep: and Asos, the sun), that point in the orbit of a planet or comet which is nearest to the sun; being the extremity of their transverse axis nearest to that focus in which the sunis placed; being thus opposed to the aphelion, which is the opposite extremity of the same axis. The perihelion distances of the several pla- nets, the mean distance of the earth from the sun being taken as unity, are as follows : Perihelion Distances of the Planets, the Mean Distance of the Earth being Unity. ELCUL Scans astassest cspenniee "1815831 VONINT Et se stencs ut aeceeh no's Wee *7164793 Cg ks a Fiat ML i gh ain : °9831468 Mars....... Pasty Qe aR ee .. 1°4305595 Vesthitiio Docs gin ahnenaee cays 2°2797800 ACT oP Mp APRESS tlle athe. path pipe - 2°4122190 LB yt See ee ae a 2°'6890660 Pallas. .cvecicthiersics stoves 2°5222080 PUD liat ee nc ctasreh ects ieee 5°1546127 SUCH cise. ccsare Cab vesepeun'y » 9°4826022 MIPS). fet. vasendt cece tA. 19°1366347 PERIMETER (from eps, about, and yerpov, measure), in Geometry, is the ambit or outward boundary of any figure; being the sum of all the sides in right-lined figures, and means the same as circumference, or periphery, in cir- cular ones. PERIOD, in Astronomy, is the time in which a planet or satellite makes one entire revolu- tion in its orbit, or return again to the same point in the heavens. It is one of the celebrated laws of Kepler, discovered by observation, and confirmed by Newton in his “ Principia,” that the squares of the periodic times of revolution are to each other as the cubes of the distance of the re- spective planets from the sun. And the same law has also place with ali the satellites revoly- ing about the same primary. For the periodic times of revolution of the several planets in our system, see ELEMENTS of the Planets. PERIOD, in Chronology, denotes an epoch, or space of time, by which the years are rec- koned. See Epocu. PERIOD, is a word also frequently used in arithmetic; as in the extraction of the square and other roots, the circulation of decimals, &e. PERIODICAL,returning at stated periods. PERIOECIANS, in Geography, are those people which inhabit the same parallel of la- litude, © PERIPATETIC Philosophy, the system of philosophy taught and established by Aristotle ; the word comes from reparemw, L walk, be- cause they always disputed while walking in the Lyceum. PERIPHERY (from wep, about, and Qepw, I bear or carry), the same as perimeter or cir- PER : cumference: perimeter, however, is more eom= monly used in speaking of right-lined figures and periphery, of curvilinear ones. 4 PERISCIL, a term formerly applied to the inhabitants of the two frigid zones, because- their shadows at times make an entire cin cumference in one day. PERITROCHIUM. See Axis in Peritrol chio. | PERMUTATIONS, the changes in the. position of things; differing from combinations” in this, that the latter has no reference to the order in which the quantities are combined; whereas in the former, this order is consider- ed, and consequently the number of permus) tations always eXeeeds the number of coms, binations. Permutation also differs from what. is simple, termed changes in this; that by changes, is commonly meant only the diffe-, rent order in which a number of things may, be arranged, taking all together; whereas per-. mutations implies a combination of a number of things into different sets, and the changes, which may then have place amongst them. In general the number of permutations may, be found by first finding the number of com= binations, and then the number of permutas' tions in each; and the product of these is the whole number of permutations. See CoMBI-, NATIONS. | PERPENDICULAR, in Geometry, is form= ed by one line meeting another so as to make, the angles on each side of it equal to each other. 4 PERPENDICULAR to a Curve, is a line per=| pendicular to the tangent of the curve at that, point. rt PERPENDICULAR Action of Gravity, is the, direction which it gives to a body falling freely, which is always in a line perpendicular to a tangent to the earth’s surface at that point, and not necessarily in a line directed to the centre of the earth. 4 a Lo erect a Perpendicular upon a given Point) C, in a given Line AB. is Fig. 1. WL B When the point is near the middle of the line. On each side of the point,C take any two equal distances Cn, Cm. From n and m, with any radius greater than cn or cm, describe ares cutting each other in s.. Through t e point s draw the line sc, and it will be th | perpendicular required. et When the point is at or near the end of the’ line. Take any point 0, and with the radi s or distance oC describe the arc mCn, cutting ab in m and C, Through the centre o, and the point m, draw the line mon, cutting the are mCninn. From the point 2 draw the line nC, and it will be the perpendicular required. From a given Point C, to let fall a Perpendicu- lar upon a given Line AB. Fie. 3. Fig. 4. i" H ? f Wen pe SOE VEE OOS Sere] cee aa ee Ph 7. - £ ee oe ee ee | lh ee ee .. “Saaceuws™ When the point is nearly opposite the mid- die of the given line. From the point C, with any radius, describe the are nm cutting A ‘Bin x» and m. From the points n,m, with the same, or any other radius, describe two ares cutting each other in S. Through the points CS draw the line CGS, and CG will be the perpendicular required. When the point is nearly opposite to the rend of the line. To any point m, in the line AB, draw the line Cm. Bisect the line Cm, or divide it into two equal parts, in the point x. ‘From x, with the radius nm, or nC, describe ‘the arc CGm, cutting AB in G. Through ‘the point C draw the line CG, and it will be ithe perpendicular required. PERPETUAL Motion, is that which pos- Sesses within itself the principle of motion ; d consequently since every body in nature, when in motion, would continue in that state, every motion once begun would be perpetual such are those of friction, resistance, &c.: and since it is also. a known principle in mecha- nics, that no absolute power can be gained by any combination of machinery, except’ there being at the same time an equal gain in an pposite direction; but that, on the contrary, there must necessarily be some lost from the wove causes, it follows that a perpetual mo- jion can never take place from any pure me- hanical combination: yet this is a problem which has engaged the attention of many in- senious men, from the earliest period to the resent time, though it has but seldom been ittempted by men of science, since the true PERPETUAL Screw. See Screw. PERPETUITY, in the Doctrine of Annui- ies, is the number of years in which the sim- le interest of any principal sum will amount o the same as the principal itself; or it is the uantity arising by dividing 100, or any other rincipal, by its interest for one year. ‘Thus ae perpetuity, at the rate of 5 per cent, is $° = 20 years; and at 4 per cent. ere 25 ears. PERSEUS, a northern constellation. See ‘ONSTELLATION. PERSPECTIVE, the art of delineating isible objects on a plane surface, such as they ppear at a given distance or height upon a ‘amsparent plane, supposed to be placed com- tonly perpendicular to the horizon between - ut for the operation of some external causes; eR ihe eye and the object. Perspective is divided into Aerial and Lineal, the former having principally a reference to the colouring aud shading of distant objects, and the latter as above defined, which relates to the position, magnitude, form, &c. of the several lines or contours of objects, &e. This part, in fact, is the only branch of perspective that properly falls within the design of the present work, and to which we shall confine our observations. This branch of mathematics, which some class under the term optics, doubtless owes its origin to painting, and was known and practised by the ancient Greeks, having been taught by Democritus and Anaxagoras, and is treated of by Euclid in his “ Optics,” and is again mentioned by Tzetzes in the twelfth century, who was well acquainted with its inj- portance in painting; and it is obvious also. from the works of Alhazen, who lived about the beginning of the twelfth century, that the Arabians were at that time acquainted with this art. But the first authors who wrote professedly on perspective, were Bartolome Bramantino, of Milan, whose book, “ Regole di Perspec- tiva, e Misnre delle Antichita di Lombardia,” is dated 1440; and Pietro del Borgo, likewise an Italian, who was the most ancient author met with by Ignatius Danti, and who it is supposed died in 1443. ‘This last writer sup- posed objects placed beyond a transparent tablet, and so to trace the images which rays of light, emitted from them, would make upon it. And Albert Durer constructed a machine upon the principles of Borgo, by which he could trace the perspective appearance of objects. Leon Battista Alberti, in 1450, wrote his treatise “De Pictura,” in which he treats chiefly of perspective. , Balthazar Peruzzi, of Siena, who died in 1506, had diligently studied the writings of Borgo; and his method of perspective was published by Sertio in 1540. To him it is said we owe the discovery of points of distances, to which are drawn all lines that’ make an angle of 45° with the ground line. Guido Ubaldi, another Italian, soon after discovered that all lines are parallel to one another, if they be inclined to the ground line, converge to some point in the horizontal line; and that through this point also will pass a line drawn from the eye paralléi to them. His “Perspective” was printed at Pisaro in 1600, and contained the first principles of the me- thod afterwards discovered by Dr. Brook Taylor. In 1583, was published the work of Giacomo Barozzi of Vignola, entitled “The Two Rules of Perspective,” with a learned cominentary by Ignatius Danti. In 1615, Marolois’s work was printed at the Hague, and engraved and published by Hondius. And in 1625, Sirigatti published his *‘ Treatise of Perspective,” which is little more than an abstract of Vignola’s. Since that time the art of perspective has been gradually improved by subsequent geo- LL, 2: PER metricians, particularly by Professor Grave- sande, and still more by Dr. Brook Taylor, whose principles are in a great measure new, and far more general than those of any of his predecessors. He did not confine his rules, as they had done, to the horizontal plane only, but made them general, so as to affect every species of lines and planes, whether they were parallel to the horizon or not; and thus his principles were made universal. And from the simplicity of his rules, the tedious progress of drawing out plans and elevations from any objects is rendered useless, and therefore avoided; for by this method not only the fewest lines imaginable are required to produce any perspective representation, but every figure thus drawn will bear the nicest mathematical examination. Farther, his sys- tem is the only one calculated for answering every purpose of those who are practitioners in the art of design; for by it they may pro- duce either the whole, or only so much of an object as is wanted; and by fixing it in its pro- per place, its apparent magnitude may be de- termined in an instant. It explains also the perspective of shadows, the reflection of ob- jects from polished planes, and the inverse practice of perspective. His “ Linear Perspective” was first pub- lished in 1715, and his ‘‘ New Principles of Linear Perspective” in 1719, which he in- tended as an explanation of his first treatise. And his methods has been chiefly followed by all others since that tinte. In 1738, Mr. Hamilton published his “ Ste- reography,” in 2 vols. folio, after the manner of Dr. Taylor. But the neatest system of per- spective, both as to theory and practice, on the same principles, is that of Mr. Kirby. There are also good treatises on the sub- ject by De Bosse, Albertus, Lamy, Niceron, Pozzo the Jesuit, Ware, Cowley, Priestley, Ferguson, Emerson, Malton, Henry Clarke; and a neat elementary treatise has been lately published by Mr. Creswell, of Trinity College, Cambridge. General Definitions and Principles of Per- spective. An original object, is that object which is to be represented, or which forms the subject of the picture. , The perspective plane, is a plane on which the objects are required to be delineated, and is commonly supposed to be situated between the object and the eye. _ The point of view, is that point from which the objects are seen, when they are delineated upon the perspective plane. The point in which the perspective plane is met by a straight line drawn perpendicular to it from the point of view, is called the centre of the picture, or the principal point. The distance between the point of view, and the principal point, is called the distance of the picture, or the principal distance. The plane which contain any original point or line, viz. any point or line, the representa- PHA — } tion of which is required, is called an original plane. 4 The base line of any original plane, is the common intersection of that plane, and the perspective plane. . Vhe point in which the perspective plane is” met bya straight line drawn through the point of view parallel to any original straight line, is" called the vanishing point of that original line, The intersection of the perspective plane, and a plane passing through the point of view parallel to any original plane, is called the vanishing line of that original plane. The point in which any vanishing line is met by a straight line drawn perpendicular to it from the point of view, is called, the centre of that vanishing line. The distance between the point of view and the centre of any vanishing line, is called the direct distance of that vanishing line. . %e, - ~ 5, Dg | These definitions will be better understood by a reference to the above figure; where O is the point of view; PL the perspective: plane; X Y an original plane, and Md its: parallel; C the centre of the picture; OC the principal distance; HL the base line of XY; Dd its vanishing line; K the centre: of Dd, and OK its direct distance. Cress-) well’s ‘‘ Elements of Linear Perspective.” PETIT (PreTeER), a respectable French mathematician, was born at Montlucon about the year 1600. He was author of several works, | viz. on Chronology; Weights and Magni- tudes of Metals; on a Vacuum; Eclipses; Comets; Heat and Cold, &c. He died in 1667. PETTY (Str WILLIAM), an eminent Eng- lish philosopher, was born in Hampshire, im 1623; to whose character, genius, and. per- severance we owe the first commencement of the Royal Society. Sir William Petty was author of numerous papers on a variety of subjects, and amongst others, a work on “* Po- litical Arithmetic.” A complete list of all his pieces is given in Dr. Hutton’s Dictionary, to which the reader is referred. Petty died in 1687, leaving at his death a very consider able property, accumulated in the discharge of many important public situations. PHANTASMAGORIA (from Qaylacpa, phantasm, and yweiew, L mock), denotes a re- markable optical illusion arising from a par ticular application of the magic lantern. Tn the exhibition of this spectacle, the audience are placed in a dark room, having a trans- parent screen between them and the lantern, which sereen ought to be let down after the lights are withdrawn, and unknown to the spectators, *s PHA - The lantern being then properly adjusted on the opposite side, the figure intended to be exhibited is thrown upon the screen, which will appear to the observers as if placed in free space, and by altering the distance of the lantern the figure may be made to appear of any size; which changes in its dimensions are attributed by the ob- servers to the distance or proximity of the image, so that at one time it appears to be at an immense distance, and at another to be exceedingly near, and over the heads of some part of the audience. Mr. Nicholson gives the following descrip- tion of the exhibition of this spectacle at the Lyceum. All the lights of the small theatre of exhi- bition were removed, except one hanging lamp, which could be drawn up so that its flame should be perfectly enveloped in a cy- lindrical chimney, or opaque shade. In this gloomy and wavering light the curtain was drawn up, and presented to the spectator a cave or place exhibiting skeletons, and other figures of terror, in relief, and painted on the sides or wall. After a short interval the lamp was drawn up, and the audience were in total darkness, succeeded by thunder and light- ning; which last appearance was formed by the magic lantern upon a thin cloth or screen, let down after the disappearance of the light, and consequently unknown to most of the ispectators. These appearances were followed by figures of departed men, ghost, skeletons, itransmutations, &c. produced on the screen ‘by the magic lantern on the other side, and ‘moving their eyes, mouth, &c. by the well- known contrivance of two or more sliders. The transformations are effected by moving ‘the adjusting tube of the lantern out of its focus, and changing the slider during the moment of confused appearance. [t must be again remarked that these figures appear without any surrounding circle of illu- Mination, and that the spectators, having no previous view or knowledge of the screen, nor ny visible object. of comparison, are each eft to imagine the distance according to their respective fancy. After a very short time of exhibiting the first figure, it was seen to con- tract gradually in all its dimensions, until it became extremely small and then vanished. ‘This effect, as may easily be imagined, is produced by bringing the lantern nearer and nearer the screen, taking care at the same time to preserve the distinctness, and at last closing the aperture altogether, and the pro- cess being (except as to brightness) exactly the same, as happens when visible objects become more remote, the mind is irresistibly led to consider the figures as if they were receded to an immense distance. Several figures of celebrated men were thus exhibited with more transformations; such as the head of Dr. Franklin being. con- verted into a skull, and these were succeeded by phantoms, skeletons, and various terrific figures, which, instead of seeming to recede and then vanish, were (by enlargement) made PHO suddenly to advance, to the Surprise and astonishment of the audicnee, and then dis- appear by seeming to sink into the ground. For another description of these interesting illusions, see notes to Gregory’s translation of Hauy’s “ Philosophy,” vol. xi. p. 390. PHASES (from Gass, ZF appear), in Astro- nomy, denote the various appearance of the moon at different ages, being at one time a crescent, then a semicircle, then gibbous, and lastly full; after which the same phases re- turn again in the same order. Venus and Mercury have the same phases as the moon, and Mars partakes of them, in some measure, being at times gibbous; the same must also have place in a less degree with the other superior planets. The same term is also ap- plied to denote the appearance of the moon or sun when eclipsed. PHENOMENON (from Qaivoues, £ appear), is strictly an appearance, but more commonly confined to those only of an extraordinary nature, particularly as relatiag to the heavens, or heavenly bodies ; as comets, meteors, shoot- ing stars, &c. We also speak of the pheno- mena of the magnet, of electricity, &c. PHILOLAUS, a celebrated Grecian phi- Josopher and astronomer, contemporary -with Plato, who is said by some to have been the author of the true solar system, as revised by Copernicus ; but this appears to be erroneous, as there seems undoubted proof of Pytha- goras having taught this system in his school, having derived his knowledge of it from the Egyptians. Philolaus flourished about 400 years before Christ. PHILOSOPHER, a person well versed in philosophy, or one who makes profession of, or applies himself to the study of those sci- ences. PHILOSOPHY, the knowledge or study of nature or morality, founded on reason and experience, the word originally implying a love of wisdom. The only part of philosophy, however, which belongs to a work of this kind, is that which is called natural or expe- rimental philosophy, and which may be gene- rally defined that branch of science which derives its data from experiments and obser- vations, on which the whole system is sup- ported, as is that of geometry upon axioms and definitions. See PHysics. PHCENIX, a southern constellation. CONSTELLATION. PHONICS; the same as AcoustTIcs. PHOSPHORUS (from ws, light, and Pzew, T bear),.a substance which shines by its own light. The discovery of this singular sub- stance was accidently made in 1677, by an alchymist of Hamburgh, named Brandt, when he was engaged in searching for the philo- sopher’s stone. Kunkel, another chemist, who had seen the new product, associated himself with one of his friends, named Krafft, to purchase the secret of its preparation; but the latter de- ceiving his friends, made the purchase for himself, and refused to communicate it. Kun- kel, who at this time knew nothing further see PHO of its preparation than that it was obtained by certain processes from urine, undertook the task, and succeeded. It is on this ac- count that the substance long went under the name of Kunkel’s phosphorus. Mr. Boyle is also considered as one of the discoverers of phosphorus. He communicated the secret of the process for preparing it to the Royal Society of London in 1680. It is asserted, indeed, by Krafft, that he discovered the secret to Mr. Boyle, having in the year 1678 carried a small piece of it to London to show it to the Royal Family; but there is little probability that a man of such integrity as Mr. Boyle would claim the discovery of the process as his own, and communicate it to the Royal Society, if this had been the case. Mr. Boyle communicated the process to Godfrey Hankwitz, an apothecary of Lon- don, who for many years supplied Europe with phosphorus, and hence it went under the name of English phosphorus. In the year 1774, the Swedish chemists, Gahn and Scheele, made the important discovery, that phos- phorus is contained in the bones of animals, and they improved the processes for procur- ing it. The most convenient process for obtaining phosphorus, seems to be that. recommended by Fourcroy and Vanguelin, which we shall » transcribe. Take a quantity of burnt bones, and reduce them to powder. Put 100 parts of this powder into a porcelain or stone ware bason, and dilute it with four times its weight of water. Forty parts of sulphuric acid are then to be added in small portions, taking care to stir the mixture after the addition of every portion. A violent effervescence takes place, and a great quantity of air is dis- engaged. Let the mixture remain for twenty- four hours, stirring it occasionally to expose every part of the powder to the action of the acid. ‘The burnt bones consists of the phos- phoric acid and lime; but the sulphuric acid has a greater affinity for the lime than the phosphoric acid. ‘The action of the sulphuric acid uniting with the lime, and the separation of the phosphoric acid, occasion the effer- vescence. The sulphuric acid and the lime combine together, being insoluble, and fall to the bettom.. Pour the whole mixture on a cloth filter, .so that the liquid, part, which is to be received in a porcelain vessel, may pass through. A white powder, which is the inso- luble sulphats of lime, remains on the filter. After this has been repeatedly washed with water, it may be thrown away; but the water is to be added to that part of the liquid which passed through the filter. Take a solution of sugar of lead in water, and pour it gradually into thé liquid in the porcelain bason. A white powder falls to the bottom, and the sugar of lead must be added so long as any pre- cipitation takes place. The whole is again to be poured upon a filter, and the white powder which remains is to be well washed and dried. The dried powder is then to be mixed with one-sixth of its weight of charcoal powder. Put this mixture into an earthen ware retort, PHO and place itin a sand bath, with the beak plung- ed into a vessel of water. Apply heat, and let — it be gradually increased till the retort be- comes red-hot. As the heat increases, air- bubbles rush in abundance through the bea of the retort, some of which are inflamed whet they come in contact with the air at the sur-_ face of the water. A substance at last drops out similar to melted wax, which congeals under the water. This is phosphorus. 'To” have it quite pure, melt it in warm water, — and strain it several times through a piece of shamoy leather, under the surface of the water. To mould it into sticks, take a glass" funnel with a long tube, which must be stop-_ ped with a cork. Fill it with water, and put — the phosphorus into it. Immerse the funnel in boiling water, and when the phosphorus is _ melted and flows into the tube of the funnel, — then plunge it into cold water, and when the phosphorus has become solid remove the cork, and push the phosphorus from the mould | with a piece of wood. Thus prepared, it must be preserved in close vessels containing pure water. When phosphorus is perfectly pure it is semi-transparent, and has the consistence — of wax. Itis so soft that it may be cut with a knife. Its specific gravity is from 1.77 to 2.03. it has an acid and disagreeable taste, and a peculiar smell, somewhat resembling garlic. When a stick of phosphorus is broken it” exhibits some appearance of crystallization. The crystals are needle-shaped, or long oetas_ hedrons; but to obtain them in their most perfect state, the surface of the phosphorus, | just when it becomes solid, should be pierced, that the internal liquid phosphorus may flow _ out, and leave a cavity for their formation, — When the phosphorus is exposed to the light it becomes of a reddish colour, which appears — to be an incipient combustion. It is there-_ fore necessary to preserve it in a dark place. At the temperature of 99° it becomes liquid, — and if air be entirely excluded, it evaporates at 219°, and boils at 554°. At the tempe-_ rature of 43° or 44° it gives out a white smoke, — and is luminous in the dark. This is a slow combustion of the phosphorus, which becomes — more rapid’ as the temperature is raised. When phosphorus is heated to the tempera-_ ture of 148° it takes fire, burns with a bright | flame, and sends out a great quantity of white smoke. Phosphorus enters into combination — with oxygen, azote, hydrogen, and carbon. Phosphorus is soluble in oils, and when thus dissolved forms what has been called liquid — phosphorus, which may be rubbed on the face — and hands without injury. It dissolves too — in ether, and a very beautiful experiment con-_ sists in pouring this phosphoric ether in smal a portions, and ina dark place, on the surface of hot water. The phosphoric matches con=_ sists of phosphorus extremely dry, minutely divided, and perhaps a little oxygenized. The simplest mode of making them, is to put a_ little phosphorus, dried by blotting paper, — into a small phial; heat the phial, and whet the phosphorus is melted turn it round, s that the phosphorus may adhere to the side “~ 2 al & Cork the phial closely, and it is prepared. On putting a common sulphur-match into a bottle, and stirring it about, the phosphorus will adhere to the match, and will take fire when brought out into the air. British Ency- clopedia. PHYSICAL, any thing relating to physics. PHYSICO-Mathematics, is the same as mixed mathematics, being those branches of this science which investigates the laws and ‘actions of bodies, and their combinations, by means of certain data drawn from observation and experiment. See Mixed MATHEMATICS. PHYSICS, is a term denoting the same as experimental or natural philosophy; being the doctrine of natural bodies, their pheno- mena, causes, and effects, with their various affections, motions, and operations. Experimental Puysics, is that which en- quires into the nature and reason of things by experiments, as in hydrostatics, pneu- matics, optics, chemistry, &c. Mechanical Puysics, explains the appear- ances of nature from the matter, motion, struc- ture, and figures of bodies, and their several parts, according to the established laws of nature. PLASTER, a Spanish coin of the value of 4s. 6d. sterling. PICARD (Joun), a celebrated French ma- thematician and astronomer of the 17th cen- tury. He was the first who applied the teles- cope to astronomical instruments, and com- _ menced the publication of the ‘ Connaissance des Tems,” which he calculated from 1679 to 1683. He also first measured the length of a degree of the meridian in France, and gave a map of that country. The time of his birth is not known, but he died in 1682 or 1683. Picard was author of several works, on Level- ling, Dialling, Dioptrics, Discharge and Men- suration of Fluids, Astronomy, &c. the whole of which are given in the 6th and 7th volumes of the Memoirs of the Academy of Sciences. PIERS, in the theory of Bridges, are the walls built to support the arches, and from which they spring as bases. PILES, in Building, are large stakes or beams sharpened at the end, ‘and shod with iron, to be driven into the ground for a foun- dation to build upon in marshy places. PILE-Engine, is an engine used for the purpose of driving piles.. Of the principles of ‘the operation of this engine, see PERCUSSION. PILE, in Artillery, denotes a collection or heap of balls or shells, piled up in a pyramidal form, the base being some regular figure, as an equilateral triangle, square, or rectangle, and. the whole pile a series of such figures, the side of each successive row diminishing by one from the bottom upwards. Therefore the whole number of balls is -equal to the sum ofa series of triangular. num- bers, squares, or rectangles, according to the figure of the pile: and may be expressed by the following formule, Triangular pile = n(n Be + 2) 6 PA _ n(n + 1) (Qn + 1) vg 6 m(m +1)(8n—m + 1) 6 Where n in the two first formule denotes the number of balls in the side of the base; and in the latter, x is the number of balls in the length of the base, and m the number of those in the breadth. These formule are difficult to remember upon an emergency ; and therefore the follow- ing general rule, for this purpose, which is not commonly known, is deserving the notice of artillery, officers. Rule. In every pile there may be found three parallel lines, the sum of which, multi- plied by the number of balls in the triangular face of the pile, and divided by 3, is the num- ber of balls. In the rectangular pile the three parallel lines, are the two bottom rows in length, and the upper ridge of the pile; and the face the triangular end. In the square pile any two opposite sides of the square base, and the upper ball, are three parallel lines. And in the triangle pile one side of the bottom row, the opposite ex- treme ball, and the upper ball, are the three parallel sides; the face in both these cases being any of the equal slant sides of the pile. I am indebted for this rule to Captain Aytoun, of the Royal Artillery. PINION, in Practical Mechanics, is any small wheel working in the teeth of a larger wheel. PINT, an English measure of capacity, being the 8th part of a gallon. PISCES, the Fishes, the last of the zodiacal constellations, denoted by the character x. See CONSTELLATION. PISCIS Australis, the Southern Fish. CONSTELLATION, Piscis Volans, the Flying Fish. See Con- STELLATION. PISTOLE, a Spanish gold coin. See DouBLoon. PISTON, in Mechanics, denotes a short cylinder working within another hollow cy- linder, as in water and air-pumps, and some other machines. See PuMP. PLACE, in Philosophy, that part of im- moveable space which any body possesses or occupies, having to space the same relation that time has to duration. ° PLAcE has various denominations, as adso- lute, relative, primary, &c. which will be rea- dily comprehended without any formal defi- nitions. ‘ Piace, in Geometry. See Locus. PLAIN. See PLANE. PLAN, a representation of something drawn on a plane, such as maps, charts, &c. PLANE, or PLAIN, in Geometry, denote a surface or superficial extension, lying evenly between its bounding lines; being such, that if aright line touch it in two points, it will touch through its whole extent. Square pile Rectangular pile = See PLA The same term is also frequently used in astronomy for an ideal plane passing through certain parts or points of the heaven, as the plane of the horizon, of the ecliptic, equator, &ec. by which is to be understood certain ideal planes passing through those circles of the sphere, or on which they are supposed to be described. And on similar principles we say a vertical plane, horizontal plane, to denote planes passing in those directions. PLANE, in Perspective. See PERSPECTIVE. Inclined Puane. See Inciinep Plane. PLANE Angle, Chart, Mirror, Triangle, Sailing, Scale, Trigonometry, &c. See the re- spective substantives. PxLaneE Problem. See Locus and PROBLEM. PLANE Table, in Surveying, a very simple instrnment, whereby the draught of a field is taken on the spot, without any future pro- traction. It is generally of an oblong rectan- gular figure, and supported by a fulcrum, so as to turn every way by means of a ball and socket, It has a moveable frame, which serves to hold fast a clean paper; and the sides of the frame, facing the paper, are divided into equal parts every way. It has also a box with a magnetic needie, and a large index with two sights; and lastly, on the edge of the frame, are marked degrees and minutes. To use this instrument, take a sheet of paper which will cover it, and wet it to make it expand; then spread it flat on the table, press- ing down the frame on. the edges to stretch it, and keep it fixed there; and when the paper is become dry, it will by contracting again, stretch itself smooth and flat from any cramp and unevenness. On this paper is to be drawn the plan or form of the thing mea- sured. Thus, begin at any proper part of the ground, and make a point on a convenient part of the paper or table, to represent that place on the ground ; then fix in that point one leg of the compasses, or ofa fine steel pin, and apply to it the graduated edge of the index, moving it round till, through the sights, you perceive some remarkable object, as the corner of a field, &c.; and from the station point draw a line with the point of the compasses along the graduated edge of the index, which is called setting or taking the object; then set another object or corner, and draw its line; do the same by another, and so on, till as many objects are taken as may be thonght fit.. Then measure from the station towards as many of the objects as may be necessary, but not more, taking the requisite offsets to corners or crooks in the hedges, laying the measures down on their respective lines on the table. Then, at any convenient place measured to, fix the table in the same position, and set the objects which appear from that place, and so on as before. And thus continue till the work is finished, measuring such lines only as are necessary, and determining as many as may be, by intersecting lines of direction, drawn from different stations. PLANET (from araynrne, wanderer), 2 PLA wandering star, as distinguished from the — fixed stars, which always preserve the same . relative position with respect to each other. — Hence it follows that comets and satellites . are included, according to the original — signification of this term, under the same general denomination; in fact the early as- tronomers had no idea of comets being per- manent bodies, and as they were also unac-_ guainted with any satellite but the moon, which, with the sun, was supposed to revolve _ about the earth, it was natural for them -to— class both under the same general appellation. But modern astronomers, in order to make_ a distinction between these, define a planet. to be acelestial body revolving about the sun, — as a centre with-a moderate degree of eccen- tricity; thus, excluding comets, the eccen- tricity of whose orbits is very considerable ; and the satellites which revolve about their — primaries as the primaries do about the sun. — These last are, however, sometimes called secondary planets. | The planets belonging to our system, as included under this definition, are therefore Mercury, Venus, the Earth, Mars, Juno, Pallas, Ceres, Vester, Jupiter, Saturn, and Uranus, or the Georgium Sidus. Of these, Mercury, Venus, Mars, Jupiter, - and Saturn, have been known from the high- | est antiquity; but the other five have been discovered within a very few years; viz. Ura- | nus, by Dr. Herschel, March 13, 1781; Ceres, by M. Piazzi, January 1, 1801; Pallas, by Dr. Olbers, March 28, 1802; Juno, by Mr. Harding, September 1, 1804; and lastly, Vester, by Dr. Olbers, March 29, 1807, being the second planet discovered by this cele- — brated observer. Astronomers use certain characters to de- signate the different planets, which are as— follows: viz. Mercury 3% ; Venus 9 ; Earth — @; Mars 3; Jupiter 2¢ ; Saturnh ; CeresC€ ; Pallas 9; and Uranus Hl; but the other | new planets have not at present, we believe, received any such distinguishing character. We have given under the several articles EccEnTrRIcITY, DisTancE, Orbit, PERIOD, &c. the several particulars included under those denominations, as also under the names of the several planets, the respective elements of each; and we have, therefore, in this place, only to state a few popular observations rela- tive to the probable nature, motion, appear- ances, &c. of these celestial bodies; and to | give a general view of the elements and other - particulars of the planetary system. | First then it may be observed that all the — planets perform their revolutions in elliptical orbits about the sun, which is situated in one of the foci of the ellipse; that the orbit of each planet lies in a plane, which passes through the centre of the sun;- that those which are nearest the centre move with a greater velocity than those that are more remote; that the same planet is also quicker — or slower according as it is in that part of its orbit which is nearer or more remote from | PLA he central body; and that all their motions re performed in the same order from west to ast, or according to the order of the signs. And farther, the motions and distances of ie several planets are related to each other y certain and invariable laws, viz. that the ubes of their periodic times of revolution, re as the squares of their distances from the un; and that equal areas are described by 1€ same planet in equal times; that is, if we appose a line drawn from the sun to a planet, ad to move about it as a centre, that line ‘ill pass over or describe equal areas in equal mes; and therefore since the periods of the om observation, we may hence find the xact proportional distance of all' these bodies. ee KepLer’s Laws. It has also been discovered by means of artain spots, observable on the discs of the lanets, that they have each a rotatory motion s0ut their own axis, the time of which, how- yer, seems to follow no particular law either ith regard to their magnitude or distance om the sun. Hence then it appears that these bodies we each a diversity of seasons, their spring, mmer, autumn, and winter, resembling ‘ose in our planets; that they have likewise same alternations of day and night; in ort, that they are equally fitted to the ac- mmodation of inhabitants, and that in all obability millions of beings are placed upon em, _“ With constitutions fitted to the spot Where Providence all-wise has fixed their lot.” “The excessive heat and cold experienced | those regions, in consequence of their prox- lity or remoteness from the sun, as have ‘en supposed and even computed by some thors, appear to be wholly imaginary. We ye no reason, for example, to suppose that yen Uranus, the most distant planet in our stem, has a temperature at all different ym us, or even from Mercury who revolves ach nearer to the sun than ourselves. — The light and heat received and experi- ced on our globe, seems, to depend much jore upon the constitution of the atmosphere jan upon any other cause; or why have we \Quntains covered with perpetual snow; and 1ence that diminution of light and heat ex- rienced by aérial voyagers; but in conse- ence of the extreme rarity of the air in the per regions? And if this be granted, it lows immediately that a simple modifica- m in the atmospheres of the several planets, puld render the temperature of each sup- jrtable even by terrestrial inhabitants. |The idea of the temperature of the several janets depending upon their distances from ‘e sun, arises from considering that body, tsimply as the cause of heat, but as an im- jense mass of fire possessing in itself, inde- /ndent of any other agent, the power of heat; ‘aereas there is every reason to conclude jat it is only by a combination of the solar A/S with certain parts of our atmosphere that | lanetary revolution are accurately known PUA the effect is produced. Water poured upon unslaked lime generates heat in the combined mass, and if we could imagine a being ex- isting in such a mvass, we should have no difficulty in conceiving, that he would attri- bute that quality to the water which we attri- bute to the sun, although in this case the contrary is evident. We cannot doubt then without presump- tuously limiting the power and wisdom of the Deity, that each of these planets are peopled with millions of beings engaged like us in the anxious vicissitude of life, and probably, each having its philosophers and astronomers con- templating this immense globe as a mere speck in the starry firmament. Nor must we stop here; it is also highly pro- bable that every fixed star is another centre, about which planets are revolving, as those of our system do about the sun. Instead therefore of one sun and one world, as the ignorant imagine, reason and contemplation poiat out to us millions of suns and millions of worlds, each peopled by myriads of inha- bitants dispersed through infinite space, to which our system appears but as a mere point or atom, and is almost lost in the im- mensity of the creation. To adopt therefore the words ofa celebrated author: “ What an august, what an amazing conception, if human imagination can con- ceive it, does this give of the works of the Creator! Thousands of thousands of suns, multiplied without end, and ranged all around us, at immense distances from each other, at- tended by ten thousand times ten thousand worlds, all in rapid motion, yet calm, regular, and harmonious, invariably keeping the paths prescribed them; and these worlds peopled with myriads of intelligent beings, formed for an endless progression, in perfection and feli- city. ‘“‘ If so much power, goodness, and magni- ficence be displayed in the material creation, which is the least considerable part of the universe ; how great, wise, and good must he be, who made and governs the whole!” Elements of the PLANETS, are certain quan- lities which are necessary to be known in order to determine the theory of their elliptic motion. ‘ » Astronomers reckon seven of those quanti- ties, of which five relate to the elliptic motion, viz. 1. The duration of the sidereal revolution. 2. The mean distance, or semi-axis major. 3. The eccentricity, from which is derived the greatest equation of the centre, 4. The mean longitude of the planet at any given epoch. 5. The longitude of the perihelion at the same epoch. The two other elements relate to the posi- tion of the orbit, and are, 1. The longitude at a given epoch of the nodes of the orbit with the ecliptic. 2. The inclination of the orbit to this plane. These several elements, as given by Laplace, “ Systeme du Monde,” 3d edition, are contained in the following table, which presents also a general view of the planetary system. PLA 3 , PLA Elements and general View of the Planetary System. ° * Ratio of the 3 Names of the | Duration of a Sidereal} Mean Distance hall the. May to}. Mean Longi- i Planets, Revolution. from the Sun. {half the Major|tude,Jan.1,1801. Inclination ; the Orbit t the nl Mean Longi-| Longitude of| tude of the |the ascending Perihelion. Node, Days. 8796925804 | 3870981 }-2 ny Seas 22470082399 | °7233323 |): 365°25638350 | 1:0000000 |: 686°97961860 | 1°5236935 | Na 1335°20500000 | 2°3730000 | jos huted 1590°99800000 | 2°6671630 |°254 1681°53900000 | 2°7674060 |-07834860 |294° 16820}162°9565! 89°9083'11°8068 | 1681-70900000 | 2°7675920 |24538400 |280°68580]134° 7040]191°7148 38° 465 1TEK seks 4332'59630760 | 5°2027911 |:04817840 |124°67781| 12°3812|109°3624' 194603 Wh Fads 10758°96984000 | 9°5387705 |-05616830 |150°38010] 99°0549 124°3662 2°7710 a 30688°7 1268720 |19°1833050 |-04667030 |197°54244] 185° 9574) 8029488) 0°859 Naites of the. [Mean Diam. Mean Distance from| Mean apparent |Mean Diam.| Densities, | Proportiona i Diarnal Rotate Piinets, fining. mies) * Suny millions | Diameter as seen [as seen from [chat of WaterlQuantties of ha oa Reap The Suns 1898246. |oc.ececcosceeee| BYU"S deccccocce 12, | 333928 los 14 Mercury....) 3224 37,000000 10 16” 92 0°1654 114 24 Venus. ..... 7687 | 68,000000 58 30 51x | 0-8899 | 0 23 2 The Earth | 7911 95,000000 | ss... Geryeaebeeg 172 | 4138 |] nee The Moon| 2180 | 95,000000| 31’ 8 46 | 3% | 0025 a9 17 4 Mars........- 4189 | 144,000000 27 10° 32 | 00875 | 0 243 163 1 3 Ceres. a4 t 263,000000 Se ' 80 “ Pallas........ Saab . 265,000000 he . HR 28 JUNO v.essvess 1425 | 252,000000 3 ee Vesta... 238. | 225,000000 OS: 3 dun Lad AOR Gee ee Jupiter...... 89170 490,000000 39 37 1g Saturn....... 79042 | 900,000000 18 16 ire Jranus w..... 1,800,000000 1 Names of the pha on of| Taeiination of Opbik Place of Aphetion Motion of {Motion of the|Eccentricities, de Aph Greatest Equationp Planets. Orbits. to the Ecliptic. in Jan. iG yearn war Hg Laue eon, of the Centre, J The ‘Sin... [p27 se Or. ctsncvnssecdo teretrres sts elk Norell a a Mercury....|-.s00-«1 72 0! 0”| 8§ 14°20’ 50”(1°33! 45”|19 12! 10"| 79554 | 23°40" O" ! POMS. 11. escack 3 23 25 110 7 59 1 |i 21 010 51 40 | 498 47 20 {| The Earth |66 32} 0 0 9 8 40 12 10 19 35 |......6..8. 1681°395| 1 55 309] The Moon ji88 17 ee Ris Cotte oe as, | ME Et cm 0 3 | Mars........,59. 22] 151 9 | 5 224 4/1 58 40 lo 46 40 14183-7 10 40 400 0 40 TSO emer ee nese seresl secseeereerasesacer! seeese tensor! seccccee Canes ate. tes 95552-10110 370! OSGI Tne areas ee 8141° 920 8 Pallas....... |-- Bb a Sites 34 50 IO De. Oa a 2 ee 28 25 07 1 PTR ceLUOE Adds Saviden sees 21 0 7, 89 495994 07. OS. Soma 25096 wa Vestal ac... :lies slices ot 8 4092.9 242 58 |... WAAL gee nt ee ae Jupiter...... 90 nearly 1 856 |611 8 20 |1 34 33 |0 59 30 | 250133 | 5 30 Saturn,...... 60 probably} =2 29 50 | 8 29 411 |1 50 7 (IO 55 30 | 53640°42| 6 26 Uranus......|...ecceseees-| 0 4 129 2/1 44 35 | 90804 5 27 6 20 {11 16 30 31 * These angles, as also those in the pr eceding columns, answer to the French division of the circle reds same according to the usual division are given in Table III. as. copied from the Edinburgh Ency lo pedia. a) Te, Os aeyrsenty pet se NRA Ul. LLL ——<—<————— ———— a ————— Engraved by Sant Lacey - 1613.by6eS Robinson Lucrnoster Row. kthie rest of the Lroprietwors. ee ’ 3 gk obi , A « See : ba a & or 6 ey deer (Dae a's 6% 7 . 3 t Nooo MMMM 4 parca MMMM Y YY EL ETELITOLCE LUAU OTELVEDUCUUEE itll MR \ With UOT Cuthbertsons Air Pump. WMMMMMMMMmMtétbr 7p \\ \N ni ti NVWW[ Awa Ud MMMM AAAAAWANAAAARS SARA EN S SSS RANEY ni MMMM RIS WISSSSS S MVMMMMMMMMMMMlbtttdldbbdbbbbd Md “ N me sis AN ARRAN A BRK CW We = E = = 2 WN WOE a | : ~ > SSN N = = = ve" NS Engraved by Sam? Lacey. SS Ny . ¥« LondonPublithedMay 311813. by G&S Robinsontaternoster low & the restr hel repr velo s. 7 , #. a oF a eh _ PLA PLANETARIUM, an astronomical ma- chine, contrived to’ represent the motions, orbit; &c. of the planets, as they really are in 1ature, according to the Copernican system. Che larger sort of them are called orreries. PLANETARY, something that relates to the planets. PLANETARY System, is the system or assem- alage of the planets, primary and secondary, noving in their respective orbits, round their 2ommon centre the sun. PLANIMETRY, that part of geometry which consider lines, and plane figures, with- yut any regard to heights or depths. Plani- meiry is particularly restricted to the mensu- ‘ation of pianes and other surfaces; as con- radistinguished from stereometry, or the nensuration of solids, or capacities of length, wreadth, and depth. PLANIMETRY, is performed by means of the squares of long measure, as square inches, square feet, square yards, &c.; that is, by iquares whose side is an inch, a foot, a yard, xc. So that the area, or content, of any sur- ace, is said to be found, when it is known how nany such square inches, feet, yards, &c. it con- ains. \See MENSURATION and SURVEYING. PLANISPHERE, a projection of the iphere, and its various circles, on a plane; as ipon paper or the like. In this sense, maps of the heavens and the earth, exhibiting the neridians and other circles of the sphere, may ve Called planispheres. PLANISPHERE is sometimes also considered i$ an astronomical instrument, used in ob- erving the motions of the heavenly bodies ; veing «a projection of the celestial sphere upon , plane representing the stars, constellations, ve. in their proper situations, distances, &c. As the astrolabe, which is a common name or all such projections. In all planispheres, the eye is supposed to in a point, viewing all the circles of the phere, and referring them to a plane beyond hem, against which the sphere is as it were lattened ; and this plane is called the plane f projection, which is always some one of he circles of the sphere itself, or parallel to ome one of them. Among the infinite number of pianispheres hich may be furnished by the different planes f projection, and the different positions of he eye, there are two or three that have been referred to the rest. Such as that of Ptolemy, there the plane of projection is parallel to ae equator: that of Gemma Frisius, where 1e plane of projection is the colure, or sol- litial meridian, and the eye of the pole of the 1eridian, being a stereographical projection ; r that of John de Royas, a Spaniard, whose lane of projection is a meridian, and the eye laced in the axis of that meridian at an in- nite distance; being an orthographical pro- ection, and called the anelemma. PLANO Concave, and Convex Lens. 4ENS. PLATO, one of the most celebrated of the acient philosophers, being the founder of the eet of the Academics, flourished at Athens See dle 9) about 250 years before Christ. He was the - favourite pupil of Socrates, and the tutor of Aristotle, Xenocrates, and Spusippus. Plato was a great admirer of geometry, a knowledge of which was necessary to be admitted into his school; over the door was written, “ Let no one enter here who is ignorant of geome- try.” Itis, however, as a moral philosopher, that he is more distinguished than for any remarkable discoveries in science, though . . 5 these were not inconsiderable. His works are all written in the way of dialogue, ina beautiful elevated style; the first Latin edi- tion of which was printed by Marcilius Fi- cinus, in 1491; and another edition in the original Greek, by Aldus, at Venice in 1513. Another edition was published in 1588, and again in 1602, beside other editions at differ- ent times and places, but one of the most correct and clegant is that printed by Ste- phens at Paris in 1578, in 3 vols. folio. PLATONIC Bodies, the same as Regular Bopies, which see. PLaTonic Year, or the great year, is the period of time determined by the revolution of the equinoxes, upon a supposition of the precession going on uniformly till they have made one complete revolution. See PREcEs- SION. PLATONISM, the philosophy or doctrine of Plato. PLATONISTS, the followers of the doc- trine of Plato. PLEIADES, an assemblage of seven stars in the neck of the constellation of Taurus; of which, however, there are but six now visible to the naked eye. PLENIUM, in Piilosophy, is that state of things in which every part of space, or exten- sion, is supposed to be full of matter; in op- position to a vacuum, which is a space devoid of all matter. See Vacuum. PLOTTING, in Surveying, is the art of laying down on paper, &c. the several angles and lines of a tract of ground surveyed by a theodolite, &c. and chain. In surveying with the plain table, the plotting is saved; the several angles and distances being laid down on the spot as fast as they are taken. But, in working with the theodolite, semicircle, or circumferentor, the angles are taken in de- grees; and the distances in chains and links, so that there remains an after operation to reduce these numbers into lines, and so to form a draught, plan, or map; this operation is called plotting. Plotting then is performed by two instruments, the protractor and plot- ting scale. By the first, the several angles observed in the field with a theodolite, or the like, and entered down in degrees in the field- book, are protracted on paper in their just | quantity. By the latter the several distances — measured with the chain, and entered down in the like manner in the field-book, are laid down in their just proportion. PLottTineG Seale, a mathematicalinstrument, usually of wood, sometimes of brass, or other matter; and either a foot or half a foot long. On one side of the instrament are scycral! POI scales or lines, divided in equal parts. The first division of the first scale is subdivided into ten equal parts, to which is prefixed the number of 10, signifying that ten of these subdivisions make an inch; or that the divi- sions of that scale are decimals of inches. The first division of the second scale is like- wise subdivided into 10, to which is prefixed the number 16, denoting that sixteen of these subdivisions make an inch. The first division of the third scale is subdivided in like manner into 20, to which is prefixed the number 20; to that of the fourth scale is prefixed the number 24; to that of the fifth 32; that of the sixth 40; that of the seventh 48; denoting the number of subdivisions equal to an inch in each respectively. ‘The two last scales are broken off to make room for two lines of chords. ‘There are also on the back side of the instrument-a diagonal scale. As to the use of the plotting seale, if we were required to lay down any distance upon paper, suppose 6 chains 50 links: draw an indefinite line; then sitting one foot of the - compasses at figure 6 on the scale, e. gr. the scale of 20 in an inch, extend the other to 5 of the subdivisions, for the 50 links; this dis- tance, being transferred to the line, will ex- hibit the 6 chains 50 links required. If it be desired to have 6 chains 50 links, take up more or less space, take them off from a greater or lesser scale, 2..e. from a scale that has more or fewer divisions in an inch. To find the chains and links contained ina right line, or one thatis just drawn, according to any scale e. gr. that of 20 in aninch. Take the length of the line in the compasses, and applying it to the given scale, you will find it extend from the number 6 of the great divi- sions to 5 of the small ones: hence the given line contains 6 chains 50 links. PLUMBLINE, a line having a plummet or weight attached to it in order to finda erpendicular. PLUNGER, the solid brass cylinder used as a forcer in forcing pumps. PLUS, in Algebra, the affirmative or posi- tive sign + which signifies addition. ‘The “ncient algebraists used the word plus at full Jength, after which the initial was employed, and finally the sign +, which was first intro- duced by Stefelius in his arithmetic. PLUVIAMETER. See Rain Gage. PNEUMATICS (from avevye, breath, or air), is that branch of natural philosophy which treats of the weight, pressure, elasticity, &e. of elastic fluids, bat more particularly of the air, the history and principles of which will be found under the articles AEROSTATION, Arr, ATMOSPHERE, BAROMETER, &c. POINT, in Geometry, according to Euclid’s definition, is that which has no parts or di- mensions, neither length, breadth, nor depth; and therefore marks position only. Point receives also various denominations in the doctrine of curves, optics, perspective, &c. which are all defined under the respective adjectives. Physical Point, is the smallest or least POL sensible object of sight, and is thus distin euished from a geometrical point, which has. only position, being of no magnitude or di- mension. | ; te Points of the Compass, are the 32 princip; 1 pvints, into which the compass card is divided, or the points of the horizon towards whieh these are directed. See Compass. f POLAR, something relating to or situated near the poles. 4 Povar Circles. See CIRCLE. st POLARITY, the quality of a thing having poles, or a tendency to turn itself into a cer tain position, but more particularly used with reference to the magnet. POLE, in Astronomy, one of the extres mities of the imaginary axis on which the sphere is supposed to revolve. These twe points are each 90 degrees from the equator that towards the north being called the north pole, and the other the south pole. is Poe, in Geography, one of the points on which the terraqueous globe turns, cach.@ them being 90 degrees distant from the equai tor, and are denominated the north or souti pole, according as they point towards th north or sonth points of the heavens. A In consequence of the inclination of thy terrestrial axis to the plane of the ecliptic and its parallelism during its annual motiol in its orbit, these parts of the world have onk one day and one night throughout the yeat each continuing for about six months. a It is singular that though the poles have; greater portion of light than any other part of the globe, yet the name by which they ar denoted in most languages, both ancient am modern, is derived from terms signifying dark ness and obscurity; but though they reall enjoy more light upon the whole than am other parts, yet in consequence of the obhi quity with which the rays of the sun fall upo them, and the great length of winter night the cold is so intense, that those parts of th giobe that lie near the poles have never bee fully explored, though the attempt has bee repeatedly made by the most celebrated née vigators. Elevation of the Poe, is an angle subtende between the horizon of any place and a lin drawn from thence to the pole, which 3 always equal to the latitude of the place. | Pots, in Spherics, a point equally distar from every part of the circumference of a greé circle of the sphere; or it is a point distant from the plane of a circle, and im line, called the axis, passing perpendicular through the centre. The zenith and nad are the poles of the horizon; and the polesi the eqnator are the same with those of th sphere. 4 Potes of the Ecliptic, are two points on I surface of the sphere, 23° 30’ distant from fl poles of the world, and 90° distant from ever part of the ecliptic. .= Po.es, in Magnetism, are two points of loadstone, corresponding to the poles of i world; the one pointing to the north, tf other to the south. See MAGNETISM. "y G | aa POL Pour, in Land Measure, is a linear measure ual to 5£ yards; or a square measure of 304 uare yards. Po te, or Polar Star, is a star of the second agnitade, the last in the tail of Ursa minor. slongitude Mr. Flamsteed makes 24° 14’ 41”; latitude, 66° 4’ 11". The nearness of this tr to the pole, whence it happens that it ver sets, renders it of vast service in navi- tion, &c. for determining the meridian line, 2 latitude, &e. POLEMOSCOPE, in optics, a kind of re- icting perspective glass invented by Heve- 's, who commends it as useful in sieges, &c. ‘ discovering what the enemy is doing, tile the spectator lies hid behind an ob- vcle. POLISHING in general, the operation of ving a gloss or lustre to certain substances, metal, glass, &c. The operation of polishing optic-glasses, er being perfectly ground, is one of the vst difficult points of the whole process. ‘fore the polishing is begun, it is proper to etch an even well-wrought piece of linen ar the tool, dusting upon it some very fine voli. Then taking the glass in your hand, in it round forty or fifty times upon the tool, itake off the roughness of the glass about * border of it. This‘cloth is then to be re- iwed, and the glass to be polished upon the xed tool, with a compound powder made four parts tripoli mixed with one of fine le vitriol; six or eight grains of which mix- 'e are sufficient for a glass five inches broad. lis powder must be wetted with eight or « drops of clear vinegar, in the middle of | tool; being first mixed and softened roughly with a very fine small mullet. fen with a nice brush, having spread this ture thinly and equably upon the tool, |}@ some very fine tripoli, and strew it thinly i equably upon the tool so prepared, after lich, take the glass to be polished, wiped 'y clean, and apply it on the tool, and move gently twice or thrice in a straight line ikwards and forwards; then take it off, and ierve whether the marks of the tripoli, king to the glass, are equably spread over whole surface; if not, it is a sign that ier the tool or glass is too warm, in which e you must wait awhile and try again, til | find the glass takes the tripoli everywhere te. ‘Then you may safely begin to polish, re being no danger of spoiling the figure the glass, which in the other case would ulibly happen. This is Mr. Huygens’s thod; but it ought to be observed, that iost every operator has a peculiar one of own, and of which some of them make a at secret. ir Isaac Newton no where expressly de- bes his method of polishing optical glasses; his method of polishing reflecting metals thus describes in his Optics. He had two nd copper plates, each six inches in diame- ' the one convex, the other concave, ground ytrue to one another. Onthe convex one POL he ground the object-metal, or concave, which was to be polished, till it had taken the figure of the convex, and was ready for a polish. He then pitched over the convex very thinly, by dropping melted pitch upon it, and warm- ing it to keep the pitch soft, whilst he ground it with the concave copper wetted, to make it spread evenly all over the convex, till it was no thicker than a sixpence; and after the convex was cold he ground it again, to give it as true a figure as possibie. He then ground it with very fine putty, till it made no noise; and then upon the pitch he ground the object- metal with a brisk motion for two or three minutes; when laying fresh putty upon the pitch, he ground it again till it had done making a noise, and afterwards ground the object-metal upon the pitch as before; and this operation he repeated till the metal was perfecily polished. For other ingenious methods of grinding and polishing lenses and specula, see Brew- ster’s edition of Ferguson’s Lectures, vol. ii. p. 452, 461. POLITICAL Arithmetic, is the art of rea- soning by figures upon matters relating to government, such as the revenues, number of people, extent and value of land, taxes, trade, &e. in any nation. These calculations are generally made with a view to ascertain the comparative strength, prosperity, &c. of any two or more nations, and is otherwise called STATISTICS, which see. POLLUX, in Astronomy, one of the Twins in the constellation Gemini, also a fixed star of the second magnitude in that constellation. See Castor and GEMINI. POLYACOUSTIC, any thing that mul- tiplies sound. POLYEDRON. See PoLyHEepRON. POLYGON (from moruc, many, and yur, angle), in Geometry, a multilateral figure, or a figure whose perimeter consists of more than four sides, and consequently having more than four angles. Ifthe angles be all equal among themselves, the polygon is said to be aregular one; otherwise, it is irregular. Polygons also take particular names according to the num- ber of their sides; thus a polygon of 3 sides is called a trigon, 4 sides is called a tetragon, 5 sides is called a pentagon, 6 sides is called a hexagon, &c. and a circle may be considered as a polygon of an infinite number of small sides, or as the limit of the polygons. -Polygons have various properties, as below: 1. Every polygon may be divided into as many triangles as it has sides. 2. The angles of any polygon taken toge- ther, make twice as many right angles, want- ing 4, as the figure hath sides; which pro- perty, as well as the former, belongs to both regular and irregular polygons. 3. Every regular polygon may be either in- scribed in a circle, or described about it; which is not necessarily the case if the poly-. gons be irregular, POL But an equilateral figure inscribed in a circle is always equiangular ; though an equi- angular figure inscribed in a circle is not alw ays equilateral, but only when the number of sides is odd. Vor if the sides be of an even number, then they may either be all equal, or else half of them may be equal, and the other half equal to each other, but different from the former half, the equals being placed al- ternately. 4, Every polygon, circumscribed about a circle, is equal to a right- angled triangle, of which one leg is the radius of the circle, and the other the perimeter or sum of all the sides of the polygon. Or the polygon is equal to half the rectangle under its perimeter and the radius of its inscribed circle, or the perpen- dicular from its centre upon one side of the polygon. Hence, the area of a circle being less than that of its circumscribing polygon, and greater than that of its inscribed polygon, the circle is the limit of the inscribed and circumscribed polygons: in like manner the circumference of the circle is the Jimit between the perime- ters of the said polygons. See CIRCLE. 5. The following table exhibits the angles and areas of all the polygons, up to the dode- cagon, viz. the angle at the centre, the angle of the polygon, and the area of the polygon when each side is 1 ¥: at|Ange.(. of Ang. Polygon. Area, Polygon. 0:4330127 1:0000000 1:7204774 2:5980762 36339124 4°8284271 61818242 76942088 9°3656399 11°1961524 Trigon Tetragon Pentagon Hexagon Heptagon Octagon Nonagon Decagon Undecagon Dodecagon 147.3, 150 Therefore to find the area of any regular polygon not exceeding 12 sides, square the side, and multiply that square by the cor- responding tabular number in the preceding table. Or generally if s represent the length of one of the equal rapt and 2 the number of 7 90n — 180 them; then s* x 7 tang. (——) = -= area of the polygon. To inscribe a Polygon within, or to circumscribe a Polygon about a given Circle. Bisect two of the angles of the given poly- gon A and B by the right lines AO, BO; and from the point QO, where they meet, with the. radius AO, describe a: circle which will cir- cumscribe the polygon. POL Next to circumscribe a polygon, dived 36 by the number of sides required, to find 1 angle AOB; which set off from the cen as and draw the line AB, on which construct polygon as in the following problem. 2. given line to describe any given regular gon. Find the angle of the polygon in tl table, and at A set off an angle equal theret then drawing CA = AB, through the poi CAB, describe a circle, and in “this appl yir the given right line as often as you can, Uf) polygon w ill be described. B Otherwise. To inscribe a Polygon in a sy cle.—Draw the diame- ~& ter AB, which divide into as many equal parts as the figure has sides. From the points A, B, as centres with the radius AB, de- seribe ares crossing each other inC. From the point C, through the second division of the diameters, draw D™— the line CD. Join the points A, D, and tl line A D will be the side of the polygon ' quired. Note. In this construction A D is the a of a pentagon. . Another method, something more accural is by erecting a perpendicular from the ce tre, of such a length that the part witho the circle shall be “equal to 4 of that witli and drawing a line from its extremity throg the second division as before. In the preceding part of the attiole if observed, that any regular polygon may | inscribed in, or circumscribed about a cirel but this must be understood under certé modifications; all that is meant is, that the is nothing in the nature of the problem | render it impossible; and not that any polyg may be geometrically inscribed. In fact, the number of polygons that ady of a geometrical construction is very limit viz. the equilateral triangle, the square a pentagon, and those figures whose number sides are some multiples of these; to whi Gauss has lately added the 17 sided polyg« and its multiples, and some others, viz. | those polygons whose number of side is prime, and of the form 2” + 1. It is obvious that the side of a polyg: inseribed in a circle, corresponds to the che of the angle of that polyg gon at the centre, to twice the sine of half “that angle, and tl the perpendicular falling from the centre that side also answers to the cosine of 1 same half angle; if therefore, the sine or « sine of the angle can be found by a quadra equation, the polygon itself may be constru ed geometrically and not otherwise. It is also known from the Cotesian Tl orem, that if a circle be divided into n eq parts, the double cosines of each of t ungles thus formed are the sums of the pa POL naginary roots belonging to the equation -1=0. That is, all the imaginary roots ie equation a — 1 = 0, are comprised in yeneral ker 7 ing any integer not divisible by x, and x esenting the semicircumference. nd since the coefficient of the second term ny equation is equal to the sum of all its s with their signs changed, it follows that pi yi x+1-—0 \s. ate is equal to the sum of two of the ginary roots of the equation ax»— 1 = 0; consquently the possibility of exhibiting cosines of these angles analytically, de- ds upon the solution of the equation 2”—1 * therefore it were required to find the € oO ne of sl we should have n = 8, or #3— 0, Here the real root being 1, we have 3 ml en Pg Ya Hae \ x—l imaginary roots of f# = —i +4 v/—8 which are ie ——i-—iy—3 ‘their sum — — 1, that is twice the cosine 20° = — 1, or the cosine = — i. be therefore since every equation 2” — 1 ' may always be reduced to half its inal degree if even, or to half the original ee minus 1 if odd, (see Reciprocal Equa- is) it follows, that there is no difficulty in solution of x5 — 1 = 0, or in the geome- il construction of a pentagon. But if '7, or when the figure is a heptagon, the ation only reduces to a cubic; and as we aot give a geometrical construction of a ie equation, so we cannot give a geome- u construction of a heptagon. tauss, however, in his “ Disquisitiones ihmeticz,” published in 1800, has shown lethod of solving any equation x” — 1 — 0, nz is a prime, by reducing it to other oler equations; thus if mis a prime, and +1 be resolved into its factors n — 1 = 5g, ev, &e. then the solution will be effected ” equations of the degree a; @ of the de- > b; y of the degree c, &c. rhus, if n = 7, then since n —1—6—= 3*, the solution will be obtained by one 1 Smet and one cubic equation; ifn — 13, in -— 1 = 12 = 2? 3', the solution is ob- ed by means of two quadratics and one ie; but if n = 17, then since 17 — 1 = = 2+, the solution will be obtained by ims of four quadratic equations, and con- nently such a polygon may be inscribed metrically in a circle; and the same has ‘© When nm is any prime number of. the oe + 1. ty this means Gauss has found the cosine of 2 ix f | ' POL —tt+ivin+ivia7—vint+ —tF3VI7) +3v507—V1N f[— —ttivld+ivid7+viz7)t Now 5, 17, 257, 65537, are prime numbers of this form, and therefore each of these ad- mits of a geometrical censtruction. We know also from other principles, that if any two polygons of an unequal number of sides, prime to each other, can be inscribed geome- trically in a circle, that the polygon, the num- her of the sides of which is equal to the pro- duct of these two, can also be inscrilied eeometrically; also all polygons of which the number of sides is any power of 2, may be inscribed by continual bisections; and again, all those whose number of sides is equal to any inscribable polygon into any power of 2. Hence we have the following series of poly- gon, each of which admits of a geometrical construction. Polygons of less than 100 sides admitting of geometrical construction. die vis NS wae a ~—, Nit pie Dl LEN PF No. of Sides. No. of Sides. + ew > OU 22 ZAG 4 oF a2 2 ts Cee Aatare 10d | 34 — 2.17 Ge 2.3 40 — 27.5 $24 48 == 29'S 10 = 2.5 rs A Diem oh hed Bota 2 GO =" 3" 1S 1b 35 64 — 2° 16 — 2+ 68 = 27.17 7W=—2*+1 80 — 24.5 ae 86 = BAF Ss 27733 96 = 23.3 And to the above we may add the three con- secutive polygons 255, 256, 257 ; each of which is inscribable in a circle for 255 — 3. 5. 17; 256 — 2°, and 257 = 2° + 1. The next three consecutive polygons that admit of a geometrical construction are; . 65535 — 255,257 5036 —"2"*5 Goes eise 2 aks For more on this subject see Gauss’s “ Dis- quisitiones A rithmeticze,”’or the French transla- tion of the same, entitled ‘“‘ Recherches Arith- metiques;’’ Le Gendre’s “¢ Essai sur la Theorie des Nombres;’”’ and Barlow’s “ Elementary Investigation of the Theory of Numbers.” POLYGONAL Numbers, are those that are formed of the sums of different and indepen- dent arithmetical series, and are termed Na- tural, Triangular, Quadrangular, Pentagonal, Hexagonal, &c. Numbers; according to the series from which they are generated. Lineal, or Natural Numbers, are formed from the successive sums of a series of units; thus LPR. 1, Parr wee Nat. num........... 1, 2, 3, 4, 5, 6, &e. Triangular Numbers, are the successive sums POL of an arithmetical series, beginning with unity, -the common difference of which is 1; thus Arith, series.... 1, 2,3, 4, 5, 6, 7, &c. Trian. num..... 1, 3, 6, 10, 15, 21, 28, &c. Quadrangular, or Square Numbers, are the successive sums of an arithmetical progres- sion, beginning with unity, the common dif- ference of which is 2: thus Arith. series.... 1,3, 5, 7, 9, 11, 13, &c. Quad. or squa. 1, 4, 9, 16, 25, 36, 49, &e. Pentagonal Numbers, are the sums of an arithmetical series, the common difference of which is 3; thus Arith. series.. 1, 4, 7, 10, 13, 16, 19, &e. Pentagonals.. 1, 5, 12, 22, 35, 51, 70, &e. And, universally, the m,gonal Series of Num- bers, is formed from the successive sums of an arithmetical progression, beginning with unity, the common difference of which is m — 2. The general form, which includes every order of Polygonal Numbers, is (m — 2) n? —(m — 4)n 2 where m is the denomination of the order; therefore, making successively m — 3, 4, 5, ke. we have the following results as to the forms of polygonals, 2 é s m2 ui m = 3 triangular num. = ur 2 2n* —On m = 4 square = —_——_ =n’ 2 3n* — n m = 5 pentagonals = tages __ 4n* — 2n m = 6 hexagonals = 2 By means of the above general form, any polygonal number, of which the root 2 is given, may be readily ascertained. ‘Thus, by making m — 3, m = 4, m — 5, &c. and in each series n = 1, 2, 3, 4, 5, &c. we shall ob- tain the same numbers as given above, under each respective denomination. ; Also any polygonal number and its denomi- uation being given, the root of the polygon is readily obtained. Yor let __ (m — 2) n* —(m— 4) n rr 2 represent any given polygonal, of which the denomination mis known; then (m — 2) n? —(m — 4) n = 2p . PERE © 1, Neceaga *) bie Qp rts th v.92 (A= n ees | whencen — m— 44 V (2p (m —2) + (m—4)?) Racuy pid Babiateadea he which is a general form for the root of every polygonal number. Fermat, at page 15, in one of his notes to prop. 9 of Diophantus on Multangular Num- bers, has given particular rules for finding the roots of given polygonal numbers, without the extraction of the square root; but they are of little or no use, and therefore we shall hot enumerate them. ~ “Son ae We may also find the sum of any se es polygonals by means of the foregoing gene formula, for representing still the deno tion of any order of polygonals by m, and abridging make m — 2 = d the common ference of the series from which they ; generated ; and let x be the number of t in the series whose sum is required ; the ; shall have 2 2 : en's ae | (< edt na (™ - *(m—2) 425 for the sum of the » terms sought. = Hence, substituting successively the nu bers 3, 4,5, &c. for m, there is obtained 1 following particular cases or formule; vizs ne +3” +2 J | Triangulars........ 6 : Squares....-....0.00 ant ont t a L. 2 Pentagonals....... 2-2 — a ») 2 _— Hexagonals....... an — : &e. &e. The denomination of polygonals seems have been given to these class of num from the circumstance, that they may be presented by the particular figures, the na of which. they bare; and the side of the fi is the same as what we call the root of. polygon. 'Thus | Triangles. Squares : _ See some particular rules for arranging points, in a note subjoined to the Eng] edition of Euler’s “ Algebra.” POLYGONOMETRY, is an extensior the science of trigonometry, having the sa reference to polygons in general, as trigo metry has to triangles in particular. We owe this extension of the rules of gonometry to L’Huiller, who publishec treatise on this subject at Geneva, in 17 which, with the exception of a chapter the third volume of Dr. Hutton’s “Cou of Mathematics,” is, we believe, the only w: on polygonometry at present before the p lic; it will, therefore, not be amiss to stat few of the principal propositions, but for demonstrations we must refer the readers the works above mentioned. 1. In any polygon, any one side is equal the sum of all the rectangles of each of other sides, drawn into the cosines of ° angles made by those sides and the propo side. * Thus, if ABCDE be any polygon, and the ides ED, DC be produced to meet the base so produced; then AB—EA.cos. EAB + ED.cos. DFA + DC.cos. DGB + CB cos. CBA. Where it is only necessary to observe, that vhenany of the angles are obtuse the cosines fthem become negative, and the terms into yhich they enter must therefore be subtracted. 2. The perpendicular let fall from the high- st point or summit of a polygon, upon the pposite side or base, is equal to the sum of he products of each of the sides on either ide comprised between that summit and the ase, drawn into the sines of their respective iclinations to the base. Thus, in the preceding figure, 1K — DE.sin. DFK + EA.sin. EAK, or 1K —DC.sin. DGK + BC.sin.CBK. Cor. Since then these two quantities are qual, it is obvious that if all the sides be iven but one, that one may be found ; for DE = 1C.sin. DGK + BC. sin. CBK—EA.sin EAK sin. DF K id in the same manner may any other side » determined. 3. The square of any side of a polygon is yual to the sum of the squares of all the her sides, minus twice the sum of the pro- icts of all the other sides, multiplied two id two by the cosines of the angles included stween them. A B Thus, calling AB = a, BC =}, CD =e, A =d, and denoting the angles by the cor- sponding letters, a’ = b* + ¢* 4 d* — (be.cos.C + bd.cos. E + edcos. D) like manner e* = a? + b* + d*— (ab. cos. B +. ad. cos. A + bd. cos. BE) ‘dso on for any other side. 4. T'wice the area of any polygon is equal the sum of the rectangles of its sides, ex- pt one, taken two and two, by the sines of sum of the exterior angles contained by dse sides; that is, the angles formed by oducing those sides. com in the trapezium A BCD, c aa A. B AB.BC. sin. B -DC. sin. (B+C) twice the area = -ODC.sin. C + BC Again, in the pentagon, D A AB. + AB. . sin. D and the same law obtains in all polygons, whatever may be the number of their sides ; and the advantage of it in practice is, that the area is found without the error that may arise in constructing the figure. For more on this subject, see L’Huillier’s “Treatise on Polygonometry ;” and chap. vi. vol. 3, Dr. Hutton’s “ Course of Mathematics.” POLYGRAM, a figure consisting of many ines. POLYHEDRON, or Potyepron (from woAus, many, and sdpa, seat), a body or solid contained by many rectilinear planes or sides. When the sides of the polyhedron are re- gular polygons, all similar and equal, then the polyhedron becomes a regular body, and may be inscribed in a sphere; that is, a sphere may be described about it, so that its surface shall touch ail the angles or corners of the solid. There are but five of these regular bodies, viz. the tetrahedron, the hexahedron or cube, the octahedron, the dodecahedron, and the icosahedron. See each of those articles. POLYHEDRON, Gnomonical, is a stone with several faces, on which are projected various kinds of dials. Of this sort, that in the Privy Garden, London, now gone to ruin, was esteemed the finest in the world. POLYHEDRONS, Consisting of many sides. POLYNOMIAL, in Algebra, a quantity consisting of many terms. Sce MULTINOMIAL. POLYSCOPE (from gous, many, and confirm the truth of this theory of the origin of porisms, or at least the justness of the notions founded on it, I must add a quo- tation from an essay on the same subject, by a member of this society, the extent and cor- rectness of whose views make every coinci- dence with his opinions peculiarly flattering. 4n a paper read several years ago before the POS Philosophical Society, Professor Dugald Stew- _ art defined a porism to be ‘a proposition af firming the possibility of finding one or more | of the conditions of an indeterminate theorem! — Where, by anindeterminate theorem, ashe had _ previously explained it, is meant one which | expresses a relation between certain quan- . tities that are indeterminate both in magni- | tude and in number. The near agreement of | this, with the definition and explanation which | have been given above, is too obvious to re- | quire to be pointed out; and I have only to- observe that it was not long after the publi-_ cation of Simson’s posthumous works, when being both of us occupied in speculations con- eerning porisms, we were led separately to the conclusions which I have now stated.” For more on this subject see Hutton’s ‘ Mathematical Dictionary,” article PoRIsM } also a very ingenious paper on porisms, by Henry Brougham, Esq. in the Phil. Trans. for 1798, or new abridgment, vol. xviii. p. 345, 355. ; PORISTIC Method, in Mathematics, is that which determines when, by what means, and how many different ways a problem may be resolved. r PORTA (JoHN Baptista), a Neapolitan, eminent for his learning. As he admitted a society of learned friends into his house, he was accused of magical incantations, and ex- posed to the censures of Rome. He died 1515, aged 70. He invented the camera obscura, improved afterwards by Gravesande, and formed the plan of an encyclopedia. He wrote a Latin treatise on natural magic, 8vo. } another on physiognomy, mixed with astro- logy, &c.; de occultis literarum notis; phy- siognomica, folio, &c. “fi POSITION, in Astronomy, relates to the sphere. The position of the sphere is either right, parallel, or oblique ; whence arise the in- equality of days, the difference of seasons, &e, Circles of Position, are circles passing through the common intersections of the hori- zon and meridian, and through any degree ol the ecliptic, or the centre of any star, or other point in the heavens, used for finding out the position or situation of any star. These are usually counted six in number, cutting the equator into twelve equal parts, which thé astrologers call the celestial houses. ' PosITIon, in Arithmetic, called also false position, or supposition, or rule of false, is¢ rule so called, because it consists in calculat ing by false numbers supposed or taken a random, according to the process describet in any question or problem proposed, as 1 they were the true members, and then fron the results compared with that given in th question, the true numbers are found. Iti sometimes also called trial and error, becausi it proceeds by trials of false numbers, ant thence finds out the true ones by a compa rison of the errors. Position is either single or double. Single Position, is that which has plaé . hf POS when the results are proportional to their suppositions, ‘and therefore only one suppo- sitionis necessary. Rule. Assume any number for that required, and perform the operations described in the question with it. ‘Then say, as the result ob- tained is to the number assumed, so is the result in the question to the answer. Exam. A person after spending 3 and 7 of his money, has yet remaining 60/.; what had he at first? Suppose he had at first 1200. Proof. Now § of 120 is 40 tof 144is 48 Lofitis 30 4 of 144 is 36 their sumis 70 their sum 84 which taken from 120 taken from 144 Jeaves 50 leaves 60 as per question. Then 50: 120:: 60: 144, the answer. Double Position, is a method of answering questions similar to those in single position, but which have not their results proportional to their positions. ‘The rule for this purpose given by Mr. Bonnycastle in his Arithmetic, Svo. edition, is as follows: Take any two convenient numbers, and proceed with them separately, according to the conditions of the question, noting the results obtained from it. Then as the difference of these results is to the difference of the supposed numbers, so is the difference between the true result and either of the former, to the correction of the number belonging to the result used; which correction, being added to that number, when it is too little, or subtracted from it when it is too great, will give the answer required. Exam. What number is that which being multiplied by 6, the product increased by 18, and the sum divided by 9, the quotient shall be 20. Let the two supposed numbers be 18 and 30 Then 18 30 6 6 108 18U0 18 18 9)126 9)198 14 the Ist result 22 the 2d result Then 22 — 14:30 — 18:: 20— 14 Or S212 ea * 9 the correction to Ist supposition ; therefore 9 + 18 = 27 the number sought. This rule, which is much more simple than that commonly given, was first published by Mr. Bonnycastle in his 8vo. Arithmetic. Centre of Position. See CENTRE. Given in PosiT10Nn, in Geometry, is an ex- pression made use to denote that the position or direction of a line is given or known. Geometry of PosiTIoN, is a species of geo- metry first treated of by Carnot, the object of which is to investigate and determine the re- lation that has place between the position of the different parts of a geometrical figure with -yegard to each other, or with regard to some POW determinate line or figure first fixed upon as a term of comparison, and which is called the primitive system, while that compared with it is denominated the transformed system; and as long as the different parts of the trans- formed system have the same directions or positions with regard to each other, their re- lation is said to be direct, but when they are different inverse. These positions in the analytical representa- tions of geometrical figures are commonly indicated by the signs prefixed to the letters or symbols representing these lines or quanti- ties, and the geometry of position at the same time that it gives very great facility to the investigation of many very interesting re- searches, sets in a clear and indisputable light all those apparent mysteries and anomalies connected with the introduction of the nega- tive sign into analytical investigations. ‘The subject however is of such a nature that it is impossible to throw any light upon it, within the narrow limits of this article, and we can therefore only refer the reader for a complete illustration to the two works of the above- mentioned author on this subject, viz. “ De la Correlation des Figures de Géométrie ;” and the more ample work entitled, “‘ Géomé- trie de Position.” POSITIVE Electricity. In the Franklinian system, all bodies supposed to contain more than their natural quantity of electric matter are said to be positively electrified ; and those from whom some part of their electricity is supposed to be taken away are said to be electrified negatively. ‘These two electricities being first produced, one from giass, the other from amber or rosin, the former was called vitreous, the other resinous, electricity. PositivE Quantities, in Algebra, are those which are affected with the sign + being affirmative or addative, in contradistinction to negative quantities which are to be sub- tracted. POSTULATE, in Geometry, a demand or petition, or a supposition so easy and self evidently true, as needs no explanation or illustration; differmg from an axiom only in the manner in which it is put, viz. as a re- quest instead of an assértion. POUND, an English weight of different denominations, as Avoirdupoise, Troy, Apeo- thecaries, &c. The pound avoirdupoise is 16 ounces of the same weight, but the other pounds are each equal to 12 ounces. 'The pound avoirdupoise is to the pound troy as 5760 to 69994, or nearly as 576 to 700. Pounp is also the highest denomination used by the English in their money accounts, being equal to 20 shillings. POWER, in Arithmetic and Algebra, that which arises by the successive multiplication of any number or quantity into itself, the degree of the power being always denomi- nated by the number of equal factors that are employed; thus ; DES aeees tkeneeke seooe 2, LSt pOWer of 2. ZX @ H asre sessrovee Sy 2A POW OF square. POrW..- 2x 2X 2 ....... 23, 3d power or cube. DX BRK Bioeedrrns 2*, 4th power, Ke. &e. So also UV cevsrecveeaeceerereee V7, LSt power. Fae = | ee x*, 2d power. LEAR eS os ove x3, 3d power. BP ET IE XK! li. x*, 4th power, ke. &e, Hence it appears that the idex which de- notes the degree of any power is always equal to the number of equal factors from which that power arises; or one more than the num- ber of operations. See Exponent and In- VOLUTION. The early writers on arithmetic and algebra, gave names to the several powers, which we now denote by their indices. Thus the 2d was called the square, and the 3d the cube, as we do at present; but they carried this much farther, calling the 4th power the quadrato-quadratum ; the dth, qua- drato-cubus ; 6th, cubo-cubus ; 7th, quadrato- guadrato cubus, &c.; but these denominations are now rejected for the more simple and ob- vious denominations by means of the ex- ponent. The powers of numbers have several curious properties, as to their forms, divisors, &c. of which the following are the most remarkable: 1. The difference of any two equal powers of different numbers, is divisible by the dif- ference of their roots, that is x” — y” is divisible by 2 — y. This is true, whether n be even or odd. 2, 2” — y" is divisible, both by « — y and x + y, ifn be even. 3. a” + y" is divisible by x + y if n be odd. Thus also 4, x”— x” is divisible by x — 1, and by x -+ lif m— mn be even. 3. a” 2" is divisible by x — 1 if m—n be odd. 6. 2” + x" is divisible by x + 1 if m—n be odd. 7. Neither the sum nor difference of any two equal integral powers above the second, can be equal to a complete power of the same denomination. 8. If m be a prime number, and x any num- ber not divisible by m, then will the remainder arising from the division of x by m, be the same as that from the division of 2” by m; and consequently «”—!—1, is always divisible by m. 9. Hence is obtained the forms of several powers, tlfat is, the remainders that they will leave when divided by given numbers; thus x”, a3, at, &e. denoting the powers of any numbers whatever we have. Forms Forms. booe * 2... 30 or! Siviek 5n,or 5n + 1. eM, Sede aS the. Ooh ak 7n,0r4 770. 1, Cte. On, tr Shed Di. JP ce ee AAS... eR. lin, or lln + 1. x°... 7m, or 7n +11| 13n, orl38n+ 1. PRA Forms Forms. at eae Sd as sa 17n, or 17n + 1. = fh pee | wh} Toh whe hee heek 19x, or 19% + 1. reg Wl er Wo te en ‘ (oe yn tare t tina dee 23n, Or 23n + 1. ead ee fs RE oa bs pe ay! ea aad coe mae And generally if 2” be any power, then if m + 1 be a prime number az” is of the form (m +1), or (m+1)n+1 and if 2m + 1 be a prime, then x” is of the form (2m-+-1) n, or(2m+1)n}. That is, when divided by 2m + 1, the re- mainder will either be 0, or + 1. 10. ‘The nth differences of any number of consecutive nth powers is constant, and equal to 1.2.3.4, ke. n. be any conse- Thusilet 1 5 ao Ae yiire Square. Ist diff. 3 3 ‘is 2d diff. 2 2 Pe at BR ' , § consecutive Again1l 8 27 64 1254 culsia, ist. diff. 7. 19) 87° V61 2d diff. 12 18 24 3d diff. 6 6 Pim | i BES And in the same manner the 4th difference of 4th powers = 1.2.3.4 24 of Sth powers = 1.2.3.4.5—= 120 and so on for any higher powers. See the demonstration of these and several other cu- rious properties of powers in chap. vi. Bar- low’s Theory of Numbers. Power of the Hyperbola, is the 4th power of its conjugate axis. Power, in Mechanics, denotes some force which, being applied to a machine, tends to produce motion; whether it does actually pro- duce it or not. In the former case, itis called a moving power; in the latter, a sustaining power. Power is also used in Mechanics, for any of the six simple machines, viz. the lever, the ( balance, the screw, the wheel and axle, the | wedge, and the pulley. PowER of a Glass, in Optics, is by some used for the distance between the convexity — and the solar focus. PRACTICE, is an arithmetical rule, prin- | cipally employed in those questions in which | the amount of a certain number of things is . required, the price of each being given; be-_ ing a more ready and expeditious method — than Compound Multiplication; by which rule _ the same questions may always be resolved, Practice is commonly divided into several cases, which by some authors are so much multiplied, as to become very burdensome to | the memory, an inconvenience that more than counterbalances the advantages arising from this subdivision: in fact, the whole of the - cases that are worth retaining, may be classed under the following heads: 1. When the price is less than a penny. 2. When the price is less than a shilling. 3. When the price is less than a pound. 4. When the price is more than a pound. And the general rule for all these cases is this. PRE Rule. Take such aliquot parts of the given imber of things, as the given price is of the xt superior denomination. Note. In the last case, multiply first by the unber of pounds; and for the shillings, pence, id farthings, proceed by the above rule, and ld the result to the preceding product. EXAMPLES. = 4/4643 at id. 4d. = $/5648 at 42d. = 4/23214 Id, — HIS882 8 1160 Id, —4| 235 4 12/34824 117 8 2,0/223,5 8 2,0) 29,0 24 £111 15 8 Ans, £.14 10 24 Ans. 5s. — 34186 at 6s. 72d. 1s. = 11046 10 6d. —1] 209 6 lid. =1| 104 13 2 33 £.1386 12 3 Ans. 6s. 8d. = 4, 7416 at 31. Gs. 8d. £.24720 Ans. The same method may be employed in ights and measures of every description, mugh the rule is generally limited to money cerns. ?RECESSION of the Equinoxes, in Astro- ny, is a slow retrograde motion of the equi- stial points, viz. from east to west, or con- ry to the order of the signs. ‘The sun every w crosses the equator in two points, which called the equinoctial points; and as it is ural to compute the course of the year from ‘moment when the sun is in one of those nts which gives equal days and nights to parts of the globe, it became very early an dortant problem in practical astronomy to ermine them with some degree of accuracy ; ich was by no means difficult, as it did not uire that accurate kind of observation that io indispensable in many other cases. It ynly necessary for this purpose, to observe sun’s declination on the noon of two or 2e days before and after the time of the inoxes. On two consecutive days of this nber, his declination must have changed n north to south, or from south to north. iis declination on one day was observed to 21’ north, and on the next 5 south, it fol- s that his declination was nothing, or that Was in the equinoctial point about twenty- 2€ minutes after seven in the morning of second day. Knowing then the precise ments, and the rate of the sun’s motion in ecliptic, it is easy to ascertain the precise nt of the ecliptic in which the equator in- sected it. sy a series of such observations made at Xandria, between the years 161 and 127 . PRE before Christ, Hipparchus, the prince of as- tronomers, found that the point of the autum- nal equinox was about six degrees to the east- ward of the star called Spica Virginis. Eager to determine every thing by multiplied obser- vations, he examined all the Chaldzean, Egyp- tian, and other records, to which his travels could procure him access for observations of the same kind; but he does not mention his having found any. He discovered, however, some observations of Aristillus and Timo- chares, made about one hundred and fifty years before. From these it appeared evident, that the point of the autumnal equinox was then about eight degrees east of the same star. He discusses these observations with great sagacity and rigour; and, on their authority, he asserts that the equinoctial points are not fixed in the heavens, but move to the west- ward about a degree in seventy-five years, or somewhat less. This motion he called the precession of the equinoxes, because by it the time and place of the sun’s equinoctial station precedes the usual calculations; a fact which is fully con- firmed by all subsequent observations. In 1750, the autumnal equinox was observed to be 20° 21’ westward of Spica Virginis. Sup- posing the motion to have been uniform dur- ing this period of ages, it follows that the annual precession is about 50"4; that is, if the celestial equator cut the ecliptic in a par- ticular point on any day of this year, it will on the same day of the following year cut it in a point 504” to the west of it, and the sun will come to the equinox 20/23” before he has completed his circle of the heavens. Thus the equinoctial or tropical year, or true year of seasons, is so much shorter than the revo- lution of the sun or the sidereal year. It is this discovery that has chiefly immor- talised the name of Hipparchus, though all his astronomical researches have been con- ducted with the same sagacity and intelli- gence. It should be observed, that since the equator changes its position, and the equator is only an imaginary circle, equidistant from the two poles or extremities of the axis; these poles and this axis must equally change their positions. And. since the equinoctial points make a complete revolution in about 25745 years, the equator being, during that time, in- clined to the ecliptic in nearly the same angle. Therefore the poles of this diu:mal revolution must describe a circle round the poles of the ecliptic, at the distance of about 234 degrees in 25745 years; and in the time of 'limochares, the north pole of the heavens must have been thirty degrees eastward of where it now is. It has been said that Hipparchus was not the original discoverer of this retrograde mo- tion of the equinoctial points, but that it was known to the Indians, Chinese, and the Egyptians, long before his time; indeed it is certain, that he had some idea of it, and had even stated the quantity of it, yet there is no reason to conclude that Hipparchus borrowed his notion of it from them; but, on the contrary, there is every reason to suppose PRE that his knowledge of it arose out of the com- parison of his own observation with those of more early astronomers. This motion of the stars in antecedentia was jong considered as an inexplicable mystery in the science of astronomy; but it is now as clearly understood and accounted for, as any of the other celestial phenomena. It is well known, that while the earth re- volves round the sun from west to east, in the plane of the ecliptic, in-the course of a year, it turns round its own axis from west to east in 235 56’ 4”, which axis is inclined to this plane in ‘an angle of nearly 23° 28’: and that this axis turns round a line perpendicular to the ecliptic in 25745 years from east to west, keeping nearly the same inclination to the ecliptic.—By this means, its pole in the sphere of the starry heavens describes’a circle round the pole of the ecliptic at the distance of 23° 28’ nearly. The consequence of this must be, that the terrestrial equator, when produced to the sphere of the starry heavens, will cut the ecliptic in two opposite points, through which the sun must pass when he makes the day and night equal; and that these points must shift to: the westward, at the rate of 503 se- conds annually, which is the precession of the equinoxes. But as to the physical causes of this altera- tion in the position of the earth’s axis, we cannot in this place enter into any detailed calculation; all we can do is, to give a popu- lar view of the subject, showing it to depend upon the protuberance of the terrestrial equi- torial regions, and the inclination of the earth’s axis to the plane of the ecliptic, and the con- sequent inequalities in the action of gravity on those parts. Newton’s illustration and computation, which though now admitted to be in some points erroneous, is still better calculated for giving a general view of the theory, to those who are unacquainted with the subject, than the more abstruse calculations of the authors quoted in the subsequent part of this article; we shall, therefore, adopt the principles of Newton, and refer the reader who is desirous of entering into the depth of this problem, to the works of the other authors mentioned below. Let S be the sun, 1x E the earth, and M the moon, movingin w, the orbit NMCDz, whichcutstheplane — |}, of the ecliptic in the line of the nodes Na, and has one half raised above it, as represented in the figure, the other halt being hid be- low the ecliptic. Suppose this orbit folded down; it will coincide with the ecliptic in the circle / N mcdn. Let EX represent the axis of this PRE i orbit perpendicular to its plane, and therefo inclined to the ecliptic. Since the moon vitates to the sun in the direction MS, w is all above the ecliptic, it is plain that gravitation has a tendency to draw the me towards the ecliptic. Suppose this force’ be such that it would draw the moon dow; from M toi in the time that she would hay moved from M to ¢, in the tangent to | orbit. By the combination of these motion the moon will desert her orbit, and describ the line Mr, which makes the diagonal of th parallelogram ; and if no farther action of th sun be supposed, she will describe anothe orbit M3n', lying between the orbit MC ) and the ecliptic, and she will come to : ecliptic, and pass through it in a point 4 nearer to M than xz is, which was the formy place of her descending node. By this cham of orbit, the line EX will no ionger be pendicular to it; but there will be ano line Ea, which will now be perpendicular the new orbit. Also the moon, moving fr M to 7, does not move as if she had com from the ascending node N, but from a po N lying beyond it; and the line of the nod of the orbit in this new position is N’ n’. Al the angle M N’m is less than the angle M Na Thus the nodes shift their places in a diz tion opposite to that of her motion, or m¢ to the westward; the axis of the orbit chang its position, and the orbit itself changes” orbit, according to the position of the line move to the eastward. But, in general, 1 into quadrature, after which it diminishes after the quadratures, and while the m recess exceeds the advance in every reve What has been said of one moon woule 0 compose a flexible ring, which would & inclination to the ecliptic. These moment the nodes. Sometimes the inclination of inclination increases from the time that the nodes are again in syzigee. The | passes from quadrature to the node, and tion of the moon round the earth, and, on true of each of a continued ring of mo be flat, but waved according to the differe = | changes are different in different parts of orbit is increased, and sometimes the ne nodes are in the line of syzigee, till they advance only while they are in the octa recede in all other situations. Therefore whole, they recede. surrounding the earth, and they would t forces acting on its different parts. pose these moons to cohere, and to fon rigid and flat ring, nothing would remail this ring but the excess of the contrary” dencies of its different parts. 7Its axis wi be perpendicular to its plane, and its posi in any moment will be the mean positie all the axis of the orbits of each part of flexible ring ; therefore the nodes of this? ring will continually recede, except wher plane of the ring passes through the sun; is, when the nodes are in syzigee: and( Newton) the motion of these nodes wil the same with the mean motion of the PRE ‘the orbit of one moon. ‘The intlination of iis ring to the ecliptic will be equal to the ‘ean inclination of the moon’s orbit during ry one revolution which has the same situa- on of the nodes. It will, therefore, be least > all when the nodes are in quadrature, and ill increase till they are in syzigee, and then iminish till they are again in quadrature. “Suppose this ring to contract in dimensions, ie disturbing forces will diminish in the same portion, and in this proportion will all their fects diminish. Suppose its motion of revo- ition to accelerate, or the time of a revolu- yn to diminish; the linear effects of the dis- bing forces being as the squares of the times f their action, and their angular effects as 1e times, those errors must diminish also on tis account; and we can compute what those ‘rors will be for any diameter of the ring, and r any period of its revolution. We can tell, verefore, what would be the motion of the odes, the change of inclination, and devia- on of the axis, of a ring which would touch 1e surface of the earth, and revolve in twenty- mur hours; nay, we can tell what these mo- ons would be, should this ring adhere to the arth. They must be much less than if the ng were detached; for the disturbing forces f the ring must drag along with it the whole lobe of the earth. The quantity of motion hich the disturbing forces would have pro- uced in the ring alone will now (says Newton) e produced in the whole mass; and therefore 1e velocity must be as much less as the quan- dy of matter is greater: but still all this can e computed. ' Now there is such a ring on the earth: for ae earth is not a sphere, but an elliptical oheroid. Sir Isaac Newton therefore engag- d in a computation of the effects of the isturbing force, and has exhibited a most eautiful example of mathematical investiga- on. He first asserts, that the earth must be h elliptical spheroid, whose polar axis is to $ equatorial diameter, as 229 to 230. Then e demonstrates, that if the sine of the incli- ation of the equator be called x, and if t be ae number of days (sidereal) in a year, the mnual motion of a detached ring will be poi 2 fhe ov¥0 =) He then shows that the ffect of the disturbing force on this ring is to 's effect on the matter of the same ring, dis- ‘ibuted in the form of an elliptical stratum jut still detached) as 5 to 2; therefore the 3./(1—2’) 10¢ r 16’ 16" 24” annually. He then proceeds 9 show, that the quantity of motion in the here is to that in an equatorial ring revolv- ag in the same time, as the matter in the here to the matter in the ring, and as three imes the square of a quadrantal arch to two quares of a diameter, jointly: then he shows, hat the quantity of matter in the terrestrial phere is to that in the protuberant matter of he spheroid, as 52900 to 461 (supposing all 10tion of the nodes will be 360° x “ar at homogeneous). From these premises it fol- lows, that the motion of 16’ 16” 24'", must be diminished in the ratio of 10717 to 100, which reduces it to 9” 07” annually. And this (he says) is the precession of the equinoxes, occa- sioned by the action of the sun; and the rest of the 503’, which is the observed precession, is owing to the action of the moon, nearly five times greater than that of the sun. This appeared a great difficulty: for the phenomena of the tides show that it cannot much exceed twice the sun’s force. Nothing can exceed the ingenuity of this process. Justly does his celebrated and can- did commentator, Daniel Bernoulli, say (in his “‘ Dissertation on the Tides, which shared the prize of the French Academy with Maclaurin and Euler), that Newton saw - through a veil what others could hardly dis- cover with a microscope in the light of the meridian sun. His determination of the form and dimensions of the earth, which is the foundation of the whole process, is not offer- ed as any thing better than a probable guess, inre difficillima ; and it has been since demon- strated with geometrical rigour by Maclaurin. The most elaborate and accurate disserta- tions on the precession of the equinoxes are those of Siverbelle and Warmsley, in the Phil. Trans. and published in 1754; that of Thomas Simpson in his Miscellaneous Tracts ; of Frisius, in the Memoirs of Berlin, and after- wards published with great improvements in his Cosmographia; that of Euler, in the Memoirs of Berlin; and of d’Alembert, in a separate dissertation; and that of La Grange, on the Libration of the Moon, which obtained the prize of the Academy of Sciences, in 1769. See also Laplace’s Mechanique Celeste, and System du Monde; a paper by Dr. Milner, and another by Dr. Robertson, in the Phil. Trans. for 1807. To find the Precession in right Ascension and Declination. Put d= the declination of a star, and a its right ascension ; then their annual variations of pressions will be nearly as follow: viz. 20” 084 x cos. a = the annual precession in declination, and 46” 0619 + 20” ‘084 x sin. a x tang. d= that of right ascension. See the Connoissance des Temps for 1792, p. 206, &e. PRESS, in Practical Mechanics, a machine made of iron or wood, serving to compress bodies close, and therefore into a less com- pass. The common presses consists of six parts or members, viz. two flat smooth plants or boards, between which the compression takes place, two screws or worms fastened to the lower plank and passing through two holes in the upper, and two nuts, serving to drive the upper plank, being moveable, against the lower one which is fixed. Hydrostatic Press. ‘The frame of this is like 2 common press; the bed A (fig. 5, Plate VI.) is fastened to the piston B, of a stout PRE brass barrel D, the lower end of which com- municates by a pipe E, with a forcing pump within the cylindric vessel F; the piston ff of this is cut hollow, and has the connecting rod g jointed within side of it. The lever 6, which works the pump, is jointed to the lower end of this rod, so that the circular motion of the lever is allowed by the connecting rod g¢ moving in or out of the hollow in the piston rod, and the parallelism of the piston is pre- served by a collar H. Fig. 6 and 7 explain the construction of the pump within the cistern F; I is the barrel of brass, this has the piece J screwed into it lower, and this piece J is screwed into the end of the pipe E, and contains within it a valve opening downwards. To the part L of the barrel, the piece K containing a valve open- ing inwards is screwed; this is open to the water, oil, &c. contained in the cistern F, when the lever G is raised the barrel fills with water through the valve K, and when it is pushed down the valve K shuts, J opens, and the water is forced through the pipes E into the large barrel D, and by pushing out its piston B presses the goods laid upon the bed A of the press. When the goods are sufli- ciently pressed the lever G is pushed down, and the lower end of the piston opens the valves J K, and the dettent & pushes towards the piston, opens the valves K, which allows the water to pass back into the reservoir, the bed of the press falls down, the valves are composed of a small brass cone, which ex- actly fits its seat, and is kept in its place by a wire fastened to it; this wire is cut flat on one side to allow the water to pass through when the valve is open, and a small spiral spring closes it. PRESSURE, in Piysies, is properly the action of «a body which makes a continual effort or endeayour to move another body on which it rests; such as the action of a heavy body supported by a horizontal table, and is thus distinguished from percussion or momen- tary force of action. Since action and re- action are equal and contrary, it is obvious that pressure equally relates to both bodies, viz. the one which presses and that which re- ceives the pressure. See a few remarks on the difference between percussion and pres- sure, under the article PERCUSSION. Pressure of Fluids, is of two kinds, viz. of elastic and non-elastic fluids. Pressure of Non-elastic Fluids. The upper surface of a homogeneous heavy fluid in any vessel, or any system of communicating ves- sels, is horizontal. This is a matter of universal experience ; and, as it is easily observed, may be taken for the distinguishing property of fluids. Thus, if ABC DEF (fig. 4, Plate VIII.) be a vessel in which the branches CDH, EFG, have a free communication with the part AB; then if water, or mercury, or wine, or any other fluid commonly reckoned non-elastic, be pour- ed in either at A, C, or E, and when the whole is at rest, the surface of the fluid stands at 1K . raise the lower paris till the whole comes t PRE 3 in the larger trunk; if the line LIKM be drawn parallel to the horizon, the surface o the fluid will stand at L in the branch EF and at M in the branch CD ; and this whateve are the inclinations of those branches, or th angles at F and D, G and H. t This is usually explained by saying, that since the parts of a fluid are easily moveable in any direction, the higher particles will de scend by reason of their superior gravity, ane rest in a horizontal plane. Now what is calle the horizontal plane is, in fact, a portion of spherical surface, whose centre is the centre of the earth: hence it will follow, that if % fluid gravitate towards any centre, it will dis pose itself into a spherical figure, the centr of which is the centre of force. | Prop. If a fluid, considered without weigh is contained in any vessel whatever, and aj orifice being made in the vessel, any pressur whatever be applied thereto, that pressur will be distributed equally in all directions. Through any point N (fig. 4, Plate VII taken at pleasure below the surface of the flui LIKM, imagine the horizontal plane PNO¢ to pass. It is obvious the weight of the flui contained in the vessel below PNOQ contr butes nothing to the support of the column LP,1O, MQ; so that the equilibrium would ob tain in like manner, if the fluid contained in tha part of the vessel below PN OQ had lost it weight entirely. We may, therefore, regar this fluid as being solely a mean of communi cation between the columns LP, 10, MQ in such manner that it will transmit the pre sure resulting from the columns LP, MQ, t the column IO, and reciprocally. If now, in stead of the columns LP, IO, MQ, ofthe fluic pistons were applied to the surfaces P, N, C and @, and were separately urged by pres sures respectively equal to the pressures ¢ the columns LP, 10, MQ, the equilibriur would manifestly obtain in like manner. if a pressure equal that of the column MQ b applied at Q, while the columns LP, IO re main, the equilibrium will still obtain; an this, whatever are the directions of the sever branches, and their sinuosities at D, F, & whence the proposition is evident. Cor. Not only is the pressure transmitte equally in all directions, but it acts perpend cularly upon every point of the surface of th vessel which contains the fiuid. For, if the pressure which acts upon th surface were not exerted perpendicularly, | is easy to see that it could not be entirely a nihilated by the re-action of that surface; th surplus of force would, therefore, occasio fresh action upon the particles of the fluic which must of consequence be transmitted i all directions, and thus necessarily occasion: motion in the fluid; that is, the fluid coul not be at rest in the vessel, which is contra to: experience. Cor. 2. Hence also, if the parts of a flui contained in any vessel ABC D (fig. 5), ope towards the part AB, are solicited by an PRE rees whatever, and remain notwithstanding i equilibrio, these forces must be perpendi- ilar to the surface AB. For the equilibrium ould obtain in like manner, if a cover or piston of the same figure as the surface AB ere applied to it; and it is manifest that in is latter case, the forces which act at the face, or their resultant, must be perpendi- ular to that surface. Cor. 3. If therefore the forces which act yon the particles of the fluid are those of vavity, we shall see that the direction of gra- ty is necessarily perpendicular to the sur- ce of a tranquil fluid; consequently the sur- ce of a heavy fluid must be horizontal to be -equilibrio, whatever may be the figure of e vessel in which it is contained. Cor. 4. If a vessel, as ABCD (fig. 5), closed roughout, except a small orifice O, is full of fluid without weight; then if any pressure applied at O, the resulting pressure on the une surface or bottom CD, will neither de- nd upon the quantity of fluid in the vessel, ron its shape; but since the pressure ap- ed at O, is transmitted equally in all direc- ins, the actual pressure upon C D will be to > pressure at O, as the area of CD is to that ithe orifice. Cor. 5. In the same manner will the pres- re applied at O be exerted in raising the > AB of the vessel; so that if the top be a ne, of which O forms a part, the vertical ssure tending to force AB upwards will to the force applied at O, as the surface B to the areaO. See Hyprostatic Bel- 08. Prop. The pressure of a fluid on the hori- ital base of a vessel in which it is contain- _ is as the base and perpendicular altitude, atever be the figure of the vessel that con- as it; the upper surface of the fluid being »yposed horizontal. Fig. 1. 4et any horizontal plane GH be supposed ‘wn, and conceive the fluid contained in part GCDH of the vessel to be void of ight; then as itis evident, from cor. 3 of foregoing proposition, that any vertical fila- nt whatever, EI of the heavy fluid ABHG, rts at the point I a pressure which is dis- uted equally through the fluid GCDH; | that this pressure acts equally upwards yppose the action of each of the other fila- ats which stand vertically above GH; refore the filament EI alone keeps in equi- io all the other filaments of the mass ‘HB; consequently the mass GC DH be- still supposed without weight, there will result any other pressure on the bottom ‘than that of asingle filament EI; which, ig transmitted equally to all the points of PRI CD, will make the pressure upon CD to that upon the base I of the filament EI, as the area of CD to the areal. If, therefore, we Fig. 3. Gee imagine a heavy fluid contained in ACD B to be divided into horizontal lamine, the upper Jamina will communicate to the bottom CD no other action than would be communicated by the single filament ab; and the same thing obtaining with respect to each lamina, the bottom therefore is pressed in the same de- gree as it would be by the combined opera- tion of the filaments ab, be, ed, &e. Whence, as this pressure is transmitted equally to all points of CD, it will be equal to the product of CD into the sum of the pressures which the filaments ab, bc, cd, are capable of exer- cising on the same point, or it will be propor- tional to CD x (ab + be + ed +-, &c.) Gre- gory’s Mechanics, article 384, &c. Centre of PRESSURE. See CENTRE. PRICE (Dr. Ricuarp), a celebrated Eng- lish mathematician, Fellow of the Royal So- ciety, and of the Academy of Sciences, New England; was born at Tynton, in Glamorgan- shire, in 1723, and died in London, in 1791, in his sixty-eighth year. Dr. Price was au- thor of a variety of works on Morals, Politics, and Religion, beside others on scientific sub- jects; but the only ones we have to record here, and by which indeed he is most distin- guished, are “Observations on Reversionary Payments,” &c. 8vo. 1771. “Appeal on the National Debt,” &c. 8vo. 1773; from which originated the plan of the sinking futtd. “ On Annuities, Assurances, and Population,” 8vo. 1779. “ On the Population of England,” 1780. “Qn the Public Debts, Finances, Loans,” &c. Svo. 1783. A new work on “ Reversionary Payments,” 2 vols. 1783. Besides a variety of papers in the Phil. Trans. on astronomical and philosophical subjects. PRIMARY Planets, are such as revolve about the sun as acentre; such are Mercury, Venus, Terra the Earth, Mars, Vesta, Juno, Pallas, Ceres, Jupiter, Saturn, and Uranus or the Georgium Sidus; being thus called in contradistinction to the secondary planets or satellites, which revolve about their respective primaries. See PLANET. PRIME Numbers, are those which have no divisors, or which cannot be divided into any number of equal integral parts, less than the number of units of which they are composed ; such as 2, 3, 5, 7, 11, 13, 17, &c. These num- bers have formed a subject of investigation and inquiry, from the earliest date down to the present day; and arule for finding them is still amongst the desiderata of mathema- ticians. Eratosthenes invented what he call- ed a sieve for this pupose, because by this he fede sifted away, or separated, those numbers that were not primes from those that were, and by this means ascertained the latter; which is an indirect method, and cannot be applied to a particular case, without a general calculation. The principle of this method, which is the same that has since been employed by mo- dern writers, for ascertaining those numbers, is as follows: Having written down in their proper order all the odd numbers from 1 to any extent re- quired; as, Pe Rea hin oO ae 13. Lb )eaeO 5] 23. 25 27 29.31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61. 63..65.. 67. 69.5 °72.. 931976. 97779 81 83 85 87 89 91 93 95 97 99 ‘ We begin with the first prime number 3, and over every third number, from that place, we put a point, because all those numbers are divisible by 3; as 9, 15, 21, &c. Then from 5, a point is placed over every fifth number, all these being divisible by 5, such are 15, 25, 35, &c. Again, from 7, every 7th number is pointed in the same manner, such as 21, 35, 49, &c. And having done this, all the numbers that now remain without points are prime num- bers; for there is no prime number between 7 and ./100; because if a number cannot be divided by a prime number less than the square root of itself, it is itselfa prime number ; add- ing therefore to the above, the prime number 2, which is the only even prime we have, 9.3.6; 7%, 11, 18,017, 19, 28,98, :31; 87, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, which are all the prime numbers under 100. The method of finding a prime number be- yond a certain limit, by a direct process, is considered one of the most difficult problems in the theory of numbers; which, like the ~ quadrature of the circle, the trisection of an angle, and the duplication of the cube, have engaged the attention of many able mathe- maticians, but without arriving at any satis- factory result. It was generally supposed, that this might be done by some formula, as 42x? 4+.17, 327 + 52 + 19, &c.; in which, by assuming any number whatever for a, the whole formula would be a prime: but it may be demonstrated, that no such formula can exist; still, however, though no rule for find- ing these numbers has yet been discovered, yet many very interesting properties of them have been demonstrated, the principal of which are as follow: 1. If anumber cannot be divided by another number less than the square root of itself, that * number is a prime. PRI -- 2. All prime numbers are of oné of th forms 4n + 1, or 4n — 1; that is, if a prim number be divided by 4, the remainder wi be +1. So alsoall prime numbers are of on of the forms 6n + 1; and various other form may be given, but the converse, as observe above, has not place; namely, that every nun ber in these forms is a prime. wb 3. The number of prime numbers is infinit 4. There cannot be three prime numbers i arithmetical progression, unless their comme difference be divisible by 1 x 2 x 3; orwW less the first of these primes be the number in which case there may be three prime nut bers in such an arithmetical progression; b there cannot be more than 3. 3 5. If n be a prime number, then will 1.2.3.4.5, ken, +1 be divisible by n. 6. Every prime number of the form 4m + is the sum of two squares; as 5 = 2? + 13 = 37 4+ 27, 17 = 4 + 1’, &e.; but a pry number, not of this form, cannot be resoly into two squares. | 7. Every prime number of the form 8” + is of the three forms y? + 2*, y*7 +2 y” — 82". Thus 415? + 4° =—3? +2.4¢=—7—24 And 73228? 4+37=1°74+2.6 —9 —2.2 8. Every prime number 8n + 7, is of } form y* — 22”. Thus 31 —7* —2.37; and 47 —7*—2.] 9. Every prime number 8n + 3, is of | form y* + 22. For example, 11, 19, and 48, are primes this form; and 1137+ 2.1’, 19=1*4 2, and:43*=— 57 ‘+ 2.37. 10. If n be a prime number, and ¢ any nt ber whatever not divisible by »; then will when divided by n, leave the same remain as r divided by n. 11. Supposing still » andr, as in the } ceding case, r”—! — 1 will be divisible by | 12. The square of every prime numbe; the form 4n + 1, is of the form y* + 252 - For the demonstration of these, and vari other properties of prime numbers, see } low’s ‘‘ Elementary Investigations,” &c. As it is frequently useful to know whe a given number be a prime or not, many thors have been induced to form table; those numbers, the most extensive of whi¢ that published in vol. ii. of Vaga’s “ Mathe tical Tables,” in Latin and German, whic carried to 400000; but the most exten table in English, is that givenin Bonnyeas octavo “ Arithmetic.” The following t exhibits all prime numbers under 5000 w will be useful as a reference, in many cas PRI | ‘ 2 | 203 | 677 | 1097 | 1553 3 | 307 | 683 | 1103 | 1559 : 5 | $ll 691 | 1109 | 1567 2.1, 318 |. -7OL .|..1117 |. 1671 ie 21 | 317 | +709 | 1123. | 1579 } 613 | 331 7i9 | 1129 | 1583 17 | 337 727. | 1151 | 1597 19 | 347 | 733 | 1153 | 1601 | 23 | 349 | 739 | 1163 | 1607 29 | 353 | 743 | 1171 |. 1609 31 | 359 | 751 | 1181 | 1613 37 | 367 | . 757 | 1187 | 1619 41 | 373.| 761 | 1193 | 1621 43 | 379 | 769 | 1201 | 1627 47 | 383 | 773 | 1213 | 1637 53 | 389 | 787 | 1217 | 1657 59 | 397 | 797 | 1223 | 1663 61 | 401 809 | 1229 | 1667 67 | 409 | 811 | 1231 | 1669 71 | 419 | 821 | 1237 | 1693 73 | 421 823 | 1249 | 1697 79 | 431 827 | 1259 | 1699 83 | 483 | 829 | 1277 | 1709 89 | 439 | 839 | 1279 | 1721 ‘i. 97 | 443 | 853 | 1283 | 1723 101 | 449 | 857 | 1289 | 1733 103 | 457 | 859 | 1291 | 1741 107 | 461 863 | 1297 | 1747 } 109 | 463 | 877 | 1301 | 1753 113 | 467 | 881 | 1303 | 1459 127 | 479 | 883 | 1307 | 1777 131 | 487 | 887 | 1319 | 1783 137 | 491 907 | 1321 | 1787 139 | 499 | 911 | 1327 | 1789 149 | 503 | 919 | 1361 | 1801 151 | 509 | 929 | 1367 | 1811 157 | 521 937 | 1373 | 1823 163 | 523 | 941 | 1381 | 1831 167 | 541 947 | 1399 | 1847 173 | 547 | 953 | 1409 | 1861 179 | 557 | 967 | 1423 | 1867 481 | 563 |..971 | 1427 }. 1871 191 | 569 | 977 | 1429 | 1873 193 | 571 | 983 | 1433 | 1877 | 197 | 577 | 991 | 1439 | 1879 4 199 | 587 | 997 | 1447 | 1889 211 | 593 | 1009 | 1451 | 1901 223 | 599 | 1013 | 1453 | 1907 227 | 601 | 1019 | 1459 | 1913 229 | 607 | 1021 | 1471 | 1931 233 | 613 | 1031 | 1481 | 1933 239 | 617 | 1033 | 1483 | 1949 241 | 619 | 1039 | 1487 | 1951 251 | 631 | 1049 | 1489 | 1973 257 | 641 | 1051 | 1493 | 1979 963 | 643 | 1061 | 1499 | 1987 269 | 647 |} 1063 |} 1511 | 1993 271 | 653 | 1069 | 1523 | 1997 277 | 659 | 1087 | 1531 } 1999 281 | 661 | 1091 | 1543 | 2003 : 283 | 673 | 1093 | 1549 | 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089 2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293 2297 2309 2311 2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411 2417 2423 2437 2441 2447 2459 2467 2473 TABLE Of Prime Numbers to 5000. 2477 2503 2521 253 1 2539 2543 2549 2051 ° 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663 2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851 2857 2861 2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 2963 2969 2971 2999 3001 3011 3019 3023 3037, 3041 3049 3061 3067 3079 3083 3089 3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221 3229 3251 3253 | 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331 3343 3347 3359 3361 3371 3373 3389 3391 3407 3415 3433 3449 3457 PRI PRI PrimME Vertical, is that vertical circle, or azimuth, which is perpendicular to the meri- dian, and passes through the east and west points of the horizon. PRIME Verticals, in Dialling, or Prime Ver- tical Dials, are those that are projected on the plane of the prime vertical circle, or on a plane parallel to it. These are otherwise call- ed direct, erect, north, or south dials. PRIME of the Moon, is the new moon at her first appearance, for about three days after her change. It means also the GoLDEN Num- - ber, which see. PRIMUM Mobile, in the Ptolemaic Astro- nomy, the ninth or highest sphere of the hea- vens, whose centre is that of the world, and in comparison of which the earth is but a point. This the ancients supposed to contain all other spheres within it, and to give motion to them, turning itself, and all of them, quite round in twenty-four hours. PRINCIPAL, in Arithmetic or in Commerce, is the sum lent upon interest, either simple or compound. See INTEREST. PRINGLE (Sir Joun), a very distinguish- ed physician and philosopher, was born in Roxburghshire in 1707, and took his degree of M.D. at Leyden in 1730; and there pub- lished his “ Dissertatio de Marcore Senili,” in 4to. In 1766 he was elected Presidént of the Royal Society, an honour which he re- signed in 1778, and died in 1782. He was author of numerous and valuable papers in the Philosophical Transactions ; an account of which, with many particulars of his life, may be seen under the article PRINGLE, Dr. Hutton’s Dictionary. PRISM, in Geometry, is a body, or solid, whose two ends are any plain figures which are parallel, equal, and similar; and its sides connecting those ends are parallelograms. Hence, every section parallel to the base, is equal and similar to the base; and the prism may be considered as generated by the paral- lel motion of this plane figure. Prisms receive particular names, according to the figure of their bases; as a triangular prism, a square prism, a pentagonal prism, a hexagonal prism, and soon. And hence the denomination prism comprises also the cube and parallelopipedon, the former ‘being a square prism, and the latter a rectangular one. And even a cylinder may be consider- ed as a round prism, or one that has an infi- nite number of sides. Also a prism is said to be regular or irregular, according as the figure of its end is a regular or an irregular polygon. The axis of a prism, is the line conceived to be drawn lengthways through the middle of it, connecting the centre of one end with that of the other end. Prisms, again, are either right or oblique. A right prism is that whose sides and its axis are perpendicular to its ends, like an upright tower. And An oblique prism, is when the axis and sides are oblique to the ends; so that, when PRI set upon one end, it inclines on one hand, mc than on the other. ' | The principal properties of prisms are, 1, ‘That all prisms are to one another int ratio compounded of their bases and height 2. Similar prisms are to one another int triplicate ratio of their like sides. | 3. A prism is triple of a pyramid of eq base and height; and the solid content of prism is found by multiplying the base by t perpendicular height. : 4. The upright surface of a right prism equal to a rectangle of the same height, | its breadth equal to the perimeter of the bs or end, And therefore such upright surfe of a right prism, is found by multiplying 4 perimeter of the base by the perpendicu height. Also the upright surface of an obliq prism is found by computing those of all parallelogram sides separately, and addi them together. And if to the upright surface be added areas of the two ends, the sum will bet whole surface of the prism. : PRISM, in Optics, is an instrument empl ed for showing the properties of solar lig) and consists merely of a triangular prism glass, which separates the rays of light their passage through it, in consequence the different degrees of refrangibility that } place in the component part of the same ra’ It is, for instance, by means of this inst ment that the origin of colours is shown to owing to the composition which takes pl in the rays of light, each heterogeneous consisting of innumerable rays of differe colours. Thus, a ray being let into a darkened room, through a_ small round aperture z, andfalling on a tri- angular glass prism Pe x, is by the refrac- — tion of the prism a considerably dilated, and will exhibit on 1 opposite wall an oblong image ad, called spectrum, variously coloured, the extremi of which are bounded by semicircles, and sides rectilinear. The colours are commol divided into seven, which, however, h various shades, gradually intermixing at th juncture. Their order, beginning from 1 side of the refracting angle of the prism, red, orange, yellow, grecn, blue, purple, a violet. The obvious conclusion from this € periment is, that the several component pa of solar light have different degrees of } frangibility, and that each subsequent ray: the order above mentioned, is more refft gible than the preceding. ‘a As a circular image would be depicted the solar ray unrefracted by the prism, so ea ray that suffers no dilation by the prism wo inark out a circular image O. Hence, ité pears that the spectrum is composed of inn merable circles of different colours. The mi ture, therefore, is proportionable to the nul i PRI ber of circles mixed together; but all such circles lie between those of two contingent circles, consequently the mixture is propor- tionable to the interval of those centres ; viz. to the breadth of the spectrum. Con- sequently, if the breadth can be diminished, retaining the length of the rectilinear sides, the mixture will be lessened proportionably ; and this is done by the following process: At a considerable distance from the hole z, place a double convex lens A B, whose focal length is equal to half that distance, and place the prism x behind the lens; then at a dis- tance behind the lens, equal to the distance rom the hole, will be formed a spectrum, the ength of whose rectilinear sides is the same is before, but its breadth much less; for the mdiminished breadth was equal to a line tubtending, at the distance of the spectrum rom the hole, an angle equal to the apparent liameter of the sun, together with a line equal 0 the diameter of the hole; but the reduced weadth is equal to the diameter of the hole mly: the image of the hole formed by the ens, at the distance of double its focal length, s equal to the hole; therefore its several mages, in the different kinds of rays, are poe to the same; viz. the breadth of the educed spectrum is equal to the diameter of ae hole. Tt is also known from experiment that a rism placed in an horizontal position will roject the ray into an oblong form, but if nother horizontal prism be applied, similar ) the former, to receive the refracted light ‘merging from the first, and having its re- ‘acting angle turned the contrary way from iat of the former, the light after passing rough both prisms, will assume a circular rm, as if it had not been at all refracted. ut if the light after emerging from the first ‘ism be received on another prism, perpen- enlar to the former, it will be refracted by is into a position inclined to the former, but 3 breadth will remain the same. In order now to show that the different ours suffer no manner of change from any umber of refractions, let there be placed Ose to the prism a perforated board, and let e refracted light transmitted through the le be received on another board parallel to 'e former, and likewise perforated with a aail hole ; and behind this hole place another ism with its refracting angle downwards, id turn the first prism slowly about its axis, be the light will then move up and down the | PRO second board; let the different colours be turned successively, and mark the place of the different coloured rays on the wall after their refraction at the second prism ; it will then be found that the red is seen the lowest, and the violet the highest, and the rest in the intermediate space in their order. From these experiments, aided by some others which our limits will not admit of de- tailing, the following conclusions have been drawn, viz. The solar rays may be resolved into different colored rays; these coloured rays are immutable, either by reflection or refraction. ‘That from the mixture of these coloured rays in due proportion solar light may be produced; and consequently that the differently coloured rays exist in solar light, though when blended together in their natural proportions it exhibits no traces of colour. See CoLour. PRISMOID, a figure resembling a prism. PROBABILITY of an Event, in the doc- trine of Chances, is the ratio of the number of chances by which the event may happen, to the number by which it may both happen and fail. So that, if there be constituted a frac- tion, of which the numerator is the number of chances for the events happening, and the denaminator the number for both happening and failing, that fraction will properly express the value of the probability of the event’s happening. PRopABILITy of Life. See EXPECTATION. PROBABILITIES, the same as CHANCES. PROBLEM, in Geometry, is a proposition wherein some operation or construction is required ; as to divide a line or angle, erect or let fall perpendiculars, &c. See Grome- TRY. PROBLEM, in Algebra, is a question or pro- position which requires some unknown truth to be investigated, and the truth of the dis- covery demonstrated. Kepler’s PROBLEM, in Astronomy. See Krp- LER’S Problem. ProsLem, Determinate, Diophantine, Inde- terminate, Limited, Linear, Local, Plane, Solid, Swsolid, and Unlimited. Sée the adjectives. Deliacal PRos_em, a term anciently applied to the problem of the Dup.icaTion of the Cube. ProsLem of the Three Bodies, is the term by which is denoted the celebrated problem of finding the inequalities of the lunar orbit. This problem in all its generality is as fol- lows: three bodies of given magnitudes, as the sun, the earth, and moon, being projected into space with given velocities, and in given directions, and attracting each other accord- ing to a given law (the inverse ratio of the squares of their distances from each other, and directly as their masses); it is required to determine the nature of the curve that one of them, as the moon, describes about one of the others as the earth. Such is the general state of the problem, but in the case in ques- tion, there are certain conditions which render NN PRO it less difficult, xz. 1. That one of the bodies, the sun, is very great in comparison with the other two, and nearly at rest. 2. Its distance from the other two is so great that it may be considered the same for both. 3. The orbit of the moon about the earth is nearly an el- lipse, and the aberrations from which is there- fore all that is required. The problem even with these limitations is sufficiently difficult, and has engaged the attention of the most celebrated analysts of modern times. PROCLUS, a Greek philosopher and ma- thematician, was born in Lycia, and lived about the year 500. He was the disciple of Syrianus, and had a great share in the friend- ship of the emperor Anastasius. It is said, that when Vitalian laid siege to Constan- tinople, Proclus burnt his ships with large brazen speculums. This philosopher was a pagan, and wrote against the christian religion. — There are still extant his Commentaries on some of Plato’s books, and a tedious com- mentary on the first two books of Euclid’s Elements. 'The latter work has been trans- lated into English by Mr. Thomas Taylor. PROCYON, a fixed star of the second magnitude in the constellation Canis Minor. PRODUCT, in Arithmetic and Algebra, is the quantity arising from the multiplication of two or more factors together. Some of the most remarkable properties of products as to their forms with reference to the factors whence they were produced, may ' be stated as follows: viz. 1. The product of the sum of two numbers by their difference is equal to the difference of their squares; that is, (x + y) (@— y) = x? — y?. 2. The double of the sum of two squares is also the sum of two squares; that is, (+ y)x2®o@t+y? +@—y) Consequently the sum of two squares, mul- tiplied by any power of 2, is the sum of two squares; thus for example: Girth 2? ob Fe iG Muse Ol BF bo bY 10 x 2 =20=— 4 + 27; 40=67 + 2’, ke, 3. The product of the sum of two squares, by the sum of two squares, is itself the sum of two squares; for e eer 2 ty ata tae § (aa! +yy')* +(cy'—a'y)* or 5 = 2% + 1? 13°'='3* +4 2? product 65 = 8* + 1° or 7 + 4’ Thus 4. The product of the sum of four squares, by the sum of four squares, is also the sum of four squares; for (w+ a® + y? + 27) X (Ww? 4 2% 4+ y2%42%= (ww! + aa! +yy! +22’) + (w2'—aw! +y2z'—4'z) + (wy'—a2'—yw' +22") +(w2! -ary’—ya'—zw')* 5. The product of two numbers, or formule re the form x” 4+ ay’, is also the same form; or | 4 B-KRG a | Lay Va!? ay sew Hayy’? + a(ay' yale (x +ay Ya’ +-ay = (ax ayy’? haley! + ye 6. The two formule 2* + y* + z*, and a® + y' + 22’ are such, that the double of either gives the other thus: | aks, 2 ants OF Fey +227 . PFET PORE ty toy) 422% Again, | 2 12, 12 __§ 2a? + 2y” a 4e* — | (a +y +22 x2) atcha PL elf Thus 144—= 3? + 2? + 1? mult.by 2 | _ § (38 +2)? + (8—2) + 2.17 the prod. 28 Sear al 12-04 Again, 1= 3* +2? + 2.1? | mult. by 2 | 30 — § 8+ 2° + B27 42'S 2 52 + 12 + DQ? See Barlow’s ‘Theory of Numbers, p. 180: PROGRESSION, in Arithmetic and AL gebra, a series of numbers advancing or pro- ceeding in the same manner, or according to a certain law, &c. Progression is either arithmetical, or geo- metrical. ‘ Arithmetical PROGRESSION, is a series three or more quantities that have all th same common difference ; as3, 5,7, &c. whic have the common difference 2; and a, a + @ a + 2d,&c. which have all the same dif ference d. : In an arithmetical progression, the chie properties are these: Ist, The sum of any twe terms, is equal to the sum of every other twe that are taken at equal distances from the twe former, and equal to double the middle term when there is one equally distant betweer those two: so, in the series 0, 1, 2, 3, 4, 5, &e.0+6=-14+522+4 4= twice 3 or 2d. The sum of all the terms of any arithme tical progression, is equal to the sum of a many terms of which each is the arithmetica mean between the extremes; or equal to hal the sum of the extremes multiplied by thi number of terms: so, the sum of these te terms 0, 1, 2,3, 4,5, 6, 7, 8, 9, is 2 um 2 x or 9 x 5, which is 45; and the reason of thi will appear by inverting the terms, settin them under the former terms, and adding eae) two together, which will make double th same series ; thus 0, 1, 2, 3, 4, 5, 6, 7, inverted 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, sums 9, 9, 9, 9, 9, 9, 9, 9, 9,9: ri where the double series being the same num ber of nines, or sum of the extremes, the sing] series must be the half of that sum. 3d. Th Jast, or any term, of such a series, is equal t the first term, with the product added of th common difference multiplied by 1 less tha the number of terms, when the series ascen or increases; or the same product subtract when the series descends or decreases: so, ¢ rie 8, 1 PRO the series 1, 2,3, 4, &e. whose common dif- erence is 1, the 50th term is 1 + (1 x 49), rl + 49, that is 50; and of the series 50, 19, 48, &e. the 50th term is 50 —- 1 x 49, or 10 — 49, which is 1. Also, if we denote by tthe least term, = the greatest term, d the common difference, n the number of the terms, and s the sum of the series; hen the principal properties are expressed by hese equations, viz. z—=a-+d.(n—1) a= z—d.(n—1) $= (a+ z)in, $= (z — td.n— 1)n, , = (a + td.n— 1) n. ' Moreover, when the first term a, is 0 or othing; the theorems become z — d(n—1) and s =Lzn, _ See Bonnycastle’s Arithmetic and Algebra. | Geometrical PROGRESSION, a progression in hich the terms have all successively the ime ratio; as 1, 2, 4, 8, 16, &c. where the ymmon ratio is 2, |The general and common property of a 2ometrical progression is, that the product ‘any two terms, or the square of any one ngle term, is equal to the product of every her two terms that are taken at an equal Stance on both sides from the former. So ‘these terms, 1, 2, 4, 8, 16, 32, 64, &e. 1x 64=2x32=4x16=8 x 8 —64. In any geometrical progression, if lenote the least term the greatest term, the common ratio, the number of the terms, the sum of the series, or all the terms ; en any of these quantities may be found m the others, by means of these general lues or equations, viz. 1 — ( ee tapes, -_- a fiex2" =", V4 “~ a A} ye i? “a _log.r + log. z — log. a ae ae log. r ; oy) IA is ea r2— a et" Slt fest r—l When the ‘series is infinite, then the least Tz r—1L n any increasing geometrical progression, Series beginning with 1, the 3d, 5th, 7th, terms will be squares; the 4th, 7th, 10th, cubes; and the 7th will be both a square tacube. Thus in the series 1,7 7’, 7,7, r, 77, 78, 79, &e. 17, 7% 79, 78, are squares ; ‘r®, 79, cubes; and r° both a square and a le, ma is nothing, andthe sums — PRO PROJECTILES, is that branch of me= chanics, which relates to the motion, velocity, range, Ke. of a heavy body projected into void space by any external force, and then left to the free action of gravity, by which it descends to the earth. Till the time of Galileo the true path of a projectile was unknown, it was supposed that its first motion was in a right line in the di- rection of the impelling force, and that it afterwards described a curve; but the na- ture of that curve was wholly unknown. Tartaglia, however, pointed out the error of Supposing any part of the path to be a right line; but it was Galileo who first demon- strated that, on the supposition of gravity acting in parallel lines; the body would de- scribe the curve ofa parabola. This, it is true, is not theoretically correct, for the action of gravity, it is well known, is not made in lines parallel to each other, but in lines perpen- dicular to the earth’s surface : the error, how- ever, is so trifling as to be of no importance, | and this supposition of its parallelism is there- fore universally adopted. There is no branch of mixed mathematics that presents a more elegant and interesting theory than projectiles; but it unfortunately happens, also, that there is none in which theory and practice are more at variance, a circumstance which is principally to be attri- buted to the resistance of the air, which is very considerable, whereas in theory the mo- tion is supposed to take place in void space, or, as it is commonly said, in a non-resisting medium. Lheory of Projectiles in non-resisting Mediums. Prop. l. Ifa heavy body be projected into a non-resisting medium, either parallel or in any way inclined to the horizon, it will by this motion, combined with that resulting from the action of gravity, describe the curve of a parabola. I pret P cit E Sante | A 1 Let a body be projected from A in the di- rection AD, with a uniform velocity, which would carry it over the equal spaces AB, BC, CD, &c. in equal portions of time; and let BE, CF, DG, &e. be the spaces, through which a body would descend by the force of gravity in those times. Then jt follows from the principles of the composition of motion, that the body will be found at the end of these portions of time, in the points E, I’, G; or thatits path will be the curve AB FG Hi passing through those points, NN2 PRO and which is to be demonstrated to be a pa- rabola. Let t, T represent the times the body would be in passing over any two of those spaces AB, AD; then, since the motion in this di- rection is unform, we have AR. 2 AD sce tyr therefore A B* : AD? :: ¢? : T? Ea epee REE iy ey by the laws of falling bodies ; therefore AB*: AD’ :::BE:DG, which is a known property of the parabola. Cor. 1. The velocity in the direction AD being uniform, the horizontal velocity is also uniform, and it is to the former as radius to the cosine of the angle H A D of elevation. Cor. 2. 'The velocity in the direction of gra- vity at any point I’, is to the first projectile velocity, as2CF is to AC. For this velocity would carry the body over 2CF in the same time, by the law of falling bodies, but in the same time the projectile velocity would carry the body over the space AC; therefore the former velocity is to the latter, as 2CF: AC. Cor.3. Let P A be the height due to the first projectile velocity, or that space through which a heavy body must fall to acquire a velocity equal to that of projection; and let DG be any ordinate equal to P A, then the velocity in the direction of gravity at G, will be equal to the projectile velocity at A; butit has been shown that these velocities are to each other as 2DG is to AD, therefore being equal at this point, we have 2DG or 2PA — AD, whence AD? —2DG x 2PA =4PA.DG; that is, A D* is a certain multiple of P A, and since the squares of all these distances, viz. AB’, AC*, AD*, AT’, &c. vary as BE, CF, DG, IH, &c. from what has been already demonstrated, it follows that each of those squares are the same multiple of their cor- responding ordinates, vz. AB? = 4AP.BE AC* =4AP.CF AT? =—=4AP.1H And from this property is readily drawn the geometrical consiruction, for the determina- tion of the range, elevation, &c. of a projec- tile upon either a horizontal or inclined plane. Prop.2. Having given the impetus, or height due to the velocity, and the angle of projec- tion to find the range upon any given plane either parallel or oblique to the horizon. i i Let P A be the height due to the velocity, Al the direction of projection, and AH the given plane. From P draw PQ making the PRO angle APQ = to the angle QAR; take A. — 4AQ and draw H I parallel to P A, s shall AH be the range required. For the triangles AP Q and ATH are si milar, and therefore PA: AQ = fAI:: A. : TH, hence LAT? — PA, IT Hior'Ad* =4PA 1H therefore the parabola passes through H, b cor. 3 above. Cor. If AH be bisected in x and nm bi drawn parallel to TH, it will pass through thy vertex V of the parabola, and will be bisecte in that point by the property of the cura therefore n V = 11H = QR, is the greates! height of the projectile above the plane On ihe same principle may be found th elevation when the range and velocity is give}| or the velocity when the range and elevatio! are known. Prop. 3. Giving the range and impetus 1 find the angle of elevation. On the impetus PA describe a segment ¢ pable of containing the angle ARQ, or PA take AR = 1 AH, and diaw RQ parallel 1 P A, cutting the circle in QQ’, so shall A} or A Q' be the direction required. 4 For since the angle PQA = the angle AR and < PAQ = < AQR because of the pi rallels, it follows that the angle APQ = t angle QAR, andsince AR=4 AH, AQ: 1 AT; therefore A Q is the direction require as is obvious from the preceding proposition Cor. 1. Hence it follows that there a) always two angles of elevation which give tl same range, except when the two points Q) coincide, or when the angle P A H is bisecte| in which case the range is the greatest pc sible. . Cor. 2. Hence also the impetus may || found when the direction and range are give) viz. Draw P A perpendicular to the borizont) line AB; take AR = 1 AH, also dra) RQ parallel to AP, and from Q draw Q) making the angle AQP = the angle AQ) and the point P, where this line cuts the pé pendicular, will give the height due to 1) projectile velocity. The above are the principal geometric properties relating to projectiles, and whi are general both for horizontal and obliq} planes; but the results are obtained me generally from analysis, as follows: - | Let AP the impetus = m; the velocity! projection = v = 2 vgm, the time of flig —t; the decent of gravity in one seco 163, = And call the angle of eclevati BAL=A, the angle of inclination of 1 plane = B, and the sum of these two Hi — C; then since the velocity is uniform, 7 have in the first place AI = tv. Again, As sin. AHI (= cos. B): sin. C 33) (= tv): LH, ; whence IH = C tv=gt—th, hbei cos. B S| the greatest height of the projectile, | cos. B sin. C whence - vy = — tand ¢ = ————. > sin. C® cos. B. g PRO - Again, As sin. AHI (= cos. B): sin. HIA (= cos. A): AIT: AH, cos. A cos. A mhence A H — ———— bane : cos. B cos. B Now substituting in this last expression, for the range, the values of v, ¢, &e, as found above, we have tv COs. : = oe sin. C cos. A (os B cos.A.sin.C »* — cos.A.sin.C ane er hn cos.*B gg cos. *B /* cosa A ee ee And from these we readily draw the follow- ng values of v, ¢, and p. cos. B r or : Sent COS if Se »— J cos.A zt cos. A.sin. C a cos. B 2 cos. B ——,, ¢¢ =<————— Yoh Y4m ie he sin. C ° ith Oo S 2sin. C m__sin,C v__ sin.Cr | < > RE a, ‘ a aia Aa a”) cos. B g cos.Bg cos.A g | sheng Wa. h__cos. B r | bi g cos. Av © sing °C! oF sing we? ~ cos.2B 42” cos.A 4” 4 _And from these formula it is easy to find ‘xpressions for the sine or cosine of the angle if elevation, when these are to be determined. _ When the plane is horizontal these expres- ‘ons become more simple, for in that case os. B = 1, and the angles A and © become ae same, therefore changing sin. C, in the dove, into sin. A, and cos. B into 1, we have w the horizontal plane cos. ; | 908. A.tv=— bs gt” —cos.A.sin. A. Dae sin. A wea vy é Ss. ) —— cos. A.sin.A.4m— 8 A 4h g sin. A - r r t ARE ge aie a : cos. A.t cos. A. sin. A sin. A 2/ah ) = DY mee ) sin. A vee 2sin. A / ™—Sin-A.v_ Ay sin. A.7 pe g g cos.A.¢ aA oun 7 as g cos. A.v Zz BS ain 24 m —Sin.*Av* __ sin. Avr _ gt a Ppa. A Cos. A) Ns And these again by substituting sin. A .cos. : sin. A = + sin. 2A and cos. A red still more simple, and particularly those nich relate to the angle of elevation. ‘We have thus, = tan. A, are ren- ot” : vy J he a = sin. 2A — = sin. 2A.2m an. ‘ ; 2° ; v ae SS — oom v one 4h i get ot 2m mes ge r r r one 2 Ee __2gr “ . vw 2h Tie i which last value of r it is obvious Yi & that the range varies as the sine of the double angle of elevation, and therefore that it is the greatest when this angle is 45°, or when the angle subtended by the plane and im- petus is bisected, and the same js also true in the latter sense, for the oblique plane. 2. And in both cases all angles, equally above and below, that which gives the max- imum range, will give equal ranges, as may be readily deduced from the preceding formule. 3. All things being the same, the range varies, as v, or as the square of the velocity ; as does also the greatest height of the pro- jectile. 4, The time of flight varies as the velocity, or as the square root of the impetus. On this subject see Robin’s “ Gunnery ;” Hutton’s and Simpson’s “ Tracts.” PROJECTION, in Mechanics, the art of giving a body its projectile motion. See PRro- JECTILES. PROJECTION, in Perspective, denotes the appearance or representation of an object on the perspective plane. See PERSPECTIVE. PRoJecTION of the Sphere in Plano, is a representation of the several points or places of the surface of the sphere, and of the circles described. upon it, upon a transparent plane placed between the eye and the sphere, or such as they appear to the eye placed at a given distance. The principal use of the projection of the sphere is in the construction of plani- spheres, maps, and charts, which are said to be of this or that projection, according to the several situations of the eye, and the perspec- tive plane, with regard to the meridians, pa- rallels, and other points or places so repre- sented. The most usual projection of maps of the world, is that on the plane of the meridian, which exhibits a right sphere, the first meri- dian being the horizon. The next is that on the plane of the equator, which has the pole in the centre, and the meridians the radii of a cirele, &e. The projection of the sphere is usually di- vided into orthographic and. stereographic; to which may be added gnomonic. Orthographic PROJECTION, is that in which the surface of the sphere is drawn upon a plane cutting it in the middle; the eye being placed at an infinite distance vertically to one of the hemispheres. , Stereographic PRosEecTION of the Sphere, is that in which the surface and circles of the sphere are drawn upon the plane of a great circle, the eye being in the pole of that circle. Gnomonical PRoJECTION of the Sphere, is that in which the surface of the sphere is drawn upon an external plane commonly touching it, the eye being at the centre of the sphere. Laws of the Orthographie Projection—1. The rays coming from the eye, being at an infinite distance, and making the projection, are parallcl to each other, and perpendicular to the plane of projection. PRO PRO 2. A right line perpendicular to the plane by a plane RS, passing through the centre of projection, is projected into a point where perpendicular to the diameter EH, drawn that line meets the said plane. from E the place of the eye; and let the se ; c tion of the sphere by the plane RS be t £ 4 circle CF DL, whose poles are H and E . Suppose now AG Bisa circle on the spher AL B \ to be projected, whose pole most remote fron 7 B 3 the eye is P; and the visual rays from t at - G| B circle HGB meeting in E, form the cone AGBE, of which the triangle AEB is a i section through the vertex E, and diameter of > the base A B, then will the figure ag bf, whieh 3. A right plane, as AB, or CD, not per- is the projection of the circle AGB, be itselfa pendicular, but either parallel or oblique to circle. Hence the middle of the projected the plane of the projection, is projected into a diameter is the centre of the projected circle, tight line, as EF or GH, and is always com- and this is true whether it be a great circle 01 prehended between the extreme perpendi- @ small one: also the poles and centres of al culars A E and BF, or CG and DH. circles parallel to the plane of projection, fal 4, The projection of the right lime AB is ™ the centre of the projection. _ the greatest, when A B is parallel to the plane 3. The projected diameter of any cirel of the projection. subtends an angle at the eye equal to the 5. Hence it is evident, that a line parallel distance of that circle from its nearest pole to the plane of a projection is projected into taken on the sphere 3 and that angle is biseet a right line equal to itself; but aline that is ed by a right line joining the eye and tha oblique to the plane of projection, is projected pole. Thus, let the plane RS cut the spher into one that is less than itself. 4 6. A plane surface, as ACBD, perpendi- . 5 cular to the plane of the projection, is pro- jected into the right line, as A B, in which it j cuts that plane. Hence it is evident that the ‘4 circle AC BD perpendicular to the plane of | ; projection passing through its centre is pro- jected into that diameter AB, in which it cuts the plane of the projection. Also.any arch as Ce is projected into O 0, equal toca, the right sine of that arch; and the complemental arc cB is projected into o B, the versed sine of the same are cB. _ 7. A circle parallel to the plane of the pro- H FEC through its centreI; and let AB jection, is projected into a circle equal to pe any oblique great circle, whose diamet itself, having its centre the same with the 4 C jg projected into ac; and KOL af centre of the projection, and its radius equal small circle parallel to A BC, whose diameti to the cosine of its distance from the plane. XK L, is projected in kl. The distances of tho And a circle oblique to the plane of the pro- circles from their pole P, being the arcs AHI jection, is projected into an ellipsis, whose KHP, and the angles aEc,kE/, are the a greater axis is equal to the diameter of the les at the eye, subtended by their projects circle, and its less axis equal to double the djameters, ac and kl. Then is the angle al cosine of the obliquity of the circle to a radius SI gay by the arc AHP, and the ang equal to half the greater axis.~ etek kEL measured by the arc K HP, and tho Properties of the Stereographic Projection. ances are bisected by EP. 1. In this projection a circle perpendicular to 3. Any point of a sphere is projected the plane of projection, and passing through the sych a distance from the centre of projecti¢ eye, is projected into a line of halftangents. 2. The projections of all other circles, not as is equal to the tangent of half the arc int passing through the projecting point, whether cepted between that point and the pole op parallel or eblique, are projected into circles. site to the eye, the semi-diameter of A Phus, lee AC EDB represent a sphere, cut sphere being radius. Thus, let CHEB by PRO great circle of the sphere, whose centre is ¢; GH the plane of projection cutting the dia- meter of the sphere in 6 and B; also E and C the poles of the section by that plane, and a the projection of A. Then ca is equal to the tangent of half the arc AC, as is evident by drawing CF — the tangent of half that arc ‘and joining ¢ F. : 4. The angle made by two projected circles is equal to the angle which these circles make on the sphere. For let [A CE and ABL be two circles on a sphere intersecting in A; E the projecting point; and RS the plane of projection, in which the point A is projected in a, in the line IC, the diameter of the circle ACE. Also let DH and FA be tangents to ‘the circles ACE and ABL. Then wil! the projected angle daf be equal to the spherical angle BAC. 5. The distance between the poles of the primitive circle and an oblique circle, is equal to the tangent of half the inclination of those circles; and the distance of their centres is equal to the tangent of their inclination, the ‘semi-diameter of the primitive being radius. For let A C be the diameter of a circle, whose poles are P and Q, and inclined to the plane of projection in the angle AIF; and let a, c,p, be the projections of the points A, C, P; also let HaE be the projected oblique circle, whose centre is g. Now when the plane of projec- tion becomes the primitive circle, whose pole is I, then is Ip = tangent of half the angle ATF, or ofhalf the arch AF; and Ig = tan- gent of A F, or ofthe angle FHa = AIF. 6. If through any given point in the primi- ‘tive circle an oblique circle is described, then ‘the centres of all other oblique circles passing \through that point will be in a right line drawn ‘through the centre of the first oblique circle, ‘and perpendicular to a line passing through PRO primitive circle, and AD ETI a great circle described through D, its centre being B. HK is a right line drawn through B perpendicular to a right line CI passing through D and B and the centre of the primitive circle. Then the centres of all other great circles, as FD H, passing through D, will fall in the line H K, 7. Equal ares of any two great circles of the sphere will be intercepted between two other circles drawn on the sphere through the re- motest poles of those great circles. For let P BEA bea sphere on which AGB and CFD are two great circles, whose remotest poles are E and P; and through these poles let the great circle P BEC and the small circle PGE be drawn, cutting the great circles AGB and CFD in the points B,G, D, F. Then are the intercepted arcs BG and DF equal to one another. 8. If lines are drawn from the projected pole of any great circle, cutting the periphe- ries of the projected circle and plane of pro- jection, the intercepted arcs of those peri- pheries are equal; that is, the arc BG =df. 9. The radius of any lesser circle, whose plane is perpendicular to that of the primitive circle, is equal to the tangent of that lesser circle’s distance from its pole; and the secant of that distance is equal to the distance of the centres of the primitive and lesser circle. For let P be the pole and A B the diameter of a lesser circle, its plane being perpendicular to that of the primitive circle, whose centre is C ; then d being the centre of the projected lesser ‘that centre, the given point, and the centre of circle, da is equal to the tangent of the arc : : ‘the primitive circle. Thus, let GAC E be the PA, and dC = the secant of PA. PRO The above is merely a general view of this subject: they who want to go farther into it may advantageously consult Emerson’s “'Trea- tise on Projection,” that in Robertson’s “‘ Na- vigation,” Bishop Horsley’s ‘“ Elementary Treatises,” and Lacaille’s and Vince’s “ As- tronomy.” There is also a very elegant one in Puissant’s “ Traité de Topographie, d’Ar- pentage, et de Nivellement;” as likewise in Bonnycastle’s “ Trigonometry.” PROLATE, in Geometry, a term applied to a spheroid produced by the revolution of a semi-ellipsis about its transverse diameter ; and is thus distinguished from an oblate sphere, which is produced by the revolution of the ellipse about its conjugate diameter. PROPORTION, in Mathematics, is an equality or similitude of ratio; thus if the ratio a to b is the same as that of ¢ to d; that ese AAO tod which is denoted by placing the quantities thus; a: b::c¢:d,ora:b =e: 4d, and is read as ais to b, so isc tod. Euclid, in the 5th definition of his 5th book, gives a general definition of four proportionals, or when, of four terms, the first has the same ratio to the 2d, as the 3d has to the 4th, viz. When any equimultiples whatever of the first and third being taken, and any equimultiples whatever of the 2d and 4th; if the multiple of the first be less than that of the 2d, the mul- tiple of the 3d is also less than that of the 4th; or if the multiple of the 1st be equal to that of the 2d, the multiple of the 3d is also equal to that of the 4th; or if the multiple of the 1st be greater than that of the 2d, the multiple of the 3d is also greater than that of the 4th. And this definition is general for all kinds of magnitudes or quantities whatever. Also, in the 7th book, Euclid gives another definition of proportionals, viz. when the first is the same equimultiple of the 2d, as the 3d is of the 4th, or the same part or parts of it. But this definition appertains only to numbers and commensurable quantities. See Ratio. PROPORTION, though sometimes confounded with ratio, differs from it in this, that ratio has only a relation to two quantities of the same kind, whereas proportion relates to the comparison of two such ratios. Proportion differs also from progression, in this, thatin the former it is only required that there should be an equality between the ratio of the Ist and 2nd term, and that of the 3d and 4th, whereas to constitute a progression there must be the same ratio between each two adjacent terms, these two cases, however, are sometimes distinguished by the terms discrete and continued proportion. PROPORTION, is also direct and inverse, or reciprocal, alternate, &c. Thus if the ratio of a to 6 is equal to the ratio ¢ to d, then, . is, if , then a, b, c, d are in proportion, direct a:bs:e:d inversion Ob:a::d:c altemate a:c::b:d PRO composition a+ 6: b:: conversion a+6:a:: e+d:d ~f c+di:ic Ke §a—b:a::c—dic fi ta—b:b::0e—did d PROPORTION, is again distinguished into” arithmetical, geometrical, and harmonical. Arithmetical Proportion, is the equality of two arithmetical ratios or differences. As in the numbers 12, 9,6. where the diiference between 12 and 9, is the same as the differ- ence between 9 and 6, viz. 3. ' And here the sum of the extreme terms is equal to the sum of the means, or to double the single mean when there is but one. As 12+6=—94+9-=18. Geometrical PROPORTION, is the equality between two geometrical ratios, or between the quotients of the terms. See the preceding: article. Harmonical Proportion, is when the first term is to the third, as the difference between the Ist and 2d is to the difference between the 2d and 3d; or in four terms when the 1st is. to the 4th, as the difference between the Ist and 2d is to the difference between the 3d. and 4th; or the reciprocals of an arithmetical proportion are in harmonical proportion. As, 6,4,3; because 6:33: 6—4—-2:4—3=—% or because Z, }, 3, are in arithmetical propor- tion, making £+%3=141=14. Also the four 24, 16, 12, 9 are in harmonical proportion, because 24 :9::8:3. Compass of PRopoRTION, a name by which the French, and some English authors, call the sector. x4 Rule of PRoportion. See Rue of Three. : PROPORTIONAL, relating to proportion, as proportional compasses, parts, scales, spvral, &c. for which see the respective terms. | PROPORTIONAL also denotes one of the terms of a proportion, which receives particular de- nominations according to the place it holds in the proportion, as a mean proportional, a third, fourth, &c. proportional. Mean PRoportionaL, is the middle term of three continued geometrical proportionals. See Mran Proportional. Fourth PRoportionat, is the fouth term of, a geometrical proportion, which is found arithmetically by dividing the product of the second and third terms by the first. division \ Lo find a fourth Proportional to three given Lines, A, B, C. 7 i } ly & A B ES PRE D From any point D draw two lines, making any angle GDH. In these lines take DF equal to the first term A, DE equal to the second B, and DH equal to the thirdC. Join FE, and draw HG parallel to it, and DG@ will be the fourth proportional required. That is, . DF (A): DE(B): :DH(C): DG. PTO Third Proportional, is the third of three rms in continued proportion, and is found ithmetically by dividing the square of the cond term by the first. ) find a third Proportional to two given Lines, A and B. | ‘ G. | E A B Cc AR ‘From any point C draw two right lines, iuking an angle CG. In these lines take E equal to the first term A, and CG, CD, ich equal to the second term B. Join ED, d draw GF parallel to it; and CF will be 2 third proportional required. ‘That is, CE (A): EG (B):: CD (B): CF. ‘To cut a Line in extreme and mean Propor- in; thatis, so that the whole line may be to > greater part, as the greater part is to the t Let AB be the given line, d draw BC perpendicular to and equal to halfof it. I'rom » centre C with radius CB scribe the circle D BF ; join ‘C, and with AD as radius id A as a centre, describe the + DE cutting AB in E, so iil the line A B be divided in ‘treme aud mean proportion ‘the point E. aud PROPOSITION, in Mathematics, is either ne truth advanced, which is to be demon- rated, or some Operation proposed which is be performed and shown to be that which fs required. Being, in the former case, led a theorem, and in the latter a problem. PROSTHAPHERESIS, is the saine as the UATION of the Centre. PROTRACTOR, is the name of an instru- int used for protracting or laying down on Foer the angles of a field or other figure. e protractor is a small semicircle of brass, ‘other solid matter, the limb or circumfer- xe of which is nicely divided into one hun- fd and eighty degrees; it serves not only to w angles on paper, or any plane, but also examine the extent of those already laid wn. For this last purpose let the small point tthe centre of the protractor be placed above f angular point, and make the side coincide fh one of the sides that contain the angle pposed ; then the number of degrees cut off the other side, computing on the pro- ctor, will show the quantity of the angle t was to be measured. *rotractors are now more usually made in form of a parallelogram, and properly gra- ted at the upper edge. PTOLOMAIC, something relating to Pto- ‘ ly. |}?TOLOMY, or Protemy (CLauptvs), a y celebrated geographer, astronomer, and : Fe mathematician, was born in Egypt about the year 70:of the Christian era, and died, it has been said, in the seventy-eighth year of his age, aud in the year of Christ 147. He taucht astronomy at Alexandria in Egypt, where he made many astronomical observations, and composed his other works. Ptolomy has always been reckoned the prince of astronomers among the ancients, and in his works has left an entire body of that science. He has preserved and trans- mitted to us the observations and principal discoveries of the: ancients, and at the same time augmented and enriched them with his own. Efe corrected Hipparchus’s “Catalogue of the fixed Stars; and formed tables, by which the motions of the sun, moon, and planets might be calculated and regulated. He was, indeed, the first who collected the scattered and detached observations of the ancients, and digested them into a system; which he sets forth in his Meyaan Dwvrakic, or Magna Constructio, divided into thirteen books. He adopts and exhibits here the an- cient system of the world, which placed the earth in the centre of the universe; and _ this has been called from him the Ptolomaic Sys- tem, to distinguish it from those of Copernicus and ‘Tycho Brahe. About the year 827, this work was trans- lated by the Arabians into their language, in which it was called Almagestum, by order of one of their kings; and from Arabic into Latin, about 1230, by the encouragement of the Emperor Frederic the Second. There were also other versions from the Arabic into Latin ; and a manuscript of one, done by Gi- rardus Cremonensis, who flourished about the middle of the fourteenth century, which Fabri- cus says, is still extantin the library of AllSouls’ College in Oxford. The Greek text of this work began to be read in Europe in the fif- teenth century; and was first published by Simon Gryneeus, at Basil, 1538, in folio, with the eleven books of commentaries by Theon, who flourished at Alexandria in the reign of the elder Theodosius. In 1541 it was reprinted at Basil, with a Latin version by George Tra- pezond; and again at the same place in 1551, with the addition of other works of Ptolomy, and Latin versions by Camerarius. We learn from Kepler, that this last edition was that used by Tycho. Ptolomy was also author of another impor- tant work; wiz. his “Geography,” in seven books; in which the places are laid down ac- cording to their latitude and longitude, an improvement in geography which we owe solely to him. The Greek text of this work was published first at Basil in 1533, in 4to.; a Latin version, by Gerard Mercator, at Am- sterdam in 1605, and again in 1618, in folio. We have also some other works of Ptolomy’s of less importance, on Judical Astrology, Chronology, and different pieces on Astro- nomy. His ‘ Chronologica Regum,” and “ De Hypothesibus Planetarum,” was pub- BIULL lished. by Bainbridge, in 4to. 1620. And his “‘ Apparentiz Stellarum Innerrantium,” by Petavius, at Paris, with a Latin version, in 1680, folio; and again in vol. iii. of the works of Dr. Wallis. > PULLEY, in Mechanics, one of the mecha- nical powers, consisting of a small wheel hay- ing a groove around it, and turning on an axis; and hence by means of a rope it is em- ployed to raise weights, or to draw them in any direction. “A System of Pulleys, or Polyspacton, is an assemblage of several pulleys combined toge- ther, some of which are in a block or case, which is fixed ; and others in a moveable block, that rises or falls with the weight. The move- able wheel is called the sheave, or shieve; the axis on which it turns, the gudgeon ; and the fixed piece of wood or brass, in which the pul- ley is put, the block. Mechanical properties of the Pulley.—tif the equal weights W and —— P (as in the second figure adjacent) hang by the cord BB upon the pulley A, whose block 4} is fixed to the beam H, they will counterpoise each other justinthe same |, manner as if the cord ©) was cut in the mid- ica dle, and its two ends 4 [| jo hung upon hooks fix- i ed in the pulley at Ps» A and A, equally dis- tant from its centre. Hence, a single pul- Ww Jey, if the lines of direction of the power and the weight be tangents to the periphery, : neither assists nor impedes the power, but only changes its direction. The use of this pulley, therefore, is when the vertical direction of a power is to be changed into an horizontal one; or an ascend- ing direction into a descending one; and on the contrary. This is found a good provision for the safety of the workmen employed in drawing with the pulley. This change of direction, by means of a pul- ley, has this farther advantage; that if any power can exert more force in one direction than another, we are here able to employ it in its greatest force. Thus, e.gr. a horse cannot draw in a verti- cal direction, but draws with all its advantage in a horizontal one. By changing the vertical draught, therefore, into a horizontal one, a ‘horse becomes qualified. to raise a weight. But the grand use of the pulley is, where se- veral of them are combined; thus forming what Vitruvius, and others after him, call polyspacta; the advantages of which are, that the machine takes up but little room, is easily removed, and raises a very great weight with a moderate force. PUL If a weight W hangs at the lower end the moveable block p of the pulley D, and 4 cord GF goes under the pulley, it is pla that the half G of the cord bears one-half the weight W, and the half F the other; f they bear the whole between them. Ther fore, whatever holds the upper end of eith rope, sustains one-half of the weight; and: the cord at F be drawn up so as to raise ft] pulley D to C, the cord will then be extend to its whole length, except that part whic goes under the pulley; and consequently tl power that draws the cord will have mow twice as far as the pulley D with its weig W rises; on which account, a power who intensity is equal to one-half of the weigl will be able to support it, because if the pow moves (by means of a small addition) its | city will be double the velocity of the weigh as may be seen by putting the cord overt fixed pulley C (which only changes the dir tion of the power, without giving any “ct tage to it), and hanging on the weight ' which is equal only to one-half of the weig W; in which case there will be an equi brium, and a little addition to P will cause to descend, and raise W through a spa’ equal to one-half of that through which P ¢ scends. J Hence, the advantage gained will be alwa equal to twice the number of pulleys in t moveable or undermost block. So th when the upper or fix- S——— . ed block wu contains two pulleys, which only turn on their axes, and the lower or move- able block U contains two pulleys, which not only turn upon their axes, but also rise with the block and weight, *£ the advantage gained by this is as four to the working power. Thus, if one end of the rope KMOQ be fixed to a aN) hook at I, and the UN rope passes over the ¥ pulleys N and R, and under the pulleys L { wd andP,andhasaweight » r T of one pound hung to its other end at this weight will balance and support a weis W of four pounds, hanging by a hook at 1 moveable block U, allowing the said block a part of the weight. And if as much mi power be added, as is sufficient to overeo the friction of the pulleys, the power 14 descend with four times as much velocity the weight rises, and vonsequently thro four times as much space. ‘The two pull in the fixed block X, and the two in the mo able block Y, are in the same case with th last mentioned ; and those in the lower bl give the same advantage to the power. PUL _ Another very useful combination of pulleys s that in which a weight SS w force at P will balance mother at W of sixteen mes its magnitude. Ifa ower move a weight by * neans of several pulleys, he space passed over by he power will be to the space passed over by the weight, as the weight to he power. Hence, the smaller the ‘orce that sustains a weight 4 »y means of pulleys is, the prt slower is the weight raised : (1 so that what is saved in 4 orce, is spent in time. The common methods of wranging pulleys in their seed ylocks may be reduced to two. The first consists in placing them one by the side of another, upon the same pin; the other, in dlacing them directly under one another, upon separate pins. Each of these methods is liable ‘0 inconvenience. Mr. Smeaton, in order to woid the impediments to which these com- yimations are subject, proposes to combine hese two methods in one. Accordingly, the ulleys are placed in each block in two tiers ; several being upon the same pin, as in the irst method, and every one having another mder it, as in the second; and so that, when he tackle is in use, the two tiers that are the ‘emotest from one another, are so much larger n diameter than those that are nearest, as to Mlow the lines of the former to go over the ines of the latter without rubbing. From this vonstruction arises a new method of reeving he line upon the shieves; for here, whatever ye the number of shieves, the fall of the tackle vill be always upon the middle shieve, or on hat next to the middle, according as the aumber of pulleys in each pin is odd or even. fo do this, the line is fixed to some conve- tient part of the upper block, and brought ound the middle shieve of the larger tier of he under block, from thence round one of the same sort next to the centre one of the ipper block, and so on, till the line comes to he outside shieve, where the last line of the arger tier falls upon the first shieve of the Maller, and being reeved round those, till it omes at the opposite side, the line from the ast shieve of the smaller tier again rises to he first of the larger, whence it is conducted ound, till it ends on the middle shieve of the ipper block on the larger tier. As a system of pulleys has no great weight, nd lies ina small compass, itis easily carried bout, and can be. applied for raising weights, & a great many cases, where other engines annot be used. But they are subject to a reat deal of friction, on the following ac- ounts ; viz. Ist, because the diameters of their is bear a very considerable proportion to heir own diameters; 2d, because in working oe wi 4 \ ‘i PUM they are apt to rub against one another, or against the sides of the block; 3dly, because of the stiffness of the rope that goes over and under them. PUMP, an hydraulic engine used for raising water sometimes by means of the pressure of the atmosphere, sometimes by forcing, and at others by a combination of both. Machines for raising water, it is pretty cer- tain, must have been employed from a very early date; but the pump, as now constructed by the moderns, was not known to the ancient Greeks or Romans, nor has any thing ap- proximating towards it been discovered in any of the remote parts which the enterprising spirit of Europeans has of late years ex- plored on the continents of Asia, America, &c. Vitruvius ascribes the invention of the pump to Ctesibius of Athens, or as some say of Alexandria, about 120 years before Christ; but the pump of this Grecian is still very dif- ferent from those now in common use, which may be divided into three kinds, viz. the forc- ing pump, the lfting pump, and the sucking pump, though the two latter are not indeed essentially different from each other. Foreing Pume. This con- sists of a hollow cylinder, E A Cea, called the working p barrel, open at both ends, 7% and having a valve G at the bottom, opening upwards. L/- This cylinder is filled by a solid piston Ef, covered externally with leather or tow, by which means it fits the box of the cylinder ex- _ actly, and allows no water == to escape byits sides. There == is a pipe K HD, which com- 2====s- municates laterally with this = Se— cylinder, and has a valve at some convenient place H, as near as possible to its junction with the cylinder. This valve also opens upwards. This pipe, usually called the rising pipe, or main, terminates at the place D, where the water must be delivered. Now suppose this apparatus set into the water, so that the upper end of the cylinder may be under or even with the surface of the water AB; the water will open the valve G, and after fillmg the barrel and lateral pipe, will also open the valve H, and at last stand at an equal height within and without. Now let the piston be put in at the top of the working barrel, and thrust down to K. It will push the water before it. This will shut the valve G, and the water will make its way through the valve H, and fill a part Bd of the rising pipe, equal to the internal capacity of the working barrel. When this downward motion of the piston ceases, the valve H will fall down by its own weight and shut this passage. Now ict the piston be drawn up again: the valve H hinders the water in the rising pipe from returning into the working barre]. But now the valve G is opened by PUM the pressure of the external water, and the water enters and fills the cylinder as the piston rises. When the piston has got to the top, let it be thrust down again: the valve G will again be shut, and the water will be" forced” through the passage at H, and rise along the main, pushing before it the water already there, and will now have its surface at L. Repeating this operation, the water must at last arrive at D, however remote, and the next stroke would raise it still farther; so that during the next rise of the piston the water will be running off by the spout D. Lifting Pump. The simplest form and si- tuation of the lifting pump is represented by the annexed figure. ‘The pump is immersed in the cistern till both the valve G and piston F are under the surface A B of the surround- ing water. By this means the water enters the pump, opening both valves, and finally stands on a level within and without. Now draw up the piston 2 to the surface A. It-must lift up the water which is above it, because the valve in the piston remains shut by its own weight; so that its surface will now be at a. Aa being made equal to AF. In the mean time, the pres- sure ofthe surrounding water A forces it into the working = barrel, through the valve G; and the barrel is now filled with water. Now, let the pis- = ton be pushed down again; the. tale G imme- diately shuts by its own weight, and in oppo- sition to the endeavours which the water in the barrel makes to escape this way. This attempt to compress the water in the, barrel causes it to open the valve F in the piston: or rather, this valve yields to our endeavour to push the piston down through the water in the working barrel. By this means we get the piston to the bottom of the barrel; and it has now above it the whole pillar of water reaching to the height a. Drawing up the piston to the surface A a second time, must lift this double column along with it, and its surface now will be at b. The piston may again be thrust down through the water in the barrel, and again drawn up to the surface ; which will raise the water to c. Another re- petition will raise it to d; and it will now show itself at the intended place of delivery. Another repetition will raise it still farther, but while the piston is now descending to make another stroke, the water will be run- ning off through the spout D; and thus a stream will be produced, in some degree con- tinual, but very unequal. 'To remedy this in- convenience it is therefore usual to terminate the main by a cistern LM NO, and to make the spout small. By this means the water brought up by the successive strokes of the pis- ton rises to such a height in this cistern, as to produce an efflux by the spout nearly equable. PUR Sucking Pump. This, as we before ob served, does not differ essentially from the lift ing pump above described, except that in the former the piston F is supposed to work ii the water of the reservoir, which is in m cases very inconvenient on account of thi length of the rod necessary for such a con struction. In the sucking pump therefor the piston is situated considerably above the surface, and in most cases in the very body 0 the pump if this is not more than 32, or 3 feet above the surface of the water in the well Inthisconstruction,sup- . sy pose the piston to be foreed T wi down to Na, then the elas- ticity of air will shut the valve at G, and open that of the piston I’, and a cer- tain quantity of air will es- cape. Now let the piston be drawn up, and the valve F will be shut, and the air in the barrel will expand itself so as to occupy the whole .space, but its elas- ticity being thus diminish- ed, the pressure of the ex- ternal atmosphere on the surface of the wail in the well will open the valve G, and fore into the barrel a quantity of water sufficien to establish an equilibrium, between the it ternal and external air. Let us suppose th water thus admitted rises to S, then repeatin the stroke of the piston, another portion of a will be excluded, and on raising it agail another portion of water will be admitted § as to raise it in the barrel to T; and thus b repeated strokes it will finally arrive at th piston F’, if this do not exceed 32 or 33 fee after which the operation will be precisely th same as in the former case. As to the limit of 33 feet, it follows nece sarily from what has been observed under th articles AiR and ATMOSPHERE, where it shown that a column of water of 35 feet about equal to a column of atmosphere of equi base; and hence it follows that this constru tion can only be made use of when the di tance between the surface of the water an the sucker is within the above limit. Air Pump. See Air Pump. PUNCHEON, a measure for liquors co taining 84 gallons. PUNCTATED Hyperbola, is that who conjugate oval vanishes into a point. PUNCTUM ex Comparatione, is a ter used by Apollonius to denote either focus the ellipse or hyperbola. See Defini Punctum Duplex Triplex, &c. tions, article CURVE. PURBACH (GEorGE), a very emine’ mathematician and astronomer, was born Purbach upon the confines of Bavaria at Austria in 1423, and died at Vienna in 14 in the 39th year of his age. He had for a pupil the celebrated Regi montanus, between whom and Purbach PUR rict friendship and acquaintance continued 1 the death of the latter; and but for his emature death there is every reason to sup- yse that their joint labour would have been ghly advantageous to astronomy. Purbach was author of a variety of pieces 1 mathematical and philosophical subjects, z. An Introduction to Arithmetic; a Treatise 1 Gnomics or Dialling; Astrolabic Canons id Tables relating to different branches of ftronomy. To which may be added a new table of the ced stars, with the longitude by which each ar had increased since the time of Ptolomy ; ‘whose great work, the Almagest, he gave a wrected Latin edition. He also made great improvements in tri- mometry, having first intreduced the table ‘sines by a decimal division of the radius; : likewise constructed a new Table of the anets, and finally, a New Theory of the anets, which Regiomontanus afterwards tblished, the first of all the works executed his new printing-house. See MULLER. PURSUIT, Curve of, is one generated by e motion of a point which is always directed wards another point also in motion along a ht line, the velocity of the two points bear- x any determinate ratio to each other. B 4 A D » of Pennsylvania, had recourse to a similar pedient; for which reason some gentlemen that colony have ascribed the invention of is excellent instrument to him. The truth vy probably be, that each of these gentle- m discovered the method independent of e another. See Abr. Phil. Trans. vol. viii. 366; also Trans. of the American Society, i. p.21; Appendix, Hutton’s Dictionary. Before we attempt to describe the exact istruction of this instrument, it may be of ne use to the reader to illustrate the prin- les on which it rests, by a less complex ‘we than that of the quadrant itself. Though Ss instrument is Se oe S tally called a qua- mt, it is in fact fact | » tan octant, or 8th t of a circle, as 3R, having a label ‘index BM moy- le about B as a itre, and on this lex, and in the ‘ne direction, is fix- a plane reflecting ror, and on the e BR is placed ‘ther parallel to other side BQ, _ this is silvered y half way up, so that an object O may be n directly through the plain part of the $s, from E the sight vane. Now, an ob- ver wishing, for example, to measure the le subtended at H by the two objects OS, ks through the sight vane at E, and moves ‘index BM about its centre, till the re- ted image of S is seen in the other reflector previously in conjunction with the object as seen through the plain glass; then the ‘le QBM, or the arc QM =! < SHO, as y be thus demonstrated. ince the angle of incidence is equal to the le of reflection, < ABS = < DBG = HBG; therefore BG bisects the angle }D. Again, for the same reason, < CDB < HDG = < ODC; therefore DC bi- is the angle OD B. Now, «4 V (20% —x*) os QUA ; Le arr ~~ / (ax — 2x”) fluxion of the area and the fluent of Here we have yx —a xX circular are whose radius is 4.4, an versed sine x—a /(ax—x’*) which vanishe when = 0, and when 2 =a, it becomes a » semi-circumference, whose radius =4a=t area of two circles of radius = da. Exam. 5. To find the area comprehended between any two right ordinates and the ine tercepted part of the asymptote, and curve an hyperbola. nt! C—Ce I W . Let DEF be an hyperbola, of which the asymptotes are C Mand CN; to find the area EG EH F comprehended between the ordi nates G E.and FH. e Let CG =a, GE—}, GH=z, FH then by the property of the hyperbola, CG GE=CH x HF,orab=a—xa)y, ory = x . a ; and therefore ya = a ; the nal ata ©. of which is ab x hyp. log. (4 + x) which flue however requires a correction, for when « =9, the area = 0, but the above expression, when 2 —=ois ab x hyp. log. of a, therefore the correction — ab x hyp. “log. ofa; that is, th correct fluent which expresses the area, ab x hyp. log. (a +2)—ab x hyp. log. a, ate m] e bed a « 1+ a he area EGFH = ab x hyp. log. If CG andGE each=1; ya = fluent of which is hyp. log. (1 +2) which re quires no correction. | Exam. 6. To find the area of a cycloid. _ ‘ F M A ign Yak R & D L@ Let CAL be a cycloid, AD the axis, ABE the generating eircle, AF a tangent at the vertex, CF parallel toAD. Take any poin P in the arc, and draw P M perpendicular t AM. Then the fluxion of the external are AMP=PM x the fluxion of AM. i) Let AE =,2, AD = 2a, then BE v4 /(2ax — 2) and the fluxion -of BE= ' Q U A Also P B =the are BA; therefore the flux- on of PB = wnat ; and the fluxion / (2ax— 2°) , : Wie | a2 x f PB + BE, or of MhidepiT0 7 CE By herefore the fluxion of the area APM, or ‘ . Pll 7's ys ot te PM x by the fluxion AM = Gis r /(2ax — x7). But the fluent of this fluxion is the same as hat found above for a circle, whose radius is a, and versed sine x; thatis, the area ABE; und therefore when x — 2a, the whole exter- aal area C F A is equal to the area of the semi- uircle ABD. But CD being equal to the semi-circumference A D, the whole rectangle CDAF = four times the semicircle A BD, und consequently the internal area ACD = three times the semicircle A B D ; or the whole wea of the cycloid equal three times the area of its generating circle. Quadratures of Spirals. In spirals, or curves ‘eferred to a centre, as e the spiral CA NR to the centre C; lety any vadius CR, x= BN, the we of a circle described about the centre C, at | any distance CB =a; and C7 another ray in- definitely near CN R. [hen 1 CN xX Nn = 3 ax = CN; and by similar figures C N? : CR’, or a? ty? 2: CNn: gd = CRr, the fluxion of the area described py the revolving ray C R, then the fluent of this, for any particular case, will be the qua- drature of the spiral. _ Thus in the spiral of Archimedes, in which ¢:y::m:na constant ratio, or wherenx = 2 Z nm x ny we have y* =——-; whence CRr = m rn xx \ 7a? ——_j— Qa 2am the fluxion of the area. The 2 263 a ue —*Y which is the ge- 6Gam*> 6a ieral quadrature of the spiral of Archimedes. QUADRILATERAL, a figure of four sides, and therefore of four angles, under which reneral term is included the RECTANGLE, SQUARE, RHomBUS, RHOMBOID, PARALLELO- 3RAM, TRAPEZIUM, and T'RAPEZOID ; for the yroperties of which see the several articles. All quadrilaterals have the following pro- erties. The sum of their four angles is equal ‘o two right angles; andif the sum of each air of opposite angles be equal to two right angles, the figure may be inscribed in a circle otherwise it cannot; and in all such quadri- aterals the sum of the rectangles of the oppo- site sides is equal to the rectangle of the two diagonals. See TRAPEZIUM. QUADRILLION, according to English arithmeticians, is the 4th power of a million ; but according to the French only the square luent of this is (2% Of Na of a million, or the fourth power of 1000. See BILLION. QUADRUPLE, four fold. QUALITY, is defined by Mr. Locke, to be the power in a subject of producing any idea in the mind; thus a snow-ball having the power to produce in us the ideas of white, cold, and round, these powers, as they are in the suow-ball, he calls qualities ; and as they are sensations, or perceptions in our understand- ing, he calls them ideas. It has been demon- strated that every quality that is propagated from a centre, such as light, heat, cold, odour, ke. has its intensity either increased or decreased, in the duplicate ratio of the distances from the centre inversely. So at the double distance from the earth’s centre, or from a luminous or a hot body, the weight, or light, or heat, is but a fourth part, and at three times the distance, is butva ninth. And Newton has laid it down as one of the rules of philosophy, that those qualities which are in- capable of being increased or diminished, and which are found to obtain in all bodies upon which experiments could be tried, are to be esteemed universal qualities of all bodies. QUANTITY, any thing capable of esti- mation or mensuration; or which, being com- pared with another thing of the same kind, may be said to be greater or less than it, equal or unequal to it. Mathematics is the science or doctrine of quantity, which being made up of parts is capable of being. made greater or less. It is increased by addition, and diminished by subtraction; which are therefore the two primary operations that relate to quantity. Hence it is that any quantity may be supposed to enter into alge- braic computations two different ways, which have contrary effects, viz. either as an incre- ment or decrement. A quantity which is to be added is called a positive quantity, and a quantity to be sub- tracted is said to be negative. Quantitics are said to be like or similar, that are of the same denomination; they are represented by the same letter or letters equally repeated; but quantities of different denominations, or re- presented by a different letter or letters, are said to be unlike or dissimilar. A quantity consisting of more than one term is called a compound quantity ; whereas that consisting of one term only is denominated a simple quantity. The quantity of matter in any body is the product of its density into its bulk; or a quan- tity arising from the joint consideration of its magnitude and density ; thus if a body be twice as dense, and takes up twice as much space as another, it will be four times as great. This. quantity of matter is best discoverable by the absolute weight of bodies. The quantity of motion in any body is the factum of the velocity into the mass, or it is a measure arising from the joint consideration of the quantity of matter, and the velocity or the motion of the body, the motion of any whole being the sum or aggregate of the mo- tion in all its several parts. Hence in a body twice as great as another moved with an equal velocity, the quantity of motion is double; if the velocity be double also, the quantity of motion will be quadruple. Hence the quan- tity of motion is the same with what we call momentum or impetus of a moving body. QUART, an English measure of capacity, veing the fourth part of a gallon. QUARTER, the fourth part of a thing. QuarTER, in Weight, is equal to 28ibs. or the fourth part of a hundred weight. QUARTER, in Measure, is 8 bushels corn measure, and in coal measure it is the fourth of a chaldron. QUARTER, in speaking of the Moon’s Age, is a fourth part of one lunation. QuARTER Point, in Navigation, is the fourth Ravdianr Point, or RapiaTine Point, is any point from which rays proceed. RADIATION, shooting forth of rays from a centre. RADICAL Stgn (from radix, root), in Al- gebra, is the character by which the root of a quantity is expressed, and is formed thus ./, while the particular root is indicated by a figure on the left of the sign: thus 7/ a,?/ a,4/ a, &c. denote the square root, cube root, and biquadratic root of the quantity a, or of any other quantity placed under the like signs. When it is a compound quantity whose root is to be expressed, it is put in a parenthesis, and the sign prefixed ; thus, ?/ (a? +67), means the cube root of the sum of a” plus b*, or it is otherwise indicated by a line thus, ?/ a+ + 6? ; the characteristic * is generally omitted in the square root, so that instead of writing 7 a for the square root of a, we merely write a, by which the square root is always to be under- stood. RADII, the plural of Ranptus. . RADIOMETER, a naine sometimes given to the Fore Staff. RADIUS, in Geometry, the semidiameter of a circle, or a right line drawn from the centre to the circumference. It is implied in the definition of a circle, and it is apparent from its construction, that all the radii of the same circle are equal. ‘Phe radius is some- times called, in trigonometry, the sinus totus, or whole sine. The length of the radius of any circle is equal to that ofan are 0f57°2957795 degrees of the same circle. Rapius, in the Higher Geometry. Radius | RAD. part of the measure of one of the principal points, or of the arc intercepted between adjacent points. QUINQUEANGULAR, having five an- les. iat QUINTAL, a weight of 100 pounds in most countries, but in England it is 112Ibs. QUINTILE Aspect, a fifth part of the cir cle, or 72°. QUINTILLION, the fifth power of one million. i QUINTUPLE, five fold. | QUOTIENT, the quantity which arises b dividing one number or quantity by another, See DIvIsION. Ld of the Evoluta, Radius Osculi, called also thé Radius of Coneavity, and the Radius of Curva ture, is the radius of a circle having the same curvature, in a given point of the curve witl that of the curve in that point. See Curva TURE, CURVE, and EvoLure. Rapius, in Optics. See Ray. Raptius Veetor,is used for aright line drawy from the centre of force in any curve in whiel a body is supposed to move by a centripeta force, to that point of the curve where th body is supposed to be. In the elliptica orbit ofa planet let a@ = semi-axis major, ae = distance from the centre to the focus, or e= eccentricity to semi-axis major = 1, v = tru anomaly, and u — eccentric anomaly, then th radius vector r is expressed by either of th following formule r= a (1 + cos. u), orr= a (1 —e?) | 1—ecos.v Rapvtius of Curvature. See CURVATURE. © Ravius Astronomicus, the same as Rapio METER. | RADIX, the same as root, but used in” different sense by different authors; thus w Say radix of a system of logarithms, a systet of notation, &c. meaning the fundamenti quantity on which the system is constructet or by which all the others are compared. Rapix of a System of Logarithms, is thé number which involved to the power, denote by the logarithm of a number is equal to thi number, ‘Thus, under the article logarithr we have shown, if rz = a, that a is the log! rithm of a, and r is called the radix of th system. ‘Phis radix in the Common or Briggs, RAT ogarithms is 10, and in the Naperian or yperbolic Logarithms, it is 2°71828182, Ke. id generally the radix in any system of loga- hins, is that number whose logarithm in at system is unity. Raptx of a System of Notation, is that num- wv which indicates the local value of the ures, and is in all systems represented r aunit and cipher (10), which is ten in the ymmon system, ¢wo in the binary, three in eternary, &c. See NovaTIon. Raptix is also used as a term of comparison ‘tween any finite function and its expansion development; thus we know that me: =l—rtr*?—rs + r+¥— Ke. i+r which case . , is sometimes called the i+r dix of the series 1— rt 7? — 73 + rt — Xe. RAIN, water that descends from the atmos- ere in the form of drops of greater or less nitude. The quantity of rain which have fallen in ferent places have been accurately observ- , and from which it appears that much pends upon local situation. The quantity rain which fell at Paris in the course of a ar, taken at a medium of six years, was 19 inches; and in London the medium antity per annum, for the same number of ars, was 23°001. Much, however, depends on thecheight of the rain-gage from the sur- ie of the earth, more than upon the compara- e altitudes of it with reference to the surface the sea, or any fixed point; the rain-gage on » top of a mountain giving nearly as much that in the plain beneath; whereas, one re placed on the top of a house or church, d another below, give very different quan- es. ‘The following table exhibits the re- ts of several very accurate observations de in the years 1766 and 1767, on three res, one at the bottom of a house, another the top of the same, and a third on West- nster Abbey, the greatest care being taken .t none of the water should evaporate after entered the gage, by passing it through a ‘row tube into a bottle well stopped below. from these results it will appear, that there . below the top of a house above a fifth ‘t more rain than what fell in the same ice above the top of the same house. And t there fell upon Westminster Abbey not ch above half what was found to fall in the ne space below the tops of the houses. ‘This seriment has been repeated in other places th the same result; and notwithstanding cause of this extraordinary difference has : yet been discovered; it is at the same ie useful to be apprised of it, to prevent / inaccurate conclusions from a comparison lifferent gages, RAI The Quantity of Rain which fell in London Srom July 7, 1766, to July 6, 1777. Lower page yottom of a house. Upper gage on Westmin- ster Abbey. Middle gage, Months, top of a house, Inches, 3991 Inches, 2311 Inches. irom July ? 3°210 the 7th to > the 3ist Lugust september October November December January Vebruary Viarch April Vay June July 7. 0.479 0344. 2061 0°842 1:258 1455 2°494 1°303 1°213 1°745 1426 ) 0°309 4 0°558 0:42] 2°364 1-079 1°612 2°071 2861 1:807 1:437 2°432 1:997 0395 22-608 _ 0'508 1416 0-632 0'994 1°085 1°335 0°587 0°994 1142 1145 12099 The following TABLE exhibits similar Evpe- riments, made on two Gages, one onthe Top of Mount Rennig, in Wales, and another in the Plain 1350 Feet below the other. Bottom ofthe| Top of the mountain, mountain. Inches. Inches, From July 6 to 16 0°709 0°648 July 16 to 29 2185 2°124 July _29 to ne hae, 0610 | 0-656 Sep. 9, both bottles had run over From Sept. 9 to 30 3°234 2'464 Oct. 17, bottles run over From Oct. 17 to 22 0°747 0°885 Oct. 22 to 29 1281 1388 8165 _—_ RSet ai et 8:766 TEL TEN EE ES ERY TOTO EEE LET EOS LO IT SE ATES YE OR I These experiments justify the assertion made above, wiz. that the quantity of rain, in any place, depends principally upon its altitude above the surface of the earth, and not much upon the comparative altitude of two places with regard to the surface of sea, and consequently not upon the rarity or den- sity of the atmosphere, as was for a long time supposed. RAINBOW. The rainbow is a circular image of the sun variously coloured. It is thus produced: the solar rays, entering the drops of falling rain, are refracted to their far- ther surfaces, and thence, by one or more re- flections, transmitted to the eye. At their emer- gence from the drop, as well as at their en- trance, they suffer a refraction, by which the RAI rays are separated into their different colours, and thus, therefore, are exhibited to an eye properly placed to receive them. ‘That this is the true account of the formation of the rain- bow appears from the following considera- tions. 1. That a bow is never seen but when * rain is falling, and the sun shining at the same time, and that the sun and bow are always in‘opposite quarters of the heavens ; this every one’s experience can testify. 2. That the same appearance can be artificially represent- ed by means of water thrown into the air, when the spectator is placed in a proper position with his back turned to the sun: LetABbea cg drop of water, p— and C D a pen- cil of solar rays incident there- on; if all the rays of any one colour, as red, belonging to the pencil CD, be refracted to the same point G, and thence re- flected, they will fall on the space RQ, with the same obliquity, and at the same distances from each other, as the refracted rays, if pro- ceeding backward from G, would fall on the space T'S; but these at their refraction would emerge into TD, CS, &c. parallel to each other, the ray G R, GQ, will emerge from the drop parallel to each other, and therefore will enter an eye properly placed copiously enough to cause a sensation; a red colour will there- fore appear in the direction of these rays, and so of others. But if the refracted rays do not meet in the same point, the reflected rays will not fall on the surface, at the same distance from each other, as PT and IS do, though: their obliquity to the sur- face be equal to that of the latter; therefore the refractedrayswillemerge, p diverging from each other, and consequently will not enter the eye copi- ously enough to cause a perception of their colour. It is plain that where the rays of any colour emerge parallel, all these emerging rays will be inclined to the incident rays in the same angle. And by calculation it is found, that the red rays when they emerge parallel to each other, make with the incident rays an _ - ~ ~ - - “we. -- i, Pee. ae? a pate s«- ae Loe ooo ne, er eg RF ay woaee bye angle, ABO of 42° 2’ and the violet an angle, ACO, of 40°17/ and the rays of the other colours, angles greater than the latter, and Jess than the former. Ifthrough the eye which receives the emerg- ing rays, there be drawn a line A X, parallel RAI to the incident rays, it will make with th emerging rays of each colour, angles RAX and VAX, &c. equal to the above. This line, AX, is called the axis of vision. several drops placed in the lines A R, A V,& will exhibit to the eye at A, the several pr matic colours respectively, as appears fro what has been. said; and if those lines supposed to revolve with a conical moti round the axis of vision, it is evident, for th same reason, that all the drops placed in eae] of the conic surfaces so generated, will trans mit the rays of each colour respectively to t eye, and therefore that a number of cireula concentric arches of the prismatic colours aé joining to each other, will be exhibited to th eye. ‘This explanation relates to the interi¢ bow,whose colours, beginning from the outsid are red, orange, &c, as in the prismatic spe trum, which bow can never be seen if the § be elevated more than 42° 2’ above the hor here stated, the line A Q, mark- ing the vertex of a@_ arainbow, would _———— fallentirely below * the horizon. 4 As the interior bow is formed by a flection and two refractions, so the exter, a gs . a bow is formed by two reflections and two} fractions at the surfaces of the drops of 1 in grain. If the red ray of any pencil, C, ofsolar rays,after u Fe refraction, inter- “7S sect each other | | at R, so that \! when reflected “\. 4 at: (E. Viogthey ait : may proceed parallel within the drop, af a second reflection at XQ, they will p ceed to LM, intersecting each other at, equally distant from X Q, as R is from T and as the rays, Q T, X V, if they p ceeded backward, would, after reflect so fall on the surface, N O, as to be refrac; into air parallel to each other; so \ M, Q falling on the surface precisely in the sa circumstance, shall be refracted to the 4 parallel to each other, and therefore will ex it copiously enough to cause a perceptiol their colour, (and so of the rest). The} rays, when emerging parallel after two ref tions, are by calculation found to make ¥ the incident rays, and therefore with the @ of vision, an angle of 50° 57’. The vil rays, when emerging parallel, are found make with their incident rays, and there) with the axis of vision, an angle of 54° the other emerging rays meet the axis of sion in the intermediate angles. And he it is easy to explain the generation of exterior bow in the same manner as tha, the interior. It is to be remarked that order of colours in the exterior bow is reverse of that in the interior, and the reé Cc RAI this appears in the above explanation, Vor £, 3d figure above, which marks the direc- 0 of the violet rays in the outer bow, con- ms with A X, the axis of vision, a greater rle than A D, (which marks the direction of s red rays), contains with the same axis. id the reverse is the case in the interior bow. 's evident, (for a reason similar to that given ithe case of the interior bow) that an ex- jor bow cannot be seen when the elevation ihe ‘sun is above 54° 7, Lunar Ratnepow. The moon sometimes 0 exhibits the phenomenon of an iris by the raction of her rays, in drops of rain in the tht-time. Marine Rainzow, the sea-bow, is a pheno- non sometimes observed in a much agitat- ‘sea, when wind sweeping part of the tops of | waves, carry them aloft, so that the rays of i sun are refracted, &c. as in a common wer. RAIN-GA GE, or PLUVIAMETER, a machine /measuring the quantity of rain that falls. lere are various kinds of rain-gages; that sd at the apartments belonging to the Royal siety at Somerset-house is thus described. e vessel that receives the rain is a conical m, strengthened at the top by a brass ring alye inches in diameter. ‘The sides of the mel and inner lip of the brass ring are in- hed to the horizun, in an angle of more in 65°, and the outer lip is an angle of more in 50°, which are such degrees of steepness t there seems no probability either that train which falls within the funnel, or on inner lip of the ring, shall dash out, or 't which falls on the outer lip shall dash 9 the funnel. The annexed figure repre- its a rain-gage of the best con- iction. It consists of a hollow inder, having within it a cork lattached to a wooden stem which ses through a small opening at (top, on which is placed a large mel. When this instrument is ced in the open air ina free place, rain that falls within the circum- mee of the funnel will run down » a tube and cause the cork to t; and the quantity of water in ===> tube may be seen by the height vhich the stem of the float is raised. 2 stem of the float is so graduated as to w by its divisions the number of perpen- ular inches of water which fell on the sur- > of the earth since the last observation. It ardly necessary to observe, that after every ervation the cylinder must be emptied. i very simple rain-gage, and one which wers all practical purposes, consists of a per funnel, the area of whose opening is ictly ten square inches ; this funnel is fixed i botile, and the quantity of rain caught is ertained by multiplying the weight in ices by 173, which gives the depth in inches {parts of an inch. In fixing these gages ® must be taken that the rain may have access to them; hence the tops of build- RAR * ings are usually the best places. When quan- tities of rain collected in them at different places are compared, the instruments ought to be fixed at the same heights above the ground at both places, because, at different heights, the quantities are always different, even at the same place. See Rain. RAM. See ARiEs. RAMPHOID, a particular point of retro- gression. See INFLECTION. RAMUS (Perer), a celebrated French mathematician and philosopher, was born in 1515, and fell a sacrifice to his religious opi- nion, on the massacre of St. Bartholomew’s Day, 1572, in his 57th year. He was author of several works, of which those relating to mathematical subjects are as follow: Scholarum Mathematicarum, libri 31; Arithmetice, libro duo; Algebree, libri duo; Geometricz, libri 27. The latter was published in English by W. Bedwell, 1636, London, 4to. 3 RANGE, in Projectiles, is the distance to which a body is projected. Sce PROJECTILES. RARE, in Piysies, a relative term, the re- verse of dense, being used to denote. a consi- derable porocity, or vacuity between the par- ticles of a body; as the word dense implies a contiguity or closeness of the particles. RAREFACTION, in Piysies, is the mak- ing a body to expand or occupy more room or space without the accession of new matter, lt is by rarefaction that gunpowder takes effect; and to the saire principle also we owe oclipiles, thermometers, &c. ‘The degree to which air is rarefiable exceeds all imagina- tion ; perhaps, indeed, its degree of expan- sion is absolutely beyond all limits. Upon the rarefaction of the air is founded the me- thod of measuring altitudes by the barometer ; in all cases of which the rarity of the air is found to be inversely as the force that com- presses it, or inversely as the weight of all the air above it af any place. The open air, in which we breathe, says Sir Isaac Newton, is 8 or 900 times lighter than water, and by consequence 8 or 900 times rarer. And since the air is compressed by the weight of the incumbent atmosphere, and the density of the air is proportionable to the compressing force, it follows, by compu- tation, that at the height of about seven English miles from the earth, the air is four times rarer than at the surface of the earth; aud at the height of 14 miles, it is 16 times rarer than atthe surface of the earth; and at the height of 21, 28, or 35 miles, it is respec- tively 64, 256, or 1024 times rarer, or there- abouts ; and at the height of 70, or 140, and 210 miles, it is about 1,000,000, 1,000,000,000,000, or 1,000,000,000,000,000,000, &c. Mr. Cotes has found, from experiments made with a thermometer, -that linseed oil is rarefied in the proportion of 40 to 39 with the heat of the human body; in that of to 15 to 14, with that degree of heat wherein water is made to boil; in the proportion of 15 to 13 in that degree of heat wherein melted tin begins . RAT —¢o harden; and finally in the proportion of 23 to 20, in that degree wherein meited tin arrives at a perfect solidity. ‘The same author discovered that the rarefaction of the air, in the same degree of heat, is ten times ereater than that of linseed oil; and the rarefaction of the oil about fifteen times greater than that of the spinit of wine. RARITY, lightness, thinness, the reverse of density. RATIO, is the relation of two quantities of the same kind with respect to quantity, and is by some authors divided into arithmetical and geometrical ratio: viz. arithmetical when the term is used with respect to the difference of the two quantities, and geometrical when it relates to the number of “times in which the one of those quantities is contained -in the other; thus the ratio of 6 to 3 is . ed, Te leading term of the ratio being called the antecedent, and the latter the consequent ; also the quotient or division of the former by the latter, is called the index or exponent of the ratio. ‘The equality of ratios constitute pro- portion. See PROPORTION. Ratio is also distinguished by some authors, (principally of the old school), into a variety of denominations, many of which are totally use- less; but which cannot notwithstanding be passed over in the present article. Trrational Ratio, is when one of the terms of the ratio, at least, is an irrational quantity, such is the ratio of 3 to 1. Rational Ratio, is when there is no irra- tional quantity enters, or when the same irra- tional quantity enters into both terms, thus /6 to /24 — /6to 2,/6 = 1 to 2. Ratio of Equalty, is when the terms ex- pressing the ratio are equal, and therefore the exponent — 1. Ratio of Greater Inequality, is when the antecedent exceeds the consequent; and ratio of less inequality, is when ‘the latter excecds the former. For several other distinctions, as Compound Duplicate, Subduplicate, Ty iplicate, Subtripli- cate, &c. see the several articles. Reduction of Ratios, is the reducing them to less terms, the ratio of 836 to 24 = 6 to4= 3 to 2. Sometimes, however, when the terms of the ratio are very large it is difficult to reduce them in this manner, and then recourse must be had to the method of finding the greatest common measure or common divisor, which also fails if the two terms are prime to each other. In this case though the exact ratio cannot be found in less terms, it is frequently desir- able to find an approximate ratio expressed in less terms, and recourse must then be had to continued fractions; viz. convert the given ratio into a continued fraction, and thence find the series of converging fractions, each of which will be an approximate ratio of the proposed one, and so much the more accurate as it is farther advanced in the series. See CONTINUED Fractions. RAT Let 1103 to 887 be a ratio, to, which required to find an approximate ratio i terms. . oy ‘b 887)1103(1 4 216)887(4 4 23)216(9 oh 9)23(2 . 5)9(1 i 4)5(1 . 1)4(4 Hence by the rule in continued fractions, quoto.l 49 2 1 ] ai 5 2 app. ratiGb. os 5 46 97 143 240 1108 1’ 4 37 78 115’ 193’ 887 Each of which fractions approximates neal to the true ratio than any of those which P cede it, and each nearer than any ratio can be expressed in less terms. if On this subject see Lagrange’s addition Euler’s “ Elements of Algebra.” P| On the doctrine of ratios in general, sce tract published by Dr. Robertson on this ject, in 1789. | RATIONAL Fractions, is the term ¢ monly used to express those fractions wh may be decomposed into other fractions, 7 sum of which is equal to the given tract this cannot in all cases be effected, but wi it can such fractions are called rational, fi tions. In simple numerical fractions, the deet position, when it can be effected, is d from the indeterminate analysis. Thus, I q = Of be proposed to resolve the fraction =i two other fractions. Since 35 = 7.5, let two required fr ctions be 2 and ; then #) } sum ott a ne whence 54 4+7y = which is an indeterminate equation of first degree, and the solution of it gives x; and y = 2; therefore 19_! + = 35 7 9) Ifthe denominator is a prime number, decomposition is impossible, as it is als some other cases; but if x and y are pr factors of the denominator, and the nu rator is greater than xy — «— y, then the composition is always possible. If the denominator consists of three more pe. factors, then making it equa - des Pea — + &c. the decomposition i still td effected by means of an indetermi equation of the first degree. But the principal use of this decomposi is in the inverse method of fluxions, or integral calculus, for which purpose they 1 first ‘investigated by Leibnitz, and have | sincemuch extended bythe researches of C Euler, Simpson, Lagrange, &c. the forme par ticular has treated the subject at le in his “‘ Harmonia Mensurarum,” and E has a very elegant chapter on this subjec RAT “ Analysis Infinitorum,” vol. i. to which ks the reader is referred, as we can only a slight sketch of this theory in the pre- articles. (2™ + ax™—! + ba™—-? + &o.) & a" + pxr—! + ga*—?2 + &e. ny proposed fluxion of which the fluent 1s tired. ‘This, in the first place is the same x” 2 a* + px +: Q2"* >? + XC. m—t1.,, the H x tes meet, px?! + gz"—3 + &os m—2 4 ‘ , bx x | i. ar a rose “1 kets 4- | Sty eid a eC, nd the whole fluent will be equal to the ral fluents of these parts, each of which is aded in the general form | RESEEE OR Ree” hid + px! + G23? + &e. our object is now to decompose this first wr into its simple fractions. ow the denominator of this fraction, from xnown theory of equations, may be con- red as made up of the factors (ew—«) (a— 8) +y) (wx — 9), &e. w, B, y, 8, being the roots le equation arising from making the de- inator equal to zero, and which thus be- 2 known. et us therefore suppose the above frac- to be equal to | sf A B 4. ne AS, 3 au D + &e. -—“e «x — x—y x«w-—d ‘eduction of which to a common denomi- r, and comparing the numerators, gives A («—f) (x—y) (w—9), et X Orex B (x —a) ( — y) (x — 3), Ke. 'C (x— a) (x — 8) (x — 3), &e. ‘making successively « = @, B, y, 2, &e. eh will not alter the value, these being oots of the equation) gives A (a— 8B) (a—y) (2 — 8) Ke. = 1 | B (@—a) (8 —y) (B—3) &e. = 1 C (y —a) (y—B) (y— 2) &e. = 1 &e. &e. 4 | is! 1 | ~ (@—B)(%—y)(w—8) &e. >i Naeger Ba) B—y) (8 — 2) &e. jc} ions ! (—a)(y —B) (y—9) Ke, ice the values of A, B, C, &c. become im, and therefore the general fluxion 1 - | Uy wv aa + gx*=? + ra*—3 + &e. x Oxy x scomposed into the following particular 4 Cat x Toe —— &e. k&— a x—B x«x— ia fluents of which are easily obtained. i ‘ u—1 3. = Oas-le + Qanu-2 xz + &e, oe * we item RAY the fluents of which becomes 1 one t ee + &¢.+ 04-1 hyp. log. (a —a) This method admits of considerable abridg- ment, but the limits of our article will not admit of farther explanation; and we must therefore refer the reader who is desirous of information on this subject, to the “ Analysis Infinitorum” of Euler, as above mentioned ; and to Simpson’s ‘ Fluxions,” vol. i.; and shall in this place merely give one example by way of illustration. Let it be proposed for example to find the fluent of xr 23§— 6a* + lla—6 Here the equation 23 —62? + llxa—6=0, gives x = 1,2, and 3; thatis,a=1, 6=2, y = 3, and the fraction is therefore decom- composed into Ax Ba Cx Ax Be Cx — ———, or-——— — X— o% yer: + x—sy xz—l x— 2 + z—-3 eds; GU: Sy CL ae — 1 — 1: ae BNC Le) ae ; 1 where A = 1 Ae) oy ie therefore ate eens Bate hes Oe Ce heat a3—6x*+4+1lx—6° 2(a—l) 2-2 Wa—3) the fluents of which are = hyp. log. (a — 1) — hyp. log. (a — 2) + § hyp. log. (w — 3), or fluent of x J (a? —44+3) x? —62*+11—6 x—3 RATIONAL Quantities, in Algebra, are those which are expressed without any radical signs, being equivalent to integers, or fractions, in arithmetic, which are called rational numbers, or quantities, in contradistinction to irrational or surd quantities. See SuRDS. RATIONAL Horizon. See Horizon. RAY, in Optics, a beam of light propagated from a radiant point. | If the ray comes direct from the radiant point to the eye itis said to be direct ; if it first strike upon any body and is hence trans- mitted to the eye, it is said to be reflected, and the ray itself is called a reflected ray ; and if the ray in its passage to the eye be bent or turned out of its direct course by passing through any medium, it is, said to be refracted, and is thence called a refracted ray. When two or more rays proceed in directions parallel to each other, they are called parallel rays. If they converge towards each other, they are called converging rays, and if they diverge, diverging rays; and those which pass direcily to the eye in any case are called visual rays. s : Among other qualities of rays it has been found by experiment that there is a very great difference in the heating power of the different rays from the sun. i It appears from the experiments of Dr. — hyp. log. REA Herschel, that the heating power increases from the middle of the spectrum to the red rays, and is greatest beyond it where the rays are invisible. Hence it is inferred that the rays of light and caloric nearly accompany each other, and that the latter are in different proportions in the different coloured rays; these are e asily separated from each other, as when the sun’s rays are transmitted through a transparent body, the rays of light pass on scemingly undiminished, but the rays of ca- loric are intercepted. When the sun’s rays are directed to an opaque body, the rays of light are reflected, and the rays of caloric are absorbed and retained. ‘This is the case with the light of the moon, which; however much it be concentrated, gives no indication of being accompanied with heat. It has also been shown that.the different rays of light produce different chemical effects on ihe metallic salts and oxyds. These effects in- crease on the opposite direction of the spec- trum, from the heating power of the rays. From the middle of the spectrum, towards the violet end, they become more powerful, and produce the greatest effect beyond the visible rays. From these discoveries it ap- pears that the solar rays are of three kinds: 1. Rays which produce heat. 2. Rays which produce colour ; and 3. Rays which deprive me- tallic substances of their oxygen. 'The first set of rays is in greatest abundance, or are most powerful towards the red end of the spectrum, and are least refracted. The second set, or those which illuminate objects, are most powerful in the middle of the spectrum. And the third set produce the greatest effect to- wards the violet end, where the rays are most refracted. ‘The solar rays pass through trans- parent bodies without increasing their tempe- rature. ‘The atmosphere, for instance, re- ceives no increase of temperature, by trans- mitting the sun’s rays, till these rays: are reflected from other bodies, or are commu- nicated to it by bodies which have absorbed them. ‘This is also proved by the sun’s rays being transmitted through convex lenses, producing a high degree of temperature when they are concentrated, but giving no increase of temperature to the glass itself. By this method the heat. which proceeds from the sun can be greatly increased. Indeed, the inten- sity of temperature produced in this way is equal to that of the hottest fur nace. This is done either by reflecting the sun’s rays from a concave polished mirror, or by concentrat- ing or collecting them by the refractive power of convex lenses, and directing the rays thus concentrated on the combustible body. See ‘Burnine Glass. For various other properties of the rays of light, see CoLourR and Prism; see also RE- FLEXION, REFRACTION, &c. REAUMUR (RENE ANTOINE), a celebrated French philosopher, was born at Rochelle in 1683. He was author of several works, but “none that requires any mention in this article. REC , He invented the thermometer which bears } name, a description of which is given unde that article. Reaumur died in.1757. a ReEAuMuUR’sS Thermometer. See THERM METER. | RECEIVER of an Air Pump, is the lay glass vessel placed over the pipe or valve order to be exhausted of air, being thus oa from its being the recipient of those th | on which experiments are made. See A Pump. RECESSION. See Precession. ; RECIPROCAL, in Arithmetic and Algeh is the quotient arising from dividing unity ] . Pe | any quantity; thus1+ = — % is the re y + ) procal of the fraction —-. & y % RECIPROCAL Equations, are those whi contain several pairs of roots, which aret reciprocal of each other. Thus an oquati ; Fi = Cy =, Ke. is call a. reciprocal daisies ; the ieabe of whi always depends upon the solution of an eqt tion of half its dimensionif it be even, or W half the dimension minus 1, if it be odd. 4 Thus far, in fact, it differs in no resp€ from any other equation in which a sin relation is known to have place between! roots; but what is the most characteristié these equations is, that they are known td reciprocal as soonas they appear by the o and signs of their co-efficients ; that is, { terms “equally distant from either extre have the same co-efficient, thus, 1 we + Sat +703 +72? + Sz + lL oe is a reciprocal equation. 4 And therefore when an equation appe under this form, it is immediately kno be a reciprocal one, and may thence be pressed to an equation of half its dimension Let ae” + pan} + gan? + &e. qe? +petlh be any Sanat ig equation, whose roots 1 . : ,«e. Then from the the 1 whose roots are a, mi b up of the. factors (a — a) (« aa *)e — »(«— 5 )e- O(a &e Or putting a + - =m,b + : =n, ' —r, &c. these hataie (x* + ma + 1) (2? 4+nx+1)(2*+re+1) & If therefore we really perform this mul cation, and equate the co-eflicients, it is vious, since the multiplication is reduced half the number of factors, the equation which the values of m, 2, 7, &c. are obté will be of only half the dimension of the ginal equations and having found these, si xa +me+1=—0 | a+ nzxa+1—0 x* + rx+-1—0ke. me. WEE 2 shall have « = — > (8) ,; RS de Ke, &e. Thus for example let there be proposed the uation gt +523 +72? +452 +10 alt. together x” +-ma+1 and 2 + na+1 x*-+m 9 xi mn x*-+-m | Hie Saad eee FT ‘Comparing the co-efficients, we have m+n 5, and mn +2 = 7. i- — Hence m = line and n = ove F srefore x — 2 Y* f PD . af 304+10V5_, 4 16 a ‘ 9 and z= aati Og / Ce) 4 16 lich are the four roots of the proposed equa- n. If the equation be of odd dimensions, then her + 1 or — 1, is one of the roots which ii depend upon the signs of the co-effici- ts, and may therefore be determined imme- ttely from inspection, and the whole equa- n thus reduced to another one degree less, t still reciprocal; and which may then be duced to half its new dimension according the preceding rule. On this subject see nnycastle’s ‘“‘ Algebra,” vol. ii. RECIPROCAL Figures, in Geometry, are ch as have the antecedent and consequents the same ratio in both figures. RecipROcAL Proportion, is when the reci- yeal of the two last terms have the same lo as the quantities of the first terms, or ien the antecedents are compared with the iprocals of the consequents, thus, Bri Berea P15 a reciprocal proportion, because 5: 8 :: a: 445, or Ot ee tte BY ss ReciPROcAL Ratio, is the cals of two quantities. RECIPROCALLY, the property of being iprocal ; thus we say, that in bodies of the ne weight, the density is reciprocally as the gnitude ; viz. the greater the magnitude the sis the density; and the less the magni- fe the greater the density. ‘So again, the ice being given, the velocity is reciprocally the time. RECIPROCITY. The law of reciprocity a term employed by Legendre in his ‘“‘ Thé- e des Nombres,” to denote a reciprocal v that has place between prime numbers of ferent forms, which is this, that m and ing prime odd numbers, i Oe . ratio of the reci- n—1 the remainder of m2 divided by n = m—1 the remainder of n 2 divided by m mand n are not both of the form 42-1, REC and if they are both of this form, then n—1 the remainder of m™? divided by n = — m— 1 the remainder of n™2 divided by m but with a contrary sign. See “ Essai sur la Théorie des Nombres,” part 3. RECKONING, in Navigation, is that ac- count whereby, at any time, the latitude and longitude of a ship becomes known, and hence the course she ought to steer to gain the desired port. ‘This is sometimes made from observa- tions and sometimes from the logboard, in which latter case it is called the dead reckoning. RECLINER, in Dialling, is used for any dial whose plane reclines from the perpendi- cular; and if besides reclining it also declines from any of the cardinal points, it is called a reclining declining dial, and the quantity or angle at which it declines or reclines is called its reclination or declination. RECOIL or ReEsounp, of a Piece of Ord- nance, is that flying backwards which is always observed on the discharge taking place, and which arises from the exploded powder acting equally upon the ball and the gun, so that the momentum of the gun, with its carriage, is equal to the momentum of the ball, or rather to the momentum of the ball and half the powder. RECORDE (Roperr), an eminent Eng- glish mathematician of the 16th century, au- thor of several works on Arithmetic, Algebra, Geometry, &c. and which were principally written in the form of dialogues between a master and his pupils, wz. ‘The Pathway to Knowledge, containing the first Principles of Geometry,” London, 4to. 1551; “ the Ground of Arts, or the. Practice of Arithmetic,” 8vo. 1552, of which several editions were after- wards published by Dee, Mellis, Norton, Hart- well, &c.; ‘‘ the Castle of Knowledge, relat- ing to the Doctrine of the Sphere,” London, folio, 1556 ; “ the Whetstone of Witte,” being the second part of his Arithmetic, and relating to various algebraical subjects, London, 4to. 1557. Beside which, he published some others on cosmography, the use of the globes, &ec. | RECTANGLE, in Geometry, is a figure having all its angles right angles, being a par- ticular species of parallelogram, and conse- quently possessing all the properties belong- ing to the latter figure ; besides which, it has the following ones peculiar to itself, vz. If from any point in the plane of a rectangle, lines be drawn from any point either within or without, or in any of its sides to the four angles of the figure, the sum of the squares of two of those lines going tothe opposite angles of the figure, ix equal to the sum of the squares of the lines joining the other opposite angles; thus, REC AO? +0D* = OC? + OB’, and the same is true wherever the point O may be as- sumed, © To find the area of a rectangle multiply its jength by its breadth, and the product will be the area. RECTANGULAR Figures and Solids, are those which have one or more right angles. With regard to solids they are commonly said te be rectangular when their axis are perpen- dicular to the planes of their bases. RECTANGULAR Section of a Cone, was a term used by the ancients, before Apollonius, for the parabolic section. RECTIFICATION, in Geometry, is the finding of a right line equal to a proposed curve; a problem that even in the present state of analysis is, in many cases, attended _ with some difficulty, and was in all totally be- yond the reach of the ancient geometers, who were not able to assign the length of any curve line whatever, though they could in a few cases assign the area of a curvelinear space. See QUADRATURE. The first rectification of a curve was effcct- ed by Mr. Neal, as we are informed by Dr. Wallis, at the conclusion of his Treatise on the Cissoid. 'This was the semi-cubical parabola, and Neal’s rectification of it was published in July or August, 1657, and two years after the same was done by Van Haureat, in Holland, as may be seen in Schooten’s Commentary on Des Cartes’s “‘ Geometry.” It is, however, to the doctrine of fluxions that we owe the complete rectification of curve lines, in finite terms, in all cases where they admit of it, and in others by means of infinite series, circular ares, logarithms, Kc. of which method we shall give a general view in the present article. Let ACe be any curve line, AB an absciss, and BC a_ perpendicular erdinate; also be an- other ordinate indefi- nitely near to BC; and * Cd drawn parallel to the absciss AB. Put the absciss A B = z, the ordinate BC = y, and the curve AC =z; then Cd = Bb= az, the fluxion of the absciss AB; and ced=y, the fluxion of the ordinate BC, and Ce = z, the fluxion of the curve AC. Now the triangles Cde being right angled at d, we have Cc = (Cd? + cd)’, or z = /(x* + y7)*; that is, the fluxion of the curve is equal to the square root of the sum of the squares of the fluxions of the absciss and or- dinate; and therefore substituting the value - of one of these in terms of the other, drawn from the equation of the curve; the fluxion of the curve will be given in terms of one vari- able quantity, the fluent of which will be the length of the curve in terms of the same quan- tity, and which will be general for all values of that variable quantity. Taking then an known value of this quantity, the length of the curve will also become known. BG REG a Exam. To find the length of the semi-cu bical parabola, of which the equation is : 3 yz ax” = y', or x = —, and | 6 ae ] chinks on : By y +5 Bie : consequently « = —— or x* = +, Dar a 2 | substituting, therefore, this value of «* in th general expression z=. / (x? + y*),we have Oy. hai? | evi ( -— +7 )= — iy / (Dy +40 ¢ Zaz ihe fluent of which is 3 x (Dy 4+ 4a)* + corr. 27 az Now when the are = 0, then y= 0; .. 03 3 4az aS 27 ar RECTION. —8ai 27 a2 gen Whence the complete fluent is _ (Oy + 4a)z hhh 8a2 a 27 a2 Again, in the common parabola, Here the equation is ax = y*, or 2ba =y w —Aaz +C,or€ = . See Co} a ~ Zz putting a —=26, whence x = oe fy? . mh, 2 Ge Substitute this value of a the general expression 2 = /(«? + y”), a we have he < ro/ (Ghee) at ve +h the fluent of which is | and. = ares Ly 407)? + Lb x hyp.log. y+ Vy + + corr. | Now when z=0, y = 0; and we have the fore ) = ib x hyp.log. b + corr. or corr. = — 4) hyp. log. b Whence Zz Zz =i (y” 407)? + 1b.hyp. log. viv Instead of reasoning in all cases from expression z = /(a* + y?) it is sometit useful to adopt other methods according the nature of the curve, as in the follow example, 3. Vo find the Length of the Cycloidal Ari ABC. | C Take BD—a,BE=2,BG=2z,Gk the direction of a tangent atG =z; HIK parallel to GE, and let EK = 2,1 by similar triangles BEF and GIH, BE: BF ::G1:GH_ ? ae RECS. or since BF = “(DB.BE) = Vaz, we have ii 3 tag ‘ é: azz + phe BVA se Ue eS i omit) aD ve the fluent of which is 22a? x2 —.2./ax, which needs no correc- tion, for when z = 0, x=0, and the fluent itself becomes zero, as it ought to do. That is, the cycloidal are BC = 2BF, and therefore BA = 2BD, or the whole arc = ‘our times the diameter of the generating tircle, 4. To find the Length of a Circular Are. This may be expressed either in terms of he sine, cosine, versed sine, or any other rigonometrical line, as follows : First let the versed sine = x, the sine = y, adius =r, and arc = z, then by the property f the circle y” = 2ra — x ~ ~ 2 42 : ree or putting tangent —¢; y* = ——, r~+t g?—y2 and secant = s gives, y* = 7 = $ are readily deduced from the known pro- erties of the circle. , Now by means of these values of y’, or of rz — x and the general equation z = Ma? + y*) we readily draw the following uues of 2, viz. ra ad ry is si VV (Qra—2x’)” V(r?7—y?) i — rt wa ce y* + {* a / (s* a r*) e fluents of which can only be found in ries, which are as follows; making radius lee 2, viz. xv 32:2 3.523 ) PR oe BPO ea iy =G+55+5 45 *34.67+ hy g. Syt* 3.5y° & ) “+5 PA S467 ; i t+ t© t® ) 4 RRO Tg RR OT ek var (i 3 + 3 ¥ + Y 3 _fs—r , si3—r3 3(s>5—r3) ) as see a ead ——_——_— & . ja e( RAD Cy aan wR Gig al dol ere 7 is retained in the latter for sake of uogy. It is obvious therefore that the are y be computed by any of these in terms, rer of the sine or versed sine, tangent or ant, and consequently also in terms of the ine, cotangent, cosecant, &c. “hus in the first taking « = 4, which is the sed sine of 60°, we have yt 1 3 3.5 \ rc 60°=—{ 1 +——_ + —_ 4 — Jy 1, ee ( ts3taget aaa”? n the second, assuming y = 4 = sin. 30°, have i P(A ] 3 3.5 i 8 targa a8 4.67 . n the third, assuming ¢ = 1 = tang. 45°, iobtain ire 45°=(1—F + 2 — 4— i + &e.) ee i I the fourth, assuming x, i+ —— 38 3(25—15) 2°, 4.5. : oo re 60° A ta + +&e.) | . REC Then multiplying the numbers obtained from these series by the number of times that the are is contained im the whole cireum- ference, will give the circumference required. But no one of these series are sufficiently convergent for ascertaining the circumference of the circle to a great degree of accuracy, and therefore other methods have been con- trived, in order to produce series better cal- culated for this purpose, of which that of Machin has been the most popular; thouge it does not appear that he employed it in his celebrated quadrature or rectification, in which he found the circumference to 100 places of figures. In order to render these series more con- verging, it is obvious that less arcs must be assumed, and the difficulty consists only in finding the tangents for example, (using that series) of a small are, which may be expressed im numbers that are tolerably managable in the general series. For this purpose, Machin knowing the tan- gent of 45° to be 1, and that the tangent of an are being known, any multiple of it is readily found, considered, that if there were assumed some small simple number for the tangent of an arc, and then the tangent of the double are were continually taken, until a tangent was found nearly equal to 1, the tan- gent of 45°; by taking the tangent of this small difference between 45° and the multiple _ are, there would be had two very small tan- gents, the one of the first arc, and the other of this difference. Then computing the are to these tangents, whether the measure of them in degree, &c. were known or not; the whole arc of 45° would become known, viz, by multiplying the first by the assumed mul-. tiple, and adding the last are to the product; if the tangent of the multiple are were less than 1, or the arc itself less than 45°; but subtracting it if greater. Having thus laid down his plan of opera- tions, by a few trials he hit upon a num- ber well suited to his purpose, viz. knowing the tangent ~ of 45°, or of 11°15’, to be very nearly equal to 2, he assumed for his first are, that whose tangent is 4; then since tan. 24 = 2 tan. a T — tan. 7a tangent of his double are; and again 122 for the double of this, or of quadruple the first, and this being a little greater than the tans gent of 45° was well adapted to his views; for by a known trigonometrical property tan, radius being 1, he had |, for the tan. a — 1 ’ (a—45) = Re RE: that is, the tangent of the small arc, which is equal to the excess of IZO __ his multiple above 45°, was aaa = zh IYO He had therefore two ares to compute, the one whose tangent was +, and the other having for its tangent .4,; and then 4 times the first arc minus the latter, would evidently give the exact are of 45°, and both these numbers being such as converge yery well in PPB REC a REC the gencral series, the difficulty attending the then» +» +6 is the scale of relation, which | usual approximations was avoided, bat others — is of three terms, and so on of any other. still more converging than these have since Now referring to the first of the preceding been discovered. See Dr. Hutton’s Men- set of formule as the simplest, and calling the suvation, p 90, and vol. i. of his Tracts; the infinite sum of the terms a + B + y + 0, &@, | Treatises on Fluxions,by Simpson, Maclaurin, = s, it is obvious that we shall have | T+ « + . a q 1 F we “y \ api Dealtry, &c.; see also Phil. Trans. s—at+B + pa(s—a) + yX*.8, oF vol. Ixvi. 2 4 : ees ; $— pX.s —yv2?.s ma — aux, or RECTIFIER, in Navigation, an instru- ¥ 18 r 7? gg 4 ment consisting of two parts, which are two s = aT eK the sum of the series. | circles, either laid the one upon the other, or De eee: | the former let into the latter, and so fastened And in a similar way we may find the sum - together at their centres, that they represent when the scale of relation is of three, four, &e, | two compasses, one fixed the other moveable, terms, but the resuit must be necessarily a each of them being divided into 32 points little more complicated. corresponding with the points of the compass, This method of series is the invention of and also into 360°, and numbered both ways De Moivre, and is treated of by him, both in from north to south, terminating in the east his Miscel. Analyt. and in his ‘ Doctrine of and west in 90°. The use of this instrument Chances,” and has since been investigated "is to rectify the ship’s course, by determining and improved by Euler, Colson, Stirling, &e.- the variation of the compass. Let us illustrate this by a few examples. RECTIF YING of Curves. Sce ReEctiFI- 1. Required the sum of n terms of the series, CATION. 1 + 2a + 3x7 + 423 + Ke. na". RectiFyinc the Globe, is a previous adjust- _‘ First, it may be observed that the formula ment of it to prepare it for the solution of any above given extends to the infinite sum proposed problem. whereas it is here only required to find the RECTILINEAL, or Rectilinear, consisting sum of the first 2 terms, we must therefore of, or being bounded by, right lines. first find the infinite sum of the whole series RECURRING Series, is a series so con- and then the infinite sum of all the terms be stituted, that each succeeding term is con- yond na", and the difference of these wil nected with a certain number of the terms evidently be the sum required. immediately preceding it by a constant and Now, in the first place, it is obvious tha invariable law; as the sums or differences of each term is equal to 2x times the precedin some multiples of those terms, &c. term minus 2? times that preceding the latte Thus the series that isp = 2, andy —= —1; ranks” haere Va degeat rat Pe b,c, therefore 1, 32, 527, 7x3, 92*, 1125, &c. _atB—aur _142x—2r.. 1 _ is a recurring series, for these terms being ac 1x —va® 7 1— 2 + x* > (1 —@ 1 espectively represented by «, B,y,2,&¢.We — Inthe second case, viz. of the terms beyor ABR Rae eye nx", the series is }=2xy — 278 (w+ Va" + (w+ Qa" + (a + 8), & ¢ = 2rd — xy and in the formula s = oe DoE ) Ke. Ke. 1—pr—ver that is, each term is equal to 22 times that have only to substitute a = (n + 1) 2”, a which precedes it, minus x? times the one 8 —=(n + 2)2”t' instead of a= 1, and 6 = 2 preceding the last. as in the former. Hence we have Or generally let (n+ Da 4 (n 4 2) a1 —2(n 4 Vat! § a, B, yf é Ey 6, &e. ee 122 + 2x" ? ’ b] Vax, bx?, cx3, dxt, ex, fx®, &e. A n m+1 be any series whose terms are denoted as s- @ 41) +t a above by a, B, y, 0, &e. and let p, », 9, &e. (1 — 2)” represent the successive multiples of the pre- And therefore s — s’, or the sum of 3 ceding terms, then if we have first n times, is equal to | % — & 1 —(n + 1) 2" + nat é, ai e x B -+- a ; 1—2 , dans§ apis tel fs 2. Required the sum of » terms in o — Pe, in y a*.B series “s . Ee — pee. yu -y : a 3 ; * then this series is recurring, and jz + » is call- j : + a ae ace oe Lin , ed the scale of relation, which is here only of Here by trial we find the scale of relat : ; ’ ; to be w= 2, andy = — I, as before ; thi two terms, and this subtracted from unity is api ; fore the infinite sum is called the differential scale. be acai a Tee tn ace Sut if — — ae Sa pe.y tya*.B + pr>.e 1— px — ya" 1—2r+2* (1-| e—pe.p xk oy + 943.6 After n terms the series becomes 9 —pase + yx*.0 + pasiy (2n+4+1)a"+ (2n+8)a"t) + (2n+5)a"t? +, Ox Ses which therefore arises from the fraction RED Qn + 1)2" + (2n +3) a"! +2(2n41)a"t! bi wal 1—2e +2° : (2m + 1) a®— (2n — 1) a™! NN GESa Cane aren Whence the sum of n terms is 1 +e — (20 + 1) a” + (20 — 1) at! (1 — x)* 3. Required the sum of » terms of the series (nm —1) x + (n — 2) a? + (n — 1) 83 + &e. Here again the scale of relation is 2, — 1, jerefore the infinite sum is (a— 1) x + (n— 2) 2*?*— 2(n— 1) 2? , (1 — x)* we (xn — 1) » — na Oi ae After 2 terms it becomes — yt! — Qynt2 - &c. the sum of which is found in the same ianner to be ; +1 ; therefore n terms of —— v~ ie proposed series is (n — 1) x — na” + gt! (L— x)? Hence also x terms of the series i— lax , (n—2)a? |, (n—3) 23 —— > +- n n oT + &e. is (mn — 1) 2 — na® + art! n (1 — 2)* In a similar manner the sum of 1 terms of e series 1* + 2%a 4+ 3*2? + 4723 + Ke. found to be ba—(n+1)?ar 4+ (2n* +2n—1) at! — n24r+2 1 — x)3 | e scale of relation being », + », + p, or 3,—3, that is, » = 3,» —=— 3, andp=1. On is subject the reader should consult De oivre’s Misc. Analyt. and his Doctrine of iances ; Euler’s Analysis Infinitorum; Col- n’s Comment on Newton’s Fluxions; Stir- g’s Methodus Differ.; Cramer’s Analysis s Lignes Courbes; Bernoulli de Serieb. if.; see also a chapter on this subject in l. ii. Bonnycastle’s Algebra. ReEcURRING Decimals, those which are con- ually repeated in the same order, at certain ‘ervals, as 4 — 6666, and 3, = ‘272727, &e. e CIRCULATING Decimals, and REPETEND. {t is a singular property of all fractions ving a prime denominator (n), that the mber of places in the repetend is always ial to (n — 1), or $ (nm — 1), or (n — 1), .; and when it is equal to nm — 1, the same retend will always recur in the same order, atever may be the numerator of the frac- a, but the period of commencement will be erent in all; thus in the number 7, 1 = *1428-57°142857 2 = +285714-285714 $ = :42857°142571 # = 571428571428 1 § = °714285'714285 i 5 == *857142°857142 RED, in Physics, or Optics, one of the sim- } or primary colours of natural bodies, or RED the least refrangible of all the rays of light. And hence, as Newton supposes the different degrees of refrangibility to arise from the dif- ferent magnitudes of the luminous particles of which the rays consist; therefore the red rays, or red light, is concluded to be that which consists of the largest particles. See CoLour and Ligur. Authors distinguish three general kinds of red, one bordering on the blue, as colombine or dove colour, purple, and crimson; another bordering on yellow, as flame colour and orange; and between these extremes is a medium, which is that which is properly called red. REDUCING Scale, or Surveying Scale, is a broad thin slip of box, or ivory, having several lines and scales of equal parts upon it; used by surveyors for turning chains and links into roods and acres by inspection. It is used also to reduce maps and draughts from one dimension to another. REDUCTION, in general, is the convert- ing, or changing, a quantity from one deno- mination or state to another, without altering its absolute value. REDUCTION, in Avithmetic, is, by some au- thors on this subject, distinguished into re- duction ascending, and reduction descending. The former relating to the conversion of a quantity from a lower denomination to a higher; and the latter, when the quantity is to be reduced from a higher denomination to a lower. 'These cases may, however, be both included under the following rule. To reduce a Quantity from one Denomination to another. Rule. Consider how many of the less de- nominations make one of the greater, then multiply the higher denomination by this number, if the reduction be to a less name, or divide the lower denomination by it, if the reduction be tq.a higher name. EXAMPLES. 1. Reduce £171. 13s. 8d. to farthings. i.) . ae 171 13 8 20 3433 shilling's 12 41204 pence Oo 164816 farthings 2. Reduce 1469861 farthings to pounds sterling. 4\1469861 12) 367465 20} 30622 z 4 1k £1531 2 1 t Ans, Revuction of Fractions, may be divided ‘her of the rays of light. ‘The red rays are into the following cases. | : ! PP2 RED. Case 1. To reduce a Fraction to its simplest Terms. Rule. Divide both the numerator and de- nominator of the given fraction, by any num- bers that will divide them both without a remainder, so will the fraction be reduced to its simplest form. Note. If the common divisors of the nu- merator and denominator do not appear from imspection, the greatest common divisor of both must be found by the proper rule for that purpose. See Divisor. EXAMPLES. 1. Reduce 144 to its lowest terms. Here 444 = 72, = 38° 2 answer. as o —— 12906 —— 144 - 2 Again 3434 = 4235 = 447 answer. It may be observed, that this method of reduction is frequently much facilitated by the properties of divisors, mentioned under the article Divisor. Case 2. To reduce a mixed Number to an im- proper Fraction; and the converse. Rule. Muitiply the integral part by the denominator, and to the product add the nu- merator, which sum placed over the deno- minator will be the fraction required. And conversely, divide the numerator by the denominator in the second case. EXAMPLES. a's = ee ay 5x11+6 2. or ethene pana 2 3. Conversely 4 = 82; 4+ = 158, &e., Case 3. To reduce a Compound Fraction to a simple one. Rule. Multiply all the numerators together for a new numerator, and the denominators together for a denominator, and it will be the fraction sought. Note. If any of the proposed quantities be integral, or mixed numbers, they must be first reduced to improper fractions. And if there are any factors, common in the numerators and denominators, they may be omitted. EXAMPLE S. 1. Thus %o0f3 = 3% x 3= 4 answer. 2. Again 3 offo0f$=%4x2xi=1 ans 3. So alsozg,0f 125 =73, x 9° =2Zasreq Case 4. To reduce Fractions, having different Denominators, to a common Denominator. Rule. Multiply each numerator by all the denominators, except its own, for a new nu- merator; and all the denominators together for the common denominator; or Find the least common multiple of all the denominators, and multiply the numerators accordingly. See MULTIPLE. EXAMPLES. 1. Reduce 4, 2, 3, and Z, to a common de- ROmMinater. Here the common multiple being 24, w have for the fractions required. 2. Reduce 3, 4, and 4, to a common de nominator. | Here by the first rule 3X5 xX 7— 105 4x4x7= 112 E numerator 1S 4 5 20. common X5X7T= 140 denominator. Whence 793, 112, 2S, are the fraction Cause 5. To reduce a Fraction from one Den mination to another. Rule. Consider how many of the lower de nomination make one of ihe higher; the multiply the numerator of the fraction by thi number, if the reduction be to a lower name but the denominator if it be to a higher. EXAMPLES. 1. Reduce 2, of a pound to the fraction ¢ a farthing. or X 32 x xX + = 489° ofa farthing. 2. Reduce 4, of a guinea to the fraction ¢ a pound. 5 2 I — 10 yr X 7 X a = 495 of a pound. Case 6. To reduce a Fraction to its proper integral Value. Rule. Multiply the parts in the next inferic denomination by the numerator of the fraq tion, and divide by the denominator, as i compound multiplication and division. EXAMPLES. 1. Find the value of 7 of half a guinea. Ss. ° 10 6 ‘I 8)3 13. 6 9 2 Ans. ———___.. (OR reece as. eer The reduction of complex fractions to sim ple ones, is no more than dividing one fractio by another; see Division of Fractions. Repuction of Decimals, may be divigl into the following cases. Case 1. To reduce a Vulgar Fraction to it equivalent Decimal. Rule. Annex to the numerator as man decimals as may be thought necessary ; the divide by the denominator, and point off a many decimal places in the quotient, as ther are ciphers annexed. If there be not so man figures in the quotient as are requisite, th defect must be supplied by prefixing ciphers EXAMPLES. 1. Thus zz == °208333, &e. 2. Also ~43> = 0075231. a PRED. * | Case 2. To reduce Quantities of different Deno- minations to their equivalent Decimal Values. Rule. Reduce the compound quantity to its lowest denomination, and the whole in- teger to the same denomination; then con- sidering the first of these as the numerator, and the latter as the denominator of a fraction, the reduction to decimals may be performed as in the preceding case. Or the same may be otherwise done, as follows: Write the given numbers perpendicularly under each other for dividends, proceeding in order from the least to the greatest. Then opposite cach dividend, on the left, place such a number for a divisor, as will bring it to the next superior denomination, and draw a line between them. This done, begin with the uppermost number, and divide it, as is usual, pointing of the decimals, then divide by the second number, and so on to the last, which will be the decimal sought. EXAMPLES. Reduce 15s. 93d.; and 12s. 102d. to de- mals. 4| 3 4} 2 12) 975 12/105 20) 15°8125 20)12°875 ‘(90025 decimals 64375 Yase 3. To reduce a Decimal to its equivalent integral Value. | Rule. Multiply the given decimal by the umber of parts in the next less denomina- ‘on, and point off as many places to the right- and for decimals, as there are in the original amber. Then multiply the decimals, thus at off, by the number of parts in the next ‘ss denomination, reserving again the same amber of places for decimals ; multiply these fain in the same manner, and so on; then ie figures separated on the left-hand will «press the value of the decimal required. EXAMPLES. /1. Find the value of °78546 of a pound erling. "78546 20 15°70920 12 9°51040 4 2° 4160 Ans. 15s. 92d. 2. Required the value of °7854 of a day. 7854 24 3 1416 15.708 18°8496 60 509760 Ans. 18h. 5 min. ——— RED Revuction of Circulating Decimals. Ses CircuLatine Decimals. Repuction of Algebraic Fractions, is per- formed in exactly the same manner as the reduction of common fractions ; it will, there- fore, be useless to repeat the rules; we shall, however, for the sake of illustration, give an example or two in each case. Case 1. To reduce an Algebraic Fraction to its simplest Form. xt — b+ ‘ 1. Reduce =—, to its lowest terms. x — b* x3 xt ht © (x? 4 57) (x? —b*) _ 2? + B dite ei Pa pe be) ees st x” — y” Wi ae 2. Reduce ———*- to its simplest form. epee) pai ites 23 we Pah as re sty @ +P ey) rte e In these examples, the factors of the terms are found by inspection, and the reduction is therefore very simple ; but if this be not the case, the greatest common divisor must be found by the proper rule, and both terms of the fraction divided by it. See Divisor. Case 2. To reduce a mixed aleebraical Expression to animproper Fraction; and the Converse. ax + fhe 1. Reduce x + to an improper fraction. z 2 Z Z ate xryta te ; 2 + ee as required, 7 y L haa x—3 ae 1+ 22x)5r¢—(x—3) - ox 5x 2. Again 10a* + 424+ 3 = ———__——_- as required. 5x 3. Thus also conversely ay+2y _ eh a iets uC ! mane +2y*) + @t+ty)=yt+ mee Case 3. To reduce Fractions to a Common De- nominator. See Case 4. Common Fractions. 1. Reduce docs 8 and = to a common de- 2a” 2a y nominator. Here the least common multiple of the denominators being 2a*y, we have by cay 2a’x 2a*y’ 2a*y’ 2ay for the fractions required. z z Zz 2. Reduce x =) and = ss , mon denominator. Here the least common multiple is 6a + 6x, therefore the fractions when reduced are to a com- 8a + 3x2 2a? + 2a’*x mr 6a> + 62? 6a+62’ 644+ 62 ’ Ga+6x- RepucrTIion of Surds, is the method of re- ducing a surd, or irrational quantity, to its most simple form; which may be divided into the following cases. Roe ces Case 1. To reduce Quantities having different. Tndices or Radicals, to one Common Index. Rule. Let the proposed radicals be repre- sented by their proper fractional indices, then reduce these fractions to a common denomi- nator, which will be the common index re- quired, the numerators of the fractions being made the powers of the given quantities. EXAMPLES. 1. Reduce’ Va, and Ya, to a common radical. Here the proper fractional indices, being > and 4, they become when reduced to a coin- : 1 mon denominator, 3 and 2, whence (a?)s, and (a*\z; or Y a3, and Ya’, are the common radicals required. 1 1 2. Thus also 32, and 23, = 67 27, and * 4. , : L L be 3. Again a, and bm, become (a”)m, and mG (b")nn; or ™/ a”, and ™/ b”; and so of others. 33), and (22), = Case 2. To reduce Surds to their simplest Terms. Rule. Find the greatest power contained in the given surds, by dividing it successively by all the powers between itself and unity; and set the root of this power before the quotient, with the proper radical sign between them ; or if the surd has any co-efficient, it must be multiplied by the root of the power before found, and the product placed as above. Note. If the given quantity under the ra- dical be a fraction, it must be rendered inte- gral, by multiplying both numerator and de- nominator by such a number as will make the denominator a complete power; and then the root of this denominator must be taken as a fractional co-efficient to the radical thus obtained. EXAMPLES. ee hus) 4/1250 46/9, 58, B/G 2. Again 4/243 = 4/35 x 9 = 34/9. 3. Also 98a? 2 =a4/(7a)* X2KX=—7a/ 22. 4. Thus also /(23—a’*x*) = wJ/(x — a”). 5. ; 50 rit pea 1 Again hewn ee platen a! LOT are an, V iY 1 PY ah 3 ett eden | eines ned. / 6. e761 {PDAS AL 6. And Cal =v 009. See the application of this rule to the ex- traction of roots, under the articles SQUARE and CusE Roots. * In compound surds of the form Jat Vb Jet vd the rule is, to multiply both numerator and denominator by the terms of the denominator, but connected with a different sign, which renders the denominator rational, and reduces the whole expression; thus 474+ V5 74+ 75 : /7+ /5 Il I—vV/5 VI— Vd V7 + V9 RED ” 74273545 12 +235 a — At 4 EO oe “ct vo? = 6 +786 7—9d 2 , Again, . /18 + 78 _V1I8+ V8 _ V3 + ae ¥38—V/2) ¥3—V2 V3d+ V2, OAL VE Ee a 3—2 ; REDUCTION of Fguations. See EQuATIONS DEPRESSION of Equations, &c. . ; Repucrion of Ratios. See Ratio, ane CONTINUED Fractions. | REDUCTION of a figure, Design, or Drang is properly the reducing of it from a larger t a smailer scale; though the term is soul times used indifferently, whether the copy b made larger or smaller. In this reductio the exact form and proportion of the figurr must be observed, for which purposes survey, ors and architects make use of an ins trun calied a PenraGRaPH; Proportional ComPAs) sEs are also employed for this purpose ; rl the method of using these instruments nnde the respective articles. There are also othe) methods of reduction, the best. of which ar as below. To reduce a simple rectilinear Figure by Line} Assume a point P, anywhere about thi given figure ABCDE, either within it, ¢ without it; or in one of its sides or angles but near the middle is best. From that poi) P, draw lines through all the angles, wpo} one of which, take Pa to P A, in the propose; proportion of the scales, or linear dimensions then draw ab parallel to AB; be parallel 4} BC, &e.; so shall abcde, &c. be the reduee) figure required, either greater, or smalle than the original. A To reduce a Figure by a Scale. Measure all the sides and diagonals of figure, as ABCDE, by a scale; and kf down the same measures respectively fro another scale, in the proportion required. — To reduce a Map, Figure, or Plan, by Squan Divide the original into a number of sm squares, and divide also a fair sheet of papi of the dimensions required, into the sa nuimber of squares, which will be equé greater, or less, according as the new figt is to be equal, greater, or less. ‘This done, every square of the second figure, draw wh is found in the corresponding square in t first, or original plan; so will the figure reduced as required. ; | wien) REF The cross lines forming these squares may be drawn with a pencil, and rubbed out again after the work is finished. But a more ready and convenient way, particularly when such reductions are often wanted, is to keep always ready frames of squares of several sizes; for then, by barely placing them upon the papers, the corresponding parts may be readily copied. ‘These squares may be made with four inflexible bars, strung across with horse hair, or the like. Repvuction to the Eeliptic, in Astronomy, is the difference between the argument of lati- tude, as N P; and an arc of the ecliptic NR, intercepted between the place of a planet and the node. To find this reduction, or difference, there are given in the right-angled spherical triangle NPR, the angle of inclination, and the argu- N| ment of latitude NP; to find NR, then the difference between NP and NR is the reduction sought. REDUNDANT Ayperbola, one of the higher order of hyperbolas, having more than two infinite branches. P REPLECTED Ray. See Ray. REFLECTING Circle. See CIRCULAR | Instruments. REFLECTING Dial, one that shows the time by reflection. ReFvectine Telescope. See TELESCOPE. REFLECTION, or Reflexion, in Mechanics, ‘is the return or regressive motion ofa moveable body, arising from the re-action of some other ‘body on which it impinges. The reflection of bodies after impact, is at- ‘tributable to their elasticity, and the more perfectly they possess this property the greater will be their reflection, all other things being the same. In case of perfect elasticity they ‘would be reflected back again with the same velocity, and at an equal angle with which ‘they met the plane; that is, the angle of in- idence would be equal to the angle of reflec- tion, and the velocity both before and after impact would be the same, at equal distance from the body on which they impinge. See INcIDENCE and PERCUSSION. REFLECTION of the Rays of Light, like that of other material particles, is their motion after being reflected from the surfaces of other bodies; and by which means those bodies become visible. For the laws of reflection, as it is connected with the doctrine of optics, see MIRROR. REPLECTION of Heat. In the same manner as we find the rays of light are reflected by polished surfaces, so it is found that the rays of caloric have precisely the same property. The Swedish chemist Scheele discovered, that the angle of reflection of the rays of caloric is equal to the angle of incidence, a fact which has been more fully established by Dr. Herschel. Some very interesting experiments were made ey Professor Pictet of Geneva, which proved : | REF the same thing. These experiments were conducted in the following manner: two con- cave mirrors of tin, of nine inches focus, were placed at the distance of twelve feet two inches from each other; in the focus of the one was placed the bulb of a thermometer, and in that of the other a ball of iron two inches in diameter, which was just heated, so as not to be visible in the dark. In the space of six minutes the thermometer rose 22°. A similar effect was produced by substituting a lighted candle in place of the ball of iron. Supposing that both the light and heat might act in the last experiment, he interposed be- tween the two mirrors a plate of glass, with the view of separating the rays of light from those of caloric. The rays of caloric were thus inter- rupted by the plate of glass, but the rays of light were not perceptibly diminished. In nine minutes the thermometer sunk 14°; and in seven minutes after the glass was removed, it rose about 12°. He therefore justly con- cluded, that the caloric reflected by the mir- ror was the cause of the rise of the thermo- meter. He made another experiment, sub- stituting boiling water in a glass vessel in place of the iron ball; and when the apparatus was adjusted, and a screen of silk which had been placed between the two mirrors re- moved, the thermometer rose 3°; namely, from 47° to 50°. The experiments were varied by removing the tin mirrors to the dis- tance of 90 inches from each other. The glass vessel, with boiling water, was placed in one focus, and a sensible thermometer in the other. In the middle space between the mir- rors, there was suspended a common glass mirror, so that either side could be turned towards the glass vessel. When the polished side of this mirror was turned towards the glass vessel, the thermometer rose only five- tenths of a degree; but when the other side, which was darkened, was turned towards the glass vessel, the thermometer rose 3° 5’. And in another experiment, performed in the same way, the thermometer rose 3° when the po- lished side of the mirror was turned to the glass vessel, and 9° when the other side was turned, which experiments show clearly, that the rays of caloric are reflected from polished surfaces, as well as the rays of light. ‘Transparent bodies have the power of refract- ing the rays of caloric, as well as those of light. They differ also in their refranyibility. So far as experiment goes, the most of the rays of caloric are less refrangible than the red rays of light. The experiments of Dr. Herschel show that the rays of caloric, from hot or burning bodies, as hot iron, hot water, fires, and candles, are refrangible, as well as the rays of caloric, which are emitted by the sun. Whether all transparent bodies have their power of transmitting these rays, or what is the difference in the refractive power of these bodies, is not yet known. The light which proceeds from the sun seems to be composed of three distinct sub- stanees, Scheele discovered that a glass mir- ‘ REF ror held before the fire, reflected the rays of light, but not the rays of caloric; but when a metallic mirror was placed in the same si- tuation, both heat and light were reflected. The mirror of glass became hot in a short time, butno change of temperature took place on the metallic mirror. This experiment shows that the glass mirror absorbed the rays of caloric, and reflected those of light; while the metallic mirror, suffering no change of temperature, reflected both. And if a plate of glass be held before a burning body, the rays of light are not sensibly interrupted, but the rays of caloric are intercepted ; for no sensible heat is observed on the opposite side of the glass; but when the glass has reached a pro- per degree of temperature, the rays of caloric are transmitted with the same facility as those of light. And thus the rays of light and caloric may be separated; and the curious experiments of Dr, Herschel have clearly proved, that the invisible rays which are emit- ted by the sun have the greatest heating power. In these experiments the different coloured rays were thrown on the bulb of a very deli- cate thermometer, and their heating power was observed. That of the violet, green, and red rays were found to be to each other as the following numbers: ® IDIELA occksc spence La OG ESOC dt os Ueeee ia Het. ures tlie The heating power of the most refrangible rays was least, and this increases as the re- frangibility diminishes. The red ray, there- fore, has the greatest heating power, and the violet, which is the most refrangible, the least. The illuminating power, it has been already observed, is greatest in the middle of the spectrum, and diminishes towards both ex- tremities. but the heating power, which is least at the violet end, increases from that to the red extremity, and when the thermometer was placed beyond the limit of the red ray, it rose still higher than in the red ray, which has the greatest heating power in the spectrum. ‘The heating power of these invisible rays was greatest at the distance of half an inch beyond the red ray, but it was sensible at the distance of-one inch and a half. REFLECTOIRE Curve, is a term given by Mairan to the curvilinear appearance of ihe plane surface of a bason containing water to an eye placed perpendicularly over it. In this ‘position the bottom of the bason will ap- pear to rise upwards from the centre outwards, but the curvature will be less and less, and at last the surface of the water will be an asymp- tote to it. See Memoirs of the Acad. of Paris, 1749, and Prie: tley’s Hist. of Vision, p. 752. REFLEXIBILITY, the property neces- Sary for reflection. REFLUX of the Sea. See Tipe. REFRACTED Angle, or Angle of Refrac- tion, is the angle which a refracted ray makes .with the surface of the refracting body. The complement of this angle is, however, some- times called the refracted angle, REF | REFRACTED Ray. See Ray. aa REFRACTION, in Mechanics, is the de- viation of a body in motion, from its direct | course, in consequence of the variable density} of the mediums in which it moves. This, } however, except in speaking of the rays of light, is more commonly called deflection. | Rerraction of Light. If the rays of light,| after passing through medium, enter another | of a different density perpendicular to its sur) face, they proceed through this medium in the same direction as before. * Thus the ray} PC proceeds to K, in P the same direction. But if they enter ob- liquely to the surface of a medium, either denser or rarer than what they moved in before, they change their direction in pass- ing through that me- dium. If the medium which they enter be; denser, they move through it in a direction| nearer to the perpendicular drawn to its sur-| face. Thus AC, upon entering the denser} medium HG K, instead of proceeding in the} same direction A L, is bent into the direction) CS, which makes a less angle with the per-| pendicular PK. On the contrary, when light, passes out of a denser into a rarer medium, it moves in a direction farther from the perpen- dicular. Thus if SC were a ray of jlight which} had passed through the dense medium HG K, on arriving at the rarer medium it would) move in the direction CA, which makes aj greater angle with the perpendicular. This refraction is greater or less, that is, the rays| are more or less bent or turned aside from) their course, as the second medium throughi which they pass is more or less dense than} the first. Thus, for instance, light is more} refracted in passing from air into glass, than| from air into water; glass being denser than water. And in general, in any two given} media, the sine of the angle of incidence has a constant ratio to the sine of the corres- ponding angle of refraction. Hence, when the angle of incidence is increased, the cor- responding angle of refraction is also in- creased; and if two angles of incidence be equal, the angles of refraction will be equal. The angle of deviation must also vary with the angle of incidence. Ifa ray of light A pass obliquely out of air into glass, AD, the sine of the angle of incidence, is to NS, the sine of the angle of refraction, nearly as 3 t@ 2; therefore, supposing the sines proportional to the angle, the sine of SCL, the angle of deviation, is as the difference between AT and NS, that is, as 3 — 2, or 1, whence th sine of incidence is to the sine of the angle of deviation as 3 to 1. In like manner it may be shown, that when the ray passes obliquely out of glass into air, the sine ef the angle ol incidence will be to that of deviation, as N, to AD—NS, that is, as 2 to 1. In passi from air into water, the sine of the angle ¢ m REF cidence is to that of refraction, as 4 to 8, id to that of deviation, as 4 to 4 —3, or 1; id in passing out of water into air, the sine ‘the angle of incidence is to that of refrae- om, as 3 to 4, and to that of deviation as 3 1, Hence a ray of light cannot pass out water into air at a greater angle of inci- mce than 48° 36’, the sine of which is to dius as 3 to 4. Out of glass into air the igle must not exceed 40° 11’, because the ye of 40° 11’ is to radius as 2 to 3 nearly ; nsequently, when the sine has a greater pro- rtion to the radius than that above stated, eray will not be refracted. And it may be served, that when the angle is within the nit for light to be refracted, some of the ys will be reflected. For the surfaces of all dies are for the most part uneven, which casions the dissipation of much light by the yst transparent bodies; some being reflected, d some refracted, by the inequalities on the faces. Hence a person can see through ter, and his image reflected by it, at the me time. Hence also, in the dusk, the fur- ure in aroom may be seen by the reflec- ‘in of a window, while objects that are with- tare seen through it. To prove the refraction of light, take an tight empty vessel into a dark room; make mall hole in the window shutter, so that a wn of light may fall n the bottom at a, ere you may make a tk. Then fill the on with water, with- /moving it out of its ce, and you will see t the ray, instead of img upon a, will fall at b. If a piece of King-glass be laid in the bottom of the sel, the light will be reflected upon it, and be observed to suffer the same refraction n coming in; only in a contrary direction. he water be made a little muddy, by put- jinto it a few drops of milk, and if the room filled with dust, the rays will be rendered ch more visible. The same may be proved another experiment. Put a piece of money ( the bason when empty, and walk back you have just lost sight of the money, ch will be hidden by the edge of the n. Then pour water into the bason, and will see the money distinctly, though you c at it from the same spot as before. sthe piece of money at } will appear at and hence it is that a straight oar, when ly immersed in water, will appear bent. (the rays of light fall upon a piece of flat 8, they are refracted into a direction ‘er to the perpendicular, as described ve, while they pass through the glass; but > coming again into air, they are refracted ‘uch in the contrary direction; so that ‘move exactly parallel to what they did re entering the glass. But on account of ‘thinness of the glass, this deviation is “ally overlooked, and it is considered as ing directly through the glass. REF Atmospherical Rerraction. It is evident from the nature and progression of light, that rays, in passing from any object through the atmosphere, or part of it, to the eye, do not proceed in a right line; but the atmosphere being composed of an infinitude of strata, (if we may so call them) whose density increases as they are posited nearer the earth, the lu- minous rays which pass through it are acted on as if they passed successively through media of increasing density, and are therefore inflected more and more towards the earth as the density augments. In consequence of this it is, that rays from objects, whether ce- lestial or terrestrial, proceed in curves which are concave towards the earth; and thus it happens, since the eye always refers the place of objects to the direction in which the rays reach the eye, that is, to the direction of the tangent to the curve at that point, that the apparent or observed elevations of objects are always greater than the true ones. The dif- ference of these elevations, which is, in fact, the effect of refraction, is, for the sake of brevity, called refraction; and it is distin- guished into two kinds, horizontal, or terrestrial refraction, being that which effects the alti- tude of hills, towers, and other objects on the earth’s surface; and astronomical refraction, or that which is observed with regard to the altitudes of heavenly bodies. Refraction is found to vary with the state of the atmosphere, in regard to heat or cold, and humidity, so that determinations obtained for one state of the atmosphere will not answer correctly for another without modification. Tables com- monly exhibit the refraction at different al- titudes, for some assumed mean state. 2. With regard to the horizontal refraction, the following method of determining it has been successively practised in the English Trigonometrical Survey. Let A, A’, be two elevated stations on the surface of the earth, BD the intercepted arc of “A the earth’s surface, C the earth’s centre, A H’ A/H, the horizontal lines at A, A’, produced to meet the opposite ver- tical lines CH’, CH. Let a, a’, represent the apparent places of the Cc objects A, A’, then is a’ AA’ the refraction observed at A, and a A’A the refraction ob- served at A’; and half the sum of those angles will be thé horizontal refraction, if we assume it equal at each station. Now an instrument being placed at each station A, A’, the reciprocal observations are made. at the same instant of time, which is determined by means of signals, or watches previously regulated for that purpose; that is, the observer at A takes the apparent depres- sion of A’, at the same moment that the other observer takes the apparent depression of A. In the quadrilateral AC A’I, the two angles tL a bene OS ee eee REF A A’, are right angles, and therefore the angles I and C are tog her equal to two right angles: but the three angles of the triangle IA A’ are together equal io two right angles; and con- sequently the angles A ‘and A’ are together equal to the angle C, which is measured by the are BD. If therefore the sum of the two depressions H A’a, H’A A’, or, which is equi- valent, from the angle C (which is known, because its measure BD is known); the re- mainder is the sum of the two depressions from the measure of the intercepted terrestrial are, half the remainder is the refraction. 3. If by reason of the minuteness of the contained arc BD, one of the objects, instead of being depressed, appears elevated, as sup- pose A’ to a’; then the sum of the angle a’ A A', and aA‘A will be greater than the sum TA A' + TA’A, or than C, by the angle of clevation a’A A’; but if from the former sum there be taken the depression HA’A, there will remain the sum of the two refrac- tions. So that in this case the rule becomes as follows: take the depression from the sum of the contained are and elevation, half the remainder is the refraction. 4. The quantity of this terrestrial refraction is estimated by Dr. Maskelyne at one tenth of the distance of the object observed, ex- pressed in degrees of a great circle. So if the distance be 10000 fathoms, its tenth part, 1000 fathoms, is the sixtieth part of a degree of a ereat circle on the earth, or 1’, which there- fore is the refraction in the altitude of the object at that distance. But M. Legendre is induced, he Says, by several experiments, to allow only 4! Cth part of the distance for the refraction in altitude. So that on the distance of 10000 fathoms, the fourteenth part of which is 714 fathoms, he allows only 44” of terrestrial refraction, so many being contained in the 714 fathoms. See his Memoir concerning the Trigonometri- cal Operations, &c. Again, M. Delambre, an able French as- tronomer, makes the quantity of the terres- trial refraction to be the eleventh part of the are of distance. But the English measurers, especially Col. Mudge, from a multitude of exact observations, determine the quantity of the medium refraction to be the twelfth part of the said distance. ‘The quantity of this refraction, however, is found to vary considerably with the different states of the weather and atmosphere, from the 4th to the th of the contained are. 'Trigonometrical Survey, vol. i. p. 160, 355. Iiaving thus given the mean results of ob- servation on the terrestrial refraction, it may « Rot be amiss, though we cannot enter at large REF into the investigation, to present here ae rect table of mean astronomical refractio| The table, which has been most commoy; given in books of astronomy, is Dr. Bradley, computed from the formula » = 57” x ¢ (a + 3”) where a is the altitude, » the refre tion, and 7 = 2’ 35”, when a = 20°, - | But it has been found that the refracti¢ thus computed are rather too small. Lapla) in his ‘“ Mecanique Celeste,” tom. iv. p. 4) deduces a formula which is strictly similar) Bradley's; for it is r= m x tan. (wx —4) where zis the zenith distance, &c. mand ni two constant quantities to be determined fr observation. The only advantage of the] ter formula over the one given by the Engl; astronomer is, that Laplace and his colleagi have found fore correct co-eflicients MY Bradley had. Now, if R = 57°.2957795, the are equal RR | the radius, if we make m = ae (where ki constant co-efficient, which, as well as 2, is| abstract number) the preceding equa laa y become nr R . Here the refraction 7 is always very sm| as well as the correction nr, the trigonome; —k x tan. (z — nr). for 3 thus we shail have tan. ar = k x tan. (z — mr). : But nr = tz — (Ez — nn), and z— nz; iz + (iz — nr); consequently z 2@—2nr tan. nr 2 tan.(z— nr)” tan. ( z 2 sin. z — sin. (z — nr) OOO OO k. sin. 2 + sin. (z + me) Hence sin. (z — 2nr) = sin. Z| L+hk This formula is very convenient when ; k—1 . co-efficients n and kel are known; anc has been ascertained, by a mean of many servations, that these are, 4 and ‘99765. respectively. Thus Laplace’s equation — comes sin. (z — 8r) = *99765175 x sin. z.’ And from this the following table has bi computed, where, besides the refractions, difference of refraction for every 10 mino of altitude are given, an addition which} make the table particularly useful wh great accuracy is required. See Hutto Course of Math. vol. iii, | aihitcde. Refractions, mat aaPende. Refractions. perk hey APR Refractions pe appar. Refract, 10". ao. M.| M. Be Ss. Div det Ste 8. Ss. Worm. 8. Ss. Dd. Ss. Si 33 463 7 Ol7 248 14/3 498 56 | 39-3 Bi A seal. 107 153] 2? is |s 343] 728 | 57 be es ee O83 | io 5.4 sc bi OT el tee ce TS | 01 : ‘ QO: 2 9 0) 36: Eilon so | 879 s0i6 sr7| °° llavis e5l- "ll so so (o . 32°1 er h . 4 “ie "ae 40 | 6 oi oA Tei a egeele A G0 ee [OR 0}27 22 ‘6 2 57° 33°G } i chee is 50 | 6 ae 7 Vig lo ar7 |) Il 61 4 ae D4 8° n . hog : . ee Se ging halaeml Go ci onal agen lan: ge my (oe! 24. . Ae ‘ . ‘ 0 : P ost 10/6 avi! ~? |lerleso6| 2” Il es von {O24 ; ) oa Q- le . . Q- ; Do | o> Od Mie 20 | 6 ey Gill gshlial bao! ac |p as na (Ot 29 122 3 016 20: 29-4 ) ee, he 30.16 1311 °° 1 98 12 1661 0 oles ny (020 5+ Rag gy heh a th bal eerg-9 ft Oo Ab Gg wey [22] l A PAR) 4° oy, y, er) a : : A oe 53°3 BMS OO He over (et Gag | 4 eog AN 1°5 Py: yr i : ' ,| 498 9 5 is os és 1891 °°" |l 68 oe, (2 | ? " 3 22 ! 53°6 YI6 AR: 92°5 ibs pac kaaobhonso volEe £62 oe | 0°88 ¥° 10-20 | 10 117 363 1015 47-4 27.11 53:9] - 6951 do.4) nN 43°1 o'9 0°78 9:20 | 90 | 16 53-2 | 2015 41:5 as | 1 59-2 70 | 21-9 : iis4] "| sols so8| °” |laol1 aaa OI a1 A De 6 9) . . . an. by et.” 37°4 55 0-70 POO eg 40 | 15 36-0 4015 363 30/1406] — || 72] 1¢-9 , 50115 091 > Boul Serpe, ull 31} doz bob v8 : as 2 >” 4 Lay 0 “A pay (Meme MTG ONE Toe Mors sian kiasrh oo lure ve [os if . Sed hse ahesttred 471°) | 33 }1 29-6) 08 | 25 eg [ors ae oe? 9 . 5 A i j ¥ ih see aa Ae Laman .) 1 262 | °° Il 26 oe joa { FOF o ° / ¥ee Ae 1 ie a i, Weal a ainted Vink sac tch soak Cao e 0 3 J ov 2 mae F I 12 56/7 do toro + 4 sah itachhoc line oe | ou7 50 | 12 a rin 50 ‘ Boot Naz Lagos ool 79 oe hate ) a» d 99 Pie q “ 4 0/11 at dhl ABR hata] cnc ea eae ie cig a a ieee ae Ueda | “3 . 4s of | 217 4°] 0°43 103 O17 10 | 11 26-6 10|4 47-6 39 | 1 11-8 81! 99 20:5 40 0°42 O17 20/11 61 20/4 43-6 4011 93 g2| go | 30. |10.467| °° 4 a0 eae bot teh ab Lie fess eek , ks 40 | 4 Te OPT dae ¥ [ou 0 “s 33°7 . e A Byl10 1091 50 | 4 ee Belt batten oer hoo lltas "| ol7 ; : Fs : ke : 5 Fe 166 || maney: 0°35 Bel Ris 5 0} 9 543 12 0/4 2860 44/1 08 BG: Peep | 15-9 37 || 0:34 O17 10 | 9 38-4 10/4 243 45 | 0 58:2 87) | aay st 150 36 033 0-17 | 29 | 9 234 2014 207 46 | 0 562 881 29 14:4 35 0:32 0-17 30| 9 90 30/4 17-2 47.10 543 Co en 13-7 3-4 0:31 0-17 40 | 8 553 40|4 138 48 | 0 52:4 ae ee 13-0 32 0°30 50 | 8 42:3 5014 106 49 | 0 506 12°4 31 0 29 6 0} 8 299 13 0|4 75 50/0 8-9 , "| 118 3] 0°28 10} 8 181 10)4° 4-4 51 | 0 47-2 115 3:0 | 9-27 2 | 8 66 20/4 1-4 5210 455 11-0 3:0 026 30 | 7 556 3013 584 53 | 0 43-9 10°6 2-9 0:26 40 | 7 45-0 40/3 555 54.10 42:3 10°3 2-9 0:25 50 | 7 347 5013 526 55 | 0 40-8 9°9 || 2'8 HO || er ERE ERS NE A ELSI API NMOS A TE PPE Pl MR AN I SAS STORE cng in mr OO * Barometer 29:92 Inches, Fahrenheit’s Themometer 54°. % TABLE OF REFRACTIONS. Differ. REG With respect to the refraction under dif- ferent temperatures and at different states of the barometer, the following theorems have been given, the first by Dr. Maskelyne, and the second by Dr. Brinkley, viz. Ist Refract. — —~ x tan. (z— . st Refrac 506 * an. ( ar) x 57” x 350 * 2d Refract. = ao x tan. (z—3'2r) x56".9 x 500 450 + h. Where a = the height of the barometer in inches; h = the height in degrees of Fahren- heit’s thermometer, and 7 = 57” tan. z. REFRACTION of Altitude, is an are of a vertical circle, by which the altitude of a star is increased by the refraction. REFRACTION of Ascension or Descension, is an arc of the equator by which the ascension or descension, whether right or oblique, is in- creased or diminished by refraction. REFRACTION of Declination, is the increase or decrease in the declination of a star by refraction. REFRACTION, in Latitude, is the increase or decrease in the latitude of a star from refrac- tion. REFRACTION in Longitude, is the increase or decrease in the longitude of a star from refraction. REFRANGIBILITY of Light, the pro- perty of the rays to be refracted, but more commonly employed with reference to the different degrees in which the different rays possess this property; and on which Newton has founded his whole theory of colours. For the several experiments relating to this sub- ject, see PRISM. REGEL, or RIGEL, a fixed star of the first magnitude in the left foot of Orion. REGIOMONTANUS. See MuL_Ler,(JoHN.) REGIS (PETER SYLVAIN), a French philo- sopher, was born im 1632, author of a “ Sys- tem of Phylosophy,” in 3 vols. 4to. published in 1690, and some other works. He died in 1707, at 75 years of age. REGRESSION. See RETROGRESSION. REGULAR Figure, in Geometry, is one that has all its sides and all its angles equal. If these are not both equal the figure is irre- gular. See POLYGON. REGULAR Bodies, are those which have all their sides, angles, and faces, similar and equal. Of these there are only 5, viz. the Tetraedon, contained by 4 equilateral tri- angles ; the Hexaedron or cube, by 6 squares ; the Octaedron, by 8 triangles ; the Dodecaedron, by 12 pentagons ; and the Icosaedron, by 20 triangles. For the properties of which, see the several articles; see also Hutton’s ‘ Mensuration,” sect. 2. REGULUS, a fixed star of the first magni- sii aa tude in the constellation Leo; sometime called Cor Leonis, or the Lion’s Heart. Bi the Arabians it was termed Alhabor, and bj the Chaldeans Kalbeleceid. REINHOLD (Erasmus), an eminent Ge man mathematician and astronomer, was bor in Upper Saxony in 1511, and died in 188 in his 42nd year. Reinhold was author ¢ several works, ef which only the four follow ing were published, viz. 1. Theorie nove Planetarum G. Purbachii, 8vo. 1542, and agai) in 1580. 2. Ptolomy’s Almagest, the firy book in Greek, with a Latin version, an Scholia, 8vo. 1549. 38. Prutenicae Tabul Coelestium Mutuum, 4to. 1551; 2nd editic in 1571; 3d, 1585. 4. Primus Liber Tabr Jarum Directionum, or Tables of Tangents every minute of the quadrant. And ney Tables of Climates, Parallels, and Shadow; with an Appendix, containing the secon| book of the Canon of Directions, in 4to. 1554 Beside these works Reinhold was authe of several others, which were never published RELATION, in Mathematics, is the sami as Ratio, though we sometimes use it inj more general sense, indicating any deper dence of one quantity upon another. REMAINDER, that which arises by suk tracting one quantity from another. San REPELLING Power. See REPULSION REPERCUSSION. See REFLECTION. REPETEND, in Arithmetic, denotes thi part of an infinite decimal, which is contim ally repeated ad infinitum, and is otherwis called a circulate. See CircuLaTine Dee mals, / REPULSION, in Physies, that property bodies, whereby, if they are placed just bi yond the spheres of each other’s attraction 4 cohesion, they mutually recede and fly oj Thus, if any oily substance, lighter than wate be placed upon its surface, or if a piece of ire be laid upon mercury, the surface of the flu) will be depressed about the body which is la: onit; this depression is manifestly occasioné by a repelling power in the bodies, whit prevents the approach of the fluid towar them. But it is possible, in some cases, to pré or force the repelling bodies into the sphej of each other’s attraction ; and then they w, mutually tend toward each other, as when ¥ mix oil and water till they are incorporate Dr. Knight defines repulsion to be that cau which makes bodies mutually endeavour } recede from each other, with different force at different times; and that such a cat exist in nature, he thinks evident for the fe lowing reasons. 1. Because all bodies a electrical, or capable of being made so; at it is well known that electrical bodies bo attract and repel. 2. Both attraction and & pulsion are very conspicuous in all magneti¢ bodies. 3. Sir Isaac Newton has shown fro experience, that the surface of two cony glasses repel each other. 4. The same gre philosopher has explained the elasticity of f air, by supposing its particles mutually | repel each other. 5, The particles of Ii . ‘i doe 4 © 1 : % RES e, in part at least, repelled from the surfaces ‘all bodies. 6. Lastly, it seems highly pro- ible that the particles of light mutually repel h other, as well as the particles of air. r. Knight also ascribes the cause of repul- yi, as well as that of attraction, to the im- ediate effect of God’s will; and as attraction id repulsion are contraries, and consequently mnot at the same time belong to the same bstance, the doctor supposes there are in ture two kinds of matter, one attracting, e other repelling ; and that those particles »matter which repel each other, are subject | the general law of attraction in respect of her matter. A repellent matter being thus posed,’ equally dispersed through the uni- tse; he attempts to account for many na- tal phenomena by this means. He thinks ‘ht consists of this repellant matter put into Ment vibrations by the repellent corpuscles lich compose the atmosphere of the sun and s; and that, therefore, we have no reason believe they are gulfs of fire, but, like the it of heavenly bodies, inhabitable worlds. om the same principles he attempts to ex- iin the nature of fire and heat, the various enomena of the magnet, and the cause of ! variation of the needle; and, indeed, it difficult, if not impossible, by the doctrine uttraction alone, to account for all the phe- nena observable in experimeuts made with gnets, which may be solved by admitting doctrine of a repellent fluid; but whether will be sufficient to account for all the par- ilar phenomena of nature, which is the per test of an hypothesis; time and expe- ace alone must determine. The doctor » endeavours to show that the attractions of sion, gravity, and magnetism, are the ie, and that by these two active principles, ‘attraction and repulsion, all the pheno- aa of nature may be explained; but as his enious treatise on this subject is laid down series of propositions connected together, ould be impossible to do justice to his ar- ments without transcribing the whole; we ll therefore refer the curious reader to the ‘k itself. «ecording to Gravesande and others, when t is reflected from a polished spherical face, the particles of light do not strike n the solid parts, and so rebound from n; but are repelled from the surface t small distance before they touch it, by wer extended over such polished sur- ESIDUAL Analysis, a branch of analysis mted by Landen, and applied by him he solution of those problems which more generally solved by the doctrine luxions. This method was called the dual analysis, because, in all cases where » made use of, the conclusions are ob- ed by means of residual quantities. In analysis a geometrical or physical pro- 1 is reduced to another purely algebraical ; ‘the solution is then obtained without any osition of motion, and without consider- RES ing quantities as composed of infinitely small particles. The residual analysis proceeds by the difference of the same function of a vari. able quantity in two different states of that quantity, and expressing the relation of this difference to the difference between the two states of the said variable quantity itself. This relation being first expressed generally, is then considered in the case when the differ- ‘ence of the two states of the variable quantity ist 0: Mr. Landen published the first book of his “ Residual Analysis,” in 1764, and therein ex- emplified its application to several algebraic in- quiries, as well as in determining the tangents, evolutes, ordinates, points of contrary flexure; double and triple, &c. points; asymptotes, centres, &c. of curve lines. In the second book it was intended to show the application of this analysis in a variety of mechanical and physico-geometrical inquiries: but that book was never published, ResiDUAL Quantity, in Algebra, is a bino- mial connected by the sign —; thus a—b a— 4/6, &e. are residual quantities. RESIDUE or Restpuum, that which is left after taking a part of any thing away, being much the same as remainder, the for- mer being applied to quantity in the same sense as the latter is to number. RESISTANCE, or Resistine Force, in Physics, any power which acts in opposition to another, so as to destroy or diminish its effect. Resistances are of various kinds, arising from the nature and properties of the resisting bodies, the circumstance in which they are placed, and the laws by which they are £0- verned, ‘These may be divided into the fol- lowing cases: 1. The resistance between the surfaces of contiguous solid bodies, generally denomi- nated friction. See FRIcTION and ADHESION. 2. The resistance between the contiguous particles of the same body, whether fluid or solid; for the laws of which, see CoHESION and STRENGTH. 3. The resistance that solid bodies oppose to penetration; for which see PENETRATION and REPULSION. 4. The resistance of elastic and non-elastic fluids to the motion of bodies moving in them. The principles of which we will endeavour to illustrate in the present article. The resistance that a body experiences from the fluid medium through which it is impelled, depends on the velocity, form, and magnitude of the body, and on the inertia and tenacity of the fluid. For fluids resist the motion of bodies through them. 1. By the inertia of their particles. _ 2. By their tenacity, or the adhesion of their particles. 3. By the friction of the body against the particles of the fluid. In perfect fluids the latter causes of resist- ance are very inconsiderable, and therefore taking e > RES are not commonly considered. But the first is always very considerable, and obtains equally in the most perfect, and in the most im- perfect fluids. In what follows, and in all cases of a similar description, it will be necessary to distinguish between resistance and retardation; the for- mer being the quantity of motion, and the latter the quantity of velocity, which is lost ; therefore the retardations are as the resist- ances applied to the quantity of matter, and: in the same body they have always the same constant ratio to each other. In fluids of uniform tenacity, the resistance from the cohesion of its particles is as the velocity with which the body moves. For, since the cohesion of the particles is constantly the same, in the same space, whatever may be the velocity, the resistance from this cohe- sion will be as the space described in a given time ; that is, as the velocity. In a fluid whose parts yield easily with- out disturbing each other’s motions, and which flows in behind as fast as a plane body moves forward, the resistance will be as the den- sity of the fluid; for in this case the pressure on every part of the body is the same as if the body were at rest. And on the same hypothesis, the resistance from inertia will be as the square of the velo- city. For the resistance must vary as the number of particles which strikes the plane in a given time, multiplied into the force of each against the plane, and both these quantities varying as the velocity, the resistance which is measured by this product must vary as the square of the velocity. We here suppose the plane of the body to be perpendicular to its direction; but if in- stead of being so it is inclined to it in any given angle, then the resistance of the plane in the direction of the motion will be dimi- nished in the ratio of 1, to the sine cubed of the angle of inclination. For AB being the di- A. Cc rection of the plane, and A C that of the motion, BAC the angle, whose sine is 5; the number of particles, or quantity of the fluid, which strikes B the plane will be diminished in the ratio of 1 to s, (assuming radius unity), and the force of each particle will be likewise diminished in the same ratio of 1 to s; so that theaction of the fluid in the direction AC will, by com- bining these ratios, be as 1 tos*. But this action being in the direction AC, its effect, as to the resistance of the plane, will be far- “ther diminished in the ratio of 1 to s, whence upon the whole it appears that the resistance to a plane inclined to the line of. its motion, will be to the same, when the plane is per- pendicular to its motion, as 1 10 sine cubed of the angle of inclination. Hence making the velocity =v, the area of the plane = a, the specific gravity of the fluid = n, the force of gravity 323 feet = g ; then the altitude due to the velocity v being eee memes eecencseeese RES. i | vw A ; ae ; the whole resistance or motive force ; will be expressed by the following formula: v anv” _ anv*s3 m—anxX — =—; == ‘ 2g 22 22 when the plane is not perpendicular. to t] line of its motion. If w be made to represent the weight of t] body, and_/f the retarding force ; then on t same principles we derive 2 93 fz m _ anv's w 22w If the body be a cylinder moving in ‘ direction of its axis, and the diameter of | base = d, or radius 7, and + = 3°14159, & then p= mo mend?” __ wrnr*y wo) Sew | 2rw But if the body be a cone, then the say notation remaining, only writing s for the si} of the angle of. inclination of the side of 1 cone, then ee m _ xnd?v’s* _ wenr*y* s* Nel 2 EAR ae PT ee Rp . For in this case the inclination has 5 effect in reducing the section opposed tot resistance of the fluid, this being the same} in the cylinder, and therefore will vary as gi The same notation still remaining, it is fou| from a fluxional investigation that the resi, ance to the body, when terminated with; hemispherical surface, is, fat _aend*y” _ ernr*y ~ wow l6gw ” 4gew that is, half what it is when the end is a pl surface. Hence the resistance to a sphere impel} through any fluid is equal to half the dir of the same diameter. Since 2 wd3, is the magnitude of the glo] if N denotes its density or specific gravity, | weight w—=4d3N, and therefore the } tardive force i a fa mazn 6 3nv* _@ _ wo 16g «Ndi 8gNd~ 2g where s is the space described; fur 2fgs= by the laws of accelerated or retarded moti Hence we have s = a x d; which is n space that would be described by the glo while its whole motion is generated or stroyed by a constant force which is eq! to the force of resistance, if no other fof acted on the globe to continue its moti And if the density of the fluid were equa that of the globe, the resisting force is such acting constantly on the globe without ; other force, would generate or destroy, motion in describing the space 4d, or 4 of diameter, by that accelerating or retard force. Hence the greatest velocity that a balk ™ acquire by descending in a fluid by mean!) its relative weight in that fluid, will be for by making the resisting force equal to t weight. Lor, after the velocity has arrived , RES ha degree, that the resisting force is equal the weight that urges it, it will increase no ger, and the globe will then continue to cond with an uniform velocity. Yow N and » being the separate specific vities of the globe and fluid, N—vn will be relative gravity of the globe in the fluid, \ therefore w = 4 7d* (N —n) is the weight ony d* 1l6¢ 2 Jz ance ; consequently ar = ind’ (N—n) en the velocity becomes uniform; whence obtain 4 sae v=(V/ 28 x ay hep e) 3 n ‘the uniform or greatest velocity of the ye, ‘hus for example, if a leaden ball 1 inch in neter descend inwater, and in air of the e density as at the earth’s surface; the e specific gravities being, lead = 114, which it is urged, m = is the re- er — 1, andair — ; then 2 2000 a/ (2 eRe x sat x 103) = 85944 § 36 TA per second, for the greatest velocity in ‘er; and 193 4 34 a 4X —— X— K — x —— )= 259°82 v ( CECI Mar tone per second, for the greatest velocity in ut as this velocity, all other things being same, varies as /d; it follows that a ball § ofan inch diameter, would only acquire cities ,.th of those given above. is obvious, however, that all the preced- results are deduced upon an hypothesis TABLE L. Resistances to a ball of 1:965 inches diameter, and 16 oz. 13dr. weight. Resistances to a ball 2°78 inch. diameter, and 3lb, weight. weight. ! Vel Resistances. 1 Dif, 2d Dif. Vel Res, Difs, Vel. feet. Ibs. ounces. feet. Ibs, feet. 100 | O17} 23 gt 900 39 6 1200 200 | 0°69 li l yh 53 950 41 G 1250 300 | 1°56 20 2) 6 1600 47 G 1300 400 ; 281 45 7 7 1050 53 7 1350 500 | 4°50 72 35 8 1000 60 ~ 14.00 600 | 6:69 107 44 9 1150 67 7 1450 700 | 9°44 151 54 10 1200 74 8 1500 800 | 12°81 | 205 66 12 1250 $2 9 1550 900 | 16°94} 271 “9 13 1300 91 10 1600 1000 | 21°88} 350 92 13 1350 | 101 ll 1650 1100 | 27°63] 442 | 9, | 12 || 1400} 112 | 501 |} 1700 1200 | 34:13 | 546 115 ll 1450 | 1224 10° 1750 13600 | 41°31 661 124. 9 1500 | 1322 9 1800 1400 | 49°06} 785 | 43) 7 || 1550 | 1412] 3, 1500 | 57°25| 916 135 4 1600 | 150 3 1600 | 65°69 |} 1051 1 25 0 1650 | 158 2 1700 | 74°13 | 1186 133 —2 1700 | 165 6 1800 | 82°44! 1319 128 —) 1750 | i7l 5 1900 | 90-44| 1447 | (755 | —6 -|| 1800 | 176 2000 | 98°06; 1569 ny RES which cannot obtain in real practice ; because it supposes first, that the medium in which the body moves, falls in behind the body in mo- tion as fast as this moves forward, and that the body is therefore always in equilibrio with regard to the pressure of the medium, which it is evident cannot be the case except when the velocity is very small. It has moreover no reference to any other resistance than that which arises from the inertia of the particles of the fluid. And farther, these particles are supposed to be so constituted that after the body strikes them,- the action of these par- ticles entirely cease; whereas the particles after they are struck must necessarily diverge and act upon other particles behind them. On all of which accounts therefore there must necessarily be considerable difference between the theory and practice. To remedy, therefore, in some degree, the insufficiency of the theory, by combining it with practical results, experiments have been made at different times by several eminent mathematicians, viz. by Robins, of which an account was published in his Treatise on Gun- nery, in 1742; by Borda, Mem. of the Acad. of Sciences, 1763; by Mr. Edgeworth, of which an account is published in the Phil, Trans. vol. xxii. ; and by d@’ Alembert, Bossut, and Condorcet. But those of Dr. Hutton, carried on for some years at Woolwich, viz, from 1783 to 1786, are the most numerous and accurate of any hitherto made, and of which a circumstantial account is given in vol. iii. of his Tracts, lately published ; as also in the 2nd and 3d vols. of his ‘* Course of Mathematics,’ of which, however, we can only give the following abstract, with a few of the results thence deduced. TABLE III. Resistances to a bal! 3°55 inch diameter, and 6ib, 1 oz. 8dr. TABLE ITI. RES The analogy among the numbers in all these tables is very remarkable and uniform, the same general laws running through them all. The same laws are also observable as in the table of resistances near the end of the 2d volume, particularly the Ist and 2d remarks immediately following that table, viz. that the resistances increase in a higher proportion than the square of the velocitics with the same body; and that the resistances also increase ina rather higher ratio than the surfaces, with different bodies, but the same velocity. Yet this latter case, viz. the ratios of the resist- ances and of the surfaces, or of the squares of the diameters, which is the same thing, are so nearly alike, that they may be considered as equal to each other in any calculations relat- ing to artillery practice. For example, sup- pose it were required to determine what would be the resistance of the air against a 24lb. ball discharged with a velocity ‘of 2000 feet per second of time. Now, by the first of the fore- going tables, the ball of 1-965 inches diameter, when moving with the velocity 2000, suffered a resistance of 98lb.; then, since the resist- ances, with the same velocity, are as the sur- faces, and the surfaces are as the squares of the diameters; and the diameters being 1:965 and 5°6, the squares of which are 3:86 and 31°36, therefore as 3°16 : 31°36 :: 98lb. : 796lb.; that is, the 24lb. ball would suffer the enormous resistance of 796lb. in its flight, in opposition to the direction of its motion! And in general if the diameter of any pro- posed ball be denoted by d, and 7 denote the resistance in the first table due to the proposed velocity of the 1:965 inch ball; then its will denote the resistance with the same velocity against the ball whose diameter is d; or it is nearly ¥ d’r, which is but the 28th part g greater than the former. These resistances relating to certain and determinate velocity, the next object of in- vestigation, is to determine a rule or formula for the resistance with any velocity whatever, the result of which is as follows: Resistayce = r = '00002576 v* — 003880 in avoirdupoise pounds, v being the velocity. This rule °00002576* — :00388v = r, de- notes the resistance for the ball in the first table, whose diameter is 1°965, the square of which is 3°86, or almost 4; hence to adapt it to a ball of any other diameter d, we have only to alter the former in proportion to the false of the diameters, by which it becomes = RANG 00002576v?—.-00388v) = (0000066707 — "001v)d* = (000002 »* — ‘001 v) d*, which is the resistance for the ball whose diameter is d, with the velocity v. And, in a similar manner to adapt the the- orem '00001725v* = r, for the smaller veloci- ties, to any other size of ball, we must multiply it bya ee the ratio of the surfaces, by which it becomes -00000447 d?v By means of these formula the Doctor next RES investigates the height to which a body ascend that is projected perpendicularly | wards with any given velocity, supposing ‘ air to be uniformly of the same density as) the surface of the earth, from which he fi; the greatest height is v’ — 150v + 21090d 2god x oa where d is the diameter, and v the velocity projection. Hence supposing the ball to be one belo; ing to the first table of resistances, its wel being 16 oz. 13 dr. or 1:05 Ib. and its diame; 1-965 inches s; and the velocity of the proj} tion 2000 feet per second; the height to whh it will rise is 2920 feet, whereas without Ci sidering the resistance of the air, the proj tile theory would give 12 miles for the hei} due to that velocity. If the resistance, instead of depending uy the function of the velocity stated above,¥ taken as the square of the velocity, the 4 pression for the height becomes h = hyp. log. av kw w —_— J oe Wisse re x hyp. log. | where wis the weight of the ball a—=:00002 g the force of gravity, and v = the velocity projection. When the height is given and the tim required, this is found from the formula, t=xVe — Xx arc to tang. (or “| 4 the letters ihdicating the same quantities before. For various other formule and results equi! curious and interesting, see vol. iii. of 1 Hutton’s “ Course of Mathematics,” or| vol. iii. of his “ Tracts ;” Robins’s “ Gunnell and the other works above mentioned. 1% reader will also find several problems relat to the resistances on different bodies in Di try’s “ Fluxions;” in the works of Jay Bernoulli, and in the Acta. Euriditorum, 1695, and in various other places; but ast are founded on pure theory unaided by | periments, they are of little or no use practical cases. See also Gregory’s “ ] chanics,” vol. i.; and the article RESISTAN Encyclop. Brit. Solid of Least Resistance-—This is on the simplest of the class of problems, comme called zsopertmetrical, and admits of a gen and ready solution on those principles as be seen in p. 115 of Woodhouse’s ‘“ Trea on Isoperimetry.” This problem was | proposed and solved by Newton, “ Prine p- 324, aud has since been investigated other authors, as by Euler, ‘ Methodus Ii niendi,” &c. p.51; Simpson’s “ Fluxionsy. 487 ; Emerson’s “Fluxions .’ p. 183 ; Lacre ig Calcul. Diff.” tom. ii. p. 698; Macla r * Fluxions,” &e. &e. The figure is this. Let DNG be a of such a nature, that if from any point N ordinate N M be drawn perpendicular to axis AB; and from a given point G, ther drawn GR parallel to a tangent at N, meeting the axis produced in R; then if] RES be to GRas GR’ to4 BR x BG’ a solid de- scribed by the revolution of this line about its axis AB, moving in a medium from A to B, will be the solid of least resistance. E RESOLUTION, in a general sense, de- notes the dividing or separating any com- »yound quantity or thing into its original com- »onent parts. RESOLUTION of Equations, in Algebra, is the iletermination of the values of the unknown etters or quantities of which the equation is ‘omposed ; in order to which, it is necessary, rst to exterminate or eliminate all the un- nown quantities but one out of the equation, nd then the value of the remaining quantity 3to be found by the proper rules for this pur- ‘Ose, viz. by the rules given for Simple, Qua- ratic, Cubic, or Biquadratic Equations, ac- ording to which of these it may belong; or y the general method of Approximation; for rhich see the respective articles. | But as all these cases have reference only to iis place to explain some of those methods hich are most commonly employed for re- cing equations to this state. | First it may be observed, that in any deter- inate problem, there are always as many juations independent of each other as there ‘e unknown quantities; if there are not so any, the question is indeterminate, and if ere be more it is ¢mpossible. ‘We cannot in a limited article like the pre- nt give all the methods that may be em- oyed for exterminating the unknown letters, hich are extremely various, and depending ‘ach upon the practice and proficiency of the lalyst himself, and the manner in which ‘Ose quantities are involved; but the most plicable and general methods are the three lowing, viz. 1, Find the value of one and the same un- ‘own quantity in each equation, and put all 8se values equal to each other, which will minate one of the quantities, and reduce number of equations to one less. Then the same in these new equations; and ain in the last, and so on, till there be but @ equation and one unknown quantity, the ‘ue of which must be found by the proper es, as above referred to. 2, Find the value of one of the unknown antities in one of the equations in terms of other quantities ; then substitute this value that quantity in all the other equations. fain, find the value of one of the remaining antities, and substitute its value as before, d so on, till there remain but one equation he unknown quantity, it will not be amiss in. RES and one unknown quantity, whose yalue is to be found as before. . 3. Multiply each of the equations by such a number as will render the co-efficients of one of the letters the same in all, then by adding or subtracting these equations according, as the equal co-efficients have unlike or like signs, the quantity whose co-eflicients were equal will disappear; which being repeated again upon the remaining quantities, there willultimately be found only one equation and one unknown quantity. See ELIMINATION. And it may be proper to observe, that in all these cases, if any of the unknown quantities have fractional .co-efficients, the whole equa- tions in which they are found should be mul- tiplied by such a number as will convert these fractions into integers. Thus in equations, _ fiat Ay = 10 tic + 95 Multiply the first by 15 and the latter by 2, gives om — — —- §9x +2y = 150 x+6y—= 190 The solution of which, by each of the pre- ceding rules, will be as follows. Or putting letters instead of the above nu- merical co-eflicients, in order to render the solutions more general, let there be given qe + a = to find x and y. Ist Method. ax = c — by dx =f—ey oe PERRY by transposition. yey d a de—dby =af — aey by multip. (ae — dbjy =af—de x by division. __ af—de Chae ae — db And in the same manner we find {, SRO b ~ ae — db. 2d Method. ax + by =e | de + ey av to find x and y c—by L = cae AG above con hy + ey =f by substit. de—dby + aey = af by mult. (ae—db)y= af —de Ah de y ee Seemann ae — db \ 45 before. % _ec—bf — Ge-a 3d Method. ax +by=e dx ey me to find x and y. dax +-dby = de mult. by d adx + aey = af mult. by a (db —ae) y= de — af } _de—af__af—dc I~ db ae ae—db __ ec—bf db—ae™~ ae—db This will in some measure illustrate the preceding ane which we shall not insist upon QQ as before. RES | RES any farther; but, in the following examples, (e+y)xr=a/(a+b)=a shall avail ourselves of any advantages that («+y)yry V(at+b)=)b the equations may present, in order to arrive at AL a aus b the solution in the easiest manner possible. — S(a + b) ie /(a + b) EXAMPLE lI. EXAMPLE 6, Given «* + y? Find three numbers in arithmetical pro- xy = b . to find # and y. gression, whose sum is a, and sum of their ne) ya oe squares b. Qay = 2b doubling the 2d. Let «—y, « and x + y be the numbers. a + 2ayt y —at+2bby addition Thent—y +e + x+y =3xr—a | a seme # =a—2b by subtract. Vea ste ae hat yy —3 x+y V(a+2b)) . a —% — OO | x —y = (a—2b) § by extraction. 9 3 | _ o(a +26) + V(a— 2b) Whence a* = 3b — 6y” Less ORT Pe Le 3b —a? a | 2 a i and ¥ —/ and 2 = =. _ V(a + 2b)— Va — 2) 6 teas sl co 2 And a similar substitution, vzz. one whic! 3 answers one of the conditions of the question EXAMPLE 2, may frequently be employed to great adval Given? +y =a) tage. ; xy = bS Ketan x and y belit F diet | a+ ery + y= a* by squaring Sometimes it will be convenient to substi ee As ee me 46 mult. by 4 tute for the sums and differences of number vo — oxy + ie OR. ca as in the following example: | BT aga ere AI ee extract. Given x +y =a | x + y=a Ist equation a7, to find = and y. | pie Cat / (a* — 46) porte Sie aVaieas oe og ; 2 and x—y = 2n ly=m—n | Ay SAE Te / (a* — 4b) (m + n)* + (m—n)* = J, or bing 2 Q2m* + 12m*n* + 2n* — |), ; EXAMPLE 3. but 2m = a, or m =5 therefore m is knoll Given x—y—a) and the above becomes aye b gto finds andy 2n* + 12m7*n” = b—2m* x*—2xy + y* = a’ by squaring n+ + 6m?n* = 7b—m* : ae = 4b mult. by 4 n> = —3m* + v(1b + 8m*) a + 2xey +y* =a’ + 4b by adding te mt) Ut e+ y= V(a* + 46) by extract. and 1 = }— 8m t/t 6 4, Bin") § ‘a c— y =a 1st equat. Whence m and n being known, x and y @ hype V(a*>+4b)4+a also known for x = m + n and y = m—”. 2 ‘i a EXAMPLE 8. om AS daa Sometimes itis advantageous to consider 0} of the quantities as an unknown multiple! the other; thus | eae Given cya — yo +y? to find x ang Given ar ad Viggen an: and y. Make y=zza, then these become a +y’ bs 2u7 = 2*—27 x’ x*—2ay + y* =a* by squaring Ca ae 4 ee { 2x7 + 2y* = 2b doubling z = 1— 2” from the Ist x* + 2ry+ y* = 2b— a? subtr- 24+ 2-lorz=—i+ : + y=v2b—da?’) whence z is a known quantity. Now, — = 3 z—-a-+ 232 from the 2d ie ees ts ne coe ee —J1l+23.-, l—@—t yb a av (2b—< a) 4 ‘id + Ln a erg 2 2 VY) eg eo UNDE TT Bae a ee This method of resolution will apply in y= zx—=—t++1 vY5,as required. many problems, and sometimes saves consi- These and ri 4 derable labout € and a variety of other artifices pe liar ee Sab equations, will occur to practical analyst; of which numerous _ EXAMP . fi é és, LES amples may be seen in Bland’s “ Algebra iven x +xy=aQ to find « andy. Problems ;”’ as also in Bonnyecastle’s if yo tay = by Euler’s “ Algebra.” a+ Q2xy+y*—atb by add. ResoLution of Geometrical Problems, al t+ y= v(a+6) by extract. braically. See APPLICATION. . RET Reso.uTion of Motions and Forces. PARALLELOGRAM of Forces. REST, in Physics, the continuance of a body in the same place, either absolutely or relatively; viz. its continuance in the same part of absolute space, or in the same part of relative space; and is hence denominated absolute or relative rest. It is, however, highly probable that in its most extensive sense there is no such thing as absolute rest ‘in the whole creation, at least we know of nothing in such a state. RESTITUTION, Motion of, is used by some to denote the return of elastic bodies to their natural form, after compression or bend- See | ing. ‘RESULTANT, in Mechanics, is used to denote that single force, or the line repre- sent the quantity and direction of that single force, which is equivalent to two or more ‘forces whose quantities and directions are given. RETARDATION, any force tending to diminish the velocity of moving bodies. RETARDATION may arise either from the effect of resistance, or from the action of gra- vity. For that which arises from the former, and for the distinction between resistance and retardation, see RESISTANCE. RETARDATION, from Gravity, is peculiar to bodies projected upwards, which have their velocities diminished, by precisely the same laws as falling bodies have theirs acce- lerated. Thus if a body be projected perpendicularly upwards, with a velocity which would, inde- pendently of gravity, cause it to ascend a feet per second; it will, in consequence of the action of gravity, have its velocity su diminish- ed, that, at the end of the first second, it will de only (a—32) feet, at the end of the second t will be only (a — 64) feet, &c. Hence to find the greatest height to which a dody will ascend when projected perpendicu- iarly upwards with any given velocity, the time of ascent, &c. it is only necessary to find the space through which a body must fall to gene- tate that velocity, and the time it would be in lescending through that space, which will be wrecisely the same as the height through which t will ascend, and the time of. its ascent. jee ACCELERATION. RETICULA, or RETICULE, the name of an ustrument formerly employed for measuring he number of digits eclipsed in either lumi- lary ; its construction depends on nearly the ame principles as that of the MIcRoMETER. RETROCESSION of Curves, the same as LETROGRESSION. ReEtTROCESSION of the Equinoxes. See PRE- ESSION. | RETROGRADATION, or ReETROGRES- ION, in Astronomy, is an apparent motion of he planets, by which they seem to move back- yard in the ecliptic, or in antecendia, or con- rary to the order of the signs. ' When a planet moves in consequentia, or ccording to the order of the signs, Aries, RET Taurus, Gemini, &c. it is said to be direct. When it appears for a few successive days in the same place or point of the heavens, it is said to be stationary. And when it goes in antecedentia, or contrary to the order of the signs, it is said to be retrograde. Both the superior and inferior planets are subject to this apparent irregularity in their motions, but it arises from different causes, as may be illustrated as follows: Let CDed repre- a 1 sent the orbit of the _~ earth, and A Bad that of any inferior planet,as Venus; now when the earthis ate and Venus at A, the former being moving towards d with a less velocity than the latter is towards B, it is obvious that the ap- parent motion of Venus, as referred to the heavens MN, will be from M towards N, or according to the order of the signs. But if, when the Earth is atc, Venus is at a, then her apparent place in the heavens will be at N; and when the Earth is arrived at d, Venus will have come to 4, and her apparent place in the heavens will be at M, and consequently during this time she will appear to be moving in the heavens from N to M, viz. contrary to herformer motion, which motion being inante- cedentia, or contrary to the order of the signs, is then said to be retrograde. Whence it appears that when an inferior planet is in, or nearly in its superior conjunction, its apparent motion is direct, but when in its inferior con- junction, its apparent motion is retrograde, and for a few days between these two she has no apparent motion, and is therefore said to be stationary. With regard to the superior planets it is obvious that their retrogradation must happen when they are in opposition; thus, if we now suppose A Bab to be the orbit of the earth, and CDed that of Saturn, and suppose that when the latter is at C, the former is at A, then the apparent place of Saturn in the heavens will be atn; but the motion of the Harth exceeding that of Saturn, when we are arrived at B, he will only be got to D, and his apparent place will be at m; and therefore during this interval his apparent motion will be retrograde. Whereas it is obvious that had the earth been at a, instead of being at A, as we have supposed above, that is, if Saturn had been in conjunction, its apparent motion in the heavens must have been contrary to the former, or in consequentia, or according’ to the order‘of the signs ; and for a time between these two motions, he must necessarily have appeared stationary, as is obvious without any particular illustration. The periods of retrogradation of the several planets are not always the same, but at a mean they are nearly as follows: Saturn 140 days, Jupiter 120 days, Mars 73 days, Venus 42 days, and Mercury 22 days. QQ2 ae REV RETROGRADATION of the Lunar Nodes, LipRATION. RETROGRESSION, Point of. See INFLEC- TION. REVERSION of Series, in Algebra, is the method of finding the value of the root or unknown quantity, whose powers enter the terms of a finite or infinite series, by means of another series in which it does not enter. Thus if we have, x= ay + by* + cy? + dy* + &e. and we can find y = Aw + Ba? + Cx? + Dat + Ke. the original series is said to be inverted. The reversion of series was first proposed hy Newton, in his “ Analysis per Equationes Numero terminorum Infinitas,’”’ and has since engaged the attention of many of the most profound modern analysts; and accordingly different methods have been suggested for this purpose, but that of M. Arbogast, in his * Calcul. des Derivations,” is the most com- plete, and we regret that the nature of this work will not admit of sufficient detail to enter into an explanation of his notation and ope- ration; the reader, however, will find every necessary information on this head in Wood- house’s “ Principles of Analytical Calcula- tion,” arts. 72 and 73, to which work he is therefore referred, and we shall merely give here the more simple but less general method of our own algebraists. This consists in assuming a series of a pro- per form, for the required unknown quantity, and then substituting the powers of this series, instead of the powers of that quantity in the proposed series, and finally equating the co- efficients, whereby the values of the indeter- minate or unknown co-efficient will be ob- tained; thus, Letamay + by? + cy? + &e. Assume y= Ax+ Ba?+ C2? +kc, See then ay—aAx+aBa* + aCa} 4+&e, by* = bA’ 2? + 20ABxi +&e. 5 pha cA3 23 +e. &e. &e. Now since x =ay + by” + cy? + &e. And this latter series being also found in powers of x, it follows that the sum of all the co-efficients of « must be equal to 1, and the co- efficients of the higher powers equal to 0, which depend upon a known property, viz. that if ex + Bx + ya? + &c. =0, under every pos- sible value of x, then a= 0, 8=0, y = 0, &c. Hence we have aA = 1 aB + bA® = 0 aC +2bAB +cA3? =0 3 _1, _—bA*_ —6 Consequently A = B= oe (eed ‘ C _—¢cA3 2bAB __ 26?—ae = " a err a> Whence Big Ee y = tem 5 eine “r ~~ x3, &e, Again, let x = y —ay? + by — Ke. to find y in terms of x. Assume y =Ax+ Bx? + CxS +&ce. REV they y =Axt+ Ba? + C25 +e, —aye = —aA3x3—3aA7 Bai —&e, + by = + bASx5 +&¢, And hence A =1; B—aA?=0; ~ C — 3aA*B+bA5 —0. Consequently A =1; B =a; c= 3a*—, &e. therefore y = x + ax3 4+ (3a* — b) x5 + &e, In the above examples we have only carried the reversion to three terms, the same method would, however, have led us to a fourth, fifth, &e. term; but it must be observed, that be- yond this, the co-efficients become totally unmanageable, except in particular cases, which is the great imperfection of this method, | and the obviating of this difficulty is what constitutes the excellency of that of M. Ar gobast, above alluded to. This author, by means of his peculiar notation, arrives at the law of the co-efficients of the reverted series, which may therefore be continued at pleasure,| though it must be allowed that even in this case, the absolute reversion of a given ceri is a matter of considerable difficulty. See W oodhouse’s Anal. Calcul. ; De Moivre, Phil. Trans. No. cexl.; Maclaurin’s Algebra, part ii. ch. 10; Newton’s Analysis, p. 455, and his Fluxions, p. 219; Simpson’s Fluxions;, Bonnycastle’s Algebra; and Arbogast, Calcul.| des Derivations. | REVERSION, in the doctrine of Annui-| ties,is anannuity which is not to commence till after a certain number of years; and its pre sent value is such a sum as, put out to in-. terest, will provide for the several payments of the annuity as they become due; and in order to obtain this value, we must find the present value of each reversionary payment, and the sum of them will be the total present value of the reversion. Or if we make p = _ the present value a= the annuity eb se number of years the an nag nuity is to continue ; the number of years before i becomes. payable the annual rate of interest. y , Then ag tek GQ + r)-7— (1 +.7)-0t pre ai f C=PX tO oF) co. log. ; 1 — Fr Af ry} n= —_——————————que— p log. (1 + 7) § ~. pr log. --- —log, — og rv! (1 +r) ; log. re log. (1 + 7) ; 123-(@—1)8t 2 12)—2(n? 18 7 w cece 2 ‘ In which last ) = 2y+-n+1, and B (2 3 P From which theorems any one of thos: quantities may be found when the other qua tities are given, When the reversion is ¢ perpetuity then » becomes infinite, and th | RHE quantity (1 + r)-@+™) vanishes out of the _ general expressions, which then is reduced to (1 + »)-v r pmax = present value a=pr(l+r) annuity co. log. 22 - a ie log. (i +r) 16+ G+ ~e8 Se teat ete Se ann! rate of int. 6+ 4(27+ 1)8 where 8 = tetiok flee 1. It should be observed, that these theorems ‘are formed on the supposition of yearly pay- ‘ments. If the payments are to be made quar- ‘terly, or half-yearly, then n and » must repre- ‘sent the number of periods of payment, and fi the rate of interest for such period of time. _ For the investigation of these theorems, see ‘Baily’s “‘ Doctrine of Interestand Annuities ;” where this, and every case relating to these important subjects, are very fully and ably illustrated. “REVOLUTION, the motion of a body or line about a centre, which remains fixed. Period of REVOLUTION, in Astronomy, is the time a planet, comet, &c. employs in pass- ing from any point in its orbit to the same point again. ‘This, with regard to the earth, is what determines the length of the year. See Year. And for the times of revolution of the other planets, see PeR1op and PLANET. REY NEAU, (CuarLes RENE) a reputable French mathematician, was born in 1650 at Brissac, in Anjou. He taught philosophy at Pezenas and Toulon; and in 1683 was ap- dointed to the mathematical professorship at Angers. He published ‘‘ Analyse Demon- rée,” 2 vols. 4to., a popular work, in which 1e reduced to one body the theories of New- ion, Descartes, Leibnitz, &c. He also pub- ished “Science du Calcul des Grandeur,” + vols. 4to.; and a small tract on Logic. Reyneau died in 1728, at the age of 78 years. ~RHABDOLOGY, a name given by Napier 9 his method of performing multiplication, vision, &c. by means of small bones or rods. ‘ee Napier’s Rops. ' RHETICUS (GeorGe Joacui), a distin- wished German astronomer and mathema- ‘clan, was born at Feldkirk in the Tyrol, in 514, and for some years assisted the cele- rated Copernicus in his astronomical labours; those work ‘“ De Revolutionibus,”’ was pub- shed by Rheticus after the death of its au- 10r, the latter having died the very day the cinting was completed. Rheticus also, in ‘der to facilitate astronomical calculations, ndertook his elaborate canon of sines, tan- ents, and secants, to 15 places of figures, ad to every 10 seconds of the quadrant a 2sign which he did not live to complete, but hich was at length accomplished by his isciple Valentine Otho, in 1596. His “ Narratio de Libris Revolutionum Co- = years deferred RHU pernici,” was first published at Gedunum, in 4to.in 1540; and was afterwards added to the editions of Copernicus’s work. He also com- puted Ephemerides, according to the doctrine of Copernicus, till the year 1551; besides some other works which were never published. He died in 1576, near 63 years of age. RHUMB Line, or Loxodromia, in Navica- tion, is aline prolonged from any point in a sea chart, except in the direction of any of the four cardinal points, or it is the line de- scribed by a ship while her course is con- stantly directed towards one and the same point of the compass, except the four above mentioned, that is, while she crosses all the meridians at the same angle, providing this is not aright one, and this angle is called the angle of the rhumb; and that which it makes with the equator, or a parallel to the equator, is called the complement of the rhumb. If a vessel sail either north or south, it evidently describes a great circle of the sphere, or part of such a circle, and if her course is either due east or west, she cuts all the me- ridians at right angles. But if her course is oblique to these principal points, then she no_ longer describes a circle, but a sort of spiral, the characteristic property of which is, that it cuts all the meridians at the same angle, and is thence denominated the loxodromia, or loxodromic curve, or rhumb line, which, though it continually approaches towards the pole, can never arrive at it, except after an in- finite number of revolutions. A ship’s way is, therefore, not to be com- puted, as if her course was made in a great circle of the sphere, but as made up of the successive arcs of this spiral, the principles of which may be illustrated as follows: Let P be the pole, RW the equator, AB CDEP a spiral rhumb, divided into an indefi- nite number of equal parts at the points B, C, D,&c. through which are drawn the meri- dians PS, PT, PV, &c. and the parallels FB, KC, LD, &e. also draw the parallel AN. Then 5 as a ship sails along the rhumb-line towards the pole, or in the direction ABCD, &c. from A to E, the distance sailed AE is made up of all the small equal parts of the rhumb AB + BC +CD + DE; and the sum of all the small differences of latitude AF + BC + CH + DI, make up the whole difference of lati- tude AM, or EN; and the sum of all the parallels FB + GC + HD + IE is what is called the departure in plane sailing; and ME is the meridional distance, or distance he- tween the first and last meridians, measured on the last parailel; also RW is the difference of longitude, mearured on the equator. So that these last three are all different, viz. the departure, the meridional distance, and the difference of longitude. REC If a ship sail towards the equator, from E to A, the departure, difference of latitude, and difference of longitude, will be all three the same as before; but the meridional dis- tance will now be AN instead of ME; and one of these AN being than the departure FB + GC + HD + IE, and the other ME is less than the same; and indeed that de- parture is nearly a mean proportional between the two meridional distances ME, AN. Other properties are as follows: 1. All the small elementary triangles ABP’, BCG, CDH, &e. are mutually similar, and equal in all their parts. For all the angles at A, B, C, D, &c. are equal, being the angles which the rhumb makes with the meridians, or the angles of the course; also all the angles, F, G, H, I, are equal, being right angles; therefore the third angles are equal, and the triangles all similar. Also the hypothenuses AB, BC, CD, &c. are all equal by the hypo- thesis; and consequently the triangles are both similar and equal, 2. As radius; distance run A E :: sine of course 2 A: departue FB + GC, &e. :: cosin. of course 2 A: dif. of lat. AM. For in any one ABF of the equal elemen- tary triangles, which may ‘be considered as small right-angled plane triangles, it is as rad. or sin. 2 F': sin. course A:: AB: FB:: (by composition) the sum of all the distances AB + BC + CD, &c.: the sum of all the departures FB + GC + HD, &c. And in like manner, as radius : cos. course A:: AB: AF:: AB + BC, &c.: AF + BG, &e. Hence of these four things, the course, the difference of latitude, the departure, and the distance run, having any two given, the other two are found by the above proportions. By means of the departure, the length of the rhumb, or distance run, may be connected with the longitude and latitude, by the fol- lowing two theorems: 3. As radius: half the sum of the cosines of both the latitudes of A and E:: dif. of long. RW : departure. Because RS: FB:: or cos. RA, and VW: 1E:: radius: sine of PE, or cos. EW. radius : sine of PA, 4, As radius : cos. middle latitude :: dif. of, longitude : departure. Because cosine of middle latitude is nearly equal to half the sum of the cosines of the two extreme lati- tudes. RICCIOLI (Joannes Baptist), a learned Italian astronomer and mathematician, was born at Ferrara, a city in the papal dominions, in 1598, and died in 1671, in the 73d year of his age. He was author of the three follow- ing works, viz. 1. Almagestum Novum, in 2 vols. folio, Bologna, 1651. 2. Astronomia Reformata, 1665; and 3. Chronologia Re- formata, 1671. RICOCHET Firing, in the practical part ‘ee | RIN of gunnery, is a method of firing with small | charges, and at small degrees of elevation, — viz. from 3° to 6°, in consequence of which the ball is constantly bounding and rolling | along, and thus destroying more men than | that which has a charge sufficient for pene-_ trating a column. RIGEL. See REGEL. | RIGHT Angle, Cone, Cylinder, Sphere, &c. See the respective substantives. | RIGIDITY, a brittle hardness; or that | kind of hardness which is supposed to arise | from the mutual indentation of the component | particles of a body. RING of Saturn, in Astronomy, is a broad | opaque circular body encompassing the equi-’ torial regions of that planet, at a considerable distance from him; which presents, under favourable circumstances, one of the finest telescopic objects in the heavens. An ap-' parent irregularity was first observed in the form of Saturn by Galileo, but his telescope! was not sufficiently powerful for him to be | cover the cause of it; this, however, was soon after effected by Huygens, who in cone quence published his ‘‘ New Theory of Sa- turn,” in 1659. ‘ This ring, which is very-thin, not exceeding’ 4500 miles, is inclined to the plane of the} ecliptic in an angle of 31° 19’ 12”; and re-| volves from west to east in 105 29™ 16.8,) being nearly the time of the diurnal revolution of Saturn, and which is also found, from the. laws of Kepler, to be the time in which a satellite would revolve about that planet et the mean distance of the ring; a very re-| markable confirmation of the universality of the laws of the planetary motions. This ro-) tation is performed about an axis perpendi- cular to the plane of the ring, and passin through the centre of the planet. The ring being, as we observed above, very thin, it sometimes nearly disappears, that is) when its plane coincides with, or passes) through the centre of the earth or sun, af which time it subtends an angle of not more than half a second, and can therefore only bé discovered by the most powerful telescopes) through which it has then the appearance 0 a luminous line beyond the body of the planet: And as this plane is presented to the sur twice during each sidereal revolution of thi planet, the disappearance of the ring wil) happen abont every 15 years, and at nearly the same intervals it will appear to the great, est advantage. When viewed in the most favourable posi, now ascertained to be a real separation, an that what we call the ring of Saturn consis | at least of two rings; and some astronomel have even supposed it to be still farther subd’ vided, and to consist of several circular part but this at present is little more than conjec ture. vs RIV The dimensions of this double ring, as given by Dr. Herschel, are as follows: Eng. Miles, . Diameter of the planet............ .» 76068 Inside diam, smaller ring .......0. 146345 | Outside............ ccceceeeseeeeeseeene ... 184393 Inside diam. Jarger ring ............ 190248 Oe Seas det 155i ue avishss de vatnel SUEROD Breadth of inner ring..........0+-+00 20000 - of outer ring.......c0ceeee. 7200 Space between the rings............ 2839 betweentheplanetandring 70277 _ Mean thickness of ring ........... 4500 , The intersection of the plane of the ring with the ecliptic is in 5* 20°, and 115 20°, in which points therefore it disappears, and be- itween these, viz. at 2° 20°, and 8* 20°, it ap- ypears most brilliant. As the plane of the ring coincides, or nearly seoincides with the equator of the planet, it is obvious that the ring never becomes visible in this polar regions, and even in those parts »mearer the equator it must be very frequently ‘eclipsed by the planet, which last, on the contrary, is as frequently eclipsed by the ring; hence it does not seem that it can be de- signed to supply these remote regions with light as some have supposed, as it rather serves to render them more dreary and comfortless ; well might the poet say, ~ One moment’s cold like theirs would pierce } our bone, | Freeze our heart’s blood, and turn us all to ) stone.” , RISING, in Astronomy, the first appearance _of the sun, moon, or other celestial body above (the horizon. RIVER, in Geography, a stream or current of fresh water flowing in a bed or channel from its source or spring into the sea. The doctrine which relates to the flux, re- flux, motion, and discharge of rivers, is a branch of hydraulics, and as such forms a part of the present work, though our limits will only admit of a slight sketch of the theory. Water running in open canals or rivers is ac- ‘celerated in consequence of its depth, and of the declivity on which it runs, till the resist- ance increasing with the velocity, becomes equal to the acceleration, when the motion of the stream becomes uniform. But this re- sistance it is obvious can only be determined by experiment, and hence several philosophers have undertaken different courses of experi- ments for this purpose, amongst whom Buat seems to have met with the most complete ‘success. See his “ Principes d’Hydraulique,” 2 vols. 8vo. 2d edit. Paris 1786; of which work an abridged account is given in the ‘Encyclopedia Britannica under the articles ‘Rivers and WaTerR-worKs, whence is de- duced the follownig theorems: Let V represent the velocity of the stream per second in inches; R the quotient arising from the division of the section ofihe stream by its perimeter, minus the superficial breadth, allin inches; and S the cotangent of the in- eo J clination of the slope. Then the section and - velocity being both supposed uniform, 307 v(R— +4) v= 3 vv (R-); S2—hyp. log. (S + 4$ ro ¥( Yo) or V=v(R—y)x § 7 __ -3} S2—Eh4.1(S8 +48) 10 which when R is very great, and s small, may be reduced to 307 3 Woo Ha Reger t Seb set KT § 10 From which it appears that when the slope remains the same, the velocity varies as “”(R— zs), Or as / R, when R is very great. Hence the velocity of two great rivers of the same de- clivity are as the square root of , Where bd b + 2d b and d represent the breadth and depth of a transverse section in inches. It follows, also, from what is said above, that if R = 54, or is less than that, the ve- locity is zero, which agrees with the theory of capillary attraction; also the slope may be so small that the other factor may become zero, or 307 Pe Seo bail (Si4 Oy 20 in which case likewise there will be no mo- tion; this, however, can never happen, if the declivity be not less than =4th of an inch in an English mile, as this will produce a sensi- ble motion in the water. In a river the greatest velocity is at the surface and in the middle of the stream, from which it diminishes towards the bottom and sides where it is least; and it has been found by experiment, that if v = the velocity of the stream in the middle, in inches, then vo —2 /v + 1, is the velocity at the bottom. Prin. d’Hydraul. par Du Buat, art. 67. The mean velocity, or that with which (were the whole stream to move) the discharge would be the same as the real discharge, is equal to half the sum of the greatest and least velocities, as computed by the above formula. Hence the mean velocity is v— V/v + Zz, Suppose that a river, having a rectangular bed, is increased by the junction of another river equal to itself, the declivity remaining the same; required the increase of depth and velocity. Let the breadth of the river equal 6, the depth before the junction d, and after it a; the velocity before v, and after it v'; then the quantity denoted by R, in the preceding for- mula, is R = j — before the junction, and oD : F RAs at after the junction. 1 AS Sov= eat ni and vo! = wats vk that is, by 82 S?2 supposing the breadth of the river to be such, that we may reject the small quantity suh- OB tracted from R, R’, in the above formule; whence substituting for R and R’, we have _ 207 bd i= 3} ba oa ant vy — 307 x bx a Re 6422 Now multiplying these into the areas of the section 6d, bx, we have the discharges; viz. 307 bd evant: Ae . d bdv = = x bav/ 5a an ,__ 307 : bx bav' = Ky x bx YS Now the last of these is double the former; therefore 7, Seer nes J(b +22)” V(b 4-2d) bes oan Nar 6, i: whence P+ Qe boa 4 3 d3 4bd3 ‘dl b+ 2d eat | + 2d from which equation 2 may be determined. As an example, let 6 = 10 feet, d= 1, then ys i = = aD and a2 = 1°4882, 3 3 which is the depth of the increased river; hence we have 1°488 x v' = 2v, and 1°488; 2 7.0: v,0rv0:% °: 37: 50 nearly. When the water in the river receives a per- manent increase, the depth and velocity, as in the example above, are the first quantities that are augmented, the increase in the ve- locity increases the action on the sides and bottom; in consequence of which the width ts augmented, and sometimes, though rarely, the depth also. The velocity is thus dimi- nished till the tenacity of the soil, or the hardness of the rock, affords a sufticient resist- ance to the force of the water. The bed of the river then changes only by insensible degrees, and is said to be permanent; though, in strictness, this is not applicable to the course of any river. When the sections of a river vary, the quantity of water remaining the same, the mean velocities are inversely as the areas of the sections; this being necessary to preserve a uniform discharge. Playfair’s Outlines of Natural Philosophy. RIX DOLLAR, a silver coin in different countries on the continent, and of different values, viz. s. d. Rixdollar of Basil............ 3 64 Denmark...... 4 63 Hamburgh,... 4 62 Holland........ 4 63 Lubeck........ 3 St Poland......... 4 OL BTUSSia.....:i4e 3.0 Sweden........ 4 74 Germany...... 4 8 stand.of 1566 BRAC sokhinss 5 4 2 ditto 1753 ROBERVAL (Gites PERSONNE), an emi- nent French mathematician, was born in 1602, ROD : and died in 1675, at the age of 73. He was author of-several papers in the Memoirs of the Academy of Sciences for 1666, on the theory of equations; the theory of indivisi- bles; the properties of cycloid, &e.; besides; which he published two works, viz. “ A/ Treatise on Mechanics,” and another entitled “ Aristarchus Samos.” He also gave name; to certain species of curve lines, as described in the following article. ia ROBERVALLIAN Lines, a name applie to denote certain lines which bound spaces: 1646, | 1632; and it was asserted by the abbot Gallois, in the Memoirs of the Academy for, 1693, that this was precisely the same in prin-. ciple as that published by James Gregory, im his “ Geometria Universalis,” and that the! latter owed his knowledge of the subject to} the discovery of Roberval, having obtained] his information during his visit in Italy, im 1668. To this David Gregory replied in the} Phil. Trans. for 1694, which was again any swered by Gallois in the Memoirs of the French Academy, in 1703. My ROBINS (Benjamin), a distinguished Eng- lish mathematician, was born at Bath, in| 1707, and early discovered very superior talents, particularly for mathematical and philosophical subjects, which he afterwards) pursued with the greatest possible success, He was author of several detached papers in) the Phil. Trans. and some other tracts, which were all, (wz. those relating to mathematicak and philosophical subjects), collected and pub4 lished after his death, in 1761, in 2 vols. 8yo) These works relate to various subjects, viz New Principles of Gunnery, first published! in 1742, an account of various experiments on the resistance of the air, the flight of rockets, &c. In the second volume is giver his discourse concerning the Doctrine « Fluxions, and Prime and Ultimate Ratios intended as an answer to the Analyst, be sides other subjects. ; Dr. Hutton, in summing up the character o| this author, says, it is butjustice to Mr. Robins to state that he was one of the most accurate and elegant mathematicians our language cap boast of; and that he made more real improve ments in artillery, relating to the flight ané resistance of projectiles, than all the preceding writers on that subject. His New Principles of Gunnery were translated into several othe) languages, and commented upon by several eminent mathematicians. The celebrated Euler translated the work into the Germar language, accompanied with a large and cri: tical commentary, which work of Euler’s was again translated into English by Brown, with notes, in one volume, 4to. A ROBUR Carolinum, the Royal Oak. See CONSTELLATION. | ROD, or Pole, a long measure of 16% linear feet, or a square measure of 2721 square feet, ls ROO ROEMER (Otavs),a celebrated Danish ma- ematician, was born in Jutland in 1644, but ent several years at Paris, during which time : discovered the progressive motion of light by yservations on the eclipses of Jupiter’s satel- }es; to him also we owe the theory of epyci- joids, and their application to the teeth of heels in machinery, which De La Hire after- jards wished to appropriate to himself. Roe- er was author of various papers in the yemoirs of the Academy of Sciences of Paris, ym vol. i. to vol. x. and was preparing to jblish the result of his numerous astrono- ijeal observations, when death put an end to % labours, Sept. 19th, 1710; this work, how- jer, was afterwards accomplished by his iciple Horrebow, under the title of “ Basis itronomiz ;” Copenhagen, 1753, 4to. (ROHAULT (James), a French philosopher, 18 born in 1620; and very early became a jeat admirer and advocate of the philosophy Descartes, in defence of which he wrote i) “ Physics,” which was afterwards trans- ved into Latin by Dr. Clark, with correc- ns, the best of which is that of 1718. )Rohault was also author of ‘‘ Elemens des athematiques ;” “ Traité de Mechanique ;” Entretiens sur la Philosophie,” &c. His 'sthumous works were collected and printed two volumes, first at Paris, and afterwards ithe Hague in 1690. These relate to tri- nometry, geometry, perspective, &c. He d in 1675, at the age of 55 years. ROLLE (MicuHeEv), a French mathema- tian, born in 1652, and died in 1686, in his lyear. He was author of a “ Treatise of gebra,” 4to. 1690; and a “ Method of Re- ving Indeterminate Problems in Algebra,” 99. Besides a great number of papers in Memoirs of the French Academy. ROLLING, that motion of a body which is ased by its rectilinear motion being resisted, ‘the friction of some surface or otherwise ; nereby its several parts come successively ‘contact with that plane or surface; such is ‘ motion of a carriage wheel upon the wmnd, &c. See ROTATION. ROOD, a square measure, the fourth of an e. See MEASURE. ROOT, in Arithmetic and Algebra, denotes yuantity, which being multiplied a certain mber of times into itself produces another mber, called a power, and of which power original quantity is called the root. Roots ) distinguished into squarg roots, cube roots, buadratic roots, &c. or into 2d, 3d, 4th, 5th, , roots, which depends upon the number of Itiplications necessary to generate the pro- ed power. If one multiplication only is sessary, or if two equal factors are multi- »d together, it is called the square or se- iid root; if three, the cube or third root; our, the biquadratic or fourth root, &c.; s ; 8 is the square root of 64 4 is the cube root of 64 2 is the sixth root of 64 Ke. &e, ROT For the extraction of the roots of numbers. See EXTRACTION. Roots of an Equation, are those numbers or quantittes which substituted for the un- known quantity, render the whole equation equal to zero. And of these there are always as many real or imaginary, as there are units in the highest power of the unknown quantity. So an equa- tion of the 2d degree has two roots; one of the 3d degree, three; of the 4th degree, four, &e. See EQuaTIONs. The roots of an equation are either positive, negative, or tmaginary. A Positive Root, is an absolute number affected with the sign + plus. A Negative Root, is also an absolute num- ber, but is affected with the sign — minus. An Imaginary Roor, is one to which no absolute value can be attached, one part of it consisting always of the square root of a ne- gative quantity; yet it is such, that when substituted for the unknown quantity, it is found to answer the conditions of the equa- tion. See Imacinary Roots and Quantities. ROTA, in Mechanics. See WHEEL. Rota Aristotelica, or Aristotle's Wheel, de- notes a problem in mechanics proposed by Aristotle concerning the motion. of a coach wheel; viz. that the nave of a wheel describes by its motion, (supposing it to roll along a plane) a line of the same length as the cir- cumference, by its motion on- the ground; which was long considered paradoxical, nor was it clearly understood till M. Meyran, a Frenchman, sent a satisfactory solution of it to the Academy of Sciences, the principle of which is, that each point of the circumference of the nave as it approaches the plane, is drawn forward over a space greater than itself, whereas every point and part of the circumference passes over a space exactly equal to itself. ROTATION, the motion of the different parts of a solid body about an axis, called the axis of rotation, being thus distinguished from the progressive motion of a body about some distant point or centre; thus the diurnal mo- tion of the earth is a motion of rotation, but its annual motion one of revolution. We have already treated of the several cases of rotatory motion under the articles CENTRE of Gyration, PERCUSSION, OSCILLA- TION, &c.; and it only therefore remains here to state some few general remarks with regard to this kind of motion, and to enumerate some results not contained in the articles above referred to. When a solid body turns round an axis, retaining its shape and dimensions unaltered, every particle is actually describing a circle round this axis, which axis passes through the centre of the circle, and is perpendicular to its plane. Moreover in any instant of the motion, the particle is moving at right angles with the radius vector, or line joining it with its centre of rotation; therefore, in order to ascertain the direction of any particle, we ROT may draw a line from that particle perpendi- cular to the axis of rotation. This line will be in the plane of the circle of rotation of that particle, and will be its radius vector, and a line drawn from the particle perpendicular to its radius vector, will be a tangent to the circle of rotation, and will represent the direc- tion of the motion of this particle. The whole body being supposed to turn together, it is evident, that when it has made one complete rotation, each point has de- scribed the circumference of a circle, and the whole paths of the different particles will be in the ratio of these circumferences; and therefore of their radii, and this is also true of any portion of such circumferences ; that is, the velocities of the different particles are proportional to their radii vectores, or to their distances from the axis of rotation. And all these motions are in parallel planes, to which the axis of rotation is perpendicular. Hence it follows, that when we compare the rotation of different bodies in respect of velocity, it is evident that it cannot be done, by directly comparing the velocity of any particle in one of the bodies with that of any particle of the other; for as all the particles of each have different velocities, this comparison can estab- lish no ratio. But we may familiarly compare such motions by the number of complete turns which they make in any equal portions of time. Therefore as the length or number of feet described by a body in rectilinear motion is a proper measure of its progressive velocity, so the angle described by any par- ticle of a whirling body, is a proper measure of its velocity of rotation; and in this manner may the rotation of two or more bodies be compared, and this velocity is, with propriety, called the angular velocity. In what is stated above we have had prin- cipally in view, a fixed and permanent axis of rotation, the body not being supposed at li- berty to revolve about any other; but it is obvious, that if any force is impressed upon a body, or system of bodies, in free space, (un- less that force be exerted in a direction pass- ing through the centre of gravity of the system) a rotatory motion will ensue about an axis passing through the centre of gravity of the system; and the centre, about which this mo- tion is performed, is called the centre of spon- taneous rotation. A body may begin to revolve on any line as an axis that passes through its centre of gravity, but it will not continue to revolve permanently about that axis, unless the opposite centrifugal forces exactly balance each other. Thus a homogeneous sphere may revolve permanently on any diameter, because the opposite parts of the solid, being in every di- rection equal and similar, the opposite centri- fugal forces must be equal; so that no force has a tendency to change the position of the axis. Hence also a homogeneous cylinder may revolve permanently about the line, wich is its geometric axis; as it may also about any line that bisects that axis at right ROT i 5 angles, but it can revolve permanently e no other line, because then the centri forces could not be equal, and the same true with respect to any solid of rotation. In every body, however irregular, there aj three permanent axis of rotation, at rig angles to each other, on any one of whi¢ when the body revolves, the opposite cen fugal force exactly balances, and therefo) the rotation becomes permanent. These thry axis have also this remarkable property, th’ the momentum of inertia, with respect to al of them, is either a maximum or a minimun) that is, is either greater or less than if ft) body revolved about any other axis. } This curious theorem was first proposed | Segner in 1755, and first demonstrated | : Albert Euler, son of the celebrated Leona} Euler, in a memoir presented to the Ac demy of Sciences at Paris in 1766; the sar} is also given by the father in his “ De Mo Corporum rigidorum,” prop. 27. See aj “ Frisius De Rotatione Corporum,” &c.; a Gregory’s Mechanics, vol. i. book 2, chap, i} . At present we have considered those cas of rotation that are produced by a force 1 pressed upon a body, either as supported } a fixed axis, about which, therefore, tl system must necessarily revolve, or as in fi space, in which case the system acquires spontaneous centre of rotation, and finally permanent axis of rotation; but there ; other circumstances which will produce a tatory motion that are not included in eit of the above, but which it will be proper mention before we conclude this article, st are those which arise from a body descend down an inclined plane, having a ribbon cord wound about it, one end of which fixed at the upper part of the plane, which preventing the body sliding freely cause rotatory motion; the same effect also follc from the friction of the body against plane; and the same may be imagined wl there is no plane but the body left to freely, except so far as the cord wound ab it shall produce a rotatory motion in its scent. We shall not attempt the investi tion of these cases, but merely state the sults that have been obtained, and must r the reader for the former to the sew treatises on dynamics enumerated above, under the articles DyNAMics and MECHAN Let a body have a cord wound about it, either at its cir- cumference or any other part, as BC, having one end fix- ed ata point above, as at A; thenif the body be left to de- scend by the action of gravity, it will acquire a motion of rotation by the un- winding of the cord, and the space ac- RUL illy descended by the body in this case, { be to the space descended in the same /e when falling freely, as CG to CO; O and epresenting the centres of oscillation and fation, when the point of suspension is at and the weight of the body will be to the (sion of the cord, as CO to CG; and the ‘ie ratios have place when the body de- ‘nds down an inclined plane; the forces ich generate the motion being both de- -ased in the same ratio. “he force by which spheres, cylinders, &c. caused to revolve as they move down an ilined plane (instead of sliding), is the ad- jion of their surfaces, occasioned by their jssure against the plane; this pressure is t of the weight of the body, for this weight ag resolved into its component parts, one s direction of the plane, the other per- idicular to it; the latter is the force of the ssure ; and which while the same body rolls [7m the plane, will be expressed by the co- Js of the plane’s elevation. Hence, since | cosine decreases, while the are or angle ‘eases; after the angle of elevation arrives ja certain magnitude, the adhesion may jome less than what is necessary to make | circumference of the body revolve fast }ugh, and in this case it will proceed partly sliding and partly by rolling; but the ile, at which this circumstance takes place, | evidently depend upon the degree of ad- ion between the surfaces of the body and ie. This, however, will never happen, if rotation is produced by the unwinding of -bbon, and it is on this latter supposition { the following particular cases are de- ed. wet W be the weight of the body, s the ice descended by a heavy body, falling ly, or sliding freely down a plane, then spaces described by rotation in the same e, by the following bodies, will be in these portions. . A hollow cylinder, or cylindrical surface } S = 4s,tension=4 W. » Asolid cylinder S = 4s,tension = 4 W. . Aspheric surface S = 2s, tension = + Asolid sphere S—3-s, tension = Gregory’s Mechanics, vol. 1. tOTATORY. See Rorarion. Bie, a coin of Turkey, value 1s. 54d. ‘ling. tOW NING (JouHN), an ingenious English thematician, was born about the year 1700, ‘died in 1771. He published A Compen- is System of Natural Philosophy, 2 vols. 1738, which has passed through several jons; beside some papers in the Phil. Hs, tUBLE, a Russian coin, those of 1764, le 3s. 3d.; and of 1801, value 2s. 92d. ‘ling. tUDOLPHINE Tables, a celebrated set idstronomical tables published by Kepler, {thus entitled, in honour of the emperor, ‘dolph, or Rudolphus. tULE of Three, in Arithmetic, called by 1) i 4 W. W. 2 5 2 F RUL some authors the Golden Rule, is an applica- tion of the doctrine of proportion to arith- metical purposes, and is divided into two cases, simple and compound ; now frequently termed Simple, and Compound Proportion. Simple Rule of Three, or Simple Proportion, is, when from three given quantities, a fourth is required to be found, that shall have the same proportion to the given quantity of the same name, as one of the other quantities has to that of the same name with itself. This rule is, by some authors, divided into two cases; viz. The Rule of Three Direct, and The Rule of Three Inverse ; but this distinction is unnecessary, and the two cases are now generally given under one head by all our best modern authors; but as they are still re- tained by others, it will not be amiss to point out the distinction. The Rule of Three Direct, is, when more requires more, or less requires less, as in this example; if 3 men will perform a piece of work, as for instance, dig a trench 48 yards long, in a certain time; how many yards will 12 men dig in the same time; where it is ob- vious, that the more men there are employed, the more work will they perform, and there- fore, in this instance, more requires more? Again, if 6 men dig 48 yards in a given time, how much will 3 men dig in the same time? Here less requires less, for the less men there are employed, the less will be the work that is performed by them; and all questions that are of this class are said to be in the Rule of Three Direct. The Rule of Three Inverse, is, when more requires less, or less requires more. As in this, if 6 men dig a certain quantity of trench in 14 hours; how many hours will it require for 12 men to dig the same quantity? Or thus, if 6 men perform a piece of work in 7 hours; how long will 3 men be in performing the same work? These cases are both in the Inverse Rule, for in the first, more requires less, that is, 12 men being more than 6, they will require less time to perform the same work ; and in the latter, the number of men being less, they will require a longer time. All questions of this class are said to be in the Rule of Three Inverse. These two cases, however, as we before observed, may be classed under one general rule, as follows: Rule. Of the three given terms set down that which is of the same kind with the an- swer towards the right hand; and then con- sider from the nature of the question, whether the answer will be more or less than this term. Then, if the answer is to be greater, place the less of the other two terms on the left, and the remaining term in the middle; but if itis to be less, place the greater of these two terms on the left, and the less in the middle; and, in both cases, multiply the second and third terms together, and divide the product by the first term for the answer, which will always be of the same denomination as “the third term. Ke Note 1. If the first and second terms con- RU L sist of different denominations, reduce them both to the same; and if the third term be a compound number, it is generally more con- venient to reduce it to the lowest denomina- tion contained in it. Note 2. 'The same rule is applicable whether the given quantities be integral, fractional, or decimal. EXAMPLES. If an acre of land be worth £73. 10s. 3d. ; how many acres may be bought for £250. 10s. ? Integral. As £73. 1s. : i:£250. 10s. :: 1 acre ah) 20 1464 5010 12 12 1753: GU120 -17532)60120(3 acres 52596 7524 4 17532)30096(1 rood 17532 12564 4) ea cancae (28 3,22; perches. 35064 151920 140256 LI GG 4a i nies 17632 —. *46F By Fractions. As 733, : 2502 :: 1 Ogasi@ahe: Ags, sit 20 5OL — 5010 — Whence 738, X °$' = $912 = ae. 1 rood 28 7 perches. By Decimals. As 73°05: 250°5 2 1 1 7305)25U 50(3°4291=3 ac.1 ro.28°656 p- 21915 31 350 29 220 2.1300 14610 66900 65745 11550 7305 4245 ——— Se Compound Rue of Three, or Compound Pro- portion, sometimes also called the Rule of Five, or Double Rule of Three, is the method of solving, at one operation, such questions as would require two or more statings by the common, or Simple Rule of Three. Rule. Set that term which is of the same kind with the answer on the right, and take RUL "my any two of the other given terms that are the same name, and consider from the nat of the question whether, in this partic case, the answer ought to be more or less th the above-mentioned right-hand term, arrange these two terms accordingly, as in’ last rule. Considering still the same right-hand as common to every stating, take two ot; terms that are of the same kind, and a them as above, according. as in this case, answer ought to be more or less than 4; right-hand term; and proceed in the ga manner with every pair of terms that are} the same name. ‘Then multiply all the fy terms together for a divisor, and the terms together for a dividend, and the quoti thence arising will be the answer sought. — EXAMPLE. If 248 men, in 5 days, of 11 hours each, 4 dig a trench 230 yards long, 3 wide, and deep ; in how many days of 9 hours long, | 24 men dig a trench 420 yards long, 5 wi and 3 deep? / Men 24 : 248 Hours Soe i | ‘4 Length 230 : 420\%:: 5 days common tei Width 3: 5 Depth Grete ba t Then 248 x11 xX 420x5X3x5 SR CE DX SIR CD oe gta 24x9X2B0x3X2Q zo7 CAys ‘RuLe of Five. See Compound RULE Three. Central Rute. See CentRAL Rule. |} Sliding RULE, a mathematical instram# serving to perform computations in gaugi} measuring, &c. without the use of compass merely by the sliding of the parts of the} strument one by another, the lines and di sions of which give the answer or amount} inspection. | This instrument is variously contrived < applied by different authors, particularly Gt ter, Partridge, Hunt, Everard, and Cog shall, but the more usual and useful ones those of the two latter. % Kverard’s Sliding Rue, is chiefly used cask gauging. It is commonly made of b 12 inches long, 1 inch broad, and <6 of inch thick. It consists of three parts, viz.4 stock just mentioned, and two thin slips the same length, sliding in small groves two opposite sides of the stock ; consequen when both these pieces are drawn out to th full extent the instrument is 3 feet long. — On the first broad face of the instrume are four logarithmic lines in numbers ; for properties, &c. of which, see GUNTER’S Li The first, marked A, consisting of two ra 1, 2, 3, 4, 5, 6,7, 8, 9,1; and then 2, 3, 4) &e. to 10. On this line are four brass cen’ pins, two in each radius; one in each of tht being marked M B, for malt bushel, is set 215042, the number of cubic inches in a m bushel; the other two are marked with A, | ale gallon, at 282, the number of cubic inet RUL anale gallon. The 2d and 3d lines of num- ’s are on the sliding pieces, and are exactly j; same with the first; but they are distin- tished by the letter B. In the first radius is ot marked § 1, at ‘707, the side of a square cribed in a circle whose diameter is 1. other dot, marked Sc. stands at °886, the e of a square equal to the area of the same cle. A third dot, marked W, is at 231, - eubic inches in a wine gallon. And a irth, marked C, at 314, the circumference 1 the circle, whose diameter is 1. The prth line of numbers, marked MD, to ynify malt depth, is a broken line of two ii, numbered 2, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 3,7, &c.; the number 1 being set directly |inst MB on the first radius. Jn the second broad face, marked ed, are eral lines: as, Ist, a line marked D, and inbered 1, 2, 3, &c. to 10. On this line jfour centre pins; the first marked W G, |wine gage, is at 17°15, the gage point for je gallons, being the diameter of a cylinder se height is one inch, and content 231 ic inches, or a wine gallon. The second tre pin, marked A G, for ale gage, is at ‘5, the like diameter for an ale gallon. |) third mark, MS, for malt square, is at , the square root of 2150°42, or the side of uare whose content is equal to the number jaches in a solid bushel. And the fourth, ike M R, for malt-round, is at 52°32, the eter of a cylinder or bushel, the area of se base is the same 2150°42, the inches | bushel. 2dly, Two lines of numbers on sliding piece, on the other side marked C, hese are two dots, the one marked e, at 5, the area of a circle whose circumference ; and the other marked d, at °785, the of the circle whose diameter is one. , Two lines of segments, each numbered '3, to 100, the first for finding the ullage cask, taken as the middle frustrum of a roid, lying with its axis parallel to the “on; and the other for finding the ullage cask standing. gain, on one of the narrow sides noted ce, ist, a line of inches, numbered 1, 2, 3, &c. 2, each subdivided into 10 equal parts. A line by which, with that of inches, we a mean diameter for a cask in the figure ie middle frustrum of a spheroid; it is “ed spheroid, and numbered 1, 2, 3, &c. Sdly. A line for finding the mean dia- rof a cask, in the form of the middle tum ofa parabolic spindle, which gaugers the second variety of casks; it is there- marked second variety, and is numbered B, &c. ily. A line by which is found the mean ‘eter of a cask of the third variety, con- g of the frustrams of two parabolic co- , abutting on a common base, itis there- marked third variety, and is numbered 3, Ke. ithe other narrow face, marked f, are, a line divided into one hundred equal f, marked FP M. 2ndly. A line of inches, { RUP like that before mentioned, marked I M. 3dly. A line for finding the mean diameter of the fourth variety of casks, which is formed of the frustrums of two cones, abut- ting on a common base. It is numbered 1, 2,3, &c. and marked FC, for frustrum of a cone. On the back side of the two sliding pieces is a line of inches, from 12 to 36, for the whole extent of the 3 feet, when the pieces are put endways; and against that, the correspondent gallons, and 100th parts, that any small tub or the like open vessel will contain at 1 inch deep. For the various uses of this instrument, see the authors mentioned above, and most writers on gauging. Coggeshall’s Sliding RULE, is chiefly used in measuring the superticies and solidity of tim- ber, masonry, brick-work, &c. This consists of two parts, each a foot long, which are united together in various ways. Sometimes they are made to slide by one an- other like glazier’s rulers: sometimes a groove is made in the side of a common two foot rule, and a thin sliding piece on one side, and - Coggeshall’s lines added on that side; thus forming the common or carpenter’s rule: and sometimes one of the two rulers is made to slide in a groove made in the side of the other. On the sliding side of the rule are four lines of numbers, three of which are double, that is, are lines of two radii, and the fourth is a single broken line of numbers. The first three, marked A, B,C, are figured, 1, 2, 3, &c. to 9; then 1, 2, 3, &c. to 10; the construction and use of them being the same as those on Ever- ard’s sliding rule. The single line, called the girt line, and marked D, whose radius is equal to two radii of any of the other lines, is broken for the easier measuring of timber, and figured 4, 5, 6, 7, 8,9, 10, 20, 30, &c. From 4 to dit is divided into 10 parts, and each 10th sub- divided into 2, and so on from 5 to 10, &c. On the back side of the rule are, 1st, a line of inch measure, from 1 to 12, each inch being divided and subdivided. 2dly. A line of foot measure, consisting of one foot divided into 100 equal parts, and figured 10, 20, 30, &c. The back side of the sliding piece is di- vided into inches, halves, &c. and figured from 12 to 24; so that when the slide is out there may be a measure of 2 feet. In the carpenter’s rule the inch measure is on one side, continually all the way from 1 to 24, when the rule is unfolded and subdivided into 8th or half quarters ; on this side are also some diagonal scales of equal parts. And upon the edge, the whole length of two feet is divided into 200 equal parts, or 100ths of a foot. Parallel Rute. See Paratiet Rule, RUMB. See Ruvums. RUPEE, a coin of different parts of the East Indies, of the sterling value of 2s, or a little more or less, SAI RUSPONO, a coin of Tuscany, value £1. System of Natural Philosophy,” which y, published in 1748. a | 8s. 6d. sterling. RUTHERFORD (WILtiAM), an English philosopher, was born in 1712, and died in 1771. He is principally distinguished by “‘ A SACRO-BOSCO. See HALLIFAX. SAGITTA, the Arrow, one of the northern constellations. See CONSTELLATION. SAGITTA, in Geometry, is used by some old authors, to denote what we call the versed sine of an arc, others have used it to denote the absciss of any curve. SAGITTARIUS, Sagittary, the Archer, the ninth sign of the zodiac, denoted by the cha- racter {. See CONSTELLATION. SAILING, in Navigation, denotes the act of conducting a vessel from one port to an- other, by means of the action of the wind upon her sails, being otherwise expressed by the more significant term navigating. SAILING is distinguished into different cases, according to the principles upon which the computations are founded, as Plane Sailing, Middle Latitude Sailing, Mercator Suailing, Globular Sailing, &c. Plane Saittne, is that which is performed on a supposition of the earth being an extend- ed plane surface, and by means of plane charts, in which case the meridians are con- sidered as parallel lines, the parallels of lati- tude at right angles to the meridians, and the lengths of the degrees on the meridians, equa- tor, and parallels of latitude, as every where equal. Here the principal terms are the la- titude, distance, and departure; difference of latitude and rhumb, longitude having no place in plane sailing. It is obvious, however, that calculations conducted on these principles must be too erroneous to be depended upon in any case, and therefore it would be but wasting the reader’s time to enter farther into an explanation of this case, which is now nearly if not wholly disused by navigators. Traverse SAILING, may be defined compound plane sailing, being the method of working, or calculating traverse or compound courses, so as to reduce them into one. This is used when a ship having to sail from one port to another is, by reason of contrary winds, or other obstacles, obliged to tack and sail upon different courses, which are then to be brought into one; and hence the difference of latitude, departure, and other circumstances deter- mined as in plane sailing. Globular SAiLtnc, is the method of esti- mating a ship’s motion and run, upon prin- ciples drawn from the globular figure of the earth. In this, its most extended sense, glo- bular sailing comprehends Parallel, Mercator, S SAL RYDER, a Dutch coin, value £1. 4s, 1} sterling. Middle Latitude, and Great Circle Sailin for a definition of each see the follows articles. | Parallel Sai.1na, is the sailing on a para} of latitude, or parallel to the equator, of whl there are three cases. | 1. Given the distance and difference of hh gitude; to find the latitude, which is formed by the following rule : As the difference of long. : the distane; the radius : the cosine of the latitude 2. Given the latitude and difference of , gitude; to find the distance. Rule, As radius : the cosine of the latitude the difference of longitude : the distaj 3. The latitude and distance being give! find the difference of longitude. Rule, As cosine of latitude : radius :: the distance : the difference of longit Middle Latitude SAILING, is a method 0} solving the cases of globular sailing, by m of the middle latitude between that depat from and that come to. This method is) accurate, being founded on the principle plane and globular sailing conjointly ; viz a supposition that the departure is recka as a meridional distance in that latitude, wy is the middle parallel between the lati) sailed from, and the latitude come to; wy would be correct if the cosine of a middle tude was an arithmetical mean between] cosine of two extreme latitudes; and the parture between two places on an obli@ rhumb, equal to the meridional distance the middle latitude ; but neither of these d@ obtain. Yet when the parallels are-neaili equator, or near to each other, in any lati) the error is not considerable. This method seems to have been invelge on account of the easy manner in which several cases may be resolved by the trav table, and when a table of meridional pai not at hand the computations may be mai follows; viz. i Take half the sum of the two given tudes for the middle latitude, then say, |@ 1. As cosine of mid. lat. ; the radius +) the departure : diff. of longitude. 2. As cosine of mid. lat. : tan. of cour diff. of lat. : diff. of longitude. : Mercator SAiL1NG, or more properly Wri Sailing, is the method of computing the | of sailing on the principles of Mercator’s @ which principles were first laid dow SAI ‘right in the beginning of the 17th century. ese consist in finding on a plane the motion a ship upon any assigned course that shall true, as well in longitude and latitude, as distance, the meridians being all parallel, id the parallels of latitude straight lines. In the annexed figure let A C represent e distance run, A 6 the difference of latitude, -B the difference of longitude, cb the de- irture, the angle CAB the course, and AB e meridional difference of latitude. A eee c4 B en from the similarity of triangles, when ee of these parts are given, the rest may be ind from the following analogies: Rad. : sin. course :: dist. : depart. Rad. : cosin. course :: dist. : dif. of lat. Rad. : tan. course :: merid. dif. lat. : dif.lon. means of which any of the cases of Mer- or sailing may be readily resolved. See 3RCATOR’S Chart and MERIDIONAL Parts. Yireular SaiLinG, or Great Circle SatLine, the finding what places a ship must go ough, and what courses to steer, that her h may be in an arc of a great circle on the he, or nearly so, passing through the place ed from, and that bound to. is the solutions of the cases of Mercator ing are performed by plane triangles, in method they are resolved by means of erical triangles, and present a great variety ases ; but the following is that which most amonly occurs, viz. riven the latitude and longitude of two 3s, to find their nearest distance on the ace, with the angles of position from either ®e to the other. ‘his problem involves 6 cases, viz. 1, when places lie under the same meridian; 2, m they are both under the equator; 3, m only one of them is under the equator; then they are both under the same parallel atitude; 5, when the places are both on ‘same side of the equator; and 6, when ‘are on different sides; these cases, it is ous, are all solved by means of the dif- nt cases of spherical trigonometry, and Sidereal Revolution. 1¢ 18" 27’ 33" 5 3 13 13 42 ,0 7 3 4233 ,4 16 16 31 49 ,7 14 769137788148 3 551181017849 7 154552783970 16 688769707084 SAT which it would therefore be useless to repeat in this place. On the subjects of these articles see Robert- son’s “ Elements of Navigation.” Spheroidical S a1L1NG, is computing the cases of navigation, on the principles of the true spheroidical figure of the earth; on which subject see also Robertson’s ‘“ Navigation,” vol. ii. book 8. sect. 8. SAILING is also used in a different sense, to denote the theory and practice of manoeuvring of vessels, viz. as to the best position of the sails and rudder, their effect on the motion of the ship, &c. &c. on which subject we have several learned treatises by Bernoulli, Euler, Bouguer, Borelli, Juan, &c. SAROS, in Chronology, is a period of 223 Junar months. SATELLITE, in Astronomy, certain se- condary planets moving round the primary planets, as the moon does about the earth; they are thus called because always fonnd attending them, from rising to setting, and making the tour of the sun together with them. - The satellites, in their motion round their primaries, are governed by the same laws, as these are in their revolution round the sun; viz. they describe equal areas in equal times, and those belonging to the same planet have the squares of their periodic times, propor- tional to the cubes of their mean distances from the planet. The number of satellites at present known in our system is eighteen, viz. the moon, the satellite of the earth; four belonging to Jupiter; seven to Saturn; and six to Uranus: but of these only the moon is visible to the naked eye, for the particulars of which see Moon. The satellites of Jupiter were first discover- ed by Marius, a mathematician of Branden- burg, in 1609; but the first observations re- lating to them were published by Galileo, in 1610, who discovered them in Italy, in the month of January in that year. Huygens discovered one belonging to Saturn in 1655, four others were discovered by Cas- sini; and finally, two others have been since added to that number by Dr. Herschel, who has also discovered six satellites belonging to Uranus, making in all, with the moon, eighteen, as stated above. SATELLITE of Jupiter. By the aid of a te- lescope we may discover four satellites re- volving about this planet; the sidereal revo- lution of which, with their mean distances, &e. are given in the following table, as ex- tracted from Laplace’s “Systeme du Monde,” 3d edition. Mean Distance. *0000173281 ‘0000232355 ‘0000884972 000042659 | 5°812964 9°248679 14°752401 25°946860 SAT “Sara First Satellite. The inclination of the orbit | of this satellite does not differ much from the plane of Jupiter’s orbit. Its eccentricity is [atellite. Sidereal Revolation. insensible, RNG pi Second Satellite. 'The eccentricity of the I.} 0°22437'30",1 | 0° 94271 orbit of this satellite is also insensible. ‘The II.| 1 853 8 ,7 | 1 37024 inclination of its orbit to that of its primary Iff.| 1 21 18 25 ,9; 1 88780 is variable, as well as the position of its nodes. IV.| 217 44 51 ,1 | 2 73948 Third Satellite. This satellite has a little V.| 4 12 26 11 ,1| 4 51749 eccentricity, and the line of its apsides has a VI.| 15 22 41 13 ,9 | 15 94530 direct but variable motion; the eccentricity | VII.| 79 7 54 37 ,4 | 79 32960 itself is also subject to very sensible variations. The inclination of its orbit to that of Jupiter, iq and the position of its nodes, are far from SATELLITES of Uranus. Six satellites b being uniform. volve round Uranus; which, together wa Four th Satellite. The eccentricity ofthissatel- their primary, can be discovered ‘only by » fite is greater than that of any of the other three, telescope. The following table will shy and the line of the apsides has an annual anil their sitdereal revolutions, ane mean distang direct motion of 42’ 58" ,7. The inclination jy semi-diameters of the primary. a of its orbit, with the plane of Jupiter’s orbit, forms an angle of about 2° 25'48”; but this angle, although stationary about: the middle |. Mean‘ - of pte last century, has lately begun to in- Satellite. Sidereal Revolution. tane crease very sensibly. At the same time the motion of its nodes has begun to diminish. The motions of the first three satellites are related to each other by a most singular ana- logy. For the mean sidereal or synodical motion of the first, added to twice that of the third, is constantly equal to three times the mean motion of the second. And the mean sidereal or synodical longitude of the first, J : minus three times that of the second, plus , Ail these satellites move in a plane w twice that of the third, is always equal to two 18 nearly perpendicular to the plane of right angles. planet’s orbit, and contrary to the order 9 The satellites of Jupiter are liable to be sgns/! Laplace’ s “Syst, du Monde.” eclipsed by passing through his shadow ; and, SATURN, one of the primary planets, on the other hand, they are frequently seen ing the sixth in order of distance from the to pass over his dise and eclipse a portion of of the old planets, but the tenth inclu his surface. This happens to the first and the four new ones, being the outermost o} second satellite at every revolution; the third except Uranus; and is “marked with the very rarely escapes in each revolution; but the racter hb , denoting an old man supporting, fourth (on account of its great distance and self with a staff, representing the ancient) inclination) is seldom obscured. Saturn. These eclipses are of great utility in ena- Saturn shines with but a feeble light or | bling us to determine the longitude of places count of his great distance, and his dull by their observation; and they likewise ex- colour, which latter is supposed to arise | hibit some curious phenomena withrespect to a peculiarly dense atmosphere. | light. This planet is perhaps one of the mos From the singular analogy above alluded gaging objects that astronomy presents t¢ to, it follows that (for a great number of years view, being surrounded by a double ring at least) the first three satellites cannot be seven satellites, all visible in a telescoj eclipsed at the same time; for in the simul- sufficient. magnifying power, and whie taneous eclipses of the second and third, the certain situations present one of- ‘the first will always be in conjunction with Ju- beautiful objects it is possible to conc piter, and vice versa. See Rine. ai SATELLITES of Saturn. Seven satellites Saturn has also certain obscure zone may be seen, by means of the telescope, to belts, appearing at times across his disc revolve about Saturn, the elements of which those of Jupiter, which are changeable, are but little known, on account of their great are probably obscurations in his atmospi distance. The following table, however, will which Dr. Herschel, Phil. Trans. 1790, show the duration of their sidereal revolutions, shown to be very dense. He’ revolves (@ and their mean distances in semi-diameters of an axis, which.is perpendicular to the | Saturn. of the ring, and his figure is, like the The orbits of the first six satellites appear planets, an oblate spheroid, being considé to be in the plane of Saturn’s ring; whilst the flattened at both poles. seventh varies from it very sensibly. Saturn performs his sidereal revoluti 107584 23 16’ 34” ,2, or in 29°456 Julian | | F 5* 215 25’ 20" .6 54 8925 | 131 If. 8 16 57 47 ,5 8 7068 | 17°) Ill. | 10 23 3 59-,0} 10 9611 | 19°8} IV. | 13 10 56 29 ,8} 13 4559 | 224 0 2 ws ae Vs eS er 4s oO 38 0750 | 45% VI. | 107 16 39 56 107 6944 | 91°¢ “ a | } SAU But this period is subject to some inequalities. is mean synodical revolution is performed n about 378 days. | _ His mean distance from the sun is 9°539; hat of the earth being considered as unity, vhich makes his mean distance above 890 nillions of miles. The eccentricity of his orbit is ‘0562; half he major axis being taken as unity. His mean longitude at the commencement f the present century, was in 4s 15° 20' 31” ,5. ' The longitude of his perihelion was at the ame time in 2° 29° 8’ 57” 9; but the line of 1€ apsides has an apparent motion in longi- ade, according to the order of the signs of | 9" 5 in a year, or 1° 55! 47” lina century. ‘His orbit is inclined to the plane of the tliptic in an angle of 2° 29’ 38” ,1; which is »served to decrease about 15” ,5 in a century. His orbit, at the commencement of the pre- nt century, crossed the ecliptic in 3* 21° 55’ “,9; but the place of the nodes has an ap- went motion in longitude, according to the er of the signs, of 27” ,4 in a year, or 45’ #5 in a century. The rotation on his axis is performed in’ B16 19” ,2: and the avis is inclined in an gle of 58° 41’ to the plane of the ecliptic. His mean diameter is 76068 miles; conse- ently he is nearly 1000 times as large as our tth. The axis of his poles is to his equa- jal diameter as 11 to 12. His mass, compared with that of the sun, nsidered as unity, is =715,; but his den- yis °45, that of water being one. The proportion of ight and heat received m the sun is ‘011; that received by the 'th being considered as unity. As viewed from the earth, the motion of Urn sometimes appears retrograde. The ‘am arc which he describes in this case is vat 6° 18’; and its duration is nearly 139 Ss This retrogradation commences, or shes, when the planet is distant about 108° from the sun. Tis mean apparent diameter is 17’ ,6. AUNDERSON (NicHo.as), professor of thematics in the University of Cambridge, jafellow of the Royal Society, was born Thurlston, in Yorkshire, in 1682. ° When Was but twelve months old he. lost not yvhis eye-sight, but his very eye-balls, by ‘small-pox ; so that he could retain no € ideas of vision than if he had been born d. At an early age, however, being of ' promising parts, he was sent to a free ol at Penniston, and there laid the foun- on of that knowledge of the Greek and ‘n languages, which he afterwards improy- 0 far, by his own application to the classic iors, as to hear the works of Euclid, Archi- es, and Diophantus, read in their original otwithstanding this great impediment to progress in mathematical and philoso- al pursuits, Saunderson soon became so 1ent that he was invited to Cambridge to ‘er lectures on these subjects, which he SAU did with great honour to himself and adyan« tage to his hearers, As he instructed youth in the principles of the Newtonian philosophy, he soon became acquainted with its illustrious author, though he had several years before left the University ; and frequently conversed with him on the most difficult parts of his works: he also held a friendly communication with the other eminent mathematicians of the age, as Halley, Cotes, Demoivre, &c. Mr. Whiston was at this time in the ma- thematical professor’s chair, and read lectures in the manner proposed by Mr. Saunderson on his settling at Cambridge, so that an at- tempt of this kind looked like an encroach- ment on the privilege of his office ; but, as a liberal man, and an encourager of learning, he readily consented to the application of friends made in behalf of so uncommon a person. Upon the removal of Mr. Whiston from his professorship, Mr. Saunderson’s merit was thought so much superior to that of any other competitor, that an extraordinary step was taken in his favour to qualify him with a de- gree, which the statute requires; in conse- quence he was chosen, in 1711, Mr. Whiston’s successor in the Lucasian professorship of mathematics, Sir Isaac Newton interesting himself greatly in his favour. Dr. Saunderson was naturally of a strong healthy constitution; but being too seden- tary, and constantly confining himself to the house, he became a valetudinarian: and in the spring of the year 1739 he complained of a numbness in his limbs, which ended in a mor- tification in his foot, of which he died the 19th of April that year, in the 57th year of his age. There was scarcely any part of the mathe- matics on which Dr. Saunderson had not composed something for the use of his pupils. But he discovered no intention of publishing any thing, till, by the persuasion of his friends, he prepared his “ Elements of Algebra” for the press; which, after his death, were pub- lished by subscription, in 2 vols. 4to. 1740, and of which there is also an abridgment in one vol. 8vo. He left many other writings, though none perhaps prepared for the press. Among these were some valuable comments on Newton’s “ Principia,” which were published in Latin at the end of his posthumous “ Treatise on Fluxions,” in 1756, in 8vo. SAURIN (JosepH), an ingenious French mathematician, was born in 1659, and died in 1737, at 78 years of age. None of his works were ever published, except those sent to the Mem. of the Acad. of Sciences, of which there are several from the year 1709 to 1727, SAUVEUR (Josep), a French mathema- tician, was born in 1653, and died in 1716, in the 64th year of his age. He was author of numerous papers, on music, geometry, and some philosophical sub- jects; but none of his works were ever pub- lished in a separate form. > a AG SCA SCALE, in Arithmetic, is used to denote any particular division of numbers into periods ; thus we have the binary, ternary, quaternary, &e. scale. See NOTATION. | ScALE, is also the name given to certain ma- thematical instruments, consisting of several lines drawn on wood, brass, silver, &e. Such are the Diagonal Scale, Gunter Scale, Plain Scale, &c. Diagonal Scare, is projected thus: first draw eleven parallel lines at equal distances, the whole length of which being divided into a certain number of equal parts, according to the length of the scale, by lines parallel to its length, let the first division be again subdi- vided into ten equal parts, both above and below; then drawing the oblique lines from the first perpendicular below, to the first sub- division above, and from the first subdivision below to the second subdivision above, &c. the first space shail thereby be exactly divided into one hundred egual parts; for as each of these subdivisions is one tenth part of the whole first space or division, so each parallel above it is one-tenth of such subdivision, and consequently one hundredth part of the whole first space; and if there be ten of the larger divisions, one thousandth part of the whole scale. If therefore the larger divisions be accounted units, the first subdivisions will be tenth parts of an unit ; and the second sub- divisions, marked by the diagonals on the pa- rallels, hundredth parts of a unit. Again, if the larger divisions be reckoned tens, the first subdivisions will be units, and the second subdivisions tenth parts; and if the larger divisions be accounted hundredths, the first 7 3 FO 70 wn = — “ S 6¢ > o ‘ ee) Si 2 8 _& 30 30, 4 yo 40 3.40 39 S90 ny y ~ 20 a" 10 le Se A | subdivisions will be tens, and the units, and so on. | ¥ The method of constructing the scales chords, sines, tangents, and secants, usual engraven on instruments, for practice, is ex’ hibited in the annexed figure; and is too ob! vious to require any particular explanation. * Gunter’s ScALt,»an instrument so cally from Mr. Gunter, its inventor, is generalh made of box: there are two sorts, the lat Gunter, and the sliding Gunter, having botl’ the same lines, but differently used, the forme? with the compasses, the latter by sliding. The lines now generally delineated on those in) struments are the following; viz. a line o numbers, of sines, tangents, versed sines, sin of the rhumb, tangent of the rhumb, meridi onal parts, and equal parts; which are con, structed after the following manner: wif The line of numbers is no other than th’ logarithmic scale of proportionals, wherei’ the distance hetween each division is equal t} the number of mean proportionals containe’ between the two terms, in such parts as th) distance between 1 and 10 is 1000, &c. equé) the logarithm of that number. Hence it fo lows, that if the number of equal parts € pressed by the logarithm of any number b taken from the same scale of equal parts, am set off from 1 on the line of numbers, the d vision will represent the number answerit to that logarithm. Thus, if you take .954, & (the logarithms of 9) of the same parts, ar set it off from 1 towards 10, you will have | division standing against the number 9, ] like manner, if you set off .903, &e. .845, & 778, &c. (the logarithms of 8, 7, 6) of the san equal parts from 1 towards 10, you will hay the divisions answering to the numbers 8,7, | After the same manner may the whole li be constructed. a The line of numbers being thus construc ed, if the numbers answering to the natw sines and tangents of any arch, in such pal as the radius is 10,000, &c. be found upon tj line of numbers, right against them will sta the respective divisions answering to the 1 spective arches, or, which is the same thit if the distance between the centre and fl division of the line of numbers, which e presses the number answering to the natw sine or tangent of any arch, be set off on ® respective line from its centre towards t left hand, it will give the point answering the sine or tangent of that arch; thus 1 natural sine of 30 degrees being 5000, &e. the distance between the centre of the line numbers (which in this case is equal to 10,0( &c. equal the radius) and the division, ont same line representing 5000, &c. be set from the centre, or 90 degrees, on the ii sines, towards the left hand, it will give t® point answering to the sine of 30 degre And after the same manner may the whi} line of sines, tangents, and versed sines divided. | The line of sines, tangents, and vers} sines being thus constructed, the line sine l secon¢ SCA 2 rhumb, and tangent of the rhumb, are sily divided ; for ifthe degrees and minutes swering to the angle which every rhumb ikes with the meridian, be transferred from respective line to that which is to be di- led, we shall have the several points re- ired; thus, if the distance between the ra- is or centre, and sine of 45 degrees, eqnals : fourth rhumb, be set off upon the line e of the rhumb, we shall have the point iwering to the sine of the fourth rhumb; 1 after the same manner may both these 2s be constructed. The line of meridional ts is constructed from the table of meri- nal parts, in the same manner as the line tumbers is trom the logarithms. “he lines being thus coustructed, all pro- ms relating to arithmetic, trigonometry, their depending sciences, may be solved she extent of the compasses only; and, as questions are reducible to proportions, the eral rule is, to extend the compasses from first term to the second, and the same ex- of the compasses will reach from the third te fourth; which fourth term must be so inued as to be the thing required, which tle practice will render easy. SALE of Relation and Differential Sca.e. RECURRING Series. SALENE, or ScALENoUs, is a term used istinguish any figure or solid, when the \drawn from the vertex to the centre of Dase is not perpendicelar to the base. ‘term, however, is most commonly used reference to a triangle, in other cases the oblique is employed; thus we say, an ue cone, cylinder, &c.; and very fre- tly also oblique triangle, instead of sca- triangle. APEMENT, from the French word ent, a term used among clock and makers, to denote the general contri- » by which the pressure of the wheels, th move always in one direction) and *ciprocating motion of the pendulum or ce, are accommodated the one to the / When a tooth of a wheel has given valance or pendulum a motion in one ‘ion, it must quit it, that it may get an sion in the opposite direction ; and it is Scaping of the tooth of the wheel from ance or pendulum, or of the latter from mmer, whichever we please to call it, as given rise to the general term. m the nature of a pendulum it follows requires only to be removed from the ver- sition, and then let go in order to vibrate teasure time. Hence it might seem, othing is wanted but a machinery so sted with the pendulum as to keep a r, as it were, of the vibration. It could difficult to contrive a method of doing but more is wanted, the air must be ted by the pendulum. This requires orce, and must therefore employ some ‘the momentum of the pendulam. The n which it swings occasions friction— ; | SCH the thread, or thin picce of metal by which it is hung, in order to avoid this friction, ocea- sions some expenditure of force by its want of flexibility or elasticity. These, and other causes, make the vibrations grow more and more narrow by degrees, till at last the pen- dulum is brought to rest. We must there- fore have a contrivance in the whceelwork which will restore to the pendulum the small portion of force which it loses in every vibra- tion. The action of the wheels therefore may be called a maintaining power, because it keeps up the vibrations. But this may affect the regularity of vibration. If it be supposed that the action of gravity renders alf the vi- brations isochronous, we must grant that the additional impulsion by the wheels will destroy that isochronism, unless it be so applied that the sum total of this impulsion and the force of gravity may vary so with the situation of the pendulum as still to give a series of forces, or a law of variation, perfectly similar to that of gravity. In order to accomplish this a great number of ingenious artists have at times proposed different constructions for this purpose, a description of several of which may be seen in vol. ii. Gregory’s “‘ Mechanics.” See also the works of L’Ainée, Lepaute, Le Roy, Berthoud, and Cumming’s “ Elements of Clock and Watchwork.” SCENOGRAPHY (from oxnyn, scene, ypady, I describe), in Perspective, a representation of a body on a perspective plane, or a descrip- tion thereof in all its dimensions such as it appears to the eye. . The ichnography of a building, &e. repre- sents the plan or ground-work, the orthography the front or one of the sides, and the sceno- graphy the whole building as it strikes the eye of an observer at a proper distance. SCHEINER (CuHrIsTopHER), a German mathematician, astronomer, and Jesuit, emi- nent for being the first who discovered spots on the sun, was born at Schwaben, in the ter- ritory of Middleheim, in 1575. He first dis- covered spots on the sun’s disc, in 1611, and made observations on these phenomena at Rome, until at length reducing them to order, he published them in one volume, folio, in 1630. He wrote also some smaller things re- lating to mathematics and philosophy, and died in 1650. — SCHEME, a, draught or representation of any geometrical or astronomical subjeét, and is otherwise called a diagram. SCHOLIUM, cxorsoy, a note, annotation, or remark, occasionally made on some pas- sage, proposition, or the like. ‘The term is much used in geometry, and other parts of mathematics ; where, after de- monstrating a proposition, it is common to point out how it might be done some other way; or to give some advice or precaution in order to prevent mistakes, or to add some particular use or application thereof. Wolfius has given us abundance of curious and useful arts and methods, and a good part RR 2 “SCR of the modern philosophy, the description of m thematical instruments, &e. all -by way of Scholia to the respective propositions in his ‘« Elementa Matheseos.”’ SCHONER (Joun), a noted German phi- losopher and astrologer, born in 1477, and died in 1547, at 70 years of age. He was author of several works; many of them on astrology; but he was celebrated for a set of astronomical tables, entitled “ Resolute,” and a very considerable work on Dialling ; these, with his other works, were published in folio at Nuremberg, in 1551. ~ SCIENCE, in Philosophy, denotes any doc- trines deduced from self-evident principles. Sciences may be properly divided as follows. 1. The knowledge of things, their constitu- tions, properties, and operations: this, in a little more enlarged sense of the word, may be called natural philosophy, the end of which is speculative truth. 2. The skill of rightly applying these powers. ‘The most consider- able under this head is ethics, which is the seeking out those rules and measures of hu- man actions that lead to happiness, and the means to practise them; and the next is me- chanics, or the application of the powers of natural agents to the uses of life. 3. The doctrine of signs, the most usual of which being words, it is aptly enough termed logic. “ This (says Mr. Locke) seems to be the most general as well as natural division of the objects of our understanding. For a man can employ his thoughts about nothing but cither the contemplation of things themselves for the discovery of truth, or about the things in his own power, which are his actions, for the aitainment of his own ends; or the signs the mind make use of both in the one and the other, and the right ordering of them for its clearer information. All which three, viz. things as they are in themselves knowable, actions as they depend on us in order to hap- piness, and the right use of signs in order to knowledge, being toto eclo different, they seem to be the three great provinces of the intellec- tual world, wholly separate and distinct one from another.” SCIENTIFIC, or SciENTIFICALLY, relat- ing to or conducted by the rules of science. SCIOPTIC, or ScioptRic Ball, (from cx, shadow, and orsoues, I see), a sphere, or globe of wood, with a circular hole or perforation filled with a lens, and mounted in a frame so as to be turned in any direction, being used in making experiments in a darkened reom or camera obscura. SCIOTHERICUM Telescopium, a name given to a particular sort of dial invented by M. Molyneux, and of which he has given a description in a work of his under the same title. SCORPIO, the Scorpion, the eighth sign of the zodiac, denoted by the character 1]. See CONSTELLATION. SCREW, one of the mechanical powers, or rather a combination of two of them, the in- SCR clined plane and the lever, principally w in pressing bodies together, or in lifting gr weights, which may be conceived to be ger rated as follows: “e4 Let a solid and a hollow cylinder of ¢ diameters be taken, and let ABC be a rig angled plane triangle, whose base BC is equ to the cireumference of the solid cylinder, a applied to the latter in such a manner, thatt base BC may coincide with the circumferer: of the base of the cylinder, and CA will fo; a spiral thread on its surface. At 7. ~ toe By applying to the cylinder, triangles succession similar and equal to ABC, in a manner that their bases may be parallt BC, the spiral thread may be contim and supposing the threads to have thic or the cylinder to be protuberant whe falls, the external screw will be formet which the distances between two contig) threads, measured in a direction parall the axis of the cylinder, is A B. - § Again, let the triangles be applied it same manner to the concave surface of t low cylinder, and where the thread falls xroove be made, and the internal screw Ww formed. The two screws being thus ex adapted to each other, the solid or hi cylinder, as the case requires, may be about the common axis by a lever, or in other manner as the nature of the casé require. The external screw is some called the male screw, and the intern female screw. When there is an equilibrium upo screw, then the power is to the weig) the distance between two contiguous th measured in a direction parallel to the is to the circumference of the circle dest by the power. That is, if P represents the power, weight to be lifted, or the resistance overcome, also d the distance of two: cuous threads, and / the length of the. then P : W :: d: 62832.1. calling | the circumference of a circle to radius © This results immediately, if we adm equality in the momenta of the powé weight, for the velocity of the power the velocity of the weight, as the € ference of the circle described by the! is to the distance of the threads; bi principle has been justly objected = SCR yme modern writers, and ‘other demonstra- yn adopted in its stead. The result, how- rer,is the same in all cases; and so far as re- tes to the theory it is perfectly correct; but practice, not only the weight, or resist- ice, but also the friction of the screw, is to » overcome, which in this machine is very eat, in some cases equal to the weight itself, ing frequently sufficient to sustain this after @ power is removed. Endless or Perpetual Screw, is one which orks in and turns a dented wheel, without > internal screw, and is thus called because constantly preserves its motion, while that the wheel is continued. See Plate VII. 3. n this machine, when the weight and power +in equilibrio, the product P x AB x ED W x dist. of threads x rad. of axle. is being a combination of the power of the ew and axis in perithochio. Archimedes’ ScREW, or the Water-snail, is a chine for raising water, which consists ier of a pipe wound spirally round a cy- ler, fig. 2, Plate VII. or of one or more spi- excavations formed by means of spiral pro- fions from an internal cylinder, covered by external coating, so as to be water tight. s screw is one of the most ancient, and, he same time, ingenious machines we »w, being truly worthy of the name it bears, posing Archimedes to be the real inventor. pugh simple in its general manner of ope- on, its theory is attended with some diffi- ies which could only be conquered by the lern analysis: it was first stated correctly, ar as we have been able to ascertain, by Pitot, in the Memoires de |l’Academie Jale des Sciences, and afterwards more jorately by Euler, in Novy. Comment. Pe- ol.tom. v. Later attempts by Langsdorf, is Handbuch der Maschinenlehre, and 2 other authors, are not to be relied on. {the nature of this curious machine may 4e better understood, we shal! here state ally its manner of operation. ‘If we conceive that a flexible tube is d regularly about a cylinder from one end e other, this tube or canal will be a screw iral, of which we suppose the intervals of spires or threads to be equal. The cy- ion, if we put in at the upper end of the I tube a small ball of heavy matter, which move frecly, it is certain that it will fol- il the turnings of the screw from the top te bottom of the cylinder, descending ys as it would have done had it fallen in ht line along the. axis of the cylinder, it would occupy more time in running igh the spiral. If the cylinder were placed its axis horizontally, and we again put all into one opening of the canal, it will nd following’ the direction of the first “spire ; but when it arrives at the lowest ‘of this portion of the tube it will stop, ust be remarked, that though its hea- t being placed with its axis in a vertical , SCR viness has no other tendency than to make it descend in the demi-spire, the oblique position of the tube with respect to the horizon, is the cause that the ball, by always descending, is always advancing from the extremity of the cylinder, whence it com- menced its motion, to the other extremity. It is impossible that the ball can ever ad- vance more towards the farther, or, as we shall call it, the second extremity of the cylinder, if the cylinder, placed horizontally; remains always immoveable; but if, when the ball is arrived at the bottom of the first demi- spire, we eause the cylinder to turn on its axis without changing the position of that axis, and in such manner that the lowest point of the demi-spire, on which the ball presses, becomes elevated, then the ball falls necessarily from this point upon that which succeeds, and which becomes lowest; and since this second point is more advanced to- wards the second extremity of the cylinder than the former was, therefore, by this new descent, the ball will be advanced towards that extre- mity, and so on throughout, in such a manner that it will at length arrive at the second ex- tremity, by always descending, the cylinder having its rotatory motion continued. ‘More- over, the ball, by constantly following its tendency to descend, has advanced through a right line equal to the axis of the cylinder, and this distance is horizontal because the sides of the cylinder were placed horizontally. But if the cylinder had been placed oblique to the horizon, and we suppose it to be turned on its axis always in the same direction, it is easy to see, that if the first quarter of a spire actually descends, the ball will move from the lower end of the spiral tube, and be car- ried solely by gravity, to the lowest point of the first demi-spire, where, as in the preced- ing case, it will be abandoned by this point, as it is elevated by the rotation, and thrown by its heaviness upon that which has taken its place; whence, as this succeeding point is farther advanced towards the second ex- tremity of the cylinder than that which the ball occupied just before, and consequently more elevated, therefore the baH while follow- ing its tendency to descend by its heaviness, will be always more and more elevated by virtue of the rotation of the cylinder. Thus it will, after a certain number of turns, be ad- vanced from one extremity of the tube to the other, or through the whole length of the cylinder; but it will only be raised through the vertical height determined by the obli- quity of the position of the cylinder. \ Instead of the ball let us now consider water as entering by the lower extremity of the spiral canal, when immersed in a reser- voir: this water descends at first in the canal solely by its gravity ; but the cylinder being turned, the water moves on in the canal to ‘occupy the lowest place, and thus, by the continual rotation, is made to advance farther and farther in the spiral, till at length it is SEA yaised to the upper extremity of the canal, where it is expelled. There is, however, an essential difference between the water and the ball; for the water, by reason of its flui- dity, after having descended by its heaviness to the lowest point of the demi-spire, rises up on the contrary side to the original level; on which account more than half one of the spires may soon be filled with the fluid. This is an important particular, which, though it need not be regarded in a popular illustration, must be attended to in the more particular exhi- bition of the theory. SCRUPLE, a small weight equal to 20 grains, or the 3d part of a drachm. See WEIGHTS. ScruPLE was also used by some ancient nations for a small portion of time, viz. the 1080th part of an hour. ScRUPLE, in Astronomy, the same as Dieit, which see. SCUDO, a coin of different nations and of different values, from 2 shillings to 5 shillings and upwards. SEA, in a general sense, the great reservoir of water, into which the lakes and rivers empty themselves, and from which it is again drawn by evaporation, that moisture which, falling in showers of rain, fertilizes the earth, and supplies the waste of the springs and rivers. The absolute quantity of sea-water cannot be ascertained, as its mean depth is unknown. La} lace has demonstrated that a depth of four leagues is necessary to reconcile the height to which the tides are known to rise in the main ocean with the Newtonian theory. If we suppose this to be the mean depth, the quantity of water in the ocean must be im- mense. Even on the supposition that its mean depth is not greater than the fourth part of a mile, its quantity, allowing its sur- face to be three-fourths of that of the super- ficies of the earth, would be 32,058,939 cubic miles. ' According to the most accurate observa- tions hitherto made, the surface of the sea is to the land as three to one; the ocean, there- fore, extends over 128,235,759 square miles, supposing the superficies of the whole globe to be 170,981,012 square miles. To ascertain the depth of the sea is still more difficult than its superficies: both on account of the nume- rous experiments which it would be necessary to make, and the want of proper instruments for that purpose. Beyond a certain depth the sea has hitherto been found unfathomable; and though several very ingenious methods have been contrived to obviate this difficulty, none of them has completely answered the purpose. See Sea-GaGE. We know in general that the depth of the sea increases gradually as we leave the shore; but if this continued beyond a certain dis- tance, the depth in the middle of the ocean would be prodigious. Indeed the numerous islands every where scattered in the sea de- SEA monstrate the contrary, by showing us the bottom of the water is unequal, like land; and that, so far from uniformly sinkis it sometimes rises into lofty mountains. — the depth of the sea is in proportion to 4] elevation of the land, as has generally bes supposed, its greatest depth will not excee four miles, for there is no mountain who; altitude exceeds this in perpendicular heigh The sea has never been actually sounded | a greater depth than a mile and sixty-s feet; every thing beyond that therefore res entirely upon conjecture and analogical re soning, which ought never to be admitted determine a single point that can be asec tained by experiment, because, when adm ted, they have too often Jed to false cone) sions. Along the coasts where the depth] the sea is in general well known it has alwaj been found proportioned to the height of t shore; when the coast is high and mot tainous, the sea that washes it isdeep ; wh on the contrary, the coast is low, the water shallow. Whether this analogy holds at distance from the shore, experiments al can determine. 4 In order to compute the quantity constani discharged into the sea, let us take a ri whose velocity and quantity of water known, the Po, for instance, which, accor to Riccioli, is 1000 feet (or 100 perches of logne) broad, ten feet deep, and runs at _ rate of four miles in an hour; conseque that river discharges into the sea 200, cubic perches of water in an hour, or 4,800, ina day. A cubic mile contains 125,000 cubic perches ; the Po therefore will twenty-six days to discharge a cubic mil water into. the sea. Let us now suppi what is perhaps not very far from the t& that the quantity of water which the sea’ ceives from the rivers in any country Is portioned to the extent of that country. Po from its origin to its mouth travers country 380 miles long, and the rivers w fall into it on every side rise from sow about sixty miles distant from it. The therefore, and the rivers which it rece} water acountry of 45,600 square miles. -. since the whole superficies of the dry lal about 42,745,253 square miles, it follows, our supposition, that the quantity of v discharged by all the rivers in the worl) one day, is thirty-six cubic miles. If, t fore, the sea contains 32,058,939 cubic | of water, it would take all the rivers it world 2439 years to discharge an equal | ( tity. It may seem surprising that the sea, | it is continually receiving such an imn} supply of water, does not visibly inci and at last cover the whole earth. Bu surprise will cease, if we consider tha! rivers themselves are supplied from the and that they do nothing more than | back those waters which the ocean is | nually lavishing upon the earth. Dr. F SEC as demonstrated that the vapours raised ‘om the sea and transported upon land are ufficient to maintain all the rivers in the vorld. See EVAPORATION. SEASONS, certain portions of the year, as pring, Summer, Autumn, and Winter. -SECANT, in Geometry, a line which cuts nother, whether right or curved. Secant, in Trigono- etry, is a right line rawn from the-centre of ‘circle to meet the upper » farther extremity of 1y tangent, to the same rele. Such is the line 'B, which is the secant the are A D, or of the igle ACD. An are and its supple- ent have their secants qual, only the latter one ‘accounted negative, being drawn the con- ary way to the former. And thus the se- mts in the second and third quadrants are ~gative, while those in the first and fourth iadrants are positive. The secant of an arc, which is the comple- ent of the former arc, is called the cosecant, the secant of its complement. The cose- mts in the first and second quadrants are firmative, but in the third and fourth nega- re. ‘The secant of an arc is reciprocally as the isme, and the cosecant reciprocally as the ie, or the rectangle of the secant and cosine, d the rectangle of the cosecant and sine, ? each equal to the square of the radius. An arc a, to the radius 7, being given, the sant sand cosecant c, and their logarithms, the logarithmic secant and cosecant, may | expressed in infinite series, as follows, B , a? 5a* 61a® 277 a8 a +o t+ ii Be &e. i to Sar: + F208 806477” 2 3 5 7 | ian Be 7a dla hb &e. a 6 360r¢ 15120r+ 604800r a® 17a ©, We a) 45 * o500 ~° a er Paty.) har I a at ‘6 — Mm TE7xz : nal te x (5 aa at 4 ere m is the modulus of the system of loga- ms. ‘fa be any arc, then 2 sec. *a sec. 2 a — a 2— sec. *a sec. 3 a= — se " ~ 4—83 sec..2a iets a sec. ta é ~~ §8— 8 sec. 7a + sec. ta : 2 sec. a ) Sec. 3 a= 1+seaa SEC SECANTS, Figure of. See Ficurr. SECOND, is the 60th part ofa minute, both as it relates to the measure of angles or time. See MINUTE. SECONDARY Circles, in Astronomy, are any circles that intersect one of the six great circles of the sphere at right angles. SECONDARY Planets. See SATELLITES. SECTION, in Geometry, denotes a side or surface of one body or figure cut by another, or the place where lines, planes, &c. cut each other. SEcTIONS of a Cone. See Conic Sections. SECTOR, in Geometry, is a portion of « circle comprehended between any two radii and their intercepted ares, as ABC. A. B , _ Similar Secrors, are include equal angles. | _ Lo find the Area of a Sector.—Say as 360° 1s to the degrees, &c. in the arc of the sector; so is the area of the whole circle to the area of the sector. ~ ; Or multiply the radius by the length of the arc, and half the product will be the area. See Cincu.ar Sector. SECTOR also denotes a mathematical instru- ment, of great use in finding the proportion between quantities of the same kind: as be- tween lines and lines, surfaces and surfaces, &c. whence the French call it the compass of proportion. ' The great advantage of the sector above the common scales, &c. is, that it is made so as to fit all radii, and all scales. By the lines of chords, sines, &c. on the sector, we have lines of chords, sines, &c. to any radius betwixt the length and breadth of the sector when open. ‘The sector is founded on the fourth propo- sition of the sixth book of Euclid; where it is demonstrated, that similar triangles have their homologous sides proportional. An idea of the theory of its construction may be con- ceived thus. Let the lines AB, AC, repre- those whose radii B sent the legs of the sector; and AD, AE, twe equal distances from the centre: if, now, the points CB and DE be connected, the lines SEG SEG CB and DE will be parallel; therefore the To describe on a given Line AB, a Segment ca- triangles ADE, ACB, will be similar: and, pable of containing a given rectilineal Angle C, consequently, the sides AD, DE, AB, and BC, proportional; that is,as AD: DE:: AB: BC; whence, if AD be the half, third, or fourth part of AB; DE will be a half, third, or fourth part of CB; and the same holds of all the rest. If, therefore, AD be the chord, sine, or tangent, of any number of degrees to the ra- ae AB; DE will be the same to the radius C. For a complete history of this instrument, with its construction, use, and application to a variety of problems, in trigonometry, &c. See Robertson’s “ Treatise of such Mathe- matical Instruments as are usually put into a portable Case.” Sector is also used to denote certain por- tions of ellipses, hyperbolas, &c. resembling the circular sector above described. Astronomical SECTOR, an, instrument in- vented by Mr. Graham for finding the differ- ence in right ascension and declination be- iween two objects, whose distance is too great to be observed through a fixed telescope by means of a micrometer. SECULAR Equations, in Astronomy, are corrections required to compensate such in~ equalities in the motions of the heavenly bodies as are found to obtain in the course of a century. Thus, there are secular inequali- ties in the motion of the moon, which require for their correction so many distinct secular equations; and the like may be said of other bodies. See Vince’s and Woodhouse’s ‘¢ Treatises on Astronomy ;” Biot’s “ Astro- nomie de Physique ;” and Laplace’s ‘“‘ Meca- nique Celeste.” SECULAR Year, the same as JUBILEE. SEGMENT of a Cirele, is a part of a circle bounded by an are and its chord, and is either greater or less than a semicircle. The follow- ing are the most remarkable properties of circular segments, viz. All angles in the same segment of a circle are equal to each other; if the segment is greater than a semicircle, the angle is less than a right-angle; if less than a semicircle, it is greater than a right-angle. Cc D Thus the angle ACB is greater, and the angle ADB less, than a right-angle; the twa together making up two right-angles. | ty ae D FF At the ends of the given line make angle DAB, DBA, each equal to the given angle C Then draw AE, BE, perpendicular to A BD; and with the centre E, and radius E or EB, describe a circle; so shall AF Bb the segment required, as any angle F mad in it will be equal to the given angle C. To cut off a Segment from a Circle that s contain a given Angle C. E Cc A <4 Draw any tangent AB to the given cirel and a chord AD, to make the angle DA equal to the given angle C; then DEA w be the segment required, any angle E mac in it being equal to the given angle C. For what relates to the area, &c. of s ments, see CIRCULAR Segments. SEGMENT of a Sphere, is any part of a sph cut off by a plane; the section of which, wi the sphere, is always a circle. Lo find the Superficies and Solidity of Spheri Segments. Let d denote the diameter of the sphere, the chord of half the circumference, and ¢ chord of half the arc of any segment, a a the altitude or versed sine of the same; th 3'1416d7’ is the surface of the whole sphere,a 3°1416c’, or 3°1416 ad, the surface of the ment. To find the Solidity.x—1. To 3 times 1 square of the radius of its base, add the squi of its height; multiply the sum by the hei and the product by *5236. Or, 2dly, Fro times the diameter of the sphere, subtr; twice the height of the frustum ; multiply 1 remainder by the square of the height, and product by *5236. That is, in symbols, solid content is either = '5236a x (37? + a’) or = 5236 a” x (3d — 2a); where a is the altitude of the si SER yent, r the radius of its base, and d the jameter of the whole sphere. See Hutton’s nd Bonnycastle’s Mensuration. , Line of SEGMENTS, is one of the lines on 1e Gunter scale. SEMICIRCLE, is half a circle, or the area omprehended between a diameter and the emicircumference. ; SEMICIRCLE, or Graphometer, is the name of a instrument much used in surveying; it is a wt of half theodolite,: the only difference eing, that the limb of the theodolite is an otire circle graduated from 1° to 860°, hereas in the semicircle the gradation is aly carried to 180°. See THEODOLITE. SEMICUBICAL Parabola. See PARABOLA. ~-SEMIDIAMETER, the same as Radius. ee Rapivus. SEQUIN, a coin of different countries, and ‘different values, from 9s. to 9s. 6d. sterling. SERIES, in Algebra, is a continued rank, "progression, of quantities connected toge- er by the signs + or —; usually proceeding ‘cording to a certain determinate law. Such are the following: 1 1 l+s+ + x ke. 1 1 1 a sipned elton 1G e former being the reciprocals of the odd ambers, and the latter a geometrical series, 1 ary 1 + &e. ‘which the ratio is > Series are of different forms, and arise in rious ways, but more commonly from the ‘pansion of some binomial expression; thus 2 find 1 = } ey he 1 b ! b 53 ' & +6 (a b) aaa at io 1 EL Bia I harait, © >, —@—4) ann bie toe + + &e. lich differ from each other only in the ens. Series also receive several different deno- Nations according to certain circumstances lating to their formation, the law which they serve, &c. as converging, diverging, &c. cles. (Converging SrRiEs, is one in which the mms decrease, or become successively less d less, as 1 1 1 1 Seep cad tbat sles ERT gin V+ eto tots t ke Diverging SER1ES, is one in which the terms mtinually increase, as P34 4-—8 + 164582 ee: Neutral Series, is that in which all the ‘ms are equal to each other; such is 1—1+1—1+1—1 +4 &e. ich arises from the division of 1 by 1 +1. Indeterminate Srries, is one whose terms yeeed by the powers of an indeterminate antity, as e+iar+ fait inat+ ke, d this is either ascending or descending. An Ascending Serizs, is that in which the SER powers of the indeterminate, or unknown quantity, continually increase, as 1 + ax + ba* + cx + &e. . A Descending Series, is that in which the powers of the indeterminate quantity conti- nually decrease, as L + ax + ba? + cx + &e. or a b c 1 + x ta tT Fb &e- Law of a Series, is used to denote that relation which subsists between the succes- sive terms of a series, and by which their general term may be denoted; thus the series 2 See ie ig eee 1 + rhe + wee + an fF 316 + &e. may be put under the form 2 eb RAG ; ttg* tg” ta" tate. where the law by which it may be indefinitely continued is manifest. There are beside these, various other deno- minations for particular series, or particular forms of series, as Circular Series, which is that whose sum depends upon the quadrature of the circle. Recurring SERIES, is one in which each term is some constant function of a certain number of the preceding terms. See Recur- RING Series. Summation of SERIES, is the method of find- ing the sum of a series, whether the number of its terms be finite or infinite; of. which there are several methods, as by addition, subtraction, multiplication, increments,. the differential method, recurring series, &e. of which, perhaps, the method of increments is the most general; this we have treated of under the article INcREMENTs, and recurring series under the article REcurRING Series; and of the other methods, we propose giving a slight sketch in the present article. Of the Differential Method—We have al- ready, under the article INTERPOLATION, ex- plained the method of finding the successive differences of the terms of a series, and therefore in this place we shall merely state the application of that method to finding the sum of a given series. 1. To find the nth term of any series a, }, ¢, d, e, &c. when the differences of any order become equal to each other. Let d’, d’, d", d'", dv, &c. be the first terms of the several order of differences, then will nan—I1, , (»n—1)(n—2Q) ,, SCR Ba a 1.2 aN (n nee 1) (n bgt 2) (n ti 3) dq’ — the nth term required. Thus if it were required to find the 12th term of the series 2, 6, 12, 20, 30, Ke. Here we have 2.4.6.8 + &e. 2. 6" 1209200) Be OE Ist dif. 4° 6 8. ni tO, ee 2d diff. NY Se Sms 3d diff. O° Osis SER therefore, a = 2, d’ = 4, d’ = 2, and d”—0; also, n = 12, whence m—1, , m—1)(n—2) ,, St rae ee 1.2 ¥ 2+ 11d’ + 55 d’ = 156, the 12th term. Again, required the 20th term of the series of triangular numbers 1, 3, 6, 10, 15, &e. + &c. = Ts Ole 10s 15,. &e. Ast diff. 2 3 4 5, &e. 2d diff. 1 1 iP &e. 3d diff. 0 0, &e. therefore.a = 1, d' = 2, d’ = 1, and.n = 20; whence a + “ most remarkable and useful results, but - their investigations we must refer the ider to the following works: viz. “ Traité Trigonometrie,” by M. Cagnoli, translated m the Italian by M. Chompré; Bonny- stle’s Treatise of Plane and Spherical Tri- nometry, 2d edit.; Woodhouse’s Trigono- ‘try, and the article Arithmetic of Sines in » Brewster’s Encyclopedia. Some of the most useful formule relating to les and cosines are as follows: similar ex- assions for tangents, secants, Xc. will be ind under the articles ‘TANGENT and SrE- NT. cos. (a — b) + cos. (a + b) —= 2cos.a.cos.b cos. (a —b)— cos.(e + b) = 2sin. a.sin.b sin. (a + db) + sin. (a— 6) = 2 sin. a.cos.b sin. (a + 5) — sin. (a — b) = 2cos.a. sin. b writing a=3(p + q) and 6 = 4 2 (p— q) q + Cos. p =2cos, i(p+q). cos. 2 (p— q) gq — cos. p — 2sin. $(p + q).sin, 3 (p—q) +p + sin. q = 2sin.2(p + q).cos. 3 (p—q) -p —sin..g = 2c0s.7(p + ¢).sin. 3 (p—q) Or writing a = np, andb=p., Bal + cos.(a+ 1) p= 2.008, p.COs. np (n—1)p — cos.(n + 1) p =2sin. p.sin. np Gln + 1p + sin. (n— 1)p = 2 cos. p. sin. np .(n + 1)p—sin. (n—1)p = 2sin. p.cos. np From which are readily obtained the sine A cosine of any multiple are, vz. by making = a, sine a — x, cosine a= y, and xn = ‘ 3, &e. we readily obtain the following for- ‘lee: cos. 2a = 2x cos. la — cos. 0a cos. 3a = 2x cos. 2a — cos. la cos. 4a = 22 cos. 3¢ — Cos. 2a cos. 5a = 2x cos. 4a — cos. 3a &e. &e. Where the law is obvious, and which may brought under the following form: wla=e mee — 2 — yf? .38a—=23— 32ry” _*, Gary" + y* : eee = 2° 5 — 10a3y* + Say" 6a =x °— lbaty” + 15x27 y* — y® na = x — ry"—2y” + sa%—4y* + ta" Sy°,&e. Where 7, s, t, &c. are the alternate a a nts of a binomial raised to the nth power. {nd in a similar manner we have from for- la 3, jin. 2a = 2x sin. la — sin. Oa jin. 3a — 2x sin. 2a — sin. la tin. 4a — 2a sin. 3a — sin. 2a Wn. 5a — 22 sin. 4a — sin. 3a &e. &e. in. lay Min. 2a = 2ary m.3a—3a°y— y? in. 4a = 4a3y — aay" fin. 5a = 5aty — 10a*y3 + y? SIN Sin, 6a = 625 y — 2aty3 + 6y® Sinena = na*—! ya per-> yas qae—s —ke. where », p,q are the alternate even terms of a binomial raised to the power n. These general formula were first given by John Bernoulli without demonstration, in the Leipsic Acts for 1701. Some other formule for the sines and co- sines of inultipie arcs, are as follows: (2z)*—n(Qayr2 4 MA __ n(n—4)(n—5 1D oat n — n —2 at (2x)"-* Cos. na = 4 cone + Xe. ; ((2 x) u—1 | Mais) in eas gs Tears Le a (u—-4) (n—5)(n—6 6) ee EEN Ry + &e. i n(n? —I ——1) a A, —1)\(n?—9) ‘bio. 1.2.3.4.5 — nh Oy a a) 1.2.3.4.5.6.7 x7 + &e. the ambiguous sign being plus when n is of the form 4m + 1, and minus when it is of the form 4m +. 3. 3 _ nl 2, (n*—1)(n* od a yak TU niaignbis ph Tae ie weg _ OD @* 9)? — 25) 6 1.2.3.4.5.6 —— (2 ip n—3 + Sin. wa =y (Qx)r-7 | ( nx— car Ne Te : x Sin. nat + &e. If n be even, or of the form 2m, then es rae (n* — 4) sa 2 1.2.3.4 __ 2° (nt —4) (n?—~16) 6 1.218.451) ee n (n* — 4) {rz— Tag + t n(n” — 4) (n? = 16) | 1.2.3.4.5 <8 Another elegant expression for the cosines of multiple arcs, is as follows: Let 2 cos.a =a + Cos.na = Sin.na = ; then we have a4—2 ie x 2 cos. 2a = a* + + x 2 cos. 2 cos. 3a = x3 +3, 2 cos. 4a = a+ + 4 = ee < The preceding are the ad general and useful expressions for the sines and cosines of multiple ares; let us now investigate a 2 cos. na ‘few formule for the powers of the sines and cosines: for this purpose we shall repeat again table 3 of formule; ouly writing uniformly ¢ instead of p; viz. Ss SIN % cos. a.CO8. na = Cos. (n—1) a+ coSs.(n+1)a 2 sin. a. sin. na= cos. (n—1)a—cos.(n+1)a 2 cos. a. Sin. na = Sin. (vn +1) a+sin, (n—l)a 2 sin. a. Cos. na = sin. (n +1) a—sin.(n+1)a Here in formula 1, making » = 1, we have 2 cos. 7a = 1+ cos. 24. Multiply both sides by 2 cos. a, and we have 4 cos. 3a == 2 cos. a + 2 COS. a. COS. 2a. But making n = 2 in formula 1, it becomes 2 cos. a. cos. 2a = cos. a + cos. 3a; therefore 4 cos. 3a — cos. 3a + 8 cos. a And proceeding in a similar manner we ob- tain, COs. @—= cos. a 2 cos. 7a = cos. 2a + 1 4 cos. 3a = cos. 3a + 3 cos. la 8 cos. 4a = cos. 4a +4 cos. 2a+ 3 16 cos. 5a = cos. 5a+ 5 cos. 3a +10 COS. a oo 2 2” cos. xa = 2 cos, na +2ncos.(n—2)a + ——— a = 1) 2n (n — —2 cos. (n—4) a + n (n—1) (n ie 1.2.3 (n—6) a+ &e. ‘The formula ending in 2n (n— 1) (n—2) &e. 1.2.3, &e. 2n (m— 1) (n — 2) 1.2.3 According as 7 is an even or an odd number, and in the former case, half the expression is to be taken for the last term. From the third of the preceding formule the solution of cubic equations may be ob- tained, for since 4 cos. 3a = cos. 3a + 3 cos. a we have cos 74 — 3 cos. a = cos. 3a; which it is obvious may be easily assimilated with the cubic equation of the form x3 —ax = b. And in a similar manner we obtain from the 2d general formule, viz. 2 sin. a. sin. na=Cos. (nw — 1) a—cos. (n+])a general expressions for the powers of the sine of an arc, viz. Sin. a= sina 2 sin. *a =— cos. 2a +1 4 sin. 3a —=— sin. 3a +3 sin. a 8 sin. 3a = cos. 4a—4 cos.2a+ 8 16 sin. +a = sin, 5a —5 sin. 3a + 10sin.a@; and generally 2” sin. “a4 = + 2 sin. na = Qn sin. (n— 2)a cos. (nm —n) a or, in COS. @ en aa Aran sin. (n—4) a 1.2 _ 2n (n — 1)\(n—2) .. r Sane eee ASIN. (92 — G 3 1.2.3 Marae act &e, Or, 2 sin. “a = 2 cos, na | Wn cos. (n— 2) a ae 2 BY) cos. (n— 4) a 2n (n—1}) (n— 2) sre COS. (1 — ; 1.2.3 ¢ e @ch &e. SIN | bd In the first series the upper sign is to] h used when n is of the form 4 m+ 1, and th lower when it is of the form 4m — 1. Andi the second the upper sign is to be used wh e nis of the form 4m, and the lower when it of the form 4m + 2. In all the preceding formule we have eo sidered radius as unity; but in case it is_ cessary to introduce it, it is only necessar y make it enter into each term with such a pow as will render all the terms homogeneous. _ ‘The following are some of the simples cases in which the sines of certain angles at expressible in surd forms, i Sin. 0 = 0 Sin. 9° = “} / (8+ /5)— V(5b— vst ey | Sin. 15° = mat a7 eee a) Sin. 18° = 7 aC + 5) Nin. 277 = a} V(5+ /5)— 43 — vy \ Sin. 30° —” : > £3 Sin. 36° = Z /(0—24/5) : Sin, 45° = 5 v5 a Sin. 54° = r (1 + V5) Sin. 60° = 5 Fae | si il sk : V(5 + V5) + V8—v 5) Sin. 75° = — jc + 4/2) | Sin. 81° = 7) ¥(3+ 75) + BGP v8 Sin, 90° = r. | ‘These expressions for particular arcs, giv merely for the convenience of reference, a readily drawn from the preceding general fi mule, for multiple arcs, and the following | Pat 9 the half are, viz. sin. a = id —— a r : from the known sines of 30°, 45°, 60°, 90°. See Bonnycastle’s “ ‘Trigonometr n p. 312, 2d edit. i As the sin. a. — cos. (90 —a) it is obvie how the cosines may be found to any of : above angles, Artificial Sines, are the logarithmic sin or the logarithms of the sines. Curve or Figure of Sines. See Fievaell Line of Sinzs. See Gunrer’s Seale, i SINICAL Quadrant; is a quadrant made wood or metal, with lines drawn from a side, intersecting each other, with an ind 1 divided by sines, also with 90 degrees ont * a é SIP mb, and sights at the edge. This quadrant vas formerly used for taking the altitude of the an, but it is grown into disuse since the great nprovement in the construction of other in- uments for the same purpose. ‘SIPHON, or Sypuon, is a bent tube used 1 drawing off wine, liquors, and other fluids; ie principle of which may be illustrated as lows. ‘Tf one end of the phon or bent tube (NO be put into a sssel of water or tid, and the other id without be lower an the surface of e water; then if e air be drawn out, @ water will ascend in the leg MN in conse- ence of the pressure of the atmosphere on 2 external surface of the fluid, from which will descend to O by its own gravity, and ‘there discharged into any vessel placed to seive it. For the pressure of the air at O, force the fluid in the direction ONM, is jal to the pressure of the air on the surface that in the vessel, which force is exerted to ve the fluid in the contrary direction MNO, least these forces are very nearly equal, the y difference being in their different alti- les, and consequently different densities of ‘air at those places, which must necessarily extremely small. But the former pressure opposed to the pressure or weight of the gmn NO, and the latter by the pressure of column MN;; the latter pressure of the therefore being less opposed than the for- the fluid must move in the direction of Jatter pressure, or in the direction MNO; the fluid will continue to run till the pres- % MN and NO become equal, or till O _M are in the same horizontal line, for a their perpendicular heights being equal r pressures will be equal, and the columns uid in the two branches wiil remain sus- ded. Instead of sucking out the air from siphon which may sometimes be incon- ‘ent or unpleasant, it may be first filled 4 the fluid, and then while the orifices are ped, inverted,-and placed in its proper tion, when the fluid will begin immediately ow as above described. ‘hen the siphon is very large, and several in height, as in drawing off water from a y or pit, it will be best to proceed as fol- . Stop the two orifices, or at least that hich the discharge is to take place, and by means of an opening made at the top 1e purpose, fill the siphon completely, and istop this opening by means of a plug, or wise, after which if the two ends be ed the water will begin to flow, and so nue till either the water be exhausted, or he surfaces of it at each end are in the horizontal line. It must be observed, ‘ver, that this cannot be put in practice :top of the siphon be more than 33 fect ethe surface of the water to be drawn off. SME This method of drawing off water is men- tioned of Hero of Alexandria, the first author of any consequence upon this subject amongst the ancients. It is said, indeed, that the water in the legs, unless it be purged of its air, will not rise to the height of 30 feet, because air will extri- cate itself from the water, which rising to the top of the siphon, will there in part counteract the atmospheric pressure ; but in small capil- lary tubes the experiment will succeed to a height somewhat exceeding that due to the pressure of the atmosphere, which anomaly is easily accounted for on the principle of capil- lary attraction. | Woartemberg S1rHon, is a peculiar construc- tion of this machine, of which a description is given in Musschenbrock Introd. ad Phil, Nat. tom. ii. See also Phil. Trans. vol. xiv. SIRIUS, the Dog Star, a very bright star of the first magnitude, in the constellation Canis Major, or the Great Dog. The Arabs call it Aschere, Elschecre, Scera; the Greeks Sirius; and the Latins Canicular. SITUS, in Algebra and Geometry, is used by Wolfius and. Leibnitz to denote the situ- ation of lines, surfaces, &c. It is, however, never used: by modern authors. SLIDING, in Mechanics, is the motion of a body along a plane, when the samé face or surface of the moving body Keeps in contact with the surface of the plane; and is thus dis- tinguished from rolling, in which the several parts of the moving body come successively in contact with the plane on which it rolls. SLIDING Rule. See Rute. SLING, a string instrument,. used in cast- ing stones, &c. which it does with great vio- lence. ‘The motion of a stone discharged from a sling arises from its centrifugal force, when whirled round in a circle. The velocity with which it is discharged, is the same as that which it has in the cirele, which much exceeds what could be commu- nicated to it in any other way by the hand; and the direction in which it is discharged is that of the tangent of the circle at the point of discharge ; whence its motion, flight, &e. may be computed the same as a common pro- jectile. ; SLUSE, or Siusius (Rene Francts Wat- TER), an ingenious mathematician, was born at Vise in 1620, and died in 1683; he was a fellow of the Royal Society of London; and has some papers in vols. vii. and viii. of the ‘Transactions. He was also author of a work entitled “Mesolabium et Problemata Solida,”’ &e. SMEATON (Joun), a very celebrated Eng- lish civil engineer, was born the 24th of Ma 1724, near Leeds, where he also died in 1792, in the 68th year of his age. Mr. Smeaton’s fame rests more upon the many excellent machines that were con- structed under his direction, and the nume- rous works which he conducted, than upon his writings, though these last were both numerous SS2 SO.L and important, most of which are inserted in different volumes of the Phil. Trans. from the year 1750 to 1776; besides which he published an account of his construction of the Hdystone Light-house, a work which alone could not fail of distinguishing him as the first of civil engi- neers. . SMOKE, or SMoAkK, a humid matter ex- haled in the form of vapour by the action of heat, cither externally or internally. Or smoke may be defined as consisting of palpable par- ticles, elevated by means of the rarefying heat, or by the force of the ascending current of air from certain bodies exposed to heat, which particles vary much in their properties accord- ing to the substances from which they are produced. SNELL (Ropotrex), an eminent Dutch philosopher, was born at Oudenwater in 1546. He was sometime professor of Hebrew and Mathematics at Leyden, where he died in 1613. He was author of several works on geometry, and various subjects of philosophy. SNELL (WILLEBRORD), son of the preced- ing, was born at Leyden in 1591, and suc- ceeded his father in his professorship of ma- thematics, in 1613. He died in 1626. He first discovered the true law of refraction of ‘ dhe rays of light. His works are numerous and very reputable for the time when they were written; the principal one of which is entitled “ Cyclometricus de Circuli Dimen- sione, 4to. 1621. SNOW, awell-known meteor, formed by the freezing of the vapours of the atmosphere. SOCRATES, a celebrated philosopher of antiquity, who flourished about 440 years be- fore Christ. SOL, a name given to the sun. SOLAR, relating to the sun, as Solar Month, Cycle, Eclipse, System, &c. for which see the several substantives. SOLID, in Geometry, is a body of three dimensions, having length, breadth, and thick- ness; being thus distinguished from a surface which has but two dimensions, and from a line which has but one. SoLip, in Physics, is that whose parts adhere to each other with a greater or less force; being thus distinguished from a fluid whose parts yield to the least external pres- sure. Regular Soxrips, are those bounded by equal and regular plane figures. See Regular Bopres; all other solids are called irregular. SoLip Angle, is that formed by three or more plane angles meeting in a point, like an angle of adie, or the point of a diamond, &c. Or, more generally, a solid angle is the angular space included between several plane sur- faces. Solid angles may be computed and com- pared with each other, as to quantity, by con- sidering the angular point as the centre of a sphere, and the portion of its surface inter- cepted between the bounding planes as the measure of the angles. Assuming then any pumber as 2000 for expressing the whole SOL spherical, surface, or 1000 for half the salma being the limit of the greatest solid angle; the measure of the angles of any solid will be known by finding the measure of the spherical surface included between its bounding planes, This idea of comparing solid angles by means of spherical surfaces seems to have been firs given by Albert Girard in his “ Invention; Nouvelles en l’Algebra,” and has been lately much extended and exemplified in vol iii, 0 Dr. Hutton’s ‘ Course of Mathematics.” Sotip Numbers, are those arising from product of three prime factors. SoLtip Problem, is that which cannot bi constructed geometrically, wz. by means ¢ right lines and circles, but requires the intro duction of some curves of a higher order, a the ellipse, parabola, and hyperbola, whie being the sections of solids gave rise to th term solid problem; it is not, however, offte employed by modern writers. » Soxip of Least Resistance. See RESISTANG Curbature of Souips. See Souipity. | Surface of Sotips. See conclusion of tf following article. SOLIDITY, in Geometry, denotes tl quantity of space contained or occupied | a solid body, called also its solid conte being estimated by the number of solid cubic inches, feet, yards, &c. which it co tains. The rules for finding the content all the most common solids are given un the respective articles in this work, and 3 shall therefore in this place confine oursely to the investigation of the general methods finding the solidity and surface of any solid rotation, viz. any one whose generation may conceived to arise from the revolution of a plane surface about a fixed line or axis 5 which amounts to nearly the same thing, the parallel motion of a circle gradually panding and contracting itself according the nature of the generating plane. a1 To find the Surfaces and Solidities of Solids Rotation. Let ABE be any solid of rota- tion, that is, such as may be con- ceived to be ge- nerated by the revolution of the plane figure BCE about the line EC as an axis. Then it is vious that the same solid may be othery supposed to be formed by a variable ei moving parallel to itself from the vertex } the base ACB, the law of its variation pending upon the relation between the abs EI and the ordinate IH of the curve E] which is to be determined in all cases the given equation of the curve. Now, as fluxion of any generated quantity is equi the product of the generating quantity, the fluxion of the line of its direction, it is vious that the fluxion of the. solidity wil | SOL squal to the area of the generating circle 31H, drawn into the fluxion of EI; and the luxion of the surface equal to the circum- erence of the same circle drawn into the luxion of the are EH. ~ Now let e= 3°1416, the circumference of a sirele whose diameter is 1, and a= °7854, the wea of the same circle; put any variable bsciss El = a, the corresponding ordinate H=y, and the arc EH =z. AlsoS the olidity, and s the surface required. Then because z = /(« + y’) (see REcTI- ICATION), we shall have from what is stated bove, ) § =2cyz =22cy V(x? + y’), also I~ s= 4ay*x ; therefore ' s = fluent of 2cy ¥ (a + y”), and S = fluent of 4ay’ x. And hence is derived the following general ile, viz. from the given equation of the curve, nd the value of x in terms of y, or y in terms fa, and substitute it for that quantity in the hove fluxional values of § or s, and the fluent that expression will be the solidity or sur- ce required. jameter is d, to find its surface and solidity. ‘1. For its solidity. By the equation to the wele we have y* = dx — 2’, and substituting lis value of y* in the fluxional equation, § = 4ay* x, it becomes § = 4adxx —4ax*x ie fluent of which is, S = 2adx* — ax general expression for any segment of the there, which, when x = d, (as is the case in e complete sphere) becomes td3 — 50 Mek me = d3 = ‘5236 d3 the soli- ty required. 2. For the surface. ‘= dx — x’; or, y =v (dx—x’); the fluxion of which gives 3 (d—2x)x _ (d—22x)x % 2/(dx—x*) 2y 2 Shale ue gd’ —4dx + 42” . ¥ ae 4 y” “ 4y? Here again we have , Whence v oe NE oy az:—# msequently «* + y* = ay and z = Dy Substituting this value of 2, in the general xional equation s = 2eyz gives s = edx, d consequently s = cdx, and when »=d $s becomes cd”, = 3:1416d* for the whole face. 3. Let a paraboloid be proposed to find its tid content. ) ake EC =p, CB=q, El=2z,1H=y; en by the nature of the parabola q°x 7 - Z lich substituted for y* in the general ex- zip3:y: gory = ‘ : iM . 5 4aq*x2x assion § = 4ay*x gives S$ = wee and Let the proposed solid be a sphere whose SOS DGQ* +» 5 , Which, when « = p the fluent S — becomes § = 2aq’p, the solidity required. 4. Let it be proposed to find the solidity and curve surface of a right cone, Put the altitude = p, the radius of the base = b, x and y being still the same as before. Thenp >: 0:22: y,0ry aap Substitute this value of y in the general equation § = 4ay* x, and we have _ 4ab* x’ x : 4ab*x3 s=-——, 3p” which when x = p, becomes S = sue p, the and the fluent S = solidity required. 5. For the surface put.the slant length of the cone = J, then again, © ly rey 7, or: tity re = Sore = “8 Substituting this in the equation § =2cyZ : i Qelyy _ vor Chie it becomes zs = ron whence’ s+ and when y = b, we have s = cly for the curve surface of the cone. On this subject the reader may consult any of our elementary works on fluxions. See also another method of finding the solid contents of bodies of revolution, under the article CEN- TROBARYCO. SOLSTICE, in Astronomy, that time when the sun is in one of the solstitial points; that is, when he is at his greatest distance from the equator; thus called because he then appears to stand still, and not to change his distance from the eyuator for some time, an appearance arising from the obliquity of our sphere. The solstices are two in each year; viz. the estival, or summer solstice, and the hyemal, or winter solstice. The Summer SousTIcek, is when the sun is in the tropic of Cancer, which is about the 2\st of June, being then the longest day. The Winter SoLsTick, is when he enters the first degree of Capricorn, about the 22d of December, being then the shortest day. This is to be understood only of the northern hemisphere, it being directly the reverse with regard to the longest and shortest days, and summer and winter, in southern latitudes; so that the terms summer and winter solstices are partial expressions; we ought rather to say northern and southern solstices. SOLSTITIAL Points, those two points in the ecliptic, which are farthest from the equa- tor and diametrically opposite to each other, being the first points of Cancer and Capricori. SoustiTiAL Colure. See COLURE. SOLUTION, in Mathematics, is the opera- tion whereby the answer of a question or pro- blem is determined. SoLuTION, in Physics, is the reduction of any solid to a fluid state by means of some menstruum. SOSIGENES, an Egyptian mathematician, SOU who flourished in the time of Julius Cesar, and was employetl by him in reforming the calendar. He was well versed in mathematics and astronomy; but his works have not been handed down to us. SOUND, in Physics, a perception of the mind, communicated by meaxs of the ear; being an effect of the collision of bodies, and their consequent tremulous motion, commu- nicated to the ambient fluid, and hence trans- mitted to the organs of hearing. To illustrate the cause of sound, it is to be observed, Ist. That a motion is necessary in the sonorous body for the production of sound. 2dly, That this motion exists first in the small and insensible parts of the sonorous bodies, and is excited in them by their mutual colli- sion against each other, which produces the tremulous motion so observable in bodies that have a clear sound, as bells, musical chords, &e. 3dly. That this motion is communicated to, or produces a like motion in the air, or such parts of it as are fit to receive and pro- pagate it. Lastly, That this motion must be communicated to those parts that are the pro- per and immediate instruments of hearing. Now that motion of a sonorous body, which is the immediate cause of sound, may be owing to two different causes ; either the per- cussion between it and other hard bodies, as in drums, bells, chords, &c. or the beating and dashing of the sonorous body and the air immediately against each other, as in flutes, trumpets, kc. Butin both these cases, the motion which is the consequence of the mu- tual action, as well as the immediate cause of the sonorous motion which the air conveys to the ear is supposed to be an invisible, tre- mulous, or undulating motion, in the small and insensible parts of the body. The sonorous body having made its impres- sion on the contiguous air, that impression is propagated from one particle to another, ac- cording to the laws of pneumatics. A few par- ticles, for instance, driven from the surface of the body, push or press their adjacent par- ticles into a less space; and the medium, as it is thus rarefied in one place, becomes con- densed in the other; but the air thus com- pressed in the second place is, by its elas- ticity, returned back again both to its former place and its former state; and the air imme- diately contiguous to itis compressed in the same manner; which afterwards by expand- ing produces another compression, and so on. Therefore from each agitation of the air there arises a motion in it, analogous to the motion of a wave on the surface of the water, which is called a wave or undulation of air. {tn each wave the particles go and return back again through very short equal spaces, the motion of each particle being analogous to the motion of a vibrating pendulum while it performs two oscillations, and most of the laws of the pendulum, with very little altera- tion, being applicable to the former. Sounds are as various as are the means that we SOU concur in producing them. The chief varietie; result from the figure, constitution, quantity &e. of the sonorous body; the manner of p cussion, with the velocity, &e. of the co quent vibrations ; ; the state and constituti of the medium; the disposition, distance! &e. of the organ; the obstacles between the organ and the sonorous object and the adja cent bodies. The more notable distinction o sounds arising from the various degrees and combinations of the conditions above mar tioned, ‘are into loud and low (or strong an¢ weak), into grave and acute (or sharp ald flat, or high and low), and into long ant short. The management of ek is the offic: of musie. Euler is of opinion, that no sound makin; fewer vibrations than 30 in a second, or mor, than 7520, is distinguishable by the humai ear. According to this doctrine the limit ¢ our hearing, as to acute and grave, is an i terval of eight octaves. The velocity of soun is the same with that of the aérial waves, an does not vary much, whether it go with # wind or against it. By the wind, indeed, ; certain quantity of air is carried from on place to another, and the sound is accelerate while its waves move through that part of th air, if their direction be the same as that. the wind. But as sound moves much swift than the wind, the acceleration it will her by receive is but inconsiderable; and _ th chief effect we can perceive from the wind; that it increases and diminishes the space the waves, so that by help of it, the sound ma be heard to a greater distance than it woul otherwise be. i That the air is the usual medium of soun appears from various experiments in rarefie and condensed air. In an unexhausted receiv a small bell may be heard to some distane but when exhausted, it can scarcely be heat at the smallest distance. When the air! condensed, the sound is louder in propor to the condensation, or quantity of air crow ed in, of which there are many iistancell Hauksbee’s experiments, in Dr. Priestley’s, ai others. Besides, sounding bodies comm nicate tremors to distant bodies; for exampl the vibrating motion of a musical string pu others in motion, whose tension and quanti of matter dispose their vibrations to keep tin with the pulses of air, propagated from # string that was struck. Galileo explains ff phenomenon by observing, that a heavy pe dulum may be put in motion by the leg breath of the mouth, provided the blasts” often repeated, and keep time exactly wi the vibrations of the pendulum, and also} the like art.in raising a large bell. i It is not airalone thatis capable of thei impre sious ofsound but water also, as is manifest by striking a bell under water, the sound m plainly enough be heard, only not so lov } and also a fourth deeper, according to ¢ judges in musical notes. And Meme says, a sound made under water is of | SOU same tone or note as if made in air, and heard under the water. Dr. Chladni, the ngenious inventor of the Euphon, engaged professor Jacquin of Vienna to make some »xperiments on those gases which constitute yur atmosphere, and serve to produce vocal sounds. Comparative experiments were made with atmospheric gas, oxygen, hydrogen, varbonic acid, and nitrous gas. The inten- ity of the sounds did not vary; but when sompared with that produced by atmospheric air, the oxygen gas gave a sound half a tone lower; azotic gas, prepared by different me- yhods, constantly gave a sound half a tone ower; hydrogen gas gave nine or eleven yones higher; carbonic acid gas one-third yower; and nitrous gas very nearly a third ower. A mixture of oxygen gas and azote, athe proportions of the atmospheric air, af- orded the same tone as atmospheric air; hat is, it was half a tone higher than either f the component parts alone. When the Wo gases were not uniformly mixed, the ound was abominably harsh. M. Paul of eneva, after having inspired a large quan- y of pure hydrogen gas, attempted to speak, ut found that the sound of his voice had vecome shockingly hoarse and shrill. | The velocity of sound, or the space through yhich it is propagated in a given time, has een very differently estimated by. authors tho have written upon this subject. Ro- erval states it at the rate of 560 feet in a second; Gassendus at 1473; Mersenne at 474; Duhamel, in the History of the Aca- emy of Sciences at Paris, at 1338; Newton t 968; Derham, in whose measure Flamstead nd Halley acquiesce, at 1142. The reason f this variety is ascribed by Derham, partly » some of those gentlemen using strings and lummets instead of regular pendulums; and artly to the too small distance between the morous body and the place of observation, id partly to no regard being had to the inds. But by the account since published y M. Cassini de Thury, in the Memoirs of ie Royal Academy of Sciences at Paris, 738, where cannon were fired at various as ell as great distances, under many varieties ‘ weather, wind, and other circumstances, ad where the measures of the different places d been settled with the utmost exactness, was found that sound was propagated on a vedium, at the rate of 1038 French feet, in second of time. But the French foot is in foportion to the English, as 15 to 16: and msequently 1038 French feet are equal to 07 English feet. Therefore the difference ‘the measures of Derham and Cassini is 35 nglish feet, or 33 French feet, in “a second. he medium velocity of sound therefore is varly at the rate of a mile, or 3280 feet, in “seconds, or a league in 14 seconds, or 13 jiles in a minute. But sea miles are to land iles nearly as 7 to 6; and therefore scund ‘oves over a sea mile in + seconds, or a sea ague in 16 seconds. Farther, it is a com- on observation, that persons in health have SPE about 75 pulsations, or beats of the artery at the wrist in a minute; consequently in 75 pulsations sound flies about 13 land miles, or 115 sea miles, which is about 1 land mile in 6 pulses, or one sea mile in 7 pulses, or a league in 20 pulses. And hence the distance of objects may be found by knowing the time employed by sound in moving from those ob- jects to an observer. For example: on see- ing the flash of a gun at sea, if 54 beats of the pulse, at the wrist, were counted before the report was heard, the distance of the gun will easily be found by dividing 54 by 20, which gives 2°7 leagues, or about 8 miles. SOUTH, one of the four cardinal points of the compass, being that directly opposite to the north. SOUTHERN Hemisphere, Latitudes, Signs, &e. those on the south side of the equatoy. SOUTHING, in Navigation, the difference of latitude made by a ship in sailing to the southward. SOUVERAIN, Stngle, an Austrian coin value 13s. 10d. sterling. SPACE, a simple idea, of which the modes pa distance, capacity, extension, duration, ee. Space, considered with regard to length only, is the same idea as we have of distance. If it be considered in length, breadth, and thickness, it is the same as capacity. When considered between the extremities of matter, which fills the capacity of space with some- thing solid, tangible, and moveable, it is call- ed extension. So that extension is an idea belonging to body only; but space, it is obvious, may be considered without it. Space therefore in the general signification, is the same thing with distance considered every way, whether there be any solid matter in it or not. Space, is usually divided into absolute and relative. ; Absolute Spacek, is that which is considered in its own nature without regard to any thing external, which always remains the same, and is infinite and immoveable. : Relative Spacg, is that moveable dimension, or measure ot A former, which our senses define by its*positions to bodies within it. Much has been said and written relative to the nature of space, by some very eminent and acute metaphysicians; and those who wish to be acquainted with all the intricacies which attend the different modes of consider- ing space should read the collection of papers which passed between Leibnitz and Dr. Clarke, in the years 1715 and 1716, relating to the principles of natural philosophy and religion. Spack, in Geometry, is used to express the surface of any figure, or that which fills the interval or distance between the lines that terminate or bound it; thus we say hyper- bolic space, parabolic space, Ke. SPECIES, in Algebra, are the letters, sym- bols, marks, or characters, which represent the quantities in any operation or equation. SPE SPECIFIC Gravity. See Specific Gravity. SPECTACLES, an optical instrument con- sisting of two lenses set in a frame, and ap- plied on the nose to assist in defects of the organs of sight. The invention of spectacles has been much disputed, though there seems no doubt that it must have happened somewhere about the 12th or 13th century. Some have attributed the invention of them to Alexander de Spina, a monk of the order of Predicants; about the year 1290; while others, particularly Mus- schenbroek, informs us that they were in- vented by Salvinus Armatus, who died in 1317, and that the circumstance is engraven on his tomb-stone. Others have given the honour of the discovery to Roger Bacon, and others again to Alhazen, so that in fact it may be said that nothing certain is known on the subfect. Theory of Spectacles.—If the objects are seen through a perfectly flat glass, the rays of light pass through it from them to the eye, in a straight direction, and parallel to each other, and consequently the object appears very little either diminished or enlarged, or nearer or farther off, than to the naked eye; but if the glass they are seen through have any degree of convexity, the rays of light are directed from the circumference towards the centre, in an angle proportional to the con- vexity of the glass, and meet in a point, at a greater or lesser distance from the glass, as it is more or less convex. "Vhis point, where the rays meet, is called the focus, and this focus is nearer or farther off according to the con- vexity of the glass; for as a little convexity throws it to a considerable distance, so when ihe convexity is much, the focus is very near. {ts magnifying power is also in the same pro- portion to the convexity; for as a flat glass scarcely magnifies at all, the less a glass de- parts from flatness the less of course it mag- nifics, and the more it approaches towards the globular figure, the nearer its focus is, and ihe more its magnifying power. People’s different length of sight depends on the same principle, and arises from more or less convexity of the cornea and crystalline humour of the eye; the rounder these are, the nearer will the focus or point of meeting rays be, and the nearer an object must be brought to see it well: The case of short-sighted peo- ple is only an over roundness of the eye, which makes a very near focus; and that of old people is a sinking or flattening of the eye, whereby the focus is thrown to a great dis- tance, so that the former may properly be called eyes of too short, and the latter eyes of too long a focus. Hence too, the remedy for the lastis a convex glass, to supply the want of convexity in the eye itself, and brings the rays to a shorter focus; whereas a concave glass is needful for the first, to scatter the rays and prevent their coming to a point too soon. ‘he nearer any object can be brought to the eye the larger will be the angle under which it appears, and the more it will he magnified. SPH Now the distance from the naked eye, wher the generality of people are supposed to se small objects best, is about six inches, conse quently, when such objects are brought neare than six inches they will become less distinet} and if to four or three, they will scarce b seen at all. But by the help of convex glasse we are enabled to view things clearly at muel shorter distances than these; for the natur of a convex lens is to render an object dis tinctly visible to the eye at the distance ol its focus; wherefore the smaller a lens i and the more its convexity, the nearer is it focus and the more its magnifying power Now it is evident that if either the corne} or crystalline humour, or both of them, b too flat, their focus will not be on the re tina, where it ought to be, in order to rende vision distinct, but beyond the eye. Cons¢ quently those rays which flow from the ol ject, and pass through the humours of the eye are not sufficiently converged to unite, an! therefore the observer can have but a ver! indistinct view of the object. This is remedie hy placing a convex glass of a proper foew before the eye, which makes the rays converg sooner, and imprints the image duly on th) retina. If cither the cornea or crystallin’ humour, or both of them, be too convex, th rays which enter in from the object will be cor) verged to a focus in the vitreous humour, ani by diverging from thence to the retina, wi form a very confused image thereon, and s| of course the observer will have as confused | view of the object as if his eye had been to flat. ‘This inconvenience is remedied by pla¢ ing a concave glass before the eye, whic glass, by causing the rays to diverge betwee! it and the eye, lengthens the focal distance, §| that if the glass be properly chosen the ray will unite at the retina, and form a distine picture of the object upon it. al SPECTRUM, in Optics, denotes principal the image formed on any white surface, ° the rays of the sun passing through a sme hole in the window-shutter of a dark roor when refracted by aglass prism. See PRisM SPECULUM, the same as Mirror, whie see. : SPHERE, or GLose, in Geometry, is a soli contained under one uniform surface, ever point of which is equally distant from a poil within, called the centre of the sphere, an may be conceived to be generated by the r volution of a semi-circle about its diamete which remains fixed, and which is hene called the axis of the sphere. ‘The principal properties of the sphere ar as follows: it 1. A sphere is equal to two thirds of ij circumscribing cylinder; or it is equal to pyramid or cone, whose base is equal to th whole surface of the sphere, and its altitud equal to half the diameter. fi 2. All spheres are similar figures, and are f each other as the cubes of their diameters ¢ circumferences. 4 3. The surface of a sphere is equal to th SPH oa of four of its great circles, or to the curve rface of its circumscribing cylinder; and erefore the surface of different spheres are to ch other as the squares of their diameters, 4. The surface of a sphere is equal to the ea of a circle, whose radius is equal to the vameter of the sphere ; and the curve surface “any segment of a sphere is equal to a circle, wying for its radius the chord of half the are / that segment. é . 5. The surface of any segment or zone of a here is equal to the curve surface of a cor- )sponding portion of the circumscribing cy- ader; that is, any two planes passing through 1e sphere and its cireumscribing cylinder arallel to the base of the latter, the surface ‘the segment of the sphere and cylinder thus jit off will be equal to each other. Most of 1ese properties we owe to Archimedes, being jyen by that celebrated geometrician in his jeatise on the sphere and cylinder. , The several rules relating to the solidities jad surfaces of spheres and their segments ‘e contained in the following formule : Let d represent the diameter, and ¢ the cir- unference of a sphere; also s the surface, id § the solidity of the same, then los ed =3°14159d? = °3183c* 2.Smisd= °6236d3 = :01688c%. Spherical Segments, the same notation re- jaining, and r being put for the radius of the ase, and / for the height of the segment. 3. surface = 3:14159dh 4. solidity = *5236h (3r7 +h’), or 5. solidity = *5236h (8d— 2h). Spherical Zones. Put R and r for the radii the ends, and h for the height, then 6. surface = 3'14159 dh 7. solidity = 1:5708 (R* +77 + 3 h?). ee Hutton’s and Bonnycastle’s “ Treatises * Mensuration.”’ Armillary SPHERE. See ARMILLARY. (Spuere of Activity of a Body, is used to inote the space through which the influence fany body, as a magnet, &c. extends. Projection of the SPHERE. See PROJECTION. “SPHERE, in Astronomy, that concave orb or xpanse which appears to invest our globe, nd in which all the heavenly bodies appear ixed; this is called the sphere of the world, nd is the subject of spherical trigonometry. The sphere, as it relates to the fixed stars, cing immensely great, the spectator always adges himself to be in its centre, and the aleulations made on this supposition deviate ot in the least sensible degree from the truth. Circles of the SPHERE, are certain circles apposed to be described about the sphere for ae better determination of the several points, ¢. about it, as the equator, ecliptic, &c. ‘These Mireles of the heavens are transferred to the arth, and the disposition of these circles, with regard to each other, is what is called the phere in geography, which varies in the dif- erent parts of the earth; and according to he position of these cireles we have a right, blique, and parallel sphere. A Right or Direct Spuere, is that which Sm FL has the poles of the world in the horizon, and the equator in the zenith and nadir. In an Oblique Spuere, the equator cuts the ho- rizon obliquely ; and ina Parallel Spuere, the poles are in the zenith and nadir, and the equator coincides with the horizon. SPHERICAL, something relating toa sphere, as spherical angle, triangle, trigono- metry, for which see the articles TRIANGLE and ‘TRIGONOMETRY. SpHeriIcaL Excess, in Trigonometry, the excess of the sum of the three angles of any spherical triangle, above two right angles, Let A, B,C, be the angles of a spherical triangle, a, b, c, the sides opposite to the angles A, Bo respectively, + = two right angles, r = the radius of a great circle of the sphere, and R“ the number! of seconds comprised in radius ; then the spherical excess may be ascertained by the following elegant theorem, first given by Simon Lhuillier of Geneva. Tan. t (A +B+C—7) = 1 Ai Stig ain emia ate—b r 4 4 4 hoe) fae et ab 4 M. Puissant gives the following theorems for « the excess: R’ 4 i Wa Meee ges (55) a? sin. 2C + (55) c* sin. 2A. R’ , P Peon 1(%) bh, where b = the base, and h — the height of the triangle. It will hence be easy to form a table, from which the sphe- rical excess of any triangle measured may be learnt from its base and height. SPHERICAL Polygon, is a figure of more than three sides, formed on the surface of a sphere by the intersection of several of its creat circles. SPHERICITY, the quality of a sphere, or that by which a thing becomes spherical. SPHERICS, the doctrine of the sphere, particularly of the several circles described on its surface, with the method of projecting the same ona plane. See PROJECTION. SPHEROID, a solid body resembling a sphere, which is supposed to be generated by the revolution of any oval figure about an axis. This is the most general definition of a spheroid ; but the term is generally meant only to denote a solid generated by the revolution of an ellipse about one of its axis, and there- fore the latter is sometimes called an ellipsoid, to distinguish it from the more general figure spheroid. When the ellipse revolve about its trans- verse axis, it is called an oblong or prolate spheroid; when about its conjugate axis, an ob- late spheroid ; and when about any other of its diameters, a universal spheroid, in which latter case its figure is somewhat resembling a heart. The principal rules for finding the solidities, surfaces, &c. of spheroids generated from an ellipse and their segments, zones, &e, are con- tained in the following formule ; . SPI Let f be the fixed axis, and r the revolving 2 axis, 3°14159, &c. = p, mil — , then 1, solidity = % fr” p di Se eA g 2. surface = frp ? Soe 73) 4.5 ‘lg Be a ke. 6.7 The upper sign having place for the prolate, and the lower for the oblate sphere, also A, B, C, &c. the preceding terms. If also we make ~ = z, V(lv at ets m = the degrees of the are whose sine is s, likewise P = ‘01745329 m, in the prolate sphere P = 2°30285 log. (s+) in the oblate sphere. 3. surface — ct es xX 314159 r, The area of a prolate spheroid, is less than four times the area of its generating ellipse, and the area of an oblate spheroid greater than four times the same, and the sphere, which is the intermediate limit of the two, is equal to four times its generating circle. For several other interesting properties of these solids, see Hutton’s “ Mensuration,” p. 267, 2nd edit. Frustums of Spheroids, or the parts cut off by two planes. Here/f, 7,q,p, denoting the same as above, and / the height of the frustrum, also Zz ae =z; then the frustum being cut off by two planes passing perpendicular to the fixed axis, one of them passing through the centre of the spheroid, we have Az 1.3Bz Aj. sc pik, MIRA = surface pr Ny F353 3 1.3.5Cz Man Timi A, B, C, representing the same as above, and the signs having place as there stated. For the solidity, make the diameter of the greater end — D, of the lessd; then 1. solidity = = (2D* + d*) hx 314159 Segments of Spheroids, or parts cut off by one plane. Here f,r, and / being still as above; then 1. When the base is parallel to the revolving axis. Solidity =F (3f 2h) i? x 5236 2. When the base is perpendicular to the re- volving axis. am Solidity = — (3r w 2h) h? x°5236 See Hutton’s “ Mensuration.” SPHEROIDAL, or SPHEROIDICAL, having the form of a spheroid. SPHERULE, a little globe or sphere. SPINDLE, in Geometry, a solid generated by the revolution of a curve about its base or double ordinate, and is farther denominated elliptic, parabolic, hyperbolic, &c. according to figure from which it is generated. See EL- LIPTIC, PARABOLIC, HYPERBOLIC Spindle. - radius is made to revolve uniformly about t SPI SPIRAL, in Geometry, a curve line of +) circular kind, which in its progress alwa recedes more and more from a point with called its centre. SF There are various kinds of spirals, viz. t| spiral of Archimedes, the Logarithmic spir, the Loxodromic spiral, the Parabolic spiral,) Helicoid, &e. according to the law by whit the point recedes from the centre. Archimedian Spirau. ‘This is generated supposing the are of a circle to be divid| into any number of equal parts in p, p, p,& then radii being drawn from the centre | these points, divide one of them into the sar number of equal parts, and make Cm, Cm, C &e. equal to 1, 2, 3, Kc. of those parts, ail the line drawn from the centre passing throu these several points will be the spiral of A chimedes. This is more particularly called the fii spiral when it has made one complete revol tion to the point A; and the space includ between the spiral and radius AC is calh the spiral space. p> Sy The first spiral may be continued to a s cond, by describing another circle with ar dius double of the first, and this second mz be again continued to the third, and so o The same spiral may be otherwise generate by a continued motion, by making a poi receded uniformly from the centre, while th same point. Hyperbolic or Reciprocal SPIRAL. one of the extremities c of the right line eF as a centre, with the distances cD,cE,cF, &c. as radii, an indefinite number of arcs Dd, Ee, Ff, &c. be described so as to be all equal in length to each other, the curve cdef drawn through their extremities, is called the hyperbolic or reciprocal spiral, ¢ which the right line c F is called the axis, a the point ¢ the centre. The principal properties of this curve are a follow: . 1. Any two ordinates we have If fro; ef, ce being drawt cf s¢e':: > Sr , whose : my” fluentis A = Tae 4+C; but wheny 0, A=a, n therefore C = 0, and consequently the area —_ my” vt Ase In the spiral of Archimedes y : wi: m:n, : . . ny or in a constant ratio ; therefore w —"Y. con- m yw _ ny? y 2Zma’ sequently A = yin whose fluent is 3 ny Pi A= eu -+- C, where again C = 0, and there- oma / \ 3 v2 fore A = Pacha! 6ma For the rectification we have t: y::yi:z= a, the fluent of which will be the length re- quired, In the Logarithmic spiral,t : yi: min, a . mz . constant ratio; hence, é = my therefore z = v2) . Zz nt MY z : : ~/, and z= J ==, which is the correct m fluent. ral. of Archimedes put —< = In the spiral of Archimedes put ~~ os) 2, uy ; hence and we have ¢ = VP Eby: SPR , Which gives 2 p2 © (yt ebay y+ Ebxh LEZ FH) 26 b SpirsL Pump. See Archimedes’ Screw. SPONTANEOUS Rotation. See Rota- TION. SPORADES, a name given by the ancients to those stars that were not included in any of the constellations. , SPOTS, in Astronomy. See MAcuL® and FAcuL2z. SPOUT. See Water-Spout. SPRING, in Natural Mistory, a fountain or source of water, rising out of the earth. Various have been the opinions of philosophers concerning the origin of springs; as, 1. That the sea water is conveyed through subterra- neous ducts, or canals, to the places where the springs flow out of the earth; but as it is im- possible that the water should be thus con- veyed to the tops of mountains, since it cannot rise higher than the surface, some have had recourse to, subterraneous heat; by which being rarified, it is supposed to ascend in vapours through certain cavities of the moun- tains. But as no sufficient proof is brought of the existence of tliese central licats, or of caverns in the mountains big enough to let the vapours ascend, supposing such heats, we shall not take up our readers’ time with a formal refutation of this hypothesis. 2. As to those who advance the capillary hypothesis, or suppose the water to rise from the depths of the sea through the porous parts of the earth, as it rises in capillary tubes, or through sand or ashes, they seem not to consider one principal property of this kind of tube, or this sort of attraction: for though the water rise to the top of. the tube or sand, yet will it rise no higher, because it is by the attraction of the parts above that the fluid rises, and where that is wanting it can rise no farther. Therefore, though the waters of the sea may be drawn into the substance of the earth by attraction, yet it can never be raised by this means into a cistern or cavity, to be- come the source of springs. 3. The third hypothesis is that of the learned Dr. Halley, who supposes the true sources of springs to be melted snow, rain water, dew, and vapours condensed. Now in order to prove that the vapours raised by the heat of the sun from the surface of the seas, lakes, and rivers, are abundantly sufficient to supply the springs and rivers with fresh water, the Doctor made the following experiment: he took a vessel of water, made of the same degree of saltness with that of the sea, by means of the hydrometer; and having placed a thermometer in it, he brought it, by means of a pan of coals, to the same degree of heat with that of the air in the hottest sum- mer. He then placed this vessel, with the thermometer in it, in one scale, and nicely counterpoised it with weights in the other; alter two hours he found that about the six- ~~ Ah y S(y? +0’) a b a SPR tieth part of an inch was gone off in vapou and consequently in twelve hours, the leng of a natural day, one tenth of an inch wou have been evaporated. From this experimer it follows, that every ten square inches of th surface of the water yield a cubic ineh water in vapour per day, every square mi 6914 tons, and every square degree (or English miles) 33 millions of tons. Now if y suppose the Mediterranean to be 40 degre¢ long, and 4 broad at a medium, which is th least that can be supposed, its surface will 160 square degrees, from whence there wi evaporate 5280 millions of tons per day in t summer time. The Mediterranean receiy water from the nine great rivers following, wi the Iberus, the Rhine, the T'yber, the Po, th Danube, the Neister, the Borysthenes, ‘Tenais, and the Nile ; all the rest being sma and their water inconsiderable. Now let suppose that each of these rivers conveys te times as much water to the sea as the Thame which, as is observed, yields daily 76,032, cubic feet, whichis equal to 203 millions ¢ tons, and therefore all the nine rivers will pre duce 1827 millions of tons; which is little may than one-third of the quantity evaporated eae day from the sea. ‘The prodigious quantity water remaining the Doctor allows to rain which fall again into the seas, and for tl uses of vegetation, kc. As to the manneri which these waters are collected, so as form reservoirs for the differentkinds of spring it seems to be this: the tops of mountains 7 general abound with cavities, and subterr neous caverns formed by nature to serve ¢@ reservoirs ; and their pointed summits, whi Y seem to pierce the clouds, stop those vapout which fluctuate in the atmosphere, and bein constipated thereby, they precipitate in wate and by their gravity easily penetrate throug beds of sand and lighter earth, till they ar stopped in their descent by more dense strats as beds of clay, stone, &c. where they form bason or cavern, and work a passage horizor tally, and issue out of the side of the mou tain. Many of these springs running dow, by the valleys between the ridges of hills, an uniting their streams from rivulets or brooks and many of these, again, uniting on the plait become a river. Springs are either such as run continually called perennial; or such as run only for time, and at certain seasons of the year, am therefore called temporary springs. OtLer again are called intermitting springs, becaus they flow and then stop, and flow and sto again ; and finaily,reciprocating springs, whos waters rise and fall, or flow and ebb, by re gular intervals. é SPRING, in Mechanies, is used to signify: body of any shape perfectly elastic. Strength or Force of a SPRING, is used fo the foree or weight, which, when the spring i wholly compressed or closed, will just preven it from unbending itself. Also the force of : spring partly bent or closed, is the force o weight which is just sufficient to keep th SPR ring in that state, by preventing it from un- nding itself any farther. The theory of springs is founded on this inciple: that the intensity is as the com- essing force; or if a spring be any way reed or put out of its natural situation, its sistance is proportional to the space by which js removed from that situation. This prin- ple has been verified by the experiments of r. Hook, and since him by those of others, wticularly by the accurate: hand of Mr. eorge Graham, ‘“ Lectures de Potentia Res- tutiva,” 1678. For elucidating this principle, on which e whole theory of springs depends, suppose spring CL resting at L against any im- oveable support, but otherwise lying in its itural situation, and at full liberty. ‘Then if is spring be pressed inwards by any force p, from C towards L, through the space of one h, and can be there detained by that force the resistance of the spring, and the force p, actly counterbalancing each other; then ill the double force 2p bend the spring rough the space of 2 inches, and the triple tee 3 p through 3 inches, and the quadruple rce 4p through 4 inches, and so on. The ace CL through which the spring is bent, | by which its end C is removed from its stural situation, being always proportional the force which will bend it so far, and will st detain it when so bent. On the other und, if the end C be drawn outwards to any ace A, and be there detained from returning ck by any force p, the space Ca, through hich it is so drawn outwards, will be also oportional to the force p, which is just able retain it in that situation. b] Fig. 1. It may here be observed, that the spring of je air, or its clastic fotce, is a power of a SPR different nature, and governed by different laws, from that of a palpable rigid spring. For supposing the line L to represent a cylin- drical volume of air, which by compression is reduced to L/, or by dilation is extended to La, its elastic force will be reciprocally as Ld or La; whereas the force or resistance of a spring is directly as Cl or Ca. This principle being premised, Dr. Jurin lays down a general theorem concerning the action of a body striking on one end of a spring, while the other end is supposed te rest against,an immoveable support. Fig. 4. A B F cl ry Thus, if a spring of the strength p, and the length CJ, lying at full liberty upon an hori- zontal plane, rest with one end L against an immoveable support; and a body of the weight M, moving with the velocity V, in the direction of the axis of the spring, strike directly on the other end C, and so force the spring inwards, or bend it through any space CB; and if a mean proportional CG be taken between ~ x CL, and 2a, where a denotes the height to which a body would ascend in vacuo with the velocity V; and farther, if upon the radius R = CG, be described the quadrant of a circle GF A: then, 1. When the spring is bent through the | right sine CB of any are GF, the velocity v of the body M is to the original velocity V, as the cosine BF is to the radius CG; that is, VEER OCGL onteiom ee R * V. 2. The time ¢ of bending the spring through the same sine CB, is to T the time of a heavy body’s ascending in vacuo with the velocity V, as the corresponding arc is to 2a; that is, t:T :: GE: 2a, or The doctor gives a demonstration of this theorem, and deduces a great many curious corollaries from it. These he divides into three classes. ‘The first, contains such corol- laries as are of more particular use when the spring is wholly closed before the motion of the body ceases: the second, comprehends those relating to the case when the motion of the body ceases before the spring is wholly closed : and the third, when the motion of the body ceases at the instant that the spring is wholly closed. We shall here mention some of the last class, as being the most simple; having first SPR. premised, that P = the strength of the spring, L = its length, V = the initial velocity of the body closing the spring, M = its mass, ¢ = time spent by the body in closing the spring, A = height from which a heavy body will fall in vacuo in a second of time, a = the height to which a body woald ascend in vacuo with the velocity V, © = the velocity gained by the fall, »m = the cireumference of a circle whose diameter is 1. Then the motion of the striking body ceasing when the spring is wholly closed, it will be 1 a yee 2MA CL Op eee a? a Ae a PLM ¥, 3MV=C SA , the first mo mentum. 3. Ifa quantity of motion MV bend a spring through its whole length, and be destroyed by it, no other quantity of motion equal to the be former, as nM x —, will close the same % spring, and be wholly destroyed by it. 4, But a quantity of motion, greater or less than MV, in any given ratio, may close the same spring, and be wholly destroyed in clos- ing it; and the time spent in closing the spring will be respectively greater or less in the same given ratio. C*PL. ? 5. The initials vis viva, or MV”, is — and 2aM = PL: also the initial vis viva is as the rectangle under the length and strength of the spring, that is, MV? is as PL. 6. If the vis viva MV? bend a string through its whole length, and be destroyed in closing it; any other vis viva, equal to the former, as 2 Vv : i «*M x —, will close the same spring, and n toP) be destroyed by it. 7. But the time of closing the spring by the 2 vis viva n*M x 7 will be to the time of closing it by the vis viva MV?, as n to 1. 8. If the vis viva MV* be wholly consumed in closing a spring of the length L, and strength P; then the vis viva n* M V? will be sufficient to close. Ist. Either a spring of the length L, and strength n?P. 2d. Ora spring of the length nL and strength nP. 3d. Or of the length »’L and strength P. 4th. Or, if n be a whole number, the number n? of springs, each of the length L and strength P. It may be added, that it appears from hence, that the number of similar and equal springs a given body in motion can wholly close, is always proportional to the squares of the velocity of that body. And it is from this principle that the chief argument, to prove that the foree of a body in motion is as the square of its velocity, is deduced. The theorem given above, and its corollaries, will equally hold good, if tho spring be sup- posed to have been at first bent through a SQU certain space, and by unbending itself press upon a body at rest, and thus to dri that body before it, during the time of its e& pansion; only V, instead of being the initi velocity with which the. body struck the sprix will now be the final velocity with which tl body parts from the spring when totally & panded. It may also be observed, that tl theorem, &c. will equally hold good, if I spring, instead of being pressed inward, drawn outward by the action of the bod The like may be said, if the spring be sw posed to have been already drawn outward | a certain length, and in restoring itself dra the body after it. And lastly, the theore) extends to a spring of any form whateve provided L be the greatest length it can ] extended to from its natural situation, and | the force which will confine it to that lengt See Phil. Trans. No. 472, Sect. 10. or vol. xlii art. 10. Hutton’s Math. Dict. Sprine Tides. See Tipes. q SQUARE, in Geometry, is a quadrilateri figure, having its four sides equal to eaé other, and its angles all right-angles; th area of which is found by multiplying its sid by itself. Geometrical SQUARE. See QUADRAT. Magic Seuare. See Maaic Square. SQuarE is also the name of an instrumen used by mechanics and artificers for what i called squaring their work; these are of variou forms, as the T’ square, normal square, &e, these are too well known to require any p ul ticular description. | SQuARE Measure. See MEASURE. 4 Square Number, is the product arising fron a number being multiplied into itself, or number formed of two equal factors. Squar numbers haye several curious properties, 0 which the following are the most remarkable 1. Every square number is of one of the forms 4n, or 4n + 1, that is, every squart number when divided by 4 will leave eithe} 0 or 1 for aremainder; and understanding thij expression still in the same way, the follow ing table will express the forms of squar numbers to the moduls 4, 5, 6, 7, 8, 9 and 10 Moduls, Forms. | 4 4n 4n+1 | 5 bn bn21 » 6 6n 6n+1 6n+3 6n4 4 “f on In +1. 7m+2y Fane 4 | 8 8n > 8n+1 8n+4 ° 9 9n 9n+1 9n+4 9n4+7 10 |10n 10n+1 10n+4 LOn+S From an examination of which there may be drawn several curious corollaries, as fol- lows: x 2. The sum of two odd squares cannot be a square. e 3. An odd: square taken from an even square cannot leave a square remainder. 4. If the sum of two squares is itself a 24 SQU iare, one of the three squares 1s divisible 5, and therefore by 25. >. Square numbers must terminate with 2 of the digits 0, 1, 4, 5, 6, or 9. 3. The area of a rational right-angled tri- sle cannot be equal to a square. *. The two following series are remarkable being such as when reduced to improper stions, the sum of the squares of the nume- or and denominator of each are complete yares, Or, Which is the same, they are the 2s of rational right-angled triangles. These jes are as follow: viz. 14, 22, 33, 44, 5, &e. I 14, 234, 325, 438, 523, &e. . The second difference of consecutive ares are all equal to each other; thus, Squares 1], 4, 9, 16, &c. Ast diff. 3 5 7 &e. “2nd diff. 2 bake sa ak or a variety of other properties of square ibers, see Barlow’s Theory of Numbers. QUARE Root, in Arithmetic, is that number th multiplied into itself will produce the TL number or square. an integral number has not an integral neither has it a fractional one, and only pproximation towards the exact root can btained, which may be done by the fol- ng rule. ule. Divide the given number into periods Wo figures each, beginning at the unit’s gand proceeding to the left-hand in in- ts, and to the right in decimals. Find sreatest square in the first period on the hand, and set its root on the right-hand of fiven number, after the manner of a quo- figure in division. Subtract the square ‘found from the said period, and to the inder annex the two figures of the next d for a dividend. Double the root above- fioned for a divisor; and find how often tontained in the said dividend, exclusive ‘right-hand figure: and set that quotient 2 both in the quotient and divisor. Mul- the whole augmented divisor by this last vent figure, and subtract the product from jaid dividend; bringing down the next d of the given number, as before, for a dividend. Repeat the same process for period, and the number thus obtained ve the root required. ve, 1. The best way of doubling the ‘to form the new divisors, is by adding ist figure to the last divisor. The root will necessarily consist of as ‘integers and decimals as there are ds in each respectively. And when the 8 of the given number are all exhausted, peration may be continued at pleasure ding ciphers, two in each period. —- STA Required the square root of 5499026. 5499025(2345 the root. 4 43)149 129 464)2090 1856 4685)23425 23425 0 \ For the Contracted Method for Extracting the Square Root, see CONTRACTION; and for taking the square root by continued fractions, see CONTINUED Fractions. SQUARING the Circle. QUADRATURE. STADIUM, an ancient Greek long mea- sure, containing 126 geometrical paces, or 625 Roman feet; corresponding to our fur- long. It must be observed, however, that the stadium was different in different places. STAFF, a name given to several different instruments, as the Cross Staff, Back Staff; Fore Staff, &c. for which see the several articles. STAR, in Astronomy, is a general name for all the heavenly bodies; but by English writers it is more commonly used to denote a fixed star, or one completely unconnected with the solar system, the others being dis- tinguished by their particular names, as planets, comets, satellites, &c. . In order to distinguish the stars one from aliother, the ancients divided the heavens into different spaces called constellations, which they supposed to be occupied by the figures of animals and other objects, which figures have beenrctained by the moderns, who, to distinguish the stars in the same constella- tion from each other, have either numbered or marked them with certain letters, of the Greek and other alphabets, by which means they are as easily distinguished from each other, as persons of the same family are, by their parti- cular names. The stars are divided into different mag- nitudes according to their apparent size, the largest being said to be of the first magnitude; the next largest of the second magnitude; and so on to the sixth magnitude, which class includes the least stars that are visible to the naked eye. All the stars beyond the sixth magnitude are called telescopic stars, and those which lie in spaces between (and therefore not included in) the constellations, are called unformed stars. ‘The various catalogue found- ed on new observations, the time in which they were published, the number of stars which they contain, and the works in which the catalogues may be found, will be seen in the following enumeration. See CIRCLE and STA A.C. No, of Stars an ae Hipparchus....... 128......... 1022 In Ptolem. Almagest. ‘ Ty GUO iss rm ees bctihe cis eneesoesi vans . 777 Astron. Instaur. Progym. . ® Prince of Hesse...... 1593 ..cis.adpeeavies Flamstead’s Historia Celestis. RieotoliukvcAeacic-- 1666... 40awr.ess Astron. Reform. Bayer .rcscessccceesereses LB7Z9...00000. 11,800 Uranometria. PESOS ripacs cx cts LGDO i ssaee 950 Prodromus Astronomize. : FlamsStead........ iia 2 TADS. scant 2884 Historia Celestis. 4 CT | a ner oe BIOS iis kcaiss 397 Astron. Fundamenta. ie La Caille.........-.+080 E77 Des 0ssive 1942 Southern, Coelum Australe Stelliferum, p. 141. Ba Caille .c...0.se50 ons B7GD,....5.,. 515 Zodiacal. ® Le Monnier...........- 1751, &e 400 ie Tobias Mayer’...:.... ITD .ccccke. 998 ene . ay toe Tnedita, and in Vince’s Astron. vol. BYBOIEY asstrsrterrns Vy oP. 28 389 Nant. Almanack, 1778. DUE ABE BLY Ge ecngee nc 1720) ccaavbes 386 * | RBI oe se Parise ope ee ” 1800.. 381) Tabul. Mot. Solis, and in Vince’s Astron. vol. | “RE idle 1800... 162§ pp. 523, 587. ROCCE cteacl. sc dsop seen tart ities 17,000 Atlas Celeste. Berlin, 1797. La Lande, sen. and 2 1789 § 50.000 ¢ Mem. Acad. Par. 1789, and 1790; and Hist. Cele jun. ORs 1) A Rotate, § Trance. tom. i. The relative positions of the stars in their respective constellations have been represent- ed on the surface of globes, or on maps called planispheres, and atlases. The best plani- spheres are those of Lenox; and the most correct celestial atlases are those of Flam- stead, Fortin, and Bode, the last of which contains twenty sheets, each sheet being twenty-eight inches by twenty. . Distance and Parallax of the Fixed Stars.— When the fixed stars are examined through a good telescope, they do not seein to have any sensible disc; and their size rather di- minishes by increasing the magnifying power of the instrument. This circumstance alone would have been a striking proof of the im- measurable distance of these celestial bodies, had we not been in possession of evidence still more convincing. If-the earth’s diameter had subtended any sensible angle at the near- est fixed star, astronomers would have been able to determine the distance by the ob- served change in its place, when viewed from the two extremities of the earth’s diameter. This base, however, being found tobe toosmall, they have attempted to discover a change in the position of the stars, when viewed from the earth in two opposite points of its orbit, or, in other words, to find the angle subtended by the diameter of the earth’s orbit at the fixed stars, which is called their annual parallax. If a star is viewed, when the earth is in one extremity of its transverse axis, we should expect that it would appear in a diflerent part of the heavens than when it is viewed from the other extremity. But notwithstanding all the attempts of astronomers to discover this change of position, or annual parallax, with the most accurate instruments, they have hitherto been unable to detect it. The ob- servations which have been made for this purpose were so extremely exact, that the parallax of the stars, if it does exist, must be less than 1”, so that if we are unable to de- termine the distance of these celestial bodies, we can at least fix the limits beyond which they must lie. If we suppose, then, that thej rallax of the stars is 1”, and that the me distance of the earth from the sun, or’? semidiameter of the earth’s orbit, is 95,000,¢ miles, we shall have a triangle whose base 190,000,000, and the angle at its vertex 1%, find its side, or the distance of the ne star, which will be 20,159,665,000,000 mil or 20 billions of miles, a distance so immens great, that light cannot pass through it inl than three years. It is very probable, he ever, that the parallax of the stars is mui less ‘than 1”, and that the nearest of th bodies is placed at a much greater distal from us than 20 billions of miles. Some them even are, perhaps, so remote, that sii they were created, the first beam of Ii which they emitted has not yet reached limits of our system; while others which h, disappeared, or have been destroyed for mi ages, will continue to shine in the heay till the last ray which they emitted has rea ed our earth. For an account of Dr. Herschel’s theor the annual parallax of the fixed stars, Phil. Trans. for 1782; see also other pa on the same subjects by Roberts, vol. xv Flamstead, vol. xxii.; Bradley, in the sé transactions for 1728; Halley, 1720; W kelyne, for 1760; see also various other pa in the Memoirs of the Academy of Scien particularly one by Cassini for 1717, | another by Clairaut for 1739. | Motion of the Fixed Stars—By compa the places of some of the fixed stars, as termined from ancient and modern obse tions, Dr. Halley discovered that they hj motion of their own, which could not @ from parallax, precession, or aberration. % remarkable circumstance was afterwards ticed by Cassini and Le Monnier, and completely confirmed by 'Tobias Mayer, ' compared the places of 80 stars, as determ by Roemer, with his own observations, found that the greater part of them ha proper motion. He suggested that the cha P| STA of place might arise from a progressive motion of the sun towards one quarter of the heavens; but as the result of his observations did not xccord with his theory, he remarks that many rxenturies must elapse before the true cause of this motion can be explained. The probability of a progressive motion of he sun was suggested upon theoretical prin- tiples by the late Dr. Wilson of Glasgow; md La Lande deduced a similar opinion from he rotatory motion of the sun, by supposing, hat the same mechanical force which gave it . motion round its axis would also displace is centre, and give it a motion of translation 1 absolute space. If the sun has a motion in absolute space, irected towards any quarter of the heavens, }is obvious that the stars in that quarter iust appear to recede from each other, while woxe in the opposite region seem gradually oproaching, in the same manner as when alking through a forest, the trees to which e advance are constantly widening, while ie distance of those which we leave behind gradually contracting. The proper motion ‘the stars, therefore, in those opposite re- Ons, as ascertained by a comparison of an- ent with modern observations, ought to wrespond with this hypothesis; and Dr. erschel has found, that the greater part of em are nearly in the direction which would sult from a motion of the sun towards the nstellation Hercules, or rather to a part of '¢ heavens whose right ascension is 250° 52 “, and whose north polar distance is 40° 22’, ugel found the right ascension of this point ibe 260°, and Prevot makes it 230°, with 2 of north polar distance. Herschel sup- ses that the motion of the sun, and the stem, is not slower than that of the earth in ‘orbit, and that it is performed round some fant centre. The attractive force capable producing such an effect, he does not sup- se to be lodged in one large body, but in ‘centre of gravity of a cluster of stars, or ithe common centre of gravity of several ters. La Lande, however, is of opinion, t there is a kind of equilibrium among all ' Systems of the universe, and that they €a periodical circulation round their com- n centre of gravity. See Mayer’s “ Opera dita,” vol. i.; Lambert’s “Systeme du ‘nde;” various papers in Phil. Trans. and he Memoirs of Paris and Berlin. Yew and variable Stars—While the stars ibit these apparent variations they are ‘ected with others of a different kind, which 'n to arise froin some great physical changes > are going on in these bodies. Several ’ stars have appeared for a time, and then ished ; some that are given in the ancient logues can no longer be found; while ‘és are constantly and distinctly visible, th have not been described by the an- ‘ts, some stars like 8 in the Whale, have tually increased in brilliancy; others like the Great Bear, have been constantly inishing in brightness, and a great num- STA ber sustain a periodical variation in their brilliancy. The new star which was seen by Tycho in 1572, in the constellation Cassiopeia, suffered very remarkable changes. Ona sud- den it became so brilliant that it surpassed in brightness even Venus and Mercury, and was visible on the meridian in the day-time. Its light then began to diminish, till it disap- peared sixteen months after it had been first observed. ‘The new star which appeared in 1603, in the constellation Serpentarius, ex- hibited similar phenomena, and disappeared after having been visible for some months. Yn order to explain these singular changes, astronomers have supposed that the stars are suns, haying parts of their surface occupied by large black spots, which, in the course of their rotation about an axis, present them- selves to us, and thus diminish the brilliancy of the star. Some suppose the black spots to be permanent; but others are of opinion that the luminous surface of these bodies are subject to perpetual changes, which some- times increase their light, and at other times extinguish it. The same phenomena have been explained, with less plausibility, by supposing that the stars are extremely flat, and that they are more or less brilliant, ac- cording as their flat or sharp side turns to the eye. The variation in the light of the stars has also been ascribed to the interpo- sition of the planets which revolve round them; but it is not probable that the planets are sufficiently large to produce any sensible effect: even when seen from the earth, the light of our own sun is not perceptibly im- paired when Mercury or Venus are passing over his disc. The following are some of the most re- markable variable stars, viz. New star of 1572 in Cassiopeia, which changes from the 1 to 0, viz. from the first magiitude to invisible; period 150 years. 9 Whale, from 2 to 0; period 334 days. f Perseus, or Algol, from 2 to 4; period 2¢ 204 48! 58"-7, Hydra, east of 7, from 4 to 0; period 494 days. New star of 1604, in the east foot of Ser- pentarius, from 1 to 0; period not known. B Lyra, from 3 to 5; period 64 9s, New star of 1670, in the Swan’s head, which has not been seen since 1672. » Antinous, from 3 to 5; period 74 4h 15’, x In the Swan’s neck, from 5 to 0; period 3694 21, Another in the same constellation, near y in the breast, from 3 to 0; period 18”. ~ do Cepheus, from 3 to 5; period 54 84 372, Double Stars. These when viewed by the naked eye, and some of them even by the help of a telescope of moderate power, have the appearance only of a single star, but viewed through a good telescope they are found to be double, and in some cases a very marked difference is perceptible, both as to their brilliancy and the colour of their light. These Dr. Herschel supposes to be so near fed hy S Tix each other as to obey reciprocally the power of each other’s attraction, revolving about or their common centre of gravity, in certain determined periods. . The two stars, for example, which form the double star Castor, have varied in their an- gular situation more than 45° since they were observed by Bradley in 1759, and appear to perform a retrograde revolution in 342 years, in a plane perpendicular to the direction of the sun. Dr. Herschel has found them in intermediate angular positions, at interme- diate times, but never could perceiye any change in their distance. The retrograde revolution of y Leonis, another double star, is supposed to be in a plane considerably in- clined to the line in which we view it, and to be completed in 1200 years. ‘The stars « Bootes, perform a direct revolution in 1681 years, in a plane oblique to the sun. The stars ~ Serpentis, perform a retrograde revo- jution in about 375 years; and those of v Virginis in 708 years, without any change of their distances. In 1802, the Jarge star ‘Hercules, eclipsed the smaller one, though they were separate in 1782. Other stars he supposes to be united in triple and quadruple, and still more complicated systems. A catalogue of the most remarkable double, triple, &c. stars, is given by Dr. Herschel in the Phil. Trans. see vol. for 1782, p. 112; 1784, p. 487; 1785, p. 40, 213; 1786, p. 467 ; 1789, p. 212; 1791, p. 71; 1795, p. 46; 1802, p- 477 ; 1803, p. 339; 1804, p. 354. See also Kant’s “ Allgemeine Naturgeschichte ;” Lam- bert’s “ Photometria;” Cassini's ‘‘ Mem. Acad.,”’ Paris, 1789, &c. Star Pagoda, an Hast Indian coin, value as. 7d. sterling. STATICS (from craros, standing), is that branch of mechanics which treats of the equi- librium, weight, pressure, &e. of solid bodies when at rest; being thus distinguished from dynamics, which is the science of moving powers, or of the action of forces on solid bodies, when the result of that action is mo- tion. See Dynamics and MecuHANIcs. See also the articles CenTRE of Gravity, Ka@ui- LIBRIUM, and PRESSURE. STATION, in practical Geometry, &e. is a place pitched upon to make an observation, or take an angle, or such like, as in survey- ing, measuring heights and distances, level- ling, &c. An accessible height is taken from one sta- tion; but an inaccessible height or distance is only to be taken by making two stations, from two places whose distance asunder is known. In making maps of counties, pro- vinces, &c. stations are fixed upon certain eminences, &c. of the county, and angles taken from thence to the several towns, vil- lages, &c. In surveying, the instrument is to be adjusted by the needle, or otherwise, to answer the points of the horizon at every sta- tion; the distance from hence to the last station is to be measured, and an angle is to be taken to the next station; which process STE . repeated includes the chief practice of sur veying, a STATIONARY, in Astronomy, the position or appearance of a planet in the same point of the zodiac, for several days. ‘This happens from the observer being situated on the earth,| which is far out of the centre of their orbits, by which they seem to proceed irregularly; being sometimes seen to go forwards, or from west to east, which is their natural direction; sometimes to go backwards, or from east to west, which is their retrogradation; and be- tween these two states there must be an in termediate one, where the planet appears neither to go forwards nor backwards, but t¢ stand still, and keep the same place in the heavens, which is called her station, and the planet is then said to be stationary. Apollonius Pergzeus has shown how to fine the stationary point of a planet, according t the old theory of the planets, which suppose them to move in epicycles; which was fol lowed by Ptolemy in his “ Almag.” lib. 12 cap. 1, and others, till the time of Copernicus Concerning this, see Regiomontanus in “ Epi tome Almagesti,” lib, 12. prop. 1; Coperni cus’s “ Revolutiones Coelest.” lib. 5, cap. 3 and 36; Kepler, in “ Tabulis Rudolphinis, cap. 24; Riccioli’s “ Almag.” lib. 7, sect. é cap. 2; Herman in “ Miscellan. Berolinens, p. 197. Dr. Halley, Mr. Facio, Mr. D Moivre, Dr. Keil, Gregory, Vince, W oodhoust and others, have treated this subject. STATISTICS, a modern term adopted 4 express a more comprehensive view of th various particulars constituting the natur and political strength and resources of country than was usually embraced by write on political arithmetic. Its principal objee are, the extent and population of a state; tl occupation of the different classes of its i habitants; the progress of agriculture, of m nufactures, and of internal and foreign tradi the income and wealth of the inhabitants, ai the proportion drawn from them for the pu lic service by taxation; the condition of t poor; the state of schools, and other instit tions of public utility; with every other su ject, the knowledge of which tends to est blish the true civil policy of the country, a consequently to promote its prosperity. STEAM, is the name given in our langua to the visible moist vapour which arises fr¢ all bodies which contain juices easily expell from them by heats not sufficient for th combustion. Thus we say, the steam of b¢ ing-water, of malt, of a tan-bed, &c. It distinguished from smoke by its not havi been produced by combustion, by not e taining any soot, and by its being condens by cold into water, oil, inflammable spirits, liquids composed of these. a We see it rise in great abundance fn bodies when they are heated, forming a wh cloud, which diffuses itself and disappears no very great distance from the body fr which it was produced. In this case the 8 rounding air is found loaded with the wa == Bs oc me ; sa Lee een V L/ WALA d | i tn yy . Ps G an DL | | I | 4 i | ——— oo i ——o————————— ot yz Y= d M , y 1 y YE = ? mm Lacey . Engraved by Sa London Lub.Janig.1614 by G&S Robinson Laternoster Row kthe rest or thelropridtors . STE or oiler juices which seem to have produced it, and the steam seems to be completely soluble in, air, as salt is in water, composing while thas united a transparent elastic fluid. But in order to its appearance in the form of an opaque white cloud, the mixture with w dissemination in air seem absolutely ne- vessary. If a tea-kettle boils violently, so hat the steam is formed at the spout in great ibundance, it may be observed, that the vi- ible cloud is not formed at the very mouth if the spoug, but at a small distance before it, nd that the vapour is perfectly transparent it its first,emission. This is rendered ‘still nore evident by fitting to the spout of the ea-ketile a glass pipe of any length, and of 8 large a diameter as we please. The steam 3 produced as copiously as without this pipe, ut the vapour is transparent through the vhole length of the pipe. Nay, if this pipe ommunicate with a glass vessel terminating Wh Svwy\ 1 another pipe, and if the vessel be kept © Afficicntly hot, the steam will be as abundantly toduced at the mouth of this second pipe as | efore, and the yessel will be quite transparent. he visibility therefore of the matter which stitutes the steam is an accidental or ex- aneous circumstance, and requires the ad- ‘ixture with air; yet this quality again leaves when united with air by solution. ars therefore to require a dissemination in ie air. The appearances are quite agreeable , this notion: for we know that one perfectly ansparent body, when minutely divided and flused among the parts of another trans- went body, but not dissolved in it, makes a jass which is visible. Thus oil beat up with jater makes a white opaque mass. low the mean time, as steain is produced, the ater gradually wastes in the tea-kettle, and Hsoon be totally expended, if we continue on the fire. It is reasonable therefore to \ppose, that this steam is nothing but water janged by heat into an aérial or clastic form. (so, we should expect that the privation of $ heat would leave it in the form of water jain. Accordingly this is fully verified by iperiment; for if the pipe fitted to the spout the tea-kettle be surrounded with cold \ter, no steam will issue, but water will (itinually trickle from it im drops; and if t+’ process be conducted with the proper ‘cautions, the water which we thus obtain ee pipe will be found equal in quantity ‘that which disappears from the tea-ket- ! ie, and especially that which is raised the volatilization of hot water, is em- iyed as a first mover in that admirable con- ance the steam engine. This steam when ed with the ordinary heat of boiling water most 3000 times rarer than water, or more n 34 times rarer than air, and then has its iticity equal to that of the common atmos- Tic air: by great heat it has been found tthe.steam may be expanded into 14000 les. the space of water, when it exerts a he of nearly five times the pressure of the It ap-_ yA STE atmosphere: and there is no redson to sup- pose this is the extreme limit; indeed some accidents which have happened prove clearly that the elastic force of steam may at least equal that. of gunpowder. The following table shows the force of this aqueous vapour, at several different degrees of temperature. emperature on Fahrenheit, Force of aqueous Japour. 0333 0453 0620 0843 ‘L110 "1446 "1913 *2473 3153 "4043 *5050 - 6335 "7880 "9613 220 230 240 250 260 270 280 290 300 310 315 320 325 330 - Sream Engine, an engine for raising water, or for producing any powerful effect in moving amachinery, &c. by the force of steam obtained from boiling water. It is often called a fire engine, on account, perhaps, of the fire em- ployed in heating the water, in order to throw ofthe steam. This is unquestionably one of the most useful, curious, and important ma- _ chines that has ever been invented; and it is thought, that without the aid of this, or some other engine adapted to the same: purpose, we should long ere this have been deprived of the benefit of coal fires; as our forefathers, full a century ago, had excavated almost all the mines of that substance as deep as they could be worked, without the aid of some engine to draw water from greater depths. The prin- ciple of. this machine is as follows: there is a forcing pump with its rod fixed to one end of a lever, which is worked by the weight or pressure of the atmosphere upon a piston at the other end, a temporary vacuum being made below it, by suddenly condensing the steam, that had been let into the cylinder in which this piston works, by a jet, of cold water thrown inte it. A partial vacuum being thus made, the weight of the atmosphere presses down the piston, and raises the other end of the straight lever with water, &c. from the mine. ‘Then immediately a hole is uncovered in the bottom of the cylinder, by which a fresh supply of hot steam rushes in from the boiler, which acts as a counterbalance for the at- mosphere above the piston, and the weight of the pump rods at the other end of the lever carries that end down, and of course raises the piston of the steam cylinder. 'The orifice for the emission of the steam is immediately shut, and the cock opened for injecting the w he STE cold water, and another vacuum is made below the piston, which is now again forced down by the action of the atmosphere, and thus the work is continued at pleasure. This will be better understood by a reference to the following figure. Cte EE ees A 2B pth, B C Let a cylinder ABCD, piaced vertically, have a piston working in it, the rod of which is fixed to the end of the beam GH turning on the axis O, and loaded at its other ex- tremity by a weight W, and when the ma- chine is at rest the beam inclines towards the side where the weight is. Then if a vacuum be any how made in a part of the cylinder under the piston, the whole pressure of the atmosphere will tend to depress the piston, and will raise the weight W, providing the momentum be not greater than a column of water 33 feet high, having for its base the section of the cylinder, and acting by the lever GH. Suppose now the steam of water from a boiler to be introduced through a valve into the cylinder, and to expel the air; and when this is done the valve to be shut, and another valve opened, by which a jet of cold water is injected into the cylinder, the steam will be quickly condensed; a vacuum will be pro- duced, and the piston forced down by the atmospheric pressure. The introduction of steam at the bottom of the cylinder will again elevate the piston, and the reproduction of the vacuum will cause the depression of it; so that the end of the lever to which W is attached, may be employed to raise any weight, or overcome any resistance, within the limits above mentioned. This, which is the most simple form of the steam engine, must be considered merely as explanatory of the principles of its operation, and by no means of its mechanism, which is very various; having undergone numerous alterations and improvements from the simple form above given to the most perfect engines of the present day, of which the following on the improved plan of Mr. Watt is, perhaps, one of the most simple and best suited toa farther illustration, Referring to Plate XIT. STE CD is the boiler in which the water is con verted into steam by the heat of the furnace DPD. It is sometimes made of copper, bu more frequently of iron: its bottom is con cave, and the flame is made to circulate roun its sides, and is sometimes conducted b means of flues even through the middle of th water, so that as great a surface as possibl may be exposed to the action of the fire. I some of Watt’s engines the fire contained i an iron vessel was introduced into the middl of the water, and the outer boiler was forme of wood, as being a slow conductor of hea When the furnaces are constructed in th most judicious manner, eight square feet | the boiler’s surface must be acted upon b the fire or the flame, in order to convert on cubic foot of water into steam, in the spac of an hour. When fire is applied to the boile the water is not converted into steam till has reached the temperature of 212° of Fal renheit, or the boiling point. And, indee when the water is pressed by air or stear more condensed than the atmosphere, a ten perature greater than 212° is necessary fi the production of steam: but the heat requ site for this purpose increases in a less rat than the pressure to be overcome. The stea which is produced in the boiler is about 18f times rarer than water, and is convey through the steam pipe CE, into the cylind G, where it acts upon the piston g, and cor municates motion to the great beam A. But before we trace the mode of transmitti! this motion, we must describe the very 1 genious method employed by Mr. Watt f supplying the boiler regularly with water, al preserving it at the same level OP; a ¢ cumstance which is absolutely necessary, th the quantity and elasticity of the steam int boiler may be always the same. The sm cistern u, plaeed above the boiler, is suppli with water from the hot well A, by means the pump z, and the pipe f. To the botte of this cistern is fitted the pipe wr, which immersed in the water OP, and is bent at lower extremity, in order to prevent the € trance of the rising steam. A crooked a ud’, attached to the side of the cistern wu, st ports the small lever a’ b’ which moves upon as a centre. The extremity 0’ of this le carries, by means of the wire 0’P, a stone piece of metal P, which hangs just below’ surface of the water in the boiler, and 1 other extremity a’ is connected by the W au with a valve at the bottom of the cist u, which covers the top of the pipe ur. it is a maxim in hydrostatics, that whel heavy body is suspended in a fluid it loses much of its weight as is equal to that of quantity of fluid which it displaces. the water OP, therefore, is diminished by conversion of part of it into steam, the u surface of the body P will be above the ft and its weight will consequently be increas in proportion to the quantity of the body1 is not immersed. By this addition to weight the stone P will cause the extrem STE B of the lever to descend, and, in consequence, by elevating the arm d‘a’, will open the valve at the top of the pipe uv, and thus gradually introduce a quantity of water into the boiler, equal to that which was lost by evaporation. This process is continually going on, while the water is converting into steam: and it is eyident, that too much water can never be ‘ntroduced; for as soon as the surface of the water coincides with the surface of the body P, it recovers its former weight, and the valve at u shuts the top of the pipe ur. In order to know the exact height of the water in the boiler, two cocks k and / are em- dloyed, the first of which reaches to within a ttle of the height at which the water would stand, and the other, 7, reaches a very little yelow that height. If the water stands at the lesired height, the cock k being opened, will rive out steam, and the cock ¢ will emit vater, in consequence of the pressure of the uperincumbent steam on the water OP, but f water should issue from both cocks, it will ye too high in the boiler; and if steam issues rom both, it will be too low. As there would be great danger of the yoiler’s bursting if the steam should become oo. strong, it is furnished with the safety ‘alve x, which is so loaded, that its weight, dded to that of the atmosphere, may exceed he pressure of the interior steam, when of a ufficient strength. As soonas the expansive orce so far increases as to become dangerous 0 the boiler, its pressure preponderates over he pressure of the atmosphere and the safety alve: the valve therefore opens, and the team escapes from the boiler, till its strength s sufficiently diminished, and the safety valve huts by the predominance of its pressure yer that of the interior steam. By opening he safety valve the engine may be stopped at leasure: and to effect this, a small rectan- ular lever, with equal arms, is fixed upon the ide of the valve, and connected with its top; one of these arms a chain is attached, which ses over a pulley, from a horizontal to a ertical direction, so that by pulling it the alve is opened, and the machine stopped. From the dome of the boiler proceeds the team-pipe CE, which conveys the steam into 1e top of the cylinder G by means of the team-valve a, and into the bottom of the ylinder by reans of the valve ec. ‘The branch f the pipe which extends from a to c is cut ff in fig. 1, in order to show the valve 6, but , distinctly visible in fig. 2, which is a view f the pipes and valves in the direction FM. ‘he cylinder G is sometimes inclosed in a ‘ooden case, in order to prevent it from being doled by the ambient air; and sometimes in ‘metallic case, that it may be surrounded ad kept warm by a quantity of steam which brought from the steam-pipe EC, through le pipe EG, by turning a cock. It is ge- erally thought, however, that little benefit is otained by encireling the cylinder with steam, 3 the quantity thus lost is almost equal to hat is destroyed by the coldness of the cy- STE: linder. After the steam, which was admitted above the piston q by the valve a, and below it by the valve ec, has performed its respective offices of depressing and elevating the piston, and consequently the great beam A B, it es- capes by the eduction valves b and d, fig. 1 and 2, into the condenser 7, where it is con- verted into water by means of a jet playing in the inside of it. The water thus collected in the condenser is carried off, along with the air which it contains, into the hot well h, by the air-pump e, which is wrought by the piston rod TM, attached to the great beam AB. From the hot well # this water is conveyed by the pump z and the pipe f into the cistern u, for the purpose of supplying the boiler. ° The water w which renders air-tight the pump e, and supplies the jet of water in the con- denser, is furnished by the pump g, which is worked by the great beam. The steam and eduction valves a, c, 6, d, are opened and shut by the spanners aM, dM, cN, 6N, whose handles M and N are moved by the plugs 1, 2, fixed to TN the piston rod of the air-pump. This part of the machinery has been called the working geer; and is so constructed that the steam and eduction valves can be worked, either by the hand or by the piston of the air- pump. The piston rod R, which moves the piston g, passes through a box or collar of leathers fixed in a strong metallic plate on the top of the cylinder. The rod is turned perfectly cylindrical, and is finely polished in order to prevent any air from passing by its sides. The top V of the piston rod R is fixed to the machinery TV, which is called the pa- rallel joint, and is so contrived as to make the rod VR ascend and descend in a vertical or perpendicular direction. When the lever or beam rises into its present position from a horizontal one, the piston rod VR has a ten- dency to move towards p, and would move towards it were the bar yy fixed inits present position; for while the point V rises, the bar nV also rises, at the same time the angle Vuy increases, and likewise the angle AVu, so that the vertex V of the angle AV, would move towards T. The bar »», however, is not at rest, but moves round the fixed point », and rises along with the point V; while yy», there- fore, rises upon yas a centre, the adjoining bar » T moves round the point T towards V, the angle T »» increases, and the point » ap- proaches to V, and keeps VR in a perpendi- cular position, so that whatever tendency the point V has towards T by the increase of the angle A Vy, it has an equal tendency in the contrary direction, by the increase of the angle Ty: but as the beam A B falls into a horizontal position, all these motions are re- versed. . When the piston rod VR rubs most upon the side of the collar of leathers nearest to a, the fixed point » must be shifted a little in the contrary direction, wz. to the right- hand of R. That the nature of this parallel joint may be better understood, it may be proper to observe, that all the bars which have been mentioned are double, as may be — | at STE seen in the figure; that they move round points ata, T, V, , and »; and that the two bars between » and V move between the bars at py. In the steam engines of Newcomen and Beighton, where the “piston was raised mcrely by a counterweight at the extremity A of the great beam, the piston rod was connected with its other extremity by means of a chain bending round the arch of a circle fixed at B; but in Mr. Watt’s s improved engines witha double stroke, in which the piston receives a strong impulse upwards as well as downwards, the chain would slacken, and could not com- municate motion to the beam. An inflexible rod, therefore, must be employed for connect- ing the piston with the beam, or the piston must be suspended by double chains like those of engines for extinguishing fire. In some of Mr. Watt’s engines the latter of these methods was adopted: he then employed a toothed rack working in a toothed sector fixed at B, and afterwards fell upon the very su- perior method which we have now been de- scribing. All the engines which were constructed before the time of Mr. Watt were employed merely for raising water, and were never used as the first movers of machinery ; except in- deed that Mr. R. Fitzgerald published, in the Transactions of the Royal Society, a method of converting the irregular motion of the beam into a continued rotatory motion, by means of a crank and a train of wheel-work connected with a large and massy fly, which, by accu- mulating the pressure of the machine during the working stroke, urged round the ma- chinery during the returning stroke, when there is no force pressing it forward. For this new and ingenious contrivance, Mr. Fitz- gerald received a patent, and proposed to apply the steam engine as the moving power of every kind of machinery, but it does not appear that any mills were erected under this patent. In order to convert the reciprocating motion of the beam into a circular motion, Mr. Watt fixed a strong and inflexible rod AU to the extremity of ‘the great beam. ‘T'o the lower end of this rod,a toothed wheel U is fastened by bolts and straps, so that it can- not move round its axis. [his wheel is con- nected with. another toothed wheel S of the same size, by means of iron bars, which per- inits the former to revolve round the latter, but prevents them from quitting each other. ‘This apparatus is called the sun and planct wheels, from the similarity of their motion to that of the two luminaries. ' On the axis of the wheel S is placed the large and heavy fly wheel F, which regulates the desultory motion of the beam. When the extremity A of the great beam rises from its lowest position, it will bring along with it the wheel U, and cause it to revolve upon the circumference of the wheel S, so that the interior part of the former, or the part next the cylinder, will act upon the exterior part of the latter, or the part farthest from the cylinder, and put it in C8 motion along with the fly F. After the wheel U has got to the top of the wheel S, the end A of the beam will have reached its highest position, and the wheel 8, along with the fly, a of the eS will also dese by: from i its high est to its lowest position, so that for every ascent or descent of the piston or the greg beam, the planet wheel U will make one ‘turn, while the sun wheel and fly will perform two complete revolutions. When the steam engine is employed to drive machinery in which the resistance is very variable, and where a determinate yes locity cannot properly be dispensed with, Mr. Watt has applied a conical pendulum, which is represented at mn, for procuring an uniform, velocity. This regulator consists of two heayy balls mn, suspended by iron rods which move in joints at the top of the vertical axis op, and is put in motion by the rope 00 which passes over the pulleys 0, 0, and round the axis 0 0} the fly. Since the velocity of the fly and sun wheel increases and diminishes with the quan tity of steam that is admitted into the cylin. der, let us suppose that too much is admitted —then the velocity of the fly will increase but the velocity of the vertical axis op wil also increase, and the balls ma will recedt from the axis by the augmentation of thei centrifugal force. By this recess of the balls the extremity p of the lever ps, moving upo1 y as a Centre, is depressed, its other extr emit) $ rises, and by forcing the cock at a to close i little, diminishes ihe supply of steam. Th impelling power being thus diminished, th; velocity of the fly and the axis op, decrease in proportion, and the balls m, x, resume thei former position. | In Mr. Watt’s improved engine, the stear and eduction valves are all puppet clacks Onc of these valves, and the method of oper ing and shutting it, is represented in fig. 4 ¢ Plate XI. Let it be one of the eductio, valves, and let A.A be part of the pipe whie conducts the steam into the cylinder, an MM the superior part of the pipe which lead to the condenser. At OO, the seat of th valve, a metallic ring, of which nn is a sectio is fitted accurately into the top of the pip MM, and is conical on the outer edge, so é to suit the conical part of the pipe. 'Thes two pieces are ground together with emen and adhere very firmly w hen the contiguot surfaces are oxydated or rusted. The ‘clac is a circular brass plate m, with a conical edg ground into the inner edge of the ring nn, § as to be air-tight, and is furnished with a © lindrical tail mP, which can rise or fall in tk cavity of the cross bar NN. To the top| the valve ma small metallic rack mF is firm, fastened, which can be raised or depressed f the portion E of a toothed wheel, moveab upon the centre D. The small circle Dr presents a section of an iron cylindrical axi whose pivots move in holes in the opposi STE ides of the pipe AA. Its pivots are fitted nto their sockets, so as to be air-tight; and he admission of air is farther prevented by crewing on the outside of the holes necks of eather soaked in rosin or melted tallow. One nd of this axis reaches a good way without he pipe AA, and carries a handle or spanner 'N, which may be seen in fig. 1, Plate XIT. tid which is actuated by the plugs 1, 2, of he red TN. When the plug 2, therefore, levates the extremity of the spanner N4@, ‘uring the ascent of the piston rod TN, the xle D, Plate XI. fig. 4, is put in motion, the ‘alve m is raised by means of the toothed acks E and F, and the steam rushes through ae cavity of the circular ring an, by the sides f the cross piece of metal OO, NN. When ae valve needs repair, the cover B, which is ustened to the top of the valve box by means f screws, can easily be removed. Having thus described the different parts of ie most improved steam engine, it will be roper to attend to the mode of its operation. 1et us suppose that the piston is at the top of ie cylinder, as is represented in the figure, nd that the upper steam valve a, and the wer eduction or condensing valve d, are pened by means of the spanner M, while the wer steam valve c, and the upper eduction alve b, are shut; then the steam in the boiler fill issue through the steam pipe C EH, and e valve a, into the top of the cylinder, and press the piston, by its elasticity, to the sry bottom. But when the piston q is brought ) the bottom of the cylinder, the extremity ‘of the great beam is dragged down by the arallel joint TV, its other extremity A rises, ad the wheel u having passed over half of ie circumference of S, will have urged for- ard the fly-wheel F, and consequently, the achinery attached to it, one complete revo- ition. When the piston q has reached the ottom of the cylinder, the piston vod 'T'N of le air-pump, by the pressure of the plug 1 yon the spanner M has shut the steam valve and the eduction valve d, while the plug 2 is, by means of the spanner, opened the tuction valve 6, and the steam valve ec. ‘The eam, therefore, which is above the piston, ishes through the eduction valve 6 into the ondenser 7, where it is converted into water 7 the jet in the middle of it, and by the cold- ass arising from the surrounding fluid w, hile, at the same time, a new quantity of eam from the boiler issues through the open eam valve c, into the cylinder, forces up the ston, and, by raising one end of the working am, and depressing the other, makes the heel U describe the other semi-circum- rence of S, and causes the fly and the ma- jinery on its axis to perform another com- fete revolution. As the plugs 1, 2, ascend ith the piston q, they open or shut the steam id eduction valves, and the operation of the igine may be thus continued for any length “time. From this brief description of the steam igine, the reader will be enabled to perceive STE the nature and appreciate the value of Mr. Watt’s improvements. It had hitherto been the practice to condense the steam in the cylinder itself, by the injection of cold water; but the water which is injected acquires a considerable degree of heat from the cylinder, and being placed in air, highly rarefied, part of it is converted into steam, which resists the piston, and diminishes the power of the en- gine. When the steam is next admitted, part of it is converted into water by coming in contact with the cylinder, which is of a lower temperature than the steam, in conse- quence of the destruction of its heat by the injection water. By condensing the steam, therefore, in the cylinder itself, the resistance io the piston is increased by a partial repro- duction of this elastic vapour, and the impel- ling power is diminished by a partial destruc- tion of the steam which is next admitted. Both these inconveniencies Mr. Watt has in a great measure avoided, by using a con- denser separate from the cylinder, and en- circled with cold water* ; and by surrounding the cylinder with a wooden case, and inter- posing light wood ashes, in order to prevent its heat from being abstracted by the ambient air. The greatest of Mr. Watt’s improvements consists in his employing the steam both te elevate and depress the piston. In the en- gines of Newcomen and Beighton, the steam was not the impelling power, it was used merely for producing a vacuum below the piston, which was forced down by the pressure of the atmosphere, and elevated by the counter- weight at the farther extremity of the great beam. The cylinder, therefore, was exposed to the external air at every descent of the piston, and a considerable portion of its heat being thus abstracted, a corresponding quan- tity of steam was of consequence destroyed. In Mr. Watt’s engines, however, the external air is excluded by a metal plate at the top of the cylinder, which has a hole in it for admit- ting the piston rod; and the piston itself is raised and depressed merely by the force of steam. When these improvements are adopted, and the engine constructed in the most per- fect manner, there is not above + part of the steam consumed in heating the apparatus; and, therefore, it is impossible that the engine can be rendered } more powerful than it is at present. It would be very desirabje, however, that the force of the piston could be properly communicated to the machinery without the intervention of the great beam. This, indeed, has been attempted by Mr. Watt, who has employed the piston-rod itself to drive the machinery; and Mr. Cartwright has, in his ¥* Even in Mr. Watt’s best engines, a very small quantity of steam remains in the cylinder, having the temperature of the hot-well p, or of the water, into which the ejected steam is converted. Its pressure is indicated by a barometer, which Mr. Watt has ingeniously applied to his engines toy exhibiting the state of the vacuum. STE engine, converted the perpendicular motion of the piston into a rotatory motion, by means of two cranks fixed to the axis of two equal wheels which work in each other. Notwith- standing the simplicity of these methods, none of them have come into general use, and Mr. Watt still prefers the intervention of the great beam, which is generally made of hard oak, with its heart taken out, in order to prevent it from warping. A considerable -quantity of power, however, is wasted by dragging, at every stroke of the piston, such a mass of matter from a state of rest to a state of motion, and then from a state of motion to a state of rest. To prevent this loss of power, a light frame of carpentry has been employed by several engineers instead of the solid beam. Cast iron beams have been adopted with great success. On the Power of Steam Engines, andthe Method of computing it. From the account which has been given of the steam engine, and the mode of its opera- tion, it must be evident that its power de- pends upon the breadth and height of the cylinder, or, in other words, on the area of the piston and the length ofits stroke. If we suppose that no force is lost in overcoming the inertia of the great beam, and that the lever by which the power acts is equal to the lever of resistance ; then, if steam of a certain elastic force is admitted above the piston q, sq as to press it downwards with a force of a little more than 100 pounds, it will be able to raise a weight of 100 pounds hanging at the end of the great beam, When the piston has descended to the bottom of the cylinder, through the space of 4 feet, the weight will have risen through the same space, and 100 pounds raised through the height of 4 feet, during one descent of the piston, will express the mechanical power of the engine. But if the area of the piston q and the length of the cylinder are doubled, while the expan- sive force of the steam and the time of the piston’s descent remain the same, the me- chanical energy of the engine will be qua- druple, and will be represented by 200 pounds raised through the space of 8 fect during the time of the piston’s descent. The power of steam engines, therefore, is ceteris paribus in the compound ratio of the area of the piston, and the length of the stroke. These ob- servations being premised, it will be easy to compute the power of steam engines of any size. Thus, let it be required to determine the power of steam engines, whose cylinder is 24 inches diameter, and which make 22 double strokes in a minute, each stroke being 5 feet long, and the force of the steam being equal to a pressure of 12 pounds avoirdupois upon every square inch. The diameter of the piston being multiplied by its circumference, and divided by 4, will give its area in square inches ; thus, ae = 452.4, the STE number of square inches exposed to the pres sure of the steam. Now if we multiply this area by 12 pounds, the pressure upon ever square inch, we will have 452.4 x 12 = 5428, pounds the whole pressure upon the pisto or the weight which the engine is capable 0 raising. But since the engine performs double strokes, 5 feet long in a minute, th piston must move through 22 x 5 x 2=:§ feet in the same time; and therefore th power of the engine will be represented b 5428.8 pounds avoirdupoise, raised throug 220 feet in a minute, or by 10.4 hogsheads water, ale measure, raised through the sam height in the same time. Now this is equi valent to 5428.8 x 220 — 1194336 pounds, @ 10.4 x 220 = 2288 hogsheads raised throug the height of 1 foot in a minute. This is t most unequivocal expression of the meche nical power of any machine whatever th can possibly be obtained. But as steam er gines were substituted in the room of horse; it has been customary to calculate their m¢ chanical energy in horse powers, or to the number of horses which could perform th same work. This indeed is a very vague e pression of power, on account of the differer degrees of strength which different horse possess. But still, when we are told that steam engine is equal to 16 horses, we have more distinct conception of its power, t when we are informed that it is capable raising a number of pounds through a certai space in a certain time. Messrs. Watt and Boulton suppose a hor capable of raising 32,000 pounds avoirdupoi 1 foot high in a minute, while Dr. Desagulie: makes it 27,500 pounds, and Mr. Smeate only 22,916. If we divide, therefore, tl number of pounds which any steam engin can raise 1 foot high in a minute, by the: three numbers, each quotient will represe the number of horses to which the engine equivalent. Thus, in the present exampl 735338° = 374 horses, according to Watt ar Boulton ; 139433° = 434 horses, according | Desaguliers; and 153433° — 521 horses, a cording to Smeaton. In this calculation it supposed that the engine works only eig hours a day; so that if it wrought during th whole 24 hours, it wonld be equivalent ° thrice the humber of horses found by th preceding rule. Before concluding this article, we sh state the performance of some.of these © gines, as determined by experiment. An e& gine whose cylinder is 31 inches in diamete and which makes 17 double strokes per m nute, is equivalent to 40 horses, working d and night, and burns 11,000 pounds of St fordshire coal per day. When the cylind is 19 inches, and the engine makes 25 strok of 4 feet each per minute, its power is eq to that of 12 horses working constantly, ar burns 3,700 pounds of coals per day. And cylinder of 24 inches which makes 22 strok of 5 feet each, performs the work of 20 horse working constantly, and burns 5,500 poun STE coals. Mr. Boulton has estimated their rformance ina different manner. He states, at 1 bushel of Newcastle coals, containing pounds, will raise 30 million pounds 1 foot rh; that it will grind and dress 11 bushels wheat; that it will slit and draw into nails swt. of iron; that it will drive 1000 cotton indies, with all the preparation machinery, th the proper velocity ; and that these ef- sts are equivalent to the work of 10 horses. rewsier’s Ferguson. STEELYARD, or Stillyard, in Mechanics, cind of balance, called also statera Romana, the Roman balance, by means of which the sights of different bodies are discovered by ing one single weight only. The common steelyard consists of an iron am in which is assumed a point at pleasure, | which is raised a perpendicular. On the orter arm is hung a scale or bason to receive e bodies weighed: the moveable weight is ifted backward and forward on the beam, Lit be a counterbalance to 1, 2, 3, 4, &e. unds placed in the scale; and the points e noted where the constant weight balances ese 1, 2, 3, 4, &c. pounds. From this con- ruction of the steelyard, the manner of using ‘is evident. But the instrument is very ble to deceit, and therefore is not much ed in ordinary commerce. 'Chinese STEELYARD. The Chinese carry is statera about them to weigh their gems, id other things of value. The beam or yard a small rod of wood or ivory, about a foot (length: upon this are three rules of mea- ‘re, made of a fine silver-studded work; they i begin from the end of the beam, whence @ first is extended 8 inches, the second 64, e third 84. The first is the European mea- re, the other two seem to be Chinese mea- res. At the other end of the yard hangs a mnd scale, and at three several Uistances mm this end are fastened so many slender rings, as different points of suspension. The t distance makes 12 or % of an inch, the cond 32, or double the first, and the third , or triple of the first. When they weigh y thing, they hold up the yard by some one these strings, and hang a sealed weight, of out {Loz. troy weight, upon the respective visions of the rule, as the thing requires. rew's Museum, p. 369. Spring STeELYARD, is a kind of portable dance, serving to weigh any matter, from 1 about 40 pounds. It is composed of a brass or iron tube, into hich goes a rod, and about that is wound a ring of tempered steel in a spiral form. On is rod are the divisions of pounds and parts “pounds, which are made by successively mging on, to a hook fastened to the other id, 1, 2, 3, 4, &c. pounds. ‘Now the spring being fastened by a screw the botiom of the rod, the greater the eight is that is hung upon the hook, the ore will the spring be contracted, and con- ‘quently a greater part of the rod will come tt of the tube; the proportions or quantities STE of which greater weights are indicated by the figures appearing against the extremity of the tube. For an account of other steelyards, espe- cially the curious and accurate one constructed by M. Paul of Geneva, see the 2d vol. of Gregory's Mechanics. STENTOROPHONIC Tube. See Speaking TRUMPET. STEREOGRAPHIC Projection of the Sphere, is that in which the eye is supposed to be placed in the surface of the sphere. Or it is the projection of the circles of the sphere on the plane of some one great circle, when the eye, or a luminous point, is placed in the pole of that circle. For the fundamental prin- ciples and chief properties of this kind of pro- jection, see PROJECTION. STEREOGRAPHY, the art of drawing the forms of solids upon a plane. STEREOMETER, an instrument lately invented in France, for measuring the volume of a body, however irregular, without plunging it in any liquid. If the capacity of a vessel, or, which is the same thing, the volume of air contained im that vessel, be measured, when the vessel contains air only, and also when the vessel contains a body whose volume is required to be known, the volume of air ascer- tained by the first measurement, deducting the volume ascertained by the second, will be the volume of the body itself. Again, if it be admitted as a law, that the volume of any mass of air be inversely as the pressure to which it is subjected, the temperature being supposed constant, it will be easy to deduce, from the mathematical relations of quantity, the whole bulk, provided the difference be-- tween the two bulks under two known pres- sures be obtained by experiment. Let it be supposed, for example, that the first pressure is double the second, or, which follows as a consequence, that the second volume of the air be double the first, and that the difference be fifty cubic inches, it is evi- dent that the first volume of the air will like- wise be fifty cubic inches. The stereometer is intended to ascertain this difference at two known pressures. STEVINUS, or Stevin (Simon), a Flemish mathematician of Bruges of the 17th century, Stevinus was the author of several useful works on arithmetic, algebra, geometry, &c. some of which, that were written in the Dutch language, were translated by Snellius into Latin, and published in 2 vols. folio; there are alse two French editions of the same, one by Albert Girard in 1634, and the other an early one in 1608. Stevinus died in 1633. STEW ART (Dr. MatTHEw), an excellent geometrician, was born in the Isle of Bute in the year 1717. He was the favourite pupil of the celebrated Dr. Simson, and from whom he imbibed those strict geometrical principles, by which he was afterwards so deservedly distinguished. He was author of some ma- thematical works, of which the most con- siderable were, his Tracts, Physical and tee Sr © Mathematical, published in 1761; On the Sun’s Distance, published in 1763; General Theorems, 1746; and another work, entitled Propositiones More Veterum Demonstrate. This consists of a series of geometrical theo- rems, mostly new, investigated first by an analysis, and afterwards synthetically demon- strated. Dr. Stewart died in 1785, at the age of 68. For farther particulars of the writings and discoveries of this learned seometrician, see vol. i. of the Edinburgh Phil. Trans. p. 57, &e. STIFELS, Stireiius (MrcHackL), a protes- tant minister, and very skilful mathematician, was born at Eslingen, a town in Germany; and died at Gena in Thuringia, in the year 1567, at 58 years of age, according to Vossius, but some others say 80. Stifels was one of the best mathematicians of his time. He published, in the German language, a treatise on Algebra, and another on the Calendar or Ecclesiastical Computation. But his chief work is the Arithmetica Integra, a complete and excellent treatise, in Latin, on arith- metic and algebra, printed in 4to. at Norim- berg, 1544. In this work there are a number of ingenious inventions, both in common arithmetic and in algebra. STILE. See STYLE. STILYARD. See SteeLyarp: STIRLING (JAMEs), an eminent English mathematician, was born about the latter end of the 17th century. He was author of a commentary on Newton’s lines of the third order, entitled ‘ Linez tertii ordinis Neuto- niane sive illustratio tractatus D. Neutoni de enumeratione linearium tertii ordinis,’’ Lon- don, 1717, Ato, ; and an excellent original work entitled, ** Methodus Differentialis seu de summatione et interpolatione seriarum,” London, 1730, 4to. STOFLER (JoHN), a German mathema- tician of some eminence at the time in which he lived, viz. in the 15th century; he is, how- ever, rather remembered as an astrologer than a mathematician. He died in 1531, 79 years of age. STONE (Epmunp), a reputable Scotch ma- thematician, was born about the beginning of ihe last century, was son of the gardener to the Duke of Argyle; and without the assist- ance of any master made himself acquainted not only with various branches of the ma- thematics, but of the Latin and other lan- guages, and was the author and translator of several useful works; viz. 1. A New Mathe- matical Dictionary, in 1 vol. 8vo. first printed in 1726. 2. Fluxions, in 1 vol. 8vo. 1730. The Direct Method is a translation from the ¥rench of ? Hospital’s Analyse des Infiniments Petits; and the Inverse Method was supplied by Stone himself. 3. The Elements of Euclid, in 2 vols. 8vo.1731. A neat and useful edition of those Elements, with an account of the life and writings of Euclid, and a defence of his Elements against modern objectors. Beside other smaller works. Stone was a fellow of the Royal Society, STR and had insertéd in the Phil. 'Trans. (vol. 4 p. 218) an account of two species of lines the third order, not mentioned by Sir I Newton, or Mr. Stirling. if STONE denotesacertain quantity or weig of some commodities. A stone of beef, London, is the quantity of eight poundss Herefordshire, twelve pounds; in the n sixteen pounds. A stone of wool (accordi to the statute of 11 Henry VII.) is to wei fourteen pounds; yet in some ‘places it more, in others less: as in Gloucestershire, teen pounds; in Herefordshire, twelve poune A stone, among horse-coursers, is the w elg of fourteen pounds. STRENGTH, in Physiology, the same w forde. Fy On the Measure and application of Ai Strength—Men may apply their streng several ws ays in working a machine. A m of ordinary strength, turning a roller by t handle, can act fora whole day against aj sistance equal to thirty pounds weight; a if he works ten hours a day, he will rais weight of thirty pounds through three feet a a half ina second of time; or if the weight’ greater, he will raise it so much less in p} portion. But aman may act, fora small ti against a resistance of fifty. pounds or mo If two men work at a windlass, or roller, th can more easily draw up seventy pounds, th one man can thirty pounds, provided t elbow of one of the handles be at right ang to that of the other. And with a fly, or hea wheel, applied to it, a man may do one-thi part more work; and for a little while he e act with a force, or overcome a continual sistance, of eighty pounds; and work a wht day when the resistance is but forty pount Men used to bear loads, such as porters, W carry, some one hundred and fifty poum others two hundred or two hundred and fi pounds, according to their strength. A m can draw but about seventy or eighty poun horizontally; for he can but apply about h his weight. If the weight of a man be o hundred and forty pounds, he can act with greater force in thrusting horizontally, at t height of his shoulders, than twenty-se pounds. As to horses: a horse is, generally speakit as strong as five men. the weight of the pris x, and = Mute also dg — the momentum of the weight, : . therefore a + dg is the momentum of 4 prism x and its added weight. In like mi ner tcl + aw is that of the former or sh, prism, and the weight that brake it; co cx? : quently a + dg = kel + aw, and # i oie f aw + sel de , 91 is the lem Cc sought, that just breaks with the weigh at the distance d. If this weight g be nothi ¥ then x = VA ot Mets 3 x 2lis the lengt c the prism that just breaks with its own wei If two prisms of the same matter, hay their bases and lengths in the same proj} tion, be suspended horizontally; it is evi that the greater has more weight than_ lesser, both on account of its length, am its base; but it has less resistance on accc of its length, considered as a longer arm \ lever, and has only more resistance on count of its base; therefore it exceeds lesser in its momentum more than it do its resistance, and consequently it must bi more easily. Hence appears the reason why, in mal small machines and models, people are a be mistaken as to the resistance and str of certain horizontal pieces, when they ¢ to execute their designs in large, by obser, the same proportions as in the small. When the prism, fixed vertically, is about to break, there is an equilibrium tween its positive and relative weight;, consequently those two opposite powers to each other reciprocally as the arms 0 lever to which they are applied, that 1 half the diameter to half the axis of the pi On the other hand, the resistance of a bo always equal to the greatest weight whi will just sustain in a vertical position, th to its absolute weight. Therefore, subs} ing the absolute weight of the resistam appears, that the absolute weight of a I STU spended horizontally, is to its relative sight, as the distance of its centre of gravity ym the fixed point or axis of motion, is to e distance of the centre of gravity of its ise from the same. ~ The discovery of this important truth, at ast of an equivalent to it, and to which this reducible, we owe to Galileo. On this stem of resistance of that author, Mariotte ade an ingenious remark, which gave birth anew system. Galileo supposes that where e body breaks, all the fibres break at once; that the body aiways resists with its whole solute force, or the whole force that all its wes have in the place where it breaks. But ariotte, finding that all bodies, even glass elf, bend before they break, shows that ‘res are to be considered as so many little nt springs, which never exert their whole ‘ce, till stretched to a certain point, and ver break till entirely unbent. Hence gse nearest the fulcrum of the lever, or west point of the fracture, are stretched less an those farther off, and consequently em- by a less part of their force, and break er. This consideration only takes place in the rizontal situation of the body: in the ver- al, the fibres of the base all break at once ; ‘that the absolute weight of the body must ceed the united resistance of all its fibres ; greater weight is therefore required here in the horizontal situation, that is, a sater weight is required to overcome their ited resistance, than to overcome their veral resistances one after another. Varignon has improved on the system of wiotte; and some later authors, as Buffon, bison, Girard, kc. have done much towards ‘fecting the theory. For more on_ this meh of the subject we refer to Gregory’s schanics, vol. i. book i. Huitton’s Course, . iii. and Girard’s treatise on the Resistance Solids. See also the article COHESION. TRIKE, or Stryke, a measure containing r bushels, or half a quarter. TRING, in Music.. See CHORD. TURM, Sturmius (JOHN CHRISTOPHER), noted German mathematician and philo- ther, was born at Hippolstein in 1635. He same professor of philosophy and mathe- tics at Altdorf, where he died in 1703, at years of age. de was author of several useful works on thematics and philosophy, the. most es- med of which are: 1. His Mathesis enu- ata, in one vol. 8vo. 2. Mathesis Juvenilis, wo large volumes 8yo. 3. Collegium Ex- imentale, sive Curiosum, in quo primaria uli superioris Inventa et Experimenta Fysico-Mathematica, Speciatim Campanz natorize, Camere obscure, Tubi Torricel- li, seu Baroscopii, Antlize Pneumaticz, armometrorum Phenomena et Effecta; tim ac aliis jampridem exhibita, partim ‘iter istis superaddita, &c. in one large ume 4to.; Norimberg, 1701. SUB STYLE, in Chronology, a partieular man- ner of counting time, as the new style, the old style. Old STYLE, is the Jnlian manner of com- puting, as instituted by Julius Cesar, in which the mean year consists of 3655 days. New STYLE, is the Gregorian manner of. computation, instituted by Pope Gregory the XII Ith, in the year 1582, and is used by most catholic countries, and many other states of Europe. See CALENDER and BissExTILE. Styue,in Dialling, the same as GNoMON. SUBCONTRARY Position, in Geometry is when two equiangular triangles A E Dd, ABC, are so placed as to have one common angle A at the vertex, and yet their bases not parallel. Consequently, the angles at the bases are equal, but on the contrary sides; vz. the 4 E—2zB,and.zD= ZC. A C B If an oblique cone with a circular base whose diameter is C B, be so cut by a trans- verse plane, as to make, in the section ABC, through the vertex A, the 2 E= z B, and 2D= zC, then the cone is said to be cut by the transverse plane in a subcontrary po- sition to the base BC; and in this case the section whose diameter is DE is always a circle when BC is. SUBDUCTION. See SusrTraction. SUBDUPLE Ratio, is when the antece- dent is equal to half the consequent. SUBDUPLICATE Ratio of two quantities, is the ratio of their square roots, as /a: /b is the subduplicate ratio of a: 6. SUBLIME Geometry,the geometry of curve ines. SUBMULTIPLE, in Geometry, &c. A submultiple number, or quantity, is that which is contained a certain number of times in another, and which, therefore, repeated a cer- tain number of times, becomes exactly equal to it.© Thus 3 is a submultiple of 21. In which sense a submultiple coincides with an aliquot part. SuBMULTIPLE Ratio, is that between the quantity contained and the quantity contain- ing. Thus the ratio of 3 to 21 is submultiple. In both cases submultiple is the reverse of SUB multiple: 21, e. gr. being a multiple of 3, and the ratio of 21 to 3 a multiple ratio. SUBNORMAL, in Geometry, or subper- pendicular, the distance upon the axis, be- tween the foot of the ordinate and the per- pendicular to the curve or its tangent. In all curves the subnormal is a third proportional to the subtangent and the ordinate; and in the parabola it is a constant quantity, being equal to half the parameter of the axis. SUBSTITUTION, in Algebra, is the put- ting or using one quantity for another to which it is equal. See ReEsoLuTION of Equa- tions. SUBSTRACTION. See Surprracrion. SUBSTYLE, in Dialling, is the right line on which the style or gnomon is erected. SUBTANGENT of a Curve, the line that determines the intersection of a tangent with the axis; or that determines the point wherein the tangent cuts the axis prolonged. The tangent, subtangent, and semiordinate, always make a right-angled triangle. It isa rule in all equations, thatif the value of the subtangent comes out positive, the point of intersection of the tangent and axis falls on that side of the ordinate, where the vertex of the curve lies, as in the parabola and paraboloids. If it come out negative, the point of inter- section will fall on the contrary side of the ordinate, in respect of the vertex or beginning of the absciss ; as in the hyperbola and hyper- boliform figures. And universally, in all paraboliform and hyperboliform figures, the subtangent is equal to the exponent of the power of the ordinate, multiplied into the absciss: thus, in the com- mon parabola, whose property is px = y’; the subtangent is in length equal to x, the absciss multiplied by 2, the exponent of the power of y; that is, it is equal to twice the absciss; and by the former rule for paraboli- form figures, it must be taken above the ordi- nate in the axis produced. Thus, also, in one of the cubical paraboloids, where px* — y?;. the length of the sub- tangent will be 3 of the absciss: and in a parabola of any kind, the general equation being a” a® = + y™t™”, the subtangent is = man yy x, or its ratio to the absciss is con- stantly that of m + nto n. See Method of TANGENTS. SUBTENSE ofan arc or angle, is the right line by which it is subtended. SUBTRACTION, in Arithmetic, is one of the first four fundamental rules of that science; and consists in finding a number equal to the difference of two other given numbers, which number is generally called the remainder. Subtraction is either simple or compound ; the first relates to numbers or quantities, which are all of the same kind or denomination ; and the latter to quantities of different denomina- tions. a SUB Simple SusTRACTION, is the method of certaining the difference between any ty numbers having the same denomination. — Rule 1. Place the less number under 4 greater, so that units may stand under uni tens under tens; hundreds under hundred and so on, and draw a line below them. Then begin at the right-hand, and tal each figure in the lower line from the figu above it, and set down the remainder. 2. But if the figure in the lower line | greater than the corresponding figure in 1 upper line, add 10 to the upper digit, and thy take the lower one from it and set down t| remainder. Carry one to the next low figure and subtract again; and so on tillt whole is finished. EXAMPLES. 71416714 64184146 34816703 51879269 36600011 remainders 12304877 71416714 proof 64184146 ‘nese fn ecienceet Proof of Sustraction. Add the remaine and the least of the two given numbers tog} ther, and if the sum be the same as the grea number from which the subtraction was ma the work is right. Compound SuBTRACTION, is the method finding the difference between two giy quantities having different denominations, | Rule 1, Place the less number under 4 greater, so that the parts that are of the sai} denomination may stand directly under ea} other. | Then begin at the right-hand, and _ ta! each number in the lower line from that abe it, and set down the respective remainders: 2. But if any of the quantities in the loy line exceed those in the upper line, then many units must be adiled to the upper ¢ as make one of the next higher denominati subtract the lower number and set do. the remainder, observing to carry one to | next number in the lower line, then subtt as before; and thus proceed to the high} denomination ; and the several remaind will be the answer required. 4 The method of proof is the same as in { last rule. | EXAMPLES, ¢ ie sx! d. yds. ft. in. 21413 42 17 °2. 54) 112 17 6 15 1 112 101 15 10% remainders 2 0 52 | 214 13 42 proof 17° 2 See SS GC — ball SUBTRACTION of Fractions, is finding | difference between any two fractions of | same, or of different denominations. Hy Rule 1. If the fractions are of different } nominations, they must be first reduced to | same denomination, 2. Reduce all mi} SUB mbers to improper fractions; and fractions ying different denominators to a common tominator. 3. Then subtract one of the merators from the other, and place the re- t over the common denominator for the nainder sought. EXAMPLES. 5 — 63 es thus ¢ — 3; —$i—#3= 32 ) pera rs 1 45° 26) $5 1 13 — 33 —3 =3 Xequired the difference between + of a nea and 2 of a pound. ; $ guinea — %3 ofa shilling, % pound = 43 ofa shilling, ian &3 — 49 — 441 "160 — 281 — 10s. as Sy cated 2 OES Nang Me 3h ilar 28° ‘or the method of reducing fractions, see DUCTION. UBTRACTION of Decimals, is finding the ‘rence between two quantities consisting ly of integers and partly of decimals, or ecimals only. tule. Arrange the quantities so that the mal points may fall directly under each r; and then subtract as in the simple , observing only to place the decimal point he remainder under those of the given bers. EXAMPLES. Required the difference between 7854321 648°4867. Required the difference between 684384 ‘675849. 7854°321 *684384 648-4867 675849 7205°8343 remainders °008535 78543210 proof 684384 ‘ BTRACTION of Circulating Decimals, is ig the difference between quantities com- I partly of integers and partly of circulat- ‘ecimals. a le 1. Reduce the decimals to their equi- t fraction, and their difference will be aswer required. See CrRcuLATING De- le2. Carry on the repetends till they me conterminous; that is, till they all ‘their periods of circulation in the same ‘and let also the circulation of each be d two figures beyond this place, then tet, as in the former case, observing set down any remainder in the first two 5, these being only necessary in order to ‘Whether 1 should be carried to the con- 1ous period. - ; EXAMPLES. juired the difference between 78'3476 28417. 78°3476 = 78:3476)47 42°8417 42°8417|41 -remainder 35:5059 this exampie the period of circulation immediately after the fourth place of SUB decimals; it is then earried on two places farther to see if any thing is to be carried to the conterminous period, which in the present example is 0, as is obvious. Note. There may arise cases in which it will be necessary to carry the circulation on to three or more places beyond the conter- minous period; that is, when the repeating digits are the same as in that case, we cannot tell whether it will be necessary to carry one or not. SUBTRACTION, in Algebra, is finding the difference between two algebraical quantities, and connecting those quantities together with their proper signs. And this is generally di- vided into the following cases: é Case 1. When the indeterminate letters are the same and affected with the same sign. Rule. Begin™on the left hand, and subtract the co-efficients of the several quantities from each other, and prefix before the remainder the proper sign whether plus or minus. Note. When the quantity to be subtracted is greater than the other, then the sign to be prefixed to the remainder must be changed from + to —; or from — to +. EXAMPLES. From 76+ 3a—4e From 192ry — 132? Take 564-6a—3c Take 17xy— 72* i tteeeieeeeenteenet et - bee ee 26—3a— ec remainders 2xry— 627 Seen eee Case 2. When the quantities are the same, but affected with different signs. Rule. Conceive all the signs of the lower line to be changed, and then add the quanti- ties together, as in addition. EXAMPLES. From 7ry—3y* + 7z Take 62y +2y?— 32 xy—5y* + 10z remainder. From 627y—3V/xy + 72 Take 3x7y +3./xy— 52? dx*y—6 /xy + 12z* remainder. SUBTRACTION of Algebraic Fractions, is finding the difference of any two fractions ex- pressed by indeterminate letters. ule. Reduce the fractions to a common denominator; subtract their numerators, and set the difference over the common denomi- nator, EXAMPLE. 1, 24 __ 34 _46ab | Ma _ (45b—21)a "76 96% 630 636° —” 636% ‘ERO Te 2) Ade OL Akal 2de “ 38y lly 33y 33y —33y 3, etsy _- be—3y } 3ab Py et qxee+4cey 15x—9y _ 3abe 3abc fae set lavas da A required. sabe SUN SUBTRACTION of Surds, is finding the dif- ference between any two surd expressions. Rule. Reduce the surds to their simplest forms; then if the surd part is the same in both, subtract the. co-efficients, and annex to the remainder the common surd. If they are not the same the subfraction can only be ef- fected by the sign minus. EXAMPLES. lL YI8—V8 =38Vv2—2V2= V2 2, f/27— f/12=—3V3 — 9 faite 3. Y108a*t — 7 32a = 3a7/ 4a—2¥ 4a = (8a—2) + 4a SupTRAcTION of Ratios, is used by some authors to denote what is more commonly called divided ratio. Thus if a3b 3 erg then by subtraction or division, a—b:a::e—d:ec a—b:b::e—d:d SUBTRAHEND, that quantity which is to be subtracted, the other quantity from which it is to be taken being called the Minuend. SUBTRIPLE, is when one quantity is the third part of another, and subtriple ratio is the ratio of 1 to 3. SUBTRIPLICATE Ratio, is the ratio of the cube root; thus 4/ a to ¥ 6b is the subtri- plicate ratio of a to 6. SUCCESSION of Signs, in Astronomy, called also consequentia, is the order in which the sun enters'them. As Aries, Taurus, Ge- mini, Cancer, &c. When a planet’s motion is according to the order of the signs, or in consequentia, it is said to be direct; but when it moves in the con- trary way, or antecedentia, then its motion is said to be retrograde. SUCCULAR, in Mechanics, a bare axis or cylinder, with staves in it to move it round, but without any tympanum or paritrochium. SUCKER, in Mechanics, a term vulgarly used for the piston of a sucking pump. SUCKING Pump, the common pump work- ing with two valves opening upwards. See Pump. SUM, the aggregate of several quantities, or that number which is found by adding several numbers or quantities together. SUMMER, is one of the quarters of the year, beginning about the 21st of June, and ending on the 22d of September. Summer is also frequently used to denote one half of the year, viz. the time in which the sun passes through the northern signs; viz from about the 21st of March, till about the 22d of Sep- tember, exceeding the winter, or the time in which he passes through the northern signs, by about seven days. See Eartn and Eaui- NOX. SUN, Sol ©, in Astronomy, the great source of light and heat to the solar system, about which, as a centre, the several planets perform their revolutions, and to the influence of which, combined with their sidereal and diurnal re- SUN volution, they owe the successive alternatio; of summer and winter, day and night. The sun is the most considerable of all t] heavenly bodies, and governs all the plan tary motions; its diameter is 111°454 tim the mean diameter of the earth; whence volume is 1384472 times greater than that the earth, but its mass is only 337086 tim creater, and its density 1s 35425) OF abow that of our globe. It is surrounded by an atmosphere of gre extent, and is frequently obscured by spe some of which have been observed so lar as to exceed the earth five or six times diameter. : The observation of these spots shows fl the sun moves on its axis, which is nea perpendicular to the ecliptic, the duration an entire sideral rotation being about 254 de and consequently it is flattened at the pol the sam@ as all the other planetary bodies, The solar equator is inclined 7° 30’ to” plane of the ecliptic. * A body which weighs one pound at thes face of the earth, would, if removed to surface of the sun, weigh 27°933 pounds ; | bodies would fall there with a velocity 334°65 feet in the first second of time. The sun, together with the planets, m round the common centre of gravity of system, which centre is nearly in the ce of the sun; and this motion changes ecliptic orbits of the planets and comets | epicycloids. , The sun appears to have a particular tion, which carries our system towards! constellation of Hercules. The apparent diameter of the sun, as from the earth, undergoes a periodical ve tion. It is greatest when the earth is if perihelion; at which time it is 32'35” ,65 it is least when the earth is in its aphe when it is 31/31” ,0. Its mean apparent meter is therefore 32’ 3” ,3. ‘4 His horizontal parallax, as determines the transit of Venus, is 82". The greatest equation of his centre is I 27" .7, which diminishes at the rate of 1 in a century. . These particulars relating to the sum extracted from Laplace’s “System du Mor 3d edition, and may therefore be consi as the most accurate and authentic W have yet been published, having been | puted by that eminent astronomer from latest and most accurate observation. When the sun is examined with a teles of sufficient magnifying power, and the § dour of his light is intercepted by the 1 position of a dark glass or otherwise, a nU of black spots, as we have said above, 0 rious forms and magnitudes, are frequ perceived upon his disc : these are some’ so large that they have been seen by the u eye. In the year 1779 Dr. Herschel pe one which was about 50,000 miles im meter, more than six times that of our and consequently presenting a disc 36) SUN ser, The solar spots, when observed for ) or three days in succession, seem to move oss the body of the sun from east to west, 1 from this motion the duration of the time rotation of this luminary has been deter- ied as above stated. Various hypotheses have been advanced as he cause of these spots, and the nature of luminary on which they appear; some ronomers consider them as scoria float- in the inflammable liquid matter, of which y conceive the sun to be composed. Ga- o, Hevelius, and Maupertuis seem all of ‘m to have entertained this opinion. Lahire | Lalande suppose them to be eminences the body of the sun, appearing at times in sequence of the flux and reflux of the tid igneous matter ; others again, suppose _ body to be surrounded with a luminous iosphere, under which, viz. between it and ‘dense body of the sun, is another darker _more dense atmosphere to intercept the sof the outermost, and that these spots are y openings in the luminous atmosphere ough which the other then appears. Dr. sschel’s hypothesis resembles this; but it aires more room to detail it than can be rded in this place ; the reader is therefore ‘rred to the Phil. Trans. for 1795, where will find many.ingenious remarks and ob- ations relating to this subject. di the above hypotheses are founded upon ipposition of the sun being in itself a hot ‘luminous body, which would be at best . gratuitous, were it not contradicted by ry well-established facts in natural philo- ay, which show that heat is produced by isun’s rays, only when they act on a calori- nedium ; they are only the cause of the pro- tion of heat, by uniting with the matter of | which is contained in the terrestrial at- phere. or substances that are heated, as er poured upon lime is the cause of the t which ensues; and one may easily imza- , that an animal living in such a me- mn, and under such circumstances, would he first instance consider the water not lely as the cause of heat but heat itself, e do the solar rays. Numerous facts, ever, may convince us of the contrary. the tops of mountains of sufficient height re clouds can seldom reach to shelter n from the direct rays of the sun, we find ms of perpetual snow. Now if the solar | themselves conveyed all the heat we on our globe, it ought to be hottest where r course is the least interrupted; w7z. on tops of those mountains, which we know 1 observation to be in a constant state of elation. The same has been observed those who have ascended in balloons, 7. e. higher they ascend the greater degree of they experience, the sun itself appears inished both in splendour and magnitude, ‘the heavens, instead of the azure or blue, th we observe, approach more and more ards a total obscurity. SUP These facts, to which we might add many others, are sufficient to expose the common notion of the sun being a globe of fire, and to show at the same time that those planets which are nearest to the sun are not necessa- rily the hottest, nor those the coldest that are more remote ; and hence many of the fanciful calculations relating to the light and heat experienced by the different planets of our system fall to the ground; as it is obvious, from what is stated above, that by certain modifications of the planetary atmospheres the light and heat might be equalized through- out the solar system. For more on the sub- ject of this article, see Horsley, Phil. Trans. 1767, p. 398; Wilson, Phil. Trans. 1774 and 1783; Marshal, Phil. Trans. 1774, p. 194; Wollaston, Phil. Trans. 1774, p. 329; and the Mem. Acad. 1776. Also Bouguer’s Traité d'Optique; Biot’s Traité de lAstronomie Physique ; Laplace’s Exposition du Systeme du Monde, p. 14, 246; Lalande and Vince’s Astronomy, &c. &e. See also some particulars relating to the nature of the solar rays, under the article Rays in this Dictionary. ; SUNDAY, the first day of the week ; being thus denominated by the ancient Pagans be- cause set apart for the worship of the sun. SunDAyY Letter. See DomINicaL Letter, SUPERFICIAL, relating to superficies. SUPERFICIAL Content of any Thing, is the number of square inches, feet, ke. contained in its surface. SUPERFICIAL Measure, the same as square measure. See MEASURE. SUPERFICIES, or Superrice,in Geometry, the outside or exterior surface of any body. This is considered as of two dimensions, viz. length and breadth, but without thickness, and therefore forms no part of the substance, or solid content, or matter of the body. The bounds or extremities of a superficies are lines, and it is said to be a rectilinear or curvilinear superficies, according as the bound- ing lines are right or eurved. SuPERFICIES are farther divided into plane, coneave, and conver. Plane Superrictes, is that in which if a right line touches it in two points, it touches it in every point; if not it is curved, and is either concave or convex. Concave SUPERFICIES, is one which sinks inwards, like the inside of a cup or hemi- sphere. ' Convex SUPERFICIES, is that which rises out- wards, as the outside of a sphere or hemi- sphere. The rules for finding the superficies of plane and curved surfaces will be found under the several heads in this Dictionary. Line of SuPERFICIES. See SECTOR and GunTEr’s Scale. SUPERPARTICULAR Ratio, an obsolete term for a ratio when the two terms differ only by unity PUREE AE Ratio, likewise obso- SUR lete, d ratio in which the difference of the terms is greater than 1, but the greater is less than double the other. SUPPLEMENT of an Are, in Trigono- metry, is what it wants of 180°. SURD, in Arithmetic and Algebra, denotes the root of any quantity; when fhat quantity is not a complete power of the dimension re- quired; thus /2, 3/4, 4/7, &c. are surds or irrational quantities. Surps are either simple or compound. A Simple Surp, is that which consists of only one term, as / 2, 7/ 6, &ce. A Compound Surp, is that which consists of two or more terms or radicals, thus /3-+ /2; /3— /2, or /(3+ /3), &e. are compound ; and when there are only two terms they are also called binomial surds. Surps, are again either irrational or imagi- nary; they are irrational when the quantity under the radical is a positive quantity, or if it be negative, and the radical be of odd di- mensions ; and they are tmaginary when that quantity is negative, and the root to be ex- tracted is of even dimensions. Surps, are also farther distinguished into finite and continued, of the former kind are all those which we have given above, and the latter are those of the form 54 V¥54+/5+ &ce.; and it is singular that the latter, though apparently so much more complex than the former, are frequently expressible in rational numbers. See Continued SuRDS. The several rules for the reduction, addi- tion, subtraction, multiplication, and division of surds are given under the respective articles in this Dictionary, and we have, therefore, in this place, only to offer a few general remarks relating to these quantities, their roots, powers, &c. which could not be properly introduced into those articles. We have defined a surd to be the root of some quantity which is not a perfect power of the dimension required by the index of the root; the common definition, viz. that it is a quan- tity incommensurable with unity is not suffi- ciently explicit, as this includes exponential and logarithmic quantities; the condition therefore of a surd being incommensurable with unity not entering into our definition, it should be stated as a proposition, and demon- strated to-be a necessary consequence of that definition ; this may be shown as follows. In order that a quantity may be commensu- rable with unity, it must be either an integer or a rational fraction; but a surd cannot be an integer by the definition, and neither can ar it be a fraction. For if possible let "VY a= — where p and q are prime to each other, or the fraction? in its lowest terms; then a = me z n but since z is inits lowest terms 2 is so like- Gg 7 ‘ Y wise, and therefore cannot be equal to an integer, consequently "V a cannot be expressed SUR by a rational fraction; that is, it is inco mensurable with unity. a Surds are, farther, not only incommer rable with unity, but they are also ineo mensurable with each other, except that wh reduced to their simplest form the index a the quantity under it are the same in bot thus the square root of /3, /12, and ¥! when reduced to their simplest form, beeo; /3, 2/3, 3/3, and are therefore comm surable with each other; but if the quanti under the radicals be prime to each oth then the surds are incommensurable, forming a species of quantity inexpressib any other form. Neither is the sum or ference of two or more surds expressible any simple surd expression of another k that is, we cannot find /a + /b= Ve, p viding a and 6 are prime to each other; by squaring we have a+2/ab4+b =e /ab—ie—ta— ib; that is the squ root of a quantity, not a square, equal to integer or fraction, which we have seen is’ possible. In the same manner it might shown that Jat vb = Yc Vdisi% impossible providing a and b are prime to & other. It is not, however, so easy to dem strate this impossibility for the higher of surds, though there is every reason to § pose it is the same in all; and hence als seems highly probable that the root of cubic equation 23 —ax —=b, which is of form 3¥ m+3/ n, is inexpressible in any of finite form, and consequently the efforts 1 have been made, or that may be made, to1 der the solution of the irreducible case tional must be totally useless. See Cha of Barlow’s Theory of Numbers. To extract the Square Reot of Binomial Su Let it be proposed to find the square ofat 7b. Assume f/(at J/b)m= Vee then by squaring 4 at Jb=atyt V4ry this gives us si es . ee he ) Whence we find x = reer ani and y del Ai ! which may therefore be considered as gen formule for this purpose. a Suppose, for example, the /(12 + 2 v was required, here a=12, and b= 140, th fore, 7 a tv (144—140) _ 12+ v4 5 5 2 a ee! 12— y(144—140) _ 12 — v4 9 2 v7 2 therefore the root is 7 + /5. To extract the Cube Root of a Binomial Si Let a-+m ¥v b be the proposed surd, % being already in its simplest form. Assume 9 (atm Vb) =aty Vd, © | y — bail os U tt By cubing rtm fba=xit 82°y Vb +3y7bat y3b Vb 23+ 3y7%bxr —a my itence } 3x fyb =m om which equations x and y may be found ; exterminating one of those quantities ; but , they must necessarily be either integral or actional when the extraction is possible, it ill commonly be the readiest way to assume weral values of y in the latter, and thence ad the value, which, when substituted for x the first equation, must give the true result, y have been rightly assumed; if not, another , ue of y must be taken, till a proper value ?a is obtained, and if this cannot be done the aantity proposed has no radical root. Exam. Let it be proposed to find the cube wot of 135478 “3. Herea = 135,6=3, ad m= 7; and the equations are x3 +9xy* = 135 sxyt+ sy° = 75 Whence it obtain y = 2, and x =3, there- re 47 (135 + 78 /3)=3 + 2/3, as required. Newton’s Rule for any Root of a Binomial * Surd, a b. Of the given quantity a+}, let a be the reatest term, and ¢ the index of the root to e extracted. Seek the least number n, whose ower n° can be divided by a*— b* without a emainder, and let the quotient beq; com- ute Y(a+b) x Vq in the nearest integer umber, which call r; divide avq by its reatest rational divisor, calling the quotient s; nd let the nearest integer number above + /(t?s°*— ive ey) ae) be the be ¢: then will ~ 2s oot sought, if it can be extracted. This rule 3 demonstrated by s’Gravesande in his Com- aentary on Newton’s Arithmetic: where he as also given many numeral examples illus- cative of it; examples may likewise be found u Newton’s “ Universal Arith.” and in Mac- aurin’s “‘ Algebra.” Dr. Maskelyne’s Rule for any Power of an imaginary Binomial Surd. Dr. Maskelyne, the late astronomer royal, jas also given a method of finding any power if an impossible binomial by another binomial if the same kind. This rule is given in his ntroduction prefixed to Taylor’s “ Tables of Logarithms,” p. 56, and is as follows. The logarithms of a and b being given, itis ‘equired to find the power of the impossible . + ? « Mm dinomial a+ /“ —Jb* whose index is —; that > n s, to fnd(at V7 3)" by another impos- sible binomial; and thence the value of a+ y—D)" +(a—v —0)"; which is al- ways possible, whether a or 6 be the greater of the two. Solution. Put _ tang. z. Ther a m m (at y—b*)" = (a? +07)" x (cos. ea f/f — sin z). v1) nr Felice (a hf — Pha 7 Pe ~ m m (a*+5°)™" x 2cos. > z = (a X sec.z" X 2 COS. 7 aah _—_ m m z = (bxcos. 2.) ms 2 cos. — 2 where the first J or second of these two last expressions is to be used, according as z is an extreme or mean b. arc; or sather because - is not only the tan- a gent of z, but also of z + 360°, z + 720°, &c.; therefore the factor in the answer will have several values, vz. 2cos. 2; 2cos.”” (z + 360°); 2 cos. — n n wt (z + 720°), &e.; the number of which, if m and n be whole numbers, and the fraction — 7% be in its least terms, will be equal to the de- nominator n, otherwise it will be infinite. By Logarithms. Put log. b+10—log.a= log. tan. z. Then log. ( (tb Vf 2)" 5 Ny Be Fy " =< x (l.a+10— 1. cos.z) + 1.241. cos. ~ z n —l0=— x (lb + 10 —I. sin. z) +12 + 10; where the first or second oe —— ~ Mm l. cos. — 7 expression is to be used, according as z is an extreme or mean arc. Moreover by taking ‘ m m successively, 1. cos. —z; 1. cos. — (z + 360°); iL n 1. cos. = (e + 720°); &c. there will arise se- veral distinct answers to the question agree- ably to the remark above. Continued SuRDS, are expressions of the form VarvVa+t /a+t &e. the infinite sum of which is always determin- -able by means of a certain equation, and in some cases it is a real integral or rational quantity. Suppose, for example, the value of the above expression were required, put ae Va+Vat JS a + &e. By squaring, a matJ/at Yate. the latter part of which is equal to the ori- ginal surd, for being infinite, one term being taken away does not alter its value; we have, therefore, U SUR x* = @ +2, Or 2? —x = 4, whence eit v(i+a) which is therefore the literal value of the pro- posed surd; thus if a= 6, then VY 64764 V64¢Ko.=1+ v (LE 46)=3,0r—2. Again, let ats b is at Vid — be permease fat ke, by squaring, 2S a + / SMe PERNT A/a + &e. b whence x’ a+ = or x? —ax—=b an equation whence x may be determined. Suppose again, a= /atVb4Vat/b dt ke squaring, aia Wid aoeky, ABE Recor a” —a=VpaV/at/b + ke. squaring again, a*+#—2ax*+a>—b+ VW a +4/b + &e. whence «+—2ax*—x=b—da’* an equation which will furnish the value of 2. For more on this subject see Emerson’s and Euler’s ‘ Algebra.” SURFACE, in Geometry. See SUPERFICIES. SURSOLID, a term given by the early algebraists to what we now more commonly call a fifth power. Surso.ip Problem, that whose solution de- pends upon the higher geometry, vz. on the conic sections and other curves; this term, however, is now seldom or never made use of. SURVEYING, in a general sense, denotes the art of measuring the angular and linear distances of objects, whereby to delineate their several positions on paper, and to ascer- tain the superficial area or space between them. This is of two kinds, Land-surveying, and Marine-surveying; the former having generally in view the measure or content of certain tracts of land, and the latter the posi- tion of remarkable objects, as beacons, towers, shoals, coasts, &c. ‘Those extensive opera- tions which have in view the determination of the latitudes and longitudes of places, and the lengths of terrestrial arcs in different lati- tudes are also placed under the general term surveying. ; Of Land-surveying.—This consists of three distinct cases, viz. 1. The measuring of cer- tain lines, angles, &c. 2. Protracting or lay- ing the same down on paper, so as to form a map of the estate or country. 3, The com- SUR putation of the superficial content, as found by the preceding operations. * Various instruments are made use of for the purpose of taking the dimensions; the principal and most indespensable of which is the chain commonly called Gunter’s chai which is 22 yards in length, and divided into 100 links: 10 of these square chains, or 100000 square links is one acre, viz. Be 625 square links is 1 perch. 4 25000 square links, or 40 perches, 1 rood 100000 square links, or 4 roods, 1 acre. — This is for taking the linear dimensions, be- side which the surveyor must provide himself with 10 small arrows for marking the several lengths, and which are successively put down by the person leading the chain, and taken up by him who follows, and whose busi- ness it is to direct the survey. These with the chain are sufficient for surveying estates of considerable extent ; but it will frequently save a great deal of labour to be furnished with proper instruments for measuring angles, The most usual for this purpose are the cir cumferentor, theodolite, semicirele, plane table, &c. ; for a description of which, see the severat articles. ¥ To measure a Line on the Ground with the Chain. —Having provided a chain, with 10 sm wrrows, or rods, to stick one into the ground, as a mark, at the end of every chains’ length; two persons take hold of the chain, one at eae end of it, and all the 10 arrows are taken by one of them who goes foremost, and is called the leader, the other being called the follower for distinction sake. A picket, or station staff, being set up in the direction of the line to be measured, if there do not appear som marks naturally in that direction, the followe stands at the beginning of the line, holdin the ring at the end of the chain in his hand till it is stretched straight, and laid o held level, and the leader, directed by the follower waving his hand to the right or lef till he sees him exactly in a line with the mark or direction to be measured to; then’ both of them stretching the chain straight, and stooping and holding it level, the leader having the head of his arrows in the same hand by which he holds the end of the chain, he there sticks one of them down while he holds the chain stretched; this : done, he leaves the arrow in the ground asa mark for the follower to come to, and ad- vances another chain forward, being directe in his position by the follower standing at the arrow as before; as also by himself now, and at every succceding chain’s length, moving himself from side to side, till he brings th follower and the back mark into a line. Hay ing then stretched the chain, and stuck down an arrow as before, the follower takes up hi arrow, and they advance again in the samé manner another chain’s length: and thus they proceed till all the 10 arrows are employed, an are in the hands of the follower; and the leade without an arrow, is arrived at the end ut - ae ‘ : oop UR » eleventh chain. The follower then sends brings the 10 arrows to the leader, who ts one of them down at the end of his ain, and advances with it as before. And 1s the arrows are changed from one to the ver at every 10 chain’s length, till the ole is finished; the number of changes of arrows shows the number of tens, to which . follower adds the arrows he holds in his id, and the number of links of another in over to the mark or end of the line; so nere have been three changes of the arrows, , the follower holds six arrows, and the end the line cut off 45 links more, the whole zth of the line is set dewn in links thus, 0 take Angles and Bearingss.—Let B and C » the following figure), be two objects, or pickets set up perpendicular; and let it equired to take their bearings, or the angle aed between them at any station A. st. With the Plain Table.—The table being ered with a paper and fixed on its stand, it it at the station A, and fix a pin or a it of the Compasses in a proper part of the er, to represent the point A: close by the of this pin lay the fiducial edge of the *x, and turn it about, still touching the till one object B can be seen through the ts; then by the fiducial edge of the index valine. In the very same manner draw her line in the direction of the other ob- C, and it is done. 1. With the Theodolite, §c.— Direct the fixed s along one of the lines, as A B, by turn- the instrument about till you see the mark rough these sights; and there screw the ument fast ; then turn the moveable index it, till through its sights you see the other turn to the left, is placed, according as you turn to the right or left hand. These directions will be understood by a reference to the field-books and figures given in the following problems : 1. To measure a Triangular Field, ABC. A B = 1321 links AP = 794 links C PC = 826 links Having set up object staves at the three angles of the field A, B, and C, (unless there are trees or other objects that will answer that purpose) measure with the chain as above directed from A towards B, till you are ar- rived at the point P in the line AB, where a perpendicular would fall from the angle C, which point is to be found by the cross by getting the two objects A and B in a line with one pair of sights, and the angle C ina line with the other sights. Note down the distance A P (suppose 794 links), and proceed to measure the perpendicular C P, the length of which is also to be written down as above; then return to P, and measure to B, noting down the whole length of the line A B, and the field work is then completed ; from which the figure may be constructed, and the con- tents computed. Or otherwise, the three sides of the field may be measured by the chain, and hence the figure may be constructed, and its area computed by the rule proper for that purpose. See TRIANGLE. The same may be done by measuring two sides and an angle, or two angles and one side ; thus, Measure two sides, as AB, AC, and the angle A between them, which is sufficient to determine both the figure and area: or mea- sure one side, as AB, and the two adjacent angles A and B, from which data also the form and area may be determined. 2. To measure a Field contained under Four right Lines. B bt) A Cc D AC = 593 FB — 316 | AF = 362 AK= 212! DE = 240 SUR Here supposing the measurement to com- mence at A, we must measure along the dia- gonal AC, till we are arrived at E, where the perpendicular D EB falls. Note down the jlength of AE, and measure the perpendicular ED, the length of which is to be entered in the right-hand column opposite to A E; con- tinue to measure along the diagonal to I; write down the distance A I’, and opposite to it, on the left, the length of the perpendicular BF;; then return to F, and finish the measure- ment of the diagonal AC; from which di- mensions, it is obvious, both the figure and area may be determined. Or the same might be found by measuring the four sides and the diagonals ; or by measuring the angles and some of the sides, &c. asin the last case. Thus, measure the diagenal A C, and the: angles CAB, CAD, ACB, ACD; ormea- sure the four sides and any one of the angles, as BAD. In which cases the dimensions may be conceived to be as follows, viz. 1st Method. 2nd Method. AG ’= h91 AB = 486 CAB 3720 BC = 394 COA Pale CD = 410 A (eB rr72? 2" DA = 462 ACD = 54° 40’ | BAD = 78° 36 The above methods are those more com- monly practised ; but others, depending upon the particular form of the field, may be some- times advantageously employed. ‘Thus in the annexed figure the dimensions may be taken as follows, viz. ~ : 2 D A—s ee AB = 1122 QD =595|AQ= 745 | PC.= 352, AP =. 107) AP, AQ, and AB, and the two perpendi- culars PC and DQ. Ay 3. To survey a Field of any Number of Sides, as — A, B,C, D, Se. with the Chain and Cross. TIL ; rf 4: C we Re nme oO EEE ASE OH = OARS ROMEO © hy : ” mo . - one - ae a. - meen, a“. penerer ---* 4 » be subtracted, as is evident. i Measure on the longest side the distances ~ SUR PREP FD = 1040 i Fp = 576|pE= 160 OG= 240|Fo0 = 412 _ Cii= 880. Fi gD = 460|C q = 304 a) Ac = 1100 Ra n B= 360{/An = 820 he . Am = 270|mG = 260 Here having set up marks at the corne where they are necessary, walk over th ground, and consider how it can best be d vided into triangles and trapeziums ; and me snre them separately, as in the last two pri blems; and in this way it will be proper 1 divide the figure into as many trapeziu and as few triangles as possible. Thus tl preceding figure is divided into the two E peziums AGCB, GFED, and the triang GDC. cyt | Then in the first trapezium, beginning at, measure the diagonal AC, and the two pe pendiculars mG andnB; in the triangle GD measure the base C G, and the perpendical qD; and lastly, in the trapezium GFE measure the diagonal F D; and the two pi pendiculars 0G and pE, which completet field work; and from these dimensions, entered in the above field-book, the figt may be constructed and the area comput The same might have been done by meas ing ail the sides and diagonals, or by m suring the sides and angles, as in the p ceding propositions. | 4. Another Method is as follows.—Meas) a base line, either within or without the figu as also the perpendiculars let fall upon from the several angles. By this means” figure is divided into a number of trapeze and triangles, of which the parallel sides perpendicular to the base line, and the a these areas will evidently give the whole a of the field when the perpendiculars all within the figure; but if any of them- without, the area of the external parts m As an example of this method, let ther proposed the figure ABC D E, &e, a fF = 470| Af = 1000 t Ae = 600/eE= bt = 3201 AG poe | eC =. 461 Ao, = 440 Fs Ab — 315/dB — 3al + Here the operation is begun at A, and % ~~ t "Ba 1 de I SA - Cor TA? - LUO = Newtontan Telescope. Y / TT AVR HHA TDAH 45 im boat i, l I} AUGUUANAGTUNNEHU ANAT ASTRA Gonnaenacunpacnenanernanecuimeerett Ce mer mi APEUELADELPREDENTH ETE EE ——seer f Engraved by Sam! Lacet. \ ‘ London,Fab.Jani31.1614.by GS Robinson LaternosterRowkthe restor the Proprietors. , SUR nued along the line AG, as indicated in xe above field-book. We have only room to add one other ex- uple, with which we must conclude this ‘ticle. Suppose it were required to find the Content of the Irregular Field, A BC DE, 330 — EA 12—hh| 300—- EA 32 — gg) 200 = ig eer) 100 = Ef >| ©4—923° E S10) Se 400 = “DD Griz = ee 300 = Dd |52—dd 900 = De 144 = ec 100 = Db |384= bb >| Oo3—84°D 304°= "BD 150 = Ba |96=aC >] ©O2 — 98° B moO VA ©1 — 85° A Here the operations are supposed to com- ence at A, which is entered in the centre jlumn at bottom, with the angle subtended y B and £, and the length of AB is inserted amediately above it, which being the con- usion of that side, a line is drawn across the sntre column. Then is entered the second ation B, with the angle subtended by A ad D. In measuring up this line, the per- endicular aC is measured, the distance Ba atered in the centre celumn, and against it, i the right-hand column, the length of that erpendicular; this being taken to the right- and, a line is then again drawn across the entre column, and the same operation is re- eated at D, and in measuring D E the seve- il perpendiculars 6b, ec, &c. are measured, nd each placed in the right-hand column gainst its respective distance. The same ;also done in measuring from E to A, but 1 this case the perpendiculars or offsets being aeasured on the left, they are placed in the oft-hand column. This completes the field rork of the survey; after which the figure is rotracted or laid down from the dimensions iven, and the content of the land computed. If no instrament for measuring the angles i, B, D, &c. had been at hand, then it would lave been necessary to measure a diagonal ither from A to D, or from B to E, which vould obviously have answered the same surpose. »vSU Rh In the above field-book the letters are placed so as to correspond to the several lines in the figure; but the reader is doubtless aware, that this is only done for the sake of illustration, and that it is not practised in the actual operations in the field. To lay down the Work.—Yhis is performed by first measuring off the line AB, and by means of a protractor or scale of chords, the two angles at A and B; then measure off the two distances A E and BD, and join AE, and if then the angles at D and E measure the same as those observed in the field, the work is right, otherwise there is some mistake, which must be corrected before proceeding any farther. Lastly, set off the several per- pendiculars or offsets, and join the lines as in the figure. — To compute the Contents.—In a small single field, such as ishere supposed, the area is better found from the lines measured in the field, viz. first of the trapezium AB DE, then of the triangle B C D, and the offsets DocdeE, and lastly of the offsets EfghA, the latter of which, subtracted from the sum of the former, will be the measure required. The area of the trapezium A BODE is found by dividing it into two triangles, the areas ef which may be computed by the rule given for the several cases under the article TRIANGLE. But when the field or track of land is large, and very irregular, such computations become very laborious; in which cases, surveyors most commonly proceed as follows: A small bow of brass wire, or whale-bone, or other elastic matter, is strung with a fine horse-hair, which is applied to each of the curving or bent sides of the figure, so as to exclude as much from the figure on one side, as is included on the other, as near as the eye can judge, which after a little practice becomes exceedingly correct; then marking these positions with the point of a tracer, join them with fine black lines; and thus the irregular figure is reduced to a rectilinear one, and the contents may then be computed by dividing it into trapeziums, triangles, &c, Thus repeating the preceding figure. sence 7" Se ehens tin ° o%t - ° - - - e ~* e* ‘ . ° ° o* . .* - ° -* - Po - -* - Si Let the four lines B A, BI’, F G, GH, be drawn, so as to exclude the part CBe— DF e, the part Da = aGE and HOA = GEB&, so shall the trapezium H BF G, be equal to the irregular figure ABCDE; and the same method may be used in figures of a much more complicated form. Our preceding remarks have all been made with reference to the measurement of single fields, or small parcels of land. We shali now sUR add a few observations with respect to the surveying of a large estate. In this case when there are many fields the work cannot well be done by measuring the fields separately, and then putting them together, nor can it be done by taking all the boundaries that en- close it, as in either of these cases any small errors would be so multiplied as to render the whole result extremely doubtful. 1. Therefore before the survey is begun, walk over the estate two or three times, in order to get a perfect idea of it, till you can carry the form of it tolerably in your mind. And to help your memory, draw an eye- draught of it on paper, or, at least, of the prin- cipal parts of it, to guide you. q 2. Choose two or more eminences in the estate for your stations, from whence you can see all the principal parts of it: and let these stations be as far distant from one an- other as possible; as the fewer stations you have to command the whole, the more exact your work will be; and they will be fitter for your purpose, if these station lines be in or near the boundaries of the ground, and espe- cially if two or more lines proceed from one station. 3. Take angles between the stations, such as you think necessary, and measure the dis- tances from station to station, always in a right-line ; these things must be done, till you get as many angles and lines as are sufficient for determining all your points of station. And in measuring any of those station dis- tances, mark accurately where these lines meet with any hedges, ditches, roads, lanes, paths, rivulets, &c. and where any remark- able object is placed, by measuring its dis- tance from the station-line, and where a per- pendicular from it cuts that line; and always mind, in any of these observations, that you be in a right line, which you will know by taking backsight and foresight, along your station line. And thus as you go along any main Station-line take offsets to the ends of all hedges, and to any pond, house, mill, bridge, &c. omitting nothing that is remarkable. And all these things must be noted down; for these are your data, by which the places of © such objects are to be determined upon your plan. And be sure to set marks up at the intersections of all hedges with the station- line, that you may know where to measure from, when you come to survey these parti- cular fields, which must immediately be done, as soon as you ‘have measured that station- line, whilst they are fresh in memory. ‘In this way all your station-lines are to be mea- sured, and the situation of all places adjoin- ing to them determined, which is the first grand point to be obtained. 4. As to the inner parts of the estate they must be determined in like manner by new Station-lines, for after the main stations are determined, and every thing adjoining to them, then the estate must be subdivided into two or three parts by new station-lines; taking inner stations at proper places where you can i SUR ba have the best view: measure these station lines as you did the first, and all their intone tions’ with hedges, and all offsets to suc objects as appear. Then you may proces to survey the adjoining fields by taking angles that the sides make with the station line at the intersections, and measuring th distances to each corner from the intersee tions. For every station-line will be a bas to all the future operations, the situation 9 all parts being entirely dependant upon them and therefore they should be taken of a great a length as possible ; and it is best fo them to run along some of the hedges 0 boundaries of one or more fields, or to pas through some of their angles. All thing being determined for these stations, you mus take more inner ones, and so continue 4 divide and subdivide, till at last you comet single fields, repeating the same work for th inner stations as for the outer ones till all b done; and close the work as often as yo can, and in as few lines as possible. An that you may choose stations the most col veniently so as to cause the least labour, k the station-lines run as far as you can alon some hedges, and through as many corne) of the field and other remarkable points é you can: and take notice how one field lik by another, that you may not misplace the; in the draught. ‘\ 5. An estate may be so situated, that tk whole cannot be surveyed together, becany one part of the estate cannot be seen fro. another. In this case you may divide it in three or four parts, and survey the parts sep rately, as if they were lands belonging to di ferent persons, at and last join them togeth 6, As it is necessary to protract or lay do your work as you proceed in it, you must ha ascale ofadue length to doit by. To get suc a scale you must measure the whole length the estate in chains; then you must consid how many inches in length the map is to b and from these you will. know how mat chains you must have in an inch; then mal your scale, or choose one already made accor ingly. 7. The trees in every hedge-row must placed in their proper situation, which is so done by the plain table, but may be done | the eye without an instrument; and ben thus taken by guess in a rough draught, th will be exact enough, being only to look a except it be such as at any remarkable plac as at the ends of hedges, at stiles, gates, and these must be measured. But all t need not be done till the draught is finish And observe in all the hedges what side tl ditch is on, and consequently to whom fences belong. 8. When you have long stations you oug to have a good instrument to take angles wit! and the plain table may very properly be ma use of, to take the several small internal par and such as cannot be taken from the ma stations, as it is a very quick and ready i strument. we ; >) SUR “his is only one of many methods that are de use of in measuring large estates ;.but limits will not admit of entering farther mm the subject in this place; we shall there- s only observe, that in practising the above aciples the accuracy of the survey depends the correctness of the instruments used in ing the angles; and therefore” to avoid ' errors incident to such a multitude of Jes other methods have of late years been 'd by some few skilful surveyors ; the most ‘etical, expeditious, and correct of which ms to be the following. ; is was advised in the foregoing method, lin this, choose two more eminences as nd stations, and measure a principal base » from one station to the other, noting ry hedge, brook, or other remarkable ob- t as you pass by it, measuring also short pendicular lines to such bends of hedges may be near at hand. From the extre- jes of this base line, or from any convenient ts of the same, go off with other lines to 1e remarkable object situated towards the xs of the estate, without regarding the jles they make with the base line, or with :another; still remembering to note every lge, brook, or other object that you pass These lines, when laid down by inter- tions, will with the base line form a grand ngle upon the estate, several of which, if d be, being thus laid down, you may pro- dto form other smaller triangles and tra- ioids on the sides of the former, and so on il you finish with the enclosures indivi- lly. See a more particular account of this od in Dr. Hutton’s Course of Math. and “rocker’s Treatise on Surveying. "rigonometrical SURVEYING. In the pre- ‘ing part of this article our views have not ended beyond the surveying of fields, es- *s, &c. the purpose of which is principally scertain the exact quantity of land con- ed in the several fields, and their relative itions with regard to each other; and with d to the latter the utmost accuracy is ‘required, and therefore no account is taken the spherical figure of the earth, but the dle survey is considered as if carried on i an extended plane surface. But of late Ts surveys of a much more extensive na- » have -been carrying on, under the pa- lage of most of the governments of Europe, icularly those of France and England. tse have been undertaken principally for accomplishment of one or other of these 8ets, viz. 1. For finding the difference of situde between two noted meridians, as instance, between those passing through observatories of Greenwich and Paris. Che accurate determination of the geogra- val position of the principal places of a , With a view to give greater accuracy faps and charts. 3, Vor the determination he length of an arc of the meridian in dif- t latitudes, from which to determine the ? figure and magnitude of the earth. Vhen objects so important as these are to SUR be attained, it is manifest, that in order to insure the necessary degree of correctness in the results, the instruments employed, the operations performed, and the computations required, must each have the greatest pos- sible degree of accuracy. In the determination of distances of many miles, whether for the survey of a kingdom, or for the measurement of a degree of latitude or longitude, the whole distance between the two extreme points is not actually measured ; for this, on account of the inequalities of the earth’s surface, the interposition of mountains, rivers, &c. would be always very difficult, and frequently impossible. Buta line a few miles in length is very carefully measured on some plain, heath, or marsh, which is so nearly level, as to facilitate the measurement of an actual horizontal line, or rather an are of a great circle of the terrestrial sphere; and this line being assumed as the base of the opera- tions, certain hills, towers, &c. are selected; at which signals can be placed so as to be distinctly visible the one from the other; the lines joining those points constituting a double series of triangles, of which the assumed base forms the first side; the angles of these, that is, the angles made at each station or signal- staff by two other signal-staves, are very ac- curately measured by a theodolite, which is carried from one station to another; and thus the distance between the two extreme points may be found by the rules of trigonometry, in which, however, the spherical figure of the earth and other minute circumstances are to be introduced. This may be illustrated E on a small scale by the annexed figure, in which P A and E are supposed to be the twoextreme points, but too distant to be seen c the one from the other. Let AB be the base line, the exact length of which is found by absolute mea- surement; conceive also A C and D to be two towers or other eminences, the former being distinctly visible from A, B, and D, and the latter from B, C, and E; and suppose the angles ABC, BAC; CBD, BCD; DCE and CDE to have been accurately taken by the theodolite. Then it is obvious, that in the triangle A BC, the base AB, and the angles at A and B being given, the sides AC and BC may be fonnd by computation. Then considering BC as a base, and having the angles at B and C given, the two sides BD, DC may be found; and lastly, in the triangle DCE, the line DC, and the angles at D and C being given, the other two sides DE and CE may be determined; and hence, also, the whole distance A E. And in order to assure the greatest possible degree of accuracy, one of the latter lines, as DE, or EC, is again submitted to absolute measurement, the same as the first, which is called the base of verifi- cation; and if this length, as found by abso- SUR ute measurement, agree with the length as determined by calculation, it is a convincing proof of the accuracy of all the observations and calculations; and if not, the whole must be again repeated. We have here given only three triangles, but it is obvious in extensive surveys that a great many more become ne- cessary; the principle, however, is the same, whatever may be the number of them, and the calculations are performed in the same manner. Thus far the principles of operation are ex- tremely obvieus, because we have supposed the whole series of triangles as formed upon an extended horizontal plane, but this is evi- dently not correct, as instead of such a plane, they form a part of a spherical surface, and therefore the computations must be carried on upon the principles of spherical trigono- metry, or the base actual measured must be reduced to the chord of its are, in which case plane trigonometry may be employed; on both of which methods we propose to offer a few observations. Measurement of the base.—The first import- ant operation to be attended to in these sur- veys is, the correct measurement of a base, because on this depend the whole accuracy of the results. The French philosophers used. for this purpose, in their measurement for the determination of the length of the metre, rulers of platina, and of copper, forming metallic thermometers. The Swedish mathematicians Swanberg and Overbom, employed iron bars covered towards each extremity with plates of silver. The English, under the direction of General Roy, began their measurement with deal rods 20 feet long, but they were soon found not to be sufficiently accurate, and glass rods of the same length were employed, the quantity of whose expansions from any in- crease of temperature, was first found by ex- periment, and afterwards adjusted in the sur- vey by means of thermometrical observations. The base thus measured was found to be 2740408 feet, or about 5.19 miles. And several years afterwards the same base was remeasured by Colonel Mudge, with a steel chain 100 feet long, constructed by Ramsden, and jointed after the manner of the internal chain of a watch. This chain was always stretched to the same tension, supported on troughs laid horizontally, and allowances were made for changes in its length by reason of the variations in temperature, at the rate of -0075 of an inch for each degree of heat from 62° of Fahrenheit’s thermometer; the result of which measurement differed only 23 inches from General Roy’s, which, considering the whole distance 5:19 miles, was a remarkable confirmation of the accuracy of both mea- sures. After the important operation of the mea- surement of a base, which is to be the side of the first triangle, (which side is a spherical arc, if the measurement be conducted by spherical triangles; or a straight line, if by plane triangles of the chords of the spherical arcs), the angular distance between the ob- SUR i jects that mark the several stations must | observed, but as the objects are probably n situated in the same horizontal plane, nors the same spherical surface, the angle obsery is an oblique angle: but as the lines, &e, be computed, are supposed to be in the gt face of the earth, horizontal angles are } quired. , Therefore the oblique angle observed mi be reduced to an horizontal angle; thus, the two objects be A, B; O the observer, his zenith, a, b, two points on the surface the earth, (supposed spherical) the obsery oblique angle is AO B, measured by AB, a the horizontal angle, which is required, equal to AZ B. Z , > The most direct formula for this redue is as follows: ( SA. ZY" sin. ; son sin. }(AB+ZB—ZA).sin.3(AB4+ZA—4 sin. ZA.sin. ZB __ sin. (a + H—A). sin. (a +h—E ma cos. H. cos. A where AB =a, Aa=H, Bb=h. Or in logarithms, ) i ton ; 20 +log.sin.4 (a +H—; log.sin.4 (a +h—F log.cos.H—log.cos But notwithstanding this process of re tion in a single case is not laborious, ye; practical purposes, when many hundred | operations are to be performed, it is not ficiently concise; on which account, formule have been invented by Dela which by means of certain tables const for the purpose, simplify these operations, render them very practicable and simple. Connoissance des Temps, 1793. But whether we use the preceding method, or Delambre’s approximate me} all observed angles must be reduced to zontal angles, which are then spherical an and therefore the sum of the three oug exceed 180°. If we could determine, independent observation, the quantity by which the of the three angles ought to exceed 180 should then be able to judge of the acet of the observations, which it is desira do, since observations made on objec: tuated nearly in the horizon, are liable to SUR sertainty. Now, the theorem concerning area of a spherical triangle, given under t article, enables us to determine the autity of the excess ; for the area is equal the difference of the sum of the three angles 1 180°, and consequently, since the sides the spherical triangle, described on the h’s surface, are nearly rectilinear, they wy, with scarcely any error, be considered rectilinear in the computation of the area. This elegant application of the above rule Albert Girard’s was first made by General yy, in the Phil. Trans. for 1790, where he es the following rule, for computing what calls the spherical excess : viz. From the logarithm of the area of the triangle en as a plane one in feet, subtract the constant -arithm 9°3267737, and the remainder is the arithm of the excess above 180° am seconds arly. This rule was communicated to General 9y by Mr. Dalby, now Professor of Mathe- itics at the Military College at High Wy- mbe, but at that time engaged with the sneral in conducting the survey. Delambre s also given a simple method of determining 2 spherical excess in the Memoires of the stitute for 1806. By this means the observer may examine e aecuracy of his observations, and in some gree correct them when necessary ; after aich he may proceed to calculate the sides the triangles by the rules of spherical tri- mometry; but these rules, although they ast necessarily give the exact results, will -yery deficient in point of expedition, which an object of great importance where many mdred such operations are to be performed. egendre has, therefore, furnished us with the lowing rule, which combines sufficient ac- racy, with all the conciseness that can be ‘pected in such cases: viz. A spherical triangle being proposed, of which esides are very small with regard to the radius the sphere, of from each of its angles, one- ‘ird of the excess of the sum of its three angles rove 180° be subtracted, the angles so diminished ay be taken for the angles of a rectilinear langle, the sides of which are equal in length those of the proposed rectilinear triangle. Triangles solved by either of these methods, z. either by the direct process, or by Legen- ‘e’s approximation, will be spherical trian- les, their sides arcs of great circles, and the mputed arc of the meridian also an arc of great circle. But this is not absolutely ne- sssary; Delambre in France, and Colonel fudge in England, do not consider the line ? the meridian as a curve, but as formed of ie chords of curves; and they have accord- ily resolved their triangles, not as spherical, ut as rectilinear triangles, the sides of which re chords of ares. This method, therefore, requires theorems ad formule different from those we have een explaining, though in some measure de- ending upon similar considerations. It re- uires, for instance, the oblique angles to be SUR reduced, either instrumentally or by calcula- tion, to horizontal angles ; but these horizontal angles being, as above stated, spherical angles, must be reduced to the angles contained by the chords of the spherical sides of the tri- angle; that is, we must resolve the following problem. Having given the angle contained by two spherical arcs of a spherical triangle, to find the angle contained by the chords of those ares; Which is resolved by the following for- mulee : Let the two given spherical arcs be denoted by a and J, the spherical angle by C, the required angle of the chords by 4; then b Oat cep cos. § = cos. ~.cos. —.cos. C + sin. ~. SIN. =. 2 2 2 a 2 In this method, therefore, Legendre’s theo- rem is not employed, nor is the theorem re- lative to the spherical excess requisite; for we have instead of it, the following criterion of the accuracy of the observations, viz. The sum of the three angles, contained by the chords subtracted from the spherical angles, ought, if the observed angles be truly taken, to be equal to 180°. ‘The other part of the calculation in this method is, by the common rules of plane trigonometry ; and the are of the meridian, if that is proposed to be deter- mined, is composed of the sides of an irregular polygon inscribed in a circle. Hence then it appears, that in a trigonome-~ trical survey, we may calculate the ares and angles, according to the exact rules of spheri- cal trigonometry, as Boscovich has done; or, we may reduce the observed angles to the angles of the chord, and calculate by plane trigonometry, such reduced angles and chords as practised by Delambre and Mudge; or, we may by a slight transformation, give to spherical triangles the properties of plane tri- angles, and resolve them as such according to the method of Legendre. . The above is only a slight sketch of the different methods that have heen proposed for conducting these important operations, and in which, therefore, no account is taken of the spheroidical figure of the earth, which, how- ever, in real practice, must be taken into the account; this, however, would have carried us far beyond our limits, and we must, there- fore, refer the reader, who is desirous of the most correct information, to Colonel Mudge’s “ Account of the Trigonometrical Survey ;” Mechain and Delambre, “ Base du Systéme Metrique Decimal;’’ Swanberg, “ Exposition des Operations faites en Lapponie ;” Puissant, “ Geodesie,” and “ Traité de Topographie d’Arpentage,” &c. SurveYING Wheel. See PERAMBULATOR. SURVIVORSHIP, the doctrine of rever- sionary payments depending upon certain con- tingencies, or contingent circumstances, as the extinction of some certain life or lives, or the survivorship of some particular life beyond SYR or after other lives. This is one of the most important parts of the doctrine of life an- nuities, which our limits will not admit detail- ing; we can, therefore, only refer the reader to our article LIFE Annuities, and the works of the several authors there enumerated. SUSPENSION, in Mechanics, the balancing or supporting any body from a point above it, which point is called the point or centre of SUSPENSION. SUTTON’S Quadrant, the name given to a quadrant invented by Mr. Sutton, but which is now seldom made use of, having been su- perseded by others of a more complete con- struction. SWAN, in Astronomy. See Cyenus. SYMETRICAL Equations, in Algebra, are those equations consisting of two or more un- known quantities, similarly involved, so that no difference will arise by writing these quan- tities reciprocally for each other. SYNCHRONISM, the being or happening of several events at the same time, as isochro- msm denotes twe or more things being per- formed in the same time. SYNODIC, or SynopicaL Month, is the period or interval of time in which the moon passes from one conjunction with the sun to another, and is, therefore, precisely one com- plete lunation, the mean duration of which is 294 125 44’ 2-8”, Synopic Revolution of a Planet, is the time between two conjunctions, or two oppositions of the same planet and the sun. SYNTHESIS, denotes a method of com- position, as opposed to analysis, or resolution. In the synthesis, or synthetic method, we pursue the truth by reasoning drawn from principles before established, or assumed, and propositions formerly demonstrated ; thus pro- ceeding by a regular process till we come to the conclusion; and is hence also called the direct method, and composition, in opposition to analysis or resolution. : Such is the method in Euclid’s Elements, and most demonstrations of the ancient ma- thematicians, which proceed from definitions and axioms, to the demonstration of theorems, problems, &c. and from those theorems to the demonstration of others. See ANALYSIS. SYNTHETICAL Method, the method by rks or composition, or the direct me- thod. SYPHON. See Sipnon. SYRINGE, in Hydraulics, a small simple machine, serving first to imbibe or suck in a quantity of water, or other fluid, and then to expel the same with violence in a small jet. The syringe is, in fact, a small single suck- ing pump without a valve, the water ascend- ing in it on the same principle as is explained under the article Pump. It consists, like the pump, of a small cylinder, with an embolus or sucker, moving up and down in it by means of a handle, and fitting it very close within. At the lower end is either a small hole, or a smaller tube fixed to it than the body of the instrument through which the fluid or the i, S.¥.5 water is drawn up, and expelled out aga Thus, the embolus being first pushed ele down, introduce the lower end of the pj into the fluid, then draw the sucker u ), the handle, and the fluid will immediate f low, so as to fill the whole tube of the syrin and will remain there, even when the pipe taken out of the fluid; but by thrusting fi ward the embolus, it will drive the wal before it; and being partly impeded by t smallness of the hole, or pipe, it will hence expelled in a smart jet, and to the grea distance, as the sucker is pushed down W the greater force, or the greater velocity. — SYSTEM, in a general sense, denotes assemblage or concatenation of principlesa conclusions: or the whole of any doct H the several parts of which are united togeth and follow or depend on each other. Aj; system of astronomy, a system of the plane a system of philosophy, a system of motion, 8 System, in Astronomy, denotes an hy thesis, or a supposition of a certain order a arrangement of the several parts of the w verse; by which astronomers explain all phenomena or appearances of the heave bodies, their motions, changes, &c. This is more peculiarly called the syste of the world, and sometimes the solar syste System and hypothesis have much the sai signification, unless, perhaps, hypothesis bi more particular system, and system a m¢ general hypothesis. Some late authors, : deed, make another distinction: an hypothe: say they, is a mere supposition or fictir founded rather on imagination than reasc while a system is built on the firmest grout and raised by the severest rules; it is found on astronomical observations and physi causes, and confirmed by geometrical deme strations. The most celebrated systems of the wo! are, the Ptolomaic, the Copernican or Pyti gorean, and the Tychonic; the principles each of which is as follows: Ptolomaic System, is so called from 14 celebrated astronomer Ptolomy. In this s tem the earth is supposed at rest in the cen of the universe, while the heavens are cc sidered as revolving about it, from east west, and carrying along with them all 1 heavenly bodies, the stars and planets, in4 space of 24 hours. The principal assertors of this system a Aristotle, Hipparchus, Ptolomy, and many the ancient philosophers, and. by philosophi in general, for a great number of ages, Jong adhered to in many universities and otl tre of the world, and the earth and plan all revolving round him in their several orbi See the article Copernican System. ; Solar or Planetary System, is usually ec TAB | to narrower bounds; the stars, by their ense distance, and the little relation they ito bear to us, being accounted no part . It is highly probable that each fixed is itself a sun, and the centre of a par- ar system, surrounded with a certain: ber of planets, comets, &c. which in dif- it periods, and at different distances, per- their courses round their respective suns. ie planetary system described under the le COPERNICAN, is the most ancient in the 1. It was first introduced into Greece aly by Pythagoras; from whom it was d the Pythagorean System. It was fol- 1 by Plato, Archimedes, &e. but it was inder the reign of the Peripatetic philo- y; till happily retrieved about the year by Copernicus. chonic SYSTEM, was taught by the ce- ted Tycho Brahe; who was born in It supposes that the earth is fixed in sxentre of the universe or firmament of and that all the stars and planets re- round the earth in 24 hours; but it dif- ‘om the Ptolomaic System, as it not only 3 a menstrual motion to the moon round arth, and that of the satellites about er and Saturn in their proper periods, ‘makes the sun to be the centre of the of the primary planets Mercury, Venus, | Jupiter, &c. in which they are carried the sun in their respective years, as the ryolyes round the earth in a solar year ; IE these planets, together with the sun, \pposed to revolve round the earth in 24 _ This hypothesis was so embarrassed yverplexed that very few persons em- lit; it was afterwards altered by Lon- tanus and others, who allowed the di- motion of the earth on its own axis, but LE, in Mathematies, is a series or system ubers methodically arranged, for the teady calculation of various problems uities, Astronomy, Trigonometry, &c. onomical TABLES, are computations of tions, positions, and other phenomena heavenly bodies. The oldest of these ‘ tables are those given by Ptolomy in nagest, but which are now of little or _as they no longer agree with the mo- fthe heavens. ‘These tables were cor- and republished in 1252 by Alphonso, * Castile, and were long held in high ion. See ALPHONSINE Tables. ? the revival of the sciences in Europe, LAB denied its annual motion round the sun. This hypothesis, partly true and partly false, is called the Semi-Tychonic System. _ SYZYGY, aterm equally used for the con- junction and opposition of a planet with the sun. On the phenomena and circumstances of the syzygies, a great part of the lunar theory depends. It is shown in physical astronomy, that the force which diminishes the gravity of the moon in the syzygies, is double that which increases it in the quadratures; so that in the syzygies the gravity of the moon is diminished by a part which is to the whole gravity, as 1 to 89°36; for in the quadratures, the addition of gravity is to the whole gravity, as 1 to 178°73. In the syzygies, the disturbing force is directly as the distance of the moon from the earth, and inversely as the cube of the distance of the earth from the sun. And at the SyZy- gies, the gravity of the moon towards the earth receding from its centre, is more dimi- nished than according to the inverse ratio of the square of the distance from that centre. Hence in the moon’s motion from the syzygies to the quadratures the gravity of the moon towards the earth is continually increased, and the moon is continually retarded in her motion; but in the moon’s motion from the quadratures to the syzygies, her gravity is continually diminished, and the motion in her orbit is accelerated. Farther, in the syzygies, the moon’s orbit about the earth is more convex than in the quadratures ; for which reason she is less distant from the earth at the former than at the latter. Also, when the moon is in this position, her apses go backward, or retro- grade, and her nodes move in antecedentia. 4% and particularly since the restoration of the true system of astronomy by Copernicus, the number of astronomical tables have been con- tinually increasing, and repeated observations, and a very refined calculus, have now brought them to an astonishing degree of perfection. We cannot, within the limits of this article, enter into a minute detail of the several im- provements that have been made in this de- partment of astronomy, at the same time we shall endeavour to enumerate a few of the most important particulars, as also a catalogue of those tables, which, from their accuracy and arrangement, are esteemed the most va- luable. TAB Copernicus, after thirty years of observation and calculation, published a new set of As- tronomical Tables in 1548, in his celebrated work ‘ De Revolutionibus orbium celestium,” which were republished in 1566, 1593, and 1617. These were again corrected and im- proved by the observations of other astrono- mers, and were certainly the most correct of any that appeared before the publication of the “ Rudolphine Tables,” the work of Tycho and Kepler; these were published at Lintz in Upper Austria in 1627, containing 115 pages of tables, and 125 of rules and precepts. See RupovpHineE Tables. The same were afterwards, viz. in 1650, changed into another form by Maria Cunitia, whose astronomical tables, comprehending the effect of Kepler’s physical hypothesis, are very easy, satisfying all the phenomena with- out the use of logarithms, and with little or no trouble of calculation; and Mercator made a like attempt in his Astronomical Institution, published in 1676. Beside these we may enumerate the fol- lowing, viz. Tabule motuum celestium, Xc.; Auct. Chr. Severino Longomontano, 1624, 1640, folio. Christiani Reinharti, Tabule Astronomice, 1630, 4to. Philippi Lansbergii Tabule motuum, &c. 1632, folio. Ismaelis Bullialdi Tabule Philolaice, &c. 1645. This great and important work contained the most perfect observations, methods, and tables, that had ever appeared prior to its publication. I. B. Riccioli S. 1. Tabula Nove Astron. &c. 1665, folio. There are here 102 tables, forming a com- plete collection of all those that astronomers have need of in their computations. Tabule Lodoicaee de doctrina eclipsium, &c. 1656. Astronomical Tables for the Meridian of London, by John Newton, 1657, 4to. Tabularum Astronomicarum, &c.; Ph. De La Hire, 1687, 4to. 1702, 1725, 1727, 1730. These tables were long regarded as the best, being superior to all those which had preceded them. Astronomical Tables for computing the Places of the Sun, Moon, Planets, and Comets, by Edmund Halley ; London, 1749, 4to.; 1752, 8vo.; beside two editions in French. ‘These tables were not excelled till the publication of Lalande’s tables in 1771. Besides the tables above enumerated, we might mention many others, as La Caille’s Tables of the Sun; Mayer’s Tables of the Moon, published by the Board of Longitude. Tables of the same luminary have also been computed by Charles Mason, which are very accurate, being used by the computors of the Nautical Almanack. Many other sets of as- tronomical tables have also been published by various astronomers and academies; and divers sets of them may be found in the se- TAN a veral treatises of astronomy and navigaty of which, however, those of Lalande, given his Astronomy, are esteemed the most per the same, with some additions, are given Professor Vince in his large work on the sa subject. Fora more particular detail of several astronomical tables, see Monty Hist. des Math. vol. iv. p. 302, et seq. , TABLES of Annuities. See ANNUITIES Tases of Logarithms, Sines, Tangents, See LOGARITHMS. a TACQUET (AnpREW), a laborious ; voluminous writer in mathematics. His we were collected and printed at Antwerp ini large folio volume in 1669. Tacquet die 1620. x e TACTION, in Geometry, the same as J gency, or touching. See TANGENT. 4 TAMUAZ, in Chronology, the fourth me of the jewish ecclesiastical year, answeri part of our months of June and July. TANGENCIES, Problem of. This neral problem in geometry furnishes the ject of one of the twelve treatises deser by Pappus, in the preface to the 7th boo his Mathematical Collections. In Dr. ley’s translation of Pappus the problem is enumerated: “‘ E punctis rectis et cir quibuscunque tribus positione datis, cire ducere per singula date puncta, qui, si possit, contingat etium datas lineas.” T naturally subdivided into ten distinct p sitions, which, if a point be represented | a line by (1), and a circle by (0), may bes very briefly according to the several da the following order: (..1), (.11), (110), (100), (..0), (.00), (000), (...), (111). The treatise on tangencies was restor Vieta, under the title of Apollonius G and many of his deficiencies were suppli Marinus Ghetaldus. These have been _ lated, with the addition of a suppleme Mr. Lawson, and a farther addition 0 mat’s Treatise on Spherical Tangencies, Leslie has given, in his Geometry, sol to all except the 5th, 7th, and 8th, of th ceding enumeration. TANGENT (from tango, to touch), i metry, is a line that touches a circle 0 curve without cutting it; thus the line a tangent to the circle BD at the p which point is called the point of contae A tangent touches a curve only ina trical point, but at that point the diree the curve is the same as that of the tan t 4 In the circle a tangent is perpendi the radius at the point of contact, that TAN erpendicular to BC: also, AB*=—AE x ’; or AB is a mean proportional between tand AF. \ll tangents drawn from the same point to ch the same circle are equal to each other. d, therefore, if a number of tangents be wn to different points of the circle, and an al length BA be set off upon each of them n the points of contact, the locus of all the nts A will be a circle having the same tre C. ‘is also a curious perty of tangents, t if there be any xe circles D, E, and tangents be wn common to h two, the three ats ofintersection, B, C, will be in and the same ight line. : draw a Tangent to a Circle through a given Point B. . When the given point is in the circum- nee of the circle join B and the centre C, draw AB perpendicular to BC, and AB be the tangent required. See fig. 1. . When the point s not in the cir- iference,join Band ‘centre C, and de- be a semicircle up- ‘BC, cutting the mn circlein E; draw }, so shall BE be tangent required. ‘or the method of drawing tangents to the se and the other conic sections, see EL- se, Hypernora, and Parasota: for the yerties, magnitudes, &c. of the angles of tact formed by different curves with their sents, see Angle of Contact ; and for the hod of drawing tangents to curves in ge- il, see Method of TANGENTS. ANGENT, in T’rigono- ‘y, isaright line touch- one extremity of an and limited between | point and its inter- jon with the secant ing through the other ‘emity: thus AGisa ‘ent to the arc BA, or 1e arc ABD; and AH tangent to the are AI, or to the are AIDK. also call these lines the tangents of the les subtended by those arcs. lence the tangents in the Ist and 3d quad- $ are positive, in the 2d and 4th negative, is, they are drawn in an opposite direction he former. Also, the tangent of 0° is , or nothing; but the tangent of 90° is iite, so likewise the tangent of 180° is , and of 270° infinite, &c. he tangent of an arc, and the tangent of E KN Ee TAN its supplement, are equal, but affected with contrary signs; the one being positive, and the other negative. Co-TANGENT of an angle, is the tangent of the complement of that angle; the letters co being only a contraction of the word com- plement. Some of the principal trigonometrical formu- le relating to the tangents and co-tangents of angles are as below; wz. si . an , ee bt in. @ —Vv(l COS.”a) cota /(1—sin.’a) COS, a SeC.a@ __Sin.a@.cos.a = 7 (sec.2a—1)= nnn Cosec.a cot.a __ Sin. @ __ COS. @.SeC. a as OC gE ole CAt: dike 1 1—sin.*a cos. a Col 4.2 paey (Pre PMte) LF Weve0s. a tan.a sin. a / (i cos.*a) 1 —_— v (sec.274— 1) Ke. &e. being the reciprocals of the former. All the above formule are immediately de- ducible from the preceding figure ; the follow- ing relating to the tangent and co-tangents of the sums and difference of arcs, as also of the multiples and sub-multiples of arcs, we have extracted from Cagnoli’s Treatise of Trigo- nometry; to which work we must refer the reader, who is desirous of following their in- vestigations. See also Bonnycastle’s Trigo- nometry. tan. a + tan. b l. Tan. @ + 6) = 1 = tan. a.tan.b 2, Cot. (a + b) pee cot. a.cot.b=1 ~ cot..b+cot.a cos.(a rb)—cos.(a+b) cos.(a+b) + cos.(ar b) cos.(a+b) +cos.(a b) cos.(a wb) —cos.(a +5) 3. Tan. a. tan. 6 = 4; Cotas. con, bc a = tan. a 2 tan. a tani 3 tan. a — tan. 3a 5. Tan. 6. Tan. 2a = Talo ¢ — fiosa ee 4 tan. a — 4 tan. 3a z = Sena 1 — 6 tan. 7a + tan. *a __ 5tan. a — 10 tan. 3a + tan. %a Dah aN 1 — 10 tan. 2a + 5 tan. +a &e. &c. 10. Cot. a—cota 2 —_— TibGot 2 eee 2 cot. a __ cot. 3a — 3 cot.a 12: Cot. 3 Cb. ais Se cotaee Pe i S cot.4a—6cot.”a+1 Ig. Cot. 4°. = ict. 2aue 4. 00t. a &e. Ke. sin.a@ =__ 1 — cos. a 14. Tan. 4 a= ———_—_—_— = . 4 1 + cos. a sin. @ MOY A bork be a 1 + cos. @ TAN 15. Co-tan. a= sin. a. my 1 tr COs. & 1 — cos. @ si. @ a rd 1 + cos. a a 1 — cos. a 16. Arc a = tan. a — § tan. 3@ 4 2 tan. § a — } tan. 7a, &c.; which, when the are is 45°, becomes are 45° —1—2 +i—F+si— Ke. radius being unity, to which we may add the following curious property of tangents, viz. 17. The sum of the tangents of any three ares which are together equal to 360°, is equal to the product of the same, divided by radius squared. We might have carried these formule to a much greater extent; but it is presumed that what are given include the most useful cases. Artificial TANGENTS, a term used by some authors to denote the logarithmic tangents, in contradistinction to natural tangents, or the absolute tangents to any given radius. Figure of the TANGENTS. See FIGuRE. Line of Tancents. See GunreEr’s Scale. Sub-TANGENT, the line lying under the tan- gent; or that part of the axis included be- tween the tangent and ordinate at the point of contact, as IB in the annexed figure. Cc gts Method of TANGENTS, is a method of draw- ing tangents to any algebraical curve, or of determining the magnitude of the tangent and sub-tangent, the equation to the curve being given. The method of tangents is nearly related to that of maxima et minima, and the same au- thors, who in the early state of algebra at- tempted one of those cases, never failed of touching also on the other. Hence we have the methods of Des Cartes, Fermat, Roberval, Hudde, &c. We have al- ready explained, under the article Maxima et Minima, the several methods of these authors relating to that subject, and as their method of tangents differ in no respect from this, we shall not repeat them in this place, but merely explain the principle which led to so intimate a connection between the two problems. Des Cartes’ Method of Tangents.—It has been shown under the article above referred to, that Des Cartes’ method of maxima and minima depended upon his making two roots of his equation equal to each other, and the same principle led him also to the solution of his problem of tangents. Let us conceive, for example, a curve A BB, described on an axis AC; and from any point in this axis, C, as a centre, let there be de- scribed a circle, which shall cut the curve at least in two points, as B, ; from which draw two ordinates, and which will therefore be necessarily common both to the circle and A B = TAN curve; let us now imagine the radius of t circle to decrease, while its centre rema fixed; and it is obvious, that thus the t points ofintersection will approach each o and finally coincide, in which case the ef will touch the curve in the point E, and tangent at that point will be common to b ( and perpendicular to the radius of the at that point. Thus the problem of deter ing the tangent to a curve, is reduced to fi ing the position of the perpendicular to curve, drawn from any point in its axis. In order to which Des Cartes sought j general manner the. points of intersectio the curve, made by a circle described wit given radius from a given point in the a| He thus arrived at an equation, which in} case of two intersections ought to con two unequal roots, expressing the distane| the two ordinates from the vertex of the cui But when the two points of intersection ¥ united in one, as in the case of the ei touching the curve ; then the two roots off equation were necessarily equal to each 0 His object, therefore, was in the equation obtained, and of which the co-efficients ¥ indeterminate to give them such values, } the two roots should be equal; for which pose he compared the proposed equation } an equation of the same degree having equal roots; and hence by equating the efficients, obtained the value of those il first equation. ? in order to illustrate this, let A B& be a 7 bola,and Bbacircle. Make AC =a, AD} the radius CB = r, then CD=a—za. | since the ordinate BD belongs to the ci we have y* = 7? — CD* = r* — (a — rv? — a” + 2ux — x”; but the same ordi belonging also to the parabola, we have. the known property of that curve y* =] being the parameter: therefore r?— a? +4 — x* = px, or x* —(p — 2a) x + (a? = 0; which being an equation of the se degree, must necessarily have two roo values of a, answering to the two abs¢ AD, Ad, for we should arrive at the conclusion if our equation had been ded with reference to the point b; and itis ob that these roots depend entirely upon t lation of the co-efficients p — 2a and a?- or upon the ratio of the quantities a, p, a to each other; and consequently such y may be given to these quantities, that thi values of x may be equal. a Now to find this ratio, Des Cartes fo an equation of the second degree having TAN 2al roots, as 27 — 2ex+e°=0; viz. (a — e) —e) = 0, and comparing this with that nd above, he obtained the equation « — a CD, = 4p, = 0; which shows that in the abola, the subnor mal is equal to half the ameter; whence it also follows, that the -tangent is equal to double the absciss, ich is the known property of the curve. Resides this method of tangents, Des Cartes posed another a little different from it in etice, though the same in principle, which ; as follows: he conceived aright line to aivé about a fixed point in the axis of the ve produced, which at first should cut the in a certain number of points; but by revolution, these points of intersections roaching each other would finally coincide, ‘thus the revolving line become a tangent ze curve. For this purpose he also first ob- ed the general equation, which he equated 1 one having two equal roots, and thus armined the several relations of bis inde- ainate co-eflicicnts exactly as in the case ve given. ermat’s Method of Peideten th —It will be id hy comparing the above method of tan- bin Des Cartes’, with that of his maxima unima, that the two ultimately depend a the same principle, viz. of making two sof the equation equal to each other; and soincidence of f'ermat’s methods for these ly treats of them as distinct cases, but 'simmediately for the solution of the case gents to that of his maxima et minima. rder, s says this author, that a line may be gent fo a curve, as for example, to the bola ABS at the point B; it is evident ‘every ordinate, except BC, will meet tangent without the curve, as ine, Thus |! 1D ae « c atio of Be* to De’, which is the same as of C D? to c D’, will be less tham that of to cb’, or than that of CA toc A; but if uppose this ratio is the same, and the uce Ce to vanish, the points B and 6 will ide, and we shali have an equation, which, ad in the same manner as in his method is et minimis, will give the ratio of oCA. to the method of tangents proposed by le, Huygens, &c. they differ from those above; ‘only in the same manner as in methods of maxima et minima; to which e we therelore refer the reader, and shall ceed to explain. rrow’s Method of Tangents—It is ob- from what is said above, and what has ated under the article Maxima et M- that both the methods of tangents and problems is still more obvious, in fact he TAN of maxima et minima, as well by Des Cartes as by f’ermat, and particularly those of the latter, were very nearly related to the present fluxional w ay of considering the same sub- jects; but with regard to tangents, a’ still nearer approach was made by Dr. Baywovel This accurate geometer considered the little triangle formed by the diflerence of the two ordinates, their distance, and the infinitely small part of the curve, as similar to that which is formed by the ordinate, the tangent, and sub-tangent. He then sought by the equation of the curve the ratio of the two sides ba, Ba, eee Bilt of the triangle ea when the difference of the ordinates is infinitely little; then said, as ba is to aB, so is the ordinate to the sub-tan- gent required. In the case of the parabola, for example, whose equation is y7 = px; supposing Pp the increase of the ordinate = e, and the corres- ponding increase ab of y =a; then the equa- tion for the ordinate pb becomes y + 2ayt+a —pex + pe, both sides » es Fas remains 2Zay + a* pe so, a being itself infinitely small, its square *may be entirely neglected, and there re~ anilts 2ay — pe; therefore a: e::p:2y; but a — ba, ande = Ba, alsoy = Vpx; there- fore, from the proposition stated above, viz. ab: aB :: ordin. : subtan, we have, as :2./pxi: /px: 2x, the sub-tangent required. The Method of Langents according to the Doctrine of Fluxions. —[iet the curve ACZ,be described by the extremity of the variable ordinate BC, which moves parallel to itself. It is required to draw a tangent to the curve at any point C, subtracting from) 2 , eS A B D Let TCV be the required tangent, draw any other ordinate Dr, and produce it to s; draw also C E parallel to BD; join C7, and produce it to ¢ and W; produce also C E to any point G, and draw Gmvn parallel to Ks. Now, let Drs be supposed to move up to BC, then by the motion of 7, the line WrCt will revolve about C, and when r coincides with C, it ceases to cut the curve between C and Z, and it does not cut it between C and A; for to cut CA, Cé must fall below C'T, and con- xX X TAN sequently CW must be above CV, or » must have passed s, which it cannot have done, as y has been continually approaching to s, and only now coincides with it; therefore, when r comes to C, the line Wt ceasing to cut the curve, must become a tangent, and conse- quently W Ct will then coincide with VCT. Now while the absciss A B by increasing be- comes AD, the ordinate BC becomes Dr; hence the increment of the ordinate BC is Er; and by similar triangles the increment CE of the absciss : the cotemporary incre- ment of Er of the ordinate :: CG: to Gm. But when r arrives at C, WC coincides with VC, and consequently m must coincide with n; hence the limiting ratio of the incre- ment Er of the ordinate is to the increment CE of the absciss, as Gn to GC; or as Es to CE, taking D Es in any situation before its coincidence with BC. Therefore if CE be taken to represent the fluxion of the absciss, Eis will represent the cotemporary fluxion of the ordinate. Put AB = x, BC =y, then BD = CE=za, and Es = y; also, as BC is parallel to Es, and T B to CK, the angle TCB = CsE, and CTB = sC#; consequently the triangles TBC and CEs are similar; hence 7 (Eis): x (CE) 7 (BC); BT — a therefore set off a. ZX, join T and C, and TC will be the tangent to the curve at C. If y decrease while x increase, then y be- comes negative, which shows that 'T lies on the other side of B. Exam. 1. Let the curve AC be any para- bola whose general equation is ax = y", to draw a tangent at the point C. Take the fluxion on both sides, and we eh y tt) ; ; x have ax =ny"—'y, hence ~ = ; there- y yx fore BT = yy" ny” =i. cone, because +c. x. y a a Ifn = 2 it becomes the common conical pa- rabola, and B'T = 22. Exam. 2. To draw a tangent to a circle, of which the equation is y* =(2axz—2”), ory = / (2ax — 2*), a being the radius. Poggi (@—2)% . ther Sait Here y = eat)" therefore BT = Yeu. (a—2x)x — ~ pe == ees ) /(2ax — x*) Zax — 2 a— x Exam. 3. To draw a tangent to an ellipse, 2 of which the equation is y” = i (2ax — x*), ory = 3 Vv (2ax — x*) a and b, representing the semi-axes. Here y = & (a—x)z Vi@am one)’ therefore BT oMegnde ck bs) het oS 6 eye aloe y amr td den mr) ° & J/(2Ux —z°) ‘es Ss TAR _ 2ax—x” = ~~, the same result as above, whieh a douna x bah? « tf shows that the value of the sub-tangent ofa | ellipse is totally independent of the conjugat, axis; and consequently all ellipses describe on the same transverse axis, and being al cut by the same right ordinate, will have thej tangents at those points termimated in On common point, or, which is the same, the sub-tangents will be equal to each other. — | Exam. 4. To draw a tangent to the cisso 3 e . * tad K of Diocles, whose equation is y° = a pa . 82°a (a—a2)+ x32, a Here ee a ia whene : < x= 2y(a—- 2)", therefore BT —22; y 3 a4x* —22x3 | 2x (a— x) ey OP psy Pol | Inverse Method of Tancents. This is’ reverse of the foregoing, and consists in findi the nature of the curve that has a given §1 tangent. a The method of solution is to put the giv) sub-tangent equal to the general expres: q —Y x : ; Y* which serves for every kind of curve; th 2 > } the equation being reduced and the flue taken, give the curve sought. . b. Thus, for example, to find the curve wh aye 2 + y! subtangent is = ae Here — 4%, her a 4 4 2yy = ax, and the fluent of this gives y* = the equation to the parabola; which is, the fore, the curve required. 4 To find the curve whose subtangen y* ib Qa x. y Here making —~—— = 7, we have | 2a—2x q ai yy = x (2a— x); the fluent of which gi * = ax — 2x’, the equation to the cit TANTALUS’S Cup, in Hydraulics, a sip so adapted to a cup, that the short leg be in the cup, the long leg may go down thro the bottom of it. 4 The bended siphon is called Tantalus’st from the resemblance of the experiment fi with an image, representing Tantalus im fable, fixed up in the middle of the cup wi siphon concealed in his body, beginnin the bottom of his feet, and ascending ta upper part of his breast, where it makes a” and descends through the other leg on W he stands, and thence down through the! tom of the cup, where it runs out, and ca the water to subside in the cup; as soon rises to the height of the siphon, or to the of the image, the water will begin to through the siphon concealed in the fi till the cup is emptied in the manner exp) ed under the article SIPHON. 7 TARE, in Commerce, is an allowance by merchants to their purchasers, for, weight of the chest, bag, &c. in whick goods are packed. . Tank and TrReET, a rule in arithmetic TAY puting what allowance ought: o be made apy quantity of goods, the quantity on a ain weight being given. ARTAGLIA, or 'TARTALEA (NICHOLAS), putable Ttalian mathematician, was born 3rescia in Italy about the conelusion of 14th century. Tartaglia was the author iat method of solution for cubic equations monly ascribed to Cardan, on which sub- several letters passed between them; an ract of one of which is given under the le IRREDUCIBLE Case. urtaglia was author of the following works: Nova Scientia Inventa; Venice, 4to. . This is a treatise on the theory and tice of gunnery, and the first work of the which was ever published on projectiles, e flight and paths of balls and shells. Quesiti et Inventioni diverse; Venice, + This is a collection of questions and "ers on various subjects. Trattato di Numeri et Misure; folio, and 1560. This is a universal treatise rithmetic, algebra, geometry, mensura- &e. \URUS, the Bull, one of the twelve signs 'e zodiac, denoted by the character &. SIGNS and CONSTELLATION. \UTOCHRONE, a term sometimes ap- to a cycloid with reference to its theore- roperty of isochronism. +YLOR (Dr. Brook), a learned English ematician, was born at Edmonton, Au- 8th, 1685, and died in London, Decem- th, 1731, in the 47th year of his age. Taylor was author of several tracts on 2matics, philosophy, and music, most of i were inserted in different volumes of hil. Trans. Beside which he was author ) complete treatises, viz. Methodus Incrementorum, published in , and 2. Principles of Linear Perspective, ‘8vo. which was much augmented; and lished, under the title of New Principles near Perspective, in 1719, 1749; and ar edition has been lately published, viz. 2. These works may be both considered inals, and do honour to the talents of learned author; for some particulars of mer, see INCREMENTS. ‘Lor’s Theorem, in the higher mathe- , is a most elegant and fertile formula, by Dr. Brook Taylor, in cor. 2, prop. 7, of his Method of Increments. It is in ‘tas follows: If x and z be any two va- Zz antities, the relation of which is given, ile x by flowing unifor aii is increased Zz at 1.2.5 “i which the values of z, Z, Bi are to be ined from the given equation. the convenience of demonstration the n may be thus expressed: nd x be cotemporaneous values of two ies any how related, and ~ and x = of x, cotemporaneous increments, of wis uniformly generated ; then will will be increased by z es bP ° mR. etzemz+5 bisaie S rtrs 20 2. sa Dassen &e. when iis series terminates or converges. Demonstration. Letx Te, UP Qos EPH be cotemporane- wl” -».>e Ous values of x zz r 9 an ee ‘ct and z. Let also a, a’, a", a”, &e. b, v, oN &e. be differences of the ce, c’, &e. respective orders. Then, by the theorem for differences, he n. as D, nn— 1)\(n—2) 4 +na+ ——— Hy Te Se, COG emp isicd Ww ere 2 is the number of successive values from x tox + 2, or from z toz + 2. Now, if m be increased sine limite, any ‘assigned number of terms of this quantity approaches to the same number of terms in the series, z+na meee nb aot TRS + &e. as its limit: b * a . or, because n = —, toits equal z + — x + =) Et 34 x M s Fe . ap aoe &e. But when 7 is so increased the limiting ratio of a to x, or the limiting ratio of the increments of z and xis the ratio of the fluxions of z and zx, and it follows therefore that when n is increased sine Limite, the limit- : a z ing value of —= +. Also, for the same rea- x x Bas mses Ah ; son = = =,-- = 5, ke. Whence the wv od x li b a! a z 1 = im. val. of — = —— =a , because == au Vax x x e é : h = of ~ = ——- = —,, because — = = Ey a] ara~ 3 we xt &e, &e. Whence the limiting value of z-+na +s n(n — 1) (mn — 2) ) gle a | creased sine limite is z + 4+ e¢ + &c. when x is in- z z og = &e. 1 ne 1.2 F And because when the former series termi- nates its value is z + 7; and when it con- verges its limit is also z + ~; therefore z + z —2z + ; + . + &c. when the series ter- minates or converges, When it does not con- verge nothing can be asserted of it, because we cannot reason concerning a limit which does not exist. The above demonstration is by Dr. Brink- ley. Other demonstrations may be seen at p. 25, Caicul Differential, &c. par Lacroix, and a syuthetical one in the works of Frisi; but that given by L’Huilier in his “ Princi- piorum Calculi,” &c. is considered the most complete. Taylor’s theorem may be more generall expressed: for if z be a quantity compose of two or more independent quantities z, y, v, x xk 2 TEB Ke. then while 2, y, v, &e. by flowing uni- formly, become x + ry + y,v + 4%, “&e. Z will become z + 5 + Ke. See the same also in another form under INCREMENTS, art. 4. For the use of this theorem in finding fluxions per saltum, in approximating to the roots of equations, &e. see an ingenious paper by Dr. Brinkley in the seventh volume of the Trans- actions of the Irish Academy. And for an ex- planation of the cases in which the theorem is defective, and for the determination of its limits, see Francoeur, Mathematiques Pures, tom. li. p 243; 254. TEBET, or Thevet, the fourth month of the civil year of the Hebrews, answering to part of our months, December and January. TEETH, in Mechanics, are certain projec- tions on the extreme parts of wheels, and by means of which motion is communicated to the different parts of a machine. Emerson in his Mechanics, prop. 25, treats of the theory of teeth, vzz. of their form, action, &e. maintaining that they ought to have the figure of epicycloids, for properly working one within the other, as was originally proposed by De la Hire, who affirmed that the pressure would be uniform if the teeth were formed into epicycloids ; and Camus, in his Course of Mathematics, has pursued the same principle, and applied it to the various cases that are likely to arise in practice- The construction, however, is subject to a limitation; on which account a second. method has been proposed, which secures the perfect uniformity of action without any such limitation. This method consists in making both teeth portions of circles. Thus, let AFLF, K E68, be the wheels to which the teeth are to be accommo- dated: the acting face ¢ GCH of the tooth a must have the form of.the curve traced by the extremity H of the flexible line FaH, as it is unwrapped from the circumference; and in like manner, the acting face of the tooth 6 must be formed by the unwrapping of a thread from the cireumference of the circle KIO. ‘The line FCE drawn to touch both circles will cut the surfaces of the two teeth in C, the peint where they touch each other; the faces of both teeth will always touch each other at a point in the common tangent to both circles, and the force arising from their mutual pressure will act in the direction of the circumferences of the wheels at Eand l. This form, by allowing the teeth to act on each other through the whole extent of the line FC EF, will admit of several tecth to be acting at the same time, and thus by dividing the pressure amongst several teeth, will diminish its pressure upon any one of them, and therefore diminish the cause of the ventions, published 1663. TEL indentations they unavoidably make | each other. Consequently, a consider; number of teeth thus formed, acting at ¢ ‘anse the communication of motion extremely smooth and regular. TELEGRAPH (from Tye, listance, yoxGw, I write), is a name very properly gi to an instrament, by means of which infor tion may be conveyed with the greatest pedition, from almost any distance. bY It has been said that the ancients wer possession of certain methods of conye intelligence to considerable distances, some ‘parts ofa Greek play seems to co H¥, id hovers the ancients might bean attain this respect, it was certainly lost to the derns, and, therefore, the present telegt may be considered as totally a modern tion, of which the first hint is given bj Marquis of Worcester in his Century of The next sketch we have on this sw was communicated to the Royal Socie Dr. Hooke, in 1684. ha In this discourse he asserts the possil of conveying intelligence from one ple another at the distance of 30, 40, 100, 12€ miles, “in as short a time almost as @ can write what he would have sent.” which characters a at one station be rendered plain and distinguishable é others. He directs, “ first, for the s at if they be far distant, it will be necessary they should be high, and lie exposed sky, that there be no higher hill, or part earth beyond them, that may hinder thi tinctness of the characters that are to ¢ dark, the sky beyond them appearing : by which means also the thick and va air near the ground will be passed oy avoided.” Next, the height of the st is advantageous, upon the account of 1 fractions or inflections of the air.” in choosing of these stations, care m taken, as near as may be, that there be that interposes between ‘them, that is; high enough to touch the vy cabig ray ; in such cases, the refraction of the air: hill will be very apt to disturb the cle pearance of the object.” “ The next tl be considered is, what telescopes will cessary for such stations.” ‘ One of telescopes must be fixed at each extret tion, and two of them in each inter one ; that a man for each glass, sitting an ing through them, may plainly discove € is done i in the next adjoining station, at his pen write down on paper the cha there exposed in their due order; there therefore, to be two persons at each e station, and three at each intermediate that, at the same time, intelligence conveyed forwards and backwards.| there must be certain times agreed on the correspondents are to. expect; " there must be set at the top of the * i. * TEL sorrespondents, for acting that day; if the be appointed, pendulum clocks may ad- the moment of expectation and observ- ' “ Next, there must be a convenient ratus of characters, whereby to com- ieate any thing with great ease, distinct- ,and secresy. And there must be either characters, or night characters.” The ‘characters “ may all be made of three deals:” the night characters “ may be 2with links, or other lights, disposed in rtain order.” The doctor invented 24 Te characters, each constituted of right , for the letters of the alphabet; and se- | single characters, made up of semicir- for whole sentences. He recommended three very long masts or poles should be td vertically, and joined at top by one g horizontal beam; that a large screen Id be placed at one of the upper corners is frame, behind which all the deal board acters should hang, and by the help of er cords should quickly be drawn for- s to be exposed, and then drawn back i behind the screen. By these means, the doctor, ‘ all things may be made so enient that the same character may be ‘at Paris, within a minute after it hath ‘exposed at London, and the like in pro- on for greater distances; and that the weters may be exposed so quick after one ier, that a composer shall not much ex- ithe exposer in swiftness.” Among the of this contrivance, the inventor specifies = “ The first is for cities or towns be- d; and the second for ships upon the in both which cases it may be practised great certainty, security, and expedition.” whole of Dr. Hooke’s paper was publish- 'Derham’s collection of his Experiments Ibservations; from which it appears that ad brought the telegraph to a state of reater maturity and perfection than M. aton’s, who attempted the same. thing t the year 1702; and, indeed, toa state ttle inferior to several which have been ised during the last twenty years. Was not, however, till the French revo- ithat the telegraph was applied to useful ses. Whether M. Chappe, who is said ve invented the telegraph first tised by french about the end of 1793, knew any of Hooke’s or of Amonton’s invention t,it is impossible to say; but his tele- | Was constructed on principles nearly w. The manner of using this telegraph as follows: At the first station, which in the roof of the palace of the Louvre at , M. Chappe, the inventor, reecived in 1g from the Committee of Public Wel- he words to be sent to Lisle, near which rench army at that time was. An up- post was erected on the Louvre, at the ‘which were two transverse arms, move- Mall directions by a single piece of me- $m, and with inconceivable rapidity. He ted a number of positions for these arms, i stood as signs for the letters of the norning, the hour appointed by either of DEL alphabet; and these, for the greater celerity aud simplicity, he reduced in number as much as possible. The grammarian will easily con- ceive that sixteen signs may amply supply all the Ictters of the. alphabet, since some letters may be omitted not only without detriment but with advantage. These signs, as they were arbitrary, could be changed every week ; so that the sign of B for one day, might be the sign of M the next; and it was only necessary that the persons at the éxtremities should know the key. The intermediate operators were only instructed generally in these six- teen signals; which were so distinct, so marked, so different the one from the other, that they were easily remembered. The construction of the machine was such, that. each signal was uniformly given in pre- cisely the same manner at all times: it did not depend on the operator’s manual skill; and the position of the arm could never, for any one signal, be a degree higher or a degree lower, its movement being regulated mecha- nically. M. Chappe having received at the Louvre the sentence to be conveyed, gave a known signal to the second station, which was at Mont Martre, to prepare. At each station there was a watch tower, where tele- scopes were fixed, and the person on watch gave the signal of preparation which he had received, and this communicated successively through all the line, which brought them all into a state of readiness. The person at Mont Marire then received, fetter by letter, the sentence from the Louvre, which he repeated with his own machine; and this was again repeated from the next height, with incon- ceivable rapidity, to the final station at Lisle. Various experiments were in consequence tried upon telegraphs in this country; and one was soon after set up by government, in a chain of stations from the Admiralty-office to the sea-coast. It consists of six octagon boards, eaeh of which is poised upon an axis in a frame, in such a manner that it can be either placed vertically, so as to appear with its full size to the observer at the nearest sta-_ tion, or it becomes invisible to him by being placed horizontally, so that the narrow edge . alone is exposed, which narrow edge is from a distance invisible. Six boards make thirty- six changes, by the most plain and simple mode of working ; and they will make many more if more were necessary: but as the real superiority of the telegraph over all other modes of making signals consists in its making letters, we do not think that more changes than the letters of the alphabet, and the arith- | metical figures, are necessary:. but, on the contrary, that those who work the telegraphs should avoid communicating by words or signs agreed upon to express sentences; for that is the sure method never to become expert at sending unexpected intelligence accurately. It has been objected to it, that its form is too clumsy to admit of its being raised to any con- siderable height above the building on which it stands; and that it cannot be made to change its direction, and consequently cannot TEL be seen but from one particular point ; and in consequence several other telegraphs have been proposed to remedy these defects, and others to which the instrument is still liable. The dial-plate of a clock would make an ex- cellent telegraph, as it might exhibit one hun- dred and forty-four signs, so as to be visible at a great distance. A telegraph on this prin- ciple, with only six divisions instead of twelve, would be simple and cheap, and might be raised twenty or thirty feet high above the building without any difficulty: it might be supported on one post, and therefore turn round; and the contrast of colours would always be the same. ‘The description of the two telegraphs in- vented by Captain Pasley of the royal en- gineers, may be seen in 115th and 116th num- bers of the Philosophical Magazine. TELESCOPE (from ana, far off, and cxorey, to look at or contemplate), an optical in- strument employed in viewing distant objects. The invention of the telescope is one of the most important acquisitions that the sciences, perhaps, ever acquired. At the same time that it unfolds to us’the wonders of the hea- vens, it enables us to make such observations as furnish the most certain data for astrono- mical and nautical calculations, by which we are enabled to trace the laws and principles of the planetary motions, and to employ them with certainty in the most important sciences of navigation, geodesia, &c. To whom we are indebted for the discovery of the powers of this noble instrument, is not positively known. According to Wolfius, John Baptista Porta was the first who made a telescope, as he mferred from a passage in the “Magia Naturalis” of that author, pub- lished in 1560; but this passage is too ob- scure to entitle him to the unqualified honour of this invention. Thirty years afterwards, vz. in 1590, a telescope, 16 inches long, was made and pre- sented to Prince Maurice of Nassau, by a spectacle-maker of Middleburg, but whose name is stated differently by different authors, some saying it was Lippersheim, and others Jansen, or, as Wolfius writes it, Hansen. The telescopes, however, that were made before Galileo undertook their construction, were but toys to those which were formed by this celebrated philosopher, these being well calculated for astronomical observations. It is said that Galileo being at Venice, was told of a sort of optic-glass, made in Holland, which brought distant objects nearer; upon which, setting himself to think how it should be, he ground two picces of glass into a form as well as he could, and fitted them to the two ends of an organ-pipe; and with these he showed at once all the wonders of the inven- tion to the Venetians, on the top of the tower of St. Mark. The same author adds, that from this time Galileo devoted himself wholly to the improving and perfecting the telescope ; and that he hence almost deserved all the honour usually done him of being reputed the inventor of the instrument, and of its being PELE Galileo hip from him called Galileo’s tube. self, in his Nuncius. Sidereus, published : 1610, acknowledges that he first heard of t] instrument from a German; and that hej Paris; he himself discovered the constrneti by considering the nature of refraction. F adds, in his “ Saggiatore,” that he was_ Venice when he heard of the effects of Prin struction; that the first night after his retu to Padua he solved the problem, and his instrument the next day, and soon af presented it to the Doze of Venice, who honour of his grand invention, gave him} ducal letters, which settled him for life in | lectureship at Padua, and doubled his sala) which then became treble of what any of) predecessors had enjoyed before. And tf Galileo may be considered as the invento the telescope, though not the first inventor, F. Mabillon, indeed, relates, in his Tray through Italy, that in a monastery of his @ order, he saw a manuscript copy of the wo of Commestor, written by one Conradus, ¥ lived in the 13th century; in the third pi of which was seen a portrait of Ptolo viewing the stars through a tube of four joi or draws; but he does not say that the ht had glasses init. Indeed it is more probal that such tubes were then used for no ot purpose but to defend and direct the sight to render it more distinct, by singling out particular object looked at, and shutting all the foreign rays reflected from others wh proximity might have rendered the image - precise. And this conjecture is verified experience ; for we have often observed} without a tube, by only looking through’ hand, or even the fingers, or a pin-hole i paper, the objects appear more’clear and! tinct than otherwise. Be this as it may, it is certain that the} tical principles, upon which telescopes founded, are contained in Euclid, and ¥ well known to the ancient geometrici and it has been for want of attention to th that the world was so long without that mirable invention; as doubtless there many others lying hid in the same prinejj only waiting for reflection, or accident bring them forth. ; TELEscopes are either refracting or ref ing; the former consist of different lei through which the objects are seen by’ refracted by them to the eye; and the la of speeula, from which the rays are refle and passed to the eye. The lens, or g turned to the object, is called the ob glass; and that next the eye, the eye-g! and when the telescope consists of more two lenses, all but that next the object called eye-glasses. . ‘Lhe principal effects of telescopes dey upon this maxim, “ that objects appear la in proportion to the angles which they | tend at the eye; and the effect is the sa! TEL iether the pencils of rays, by which objects a visible to us, come directly from the ob- sts themselves, or from any place nearer to e eye, where they may have been united, as to form an image of the object, because ey issue again from those points in certain -ections, in the same manner as they did pm the corresponding points in the object emselves. In fact, therefore, all that is ‘ected by a telescope, is first to make such image of ‘a distant object, by means of a 4s or mirror, and then to give the eye some sistance for viewing that image as near as ‘ssible; so that the angle, which it shall ‘btend at the eye, may be very large com- red with the angle which the object itself uld subtend in the same situation. This is ne by means of an eye-glass, which so re- ts the pencils of rays, as that they may erwards be brought to their several foci by 'e natural humours of the eye. But if the e had been so formed as to be able to see e image, with sufficient distinctness, at the me distance without an eye-glass, it would pear to him as much magnified as it does another person who makes use of a glass >that purpose, though he would not, in all ses, have so large a field of view. I | : ! 1. This will appear very plain from the pre- ding figure, in which AB is the object emit- ig the several pencils of rays Aed, Bed, &c. |tsupposed to be at so great a distance from e object-glass ed, that the rays of the same neil may be considered as parallel to each her; they are therefore supposed to be col- sted into their respective foci at the points and p, situated at the focal distance of the ject-glass cd. Here they form an image /and crossing each other proceed diverging the eye-glass hg; which being placed at own focal distance from the points m and the rays of each pencil, after passing through ut glass, will become parallel among them- ves; but the pencils themselves will con- e considerably with respect to one an- 4er, even so as to cross at e, very little far- sr from the glass gh than its focus; be- se, when they entered the glass, their axes *re almost parallel, as coming through the ject-glass at the point k, to whose distance @ breadth of the eye-glass in a long tele- ope bears very small proportion. So that e place of the eye will be nearly at the focal stance of the eye-glass, and the rays of each spective pencil being parallel among them- tves, and their axes crossing each other in larger angle than they would do if the ob- »t were to be seen by the naked eye, vision li be distinct, and the object will appear ignified. The power of magnifying in this telescope EN, Ce Eon Se Although no image be actually formed by the foci of the pencil without the eye; yet if, by the help of an eye-glass, the pencils of rays shall enter the pupil, just as they would have done from any place without the eye, the visual angle will be the same as if an image had been actually formed in that place. Telescopes are of several kinds, distinguish- ed by the number and form of their lenses, or glasses, and denominated from their parti- cular uses, &c. Such are the terrestrial, or land telescope; the celestial, or astronomical tele- scope; to which may be added, the Galilean, or Dutch telescope, the reflecting telescope, the achromatic telescope, &c. We shall proceed to describe some of these in order to illustrate the leading principles. 1. The refracting Telescope. This telescope differs from the microscope only in this, that the object is placed at so great a distance from it, that the rays of the same pencil flowing from thence, may be considered as falling parallel to one another upon the object-glass, and therefore the image made by that glass is looked upon as coinci- dent with its focus or parallel rays. “SER o. is as the focal length of the object-glass to the focal length of the eye-glass. Dem. In order to prove this, we may con- sider the angle AAB as that under which the object would be seen by the naked eye ; for, in considering the distance of the object, the length of the telescope may be omitted, as bearing no proportion to it. Now the angle under which the object is seen by means of the telescope is geh, which is to the other AkB, or its equal g kh, as the distance from the centre of the object-glass to that of the eye-glass. ‘The angle, therefore, under which an object appears to an eye assisted by a “ telescope of this kind, is to that under which it would be seen without it, as the focal length of the object-glass to the focal length of the eye-glass. It is evident from the figure, that the vi- sible area or space which can be seen at one view when we look through this telescope, depends on the breadth of the eye-glass, and not of the object-glass ; for if the eye-glass be too small to receive the rays gm, ph, the ex- tremities of the object could not have been seen at all: a larger breadth of the object- glass conduces only to the rendering each puint of the image more luminous by receiy- ing a larger pencil of rays from each point of the object. ‘age It is in this telescope as in the compound microscope, where we see, when we look through it, not the object itself, but only ar a image of itatC ED: now that image being inverted with respect to the object, as it is, because the axes of the pencils that flow from the object cross each other at k, objects seen through a telescope of this kind necessarily appear inverted. This is a circumstance not at all regarded AB is the object sending out the several pencils A ed, Bed, &c, which passing through the object-glass cd, are collected into their respective foci in CD, where they form an inverted image. From hence they proceed to the first eye-glass ef, whose focus being ut J, the rays of each pencil are rendered pa- rallel among themselves, and their axes, which were nearly parallel before, are made to converge and cross each other: the second eye-glass gh, being so placed that its focus shall fall upon m, renders the axes of the pencils which diverge from thence parallel, and causes the rays of each, which were parallel among themselves, to meet again at its focus EF on the cther side, where they form a ‘second image, inverted with respect to the former, but erect with respect to the object. Now this image being seen by the eye at A - oe -—-" oon wows j TE CL a a Le ae ee eee eS ee fer) Teas BA Gee eet Meh aeenatn= anew 772 AB is an object sending forth the pencils of rays ght, kim, &c. which, alter passing through the object-glass ed, tend towards e Ef (where we will suppose the focus of it to Be), in order to form an inverted image there as. before ; butin their way to it, are made to pass through the concave glass ne, so placed that its focus may fall upon E, and consequently the rays of the several pencils which were converging towards those respective fecal points e, I, f, will be rendered paralicl among themselves; but the axes of those pencils crossing each other at F, and diverging from thence, will be rei.dered more diverging, as represented in the figure. Now these rays entering the pupil of an eye, will forma large and distinct image ab upon the retina, which will be inverted with respect io the object, because the axis of the pencils cross in . The object of course wili be seen erect, and the angle under which it vill appear will be equal to that which the lin s ab, 61’, produced back through the eye- gla:s, form at I’. It is evident, that the less the pupil of the eye is, the less is the visible area seen through a telescope of this kind; for a less pupil would exclude such pencils as proceed from TEL by astronomers; but for viewing objects uy the earth, itis convenient that the teleseo should represent them in their natural p ture; to which use the telescope with th eye-glasses, as represented,is peculiarly ada ed, and the progress of the rays through from the object to the eye is as follows: ab through the eye-glass ih, affords a di representation of the object, aud under sane angle that the first image C D wor have appeared, had the eye been placed al supposing the eye-glasses to be of equal ce vexity; and therefore the object is se equally magnified in this as in the fort telescope ; that is, as the focal distance of 4 object-glass to that of any one of the € glasses, and appears erect. 4 If a telescope exceeds 20 feet, it is of use in viewing objects upon the surface. the earth; for if it magnifies above 90 or times, as those of that length usually do, 4 vapours which continually float near the ear in great plenty, will be so magnified as. render vision obscure. % 2. 'The Galilean telescope with the conea eye-glass is constructed as follows : f the extremities of the object A B, as is e dent from the figure. This is an incom nience that renders this telescope unfit” many uses, and is only to be remedied) the telescope with the convex eye-glass where the rays which form the extreme pa of the image are brought together in order enter the pupil of the eye, as explained abo It is apparent also, that the nearer the ¢ is placed to the eyc-glass of this telesco: the larger is the area seen through if;_ being placed close to the glass, as in 4 figure, it admits rays that come from A a B, the extremities of the object, which could not if it was placed farther offs The degree of magnifying in this teleset is in the same proportion with that in | oiher, viz. as the foeal distance of the obje glass is to the focal distance of the eye-gla Por there is no other difference but @ viz. that as the extreme pencils in that te scope were made to converge and form) angle geh, fig. 1; or ink, fig. 2; these) now made to diverge and form the angle a, fig. 3; which angles, if the concave glass} one has an equal refractive power with | convex one in the other, will be equal ; 4 TEL refore each kind will exhibit the object ified in the same degree. here is a defect in all these kinds of tele- pes, not to be remedied in a single lens by “means whatever, which was thought only rise from. hence, viz. that spherical glasses not collect rays to one and the same point. tit was happily discovered by Sir Lsaae wion, that the imperfection of this sort of scope, so far as it arises from the spherical nof the glasses, bears almost no propor- 1 to that which is owing to the different angibility of light. This diversity in the action of rays is about a 28th part of the dle; so that the object-glass of a telescope ot collect the rays which flow from any point in the object into less than the cir- uw space whose diameter is about the 56th of the breadth of the glass Joitwithstanding this imperfection, a diop- lal telescope may be made to magnify in | given degree, provided it be of sufficient th; for the greater the focal distance of object-glass is, the less may be the pro- fion which the focal distance of the eye- is may bear to that of the object-glass, hout rendering the image obscure. Thus, object-glass, whose focal distance is about feet, will admit of an eye-glass whose ul distance shall be little more than an 1, and consequently will magnify almost ‘imes; but an object-glass of 40 feet focus ‘admit of an eye-glass of only four inches is, and will therefore magnify 120 times ; an object-glass of 100 fect focus will jit of an eye-glass of little more than six ies focus, and will therefore magnify al- it 200 times. he reason of this disproportion in their wal degrees of magnifying is to be ex- ned in the following manner. Since the ineter of the spaces, into which rays flow- from the several points of an object are ected, are as the breadth of the object- s, itis evident that the degree of confused- Sin the image is as the breadth of that 8; for the degree of confusedness will be as the diameters or breadths of those $, and not as the spaces themselves. ¥ the focal length of the eye-glass, that is, ower of magnifying, must be as that de- , for if it exceeds it, it will render the é et ab, ab, be two red rays of the sun’s {falling parallel on the first convex lens ec. sing there was no other lens present TES confusedness sensible, and therefore it imeast> be as the breadth or diameter of the objeci-— glass. The diameter of the object-glass, whicls is as the square root of its aperture or magni- tude, must he as ihe square root of the power of magnifying in the telescope, for unless the aperture itself be as the power of magnifying, the image will want light; the square root of the power of magnifying will be as the square root of the focal distance of the object-glass and therefore the focal distance of the eye- glass must be only as ihe square root of that of the object-glass. So that in making use of an object-glass of a longer focus, suppose than one that is given, you are not obliged to apply an eye-glass of a proportionably. longer focus than what would suit the given object- glass, but such a one only whose focal distance shali be to the focal distance of that which will suit the given object-glass, as the square’ root of the focal length of the object-glass you make use of, is to the square root of the focal length of the given one. And this is the reason that longer telescopes are capable of magnifying in a greater degree than shorter ones, without rendering the object confused or coloured. 3. But the inconveniency of very long tele- scopes is so great, that different attempts haye been made to remove it. Of these the most successful have been by Dollond and: Blair and the general principles upon which these eminent opticians proceeded have been men- tioned in our article ACHROMATIC. It may be sufficient to observe, in addition to what has been already said, that the object-glasses of Blair’s telescopes are composed of three dis- tinct lenses, two convex and one concave of which the concave one is placed in the mid- die. ‘The two convex ones are made of Lon- don crown-glass, and the middle one of white flint glass; and they are all ground to spheres of different radii, according to the refractive powers of the different kinds of glass and the intended focal distance of the object-vlass of the telescope. According to Boscovich, the focal distance of the parallel rays for the con- cave lens is one-half, and for the convex olass one-third of the combined focus. When put together they refract the rays in the following manner : but that one, they would then be converged into the lines be, 6e, and at last meet in the focus g. Let the lines gh, gh, represent two TEL ‘violet rays falling on the surface of the lens. "These are also refracted, and will meet ina focus; but as they have a greater degree of refrangibility than the red rays, they must of consequence converge more by the same power of refraction in the glass, and meet sooner in a focus, suppose at r. Let now the concave lens dd be placed in such a manner as to intercept all the rays before they come to their focus. Were this lens made of the same materials, and ground to the same radius with the convex one, it would have the same power to cause the rays to diverge that the former had to make them converge. In this case the red rays would become parallel, and move on in the line 00, 00; but the concave lens being made of flint glass, and upon a shorter radius, has a greater refractive power, and therefore they diverge a little after they come out of it; and if no third lens was inter- posed, they would proceed diverging in the lines opt, opt; but by the interposition of the third lens ovo, they are again made to converge, and meet in a focus somewhat more distant than the former, as at x. By the con- eave lens the violet rays are also refracted, Thus, let ab, ab, be two rays of red light falling upon the convex lens c, and refracted into the focus q; let also gh, gh, be two violet rays converged into a focus at7; it is not necessary, in order to their convergence into a2 common focus at x, that the concave lens should make them diverge: it is suffi- cient if the glass has a power of dispersing the violet rays somewhat more than the red ones: and many kinds of glass have this power of dispersing some kinds of rays, without a very great power of refraction. It is better, how- ever, to have the object-glass composed of three lenses; because there is then another correction of the aberration by means of the third lens; and it might be impossible to find two lenses, the errors of which would exactly correct each other. It is also easy to see, that the effect may be the same whether the eoncave glass is a portion of the same sphere Let AB represent an object-glass com- posed of three lenses as above described, and converging the rays 1, 2,3,4, &c. to a very distant focus. By means of the inter- TEL . and made to diverge; but having a gre; degree of refrangibility, the same power refraction makes them diverge somewi more than the red ones; and thus, if no th lens was interposed, they would proceed, such lines as dmn, Imn. Now, as the ¢ ferently coloured rays fall upon the third | with different degrees of divergence, it is pli that the same power of refraction in that | will operate upon them in such a manner to bring them altogether to a focus very nea at the same point. The red rays, it is require the greatest power of refraction bring them to a focus; but they fall upon lens with the least degree of divergence. 4] violet rays, though they require the k power of refraction, yet have the greatest gree of divergence ; and thus all meet. gether at the point 2, or very nearly so. But though we have hitherto supposed, refraction of the concave lens to be gre than that of the convex ones, it is easy to, how the errors occasioned by the first may be corrected by it, though it should h even a less power of refraction than the ¢ vex one. with the others or not; the effect depenc upon a combination of certain circumstan of which there is an infinite variety. By means of this correction of the ef arising from the different refrangibility of rays of light, it is possible to shorten dioj telescopes considerably, and yet leave t equal magnifying powers. ‘The reason of is, that the errors arising from the ob glass being removed, those which are 0 sioned by the eye-glass are inconsidera| for the error is always in proportion to length of the focus in any glass; and in| long telescopes it becomes exceedingly ¢ being no less than 4,th of the whole ; b glasses of a few inches focus it beed trifling. Refracting telescopes, which g the name of Dollond’s, are therefore now structed in the following manner: posed lens C D, however, they are conve to one much nearer, where an image 0 object is formed. The rays diverging } thence fall upon another lens EF, wher TEL acils are rendered parallel, and an eye ced near that lens would see the object ified and very distinct. To enlarge the gnifying power still more, however, the icils thus become parallel are made to fall on another at GH; by which they are again de to converge to a distant focus; but, ‘ng intercepted by the lens IK, they are de to mect at a nearer one z; whence di- ing to LM, they are again rendered pa- lel, and the eye at N sces the object very tinctly. from an inspection of the figure it is evi- at, that Dollond’s telescope, thus construct- is in fact two telescopes combined to- ther; the first ending with the lens EF, ithe second with LM. In the first we do : perceive the object itself, but the image of ormed at the focus; and in the second we eive only the image of that image formed z. Nevertheless, such telescopes are ex- ‘dingly distinct, and represent objects so arly as to be preferred in viewing terres- il things, even to reflectors themselves. e latter, indeed, have greatly the advan- ein their powers of magnifying, but they much deficient in point of light. Much re light is lost by reflection than by re- st unavoidably suffer two reflections, a lat deal of it is lost; nor is this loss coun- balanced by the greater aperture which se telescopes will bear, which enables m to receive a greater quantity of light nthe refracting ones. The metals of re- iting telescopes also are very much subject tarnish, and require much more dexterity slean them than the glasses of refractors, ich makes them more troublesome and ex- sive, though for making discoveries in the astial regions, they are undoubtedly the y proper instruments which have been 1erto constructed. II. The reflecting Telescope. The inconveniences arising from the great sth of refracting telescopes, before Dol- W’s discovery, are sufficiently obvious ; and se, torether with the difficulties occasioned the different refrangibility of light, induced Isaac Newton to turn his attention to the ject of reflection, and endeavour to realize ideas of himself and others concerning possibility of constructing telescopes upon t principle. The instrument which he trived is represented in Plate XIII. fig. 1, sre A BCD isa large tube, open at AD closed at BC, and of a length at least al to the distance of the focus from the allic spherical concave speculum G H ved at the end BC. The rays EG, FH, proceeding from a remote object PR, rsect one another somewhere before they the tube, so that EG, eg, are those that te from the lower part of the object, and PH, from its upper part: these rays after ng on the speculum G H, will be reflected as to converge and meet in mn, where ‘7 will form a perfect image of the object. t stion : and as in these telescopes the light. TEL But as this image cannot be seen by the specs tator, they are intercepted by a small plane metallic speculum KK, intersecting the axis at an angle of 45°, by which the rays tending to mn will be reflected towards a hole LL in the side of the tube, and the image of the object will thus be formed in gS; which image will be less distinct, because some of the rays which would otherwise fall on the concave speculum GH, are intercepted by the plane speculum ; nevertheless it will ap- pear in a considerable degree distinct, be- cause the aperture A D of the tube, and the speculum GH are large. In the lateral hole LL is fixed a convex lens, whose focus is at Sq, and therefore this lens will refract the rays that proceed from any point of the image, so as at their exit they will be parallel, and those that proceed from the extreme points Sq will converge after refraction, and form an angle at O, where'the eye is placed; which will see the image Sq, as'if it were an object, through the lens LL; consequently the ob- ject will appear enlarged, inverted, bright, and distinct. In LL lenses of different con- vexities may be placed, which, by being moved nearer to the image or farther from it, would represent the object more or less mag- nified, provided that the surface of the spe- culum GH be of a perfectly spherical figure. If, in the room of one lens LL, three lenses be disposed in the same manner with the three eye-glasses of the refracting telescope, the object will appear erect, but less distinct than when it is observed with one lens. On account of the position of the eye in this telescope, it is extremely difficult to direct the instrument towards any object. Huygens, therefore, first thought of adding to it a small refracting telescope, the axis of which is pa- rallel to that of the reflector: this is called a finder, or director. The Newtonian telescope is also furnished with a suitable apparatus for the commodious use of it. In order to determine the magnifying power of this telescope, it is to be considered that the plane speculum K #1 is of no use in this respect. Let us then suppose, that one ray proceeding from the object coincides with the axis G LIA, of the lens and speculum ; fe 3 K_3 A pen TE H aL let bb be another ray proceeding from the lower extreme of the object, and passing through the focus I of the speculum KH; this will be reflected in the direction bid, parallel to the axis GLA, and falling on the lens d Ld, will be refracted to G, so that GL will be equal to LI, anddG =dI. To the naked eye the object would appear under the angle 67 = biA; but by means of the tele- scope it appears under the angle d@L=dIL, —=Idi; and the angle Idi : the angle Idi :: 61: 1d; consequently the apparent mag- nitude by the telescope is to that by the naked eye as the distance of the foous ef the spe- TEL thickness, which is equal in every part of it, remains now about three inches and a half; and its weight, when it came from the cast, was 2118 pounds, of which it must have lost a small quantity in polishing, and it mag- nifies with proper eye-glasses more than 6000 times. To put this speculum into the tube, it is suspended vertically by a crane in the laboratory, and placed on a small narrow carriage, which is drawn out, rolling upon planks, till it comes near the back of the tube; here it is again suspended and placed in the tube by a peculiar apparatus. The method of observing by this telescope is by what Dr. Herschel calls the front view ; the observer being placed in a seat C, sus- pended at the end of it, with his back towards the object he views. There is no small spe- culum, but the magnifiers are applied imme- diately to the first focal image. From the opening of the telescope, near the place of the eye-glass, a speaking pipe runs down to the bottom of the tube, where it goes into’a turning joint; and after several other inflections, it at length divides into two branches, one going into the observatory D, and the other into the work-room E. By means of the speaking pipe the communica- tions of the observer are conveyed to the assistant in the observatory, and the’ work- man is directed to perform the required mo- tions. ' In the observatory is placed a valuable sidereal time-piece, made by Mr. Shelton. Close to it, and of the same height, is a polar distance-piece, which has a dial-plate of the same dimensions with the time-piece; this piece may be made to show polar distance, zenith distance, declination, or altitude, by setting it differently. The time and polar distance-pieces are placed so that the assistant sits before them ata table, with the speaking- pipe rising between them ; and in this manner observations may be written down very con- veniently. Such of our readers as wish for a fuller account of the machinery attached to it, viz. the stairs, ladders, and platform B, may have recourse to the second part of the Trans- actions of the Royal Society for 1795: in which, by means of 18 plates and 63 pages of letter-press, an ample detail is given of every eircumstance relating to joiner’s work, car- penter’s work, and smith’s work, which at- tended the formation and erection of this telescope. It was completed on August the 28th, 1789, and on the same day was the sixth satellite of Saturn discovered. The doctor has some other excellent in- struments of smaller power placed in his garden at Slough, and by means of which most of his discoveries have been made. On the different Merits of Microscopes and Telescopes. The advantages arising from the use of microscopes and telescopes depend, in the & TEL . first place, upon this property of magnifj i the minute parts of objects, so that they e by that means be more distinctly viewed | the eye; and secondly, upon their throwi) more light into the pupil of the eye than wh is done without them. The advantages ar ing from the magnifying power would be e tremely limited, if they were not also acco panied by the latter; for if the same quanti of light is spread over a large portion of si face, it becomes proportionably diminished force ; and therefore the objects, though m nified, appear proportionably dim. Th though any magnifying glass should enlar the diameter of the object 10 times, and e¢ sequently magnify the surface 100 times, — if the focal distance of the glass was abe eight inches (provided this was possible), a its diameter only about the size of the pu of the eye, the object would appear 100 tim more dim when we looked through the gl than when we beheld it with our naked eyt and this even on a supposition that the gk transmitted all the light which fell upon. which no glass can do. But if the focal tance of the glass was only four inches, thou its diameter remained as before, the ine venience would be vastly diminished, becat the glass could then be placed twice as ne the object as before, and consequently wor receive four times as many rays as in- former case, and therefore we should se much brighter than before. Going on th still diminishing the focal distance of ¢ glass, and keeping its diameter as large possible, we shall perccive the object m¢ and more magnified, and at the same ti very distinct and bright. It is evident, ho ever, that, with regard to optical instrume of the microscopic kind, we must sooner later arrive at alimit which cannot be passt This limit is formed by the following pa culars. 1. The quantity of light lost in p ing through the glass. 2. The diminution the glass itself, by which it receives only small quantity of rays. 3. The extreme sh ness of the focal distance of great magnifie whereby the free access of the light to object which we wish to view is imped and consequently the reflection of the lig from it is weakened. 4. The aberrations the rays, occasioned by their different refs gibility. To understand this more fully, as well to see how far these obstacles can be — moved, let us suppose the lens made of su a dull kind of glass that it transmits only ¢ half of the light which falls upon it. I¢ evident that such a glass of four inches f distance, and which magnifies the diame of an object twice, still supposing its o} breadth equal to that of the pupil of the e will show it four times magnified in surfa but only half as bright as if it was seen the naked eye at the usual distance; for 1 light which falls upon the eye from the obj: at eight inches distance, and likewise the $ face of the object in its natural size, bei TEL h represented by 1, the surface of the cnified object will be 4, and the light ch makes that magnified object visible ; 2; because, though the glass receives /{imes as much light as the naked eye ‘sat the usual distance of distinct vision, one half is lost in passing through the ls. ‘The inconvenience in this respect can ‘efore be removed only as far as it is pos- je to increase the clearness of the glass, so it shall transmit nearly all the rays which fupon it; and how far this can be done not yet been ascertained. ‘he second obstacle to the perfection of roscopic glasses is the small size of great isnifiers, by which, notwithstanding their t approach to the object, they receive a Her quantity of rays than might be ex- ved. Thus, suppose a glass of only ~4th fn inch focal distance; such a glass would fease the visible diameter 80 times, and fsurface 6400 times. If the breadth of the s could at the same time be preserved as itas that of the pupil of the eye, which ishall suppose -2,ths of an inch, the object ‘Id appear magnified 6400 times, at the e time that every part of it would be as (ht as it appears to the naked eye. But if juppose that this magnifying glass is only of an inch in diameter, it will then re- fe only ith of the light which otherwise ld have fallen upon it; and therefore, 1} . . . fad of communicating to the magnified | eta quantity of illumination equal to 6400, ould communicate only one equal to 1600, the magnified object would appear four is as dim as it does to the naked eye. inconvenience, however, is still capable ieing removed, not indeed by increasing diameter of the lens, because this must ia proportion to its focal distance, but by {wing a greater quantity of light on the ob- f Thus,in the above-mentioned example, ‘ar times the quantity of light which natu- ‘falls upon it could be thrown upon the ret, it is plain that the reflection from it ‘Id be four times as great as in the natural ; and consequently the magnified image, the same time that it was as many times Mnified as before, would be as bright as rh seen by the naked eye. In transparent fets this can be done very effectually by meaye speculum, as in the reflecting mi- cope; but in opaque objects the case is ewhat more doubtful; neither do the con- meces for viewing these objects seem en- ly to make up for the deficiences of the ft from the smallness of the lens and itness of the focus. When a microscopic Maguifies the diameter of an object 40 ’8, it hath then the utmost possible mag- ling power, without diminishing the natural thtness of the object. ‘he third obstacle arises from the short- t of the focal distance in large magnifiers ; tin transparent objects, where a sufficient hitity of light is thrown on the object from -W, the inconvenience arises at last from TEL straining the eye, which must be placed nearer the glass than it can well bear; and this en- tirely supersedes the use of magnifiers beyond a certain degree. ‘The fourth obstacle arises from the different refrangibility of the rays of light, and which frequently causes such a deviation from truth in the appearances of things, that many peo- ple have imagined themselves to have made surprising discoveries, and have even pub- lished them to the world; when in fact they have been only as many optical deceptions, owing to the unequal refractions of the rays. lor this there seems to he no remedy, except the introduction of achromatic glasses into microscopes as well as telescopes. How far this is practicable, hath not yet been triéd ; but when these glasses shall be introduced. (if such introduction is practicable), micro- scopes will then undoubtedly have received their ultimate degree of perfection. With regard to telescopes, those of the re- fracting kind have evidently the advantage of all others, where the aperture is equal, and the aberrations of the rays are corrected according | to Mr. Dollond’s method ; because the image is not only more perfect, but a much greater quantity of light is transmitted than what can be reflected from the best materials hitherto known. Unluckily, however, the imperfec- tions of the glass seta limit to these telescopes, as hath already been observed, so that they cannot be made above three feet and an half long. On the whole, therefore, the reflecting telescopes are preferable in this respect, that they may be made of dimensions greatly su- perior; by which means they can both mag- nify to a greater degree, and at the same time throw much more light into the eye. With regard to the powers of telescopes, however, they are all of them exceedingly less than what we should be apt to imagine from the number of times which they magnify the. object. Thus, when we hear of a telescope which magnifies 200 times, we are apt to imagine, that, on looking at any distant ob- ject through it, we should perceive it as dis- tinctly as we would with our naked eye at the 200th part of the distance. But this is by no means the case ; neither is there any theory capable of directing us in this matter; we must therefore depend entirely on experience. — The best method of trying the goodness of any telescope is by observing how much far- ther off you are able to read with it than you can with the naked eye. But that all decep- tion may be avoided, it is proper to choose something to be read where the imagination cannot give any assistance, such as a table of logarithms, or something which consists en- tirely of figures; and hence the truly useful power of the telescope is easily known. In this way Mr. Short’s large telescope, which magnifies the diameter of objects 1200 times, is yet unable to afford sufficient light for read- ing at more than 200 times the distance at which we can read with our naked eye. With regard to the form of reflecting tele- provided they are properly matched. TEL scopes, it is now pretty generally agreed, that wher the Gregorian ones are well constructed, they have the advantage of those of the New- tonian form, One advantage evident at first sight is, that with the Gregorian telescope an object i is perceived by looking directly through it, and consequently is found with much greater ease than in the Newtonian telescope, where we must look into the side. The un- avoidable imperfection ofthe specula common to both also gives the Gregorian an advantage over the Newtonian form. Notwithstanding the utmost care and labour of the workmen, it is found impossible to give the metals either a perfectly spherical, or a perfectly parabolical form. Hence arises some indistinctness of the image formed by the great speculum, which is frequently corrected by the little one, But if this is not done, the error will be made much worse; and hence many of the Gregorian telescopes are far inferior to the Newtonian ones; namely, when the specula have not been properly adapted to each other. There is no method by which the workman can know the specula which will fit one another without a trial; and therefore there is a neces- sity for having many specula ready made of each sort, that in fitting up a telescope those may be chosen which best suit each other. The brightness of any object seen through a telescope, in comparison with its brightness when seen by the naked eye, may, in all cases, be easily found by the following formula. Let m represent the natural distance of a visible object, at which it can be distinctly seen; and jet d represent its distance from the object- glass of the instrument. Let m be the mag- nifying power of the instrument; that is, let the visual angle subtended at the eye by the object when at the distance n, and viewed without the instrument, be to the visual angle produced by the instrument as 1 to m. Let « he the diameter of the object-glass, and p be that of the pupil. Let the instrument be so constructed, that no parts of the pencils are intercepted for want of sufficient apertures of the intermediate glasses. Lastly, let the light Jost in reflection or refraction be neglected. The brightness of vision through the instru- ment wil be expressed by the fraction ( er mpd the brightness of natural vision being 1. But although this fraction may exceed unity, the vision through the instrument will not be brighter than natural vision. Vor when this is the case, the pupil does not receive all the light transmitted through the instrument. In microscopes n is the nearest limits of distinct vision, nearly eight inches. Bat a difference in this circumstance, arising from a difference in the eye, makes no change in the formula, because m changes in the same proportion with n. In telescopes, x and d may be accounted z z° 3 a equal, and the formula becomes of 2 7 fr agility. Achromatic TELEscopE, a refracting scope without colour. See ACHROMATI Catadioptrie TeLescore, the same ag flecting 'TELESCOPE. ¥ Dioptriec Tevescope, the same as ne ing TELESCOPE. TELESCOPICAL, belonging to a te scope ; seeing at a distance. TELESCOPICAL Stars, are such as are 1 visible to the naked eye, being only disee ible by means ofa telescope. Sce STAR, All'stars less than those of the sixth mi nitude are telescopic to an ordinary eye. TENACITY, in Natural Philosophy, is 1) quality of bodies by which they sustain ae siderable pressure or force without breaki being the opposite quality to brittleness TENSION, that state which a chord, stri pe is in when stretched beyond its nate length. TERM, in Geometry, is the extreme of magnitude, or that which bounds and lin its extent. So the terms of a line, are poit of a superficies, lines; of a solid, superficie TERMS, of an equation, or of any quant in Algebra, are the several members of wi it is composed, separated from one anot by the signs + or—. So, the quantity a Qhe — Sax? , consists of the three terms and 2be, and 3a’. In an equation, the terms are the p which contain the several powers of the si unknown letter or quantity: for if the s: unknown quantity be found in several m bers in the same degree or power, they f but for one term, which is called a ¢ pound one, in distinction from a simpk single term. Thus, in the equation 2 (a — 3b) x*— acx = b3, the four terms x3, (a—3b) x, acx and b3; of which second term (4 — 3b) 2” is compound, the other three are simple. TERMINAL Velocity, in the theall projectiles, is the greatest velocity w hiv ball can acquire by descending verticalls air, and ‘with which, when attained, it wi continue to descend uniformly if no solid stacle destroyed the motion. Tor the use of this in practical gunt see Hutton’s Tracts, lately published, vol, TERMINATOR, in Astronomy, a 0 sometimes given to the circle of ijlumina from its properties of terminating the bou ries of light and darkness. TERRA. See Eartu. TERRA Firma, in Geography, is somet used for a continent in contradistinctio islands. TERRAQUEOUS, in Geography, an thet applied to onr earth when consider consisting of land and water. TERRELLA, or Little Earth, is a mé of a spherical figure, so formed in ord cart some of the magnetic properties ¢ carth. cf | \\ Plate Ma i \ to Theodolite . fi Steam Engine Eduction Valve. ue - cs Siti oe ine, lk ae ee Riecar ea fi | Mi Yi 7 Engraved by S.Lacey. x London, Published Nov730%613, by C&S Robinson, Paternoster Row, & the rest of the Proprietors. oe — ve sf ¢ ai Ae ‘ 4 Wiss : - a ee Oe +. tere ae ‘ %, e oat oS gametes | rae ESTRIAL, any thing rel fing to the divided the celestial sphere into: five circles 1, as terrestrial globe, line, &e. or zones; the arctic and antarctic circles, the BRIAN, an old measure of capacity two tropical circles, and the equator. He aining 84 gallons. observed the apparent diameter of the sun, EPRALDRON, or Fetrahkedron, in Geo- whieh’ he made equal to half a degree; and y, one of the five regular or Platonic bodies” formed the constellation of the Little Bear. olids, comprehended under four equila- He observed the nature and course of eclipses, and equal triangles. See Bopy. and calculated them exactly; one in parti- a denote the linear edge or side of a cular, memorably recorded by Herodotus, as edron, 6 its whole superficies, c its soli- jt happened on a day of batile between the r the radius of its inscribed sphere, and Medes and Lydians, which Thales had fore- re radius of its circumscribing sphere; told; and he divided the year into 365 days. the general relation among ‘all these He died’ at the age of ninety years, leaving ‘pressed by the following equations, viz. behind him an excellent character, as a ma- lr V6=FZR V6=V1bV3= 7 Gey. thematician, a philosopher, and moralist. Mr y3=—8R*V38= a V3= 6 Ve. ve: a se Bible V8=7, Ri V8 = 508 V2 bv Ne Whe oh ms ae OO re aVO =F V2bV3 = ZV 4CV3. THEMIS, a name formerly given to the fR=4vV6=7, V2bV3 = 7 ¥ fe v3. third satellite of Jupiter. e Hutton’s Mensuration, p. 248, kc. 2d THEODOLITE (from @coues, to see, and on. odos, way or distance), is an instrument employed ETRAGON, in Geometry, a quadrangle, by surveyors for measuring angles, whereby figure having four angles. Such as a to compute the heights, distances, &c. of re- re, a parallelogram, a rhombus, and a mote objects. There are various forms of -zium. : this instrument arising out of the successive ‘Square is a regular tetragon; to which improvements of many eminent artists, but e the term more commonly applies. the principle of its operation is the same in ETRAGONISM, is used by some eld all, and we shall, therefore, describe that of ors to denote the quadrature of the circle. Mr. Sisson, which is one of the most usual JALES, a celebrated Greek philoso- construction. , and the first of the wise men of Greece, This instrument is represented in Plate XI. vat Miletum about 640 years before fig. 2. The three staffs which serve for its (Christian era. When he had acquired support are screwed into bell-metal joints, usual learning of his country, he tra- which are moveable between brass _ pillars, |dinto Asia and Egypt, to be instructed fixed in a strong brass plate, in which round tometry. astronomy, and natural philoso- the centre is fixed a socket with a ball move- | On his return he became a teacher of able in it, and upon which the four screws jn, and among his disciples, who were press, being intended to set the limb horizon- jerous, were Anaximander, Anaximenes, tal. Above this is another such plate, through |Pythagoras. Thales was the author of which these screws pass, and on which round (onian sect of philosophers; he was rec- the centre is fixed a frustrum of a cone of bell d, by the best historians, the father of metal, whose axis (being connected with the jK philosophy, being the first that made centre of the ball) is always perpendicular to jresearches into natural knowledge and the limb, by means of a conical brass ferril jematics. He thought water was the fitted to it, and upon which is fixed a com- liple of which all bodies in the uuiverse pass box; as also the limb, which is a strong omposed: that the world was the work bell-metal ring, upon which are three move- hd, whom he regarded as omniscicut,and able indexes; in whose plate are fixed four | ilding the secret thoughts in the heart of brass pillars, which joining at top, hold the i He maintained that real happiness centre pin of the bell metal double sextant, | sted in health and knowledge: that the the double index of which is on the centre of ancient of beiugs is God, because he is the same plate. Within the double sextant |sated : that nothing is more beautiful is fixed a spiral level, and over it the telescope. ithe world, because itis the work of God; Both the vertical and horizontal limb, are ng more extensive than space, quicker divided into degrees and smaller divisions, i spirit, stronger than necessity, wiser than and each furnished with a Vernier’s scale, for i He used to observe, that we ought the more exact measurement of the angles, \¢ to say that to any one which inay be the divisions being more or less minute in id to our prejudice ; and that we should different instruments. . (vith our friends as with persons that may In using the instrument, whether for hori- {me our enemies. In geometry he was a zontal or vertical angles, it must first be ren- derable inventor, as well as an improver, dered perfectly horizontal by means of the icularly in triangles; and all the writers adjusting screws and spiral level; then for 1s, that he was the first, even in Lgypt, horizontal angles observing one-of the objects took the height of the pyramids by their in the telescope, let it be brought to coincide ows. Elis knowledge and improvements very exactly with the centre of the cross wires stronomy were very considerable. IIe of the rete glass; and having then set the THE horizontal limb at 0, or 180, turn the telescope towards the other object, and get it exactly in the centre of the object glass as before; and the angle contained between the two objects will be shown on the limb of the in- strument. In measuring vertical angles, after having rendered the instrument horizontal as before directed, set the vertical limb at 0; then elc- vate the telescope till such time as the top of the object is seen in the centre of the cross wires; so shall the degrees, &c. cut on the line be the measure of the angle of elevation required. THEODOSIUS, a celebrated mathema- tician, who flourished in the times of Cicero and Pompey; but the time and place of his birth are unknown. Theodosius chiefly cultivated that part of geometry which relates to the doctrine of the sphere, concerning which he published three books. The first of these contains 22 pro- positions; the second 23; and the third 14; all demonstrated in the pure geometrical manner of the ancients. Ptolemy made great use of these propositions, as well as all suc- eceding writers. ‘These books were translated by the Arabians, out of the origimal Greek, into their own Janguage. From the Arabic, the work was again translated into Latin, and printed at Venice. But the Arabic version being very defective, amore complete edition was published. in Greck and Latin, at Paris, 1558, by John Pena, regius professor of as- tronomy. And. Vitello acquired reputation by translating ‘Theodosius into Latin. ‘This author’s works were also commented on and illustrated by Clavius, Heleganius, and Guari- nus, and lastly by De Chales, in his Cursus’ Mathematicus. But that edition of 'Theo- dosius’s Spherics which is now most in use was translated and published by the learned Dr. Barrow, in the year 1675, illustrated and demonstrated in a new and _ concise method. By this author’s account, Theo- dosius appears not only to be a great master in this more difficult part of geometry, but the first considerable author of antiquity who has written on that subject. THEON, of Alexandria, a celebrated Greek philosopher and mathematician, who flourish- ed in the 4th century, about the year 380, in the time of Theodosius the Great; but the time and manner of his death are unknown. His genius and disposition for the study of philosophy were very early improved by close application to all its branches; so that he ac- quired such proficiency in the sciences as to render his name venerable in history, and to procure him the honour of being the famous Alexandrian school. One of his pupils was the admirable Hypatia, his daugh- ter, who succeeded him in the presidency of the school; a trust which, like himself, she discharged with the greatest honour and use- fulness. THEOPHRASTUS, a celebrated Greek philosopher, who flourished about 300 years president of * THE before the christian era, and taught philose at Athens with great applause. Theophrastus wrote many works, the 1 cipal of which are the following. 1. cellent moral treatise, entitled, Chari which, he says in the preface, he compost 99 years of age. Isaac Casaubon has wri learned commentaries on this small trea It has been translated from the Greek French, by Bruyere; and it has also | translated into English. 2. A curious Tre, on Plants. 3. A Treatise on Fossils or Sto of which Dr. Hill has given a good edi with an English translation, and learned ng in 8vo. THEOREM, in Geometry, is a propos) in which some truth is proposed to be strated, being thus distinguished hid blem in which something is proposed 4) done. Proclus defines. A Theorem, something which is prop) to be demonstrated. A Problem, something proposed to be « and | A Porism, something proposed to bj vestigated. Geometrical THEOREMS, are sometime) vided into different classes, as Local THEOREM, is that which relates} surface. As, that triangles of the same} and altitude are equal. | Plane THEOREM, is that which relates surface that is either rectilinear or boul by the circumference of a circle. As, th angles in the same segment of a cirel| equal. | Solid THrorem, is that which consit space terminated by a solid line, that) any of the three conic sections. ‘THEOREM, in Algebra, is used sometin| denote a rule, particularly when that} is expressed by algebraical symbols, or fi) lee; such as those given under the aif ANNUITIES, INTEREST, &c. Algebraical THEOREMS, have also part denominations, either from the subje which they relate, or the names of the at by whom they were invented ; thus we ha BinomiaA Theorem, Cotes’s Theorem, Gi) Theorem, MuLtTinomMiAL Theorem, ‘TAY Theorem, &c. for which see the resp. articles. TL avail myself of this opportunity of ec ing an error into which [ have fallen the article Brnom1AL Theorem. It is | under that article, on the authority ¢ merous authors, that this celebrated th) is engraved upon its author’s monumé have since been led to examine the J ment, and cannot perceive that any th) the kind was ever represented upon | least it is not now to be seen, and there appearance of its having been obliteraty 'TH EORY, a doctrine which terming the sole speculation or consideration object, without any view to the praet} application of it. ’V'o be learned in an art, &e. the thd THE ‘ent; to bea master of it, both the theory yactice are requisite. chines often promise very wellin theory, il in the practice. > say theory of the moon, theory of the yw, of the microscope, of the camera ob- , ke. ‘gory of Numbers, is a modern branch lysis, which has for its object the inves- ion of certain properties of numbers, such iat the sum of two odd squares cannot a square number ; that every nnmber is osed of four or a less number of squares; he sum of any number of the odd num- , 3, 5, 7, &c. is equal to the square of amber of terms; that the sum of any fer of the natural cubes 1, 8, 27, &c. is a fy number, and a great variety of similar irties highly interesting to the specalative j:matician. See NUMBERS. lgory of the Planets, &c. is a system pothesis, according to which astrono- ‘explain the reasons of the phenomena yearances of them. HERMOMETER, (from Sep103, heat, lerpov, measure,) an instrument for mea- i; the degree of heat or cold in any body. cnvention of the thermometer, bke that | telescope, and indeed most other useful fments, has been claimed by different lophers. ‘Thus the invention is ascribed (rnelius Drebbet of Alcmar, about the ‘faing of the 17th century, by his country- (Boerhaave (Chem. 1, p. 152, 156), and thenbroeck, (Introd. ad Phil. Nat. vol. ii, |); Fulgenzio, in his Life of Father Paul, ‘him the honour of the first discovery; nzio Viviani (Vit. de l Galil. p. 67, Iper. di Galil. pref. p. 47) speaks of (oas the inventor of thermometers. But jorino (Com. in Galen. Art. Med. p. 736, fCom. in Avicen. Can. Fen. 1, p. 22, 78, 'xpressly assumes to himself this inven- hand Borelli (De Mot. Animal. 2 prop. ind Malpighi (Oper. Posth. p. 30) as- )t to him without reserve. Upon which Martine remarks, that these Florentine (nicians are not to be suspected of parti- ifavour of one of the Patavinian school. | whoever was the inventor, the first this instrument for measuring the de- fof heat and cold was the air-thermo- |. It is a well known fact that air ex- ‘ with heat so as to occupy more space it does when cold, and that it is con- il by cold so as to occupy less space than sWarmed, and that this expansion and (asation is greater or less according to eree of heat or cold applied. The prin- (hen on which the air-thermometer was Sucted is very simple. ‘The air was con- (in a tube by means of some coloured (; the liquor rose or fell according as the ‘came expanded or condensed. What ‘st form of the tube was, cannot now iis be well known; but the following tition of the air-thermometer will fully Inits nature. It consists of a glass tube, i ! j THE B e, connected at one end with a large glass ball, A, and at the other end immersed ia an open vessel, or terminating in a ball, de, with a narrow orifice at d; which vessel or ball contains any coloured li- quor that will not easily freeze, Aquafortis tinged ofa fine blue colour with a solution of vitriol | or copper, or spirit of wine tinged with cochineal, will ~ answer this purpose. But the ball, A, must be first moderate- ly warmed, so that a part of the air contained in it may be expelled through the orifice, d; and then the liquor pressed by the weight of the atmo- | sphere will enter the ball, de, | and rise, for example, to the middle of the tube at C, ata mean temperature of the weather ; and in this state the liquor by its weight, and the air in- cluded in the ball, A, &c, by its elasticity, will counterbalance the weight of the atmo- sphere. As the surrounding air becomes warmer, the air in the ball and upper part of the tube, expanding by heat, will drive the liquor into the lower ball, and consequently its surface will descend; on the contrary, as the ambient air becomes colder, that in the ball is condensed, and the liquor pressed by the weight of the atmosphere will ascend; so that the liquor in the tube will ascend or descend more or less according to the state of the air contiguous to the instrument. To the tube is affixed a scale of the same length, divided upwards and downwards from the middle, C, into. 100 equal parts, by means of which the ascent and descent of the liquor in the tube, and consequently the variations in the cold or heat of the atmosphere, may be observed. The air being found improper for measur- ing with accuracy the variations of heat and cold according to the form-of the |, 5 thermometer which was first adopt-- ed, another fluid was proposed about the middle of the seventeenth cen- tury by the Florentine Academy. This fluid was spirit of wine, or alcohol, as it is now generally named. The alcohol being colour- ed, was inclosed in a very fine cy- lindrical glass tube previously ex- © hausted of ‘its air, having a hollow ball at one end, A, and hermetically sealed at the other end, D. ‘The ball and tube are filled with rec- tified spirit of wine to a convenient height, as to C, when the weather is of a mean temperature, which may be done by inverting the tube into a vessel of stagnant coloured spirit, under a receiver of the air- A pump, or in any other way. When the thermometer is properly filled, the end D YY2 a HOE is heated red-hot by a lamp, and then herme- tically sealed, leaving the included air of about one-third of its natural density, to pre- vent the air which is in the spirit from divid- ing it in its expansion. To the tube is ap- plied a scale, divided from the middle, into 100 equal parts, upwards and downwards. As spirit of wine is capable of a very con- siderable degree of rarefaction and condensa- tion by heat and cold, when the heat of the atmosphere increases the spirit dilates, and consequently rises in the tube; and when the heat decreases, the spirit descends, and the degree or quantity of the motion is shown by a scale. This was evidently an improvement on the air-thermometer, but was itself not free from objections. The liquor could not easily be obtained of the same strength, and hence dif- ferent tubes filled with it, when exposed to the same degree of heat, would not cor- respond. Another defect was the want of some fixed guide as a standard to commence the graduation. Philosophers soon saw that some fixed and unalterable point must be found, by which all thermometers might be accurately adjusted. - Dr. Halley proposed that thermometers should be graduated in a deep pit, where the temperature in all seasons was nearly the same. This how- ever could not generally be practised. He thought of the boiling point of water, of mercury, and of spirit of wine, but preferred the latter. At length Sir Isaac Newton de- termined this important point, on which the accuracy and value of the thermometer de- pends. He chose, as fixed, those points at which water freezes and boils; the very points which the experiments of succeed- ing philosophers have determined to be the most fixed and convenient. Sensible of the disadvantages of spirit of wine, he tried ano- ther liquor, which was homogeneous enough, and capable of a considerable rarefaction, several times greater than spirit of wine. ‘This was linseed oil. It has not been ob- served to freeze even in very great colds, and it bears a heat very much greater than water before it boils. With these advantages it was made use of by Sir Isaac Newton, who dis- covered by it the comparative degree of heat for boiling water, melting wax, boiling spirit of wine, and melting tin; beyond which it does not appear that this thermometer was applied. 'The method he used for adjusting the scale of this oil-thermometer was as fol- Jows: supposing the bulb, when immerged in thawing snow, to contain 10,000 parts, he found the oi! expand by the heat of the human body so as to take up one thirty-ninth moré space, or 10,256 such parts; and by the heat of water boiling strongly 10,725; and by the heat of melting tin 11,616. So that reckoning the freezing point as a common limit between heat and cold, he began his scale there, mark- ing it 0, and the heat of the human body he made 12°; and consequenily, the degrees of heat being proportional to the degrees of rare- then putting the ball of his thermome THE faction, or 256 : '725 :: 12 : 34, this nu will express the heat of boiling wat by the same rule, 72 that of melting tin thermometer was constructed in 170 the application of oil as a measure ¢ and cold there are insuperable obje It is so viscid, that it adheres too stro the sides of the tube. On this accour cends and descends too slowly in ea sudden heat or cold. Ih a sudden eé great a portion remains adhering to th of the tube after the rest has subsid the surface appears lower than the co ing temperature of the air requires. — thermometer is therefore not a proper of heat and cold. All the thermomet therto proposed were liable to many veniences, and could not be consid exact standards for pointing out the’ degrees of temperature. This led to attempt a new one, an account of was published in the year 1730 in t moirs of the Academy of Sciences. thermometer was made with spirit o He took a large ball and tube, the dim and capacities of which were known; | graduated the tube, so that the spa one division to, another might contain part of the liquor; the liquor containin parts when it stood at the freezing poi adjusted the thermometer to the f point by an artificial congelation of part of the tube into boiling water, served whether it rose 80 divisions; i ceeded these, he changed his liquor, adding water lowered it, till upon should just rise 80 divisions; or if the being too low, fell short of 80 divisi raised it by adding rectified spirit to it liquor thus prepared suited his purpo served for making a thermometer of ; whose scale would agree with his sti At length a different fluid was propo which thermometers could be made fr most of the defects hitherto mentioned fluid was mercury, and scems first‘ occurred to Dr. Halley, but was not < by him on account of its having a degree of expansibility than the oth used at that time. | The honour of this invention is g given to Fahrenheit of Amsterdam, w sented an account of it to the Royal of London in 1724. That we may ju more accurately of the propriety of em mercury, we will compare its qualiti those of the fluids already mention alcohol, and oil. Airis the most ex] fluid, but it does not receive nor part. heat so quickly as mercury. Alcoh hot expand much by heat. In its @ state it does not bear a much great than 175° of Fahrenheit; but when rectified it can bear a greater degree than any other liquor hitherto employ measure of temperature. At Hudsar Mr. Macnab, by a mixture of vitrio v i Et E now, made it to descend to 69° below 0 enheit. There is an inconvenience, ver, attending the use of this liquor; it iy pssible to “get it always of the same be of strength. As to oil, its expansion {15 times greater than that of alcohol ; sa heat of 600°, and its freezing point ‘low that it has not been determined ; viscosity renders it useless. reury is far superior to alcohol and oil, iS much more manageable than air. 3 far as the experiments already made {eterminc, itis of all the fluids hitherto ity in the construction of thermome- at which measures most exactly equal ences of heat by equal differences of its | its dilatations are, in fact, very nearly rtional to the augmentations of heat wbto it. 2. Of all liquids it is the most ge from air. 3. It is fitted to mea- aigh degrees of heat and cold. It sus- pa heat of 600° of Fahrenheit’s scale, and ‘not congeal till it falls 39 or 40° below 0. jis the most sensible of any fluid to heat old, even air mot sohpulap itp Count Rum- y to the boiling ha in 58), while ook 2™ 13', and common air 10™ and | . Mercury i is a homogeneous fluid, and } portion of it is equally dilated or con- tt equal variations of heat. Any one eter, made of pure mercury, Is ceteris ae of the same properties with er thermometer made of pure mer- Its power of expansion is indeed about less than that of spirit of wine, but Feat enough to answer most of the pur- r which a thermometer is wanied. ‘ixed points, which are now universally \n for adjusting thermometers to a scale, e another, are the boiling and freez- er points. The boiling water point, it Known, is not an invariable point, but “80 me degrees according to the weight lemperature of the atmosphere. In an d receiver, water will boil with a f 98° or 100° ; whereas, in Papin’s di- it will acquire a heat of 412°. Hence ears, that water will boil at a lower . according to its height in the atmo- i:, or to the weight of the column.of air ‘presses upon jt. In order to ensure mity, therefore, in the construction of ec, it is now agreed, that the bulb | tube be plunged in the water when it liolently, the barometer standing at 30 inches, and the temperature “of the phere 55°. A thermometer made i in this ivith its boiling point at 212°, is called, » Horsley, Bird’s Fahrenheit, because rd was the first person who attended to ate of the barometer in constructing ometers. artists may be often obliged to adjust ometers under very different pressures atmosphere, philosophers have been at to discover a general rule which might d on all occasions, M. de Lue, , ae from a series of experiments, has given an equation for the allowance on account of this difference, in Paris measure, which has been verified by Sir George Schuckburg; also Dr. Horsley, Dr. Maskelyne, and Sir George Schuckburg, have adapted the equation and rules to English measures, and have reduced the allowances into tables, for the use of the artists. Dr. Horsley’s rule, deduced from De Luc’s, is this: . ee iar e680 Sik 8990000 ° Where A denotes the height of a thermometer plunged in boiling w ater above the point of melting ice, in degrees of Bird’s Fahrenheit, and z the height of the barometer in 10ths of aninch. From this rule he has computed the following table, for finding the heights to which a good Bird’s F ahrenheit will rise, when plunged in boiling water, in all states of the barometer, from 27 to3l Eng lish inches; which will serve, among other uses, to direct instrument-makers in making a true allow- ance for the effect of the variation of the ba- rometer, if they should be obliged to finish a thermometer at a time when the barometer is above or below 30 inehes; though it is best to fix the boiling point when the barometer is at that height. Evquation of the Boiling Point. Barometer. | Equation. | Difference. 31.0 4+- 1.57 0.78 30.5 + 0.79 0.79 30.0 0.00 0.80 29.5 — 0.80 0.82 29.0 — 1.62 0.83 28.5 — 2.45 0.85 28.0 — 3.31 0.86 27.9 — 4.16 0.88 27.0 — 5.04 The numbers in the first column of this table express heights of the quicksilver in the barometer, in E inglish inches and decimal parts: the second column shows the equation to be applied, according to the sign prefixed, to 212° of Bird’s Fahrenheit, to find the true boiling point for every such state of the ba- rometer. ‘The boiling point, for all interme- diate states of the barometer, may be had, with sufficient accuracy, by taking propor- tional parts, by means of the third column of differences of the equations. Construction of Fahrenheit’s Thermometer.— The method of constructing Fahrenheit’s thermometer, which is now in general use in this country, is the following: a smail ball is blown on the end ofa glass tube, of an uniform width throughout. ‘The ball and part of the tube are then to be filled with quicksilver, which has been previously boiled to expel the air. The open end of the tube is then to be hermetically sealed. The next obfect is to construct the scale. It is found, by experi. THE ment, that melting snow, or freezing water, is always at the same temperature, If, there- fore, a thermometer be immersed in the one or the other, the quicksilver will always stand at the same point. Ithas been observed, too, that water boils under the same pressure of the atmosphere at the same temperature. A thermometer, therefore, immersed in boiling water, will uniformly stand at the same point. Here, then, are two fixed points, from which a scale may be constructed, by dividing the intermediate space into eqnal parts, and carry- ing the same divisions as far above and below the two fixed points as may be wanted. ‘Thus, thermometers constructed in this way may be compared together; for if they are accurately made, and placed in the same temperature, they will always point to the same degree on the seale. ‘The fluid, as we have seen em- ployed, is quicksilver, and it is found to an- swer best, because its expansions are most equable. The freezing point of lahrenheit’s thermometer is marked 32°, and the reason of this is said to have been, that this artist thought that he had produced the greatest degree of cold, by a mixture of snow and salt; and the point at which the thermometer then stood, in this temperature, was marked zero. The boiling point, in this thermometer, is 212°, and the intermediate space, between the boil- ing and freezing points, is therefore divided into 180°. This is the thermometer that is commonly used in Britain. In determining the choice of tubes it is best to have them exactly cylindric through their whole length. Capillary tubes are preferable to others, because they require less bulbs, and they are also less brittle, and more sensible. Those of the most convenient size for common experiments are such as have their internal diameter about the fourth of a line; and those made of thin glass are better than others, as the rise and fall of the mercury may be more distinctly perceived. The length of nine inches will serve for all common occasions; but for particular purposes the length both of the tubes and of the divisions should be adapted to the uses for which they are designed. In determining the best size of the balls or bulbs, it has been usual to compare new tubes with such thermometers as are well propor- tioned. But M. Durand has proposed a formula for finding the proportion which the balis ought to bear to their respective tubes. With this view he expresses the length of the tube, measured in diameters of itself, by a; the whole capacity of the ball and tube by e; the capacity of the fundamental interval, ex- pressed in the same parts with the whole ca- pacity, by d; the number of degrees of the fundamental interval by m; the number of other degrees which the scale is to contain, besides those of the fundamental interval both above and below it, by n; and the diameter of the ball measured in diameters of the tube : 3 P by 6; and paV/ (Fe x em d x (m rae) a THRE: For two cylinders having equal base as their heights, m:n::d: se: which W : capacity of that part of the tube whie ceeds the fundamental interval, to which ing d, that interval, we have the total capi dn-+dm of the tubé = le + d, or Subt op m m i ing this from e, we shall have the capaci dn + dm _ ecm—dm— the ball = e — = ———__—— m mf If this quantity be divided by the capaeilg the tube, the quotient will show how 4 the capacity of the ball contains that 0] em — dm-j dm + d) Consequently the ball is equal to as 4 cylinders, having a diameters of the tub, their respective height, and 1 diameter fof base, as are contained in this last quot and, therefore, its cylindric solidity expré in the cylindric solidities of the tube wi poss: £45 Cait dime Pe ae) But the diar dm + dn of this ball is equal to the base of the eyli in which it may be inscribed, and the so} of this cylinder is equal to 3 the solidity q circumscribing sphere. Consequently) solidity of this cylinder will be = %) ecm—.dm—dn j ‘3 ; te ae at and the diameter of its equal to the diameter of the ball, will} tube ; and this quotient is = ] . 2,cm—dm—dny__ > / 2 cm Vv (30 dm-+di )=\ \3°axGn ‘Two unalterable points of temperaturd the former where ice becomes water, an second where water becomes vapour, | been universally adopted by the various) structors of thermometers for the gradu of those instruments; but the space betel them has been divided differently by dif persons, and this difference gives the dif names of thermometers, or rather of thei! duations; such as Reaumur’s thermor tahrenheit’s thermometer, ke. Reaun r vides the space between the above-ment two points into 80 equal parts or dee placing the 0 at freezing, and the 80th & at the boiling point. Fahrenheit, as \ remarked, divides it into 180 degrees or parts, but he places the 0 thirty-two dit below the freezing point; so that the ¥ ing point is at 32, and the boiling poin! 212 degrees. : Other persons have adopted other diy which have been suggested by suppos . vantages or fanciful ideas. | Most of those graduations are at ps out of use, but they are to be met wi various, not very recent, publications have therefore thought it necessary 1 them down in the following table, whic tains: Ist. The name of the person | ciety that has used each particular divi 2dly. The degree which has been place); each of them, against the freezing- THE . The degree which has been placed ainst the boiling-point; and, 4thly. The mber of degrees lying between those two ints. Fi ~ a | ® *r = & Big ° ass ae a 3o2 bn v¢ to ovum & = o> 5 Py — 2 os = a 2 Se bez © mL mR a” Fahrenheit’s, which is nerally used in Great ‘Reaumur’s, whichis ge- lly usedin France and her parts of the conti- Sree Centigrade, or Celsius’s, ich has been used jefly in Sweden, hence is also called the Swe- sh thermometer. It has pen lately adopted by 'e French chemists, un- x the name of cen- \rrade thermometer ...... /The Florentine thermo- leters, which were made ‘id used by the members ‘the famous academy 1 Cimento, being some {the first instruments of ‘esort, were vaguely gra- ‘ated, some having a eat many more degrees yan others. But two of eir most common gra- yations seem to be........ (The Parisian thermo- eter, viz. the ancienne lermometre of the Aca- my of Sciences, seems have been graduated BMEDNAS s. 01. Soosessscgh sees De la Hire’s thermo- ter, which stood in the rvatory at Parisabove ) years, was graduated are dastocee Caen eee s TE Soc galch chi cengenass De L’Isle’s thermome- ris graduated in an in- ‘rted order...... iSeabsce ‘Sir Isaac Newton’s....... | 2 Eres )}The Edinburgh ther- wometer, formerly used, vems to have been gra- | These ate the chief thermometers that have jeen used in Europe; and the temperatures tudicated. by the principal of them may be re- THE duced into the corresponding degrees on any of the others, by means of the following sim- ple theorems; in which R signifies the degrees on the scale of Reaumur, F those of ahren- heit, and S those of the Swedish thermometer. 1. ‘To convert the degrees of Reaumur into x9. 59—F, : ty those of Fahrenheit ; 2. To convert the degrees of Fahrenheit (F — 32) xX 4 a eae R. 3. To convert the Swedish degrees into s x9 oe oa | 5 Ps i 4. To convert Fahrenheit’s into Swedish; (F — 32) x 5 _ s a —S. 5. To convert Swedish degrees into those Ss of Reaumur; — 5 6. To convert Reaumur’s degrees into the Rix/G! S In meteorological observations, it is necessary to attend to the greatest rise and fall of the ther- mometer, and therefore attempts have been made to make them mark the greatest degree of heat and cold, in the absence of the observer. We will notice one, intended to show the greatest degree of heat. AB, is a glass tube, with a cylindrical bulb, B, at the lower end, and capillary at the other, over which there is a fixed glass ball, C. ‘The bulb, and part of the tube, are filled with mercury, the top of which shows the degrees of heat. The upper part of the tube, above the mercury, is filled with spirit of wine; the ball, C, is likewise — filled with the same liquor, almost to the top of the capillary tube. When the mercury rises, the spirit of wine is also raised into the ball, C, which is so made that the liquor cannot return into the tube when the mercury sinks; of %|{é course, the height of the spirit in the ball, added to that in the tube, will give the greatest degree of heat. To make a new observa- tion, the instrument must be in- clined till the liquor in the ball cover ‘the end of the capillary tube. J g In 1782, Mr. Six proposed ano- © ther self-registering thermometer. It is properly a spirit of wine ther- mometer, though mercury is also employed forsupporting an index: abis a thin tube of glass sixteen inches long, and five-sixteenths of an inch calibre: ede, and fgh are smaller tubes, about one-twen- into those of Reaumur ; those of Fahrenheit ; centigrade ; h aera mee. «oe NS ae tieth of an inch calibre. These three tubes are filled with highlyrectified spirit ofwine, except the space between d and g, which is filled with mercury. As the spirit of wine contracts or expands in the middle tube, the mercury falls or rises in the outside tubes. An index, such as that represented in the annexed figure, is placed on the surface, within each of these tubes, so light as to float upon it: kis a small glass tube, three-fourths of an inch long, hermetically seated at each end, and inclosing a piece of steel wire nearly of itsown length. Ateach end, /m, of this small tube, a short tube of black glass is fixed, of such a diame- ter as to pass freely up and down within either of the outside tubes of the ther- mometer, ce, or fh. From the upper 7 end of the index is drawn a spring of glass to the fineness of a hair, and about five-sevenths of an inch long; which, being placed a little oblique, presses & lightly against the inner surface of the tube, and prevents the index from de- scending when the mercury descends. These indexes being inserted one into 77 each of the outside tubes, it is easy to understand how they point out the greatest heat or cold that has happened in the observ- er’s absence. When tlie spirit of wine in the middle tube expands, it presses down the mercury in the tube, Af, and consequently raises it in the tube, ec; consequently, the index on the left-hand tube is left behind, and marks the greatest cold, and the index in the right-hand tube rises, and marks the greatest heat. The common contrivance for a self-register- ing thermometer, now sold in most of the London shops, consists simply of two thermo- meters, one mercurial, and the other of al- cohol, having their stems horizontal; the former has. for its index a small bit of mag- netical steel wire, and the latter a minute thread of glass, having its two ends formed into small knobs, by fusion in the flame ofa candle, pes The magnetical bit of wire lies in the vacant space of the mercurial thermometer, and is pushed forward by the mercury whenever the temperature rises, and pushes that fluid against it; but when the temperature falls, and the fluid retires, this index is left behind, and con- sequently shows the maximum. The other index, or bit of glass, lies in the tube of the spirit thermometer immersed in the alcohol: and when the spirit retires, by depression of temperature, the index is carried along with it, in apparent contact with its interior surfa¢ but, on increase of teniperature, the ‘| goes forward and leaves the index, whi therefore shows the minimum of temperat since it was set. As these indexes mere lie in the tubes, their resistance to motion altogether inconsiderable. ‘The steel index brought to the mercury by applying a magn on the outside of the tube, and the other duly placed at the end of the column of cohol, by inclining the whole instrument. — For an account of Mr. Keith’s self-registe ing thermometers, see Gregory’s Mechani vol. ii, and Nicholson’s Journal, vol. iii, 4 series. pe Differential THERMOMETER. _ Profess Leslie, the well-known author of a curio} and interesting treatise on heat, invented thermometer for indicating and measur very minute differences of temperature, whi lie calls the differential thermometer. i * % consists of two tubes, each terminating: in small bulb of the same dimensions, joined } the blow-pipe, and bent in the form of a jj as in the above figure, a small portion dark coloured liquor having previously bes introduced into one of the balls, as for ej ample, a solution of carmine in concentrati sulphuric acid, which is generally preferre) By managing the included air with the heat | the hand, this red liquor is made to stand at UJ required point of the opposite tube. This is tl] zero ofa scale fastened to that tube, and dividy into equal parts above and below that poim The instrument is then fixed on a stand. | is manifest that when the liquor is at rest, | points at zero, the column is pressed in oj posite directions by two portions of air equ in elasticity, and containing equal portions | caloric. Whatever heat, therefore, may } applied to the whole instrument, provide both bulbs receive it in the same degree, t liquor must remain at rest. But if the o THU receives the slightest excess of tempera- , the air which it contains will be pro- jionally expanded, and will push the liquid Jnst the air in the other bulb with a force ing as the difference between the tem- futures of those two portions of air: thus } equilibrium will be destroyed, and the {i will rise in the opposite tube. The de- hs of the scale through which it passes |) mark the successive augmentations in the )perature of the ball which is exposed to ‘greatest heat; so that this instrument is lance of extreme delicacy for comparing temperatures of its two scales. When thermometers are contrived to mea- _wery great degrees of heat by the ex- sions they produce in substances, or, on feontrary, the expansions corresponding flifferent temperatures, they are charac- is of the principal of which may be seen er the word PYROMETER. HERMOSCOPE (S:eu0; and cxomew.) An ument showing the changes happening in air with respect to heat and cold. ‘The ithermoscope is generally used indiffer- iy with that of thermometer, though there jme difference in the literal import of the ; the first signifying an instrument that ,or exhibits, the changes of heat, &e. jie eye: and the latter, an instrument that sures those changes; on which founda- the thermometer should be a more ac- te thermoscope, &c. HUNDER, the noise occasioned by the josion of a flash of lightning passing through ur; or it is that noise which is excited by iden explosion of electrical clouds, which \herefore called thunder-clouds. ie rattling in the noise of thunder, which ie8 it seem asif it passed through arches, oObably owing to the sound being excited clouds hanging over one another, and agitated air passing irregularly between (le ite explosion, if high in the air, and re- :from us, will do no mischief; but when , it may, and has in a thousand instances oyed trees, animals, &c. This proxi- l, Or small distance, may be estimated ly by the interval of time between seeing ash of lightning, and hearing the report te thunder, estimating the distance after fate of 1142 feet per second of time, or 32 M4 to the mile. Dr. Wallis observes, ‘commonly the difference between the : is about seven seconds, which, at the rate f€ mentioned, gives the distance almost But sometimes it comes in a se- or two, which argues the explosion very 4 us, and even among us. And in such i the doctor assures us, he has sometimes old the mischiefs that happened. though in this country thunder may en at any time of the year; yet the months ily and August are those in which it may Ist certainly be expected; but its time ‘uration is very uncertain; sometimes | Hed by the name of pyrometers; descrip-- THU only a few peals will be heard at any parfi- cular place during the whole season; at other” times the storm will return at the interval of three or four days, for a month, six weeks, or even longer ; not that we have violent thunder in this country, directly vertical in any one place so frequently in any year, but in many seasons it will be perceptible that thunder- clouds are formed in thé neighbourhood even at these short intervals. Hence it appears, that during this particular period there must be some natural cause operating for the pro- duction of this phenomena, which does not take place at other times. This cannot be the mere heat of the weather, for we have often a long tract of hot weather without any thunder; and besides, though not common, thunder is sometimes heard in the winter also. As therefore the heat of the weather is com- mon to the whole summer whether there be thunder or not, we must look for the causes of it in those phenomena, whatever they are, which are peculiar to the months of July, August, and the beginning of September. Now it is generally observed, that from the month of April, an east or south-east wind gene- rally takes place, and continues with littleinter- ruption till towards the end of June. At that time, sometimes sooner and sometimes later, a westerly wind takes place; but as the causes producing the east wind are not removed, the latter opposes the westerly wind with its whole force. At the place of meeting, there is naturally a most vehement pressure of the atmosphere, and friction of its parts against one another; a calm ensues, and the vapours brought by both winds begin to collect and form dark clouds, which can have little mo- tion either way, because they are pressed almost on all sides. For the most part, how- ever, the west wind prevails, and what little motion the clouds have is towards the east; whence the common notion in this country that thunder-clouds move against the wind. But this is by no means universally true; for if the west wind happens to be excited by any tem- porary cause before its natural period when it should take place, the east wind will very frequently get the better of it; and the clouds, even although thunder is produced, will move westward. Yet in either case the motion is so slow, that the most superficial observers cannot help taking notice of a considerable resistance in the atmosphere. When lightning acts with extraordinary violence, and breaks or shatters any thing, it is called a thunder- bolt, which the vulgar, to fit it for such etiects, suppose to be a hard body, and even a stone. But that we need not have recourse to a hard solid body to account for the effects com- monly attributed to the thunderbolt, will be evident to any one, who considers those of gunpowder, and the several chemical fulmi- nating powders; but more especially the as- tonishing powers of electricity, when only collected and employed by human art, and much more when directed and exercised in the course of nature. TID When we consider the known effects of electrical explosions, and those preduced by lightning, we shall be at no loss to account for the extraordinary operations vulgarly as- cribed to thunderbolts. And as stones and bricks struck by lightning are often found in a vitrified state, we may reasonably suppose, with Beccaria, that certain stones in the earth, struck by the electric fluid, and bearing this vitrified appearance, have given rise to the vulgar idea of thunderbolts. TIDES, two periodical motions of the wa- ters of the sea; called also the flux and reflua, or the ebb and flow. When the motion of the water is against he wind, it is called a windward-tide; when wind and tide go the same way, leeward-tide ; when it rans very strong, it is called a tide- gate. ” 'To tide it over or up into any place, is to go in with the tide, either ebb or flood, as long as that lasts; then to stay at anchor all the time of contrary tide; and thus to set in again with the return of the next tide. It is said to flow tide and half tide, allowing six hours to a tide, when the tide runs three hours in the offing longer than it does by the shore ; but, by longer, is not meant its running more hours; but that, ifit be high water ashore at twelve, it will not be so in the offing till three. An hour and a half longer makes tide and quarter-tide, three-fourths of an hour longer makes tide and half-quarter tide, &c. The tides are found to follow periodically the course of the sun and moon, both as to time and quantity. And hence it has been suspected, in all ages, that the tides were somehow produced by the influence of these luminaries. Thus, several of the ancients, and among others, Pliny and Ptolemy, were acquainted with the influence of the sun and moon upon the tides; and Pliny says expressly, that the cause of the ebb and flow is in the sun, which attracts the waters of the ocean; and adds, that the waters rise in proportion to the proximity of the moon to the earth. Itis, indeed, now well known, from the discoveries of Sir Isaac Newton, that the tides are caused by the gravitation of the earth towards the sun and moon. Indeed the sagacious Kepler, long ago, conjectured this to be the cause of the tides: “ If,” says he, ‘the earth ceased to attract its waters towards itself, all the water in the ccean would rise and flow into the moon: the sphere of the moon’s attraction extends to our earth, and draws up the water;”’ Kepler, Introd. ad Theor. Mart. This surmise, for it was then ne more, is now completely verified in the theory laid down by Newton, and farther illus- trated by more modern mathematicians. Phenomena of the Vides. 1. The waters of the ocean alternately approach to, and re- cede from, our shores. ‘This is produced by an elevation and subsequent depression of the surface of the ocean between determinate limits, which are those of high water and low water. The interval between two high waters ‘position of the sun and moon in respect 4 TID is about 12° 25’, the half of the moon’s d circuit round the earth; so that we have tides of flood and two of ebb in 24" 50, a riod which may be called a lunar day. — gradual subsidence of the water is such, the square of the heights are nearly as squares of the times from high water. same may be said of the subsequent riss the water in the next flood. The time of! water is nearly half way between the hours of high water; not indeed exactly, it is observed at Brest and Rochfort, that flood-tide commonly takes 10 minutes than the ebb-tide. 2. The tides have a particular reference the position and phases of the moon; for/ always high water when the moon is ¢ determinate point of the compass (S.W. ne} and the highest tides happen about the of the full or change. At Brest, wher accurate register of the phenomena of} tides was made about the beginning of last century, it was observed that the hig tide happens about a day anda half after or change. If the time of high water hay at the very time of the new or full moon, third high water after that is the highe all: this is called the Spring-tide. From period the tides gradually decrease, unti third high water after the moon’s quadra which is the lowest of all, and is calle Neap-tide. After this the tides increase, the next spring-tide, and so on contim The higher the tide of flood rises, the 1} the ebb-tide generally sinks on that The total magnitude of the tide is | mated by the difference of high water low water. At Brest the medium spring is about 19 feet, and the neap-tide about 3. The tides have also a reference t@ distance of the moon from the earth, beil much the greater as the moon is nearer. highest spring-tide happen when the mo in perigee; and the next spring-tide i smallest, because then the moon is nea apogee. This makes a difference of 23 from the medium height of the spring-ti Brest, and therefore a difference of 53 between the greatest and least. 4. The tides also depends on the sun’ tance, but not so much as on that of thet In our winter the spring-tides are greateriiill in summer, and the neap-tides smaller. | 5. The tides depend very much upoj equator. The phenomena, however, are} intricate than those we have alread | scribed. | 6. All the phenomena are modified bi latitude of the place of observation; andi happen in high latitudes, which are not } seen near the equator. In particular, | the observer and moon are both on the! side of the equator, the tide which ha} when the moonis above the horizon, is gi than that which happens on the sami when she is below it; and the contrary ha) when the observer and the moon are on! ED ‘e sides of the equator. If the observer's ‘stance from the pole be equal to the moon’s »clination he will see but one tide in the day, mtaining 12 hours flood, and 12 hours ebb. 7. All the phenomena we have described le greatly modified by local circumstances ; chas the position of the shores, the width ' the channels through which the waters iss, the extent of the seas, winds, Xc. inso- ach, that frequently particular phenomena snnot be distinguished from one another. | Theory of the 'Vines. If the earth were en- ely fluid and quiescent, or if it were a solid ileus covered with a fluid, it is evident vat its particles, by their mutual attraction pwards each other, would form the whole ese particles were to be acted upon with ual forces by any external power, the only feet resulting from such a power would be ’ draw the earth from its supposed quiescent ate, its form still remaining that of an exact shere; and this would be the case whatever ‘otion the earth took in consequence of that ition, viz. whether it were circular or recti- near, for the action on each particle, accord- g¢ to this hypothesis, being the same motion "each, resulting from this action, must be jual; in which case, therefore, we should /But from the well established theory of Aiversal gravitation we know, that the action / aremote body, as the sun or moon, is not ‘e same upon the several parts of the earth, it that this power decreases as the squares ‘the distances of the several particles from the entre of the sun or moon increase; therefore pose parts of the fluid mass which are nearest e moon, for example, are more attracted - the centre, and these again more than ‘ose diametrically opposite the former. And /msequently if we were to suppose the earth +be drawn towards the moon in consequence ‘this unequal attraction, it is obvious that, at part of the fluid nearest the moon would Il quicker than its centre, and those parts ‘ore remote would fall slower than the centre, tid consequently the mass on both these ac- yunts would be protruded, and assume a rt of spherical figure. Now the earth and moon revolving about ‘eir common centre of gravity, the motion “each may be assimilated with the falling “the one towards the other, as above sup- ‘osed. Vor their natural tendency is to | ‘oceedin the direction of the tangent of their bit, but they are constantly drawn from vis direction by their mutual attractions, and vay therefore be considered as falling con- tantly the one towards the other, and hence e see the first cause of the motion of the ‘aters of the ocean, which will, however, ‘quire a farther illustration. It appears from what is stated above, and must be carefully observed, that it is not se action of the mvon itself, but the inequa- ies in that action, which causes any varia- m from the spherical figure; and that, if | | yass into a perfect sphere; and if each of LID this action were the same in all its particles as in the central parts, and operating in the same direction, no such change would ensue. Let us now admit the parts of the earth to gravitate towards its centre; then as this gravitation far exceeds the action of the moon, and much more exceeds the difference of her actions on different parts of the earth, the effect that results from the inequalities of these actions of the moon, will be only a small diminution of the gravity of those parts of the earth which it endeavoured in the former supposition to separate from its centre; that is, those parts of the earth which are nearest to the moon, and those that are farthest from her, will have their gravity toward the earth somewhat abated; to say nothing of the la- teral parts. So that sapposing the earth fluid, the columns from the centre to the nearest and to the farthest parts must rise till by their greater height they be able to balance the other columus, whose gravity is less altered by the inequalities of the moon’s action. And thus the figure of the earth must still be an oblong spheroid. Let us next consider the earth, instead of falling toward the moon by its gravity, as projected in any direction, so as to move round the centre of gravity of the earth and moon: it is evident that in this case, the several parts of the fluid earth will still preserve their rela- tive positions ; and the figure of the earth will remain the same as if it fell freely toward the moon; that is, the earth will still assume a spheroidal form, having its longest diameter directed toward the moon. From the above reasoning it appears, that the parts of the earth directly under the moon, as at H, and also the opposite parts at D, will have the flood or high water at the same time ; while the parts, at B and I’, at 90° distance, or where the moon appears in the horizon, will have the ebbs or lowest waters at that time. Hence, as the earth turns round its axis from the moon to the moon again in 24 hours 48 minutes, this oval of water must shift with it; and thus there will be two tides of flood and two of ebb in that time. But it is farther evident, that by the motion of the earth on her axis, the most elevated part of the water is carried beyond the moon TID in the direction of the rotation. So that the water continues to rise after it has passed di- rectly under the moon, though the immediate action of the moon there begins to decrease, and comes not to its greatest elevation till it has got about half a quadrant farther. It con- tinues also to descend after it has passed at 90° distance from the point below the moon, toa like distance of about half a quadrant. The greatest elevation therefore is not in the line drawn through the centres of the earth and moon, nor the lowest points where the moon appears in the horizon, but all these about half a quadrant removed castward from these points, in the direction of the motion of rota- tion. ‘Thus in open seas, where the water flows freely, the moon M is generally past the north and south meridian, as at p, when the high water is at Z and at n: the reason of which is plain, because the moon acts with the same force after she has passed the meri- dian, and thus adds to the libratory or waving motion, which the water acquired when she was in the meridian; and therefore the time of high water is not precisely at the time of her coming to the meridian, but some time after, &c. Besides, the tides answer not always to the same distance of the moon, from the meridian, at the same places; but are variously affected by the action of the sun, which brings them on sooner when the moon is in her first and third quarters, and keeps them back later when she is in her second and forth; becanse, in the former case, the tide raised by the sun alone would be earlier than the tide raised by the moon, and in the latter case later. 2. We have hitherto adverted only to the action of the moon in producing tides; but it is manifest that, for the same reasons, the inequality of the sun’s action on different parts of the earth would produce a like effect, and a like variation from the exact spherical figure of a fluid earth. So that in reality there are two tides every natural day from the action of the sun, as there are in the lunar day from that of the moon, subject to the same laws; and the Tunar tide, as we have observed, is somewhat changed by the action of the sun, and the change varies every day on account of the inequality between the natural and the lunar day. Indeed the effect of the sun in producing tides, because of his immense dis- tance, must be considerably less than that of the moon, though the gravity toward the sun be much greater: for it is not the action of the sun or moon itself, but the inequalities in that action, that have any effect; the sun’s distance is so great, that the diameter of the earth is but as a point in comparison with it, and therefore the difference between the sun’s actions on the nearest and farthest parts, be- comes vastly less than it would be if the sun were as near as the moon. However, the immense bulk of the sun makes the effect still sensible, even at so great a distance; and, therefore, though the action of the moon has the greatest share in producing the tides, the <: a : TID action of the sun adds sensibly to it wh they conspire together, as in the full ar change of the moon, when they are nea in the same line with the centre of the eart and therefore unite their forces ; consequent in the syzygies, or at new and full moon, # tides are the greatest, being what are calle the spring-tides. But the action of the su diminishes the effect of the moon’s action j the quarters, because the one raises the wat in that ease where the moon depresses i therefore the tides are the least in the qui dratures, and are called neap-tides. . Newton has calculated the effects of th sun and moon respectively upon the tide from their attractive powers. The former } finds to be to the force of gravity, as 1- 12868200, and to the centrifugal force at tk equator as 1 to 44527. ‘The elevation of th waters by this force is considered by Newte as an effect similar to the elevation of th equatorial parts above the polar parts of th earth, arising from the centrifugal foree 4 the equator; and as it is 44527 times less, } finds it be 242 inches, or 2 feet and # an inek To find the force of the moon upon th water, Newton compares the spring-tides ¢ the mouth of the river Avon, below Bristo with the neap-tides there, and finds the pri portion as 9 to 5; whence, after several ni cessary corrections, he coneludes that tk force of the moon to that of the sun, in rai ing the waters of the ocean, is as 4.4815 to! so that the force of the moon is able of itse to produce an elevation of 9 feet 12 inel and the sun and moon together may produc an elevation of about 11 feet 2 inches, whe at their mean distances from the earth, or a clevation of about 122 feet, when the moon: nearest the earth. The height to which water is found to rise upon coasts of the ope and deep ocean, is agreeable enough to th computation. re Dr. Horsley estimates the force of the moe to that of the sun as 5.0469 to 1, in his edi of Newton’s Princip. See the Princip. lib.{ sec. 5, pr. 36 and 37 ; also Maclaurin’s Disser de Causa Physica Fluxus et Refluxus Mar apud Phil. Nat. Princ.-Math. Comment ] Seur et Jacquier, tom. iii. p. 272. And othe calculators make the ratio of the forces sti greater. ‘| 3. It must be observed that the spring-tid¢ do not happen precisely at new and full mooi nor the neap-tides at the quarters, but a da or two after, because, as in other cases, soi this, the effect is not greatest or least whe the immediate influence of the cause is greate: or least. As e.g. the greatest heat is not @ ihe solstitial day, when the immediate actie of the sunis greatest, but some time after. — That this may be more clearly understood, k it be considered, that though the actions A the sun and moon were to cease this momen yet the tides would continue to have the course for some time; for the water, whet it is now highest, would subside, and flor down on the parts that are lower, till, by th = di) bP aotion of descent, being there accumulated o too great a height, it would necessarily re- arn again to its first place, though in a less aeasure, being retarded by the resistance rising from the attraction of its parts. Thus {would for some time continue in an agita- ion like to that in which it is at present. The yaves of the sea, that continue after a storin eases, and every motion almost of a fluid, ay illustrate this. 4. 'The different distances of the moon from he earth produce a sensible variation in the ides. When the moon approaches the earth, er action on every part increases, and the ifferences of that action, on which the tides pend, increase. Tor her action increases as he squares of the distances decrease; and hough the differences of the distances them- elves be equal, yet there is a greater dispro- ortion betwixt the squares of less, than the quares of greater quantities; e. @. 3 exceeds ‘as much as 2 exceeds 1, but the square of is quadruple of the square of 1, whilst the quare of 3 (viz. 9) is little more than‘ double 7€ square of 2 (viz. 4). | Thus it appears, that by the moon’s ap- roach, her action on the nearest parts in- reases more quickly than her action on the emote parts; and the tides, therefore, in- fease in a higher proportion as the distances the moon decrease. Sir Isaac Newton aows, that the tides increase in proportion ‘the cubes of the distances decrease, so that a€ moon, at half her present distance, would toduce a tide cight times greater. 'The moon describes an ellipse about the arth, and in her nearest distances produces ‘tide sensibly greater than at her greatest istance from the earth: and hence it is, that great spring-tides never succeed each ther immediately ; for if the moon be at her earest distance from the earth at the change, ie must be at her greatest distance at the il, having in the intervening time finished alfa revolution; and, therefore, the spring- fle then will be much less than the tide at te change was: and for the same reason, if great spring-tide happens at the time of full on, the tide at the ensuing change will be SS. ‘98. The spring-tides are greatest about the me of the equinoxes, z.e. about the latter idof March and September, and least about ie time of the solstices, 7.e. toward the end "June aud December; and the neap-tides ’é least at the equinoxes, and greatest at ‘solstices; so that the difference betwixt l€ spring and the neap-tides is much less msiderable at the solstitial than at the equi- ietial seasons. In order to illustrate and mince the truth of this observation, it is ma- fest, that if either the sun or moon was in e pole, they could have no effect on the, des, for their action would raise all the water the equator to the same height; and any ace of the carth, in describing its parallel to: € equator, would not meet, in its course, ~ ith any part of the water more cleyated than TID another, so that there could be no tide in any place. 6. ‘The effect of the sun or moon is greatest when in the equinoctial; for then the axis of the spheroidal figure, arising from their action, moves in the greatest circle, and the water is put into the greatest agitation; and hence it — is that the spring-tides produced, when the sun and moon are bothin the equinoctial, are the greatest of any, and the neap-tides are the least of any about that time. But the tides produced when the sun is in either of the tropics, and the moon in either of her quarters, are greater than those pro- duced when the sun is in the equinoctial, and the moon in her quarters, because, in the first case, the moon is in the equinoctial; and, in the latter case, the moon is in one of the tropics; and the tide depends more on the action of the moon than on that of the sun, and is, therefore, greatest when the moon’s action is greatest. However, it is necessary to observe, that because the sun is nearer the earth in winter than in summer, the greatest spring-tides are after the autumnal, and before the vernal equinox. 7. Siace the greatest of the two tides hap- pening in every diurnal revolution of the moon is that wherein the moon is nearest the zenith, or nadir; for this reason, while the sun is in the northern signs, the greater of the two diurnal tides in our climates is that arising from the moon above the horizon; when the sun is in the southern signs, the greatest is that arising from the moon below the horizon. In proof of this observation, let it be con- sidered, that when the moon declines from the equator toward either pole, one of the greatest elevations of the water follows the moon, and describes nearly the parallel on the earth’s surface which is under that which the moon, on account of the diurnal motion, seems to describe ; and the opposite greatest elevation, being antipodal to that, must describe a pa- rallel as far‘on the other side of the equator; so that while the one moves on the north side of the equator, the other moves on the south side of it, at the same distance. Now the greatest elevation which moves on the same side of the equator, with any place, will come nearer to it.than the opposite elevation, which moves ina parallel on the other side of the equator; and therefore, if a place is on the same side of the equator with the moon, the day-tide, or that which is produced while the moon is above the horizon of the place, will exceed the night-tide, or that which is pro- duced while the moon is under the horizon of the place. Itis the contrary if the moon is on one side, and the place on the other side of the equator; for then the elevation which is opposite to the moon, moves on the same side of the equator with the place, and, therefore, will come nearer to it than the other eleyva- tion. ‘This difference will be greatest when the sun and moon both describe the tropics ; because the two elevations in that case de- p ig) ee scribe the opposite tropics, which are the farthest from each other of any two parallel circles they can describe. ‘Thus it is found, by observation, that the evening tides in the summer exceed the morning tides, and the morning tides in winter exceed the evening tides. The difference is found at Bristol to amount to fifteen inches, and at Plymouth to one foot. It would be still greater, but that a fluid always retains an impressed motion for some time ; so that the preceding tides affect always those that follow them. Upon the whole, while the moon has north declination, the greatest tides in the northern hemisphere are when she is above the horizon, and the reverse while her declination is south. Such are the general principles on which these interesting phenomena depend. But it may be expedient to touch upon a few of the more particular appearances, which at first sight appear to contradict some parts of the preceding theory. Now, it is evident, that to allow the tides their full motion, the ocean in which they are produced ought to be extended from east to west 90 degrees at least; because that is the distance between the places where the water is most raised and depressed by the moon. Hence it appears that it is only in the great oceans that such tides can be produced, and why in the larger Pacific ocean they ex- ceed those in the Atlantic ocean. Hence also it is obvious, why the tides are not so great in the torrid zone, between Africa and America, where the ocean is narrower, as in the temperate zones on either side ; and hence we may also understand why the tides are so small in islands that are very far distant from the shores. It is farther manifest that, in the Atlantic ocean, the water cannot rise on one shore but by descending on the other; so that at the intermediate islands it must continue at a mean height between its clevations on those two shores. But when tides pass over shoals, and through straits into bays of the sea, their motion becomes more various, and their height depends on many circumstances. To be more particular. The tide that is produced on the western coasts of Europe, in the Atlantic, corresponds to the situation of the moon already described. Thus it is high water on the western coasts of Ireland, Por- tugal, and Spain, about the third hour after the moon has passed the meridian: from thence it flows into the adjacent channels, as it finds the easiest passage. One current from it, for instance, runs up by the south of England, and another comes in by the north of Scot- land; they take a considerable time to move all this way, making always high water sooner in the places to which they first come ; and it begins to fall at these places while the enr- rents are still going on to others that are far- ther distant in their course. As they return they are not able to raise the tide, because the water runs faster off than it returns, till, by a new tide propagated from the open ocean, the return of the current is stopped, and the water begins to rise again, The tide propa- TIM gated by the moon in the German oce when she is three hours past the meridia) takes twelve hours to come from thence London bridge; so that when it is high wate there, a new tide is already come to its heig in the ocean; and in some intermediate plac it must be Jow water at the same time. Cor sequently when the moon has north declin; tion, and we should expect the tide at Londe to be the greatest when the moon is above th horizon, we find it is least; and the contra when she has south declination. At several places it is high water thr hours before the moon comes to the meridial but that tide, which the moon pushes as were before her, is only the tide opposite that which was raised by her when she wi nine hours past the opposite meridian. It would be endless to recount all the pa ticular solutions, which are easy consequence from this doctrine ; as, why the lakes and sea such as the Caspian sea and the Medite) ranean sea, the Black sea and the Balti have little or no sensible tides: for lakes a} usually so small, that when the moon is ve tical she attracts every part of them alike, | that no part of the water can be raised high} than another; and having no communicati( with the ocean, it can neither increase diminish their water, to make it rise and fal) and seas that communicate by such narré inlets, and are of so immense an extent, ¢a) not speedily receive and empty water enous, to raise or sink their surface any thing sensib} Such are the principal consequences ded cible from the Newtonian theory; and, con} dering the state of science at the time | which it was produced, it was certainly! most wonderful effort of human genius, | more intimate acquaintance, however, Wj the solar system, and a more complete dey lopement of the effects of motion and attré tion discovered by Newton, have enabll subsequent philosophers, as Maclaurin, Eul} D’Alembert, Laplace, &c. gradually to exte and perfect the theory so admirably strat out by the great British philosopher. Laplé} especially, in Mem. de ?Academie Par, 1775, 1789, and 1790, and more fully in t Mécanique Céleste, has treated this subj more completely than any of his predecessi in the same region of inquiry. ‘To whi) works we beg to refer the reader who is ¢ sirous of more particular information. TIDE Dial,an instrument invented by I Ferguson for exhibiting the state of the tid« Tipe Tables, are tables showing the time! high water at different places, as they hapyl on the days of the full and change of the mo) TIERCE, an English measure of capaci; containing 42 gallons, or the third part 0 pipe. TIMBER Measure, is the method ¢ ployed by artilicers in measuring trees, joi , beams, &c:; and as these always fall uni) one or other of the regular solids which hi been already treated of under the several ticles, it would seem unnecessary to rep TIM any rules for the mensuration of timber ; ‘the fact is, that an erroneous rule has 1: adopted by persons concerned in this ) of business, which common practice has | stablished, that it is rather to be wished expected, it should be replaced by some ir, either perfectly true, or approaching juds the truth; for according to the pre- rulea tree frequently contains one-fourth (3 timber than it is estimated at, which at modern price of that article is a matter of simportance, and merits the attention of imber grower, as wellas the merchants. ae rule as it is at present employed, by jersons concerned in the buying or selling inber, is as follows: ‘ultiply the square of the mean quarter girt, circumference, by the length of the tree, for icontent ; which when the dimensions are lz in feet, will be also feet ; and this divided 1), the number of feet in a load, will give the ver of loads. ote 1. If the piece of timber is of the same throughout, the girt any where taken is jnean girt. | If the tree tapers regularly from one jto the other, the girt taken in the middle }counted the mean git; or take half the of the girts at the two ends for the same. , But if the tree do not taper regularly, 4is unequal, being thick in some places, jsmall in others; it is customary to take Val different dimensions, the sum of which ‘ed by the number of them is accounted mean girt. But when the tree is very ‘ular, it is best to divide it into several ‘hs, and to find the content of each sepa- D1 Ly. + That part of a tree, or of the branches, eZ girt is less than 4 a foot, is not ac- ted timber. Jitis usual to make a certain allowance ‘ting a tree for the thickness of the bark, 1a is generally one inch to every foot in firt. ‘This practice, however, is unreason- | and ought to be discouraged. Elm tim- is the only kind in which any allowarice yessary, and even in this one inch out of trhole girt is quite sufficient. } an example in the preceding rule, let it (quired to find the content of a tree, the ih of which is 9 feet 6 inches, and quarter |} feet 6 inches. By Decimals By Duodecimals. 35 3—6 3°5 3-—6 175 10—6 105 1— 9 12°25 12—3 95 9—6 61 25 110—3 14025 6-14 16375 — content — feet 116 — 42 — US ae Pe Te ‘this the rule commonly used by persons Wool in buying and selling of timber, \uch we intend to make a few remarks, TIM in order to point out its inaccuracy, which is not so generally known as it ought to be. Suppose for instance we take a balk 24 feet long, and a foot square throughout, and con- sequently its solidity 24 feet. Now if this piece of timber be slit exactly in two, from end to end, making each piece 6 inches, or } a foot broad, and 12 inches, or a foot thick ; it is evident that the true solidity of each piece will be 12 feet. But by the quarter girt method, they would amount to much more ; for the false quarter girt being equal to half the sum of the breadth and thickness in this case will be 9 inches, or 3 of a foot; the. square of which is 2, and therefore 2. x 24 — 133 feet for the solidity of each part, making the two pieces together 27 feet, in- stead of 24 feet, which is the true content. Again, suppose this balk to be so cut, that the breadth of the one piece may be only 4 inches, or 4 of a foot; and that of the other 8 inches, or 3 of afoot. Here the true con- tent of the less piece will be 8 feet, and that of the greater 16 feet. But proceeding by. the other method, we have the quarter girt of the less piece 2 of a foot, and of the other piece 3 of a foot. Whence the content of the less piece will be found = 4 x 24= 102 feet, instead of 8 feet; and the content of the greater piece will be 16% feet instead of 16, making the sum of the two 272 feet, instead of 24 feet. Farther, if the less piece be cut only 2 inches broad, and consequently the greater 10 inches, the true content of the less piece would be 4 feet, and that of the greater 20 feet. Whereas by the other method the quarter girt of the less piece would be 7 inches, or 4% of a foot; and 2 x 24=82 feet, instead of 4 feet, for the content; and by the same method the content of the greater piece would be 202 fect, instead of 20, and their sum 282 feet, instead of 24, Hence it is obvious, that the greater the proportion is between the breadth and the depth, the greater will be the error, by using the false method; and the sum of the two parts, by the same method, is greater, as the difference of the same two parts is greater ; and consequently, the sum is least when the two parts are equal to each other; or when the balk is cut equally in two; and finally, when the sides of a piece of timber differ not above an inch or two from each other, the quarter girt may be used without any very sensible error. ‘To avoid therefore this incon- sistency in the result, the following method should be employed, viz. Multiply the length, breadth, and depth continually together, and the product will be the true content in all cases of this kind. With regard to round timber the error is of a different kind. We have seen in the article CirncLeE, that the area of a circle is found by squaring the circumference and multiplying that square by :07958, and there- fore, if a quarter of the circumference is used, we must multiply its square by 07958 x 16 = 1:27328. Hence, to find the true content ef TIM a piece of cylindrical timber, we ought to multiply the square of the quarter girt, by the constant number 1°27328, and that product by the length; instead of which the constant multiplier is omitted, and consequently the solidity is returned about 24, parts less than itis. But as the utmost accuracy is not ne- cessary in those cases, the following rule might be used which is as simple as can be desired, viz. Multiply the square of = of the mean girt by double the length for the content, which is not far from the truth. Another error to which timber measure is always subject, is the way in which the mean girt is assumed in tapering trees, which, as we have before stated, is done, either by taking the girt in the middie, or half the sum of the extreme girts, both of which are equally false, so obviously so that a tree of certain dimen- sions will measure more after a part of it has been cut off than it did before. This being the case it will not be amiss to show the ex- tremeinaccuracy ofthe method, and the folly in persisting in it by the solution of the follow- ing problems, whichyhave been taken from Dr. Hutton’s Mensuration. Prop. 1. To find where a tapering timber must be cut, so that the two parts measured separately shall measure the most possible, or be greater than if it were cut in any other two parts, and greater than the whole. Put G = the greatest girt, g — the least girl, x the girt at the section, z = the length of the part to be cut off, and L the whole length of the timber. ‘Then by similar firures L: zi: G—g:x —g; hence x = A Seecooent C28 tg But +2).2+G +ay. (L — z), is to be a maximwn ; which being put into fluxions and reduced gives z = # L. Therefore a tree being cut exactly in the middle into parts, will measure more than if it were cut in any other two parts, and more than the whole tree. If a tree, of which the greater girt is 14 feet, and less girt 2 feet, and Jength 32 feet, be thus.cut in two parts; the measure of the two parts will exceed the mea- sure of the whole tree by 18 feet. _ Prop. 2. To find where a tree must be cut, so that the part next the greater end may measure the greatest possible. Here, by using the same notation as in the Gz—gz —_— — last problem, we have also, x = : 185 L Pry and (G + 2)*. (LL — x) a maximum, which : ; , G—3 put into fluxions as before, gives z = G & — or : 5 x i¢ds. Therefore, from the greater girt, subtract three times the less girt, and that differeuce divided by the difference of the girts, and multiplied by } of the whole length, will be the length to be cut off. _ Prop. 3. To find where a tree must be cut, so that the part next the greater end may measure the same as the whole tree before it was cut, : TIM a Using still the same notation, and Wi besides, s for the sum of the two girts, a for their difference; we shall have s*] (L.—z)(G + x)’, or substituting instead o its value sated fa +g, or + g,we ob — 2 m) which length being cut off, the remain part will measure the same as the whole These results, which are the necessary ¢ sequence of the preceding rules, are s0 viously incongruous and inconsistent thatt speak for themselves, and therefore reg no farther comment. e TIME, a succession of phenomena in; universe; or a mode of duration, marked) certain periods or measures, chiefly by motion and revolution of the sun. Thei of time, in the general, Mr. Locke obser we acquire by considering any part of infij duration as set out by periodical measu} the idea of any particular time, or lengt) duration, as a day, an hour, &c. we aeq| first, by observing certain appearances aj} gular, and, seemingly, at equidistant peril Now, by being able to repeat those lengt ( measures of time as often as we will, wei imagine duration where nothing really end} or exists; and thus we imagine to-mor) next year, &c. Some of the latter sel philosophers define time to be the durati¢ a thing whose existence is neXher wit) beginning nor end: by which time is} tinguished from eternity. ) Time is distinguished into absolute and} tive. i] Absolute Time, is time considered in if and without any relation to bodies, or | motions. This flows equally, 7. e. never} ceeds faster or slower, but glides on | constant, equable tenor. i Relative 'Time, is the sensible measu) any duration, by means of motion. | since that equable flux of time does not ; our senses, nor is any way immediately; nizable thereby, there is a necessity for ci} in the help of some nearly equable moti) a sensible measure, whereby we may q mine its quantity by the corresponden) the parts of this with the former. H) as we judge those times to be equal ¥ pass, while a moving body, proceeding} an equable velocity, passes over equal sy so we judge those times to be equal % flow while the sun, moon, and other | naries, perform their revolutions, whit, our senses, are equal. But since the fit time cannot be accelerated nor reti whereas all bodies may move sometimes} and sometimes slower, and there is, pe} no perfectly equable motion in all natu) appears, hence, to follow, that absolute) should be something truly and really di) from motion. Tor let us suppose the he} and the stars to have remained withou) tion from the very creation; does it } TOP Now, that the course of time would have m at a stand? Or rather, would not the fration of that quiescent state have been ial to the very time now elapsed? \tince absolute time is a quantity uniformly ended, and, in its own nature, most siinple, i3 hence represented by mathematicians, to |) imagination, under the most simple, sen- /e magnitudes, and particularly right lines i (circles, with which it bears a near analogy, jrespect of its genesis, similarity, &c. tis not indeed necessary, that time should neasured by motion; any constant periodi- : appearance inseemingly equidistant spaces, e freezing of water, the blowing of a plant, i returning at set periods, might do as well. effect, Mr. Locke meutions an American ple, who count their years by the coming ‘going away of birds. e authors distinguish time into astrono- ‘and civil. | stronomical Time, is that measured by the lion of the heayenly bodies only. jie! 'Timr, is the former time accommo- vd to civil uses, and formed and distin= thed into years, months, days, &c. stronomical 'Vime, is, again, either ap- ‘nt solar, or true; mean, or sidereal. \pparent Solar Time, otherwise, thongh loperly, called True Time, is that which is inated by the sun’s passage over the me- im of any place, and which is sometimes ach as 16 minutes sooner or later than shown by a good clock. See Equation ‘ime. ean TIME, is the apparent solar time cor- Vd according to the method described i'r Equation of Time. ‘This mean time is not exactly the same as sidereal time, vart added or subtracted still partaking of inequalities it is meant to correct; it ap- ithes, however, so near the truth, that no | is to be feared, except in cases of the ( extreme delicacy. dereal Time, is that which is estimated .€ passage of the same star over the me- 4 of any place, and which (except for | trifling inequalities in the motion of the (which astronomers are able to correct i. Necessary) is the most exact measure pne we are acquainted with, the most fate observations having never detected jany inequality, though it is suspected jty be subject to some, which must, Iver, if there be any, be extremely mi- t uation of Time. See Equation of Time. 11 Keepers, in a general sense, denote ments adapted for measuring time, being |Wise called CHRONOMETERS. LD, a weight principally used in buying alling wool, it is equal to two stone or ds, - 3E,a French measure containing two sh feet. See Foor. POGRAPHY, is a description of some lular place or small tract of land, differ- TOR ing from geography as a part does from & whole. TORNADO, a sudden and violent gust of wind arising suddenly from the shore, and afterwards veering round all the points of the compass like a hurricane ; these are very fre- quent on the coast of Guinea. TORPEDO, the name of a fish possessing an electric property, similar to what is de+ scribed under the article GyMNorus. TORRENT, a violent stream of water rush- ing suddenly from mountains, &c. and fre- quently making great ravages in the planes below. TORRICELLI (LvanGeELtstp), a celebrat- ed Italian mathematician and philosopher, was born in 1608. He studied under Castelli, who had been a pupil of Galileo’s, and under whom he made great progress; which being made known to the latter, attracted the atten- tion and gained him the esteem and friendship of that eminent philosopher; and it was agreed between them that Torricelli should come and reside with Galileo to assist him in the ad- justment of his papers, &c. for the press, which his own great age and want of sight prevented him from doing for himself. He did not long, however, continue in this situation, his ve- nerable master dying about three months after this arrangement: was made. Torricelli was about returning to Rome, but the grand duke engaged him to continue at llorence, making him his own mathematician for the present, and promising him the professor’s chair as soon as it should be vacant. Here he applied himself intensely to the study of mathematics, physics, and astronomy, making many im- provements, and some discoveries. Among others, he greatly improved the art of making microscopes and telescopes; and it is ge- nerally acknowledged that he first found out the method of ascertaining the weight of the atmosphere by a_ proportionate column of quicksilver, the barometer being called from him the Torricellian tabe, and the experiment with that tube the Torricellian experiment. Great things would probably have been far- ther performed by him, if he had lived; but he died, after a few days illness, in 1647, when he was but just entered the 40th vear of his age. His principal work was entitled Opera . Geometrica, in 4to. . TORRICELLIAN Experiment, a famous experiment made by Torricelli, by which he demonstrated the pressure of the atmosphere, in opposition to the doctrines of suction, &e. which pressure he shew, was able to support only a eertain length of mercury, or any other fluid, in an inverted glass tube. See Arr, ATMOSPHERE, BAROMETER, &e. ToRRICELLIAN Tube, Vacuum, &e. Tuse, Vacuum, &c. . TORRID Zone, is the middle of the five zones into which the earth is supposed to be divided, extending to 234 degrees on each side of the equator. See Zone. TORSION, Force of, a term applied by LZ See TRA Coulomb in some of his experiments, to de- note the effort made by a thread which has been twisted to untwist itself, TOUCON, the American Goose, a southern - constellation. See CONSTELLATION. TRACTION, in Mechanics, is the drawing of one body tow ards another. TRAJECTORY, a term generally used _ for the path of any body moving either in a void, or in a medium which resists its mo- tion; or even for any curve passing through a given number of pots. Thus Newton, Prin- cip. lib. i. prob. 22, purposes to describe a tra- jectory that shall pass through tive given points. Trasecrory of a Comet, is its path or orbit, or the line it describes in its motion. This path was thought by Hevelius, in his Come- Seton to be very nearly a right line; but Dr. Halley considers it to be, as it really i is, a very eccentric ellipsis; though its place may often be well computed on the suppo- sition of its being a parabola. Newton, in prop. 41 of his 3d book, shows how to deter- mine the trajeciory of a comet from three observations; and in his last’ prop. how to correct a trajectory graphically described. TRAMMELS, in ‘Mechanics, an instrument used by artificers to describe ellipses. See Elliptic COMPASSES. The engines for tuming ovals are con- structed on the same principles with the tram- mels; the only difference being, that in the latter the boards at rest and the pencil moves, whereas in the latter the tool which supplies the place of the pencil is at rest, and the board moves against it. See a demonstration of the principal | properties of these instruments by Mr. Ludlam in the Phil. Trans. vol. Ixx. p. 878, &c. or New Abridgment, vol. xiv. p. 700- 704, TRANSACTIONS ( Philosophical ), are a collection of the principal papers and matters read before certain philosophical societies, as the Royal Society of London, and the Roy al Society of Edinburgh. These Transactions contain the several ‘discoveries and histories of nature and art, either made by the mem- bers of those societies, or communicated by them from their correspondents, with the va- rious experiments, observations, &c, made by them, or transmitted to them, &e. The Philosophical Transactions of the Royal Society of London were began in 1665, by Mr. Oldenburg, the then secretary of that society, and were continued by him till the year 1677. They were then discontinued upon his death, till January 1678, when Dr. Grew resumed ‘the publication of them, and con- tinued it for the months of December 1678, and January and February 1679, after which they were intermitted till January 1683. During this last interval, however, they were in some measure supplied by Dr. Hook’s Phi- . losophical Collections. ‘rupted for three years, from December 1687 to January 1691, beside other smaller inter- ruptions, amounting to near a year and a half They were also inter- 5 more, before October 1695, since which tf the Transactions have been carried on } gularly to the present day, with various « grees of credit and merit. t 4 Till the year 1752 these Transactions } published in numbers quarterly, and the pri ing of them was always the single act ¢ . respective secretaries till that time; but f th the society thought fit that a committee sho be appointed to “consider the papers read” fore them, and to select out of them suek they should judge most proper for publicat in the future Transactions. For this pu rp the members of the council for the time be constitute a standing committee: they m on the first Thursday of every month, and less than seven of the members of the € mittee (of which number the president, ¢ ¢ his absence a vice-president, is always fe one) are allowed to be a quorum, capal acting in relation to such papers; an question with regard to the publication of paper, is always decided by the majoriti votes taken by ballot. hi They are published annually, in two p at the expense of the society; and each| low, or member, is entitled to receive copy gratis of every part published ay | admission into the society. Vor many ¥ past, the collection, in two parts, has one volume in each year; and in the } 1832 the number of the volumes was 102, There is also a very useful abridgmel| those volumes of the Transactions w hich y published before the year 1752, when| society began to publish the Tr ansaction| their own account. Those to the end oll year 1700 were abridged, in three volume} Mr. John Lowthorpe ; those from the } 1700 to 1720, were abridged, in 2 volumes Mr. Henry Jones; and those from 17) 1733 were abridged, in 2 volumes, by} John Eames and Mr. John Martyn? Martyn also continued the abridgment of from 1732 to 1744, in 2 volumes, and of j from 1744 io 1750, in 2 volumes ; maki) all 11 volumes. 4 Another very valuable abridgment has? recently published by Drs. Hutton, Shaw} Pearson, in 18 vols. 4to. with various 1 critical, scientific, and biographical: an Thomson has since given what may bi garded as an abridged sketch of the T actions i in his very interesting History 0! Royal Society, in 1 vol. 4to. The Royal Society of Edinburgh, insti in 1783, have also published 10 volun! their Philosophical ‘l'ransactions, whiell deservedly held in the highest respect fo) importance of their contents. The Society of Arts, &c. have likewis : lished about 30 volumes of ‘Transacti¢ 1 8yo. abounding with mechanical inven) discoveries in chemistry, kc. 'There 3 Transactions of the American Society, ¢ Manchester Philosophical Society, of the} necticut Society. ‘The Irish Acai TRA {st of the foreign philosophical societies, 2 to their Transactions the title of Me- irs. PRANSCENDENTAL, is a term applied yany equation, curve, or quantity, which ‘not be represented or defined by an al- | raical equation of a finite number of terms, h nameral and determinate indices. RANSCENDENTAL Quantities, therefore, in- ide all exponential, logarithmic, and triyo- yaetrical lines, because there is no finite sbraical formule by which these quantities , be expressed. . \Tence a*, xz, log. x, sin. w, cos. x, tan. a, &e. [enscendental quantities ; and any equation i) which such quantities enter, are called prscendental equations ; and every curve de- id by such an equation is, for the same son, termed a transcendental curve. PRANSCENDENTAL Equations, are, however, jietimes defined to be such fluxional equa- fs as do not admit of finite algebraical mts, being only expressible by means of ie curve, logarithm, or infinite series; as jie case in all those relating to the trigono- }rieal lines as above stated; thus y = fluent &e. ; " M i Hos 7(a*— x") (ax —a)”’ | according to this definition, transcenden- ‘equations. nd for the same reason atry curve, of eh the equation is expressed by means of fafluents, is a transcendental curve ; thus tycloid whose equation is y = Vv (24x— a’) ‘ Ment of Be eee is a transcendental v (2ax— 2") &. such also is the CATENARY, ELASTIc ye, and several others, which will be found er the respective articles. See also the bles EXPONENTIAL, LoGaritums, &c. But seomplete development of this interesting ich of analysis, the reader should consuit ers Analysis Infinitorum, vol. i.; and ace’s Logarithmic Transcendents, 4to. don, 1809. RANSFORMATION, in Geometry, is the same as reduction, being used to ite the changing of any proposed figure different one of equal surface or solidity. RANSFORMATION of Equations, in Algebra, nethod of changing an equation to another », but of equal value. his is of various kinds, as 1. 'To transform quation to another whose roots are greater $8 than the roots of the proposed equa- _ 2. To transform an equation to another se roots shall be some multiple or quotient ae proposed equation. 3. To transform mation to another that shall wait the pnd or any other term. 4. To transform uange the signs of the roots of an equa- from + to —, or from — to +, &c. 'To transform an equation to another, se roots shall be greater or less than the sof the proposed equation. feta” + ax"—1 + ba®—? + &C. =O WHY equation; and let it be proposed to and y = fluent — Vv TRA transform it to another whose roots shall be less than those of the above by a given quan- tity d. " Assuine y to: represent the unknown quan- tity of the new equation, and as this is to be less than x by the quantity d, we have y = «—d, or «= y +d; now substituting this value for x, we obtain acm > y” a. Ww dy" a an eye + &e. ax*—— ay"'+ a(n—1) dy"*+&ce. base by" +&e, Whence by adding together the co-eflicients of the like powers of y, we have n.(n—T)> Li gt —2 = 4! ain— Dag? nits b for the equation required. If the roots of the equation are to be in- creased, then we must assume y =a + d. As an example, let it be proposed to increase the roots of the equation x? — 6a” + lla—6 = 0, by unity, or 1. Assume y =a + I, or x —y —1; then emy—3y + By— 1 — 6x77 = —6y?.+ 1l2y— 6 flle = lly — 11 — 6= — 6° by addition y3 — 9y* + 26y — 24 = 0 is the equation sought. 2. To multiply or divide the roots of an equation by any given quantify. Let 2” + aa"—! 4+ ba®—-? + &. =O be the proposed equation; which is to be transformed to another, whose roots shall be some multiple of those above. Assume y = mx, m being, the propesed multiple; then « = and substituting this value for x in the proposed equation, we have n N—1 . n~—2 ER AN REIS ARE UY CE} mn m”— ae Or multiplying by m” | yf - may" —* + m*by*—* + Kee 0; which is an equation, whose roots are equal to the given multiple m of the roots of the pro- posed equation. Hence we sce, that to mul- tiply the roots of an equation by any quantity” m, it is only necessary to multiply the several co-efiicients; beginning at the first by the terms of the geometrical progression 1, m, m’, m3, m+, &e. observing only that if any term in the original equation be wanting, it must be introduced in its proper place, having zero for: its co-efficient; thus Let it be proposed to transform the equation at + 603+ 3e74+72—0 , to another equation, whose roots are the doubles of these roots. Here, supplying the third term, which is wanting, 4 proposed equat. a+ 523 +0x"+ 3e4+ 7=—0 geom. series 1 2 4 & 16 y* + Loy? + 24y + 1120 “ZL 2 TRA which is the equation required, the unknown quantity being changed from x to 4 Ks merely for the sake of distinction. When the roots of the equation are to be eee by any quantity m, we must substitute es of x we have m” y” LL aa ei 4+ bm"™—2y®—? + or, dividing by m”, b 4 y" + < yn 4 oe + &c. = 0; that is, we must divide the co-eflicients by the same series 1, m, m*, m3, m*, &c. 3. In a similar way an equation may be transformed to another, whose roots are the reciprocals of the roots of the first. Let 2” + ax"—! + ba®—-2? + &e. — 0 be the proposed equation; and let it be re- quired to transform this to an equation, whose roots shall be the reciprocals of these roots. ~, or x — my, and ee aa this value &c. = 0; Assume y = =; Orenis : ~, and substitute y this value of x in the proposed equation; and we have Se ee bh ee, ¥y ¥ oats or 1.4.ay + by? .... wy", &c.j= 0; or, dividing by w, and ata. the order of the equation, y™ b y” 4. mak y “Pp “ =—i0 aft hens) m ion o> the equation required. That is, we must invert the order of the co-efficients, and divide by the absolute quan- tity of the proposed equation ; observing here, as in the preceding case, to supply any terms of the equation that may be wanting, by pre- fixing a cipher to them for a co-eflicient. Transform the equation x* +3054 227 +4-—0 to another whose roots shall be the reciprocal of these. Here, supplying the deficient term, and inverting the order of the co-efficients, we have 4+ 2x2 + O27? + 3823 + at; then dividing by 4, and introducing y, we have 1 fhe par pner aie - or y*t+— su a a4 +5 — — 0, as required. 4. To in the signs “of the roots of an equation. Let a” + ax®—' + ba"-2 + ea"-3 4 &e. —0 be an equation, the signs of whose roots are to be changed. Assume y = —, or r= —y, and sub- stitute this for x in the proposed equation; so shall the roots of the proposed equation be changed as required. Here it is obvious, that only the signs of the odd powers of x wiil be changed, because the even powers of a negative quantity have the same signs’as a positive one; and since, ys + By* + Oy} —7y* + By --4=9, ory? + 3yt+—7y*? + 6y —4 =i es ' the equation sought. ‘ TRA: ' also, no change takes place in the valaell ( f equation, when all the signs of it are chang we may reduce what has been said abo ve the following general rule. To change signs of the roots of an equation, supply su terms as are wanting, and then change t signs of all the even n terms, viz. the 2d, ¢ 6th, &c. from the left hand; and the equg ti will be transformed as required. Let it be required to change the signs the roots of the equation, t w—3at+ 7a? 4+ 6a +4—0.° First, supplying the third term, which Ww anting, the equation becomes . x —3at+ 023 + 707+ 62 +425 a Changing the alternate terms, and writ y for x, we ‘have . 5. To transform an equation to another t] shall want the second, or any other “| the equation. Let 2" + ax®—-1 + ba®-2 + &. = 0, be any equation which is to be tr asta into another that shall want any requi . term. | Assume x = y + d, and substitute | value for x,and we obtain n “= | | Dp | x =y" +ndy"—! 4+ ——_— “ya } Gx CD belies ao 1d pa 5 ' by-2— b yf? a Now equate the co-efficient of that por of y which is to be exterminate to zero, ¢ hence find the numerical value of d, wh substituted for d will give the equation quired. As from the nature of the binomial thean| the second term involves the quantity d ‘| the first degree, the third term in the see} degree, the fourth term in the third de : Ke. it follows, that to exterminate the see term, the equation will be a simple one, may therefore always be effected in ra numbers; but to exterminate the third tl we must solve a quadratic; and to exter } nate the fourth, a cubic, and so on. i Thus in the above equation, to extern rit! the second term, we have ee nd +a=0, ord = HE! ain} n ay To exterminate the third term, © | cok ad 74 alg i ea ib ton os og n nr —n A Sony eae a d* n See EXTE 2 or d* HE Whence d = — and so on for any Ae term. NATION and ELIMINATION. See also |] laurin’s Algebra, Simpson’s a iccboal Bonnycastle’s ‘Treatise on the same subje} TRANSIT, in Astronomy, signifies the} sage of any planet over a fixed star, or te} A T TRA +; and of the moon in particular, covering moving over any planet. ‘he transits of Mercury and Venus over sun’s disc are very interesting phenomena, merely by reason of their rare and singular earance, but because of their use in deter- fing the parallax of the sun, and thence freal dimensions of the earth’s orbit. Hence | times when these transits are to be seen ¥e been very carefully computed. he following are the days and years when he was or will be transits of Mercury, from hyears 1753 to 1894 inclusive. 138 ee May 5. | 1832......... May 5. JG ......... Noy. 6. | 1838 ......... Nov. 7. | == Nov, 9. | 1845 °.:.....:. May 8 AG ........ Noy. 2. | 1848 ......... Nov. 9. eee IOOe hee FROG. iis nasnts Nov. 11. NG .....06 May 5, | 1868 ........ Nov. 4. Hi9)....... VOWS ah. | LETS toe cve May 5. imarss-.... May 7.'| 1881 ......... Nov. 7. , ee THOW, Ged FOGE siceserts May 9. e,....... Nov. 11. | 1894 ......... Nov. 10. | ee Noy. 4. “appears from this table that the transits ar Observation. | Reduction. | OMet tea ’ h. m. s. $. A. ms. | = 0 734 |—17:398| 0 7 166 tha he 0 41 158j—143 | 041 15 recht...... 0 18 405|—16:8 | 0 18 23:7 tyden..... 01553 |—17-1 | 0 15359 | eenwich./23 58 11 j}—18-7* (23 57: 52:3 \GVon Zach observed the interior contact |’ Which must be reduced by allowing 105 for the time between the contacts; reduced to apparent time it gives 0" 41™ i for the time of observation. jie shortest distance of the centres, com- }1 with the distance at the moment of the - lis, gives the middle of the transit at 21 128'6*, and hence the time of apparent imetion at 21" 16™ 2°15: but as the aber- ia of the sun is — 202, and that of Mer- i + 18°35, the true conjunction took i: 6™ 35°3° sooner, or at 21° 9™ 26°8* ap- lit time, when the stn’s true place from ; fnean equinox, by Mayer's tables, was 7° a 13-3, or 7° 16° 17’ 85, if we take iaccount the correction to be made in the 'S which the Greenwich observations jee 5” nearly. Hence the place of the is found to be 1° 15° 58’ 56”°3. “ansits of Venus over the sun’s dise Fen much less frequently than those of , 5 . Cury, because Venus is more distant from fun. The following are all that haye or »ecur between 1631 and 2110. Lae Dec. 6. 1874 TrTTiaty) Dec. 8. SPiiis.cc, Dec. 4..| 1882. ......05 Dec. 6. at %........ June 5, | 2004 ..... w.. June 7, t V severe June 3, | 2109 ......... Dec. 10. j Longitude from} Parallax | Parallax TRA of Mercury always occur either in May or in November; but they happen much the most frequently in the latter month. The difference depends upon the position of the elliptic projection of Mercury’s orbit upon the plane of the ecliptic. This ellipse is now so placed that it presents to us its perihelion during the winter, and its aphelion during the summer; and as it is very eccentric, Mercury is much nearer the sun in the month of No- vember than in the month of May. Now if we consider the luminous cone formed by the visual rays drawn from the earth to the sun, this cone is contracted in the vicinity of the earth while it is enlarged near the sun, the disc of which serves for its base: Mercury ought therefore to cut it more readily when it is near the sun than when it is remote from it; and, consequently, the transits of Mercury ought to occur most frequently in the winter part of the year. The transit of Mercury, Nov. 8, 1802, was observed at various places, at the times spe- cified below. The table also contains the quantity of the reduction, and the effect of parallax in longitude and latitude. - Names of the Greenwich, in Long. | in Lat. Observers. S. é. 0:9781| 3°615 |Mean of all. 0'6348] 3°771 |Baron Von Zach. y fs M. Von Uten- 0:8938) 3°735 } haven Calkoen. 0'9238) 3°727 IM. Von Beack. 1°1095) 3°653 {T. Firminger. hem s. 0 9 243 043 9:2 0 20 31°4 0 17 43°6 Now the chief use of these conjunctions is accurately to determine the sun’s distance from the earth, or his parallax, which astrono- mers have in vain attempted to find by various other methods; for the minuteness of the angles required easily eludes the nicest instru- ments. Butin observing the ingress of Venus into the sun, and her egress from the same, the space of time between the moments of the internal contacts, observed to a second of time, that is, to {4 of a second, or 4” of an arch, may be obtained by the assistance of a moderate telescope and a pendulum clock, that is consistent with itself exactly for 6 or 8 hours. Now from two such observations rightly made in proper places, the distance of the sun within a 500th part may be certainly concluded; &c. See PARALLAX. ‘TRANSIT Instrument,-in Astronomy, is -a telescope fixed at right angles to a horizontal axis; this axis being so supported that the line of collimation may move in the plane of the meridian. The axis, to the middle of which the tele- scope is fixed, should gradually taper toward its ends, and terminate in cylinders well turned and smoothed; and a proper weight or balance is put on the tube, so that it may stand at any elevation when the axis rests on the supporters. Two upright posts of wood TRA er stone, firmly fixed at a proper distance, are to sustain the supporters to this instrument ; these supporters are two thick brass plates, having well smoothed angular notches in their upper ends, to receive the cylindrical arms of the axis; each of the notched plates.is con- trived to be moveable by a screw, which stides them upon the surfaces of two other plates immoveably fixed to the two upright posts: one plate moving in a vertical direction, and the other horizontally, they adjust the tele- scope to the planes of the horizon and meri- dian; to the plane of the horizon, by a spirit level hung in a position parallel to the axis, and to the plane of the meridian, in the fol- lowing manner. Observe the times, by the clock, when a circumpolar star, seen through this instrument, transits both above and below the pole; then if the times of describing the eastern and western parts of its circuit be equal, the telescope is then in the plane of the meridian; otherwise the notched plates inust be gently moved till the time of the star’s revolution is bisected by both the upper and lower transits, taking care at the same time that the axis keeps its horizontal po-— sition. When the telescope is thus adjusted, a mark must be set up, or made, at a consider- able distance (the greater the better) in the horizontal direction of the intersection of the cross Wires, and in a place where it can be illuminated in the night4time by a lantern _ hear it, which mark, being on a fixed object, will serve at all times afterwards to examine the position of the telescope, by first adjusting the transverse axis by the level. ‘To adjust a clock by the sun’s transit over. the meridian, note the times by the clock, when the preceding and following edges of the sun’s limb touch the cross wires: the dif- ference between the middle time and 12 hours, shows how much the mean, or clock time, is faster or slower than the apparent or solar time, for that day; to which the equa- tion of time being applied, it will show the fime of mean noon for that day, by which the clock may be adjusted. TRANSMISSION, in Optics, is used to denote the passage of the rays of light through transparent bodies. See TRANSPARENCY. TRANSMUTATION, in Geometry. See TRANSFORMATION. . TRANSPARENCY, or TRANSLUCENCY, in Physies, a quality in certain bodies, by which they give passage to the rays of light. The transparency of natural bodies, as glass, - water, air, &c. is ascribed by some authors to the great number and size of the pores or inter- stices between the particles of those bodies. But this account is very defective; for the most solid and opaque body in nature, that we know of, contains a great deal more pores than it does matter; surely a great deal more than is necessary for the passage of so very fine and subtle a body as light. ia Aristotle, Des Cartes’, &c. make transpa- rency to consistin the straightness or rectilineal direction of the pores; by means of which, TRA say they, the rays can make their way thre without striking against the solid parts, g thus be reflected back again. But this . count, Newton shows, is imperfect; the qu tity of pores in all bodies being suffiei ent transmit all the rays that fall upon 4h however those pores be situated with to each other. * The reason then why all bodies are transparent is not to be ascribed to theirw of rectilineal pores; but cither to the unec density of the parts, or to the nore filled with some foreign matters, or to 1 being quite empty, by means of which rays, in passing through, undergoing ag variety of reflections and refractions, are} petually diverted different ways, till at ler falling on some of the solid parts.of the bi they are extinguished and absorbed. — Thus cork, paper, wood, &c. are opac while glass, diamonds, &c. are transpar and the reason is, that in the neighbour of parts equal in density with respect to< other, in these latter bodies, the attrac being equal on every side, no reflectio refraction ensues: but the rays which ent the first surface of the body proceed @ through it without interruption, those} only excepted that chance to meet with] solid parts: but in the neighbourhood of} that differ much in density, such as the} of wood and paper, both in respect of selves and of the air, or the empty $ in their pores; as the attraction is very equal, the reflections and refractions mu very great; and therefore the rays will m able to make their way through such be} but will be variously deflected, and at le quite stopped. This, at least, is the Wi which some modern writers -have atten to account for this phenomena; it does however, strike us as being very satisfac TRANSPOSITION, in Algebra, | bringing any term of an equation overt other side of it, in which case the sign ¢ quantity thus transposed must be chai viz. from plus to minus, or from minus to jill ' For example, if @ + «2 = b, then trad | ing a and changing its sign, we have 4 — a; and if, on the other hand, we had} = b, then by transposition we Have x =a This operation is performed in ord] bring all the known terms to one side ¢| equation, and all those that are unknoy| the other side, whereby the value of | Jast became determined. See Repvil and RESOLUTION of Equations. ul "TRANSVERSE Azis, or Diameter, i) Conic Sections, is the first or principal dia: passing through both the foci of the See Conic Sections, | a Tn the ellipse the transverse is the of all the diameters, in the hyperbola it} shortest, and in the parabola it is like a) other diameters, infinite in length. ~ | TRAPEZIUM, in Geometry, a plane | contained under four right lines, of 1% neither of the opposite sides are paralle a quadrilateral have both its pairs of Pp Yr TRA § parallel, it is called a parallelogram ; if one of its pairs of opposite sides are par al- vit is called a trapezoid; and when neither ir are parallel, it is called a trapezium. Lhe trapezium has several remarkable pro- ties, of which the following are the most portant : |. Any three sides ofa trapezium are greater in the fourth side. 2, The sum of the four inward angles of a pezinm is equal to four right angles. 3. The diagonals of a trapezium divide it 6 four proportional triangles; viz. a: 6:: ¥ 4, If the four sides of a trapezium ABCD, bisected in E, F, G, 11, and the adjacent ints of bisection be joincd, it will forma rallelogram EF'GH, the sides of which are Wel to the corresponding diagonals of the pezium ; and the area of the former is equal half the area of the latter. ea AC? + BD? = 2EG* + 2FH*. The sum of the four lines formed by the fersection of the two diagonals of a tra- ‘zium, is less than the sum of any other four ‘és drawn from a point within the trapezium ‘its four angles. : 6. The sum of the opposite angles of a tra- zium, which can be inscribed in a circle, is ual to two right angles, and if one side of ch a figure be produced, the external angle equal to the internal and opposite angle. 7. Also, in this case, the rectangle of its two onals is equal to the sum of the rectan- ies of its opposite sides. To find the Area of a Trapezium.—Let AB Ibe atrapezium, AC, BD its two diagonals, d BF and DE two perpendiculars falling on AC. Make AB— cu, BC =}, DC=—c,DA=—d; ip aio AC —3, BD =?2?, BF =p, ED 1. Area trapeziwm =D (p + p) 2. Area 9 y it ‘6. 3. Area s4 ; (a* +c’) #(b’ +-d*) tan, 0. } LG I is eee If the trapezium be inscribable in a ctrele ; then 4, Area trapezium = / ; (s —a) (s —b) (s—c)(s—d) , where s is equal to haif the sum of the four sides of the figure. 5. Area — (ab + ed). sin. x; where 7 represent the angle included between a and b, ore andd. Another method of find- ing the area of a trapezium is given under the article POLYGONOMETRY. TRAPEZOID, a quadrilateral figure, havy- ing two of its opposite sides parallel; the area of which is equal to half a parallelogram, whose base is equal to the sum of the two parallel sides, and its altitude equal to the perpendicular distance between them. TRAVERSE, in Navigation, is the variation or alteration of a ship’s course occasioned by the shifting of the winds, currents, &c. or a traverse may be defined to be a compound _ course, consisting of several courses and distances. See SAILING. TREPIDATION, intheancient Astronomy, denotes a motion which in the Ptolomaic sys- tem was attributed to the firmament, in order to account for several changes and motions observed in the axis of the world,and for which they could not account on any other principle. TRET, in Commerce, is an allowance made for waste, dust, &c. in various commodities, at the rate of 4 pounds to every 104 pounds purchased. See Tare and Trer. TRIANGLE, (Latin, triangulun) in Geo- metry, a figure bounded by three sides, and consequently containing three angles, whence it derives its name. Triangles are of different kinds, as plane or rectilinear, spherical and curvilinear. A Plane or Rectilinear TRIANGLE, is that which is bounded by right lines. A Spherical TRisneve, is that which is bounded by three ares of great circles of the sphere. A Curvilinear TRIANGLE, is that which is bounded by any three curve lines. , And each of these three distinct classes of triangles receive other distinguishing deno-— minations, according to the relation of their sides and angles. Of Plane TRIANGLES. guished as follows: 1. An Equilateral TRIANGLE, is that which has its three sides equal, and consequently its angles, whence it is sometimes also called an equiangular triangle. ; 9. An JIsosceles TRIANGLE, is that which’ has only two of its sides equal. ‘The angles at the base of an isosceles triangle are equal to each other. 3. A Scalene or Oblique TRIANGLE, is that which has no two of its sidés equal. This is termed a sealene triangle with reference to its sides; and oblique with regard to its angles. Triangles also receive other denominations with reference to their angles. These are distin- Ss ee ee TRI ae 4. A Right-angied TRIANGLE, is that which has aright angle. Here the side opposite the right angle is called the hypothenuse, and the other two sides the base and perpendicular, or sometimes the legs. 5. An Oblique-angled TRIANGLE, is that which has not a right angle; and is either acute or obtuse. 6. An Acute-angled TRIANGLE, is that which has three acute angles. 7. An Obtuse-angled TRIANGLE, is that which has one obtuse angle. 8. Similar TRIANGLES, are such as have the angles of the one equal to the angles of the other, each to each. Some of the principal properties of plane triangles are as follow: 1. The greater side of a triangle is opposite the greater angle, and the less side opposite the less angle. 2. Any side of a triangle is less than the sum, but greater than the difference of the other two sides. 3. The suin of the three angles of a triangle is equal to two right angles; and the external angle formed by producing one of its sides is equal to the sum of the two internal and opposite angles. 4. In aright-angled triangle the square of the hypothenuse is equal to the sum of the squares of the other two sides. 5. Or more generally, if a, b, and ¢, repre- sent the three sides of a triangle, and C the angle contained by a and ); then if a> + b* pede eM 8 cr iG, Wilke 2 | og « a* + ab + b% = c*, the < C = 120° a’—ab+ b> =c’*,the << C = 60°. . 6. If a perpendicular be let fall from the vertical angle of a triangle upon the opposite. side or base, produced if necessary, then we have < Fig. 1. & Ep B figure 1, AC’ — AB? + BC? +2AB.BD . figure 2, AC* = AB* + BC*?—2AB.BD and in both figures é A C*:— BC* = AD? — DB’. Also, if CE be drawn: from the vertex to the middle of the base, then. AC* + BC’ = 2A EF? + 2CE?. 7. If ACB be an isosceles triangle, and E any point in its base, then es) AC’, or BC? = CE? + AE. EB. 8. In any triangle the rectangle of any two sides is equal to the rectangle ofthe perpen- dicular on the third side, and the diameter of the circumscribing circle. | 9. The square of a line bisecting any angle of.a triangle, together with the rectangle of the segments of the opposite sides, is equal to bast , the rectangle of the two sides, including bisected angle. a 10. ‘Triangles, when their bases are equ are to each other as their altitudes; and whe their altitudes are equal they are to eg other as their bases; and when neither a equal they are to each other in the compon ratio of their bases and altitudes. — a 11. Similar triangles have their like sid proportional, and their areas are to each oth as the squares of the like sides. > 12. The line which bisects any angle of triangle divides the opposite side into ty segments, which are in proportion to adjacent sides. 13. The line which is drawn parallel to 61 side of a triangle divide the other two sid proportionally. 4 14. The three lines which bisect the thn angles of a triangle intersect in one comm point, as also do the three lines which bise the. three sides perpendicularly; the form point being the centre of the inscribed cire}| and the latter the centre of the circumscribii) one. ' 15. If perpendiculars be let fall from fl three angles of a triangle to the opposite sid] they will intersect in one common point; | will also the three lines drawn from the thr} angles to the middle of the opposite sid which latter point is the centre of gravity) the triangle. 16. If through any point D within a triang| ABC, three lines EF, GH, IK, be dray parallel to the three sides of the triangle ; thy DE x DK x DH=DG x DF x DIY + H Fig. 3. Fig. 4, | B By a: aS oak Salh ] \ | | Ag Thame | ) 17. And if through any point D three lin be drawn from the three angles, as AL, BI) CN; then eR AN SU A i i cos. A + sin. A.cot.B ~ cos. C + sin. C. cot. B’ ‘i (— .sin. A AB.sin. A _ sin. B aC BC AC Si Vige wee cos. C + sin.’'C.cot.A ~ cos. B + sin. B. cot. A’ " ¢ BC. sin. C Eee BED. eb Md Sin. A= 27—AB AG te a | sin. (B + C) = sin. B.cos. C + cos. B. sin. C. f g= / (A B* — BC?.sin.2?C) — +./(AC? — BC?. sin? B _ aa Cos..A = : AB a AG — a — cos.(B + C) = sin. B. sin. C — cos: B. cos. C. | BC. sin. € BC Vs Bo) ie Tan. A — 2 =v (A B*— BC’. sin.?C) 7 + v(AC* = BC? sin B) pee, —tan.(B + C= tan. B + tan. € : ~ tan. B. tan. C — 1 i ¢ AC ‘Si A eA. em. Cu: i sin. B= 2 BG ue Ge a sin. (A + C) = sin. A.cos.C + cos. A. sin. C. y ¢ = v¥(BC? — AC’.sin.27A) _ +y(A B?— AC?. sin? Uy a a Cos. B = 2 fh HCD SAG tarry te AB —_ — cos.(A + C) = sin. A. sin. C — cos. A. cos. C. AC.sin. A ie AC.sin. C = Tan. B — 2 =v (BC*— AC’. sin.*A)~ +7(A B’—A C*. sin? C) — We} coat Sp tan. A + tan. C Ye — tan. De a ” eas Sigh 6) fan. A‘.tane aia ih. Wy os , gAB.B_ AB.sin. A _ z Wear sin. C= 2 rk Celtis: BE a sin. (A + B) = sin. A. cos. B + cos. A. sin. B. ~V/(AC*— AB?. sin? B) + v(BC* — A B?. sin? A) mi Cosi C = it Jute Diy URTaR Lan ah aaa ae — cos. (A + B) = sin. A.sin. B — cos. A.cos. RB. AB.sin. B Rr AB.sin. A A Tan.C = 2 +v(AC*—.A B*. sin. B) ~— +./(BG* — A B’. sin” A)" @ _. tan. A + tan. B } Bares OS hs Be tan. A. tan. B—1 ¥ In these formule radius is taken equal to 1; when this is not the case the radius R, or its powers, must be introduced ; which is readily done by writing such a power of R in the several parts of the formule, as will render all the terms of it, when reduced to an equation, of the same degree with regard to the number of their factors; as for example, instead of PME erste + sin. C.cos. A we have, by introducing radins Bc — AC.R?’ ~ cos,C.R + sin. C. cos. A for this, when reduced to an equation, will have all its parts composed of the same num- ber of factors, viz. three each. For other trigonometrical formule relating to sines, tangents, &c. of two arcs, or of mul- tiple and submultiple ares, see Sine and "TANGENT. | iy Spherical TRIGONOMETRY, relates to #1} solution and calculation of the sides and af of spherical triangles, which are formed. intersection of three great circles of the sf: which, like plane triangles, consist 0} parts, viz. three sides and three angles. § Spherical TRIANGLE. vi In plane trigonometry any three of tI parts of a triangle being given, excey | three angles, the other. parts may. bed but in spherical trigonometry this excel has not place, for any three of the six being given, the rest may thence be ( mined, the sides being measured or estit) by degrees, minutes, &c. the same aj angles, ail Spherical trigonometry is divided into ¢ angled and oblique-angled, or thé resol of right and oblique-angled spherical trial] When a spherical triangle has a right 2 ’ y TRI | TRI dalled a right-angled spherical triangle, portion respects sines, or tangents of the sides hen one of its sides is equal toa qua- for the sides, when the proportion respects or 90°, it is called a quadrantal triangle. tangents, &e. Tan. 1 seg. of this side = cos. adj. angle x eat Third site: the side contain-( given side. ay ed between the ¢ si Ses _ sin. lseg. x tan.ang. adj.giveni a side given angles. eapee ancl tan. ang. opp. given side + Oppo- | site to one of Third angle. them. as sines of opp. sides. 1 . : ° : . lseg. sang. = cos.giv.side x tan.adj Let fall a per. Ke) Cot. lseg. ofthis ang. = cos.giv.side x ti i}. before. Sin 2 sin. Lseg. x cos. ang. opp. givems in.2 seg. = a si cos. ang. adj. given side eee ON LL PCOS | Gi e aie Lhuukdnhe Ge The ang. opp. ( », If, | to the other 5 By the common? Sines of sides are as sines of their opposite anny) Two side. oy: 5 sides the giv. sides. Cos. 2seg ed between < dicular from the _ Cos. 1 seg. x tan. giv.sidead). giv, d included angle. v 4 one of an. 1 seg. sidereq.=cos. giv.ang. x tan.adj.§ them. aT Let falla perpen- r . . . e rhird side. cos. 1 seg. X Cos. side opp. giv. ar dicular as before. and an|Angle includ- « Let mck oe 1 seg. ang. req. — tan. giv. ang. x cos. adj.if os. 2seg. = / cos. side adj. given angle, ee aha emp ed ee eM ition et ied ap: UREN Cae an eee Tan. 1 seg. of div. side = cos. giv. ang. xX” An angle opp. j Let fall a perpen- t side opp: ane. sought. to one of the < dicular from the ° . . . fan. iv. ang. X sin. Lsé given sides. ( third angle. SD SNM ti bar : sin. 2 seg. of div. side. Tan. 1 seg. of div. side = cos. giv. ang. x tan. 0 given side. >| Tan. ang. sought = Let fall a perpen- ) T Third side. dicular on one of : : ( the given sides. cos. side not div. x cos, 2 8¢ on a eee OPS Ronen? cos. 1 seg. -of side divide: Cot. 1 seg. of div. ang. = cos. giv. side x tan. ¢ opp. side sought. tan.giv.sid. X cos. seg. div. cos. 2 seg. of divided angl Cot. 1 seg. div. ang. = cos. giv. side x tan. ob} giv. angle. Let fall a perpen- Iv. |A -side opp. dicular on the A sidefo one Piytte third side and thelgiven angles. : two ad- jacent angles. Tan. side sought = ~* Let fall a perpen- dicular from one Third angle. ) of the given an- { gles. Cos. angle sought = cos. ang. not div. x sin. 28 sin. 1 seg. div. angle. Seemienmeaaie cael Vv. Let a, b, c, be the sides; a,B, c, the angles, 6 and ¢ including The a Eee ih angle sought, and s= a al b + e . en, sin.(+s—b).sin.(4s —e Y Xs-1 three | sine of its half, sin, gaze o/ SED sin so) YET 6 haa 2 ae 2s. sin. (2m sides. sin. 6. sin. e sin. 6. sin. ¢ VI. Let s be the sum of the angles a, 8, and c: and lets and c be The e nee) ieee jacent to a the side required. 'T hen, three |54 } cos.38. cos.($8—A) sin.(3s—s).sin.(s—) of its half. (sing 24 cee = ernment . M os ee ae angles. r= — sin. B. sin. C = sil. B. Sin. C TRI 4 ee | TABLE III. For the Solution of all the Cases of Oblique-Angled Spherical Triangles, by the Analogies of Napier. : I ne er Required. | Values of the Terms required. Side opposite to the other | given angle. § By the common analogy, sines of angles as sines of oppo site sides. ; ie Two angles, -d one of their oposite sides. tan.i diff. giv. sides x sin. 4 sum opp. angles sin. $ diff. of those angles __tan.isum giv.sides X cos. }sum opp. angles cos. 4+ .diff. of those angles, . ).Tan-pfitshalt— Third side. l Third angle. ; By the common analogy. Ang. opposite to the other + By the common analogy. known side. —_ ie If. Date TF ie is mg hANK, 5: iio sides, Cot. ofits hale 2": x diff. eget na) x sin.zsum giv.sides and an sin. 4 diff. those sides. hird angle. < : Mew Misite angle T bak t __tan.tsnmofother2ange x cos.tsum giv. sides 4 cos. 4 diff. of those sides. 7 t Third side. ; By the common analogy. cot. 4 giv. ane. X sin 4 diff. giv. sides f dt he I aly as — 0% ho an. 2 their diff. = —@2——__>_____—__-______, Tit. The other iad | 4 sin. £ sum of those sides. Two sides, angles. anid thoirsuin = cot. -giv.ang. x cos. 3 diff. giv. sides F, 2 cl a a Ne ae ee and the eluded angle. cos. 4 sum of those sides. Third side. ; By the common analogy. bet ie eer ag se he) ag Ca ag i es an. 4 giv. side X sin. 4 diff. giv. angles wee t Tan. 2 their diff. = — : ‘ IV. The other es | i sin. sum of those angles. roangles, and sides. ahaa tan. £ giv. side X cos. 4 diff. giv. angles re. Tan. 4 their sum = = 3 the side cos. $ sum of those angles. tween them. : Third angle. } By the common analogy- RS LER. PA ad Oey il, aR Ek ARO: Sete © VawLene Waco Ue ne Stace Mi a LaelD JOR SB Let fall a perpen. on the side adjacent to the angle sought. Vv. Either of the § Tan.%sum or 7 diff.of) _. tan.4sum x tan.7 diff. ofthe sides theseg. of the base § — tan. 4 base ‘\e three sides. angles. A SUF Cos. angle sought — tan. adj. seg. X cot. adj. side. ne ; Will be obtained by finding its correspondent angle ina . oT . . . Me anoles eee she triangle which has all its parts supplemental to those of the \¢ Rants Te triangle whose three angles are given. ‘or more on the subject of trigonometry, the reader may consult the treatises by Simpson, uduitt, Cagnoli, Bonnycastle, and Woodhouse. | 3 ee > | a RSD TRILATERAL, three sided, a term ap- plicd to any figure of three sides. TRILLION, in Arithmetic, according to the English notation, is one million billions ; but according to the French it is a thousand billions, their billion itself being a thousand million; so that with English arithmeticians a trillion occupies the 19th place of figures, but with the French only the 18th. See BiLuion. TRINE Aspect. See Aspect. . TRINE Dimension, three-fold dimension, viz. length, breadth, and thickness. TRINODA, an ancient land measure equal to three perches. TRINOMIAL, in Algebra, any quantity of three dimensions, asa*+ay + az; ora+ b—c, &c. See MULTINOMIAL. .TRIONES, a name sometimes given to the seven principal siars in Ursa Major. TRIPARTITION, a division into three equal parts. See TRISECTION. TRIPLE, three fold, or multiplied by 3. TRIPLICATE Ratio, the ratio of the cubes of two quantities; thus the ratio of a? to 53 is triplicate of the ratio of a to b. TRISECTION, as the name imports, is the division of any thing into three equal parts, as a line, angle, &c. 'TRISECTION of Angle, is a problem which has engaged the attention of mathematicians from the earliest period of authentic history ; that is, the trisection of an angle geometri- cally, or with the ruler and compasses only, being in point of difficulty, or rather perhaps of impossibility, on a footing with the other two celebrated ancient problems, viz. the du- plication of the cube,-and the quadrature of the circle. |The ancients trisected an angle Dy means of the conic sections, the conchoid, &c. several of which methods are mentioned by Pappus in his Mathematical Collections, prop. 31, 32,33, &e. Analytically the pro- blem invoives no difficulty, for since the sin. 3a = 3 sin. a— 4 sin.3 a, we have sin.’ a — 2’sin. a=— EF sin. 8a whence by Cardan’s formula sin.a=iw ; —sin. 3a-+ / (sin.* 3a—1) f iy/ ; —sin, 8a— / (sin.? 3a — 1) t TRISPAST, aterm used by some old writers for a machine consisting of a combination of three pulleys. TROCHLEA, the Latin word for Putiry, which see. TROCHOID (from zpoxos, a wheel, and dos, Like), is the same as cycloid, that term being derived in a similar manner from xvxdos, a circle. It is, however, by some authors, used to denote exclusively the Prolate CycLoip. TRONE Weight, an ancient Scottish weight, the use of which is now forbidden by several statutes, TROPICS, in Astronomy and Geography, are two circles supposed to be drawn on each side of the equinoctial, and parallel thereto. ‘That on the north side of the line is called the tropic of Cancer, and the southern tropic TRU has the name of Capricorn, as passing thror the beginning of those signs. These are d tant from the equinoctial 23° 27' 46”, the sure of the obliquity of the ecliptic. " circles drawn at the same distance from { equator on the terrestrial globe, have same names in geography, and they ing that space or part of the sphere which called the torrid zone, because the sun is, least, twice every year perpendicular o every part of that zone. bd TROY Weight, an English weight, used) buying and selling gold, silver, and ot! articles of jewelry. See WeIGHTs. rif TRUMPET, the name of an instrum( used either for the purpose of being heard, a great distance, or to apply to the ear, for creasing sound, and rendering it distinct, former being called a a hearing trumpet. Speaking TRUMPET, is a tube of consi speaking and the lat ig C able length, wz. from 6 feet to 12, and e| more, used for speaking with, to make voice heard to a greater distance. Ina tru pet of this kind the sound in one directioj supposed to be increased, not so much byt being prevented from spreading all ei by the reflection from the sides of the trun } The figure best suited for a speaking trum is, that which is generated by the rotatio} a parabola, about a line parallel to the a) ‘The trumpet used at sea is an hollow ins ment of copper, or of tinned iron plates. | is open at both ends, and the narrow en) shaped so as to go round the speaker’s mo| aud to leave the lips at liberty within it. 4 edge of this narrow end is generally cove with leather or cloth, in order that it 1 more efiectually prevent the passing of air between the trumpet and the face of speaker, The words which are spoken thre a speaking trumpet may be heard much ther and louder but not so distinctly, as y out the trumpet. A speaking trumpet been applied to the mouth of a cur pistol, by which means the explosion been rendered. audible at a vast dista Such contrivances it has been thought be used as signals in certain cases. Hearing Trump, is an instrument ti sist the hearing of persons who. are ( Instruments of this kind are formed of tt with a wide mouth, and terminating in as canal, which is applied to the ear. The. of these instruments evidently shows they conduce ‘to assist the hearing, for greater quantity of the weak and lan pulses of the air being received and colle by the large end of the tube, are reflects the small end, where they are collected condensed; thence entering the ear in condensed state, they strike the tympa with a greater force than they could 1 rally have done from the ear alone. TRUNCATED Pyramid, or Cone, is to denote the bottom part of either of t solids when ent by a plane passing pal to its base. See FRustrum. rr wee RNTTAUSEN (Ernrroy WALTER), : ngenious mathematician, Lord of Kil- swald and Stolzenberb, in Lusatia, where vas born in 1651, and died in 1708, in the year of his age. 'Tschirnhausen was author everal papers in the Memoirs of the Aca- y of Sciences ; but we are not aware of separate work that was published by him, Hwh he was often at the expense of print- the useful works of other ingenious men. UBE, a hollow cylinder, such as the tube fhe telescope, barometer, &c. The baro- ‘sical tube is frequently called the Torri- ean tube, from Torricelli, its inventor. aupillary Tuse. See Capicvary Tube. ‘UN, an English measure of capacity, con- jing 2 pipes, or 4 hogsheads, or 252 gallons. )WILIGHT, in Astronomy, is that faint ‘twhich is perceived before the sun-rising, after sun-setting. The twilight is occa- fed by the earth’s atmosphere refracting rays of the sun, and reflecting them among : articles. ‘he depression of the sun below the ho- #1, at the beginning of the morning, and fof the evening twilight, has been variously ind, at different seasons, and by different rvers: by Alhazen it was observed to be t by Tycho 17°; by Rothman 24°; by Ste- fs 18°; by Cassini 15°; by Riccioli, at lime of the equinox in the morning 16°, ie evening 20°2; in the summer solstice jhe morning 21° 25’, and in the winter HS. Whence it appears that the cause ie twilight is variable; but on a medium, lit 18° of the sun’s depression will serve bably well for our latitude, for the begin- |; and end of twilight, and according to th it is easy to compute the duration of ight for any latitude and declination. » find the time of shortest twilight at any in place, say as radius to the sine of the fade, so is the tangent of 9° to-the sine of sun’s declination at the time required. declination of the sun and the latitude i¢ place must be of contrary kinds. Hence pout 51 or 52 degrees north latitude, the ‘ght will be shortest at about the 2d or 3d larch, and the 11th or 12th of October. WINKLING of the Stars, denotes that tre- TYM mulous motion which is observed in. the light proceeding from the fixed stars, a phenomenon which has called forth various hypotheses and theories, which it would be useless to enu- merate in this place, particularly as the prin- cipal cause is now acknowledged to arise from the unequal refraction of light in conse- quence of inequalities and undulations in the atmosphere. This twinkling must be carefully distin- guished from the irradiations of remote lumi- nous objects seen under a small angle; the latter being very different in appearance, and attributable to a very different cause. Mr. Hassenfratz published an essay on this subject in Ann. de Chim. vol. clxxii. from which the following are the inferences: 1. That the figure of luminous objects within the sphere of distinct vision is per- fectly distinguishable. 2. That these figures are altered in propor- tion as we recede from this; and that at a great distance, when these objects are seen under an angle of one or two minutes, they appear surrounded with several irradiations, two of which are in the direction of the eye- lids. 3. That these irradiations are independent of the figure of the luminous object, and are produced by the organ perceiving them. 4. That these irradiations are occasioned chiefly by the irregular figure of the surfaces of the crystalline and cornea. 5. Lastly, that this irradiation is not well distinguished, except in the dark, because the iris having then a greater opening, the irradia- tion occasioned by the irregularity of the sur- faces of the crystalline and cornea becomes more perceptible. TYCHONIC System. See System. TYMPAN, or Tympanum, in Mechanics, a kind of wheel placed round an axis, or cy- lindrical beam, on the top of which are two levers, for the more easily turning the axis in order to raise weights, &c. TYMPANUM of a Machine, is also used for a hollow wheel, wherein one or more persons, or other animals, walk to turn it, such as that of some cranes, calenders, &c. 3A2 ViArc~ wh 7 | Vv, in the Roman notation, denotes 5, and with a line over it 5000. See NoTaTIon. VACUUM, in Physics, a space empty or devoid of all matter. It has been the opinion of some philoso- phers, particularly the Cartesians, that nature admits not a vacuum, but that the universe is entirely full of matter; in consequence of which opinion they were obliged to assert, that if every thing contamed in a vessel could be taken out or annihilated, the sides of that vessel, however strong, would come together; but this is contrary to experience, for the greatest part of the air may be drawn out of a vessel by means of the air-pump, notwith- standing which it will remain whole if its sides are strong enough to support the weight of the incumbent atmosphere. Should it be objected here, that it is impossible to extract all the air out of a vessel, and there will not be a vacuum on that account; the answer is, that since a very great part of the air that was in the vessel be drawn out, as appears by the more quick descent of light bodies in a re- ceiver when exhausted of its air, there must be some vacuities between the parts of the remaining air, which is sufficient to constitute a vacuum. Indeed to this it may be objected, by a Cartesian, that those vacuities are filled with materia subtilis, which passes freely through the sides of the vessel, and gives no resist- ance to the falling; but as the existence of this materia subtilis can never be proved, we are not obliged to allow the objection, especi- ally since Sir Isaac Newton has foutid that all matter aflords a resistance nearly in proportion to its density. There are many other arguments to prove this, particularly the motions of the planets and comets through the heavenly re- gions without any sensible resistance; the dif- ferent weights of bodies of the same bulk, &c. All the parts of spaces, says Sir Isaac Newton, are not equally full; for if they were, the spe- cific gravity of the fluid which would fill the region of the air, could not, by the exceeding great density of its matter, give way to the spe- cifie gravity of quick-silver, gold, or any body, how dense soever; whence neither these, nor any other body, could descend in the air; for no body can descend in a fluid, unless it be specifi- cally heavier than thatfluid. Butifa quantity of matter may by rarefaction be diminished in a given space, why may it not diminish in infini- tum? And if the solid particles of bodies are of the same density, and cannot be rarefied, with- out leaving pores, there must be a vacuum. VAN ‘ ‘o Boileanum Vacuum, is used to express | approach towards a real vacuum, whichj arrive at by means of the air-pnmp. He any thing put in a receiver so exhauste) said to, be in vacuo; and the experiments | carried on are said to be performed in ve or in vacuo Boileano. This term seem) have been intended to indicate that was the first who produced such a vaeu an honour which himself has acknowlec does not belong to him. See Arr-PumpP.. Torricellian Vacuum, is that made in] barometer tube, between the upper end | the top of the mercury, This is perhaps n} a perfect and entire vacuum; as all fluids, found to yield or to rise in elastic vapd on the removal of the pressure of the aj sphere. See BAROMETER. i) VALVE, in Hydraulics, Pneumatics, & a kind of lid or cover to a tube, vesse} orifice, contrived to open one way; but w the more forcibly it is pressed the other } the closer it shuts the aperture, like the 4 per of a pair of bellows: so that it eithel mits the entrance of a fluid into the tub) vessel, and prevents its return, or permi| to escape, and prevents its re-entrance. | Valves are of great use in the air-pump,] other wind-machines; in which they are ally made of pieces of bladder. In hyd | engines, as the emboli or suckers of pul they are mostly of strong leather, of a re figure, and fitted to shut the apertures 0 barrels or- pipes. Sometimes they are 4 of two round pieces of leather enclose¢ tween two others of brass, having divers forations, which are covered with another of brass, moveable upwards and downw on a kind of axis, which goes througt middle of them all. Sometimes they are1 of brass covered over with leather, and nished with a fine spring, which gives} upon a force applied against it; but upo removal of which the valve returns ovell aperture. VANES, in Mathematical and Philoso Instruments, are sights made to slide andi onge cross staves, fore staves, quadr} Ci fs VANISHING Fractions, are fractiol which, by giving a certain value to the able quantity or quantities which enter! them, both numerator and denominato : 0 come zero, and the fraction itself ri VAN The idea of fractions of this kind first ween Varignon and Rolle, two French thematicians of considerable eminence, iwcerning the principles of the Differential cculus, of which Rolle was a strenuous op- er. Amongst other arguments against the ‘th of the doctrine, which had been re- wing a tangent to a certain curve, at the nt where the two branches intersect each er; and as the fractional expression for the tangent, according to that method, had h its numerator and denominator equal to 9 or 0, he regarded suclta result as absurd, { adduced it as a proof of the fallacy of this le of solution; but the mystery was soon )rexplained by John Bernoulli: and upon renewal of the dispute, still farther by in, who shew that the fraction in the case 'e mentioned had a real value. Vhese fractions were also the cause of a yio- teontroversy between Waring and Powell, (1760, when those gentlemen were candi- 2s for the mathematical professorship at pidge. Waring maintaining that the tion ean when z is 1, is equal to 4, and vell (or rather Maseres, who is commonly posed to have conducted the dispute on side) that it was = 0, or rather that it could @no value whatever ; and it must be ac- wledged that the same difference of opi- irelative to these kind of fractions still its in all its force. Woodhouse, in his “rinciples of Analytical Calculation,” in iting of these quantities, after assuming the . 2 2 ale case of ——“ to find the value of it hee as na—a, observes, the signification of this ssion is, that «7 — a’, is to be divided by a; and the result of that division is x+a, ulting 2 — a, it becomes a+a or 24. This it, however, he remarks, is no direct and aral consequence arising from the prin- 3s of calculation ; but on the contrary it is sult arbitrarily obtained, by extending a , and observing a certain order, in the pro- of calculation. . pies az 0 the question, what does ———— become nx—a; the obvious and logical answer u 2 “a were ; —— , and the question is whether in this i——a (it will admit of any farther reduction. _true if we operate upon this quantity, rding to the rules laid down in other ap- —. _ a —a nt similar cases, we shall obtain mane: + a = 2a; but here is evidently an ex- jon given toa rule beyond what was first ided: for this rule was instituted for ope- ig on real quantities, whereas in this case ‘iaye employed it on quantities having no & whatever, being in fact the division of 0 for which abstractedly no rule can be given : 7 yinated about the year 1702, in a contest , itly introduced, he propdésed an example of VAR This, however, is not acase peculiar to these fractions. It is to the same souree we must attribute the introduction of the negative sym- bol, and all the mysteries attendant upon it, in as well as to every kind of imaginary quan- tity. In vol. i. p. 219, of Bonnycastle’s Treatise of Algebra, we have the following rule for finding the value of vanishing fractions. 1. If both the terms of the given fraction be rational, divide each of them by their greatest common measure ; then ifthe hypothesis which is found to reduce the original expression to the form 7 be applied to the result, it will give the true value of the fraction in the state under consideration. 2. When any part of the fraction is irra- tional, observe what the unknown quantity is equal to, when the numerator and denoimi- nator both vanish, and put it equal to that quantity + and — 7; then if this be substi- tuted for the unknown quantity, and the roots of the surds be extracted to a sufficient num- ber of places, the result, when 7 is put equal to 0, will give the true value of the fraction. From which rule the author obtains the fol- lowing results: 2, 2 ; x — 4 1, ——— = 2a, when x= a x—a x— a> 2. ——— — 4, whenxz = 1 1—wzx b(a—. fax 3. wid ie hal dg b, whenx =a x—a ym —_ qm 4, ——__— _ ma”—!, whenzx =a r—a &e. &e. See Bonnycastle’s Algebra, and Wood- house’s Principle of Analytical Calculation. VAPOUR, in Meteorology, a thin humid matter, which being rarified to a certain de- gree by the action of heat, ascends to a height in the atinosphere, where it is suspended uatil it returns in the form of dew, rain, snow, &e. See EVAPORATION. VARIABLE Quantities, in Geometry and Analysis, denote such as are cither continually increasing or diminishing; in opposition to those which are constant, remaining always the same. ‘Thus, the abscisses and ordinates : of an ellipsis, or other curve line, are variable quantities, because they vary or change their magnitudes together. Some quantities may be variable by themselves alone, while those connected with them are constant: as the abscisses of a parallelogram, whose ordinates may be considered as all equal, and therefore constant. ‘The diameter of a circle and the parameter of a conic section are constant, while their abscisses are variable. Variable quantities are usually denoted by the last letters of the alphabet z, y, x, while the con- stant ones are denoted by the first letters a, b, ¢. . Ratio of VARIABLE Quantities—In the in- vestigation of the relation which varying and dependent quantities bear to each other, the VAR conclusions are more readily obtained by ex- pressing only two tefms in each proportion than by retaining the four. But though in considering the variation of such quantities, two terms only are expressed, it will be necessary to keep constantly in mind that four are supposed, and that the operations by which our conclusions are in this case obtained are in reality the operations of proportionals. 1, One quantity is said to vary directly as another, when their magnitudes depend wholly upon each other, and in such a manner, that ifone be changed, the other is changed in the same proportion. Let A and B be mutually dependent upon each other in such a way, that if A be changed to any other value a, B must be changed to another valuc 6, such that A: a:: B: 5, then A is said to vary directly as B, which is denoted by the symbol oc placed between the two quantities. Thus, for example, while. the altitude of a triangle remains constant, the area varies directly as the base or the area c& base. For if the base be increased or diminished, the area is in- creased or diminished in the same proportion. 2. One quantity is said to vary wversely as another, when one cannot be changed in any manner, but the reciprocal of the other is changed in the same proportion. A varies inversely as B, or A& a if when A is changed to a, B be changed to 6 in such a manner, i tay) that Acai: 5: ; or A:@::6:B. For example ; if the area of a triangle be given, the base varies inversely as the perpen- dicular altitude. Let A and a represent the altitudes, B and b, the bases of twe triangles equal in area; then Bi KB go sat 106 oD BS Pha or A x Boa xb; Liss & therefore, A:a::6: Bor A:a::=2.. Bb 3. One quantity is said to vary as two others jointly, if when the former is changed in any manner, the product of the other two are changed in the same proportion. That is, A varies as B and C jointly, or A « BC, when A cannot be changed to a, but the product BC must be changed to d ec, such that A: a >: BC: be. ‘The area of a triangle for example, varies as its base and perpendicular altitude jointly. Tor let A, B, P represent the area, base, and perpendicular of one triangle; a, d, p the same in another, then B P = 2A in the first, and 6 p = 2a in the second, therefore A Apa BP tbp. 4. One quantity is said to vary directly as a second, and znversely as a third, when the first cannot be changed in any manner, but the second multiplied by the reciprocal of the third, is changed in the same proportion, That is, B fd Bek Ac & when A ; tt qt o A,B,C; being corresponding values of the three tities. | : Vor example, the base of a triangle y, as the area directly, and the perpendic. altitude inversely. For as in the preee example oh = - or multiplying bot a es og ee B_ pa by P we have 7 = pp whence A a Ss Re aie ol) The following are some of the principal } positions relating to the ratio of variable q} tities. AY If A co B, and B « C, then A Cx If Ac B, and Bx a then A o “ Y If Ax B, and Bx, then A+ Bo v AH if Acc B, and m is any given number, j A oc mB & I a If A «x B; then A” « B”, or A? x Be If Ac B, and Mom; then AM c« am If Ace BC; then Bat and C oe If AB be constant; then A « and B. IfAc BandC « and y = Taigye Stee Therefore, "PRE o tem “ iy} . +2 meyy ty¥AtV)—_ 3545 y hey si y~ ¥y VAR —3a , 3a? a ) 16y° _ 6y uit ee aes l SOU ae OF 2y re 8y> . eat x a* a’ the variation sought. VARIATION or Declination of the Magnetic Needle, is the distance of the magnetic from the true meridian in degrees measured upou the horizon. Inclination is the angle which the dipping needle makes with the horizon, as measured on a vertical or azimuth circle. 'The changes in the declination and inclination may be represented by the words variation and alteration respectively. After the discovery of that most useful pro- perty of the magnet, or loadstone; namely, the giving hardened iron and stcel a polarity, the compass was for many years used without knowing that its direction in anywise de- viated from the poles of the world; and about the middle ofthe sixteenth century, so certain were some of its inflexibly pointing to the north, that they treated with contempt the notion of the variation, which about that time began to be suspected. However, careful observations soon discovered, thatin England and its neighbourhood, the needle pointed to the eastward of the true north; but the quan- tity of this deviation being known, mariners became as well satisfied as if the compass had none, because they imagined that the true course could be obtained by making allow- ance for the true declination. From successive observations made. after- wards, it was found that the deviation of the needle from the north was not a constant quantity, but that it gradually diminished ; and at last, about the year 1658 or 1660, it was found at London that the needle pointed due north, and has ever since been getting to the westward; and now the declination is more than 24° to the westward of the north; so that in any one place it may be suspected the declination has a kind of libratory mo- tion, traversing through the north to unknown limits eastward and westward. But the. set- tling of this point must be left to time. During the time of the said observations it was also discovered that the declination of the needle was different in different parts of the world, it being west in some places when. it was east in others; and in places where the. declination was of the same name, yet the quantity of it greatly differed. It was there- . fore found necessary, that mariners should every day, or as‘often as they had opportunity, make during their voyage proper observations for an amplitude or azimuth; whereby they might’ be enabled to find the declination of the compass in their present place, and thence correct their courses. The following table of declinations for dif- ferent times and places was given by Dr. Halley, in No. 148 of the Philosophical 'Trans- actions, and is that on which he founded his theory of the variation of the magnetic needle. VAR Observed Declinations of the Needle in divers Places, and at divers Times. : Year at Places observed at. ey ee Latitude. alg tion ob hion ee ee eet Ona Os 7 Oi LODTON Ts .05is'c0sps O O {51 $1 n/1580/11 15 e 1622} 6 Oe 1654] 4 5e 1672| 2 30 w 1685] 4 30 w yet See eee seee{ 2 25€148 51n/1640} 3 Oe 1666} 0 O 1681} 2 30w Uraniburg......... 15 Oe {55 54 n/1672! 2 35 w Copenhagen ...... 12 55e€)55 41 n/1649] 1 53 e 1672| 3 45 w DantZic ....7.. ..00% 19 0e€/54 25 n/16791 7 Ow Montpelier.... ... 4 Oe /43 37 n|1674| 1 10 w BiRSCL, shot ce 4 25 wi48 23 n|1680] 1 45 w ROmesRAl ienies 13 Oe /41 50n|1681] 5 Ow Bayonne..........- 1 20 wi45 50 1/1680] 1 20 w Hudson’s Bay..../79 40 wi51 0n|1668/19 15 w t, . Ye fy Hudson's 57 Owl61 0n/1668129 30 w rrarte yay Ss Baffin’s Bay Sir T. Smith 80 Owl78 ON/1666157 Ow Sound.,........ A SPA. os tec alt 57 Ow/l58 40 n11682| 7 30 w AUBEA.. on pcaiseat 31 50 wi43 50 0/1682] 5 30 w NP RT ata AP 42 Ow/21 0On{|1678] 0 40e storing 35 30wl28 0s |16701 5 30e fe eivec bates 53 Owl39 30s |1670]20 30e Cape Frio... i<..0i5% 41 10 wi22 40 s |1670]12 10 e Entrance of ’ Magellan’s ‘ 68 “Ow/52 308 |1670]17 Oe Straits...... #, West entrance 7s Owls3 bf of ditto....... 2 Ow? . ie TL?) 14 aeOre le alee i i. At London, the declination, oy in 1580 was 11°15’ E. . in 1622 — 6 OK in 1634 — 4 6 ne in 1658 — 0 0° in 1672 — 2 30 W. in 1692 — 6 OW. in 1723 — 14 17 W. , in 1747 — 17 40 W in 1780 — 22 41 W. So that the declination here, seems oscillat- ing about a limit. According to ithe observations of Mr. Can- ton for the year 1759 (Phil. Trans. vol. li. p. 445), it appears that the diurnal variation of declination increased from 7' 8" in January, to 13’ 2)” in June, and decreased to about 6’ 58” in December. Mr. Gilpin found (Phil. Trans. 1806, part il.) by a mean of 12 years from 1793 to 1805, Oo . e) , Baldivia....<. i... 73 Owl40 058 {1670 p Cape Aguillas.....| 16 30 e|34 50s |1622 \ 1675 At S€a...cc.cescueeel 1 00/54 30 5 [1675 fr = ee 20 Owl34 08 1]1675/10 30e JAt sea.....ree| 32 Owlet 0 |1675/10 30€ St. Helena....... «| 6 30wi6 08/1677) 0 406 /Isle Ascension....] 14 30 wl 7 50s |1678| 4 Oe Johaitiia ani.ic side 44 0e/12 158 (1675/19 Mombasa.......+6. 40 Oe| 4 Os /11675/16 UC OCALTA «ss ecedes declination compared with one another ferent points of the globe, follow different ons. But there is a fact extremely wor- fattention, that has been remarked by elebrated Hallé, on the mere inspection 2 table of declination published by Van den, whose notice it had escaped. In thle, three places are pointed out, where eedle has experienced the greatest de- lion: and these are, first, in the middle ’ Indian ocean, from 10° to 15° of south de, and from 82° to 87° of east longitude oning from ihe isle of Ferro), where the ” VAR variation, from the year 1700 to that of 1756, was from 11° to 11915". Secondly, in the Ethiopian ocean, from 5° of north to 20° or 25° of south latitude, and in the interval of 10°, 15°, and 20°, of east longitude ; the vari- ation relative to this space, during the same period of time, was from 10° to 10° 45’, prin- cipally under the line and to 5° southward. Thirdly, at 50° north latitude, and between 17° of east and 10° of west longitude ; where again in the same period, there was a varia- tion of from 11° to 11° 45’, Looking at Van Swinden’s table, Hallé per- ceived that these three places formed as it were three centres, round which the numbers indicating the quantities of variation insen- sibly decreased in proportion as we departed from each centre; so that we have bere a new order of observations, answering to the places where the variation was least in the same course of years. These places are, first, the whole American ocean, without including the gulf of Mexico, that is to say, from the western point of Africa to the farthest of the Bermuda islands. And here also we must remark, that in the ocean between Africa and North America, the vari- ation is much less towards the American than towards the African coasts. Secondly, the environs of the isle of Madagascar, and part of the coast of Zanguebar. 'I'hirdly, that part of the ocean which is to the south and south- east of the Sunda islands, between those is- lands and New Holland. And lastly, in the same sea, about the 4th degree of south lati- tude and the 97th of east longitude, that is, in the middle of the space comprised between the western angle of New fiolland and_ the southern point of Africa. In all these dif- ferent places the declination of the needle has not varied, during the whole fifty-six years, so much as one degree, (Encyclop. Méthod.). Dr. Halley published in the last century a theory of the variations of the compass. In this work he supposes tltere are four magnetic poles in the earth ; twoofwhich are fixed and two moyeable, by which he explains the dif- ferent variation of the compass at different times in the same place. But it is impossible to apply exact caleulations to so complicated an hypothesis:. M. Euler, son of the cele- brated geometrician of that name, has how- ever shown that two magnetic poles placed on the surface of the carth wiil sufficiently ac- count for the singular figure assumed by the lines which pass through all the points of equal variation in the chart of Dr. Halley. M. Euler first examines the case wherein the two magnetic poles are diametrically op- posite; second, he places them in the two op- posite meridians, but at unequal distances from the poles of the world; third, he places them in the same meridians. Tinally, he con- siders them situated in two different meryj- dians. These four cases may become equaily important; because, if it is determined that there are only two magnetic poles, and that these poles change their situations, it may VAR some time hereafter be discovered that they pass through all the different positions. Since the needle of the compass ought al- ways to be in the plane which passes through the place of observation and two magnetic poles, the problem is reduced to the discovery of the angle contained between this plane and the plane of the meridian. M. Euler, after having examined the different cases, finds that they also express the earth’s magnetism, re- presented in the chart published by Messrs. Mountaine and Dodson in 1774, particularly throughout Europe and North America, if ihe following principles are established. Between the arctic pole and the magnetic pole 14° 3’, Between the antarctic pole and the other magnetic pole 29° 23’. 53° 18’ the angle at the north pole, formed by the meridian’s passing through the two magnetic poles. 250° the longitude of the meridian, which passes over the northern magnetic pole. As the observations which have becn col- lected with regard to the variation are for the most part loose and inaccurate, it is impossible to represent them all with precision; and the great variations observed in the Indian ocean seem to require, says M. Euler, that the three first quantities should be 14, 35, and 65 de- grees, In the Memoir of MM. Biot and Humboldt “On the variations of the terrestrial magne- tism in different latiudes,” the position of the magnetic equator is determined from direct observations. 'The inclination of the plane of this circle to the astronomical equator is stated to be 10° 58’ 56”, its occidental node on that equator being at 120° 2!5" longitude W. from Paris, the other node at 59° 57/55” E. of Paris. The points where the axis of the magnetic equator pierces the earth’s surface are, the northern point at 79° 1/4” N. lat..and 30° 2! 5” W. long. from Paris; the southern point is situated in the same Jatitude south, and 149° 57’ 55" E. long. from Paris. It would carry us far beyond our limits were we to attempt to sketch the various theories of terrestrial magnetism which have been proposed: we must, therefore, refer to Birch’s History of the Royal Society, vol. iii. 131; Halley, in Phil. Trans. No. 148; Canton, in Phil. Trans. vols. xlviii. and li.; Cavallo’s Magnetism, and Lorimer’s Speculation to ditto; Montucla, Histoire des Mathematiques, vol. iv. 510; and Gregory’s translation of Haity’s Natural Philosophy, vol. ii. 105—130. Calculus of VARIATIONS, is a particular cal- culus, by which having given an expression, or function, of two or more quantities of which the ratio is expressed by a determinate law, we find what this function must become, when we suppose this law itself to experience an infinitely small variation, occasioned by the variation of one or more of the quantities by which it is expressed. This calculus offers the only general and frequently the only possible means of solving VAR those problems generally termed isope trical, and of which we have already g some account under the articles BRACE CHRONE and IsopeRIMETRY. o These problems took their risé from. Bernoullis; but the general method of tion or caleulus of variations, we owe e sively to Lagrange. John Bernoulli, i solving the ‘ curve of swiftest descent, made two elements of the curve vary, by ing the intermediate ordinate between two drawn from the extremities of : And James Bernoulli, in the solution ¢ famous problems of Isoperimeters, supp an infinitely small arc to be divided into parts by two equidistant intermediate. nates, and making these vary he found position they ought to haye, in order te swer the conditions of the problems, andh obtained his ultimate solution. But these methods, though they do honour t genius and perseverance of their authors: pend more upon an extraordinary effort ¢ mind, than upon any direct calculus y should lead them to their end by a direc infallible process. 4, Euler indeed generalised the method ¢ lution, which led him to the solution of ay ber of elegant problems contained in learned work, “‘ Methodus inveniendi J curvas,” &c. which is a chef d’cewore of anall but here the author still depended upon tain geometrical considerations, which? desirable to avoid, a defect which its authe well aware of, and which was finally rem by Lagrange, Miscell. Turin, tom. il. | paper which he calls a new method of ¢ mining the maxima and minima of inde} integrals. Bat Euler resuming the suf in the 10th volume ot the New Commen of Petersburg, p. 54, called it the Caleul Variations, a name which has since beens rally adopted. Kj It is impossible, within the narrow lin this article, to enter into any explanatt! this calculus, we shall therefore only ob that here every problem requires us ultim to find the maximum or minimum of : mula, such as /Zdx, where Z is a fan of x with constant quantities; of x and: of x, y and z, or of a greater number of ables ; or Z may even contain certain inte} as /V, or integrals of integrals, as /V Si the general solution of which, a8 we obs} above, we owe exclusively to Lagrang} though certain cases had been previously sidered, and some formule of solution al : established by Euler, in his celebrated ») “* Methodus inveniendi ecurvas, &e.” Ji more circumstantial account of the suce improvements of this calculus, as well as? illustration of its principles, and appli at | various problems, we refer the reader to W house’s ‘“ Treatise on Isoperimetrical} blems, and the Calculus of Variations,” 1810. See also Lagrange’s ‘“ 'Théori¢ Fonctions Analytiques,” and the “ Leco a Calcul des Fonctions,” of the seme at y ¥ ‘ VAU GNON (Peter), a celebrated French ematician, was born at Caen, in 1654, } died suddenly in 1722, being then in his year. He was by his father intended ihe church, but accident having thrown in jvay a copy of Euctid’s Elements of Geo- Jry, he became enamoured with the study ‘he mathematics, which he pursued with greatest success. His works, which were lished separately, were, . ichr. d’une Nouvelle Mechanique, 4to. 1687. | Des Nouvelles Conjectures sur la Pesan- { Nouvelle Mechanique ou Statique, 2 vols. 1725. Besides numerous papers pub- din the Memoirs of the Academy of meces, extending through almost all the mes to the time of his death, in 1722, AULT, in Architecture, an arched roof, so ttructed that the stones which form it ain each other. heory of Vautrs. A semicircular arch or jit, standing on two pedroits, or imposts, vall the stones that compose them, being ‘and placed in such manner as that their ts or beds, being prolonged, do ali meetin tre of the vault: it is evident that all stones must be in the form of wedges, ¢.e. jt be wider and bigger at top; by means fhich they sustain each other, and mutu- ‘oppose the effort of their weight, which rmines them to fall. The stone in the idle of the vaults, which stands perpen- War to the horizon, and is called the key ie vault, is sustained on each side by two ignous stones, just as by two inclined ies ; and, consequently, the effort it makes all is not equal to its weight. But still + effort is the greater, as the inclined planes ‘less inclined; so that if they were infinitely @ inclined, z.e. if they were perpendicular he horizon as well as the key, it will tend all with its whole weight, and would ac- dy fall but for the mortar. The second ie, which is on the right or left of the key- ie, is sustained by a third, which, by virtue figure of the vault, is necessarily more ed fo the second than the second is to first; and consequently the second, in the tt it makes to fall, employs a less part of weight than the first. For the same rea- }, the stones from the key-stone employ | aless and less part of their weight to the ; which, resting on a horizontal plane, Moys no part of its weight, or, which is the fie thing, makes no effort at all, as being irely supported by the impost. Now, in aults, a great point to be aimed at is, that ‘the voussoirs, or key-stones, make an ‘al effort towards falling. ‘lo effect this, s visible, that as each (reckoning from the * to the impost) employs still a less and ti part of its whole weight; the first, for lance, only employing one-half; the second, third ; the third, one-fourth, &c. there is other way of making those different parts a VEL equal, but by a proportionable augmentation of the whole, 7. e. the second stone must be heavier than the first, the third, than the se- cond, &e. to the last, which should be infi- nitely heavy. M. De la Hire demonstrates what that pro- portion is, in which the weight of the stones of a semicircular arch must be increased to be in equilibrio, or to tend with equal forces to fall, which is the firmest disposition a vault can have. The architects before him had no certain rule to conduct themselves by, but did all atrandom. Reckoning the degrees of the quadrant of a circle, from the key-stone to the impost, the extremity of each stone will take up so much the greater arch as it is far- ther from the key. M. De la Hire’s rule is, to augment the weight of each stone above that of the key- stone, as much as the tangent of the arch of the stone exceeds the tangent of the arch of half the key. Now the tangent of the last stone of necessity becomes infinite, and of consequence its weight should be so too; but as infinity has no place in practice, the rule amounts to this, that the last stones should be loaded as much as possible, that they may the better resist the effort which the vault makes to separate them; which is called the shoot or drift of the vault. M. Parent has since de- termined the curve, or figure, which the ex- trodos, or outside of a vault, whose intrados, or inside, is spherical, must have, that all the stones may be in equilibrio. For more particular information on this subject, the reader may advantageously con- sult Gregory’s Mechanics, vol. i.; the trea- tise on arches and vaults at the end of Bos- sut’s Mechanics; Dr. Hutton’s Treatise on Bridges, &c. in the first vol. of his 8vo. Tracts, lately published; and M. Berard’s treatise, entitled Statique des Voutes. VEADAR, in Chronology, the 13th month of the Jewish ecclesiastical year, answering to our March. VECTIS, a term formerly used instead of LEVER. VECTOR Radius. See Rapius Vector. VELIQUE, or Vetic Centre. See CENTRE, VELOCITY, or Swietness, or CELERITY, in Mechanics, is that affection of motion, by which a moving body passes a certain space in a certain time. It is always proportional to the space passed over ina given time when the velocity is uniform, or constant during that time. Vevocity, is either uniform or variable. Uniform, or equal velocity, is that with which a body passes always over equal spaces in equal times. And it is variable, or unequal, when the spaces passed over in equal times are unequal; in which case it is either acce- lerated or retarded velocity; and this accele- ration, or retardation, may also be equal or unequal, 2. é uniform or variable, &c. See ACCELERATION and MOTION. VexLociry is also cither absolute or rela- “wEN tive. Absolute velocity is that we have hi- therto been considering, in which the velocity of a body is considered simply in itself, or as passing over a certain space ina certain time. But relative or respective velocity is that with which bodies approach to, or recede from one another, whether they both move, or one of them be at rest. Thus, if one body move with the absolute velocity of two feet per second, and another with that of six feet per second; then if they move directly to- wards each other, the relative velocity with which they approach is that of eight feet per second ; but if they move both the same way, so that the latter overtake the former, then the relative velocity with which that overtakes it, is only that of four feet per second, or only half of the former; and consequently it will take double the time of the former before they come in contact together. Initial VeLociry, the velocity with which a body begins to move. Virtual VeLocity of a point solicited by _. any force, is the element of the space which it would describe in the direction of the power, when the system is supposed to have under- gone an indefinitely small derangement. Principle of Virtual Vevocities, in Me- chanies, is much employed by the foreign ma- thematicians, and is thus enunciated: if any system whatever is solicited by powers in equilibrio, and there be given to this system _ any small motion, in virtue of which every point describes an indefinitely small space, the sum of the products of each power, multiplied by the space that the point where it is applied would describe, according to the direction of ihe same power, will be always equal to zero ; regarding as positive the small spaces de- scribed in the sense of the powers; and as negative, those described in the opposite sense. This principle is due to Galileo. For its history and demonstration, see Mechanique de La Grange, p. 8. VENTILATOR, a machine by which the noxious air of any close place, as an hospital, gaol, ship, mine, &c. may be discharged and changed for fresh air. VENUS, the second planet of our system, in order from the sun, revolving between the orbits of Mercury and the earth, her mean distance being -723, that of the earth being considered as uwity; whence her distance in English miles is about 68 millions. She per- forms her sidereal revolution in 224 days, 16 hours, 49'11'-2, and her mean synodical re- volution in about 584 days. . The eccentricity of her orbit is 0069; half the axis major being considered as unity, which is the least eccentricity of all the pla- nets. Her mean longitude, at the commence- ment of the present century, was in 0s 10° 44’ 35”, and the longitude of her perthelion was, at the same time, in 4° 8° 37/0”,9. The line of her apsides has a sidereal motion, con- trary to the order cf the signs, of 4/ 27’,8 in a VEN century. But, if referred to the ecliptie, motion will appear (on account of the pre sion of the equinoxes) to proceed accor to the order of the signs at the rate of 4%", a year, or 1° 19’ 2,2 in a century. vi Her orbit is inclined to the plane of ecliptic in an angle of 3° 23/ 32,7; (w angle decreases about 4’,6 in a century); at the commencement of the present cen it crossed the ecliptic in 2° 14° 52'38”,9. the nodes have an apparent motion in Jo tude of 31”,4 in a year, or 52/20",2 ina tury. - The rotation on her axis is performer 23" 21'7",2; but the inclination of her ax not known. The diameter of Venus is 7702 miles: « sequently she is nearly as large as the ea} and her mass, compared with that of the considered as unity, is +-.,, her dent being 5£4, that of water being 1. Venus is surrounded by an atmosphere,} refractive powers of which differ very I} from those of our atmosphere. . As viewed from the earth, this is the mi brilliant of all the planets; and may Soe times be seen with the naked eye at n day. She is known as the morning and é@ ing star; and never recedes far from the | Her elongation, or angular distance, val from 45° to 48°, | Her motion sometimes appears retrogr| The mean arc, which she describes in s} case, is about 16° 12’; and her mean di tion is about 42 days. This retrograda) commences, or finishes, when she is aly 28° 48’ distant from the sun. bal Venus changes her phases, like the mil according to her position with respect to 4 sun and the earth; which causes a very siderable difference in her brilliancy. | Her mean apparent diameter is 27,0; | greatest apparent diameter is about 57,3.) Venus is sometimes scen to pass overt sun’s disc, which can happen only when is in her nodes, and when the earth is in} same longitude; consequently it can t place only in the months of June or Deel ber. Three of these transits have been! ready observed: ove in 1639, one in Ji 1761, and one in June 1769. There will | be another till the 8th of December 1 See PARALLAX and TRANSIT. | VENA Contracta, in Hydraulics, is use denote a contraction in the column of | effluent water, issuing through an apertun the side or bottom of a vessel. # What is said under the article Discuat of Fluids, of the velocity of the issuing strei is true only of the middle filament, which periences no other retardation than w arises trom the resistance of the air, and the il tual adhesion of the several particles amor} each other; but those particles which is! nearer the edges of the aperture must ned sarily experience a greater attrition, therefore suficr a greater retardation: he} it follows that the mean velocity of the f l VER ss than what the theory gives us, and that lve shall show in about the ratio of lto 72. jir Isanc Newton, who examined every ject with peculiar accuracy, found that at distance of about a diameter of the orifice nit, the vein of effluent fluid experienced onsiderable contraction, viz. nearly as 1: 4, whence he concluded that the velocity he water, after its exit from the aperture, increased in this proportion, the same tity passing in the same time through a ower space. rom the result of several experiments, as hie time of discharging different quantities ivater, he ascertained that the mean velo- of the effluent column at the aperture, equal to that which a heavy body would jiire in falling through half the height of i vessel, whereas the theory gives it the icity due to the whole height, which agreed i. the velocity of the fluid at the distance Lbout a diameter of the aperture, as above ed. Therefore, in all questions relating this discharge of finids through aper- s, the area of the effluent fluid ought to aken less than the area of the aperture in ratio of 1 to ./2, and the velocity of this wvacted vein to be considered as that due ie fall of a heavy body through the whole htofthe fluid. It must be observed, how- |, that the ratio of the area of a section of contracted vein to that of the orifice is jalways the same, nor has either theory or riment been yet able to ascertain that ))in all cases, so that that stated above, viz. 72, must only be considered as an ap- jimation, which is probably more general ,S application than any other that can be ! | LL )ERNIER, is a scale, or a division, well ited for the graduation of mathematical juments, so called from its inventor Peter gier, a gentleman of Franche Comté, who }municated the discovery to the world in a il tract, entitled La Construction, I’ Usage, #28 Proprietez du Quadrant Nouveau de thematique, &c. printed at Brussels in . This was an inprovement on the me- of division proposed by Jacobus Curtius, wed by Tycho, in Clavius’s Astrolabe, in Vernier’s method of division er divid- plate, has been very commonly, though jaeously, called by the name of Nonius ; method of Nonius being very different that of Vernier, and much less conve- hen the relative unit of any line is so di- linto many small equal parts, those parts be too numerous to be introduced, or if duced, they may be too close to one an- st to be readily counted or estimated ; for | h reason there have been various methods rived for estimating the aliquot parts of (small divisions, into which the relative tof a line may be commodiously divided ; mong those methods, Vernier’s has been ). justly preferred to all others. . For the yry of this, and other inventions of a si- VER milar nature. See Robins’s Math. Tracts, vol. il. p. 265, &e. Vernier’s method is derived from the fol- lowing principle. If two equal right lines, or circular ares, A, B, are so divided, that the number of equal divisions in B is one less than the number of equal divisions of A, then will the excess of one division of B above one division of A be compounded of the ratios of one of A to A, and of one of B to B. For let A contain 11 parts, then one of A to A is as 1 toll, or des Let B contain 10 parts, then one of B to B is as 1 to 10, or a Now 5 lie” eg 368 8 A AM AA ch | | RE i Soca TEP WW TERPS Wears Tee Te Or if B contains n parts, and A contains n J n+ Y + 1 parts; then ~ is one part of B, and 1 Lipath is one part of A. And VSR (tI) _1 ot | nxnt+l Sw n+. The most commodious divisions, and their aliquot parts, into which the degrees on the circular limb of an instrument may be sup- posed to be divided, depend on the radius of that instrument. Let R be the radius of a circle in inches, and a degree to be divided into z parts, each 1 : being —th part of an inch. Now the circumference of a circle, in parts of its diameter 2 R inches, is 3,1415926 x2 R inches. Then 360° : 3,1415926 x 2R:1°: 3,1415926 360 Or, 0,01745329 x R is the length of one degree in inches. Or, 0,01745329 x R x p is the length of 1°, in pth parts of an inch. But as every degree contains n times such parts, therefore n = 0,01745329 x R x p. The most commodious perceptible division x 2 R inches. ee 1 ‘ is — or — of an inch, 8 10 EXAMPLE. Suppose an instrument of 30 inches radius, into how many convenient parts may each degree be divided? How many of these parts are to go to the breadth of the vernier, and to what parts of a degree may an observation be made by that instrument? Now, 0,01745: x R = 0,5236 inches, the length of each degree: and if p be supposed 1 . wats about 5 of an inch for one division; then ¢ 0,5236 x p — 4,188 shows the number of such parts ina degree. But as this number must be an integer, let it be 4, each being 15’: and let the breadth of the vernier contain 31 of those parts, or 73°, and be divided into 30 parts. 1 — 3m =—y; then— x Here n = in 30 Z “1p ag VER — of a degree, or 30’, which is the least part ofa degree that instrument can show. ; ; ifn e and tie 22's 39 then : of a minute, or 20”. Serer 0 5X 36 ‘ The following table, taken as examples in the instrunents commonly made from 3 inches to 8 feet radius, shows the divisions of the limb to nearest tenths of inches, so as to be an aliquot of 60’s, and what parts of a degree may be estimated by the vernier, it being di- vided into such equal parts, and containing such degrees as their columns show. inohessid. degree. kyeraien.| roputer. | Parts observed. 3 Bex] ABb ale LB od Al 30! 6 E80, 20k Ss Ao 9 Pel ond” SOP el cde 12 an a” Sa in ee bores Wm F- 15 3 | 20 rE Oh SE WES 18 B04) Bop ae 4.0940 21 4 | 30 £ci| oQrGo 24 4 | 36 92 | 0 2% 30 5 | 30 72 | 0 20 36 Bick 80 5E. | 0 20 42 8 | 30 $8.4 0-45 48 9 | 40 45 | 0 10 60 | 10 | 36 37, 1.0 ,10 72-4 ADL 20 pie ae 84 | 15 | 40 22° | 0 6 96 | 15°] 60 4 ica 4 By altering the number of divisions, either in the degrees or in the vernier, or in both, a angle can be observed to a different degree of accuracy. ‘Thus, to a radius of 30 inches if a degree be divided into 12 parts, each being five minutes, and the breadth of the vernier be 21 such parts, or 13°, and divided into 20 ‘ts, then } x an pie ahve atin hie CAD hailey the breadth of the vernier 22, and divided 1 12) = 15": or taking into 30 parts; then ck v4 — or 10’: 12 ~ 30 — 360 Or x Ley dl es 6"; where the breadtl a Pits a BOO, fie of the vernier is 42°. VERSED Sine of an Are, in Trigonometry, is the part of the diameter intercepted be- tween the sine and the commencement of the arc. See ‘TRIGONOMETRY Definitions, and the article SINE. ; VERTEX of an Angle, in Geometry, is the angular point, or the point where the legs or sides of the angle meet. Veprex of a Figure, is the uppermost point, or the vertex of the angle opposite to the base. VERTEX of a Curve, is the extremity of the axis or diameter, or it is the point where the and plummet, and then marking two pl T° VER % diameter meets the curve; which is also. vertex of the diameter. A VERTEX of a Glass, in Optics, the same its pole. | VERTEX is also used in astronomy for point of the heavens vertically or perpen¢ larly over our heads, also called the zenith VERTICAL, something relating to’ vertex or highest point. As, ‘ VeRTICAL Angles, in Geometry, are such have their legs or sides continuations of e other. See ANGLE. VERTICAL Point, in Astronomy, is the sé with vertex, or zenith Hence a star is | to be vertical, when it happens to be in 4 point which is perpendicularly over any pl} VERTICAL Cirele, is a great circle of sphere, passing through the zenith and n} of a place. The vertical circles are also ea} aziumths. 'The meridian of any place ° vertical circle, viz. that particular one ¥ passes through the north or south point of} horizon, All the vertical circles intersect} another in the zenith and nadir. | The use of the vertical circles is to | mate or measure the height of the stars,) and their distances from the zenith, whid reckoned on these circles; and to find { eastern and western amplitude, by obser} how many degrees the vertical, in which Star rises or sets, is distant from the meri¢ Prime VERTICAL, is that vertical cireli azimuth, which passes through the pol¢ the meridian; or which is perpendicula) the meridian, and passes through the equi) tial points. \ Prime Verticats, in Dialling. See Py Verticals. “| Vertical of the Sun, is the vertical wh passes through the centre of the sun at] moment of time. Its use is, in dialling find the declination of the plane on Wi the dial is to be drawn, which is done by) serving how many degrees that verti! distant from the meridian, after marking point or line of the shadow upon the plai) any times. " VeERTICAL Dial. See Dir. VerticaL Line, in Dialling, is a lin) any plane perpendicular to the horizon. is best found and drawn on an erect an} clining plane, by steadily holding up a s} of the shadow of the thread on the plai, good distance from one another, and dra} a line through these marks. i) VERTICAL Line, in Conies, is a line di on the vertical plane, and through the vo of the cone. i) VERTICAL Plane, in Conies, is a plane } ing through the vertex of a cone, and pail to any conic section. 4 VERTICAL Plane, in Perspective. See } SPECTIVE. Ay VERTICITY, that property of the mé or of a needle touched with it, by whi! directs itself to some particular point, ¢ pole. See Macner. . VES (RU, a name formerly applied to a par- iir kind, or rather to a particular appear- | of a comet. NSPER, one of the names formerly given {3 planet Venus. ISTA, one of the four new planets dis- wed since the commencent of the present ry. From their regularity observed in the ices of the old planets from the sun, some jiomers supposed that a planet existed pen the orbits of Jupiter and Mars. The (very of Ceres confirmed this happy con- jes but the opinion which it seemed to jlish respecting the harmony of the solar jn, appeared to be completely overturned lie discovery of Pallas and Juno. Dr. i's, however, imagined that these small ial bodies were merely the fragments of je planet, which had been burst asunder jme internal convulsion, and that several i might yet be discovered between the j Of Mars and Jupiter. He therefore jaded, that though the orbits of all these ents might be differently inclined to the lic, yet, as they must have all diverged the same point, they ought to have two on points of union, or two nodes in op- | regions of the heavens through which je planetary fragments must sooner or ppass. One of these nodes Dr. Olbers i to be in Virgo, and the other in the 1; and it was actually in the latter of ij regions that Mr. Harding discovered . With the intention, there- sf detecting other fragments of the sup- ( planet, Dr. Olbers examined thrice year, all the little stars in the opposite (Hations of the Virgin and the Whale, ii) labours were crowned with success on Mth March, 1807, by the discovery of a ilanet in the constellation Virgo, to which ve the name of Vesta. ,s00n as this discovery was made known itland, the planet was observed at Black- | On the 26th April, 1807, by 8S. Groom- , Esq. an ingenious and active astro- i, Who has successfully devoted his leisure Is fortune to the advancement of astro- i He continued to observe it with his ent astronomical circle, til the 20th when, from its having ccased to become /on the meridian, he had recourse to rial instruments. On the llth August, yroombridge resumed his meridional ob- lions, from which he has computed part i elements of its orbit; and he had the fortune to observe the ecliptic opposition |, planet, on the 8th September, 1808, at i in longitude 11+ 15° 54’ 26”. His ob- ons were continued till the beginning vember, 1808, and he expected to have ‘the planet again at its opposition in lary, 1810; but from a continuance of t’ weather, and probably from errors in iments, he did not succeed. planet Vesta is of the fifth or sixth y tude, and may be secn in a clear even- the naked eye. Its light is more in- VIB tense, pure, and white, than any of the other three; and it is very similar in its appearance to the Georgium Sidus. It is not surrounded with any nebulosity; and even with a power of 636, Dr. Herschel could not perceive its real disc. The orbit of Vesta cuts the orbit of Pallas, but not in the same place where it is cut by that of Ceres. According to the observations of Schroeter, the apparent diameter of Vesta is only 0.488 of a second, one half of what he found to be the apparent diameter of the fourth satellite of Saturn; and yet it is very remarkable that its light is so intense, Schroeter saw it several times with his naked eye. M. Burckardt is of opinion, that Le Mon- nier had observed this planet as a fixed star, since a small star situated in the same place, and noticed by that astronomer, has since disappeared. The following are the elements of the orbit of Vesta, computed by Mr. Groombridge, from his own observations. RGVontion 0253 ae. BY GGT ae Place of perihelion ...... » Oy 3°) Oo "0" Place of ascending node 35 14°38’ 0” Inclination of orbit...... : 7? 8! 20" Mean distance............06 2' 163 Eccentricity in parts 2 0.0953 of the earth’s radius § The following elements are given by Buck- ardt in the Connoisance de Temps for 1809, from the most recent observations on the con- tinent. Place of ascending node 3° 139 Vo 0” Place of perihelion...:... 88 9° 42’ 53” Inclination of orbit........ 7? 8 46° Mean distance ..:.:4.....3 ‘ 2.373000 Eccentricity ........... Jer 0.093221 VIA Lactea, or Milky Way. See Gavaxy. Via Solis, the Sun’s Way, is used sometimes to denote the elliptic line or path described by the sun. VIBRATION, in Mechanics, the regular reciprocating motion of a body, as a pendulum, musical chord, &c. For the laws relating to the vibrations of pendulums, see PENDULUM and OSCILLATION; and for what relates to musical chords, see CHORD. VIBRATION is also a term used by some authors to explain certain natural phenomena. Sensation, for instance, is supposed to be per- formed, by means of the vibratory motions of the contents of the nerves, begun by external objects, and thence propagated to the brain. This doctrine has been particularly illus- trated by Dr. Hartly, who has extended it farther than any other writer, in establishing a new theory of our mental operations. The same ingenious author also applies the doctrine of vibrations to the explanation of muscular motion, which he thinks is per- formed in the same general manner as sensa- tion, and the perception of ideas. For a particular account of his theory, and the ar- guments by which it is supported, see his Observations on Man, vol. i. The several sorts and rays of light, Newton VIE eonceives to make vibrations of divers mag- nitudes, which, according to those magnitudes, excite sensations of, several colours; much after the same manner of air, according to their several magnitude, excite sensations of several sounds. Heat, according to the same author, is only an accident of light occasioned by the rays putting a fine, subtile, etherea! medium, which _pervades all bodies, into a vibrative motion, which gives us that sensation. From the vibrations or pulses of the same medium, he accounts for the alternate fits of easy reflexion, and easy transmission of the rays. In the Philosophical Transactions it is ob- served, that the butterfly into which the silk- worm is transformed, makes one hundred and thirty vibrations or motions of its wings, in one coition. VIETA (FRANCIS), a very celebrated French mathematician, was born in 1540 at Fontenai, Fontenai-le-Comté, in Lower Poitou, a province of France, and died at Paris in 1603, being the sixty-third year of his age. Among other branches of learning in which he excel- led, he was one of the most respectable ma- thematicians of the sixteenth century, or in- deed of any age. His writings abound with marks of great originality, and the finest genius; and his inventions and improvements in all parts of the mathematics were very considera- ble. He was in a manner the inventor and in- troducer of specious algebra, in which letters are used instead ofnumbers, as well as of many beautiful theorems in that science. He made also considerable improvements in geometry and trigonometry. His angular sections are a very ingenious and masterly performance : by these he was enabled to resolve the pro- blem of Adrianus Romanus, proposed to all mathematicians, amounting to an equation of the 45th degree. His Apollonius Gallus, being a restoration of Apollonius’s tract on Tangencies, and many other geometrical pieces to be found in his works, show the finest taste and genius for true geometrical speculations. He gave some masterly tracts on trigonometry, both plane and spherical, which may be found in the collection of his works, published at Leyden in 1646, by Schooten, besides another large and separate volume in folio, published in the author’s life- time at Paris in 1579, containing extensive trigonometrical tables, with the construction and use of the same, which are particularly described in the introduction to Hutton’s logarithms. ‘lo this complete treatise on tri- gonometry, plane and spherical, are subjoined several miscellaneous problems and observa- tions, such as, the quadrature of the circle, the duplication of the cube, &c. Computa- tions are. here given of the ratio of the dia- meter of a circle to the circumference, and of the length of the sine of one minute, both to a great : many places of figures. Besides which it seems that he had com- posed another work, “ Harmonicon Celeste,” VIS ‘i which was surreptitiously taken from Mersenne, to whom it had been intruste its author and irrecoverably lost. % VINCULUM, in Algebra, a mark or racter either drawn over, or including some other way accompanying a factor, sor, dividend, &c. when it is compounde several letters, quantities, or terms, to con them together as one quantity, and show they are to be multiplied, or divided, & gether. Vieta first used the bar or line 1e oval quantities, for a vinculum, thus @ + a + 0; Albert Girard the parenthesis thus (@ 4 Thus a + b x c, or (a + 0b) X ce, denote product of c, and the sum a + 6 consider! one quantity. Also /a +6, or v (a: denotes the square root of the sum @} Sometimes the mark : is set before a | pound factor, as a vinculum, especially 1 it is very lone, or an infinite series ; thus } >:1— 22a + 3a? — 423 4+ 525, &e. thi the more usual method adopted by m«¢ algebraists is to include the expression, ‘ long, i in a parenthesis; thus aI : 3a(1— 22 + 3a” —423 + 525, ke, or if it should happen, as it frequently that the quantity within the parenthesis sist itself of other parenthesis, then th ) ! ward one is made larger, or thus, ; in order to distinguish the whole factor! any of its parts; thus | 3a} 1432 (a +b) — 42° (a + 6)* +8 VINDEMIATRIX; a star of the) magnitude in the constellation Virgo. — VIRGO, the Virgin, in Astronomy, the| zodiacal constellation, denoted by the racter ny. See CONSTELLATION and SI VIRTUAL Focus. See Virtual Fock VinTUAL Velocity. See VeLociry. | VIS, a Latin word, signifying for power; adopted by writers on physij express divers kinds of natural pow faculties. The term vis is either active or passiv¢ vis activa is the power of producing mi the vis passiva is that of receiving or los The vis activa is again subdivided into and vis mortua. kind of centripetal force which is a by the motion that would be generated in a given body, at a given distance, al, pends on the efficacy of the cause prot it. Vis Acceleratrix, or Accelerating Fo that centripetal force which produces i celerated motion, and is proportional | velocity which it generates in a given) or it is as the motive or absolute force di and as the quantity of matter moved iny} Vis Impressa, is defined by Newton! the action exercised on any body to ¢ its state, either of rest, or moving unis in a right line. VIS his force consists allogether in the action ; } has no place in the body after the action eased: for the body perseveres in every }y state by the vis inertiz alone. his vis impressa may arise from various ses; as from percussion, pression, and cen- etal force. 1s Inertia, or Power of Inactivity, is de- id by Newton to be a power implanted in natter, by which it resists any change en- iyoured to be made in its state, that is, by ch it becomes difficult to alter its state, er of rest or motion. this power then agrees with the vis re- endi, or power of resisting, by which every y endeavours, as much as it can, to per- ve in its own state, whether of rest or orm rectilinear motion; which power is proportional to the body, or to the quan- of matter in if, the same as the weight or vity of the body; and yet it is quite dif- at from, and even independent of, the e of gravity, and would be and act just isame if the body were devoid of gravity. |s, a body by this force resists the same in lirections, upwards or downwards, or ob- (ely; whereas gravity acts only downwards. jodies only exert this power in changes ight on their state by some vis impressa, ree impressed on them. And the exercise his power is, in different respects, both pro and impetus; resistance, as the opposes a force impressed on it to change ‘fate; and impetus, as the same body en- rours to change the state of the resisting jacle. Phil. Nat. Prine. Math. lib. 1. he vis inertiz, the same great author else- re observes, is a passive principle, by th bodies persist in their motion or rest, receive motion, in proportion to the force Pessing it, and resist as much as they are ted. See Resistance and INERTIA. is Motrix, or Moving Force of a centri- Ml body, is the tendency of the whole body ‘rds the centre resulting from the ten- Ly of all the parts, and is proportional to notion which it generates ina given time ; )at the vis motrix is to the vis acceleratrix )e motion is to the celerity; and as the Hitity of motion in a body is estimated by oroduct of the velocity into the quantity jatter, so the vis motrix, from the vis Neratrix, multiplied into the quantity of rer. ite followers of Leibnitz use the term vis fix for the force of a body in motion in same sense as the Newtonians use vis iz; this latter they allow to be inherent body at rest, but the former of vis motrix ice inherent in the same body only while ‘otion, which actually carries it from place vace, by acting upon it always with the ‘intensity in every physical part of the which it describes. '8 Motua, or Dead Force, a term used by initz to denote the power of pressure in a ‘at rest; whereas a VIS Vis Viva, or Living Force, is used by the same author to denote the force or power of a body in motion. _ VISIBLE, any thing that is an object of sight or vision, or any thing whereby the eye is affected, so as to produce the sensation of sight. The Cartesians say that light alone is the proper object of vision. But, according to Newton, colour alone is the proper object of sight; colour being that property of light by which the light itself is visible, and by which the images of opaque bodies are painted on the retina. Philosophers in general had for- merly taken for granted, that the place to which the eye refers any visible object, seen by reflection or refraction, is that in which the visual ray meets a perpendicular from the ob- ject upon the reflecting or the refracting plane. That this is the case with respect to plane mirrors is universallyacknowledged; and some experiments with mirrors of other forms seem to favour the same conclusion, and thus afford reason for extending the analogy to all cases of vision. If a right line be held perpen- dicularly over a convex or coucave mirror, its image seems to make one line with it. The same is the case with a right line held per- pendicularly within water; for the part which. is within the water seems to be a continuation of that which is without. But Dr. Barrow called in question this method of judging of the place of an object, and thus opened anew field of inquiry and debate in this branch of science. This, with other optical investiga- lions, he published in his Optical Lectures, first printed in 1674, According to him, we refer every point of an object to the place from which the pencils of light issue, or from which they would have issued, if no reflecting or refracting substance intervened. Pursuing this principle, Dr. Barrow proceeded to in- vestigate the place in which the rays issuing from each of the points of an object, and that reach the eye after one reflection or refraction, meet; and he found that when the refracting sarface was plane, and the refraction was made from a denser medium into a rarer, those rays would always mect in a place be- tween the eye and a perpendicular to the point of incidence. If a convex mirror be used, the case will be the same; but if the mirror be plane, the rays will meet in the per- pendicular, and beyond it, if it be concave. He also determined, according to these prin- ciples, what form the image of a right line will take when it is presented in different manners to a spherical mirror, or when it is seen through a refracting medium. M. Bouguer adopts Barrow’s general max- im, in supposing that we refer objects to the place from which the pencils of rays seemingly converge at their entrance into the pupil. But when rays issue from below the sarface of a vessel of water, or any other refracting medium, he finds that there are always two different — of this seeming convergence: 3 Vers otic of them of the rays which issue from it in the same vertical circle, and therefore fall with different degrees of obliquity upon the surface of the refracting medium ; and another of those that fall upon the surface with the same degree of obliquity, entering the eye laterally with respect to one another. He says, sometimes one of these images is at- tended to by the mind, and sometimes the other; and different images may be observed by different persons. And he adds, that an object plunged in water affords an example of this duplicity of images, From the principle above illustrated, several remarkable phenomena of vision may be ac- counted for: as—That if the distance between two visible objects be an angle that is insen- sible, the distant bodies will appear as if con- tiguous: whence, a continuous body, being the result of several contiguous ones, if the distances between several visible ones subtend insensible angles, they will appear one conti- nuous body; which gives a pretty illustration of the notion of a continuum. Hence also parallel lines, and long vistas, consisting of parallel rows of trees, seem to converge more and more the farther they are extended from the eye; and the roofs and floors of long ex- tended alieys scem, the former to descend; and the latter to ascend, and approach each other; because the apparent magnitudes’ of their perpendicular intervals are perpetually diminishing, while at the same time we mis- take their distance. See Priestley’s Light and Colours. The mind perceives the-distance of visible objects, 1st, From the different configurations of the eye, and the manner in which the rays strike the eye, and in which the image is im- pressed upon it. For the eye disposes itself differently, according to the different distances it is to see; viz. for remote objects the pupil is dilated, and the crystalline brought nearer the retina, and the whole eye is made more globous; on the contrary, for near objects, the pupil is contracted, the crystalline thrust forwards, and the eye lengthened. Again, the distance of visible objects is judged of by the angle the object makes; from the distinct or confused representation of the objects; and from the briskness or feebleness, or the rarity or density of the rays. 'To this it is owing, Ist, That objects which appear obscure or confused are judged to be more remote; a principle which the painters make use of to cause some of their figures to appear farther distant than others on the same plane. 2d,'To this it is likewise owing that rooms whose walls are whitened, appear the smaller; that fields covered with snow, or white flowers, appear less than when clothed with grass; that mountains covered with snow, in the night-time, appear the nearer, and that opaque bodies appear the more remote in the twilight. The magnitude of visible objects is known chiefly by the angle contained between two rays drawn from the two extremes of the ob- ject to the centre of theeye. An object ver at more or less large according to the ang subtends; or bodies seen under a greater al appear greater; and those under a less a less,&c. Hence the same thing appears gre or less as it is nearer the eye, or farthey And this is called the apparent magni But to judge of the real magnitude 0 object, we must consider the distance: since a near and a remote object may ap) under equal angles, though the magnit) be different, the distance must necessaril} estimated, because the magnitude is erei) small according as the distance is. So } the real magnitude is in the compound . of the distance and the apparent magnit} at least when the subtended angle, or appa magnitude, is very small; otherwise, thes magnitude will be in a ratio compounds the distance and the sine of the appé magnitude, nearly, or nearer still its tan Hence, objects seen under the same ai have their magnitudes in the same rat} their distances. The chord of an are} circle appears of equal magnitude from € point in the circumference, though one ji be vastly nearer than another. Or if the be fixed in any point in the circumferd and a right line be moved round, so aj extremes be always in the periphery, if} appear of the same magnitude in ever) sition. And the reason is, because the ap it subtends is always of the same magni And hence also, the eye being placed in) angle of a regular polygon, the sides of it} all appear of equal magnitude ; being all chords of a circle described about it. 1] i magnitude of an object directly opposi the eye be equal to-its distance from the} the whole object will be distinctly see), taken in by the eye, but nothing more. ai the nearer you approach an object, thee —~s - part you see of it. The least angle al which an ordinary object becomes visil about one minute of a degree. ag The figure of visible objects is estim! chiefly from our opinion of the situatid the several parts of the object. This op! of the situation, &e. enables the mind t. prehend an external object under this 0 figure, more justly than any similitude of images in the retina, with the object, 3 the images being often elliptical, oblong} when the objects they exhibit to the min! circles, or squares, &c. 4 The laws of vision, with regard to the fig of visible objects, are, 1. That if the ef of the eye be exactly in the direction right line, the line will appear only as a {| 2. If the eye be placed in the direction surface, it will appear only as a line. 34 body be opposed directly towards the ey as only one plane of the surface can ra on it, the body will appear asa surface, remote arch, viewed by an eye in the | plane with it, will appear as a right 5. A sphere, viewed at a distance, appe VIS ‘cle. 6. Angular figures, at a great distance, lnally appear round. 7. If the eye look ob- haely on the centre of a regular figure, or a le, the true figure will not be seen; but } circle will appear elliptical, &c. VISION, the perception of external objects ‘conveyed to the brain by means of the Pans of sight. IW e have under the article Eye endeavoured jexplain the construction and form of that tieate organ, and the means by whieh ex- Haal objects are painted on the retina, but fy they are thence transmitted to the sen- sum in the brain is, it must be acknow- ized, completely out of the reach of our Hiception to determine. But that vision is s.cted after the manner there described may f and has been, repeatedly demonstrated jm experiment. ake for instance an ox’s eye while it is ish, and having cut off the three coats from } back part, quite to the vitreous humour, ja piece of white paper over that part and id the eye towards any bright object, and iwill see an inverted image of the object jm the paper. Uf the form of the eye, the situation of the eral humours, and their respective sur- fes, remain unaltered, it is manifest that se rays only, which diverge from points at articular distance, can be collected upon i retina. Thus if an image Q be formed i ' fictly, the image of S, a point farther from | eye than Q, will be formed within the 1; therefore the rays which proceed from s point, will be diffused over some space 'm the retina, and if they are mixed with | rays which diverge from other points in } object necessary to be distinguished from | former, the vision will be indistinct. And | rays which diverge from T, a point nearer ithe eye than Q, will, after refraction, con- ige to ¢, a point behind the retina; and in i case also they will be diffused over some /ce upon the retina, and the vision as before it be indistinct. $y what change in the conformation of the }, We are enabled to see objects distinctly ‘lifferent distances, is not fully ascertained; | the effect itself is sufficiently manifest, ‘ough authors differ as to the manner Which it is produced. It is supposed by te, that the general figure of the eye is ‘red; that when the object to be viewed is lr, the length of the eye measured along | axis, is increased by the lateral pressure yexternal muscles; and on the contrary, tn the object is remote, that the length of VIS the eye is diminished, by the relaxation 6f that pressure. Others suppose the effeet to be produced by a change in the place, or figure of the crystalline humour; some by an alteration in the diameter of the pupil; while others ascribe the effect to a change in the curvature of the cornea. Much stress cannot be laid upon the first of these causes, as distinguished from the last, since its existence is not proved by experi- ment; and there is no necessity for recurring to a bare hypothesis of this kind. With re- spect to the second, the ligamentum ciliare does not appear sufficiently strong to produce any, considerable change in the form or si- tuation of the crystalline humour. And as-it is clearly ascertained, Phil. Trans. vol. Ixxxv. that persons couched can see distinctly at different distances, we must conclude that the effect is not to be ascribed to any change in this humour. As change in the aperture of the pupil has some effect in rendering objects distinct at different distances ; for if the eye of a spectator be directed, first to a distant object, and then to one which is nearer, the diameter of the pupil is observed to decrease. But the principal change by which the effect is produced, seems to be an alteration in the curvature of the cornea. ‘In order to show that such a change takes place, Mr. Ramsden fixed the head of a spectator so securely, that no deception could arise from its motion, and directed him to look at a dis- tant object; whilst the eye was in this situa- tion, he placed a microscope, in such a man- ner, that the wire, with which it was furnished, apparently coincided with the outer surface of the cornea; and then directing the specta- tor to look at a nearer object, he found that the cornea immediately projected beyond the wire of the microscope. See Phil. Trans. vol. Ixxxv. p. 16. Now when the distance of an object is diminished, supposing no alteration to take place in the eye, the divergency of the ex- treme rays of the pencil incident upon. the pupil, is increased ; and therefore, if the image - of the object in the first situation be formed upon the retina, in the latter it will be formed behind it; but an increase in the curvature of the cornea will increase the convergency of the refracted rays, or bring them sooner to a focus; and thus by a proper change in this coat of the eye, the rays will again be brought to a focus upon the retina, and the object be still seen distinctly, The least distance at which objects can be seen distinctly by com- mon eyes, is about 7 or Sinches. ‘The greater distance cannot be so easily or accurately ascertained. It seems that the generality of eyes are capable of collecting parallel rays upon the retina, or somear to it, as to produce distinet vision ; and thus, the greatest distance at which objects can be distinctly viewed, is unlimited. For this reason, in adopting op- tical instruments to common eyes, and cal-+ culating their powers, we suppose the parts to be so arranged, that the rays in eack 3B2 vier” pencil may, when they fall upon the cornea, be parallel. If the humours of the eye be too convex, parallel rays, and such pencils as diverge from points at any considerable distance, are col- lected before they reach the retina; and ob- jects, to be seen distinctly, must be brought nearer to the eye. ‘This inconvenience may be remedicd by a concave glass, whose focal Jength is so adjusted as to give the rays, pro- ceeding from a distant object, such a degree of divergency as the eye requires. Wood's Optics. Direct or Simple Vision, is that which is performed by direct rays, or by rays passing directly from the object to the eye. Reflected Vision, is that which is performed by reflected rays, that is, by means of rays re- flected from mirrors or specula. See Mirror and REFLECTION. Refracted V1s10N, is that which is performed by means of refracted rays; or rays turned out of their direct course, by passing through glass, or some other medium. ; Seat of Vision, is that part of the eye on which the image of different objects are paint- ed, and thence conveyed to the brain. This is a subject which has been much disputed by philosophers, some referring it to one part and some to another. According to what we have stated under the article Viston, the image of objects are painted in an inverted order on the retina, at the same time the mind per- ceives them in their proper erect posture; and different hypotheses have been invented to account for this correction of position. But in a small book lately published by Mr. G. dIorner, a new theory of vision is started, and maintained with considerable ability and in- genuity, which goes to prove that the images of objects are reflected back again from their first position into an erect posture, and thence conveyed to the brain, so that the construc- tion of the eye corresponds exactly with the Gregorian telescope. See Eye. VISUAL, relating to sight or seeing, as visual angle, visual rays, &e. VisuaL Angle, is the angle under which an object is seen, or which it subtends at the eye. See Apparent MAGNITUDE. VisuaL Rays, are rays of light conceived to come from an object to the eye, and by which it becomes visible. VITELLIO, or Viiello, a Polish mathe- matician of the 13th century; he was author of a large Treatise on Optics, the best edition of which is that of 1572. Vitellio was the first optical writer of any importance amongst the modern Europeans. He collected all that had been given by Euclid, Archimedes, Pto- fomy, and Alhazen, though his work is now but of little use. VITREOUS Humour of the Eye. See Eve. VITRUVIUS (Marcus Vitruvius Pot- L10), in Biography, a celebrated Roman ar- chitect, of whom however nothing is known but what is to be collected from his ten books, De Architectura, still extant. In the preface Ui. 4 to the sixth book, he writes that he was e fully instructed in the whole circle of arts sciences; a circumstance which he speak with much gratitude, laying it down as ce that no man can be a complete architect out some knowledge and skill in every 0 branch of knowledge. And in the preface the first book he informs us, that he known to Julius Caesar; that he was aij wards recommended by Octavia to her | ther Augustus Cesar; and that he wag favoured and provided for by this empero' to be out of all fear of poverty as long a} might live. It is supposed that Vitruvius} born either at Rome or Verona, but it is known which. His books of architecture addressed to Augustus Cesar, and not show consummate skill in that partie; science, but also a very uncommon ge} and natural abilities. The best edition of works of Vitruvius is that published at j sterdam in 1649. Perault gave an excel) French translation of the same, and aq notes and figures: the first edition of wi was published at Paris in 1673, and the cond, much improved, in 1684, Mr. Wil Newton too, an ingenious architect, publis in 1780, &c. curious commentaries on V) vius, illustrated with figures; to whie added a description, with figures, of the’ tary machines used by the ancients. ¥ VIVIANI (VINcENTIO), a celebrated lian mathematician, was born at Florene 1621 or 1622, and died in 1703, at the} vanced age of 81 years, Viviani was a diseiple of the illustr Galileo, having studied under that great i ter for three years, viz. from the age of 1 20. His principal works are as follows: 1. De Maximis et Minimis Geome Divinatio in quintum Conicorum Apol Pergwei, folio, 1659. 2. Enodatio Problematum, &c. 3. De Locis Solidis secunda Divinatio¢ metrica, &c. folio, 1673; 2d edition at Flore 1701. | ULLAGE of a Cask, in Gauging, is wht wants of being full. ULTIMATE Ratios. To avoid bot tediousness of the ancients and the inaceu of the moderns, Sir Isaac Newton intrody what he called the method of prime and mate ratios, the foundation of which is) tained in the first lemma of the first bot} the Principia. 'Taking this first lemma 4 definition, it may be explained in the fol ing manner. . Let there be two quantities, one fixed the other varying, so related to each off that first the varying quantity continually) proaches to the fixed quantity.” Secor that the varying quantity does never reac) pass beyond that which is fixed. Thirdly,} the varying quantity approaches nearer t¢ fixed quantity than by any assigned differe then is such a fixed quantity called the of the varying quantity; or in a looser wa speaking, the varying quantity may be 82 UMB ultimately equal to the fixed quantity. sse three properties may be otherwise ex- issed more distinctly thus. First, the dif- snee between the varying quantity, and fixed quantity, must continually decrease. ondly, this difference must never become er nothing or negative. ‘Thirdly, this dif- 1 : j d quantity than by any assigned ratio; or | difference between the two quantities st become a less part of the fixed quantity ha any fractional part that is assigned, how } Il soever the fraction expressing such part 4) be. Wherever these properties are found, | fixed quantity is called the limit of the ifing quantity, or the varying quantity is ». to be ultimately equal to the fixed quan- i This last phrase must not be taken in yibsolute literal sens¢c, there being no ulti- je state, no particular magnitude that is | ultimate magnitude of such a varying yatity. Under the word quantity in this nition must be included not only numbers, is, &e. but more especially ratios consi- ed asa peculiar species of quantity. i agnitudes thus considered do not consist iidivisible parts, but are imagined generated hiotion. Lines, for instance, are described, jin their description are generated not by apposition of parts, but by the continual ion of points, surfaces by the motion of 1s, solids by the motion of surfaces, angles jhe rotation of their sides, time by a con- fal flowing, and so in other things. The ie or ultimate ratios of magnitudes, thus srated, are investigated by observing their e increments or decrements, and thence | ng the limits of the ratios of those variable )nitudes; not of the ratios to which the jnitudes ever arrive, but those limits to ith the ratios of magnitudes perpetually loach. ) has been asserted by some that quantities 10t have a first and a last ratio. ow, if they have not a first ratio, they i: not a second nor a third ratio, &c.; there- iithey have no ratio in the time T. But in fime T they are quantities; and therefore must have a ratio; that is, they have a }), and they have not a ratio in the time T; this absurd. ‘Therefore they have a first = ey cease to exist at the end of the time y supposition: therefore, after the end of ime T' they are nothing ; consequently, at end of the time 'T they have no ratio. tin the time T they had a ratio; and after iond of the time they have no ratio: there- they had a last ratio. e Newton’s Principia, lib. i.; Smith’s sions; Ludlam on Ultimate Ratios, &c. LTRAMUNDANE, beyond the world, is , part of the universe which is supposed to seyond the limits of our system. MBLLICUS, or Umbilical Point, the same OCUS, MBRA. See Suapow. ance must become less in respect to the v OY, UNDECAGON, a polygon of eleven sides. See PoLyGon. UNDULATORY Motion, is applied to de- note a motion in any fluid, by which its parts are agitated like the waves of the sea, and is particularly applied to the motion of the air in the propagation of sound. UNEVEN Number, or Odd Number, that which cannot be divided into two equal in- tegral parts. See NuMBER. UNGULA, is the bottom part cut off by a plane passing obliquely through the base of a cone or cylinder, being thus called from its re- Prime to the ungula or hoof of a horse, ce, The rules for measuring conical and cylin- drical ungula will be found under the articles Cone and CyLinper, and their investigations at p. 218 and 246, 2d edition of Hutton’s Mensuration. UNICIA, the 12th part of a thing. UNICL#, the same as co-efficients. CoO-EFFICIENT. UNICORN. See Monoceros. UNIT, or Unity, is the representation of any thing considered individually, without regard to the parts of which it is composed. UNIVERSE, a collective term signifying the assemblage of heaven and earth, with all things in and upon them. UNLIKE Quantities, in Algebra, are such as are expressed by difierent letters or dif- ferent roots or powers of the same letter. See Definitions, article ALGEBRA. UNLIMITED Prodlem, that which admits of an indefinite number of solutions. See INDETERMINATE Problem. VOID Space. See Vacuum. VOLCANO, (from Vuleanus, Latin, the god of fire,) a burning mountain, hollow below, and communicating, perhaps, with cavities still deeper than its own, from which it is supplied with fire, and ignited materials, which it usually throws up, after uncertain intervals, through one or more external aper- tures or spiracles. Volcanoes constitute, without doubt, the most striking and formidable geognostic phee- nomenon which nature has presented to our view. They are not, indeed, so destructive to the lives of the human race as earthquakes ; but they offer to the eye something much more terrific. Their number is very consider- able, nearly two hundred having been reckon- ed by different writers. There is an immense range of them running from north to south on the continent of America, and oceupying the summits of many of the Andes, as well as of the Mexican and Californian ridges. There is also a considerable number which spread along the eastern coast of Asia, and sprinkle the Indian islands. Iceland alone. contains eight voleanoes. One of the loftiestis the Peak of ‘Teneriffe, though at present less frequent in ifs eruptions than many others. Several indeed of those that may perhaps be regarded as the most ancient appear to have spent See VOL themselves altogether; while others have been rising, even in our own days, as ‘it were to supply their places. Among the latter we have a very interesting account, in the Phil. Trans. for 1708, of a volcano that. burst out, for the first time, in the Archipelago, near the island of Erini, in the beginning of May in- the preceding year, raising, at the same time, a new island out of the bosom of the deep. Similar eruptions, succeeded by similar islands, haye occurred in various in- stances, in the group of the Azores, and the Sicilian seas; and captain Tillard, of his ma- jesty’s sloop Sabrina, has given a very valuable description, in the Phil. Trans. for 1812, of a like phenomenon, which took place no longer since than June, 1811, in the vicinity of St. Michael’s: the island then thrown up having been called by him Sabrina, after the name of his own ship. ‘The two volcanoes with which we are best acquainted are those of Etna and Vesuvius. The former has been burning as far back as the records of European history extend. We have an account of an eruption during the expedition of the Argonauts, which took place at least twelve centuries before the com- mencement of the Christian era. The follow- ing dates of the most remarkable eruptions of this volcano, to the middle of the seven- teenth century, are taken from an article in the Phil. Trans. for 1669. 476 years before Christ. Mentioned by Thucydides. f 40 years after Christ. During the reign of Caligula, 812 years after Christ. During the reign of Charlemagne. 1284, 1329, 1444, 1536, 1633, 1650. ‘The two writers who have chiefly signalized themselves upon the phenomena of this and the volcanoes in the neighbourhood are Sir William Hamilton and the Abbé Spallanzani. ‘To the former we are chiefly indebted for their history and effects, and to the latter for their possible causes and geognosy. _ Lunar Voicanors. As the moon has on its surface mountains and valleys in com- mon with the earth, some modern astronomers have discovered a still greater similarity, viz. that some of these are really volcanoes, emit- ting fire as those on earth do. An appearance of this kind was discovered some years ago by Don Ulloa in an eclipse of the sun. It was a small bright spot like a star, near the margin of the moon, and which he at that time supposed to have been a hole with the sun’s light shining through ‘it. Succeeding observa- tions, however, have induced astronomers to attribute appearances of this kind to the erup- tion of volcanic fire; and Dr. Herschel has particularly observed several eruptions of the Junar volcanoes, the last of which he gives an account of in the Phil. Trans. for 1787. “ April 19, 10° 36™, sidercal time. I perceive {says he) three volcanoes in different places of the dark part of the new moon. Two of voR them are either already nearly extinet, otherwise in a state of going to brea which perhaps may be decided next lw tion; the third shows an actual eruption, fire, or luminous matter. I measured _ distance of the crater from the northern li of the moon, and found it 8’ 57°3” ; its ligh| much brighter than the nucleus of the con} which M. Mechain discovered at Paris | 10th of this month. at ** April 20, 10" 2™ sidereal time. They cano burns with greater violence than § night. Its diameter cannot be less than} by comparing it with that of the Georg) planet: as Jupiter was near at hand, I turk the telescope to his third satellite, and & mated the diameter of the burning part of; volcano to be equal to at least twice thal the satellite; whence we may compute 4 the shining or burning matter must be ab) three miles in diameter. It is of an irreg) round figure, and very sharply defined ona edges. The other two volcanoes are m farther towards the centre of the moon, resemble large, pretty faint nebulee, that 4 gradually much brighter in the middle; ] no well defined luminous spot can be disea ed in them. ‘These three spots are plainky be distinguished from the rest of the mat upon the moon; for the reflection of the sv rays from the earth is, in its present situafy sufficiently bright, with a ten feet reflector show the moon’s spots, even the darkes) them; nor did I perceive any similar ey mena last lunation, though I then viewed} same places with the same instrument. “'The appearance of what I have called) actual fire, or eruption of a volcano, exal resembled a small piece of burning char@ when it is covered by a very thin coat white ashes, which frequently adhere td when it has been sometime united, and ith a degree of brightness about as strong as ih with which such a coal would be seen to gy in faint daylight. All the adjacent part the volcanic mountain seemed to be fail illaminated by the eruption, and were # dually more obscure as they lay at a grej distance from the crater. This eruptions sembled much that which I saw on the 4¢¢ May, in the year 1788, but differed consid ably in magnitude and brightness; for} volcano of the year 1783, though much brig than that which is now burning, was { nearly so large in the dimensions of its € tion; the former seen in the telescope rest bled a star of the fourth magnitude a appears to the naked eye; this, on the 4 trary, shows a visible disc ofluminous maf} very different from the sparkling brighth of star-light.” , VOLTAISM, the same as GALVANil which see. ‘@ VORTEX, or Whirlwind, in Meteorol a sudden, rapid, violent motion of the aill circular whirling directions. q Vortex, is also used for an eddy or wi) URA ol, or a body of water, in certain seas and ers, which runs rapidly round, forming a + of cavity in the middle. ‘ORTEX, in the Cartesian Philosophy, is a tem or collection of particles of matter wing the same way, and round the same ry uch yortices are the grand machines by ich the Cartesians solved most of the mo- as and other phenomena of the heavenly lies. And accordingly, the doctrine of these tices makes a great part of the Cartesian losophy. See Cartesian Philosophy. WOUSSOIRS, Vault Stones, are the stones ich form immediately the arch of a bridge, ilt, &c. being cut somewhat in the form of scuncated pyramid, their under side forming » jntrados, to which their joints or ends yuld be every where in a perpendicular a } he length of the middle voussoir, or key- pne, and which is the least of all, should be put ~,th or ~4th of the span of the arch, sence these stones should be made to in- | ase in size all the way down to the impost ; jorder that they may the better sustain the vat weight which rests upon them without tng crushed or broken, and that they may o bind the firmer together. See ARCH. RANIBURGH, the name given to the ce- trated observatory of Tycho Brahe, founded him in the little island of Weenen in the S41hHa. RANUS, Herschel, or Georgium Sidus, » name of the new planet discovered by 1. Herschel the 13th of March, 1781. ‘From certain inequalities in the motion of Joiter and Satarn, which could not be ac- cunted for from the mutual action of these ynets, it was inferred by some astronomers tt there existed beyond the orbit of Saturn ther planet, by whose action these irregu- ities were produced, which happy conjec- Le was confirmed by Dr. Herschel on the iy above stated, and which, in compliment this present majesty George III. he calied bb Georgium Sidus, though on the continent i's more commonly, for the sake of analogy, ied Uranus, and frequently the Herschel. lis new planet, which is supposed to have en formerly observed as a small star by F'lam- ad, Mayer, and Le Monnier, and introduced to their catalogues of the fixed stars, is si- ited beyond the orbit of Saturn, at the dis- ice of 1800 million miles from the centre of # solar system, and performs its sidereal re- jution about the sun in 83 years, 150 days, (118hours. Its diameter is about 45 times Iger than that of the earth, being nearly {112 English miles. When seen from the rth its apparent diameter, or the angle it WUE subtends at the eye, is 3” 32”, and its mean diameter, as seen from the sun,is 4’... As the distance of this planet from the sun is double that of Saturn, it can scarcely be perceived with the naked eye; when, however, the sky is serene, it appears like a fixed star of the sixth magnitude with a bluish white light, and a brilliancy between that of Venus and the moon, but with a power of 200 or 300 its disc is visible and well defined. Uranus is attended by six satellites, all of which were discovered by Dr. Herschel. ‘The first is 25’"5 distant from its primary, and re- volves round it in 5421525’, The second sa- tellite is 33-9 distant from the planet, and performs its revolution in 8417 1'19", ‘The istance of the third satellite is 88"57, and its periodic time 10423" 4’. The distance of the fourth satellite is 44’-22, and the time of its periodic revolution 134 115 515. The dis- tance of the fifth satellite is 1/ 28°44, and its revolution is completed in 384 1° 49’. 'The sixth satellite is placed at the distance of 2’ 56:38, from the primary, and will therefore require 1074 16 40’ to complete one reyo- lution. | The second and fourth of these satellites were discovered by Dr. Herschel, on the 11th of January, 1787; and the other four in the years 1790 and 1794; but their distances and periodic times have not been so accurately ascertained as the other two. It is a remark- able circumstance with regard to these satel- lites, that they all move in a retrograde order, and in orbits all lying in the same plane and almost perpendicular to the ecliptic. According to Dr. Herschel’s computation they will be eclipsed in 1818, when they will appear to ascend through the shadow of the planet, in a direction almost perpendicular to the plane of the ecliptic. Laplace supposes that the first five satellites of this planet may be retained in their orbits by the action of its equator, and the sixth by the action of the interior ones, and hence he concludes that Uranus revolves about an axis very little in- clined to the ecliptic, and that the time of its diurnal rotation cannot be much less than that of Jupiter and Saturn. Schroeter suspects, from the appearance of the disc of this planet at particular times, that considerable changes are going on in its atmo- sphere. See Herschel, Phil. Trans. 1781, 1783, 1788; Schrocter, Lilienthalische Beobachtun- ven der neu entdecken planeten, Ceres, &c. Laplace, Mechanique Celeste, tom. iil. URSA Major and Minor, the Great and Litile Bear, two northern constellations. See CONSTELLATION. VULPECULA et Anser, the Fox and Goose. See CONSTELLATION. WAR eatin’ W ‘j WAGGONER, a popular name given to the constellation Ursa Major. WALES (WILLIAM), a respectable mathe- matician, who accompanied Cook in his first voyage round.the world, as astronomer, and was afterwards appointed mathematical mas- ter at Christ’s hospital. He was author of an Account of Astronomical Observations in the Southern Hemisphere, 4to.; Remarks on Forster’s Account of Cook’s Voyage ; Inquiry into the Population of England and Wales; Robertson’s Elements of Navigation improv- ed; an Essay on finding the Longitude by Time-keepers, &c. Wales died in 1799. WALLIS (Dr. Joun), and eminent Eng- lish mathematician, was born at Ashford. in Kent, in 1616, and died at Oxford in 1703, in the 88th year ofhis age. Dr. Wallis was the au- thor of several ingenious and learned works on various branches of the mathematics, which he collected and published, in 1657, in two parts, entitled Mathesis Universalis, in 4to. ; and in 1658, Commercium Epistolicum de Questionibus quibusdam Mathematicis nuper habitum, in4to.; which was a collection of letters written by many learned men, as Lord Brounker, Sir Kenelm Digby, Fermat, Schoo- ten, Wallis, and others. In 1670 he published his Mechanica; sive de Motu, 4to. In 1676, he gave an edition of Archimedis Syracusani Arenarius et Dimensio Cireuli: and, in 1682, he published from the manuscripts, Claudii Ptolemei Opus Harmo- nicum, in Greek, with a Latin version and notes; to which he afterwards added, Ap- pendix de veterum Harmonica ad hodiernam comparata, &e. , In 1685, he published his History and Prac- tice of Algebra, in folio; a work full of learned and useful matter. His Arithmetic of Inf- nites is also a book of genius and invention. He had also numerous papers in the Philoso- phical-Transactions, in almost every volume, from the first to the twenty-fifth volume, In 1697, the curators of the university press at Oxford collected the doctor’s mathematical works, which had been printed separately, some in Latin, some in English, and pub- Hshed them all together in Latin, in three vo- lumes, folio, 1699. WARD (Sevn), an able English mathema- tician, was born in 1617, and died Bishop of Salisbury in 1671. He was author of a phi- losophical Essay on the Being and Attributes WAT of God; Exercitatio Enpistolica in Hok Philosophiam, 8vo.; an Idea of Trigonome Geometrical Astronomy, &e. WARGENTIN (Perer), a Swedish mat) matician, was born in 1717. Yn this counh he is best known from his tables for comp. ing the eclipses of Jupiter’s satellites. § died at the Observatory at Stockholmy 1783. . WARING (Epwarb),a very eminent E lish mathematician, was born about the yj 1735, He was author of many works on most abstruse branches of mathematies, « considers himself as the inventor of 400 n) propositions in various parts of analysis. | In 1762, he published his Miscellanea A\ lytica, and his Proprietates Algebraicar} Curvarum, in 1772; Meditationes Algebrain in 1770; and the Meditationes Analyticae, 1776. | Besides which he published a variety) papers in the Philosophical Transactions, ¢ vol. lit. p. 294, Mathematical Problems ; } 193, New Properties in Conics; ly. 143, Theorems in Mathematies; Ixix. Problel concerning I[nterpotations ; 86, a General } solution of Algebraical Equations; Ixxvi.} on Infinite Series; Ixxvii. 71, on Finding | Values of Algebraical Quautities by Ci verging Serieses, and demonstrating and § tending Propositions given by Pappus a others; Ixxvili. 67, on Centripetal Fore 2b. 588, on some Properties of the Sum of 4 Divisors of Numbers ; [xxix. 166, on the ¥ thod of Correspondent Vaines, &c.; ib. Uh on the Resolution of Attractive Powe Ixxxi. 146. on Lifinite Serieses ; Ixxxiv. 38% general term is a determinate function of the distance of the term of the series, W ing died in 1797. | WATER, a transparent fluid, without) lour, smell, or taste; in a very small deg compressible; when pure, not liable to spr taneous change liquid in the common te perature of our atmosphere, assuming a sc form at 32° Lahrenheit, and a gasseous} 212°, but returning unaltered to its liquid st ov resuming any degree of heat between th: points; it is capable of dissolving a greal number of natural bodies than any other fh Whatever, and especially those known by > name of the saline ; performing the most i portant functions in the vegetable and anit} tw WAT oms, and entering largely into their qposition as a constituent part. iter exists, therefore, in three different es; in the solid state, or state of ice, in liquid, and in the state of vapour or steam, \ssumes the solid form, as observed above, on cooled down to the temperature of 32°, vhich state it increases its bulk, and hence rts a prodigious expansive force, owing: (he new arrangement of its particles, which ime a crystalline form, the crystals cross- each other at an angle of 60° or 120°. 2 specific gravity of ice is therefore less that of water. When ice is exposed to mperature above 32°, it absorbs caloric, ‘ch then becomes latent and ‘is converted » a liquid state, or that of water. At the perature of 42°°5 water is at its maximum lensity ; and according to some accurate eriments upon water in this state, a French ic foot of it weighs 70 pounds 223 grains neh, which is equal to 529452°9492 troy ms. An English cubic foot, at the same tem- jature, weighs 437 102-4946 grains troy. By \fessor Robinson’s experiments it is ascer- ved that a cubic foot of water, at the tem- jature of 55°, weighs 998°74 avoirdupoise ces, of 437°5 grains troy each, or about 12 ice less than 1000 ounces avoirdupcise, ich Jatter, however, 1s the usual estimate. Vhen water is exposed to the temperature 1212°, it boils; and if this temperature be itinued the whole is converted into elastic jour or steam. In this state it expands bout 1800 times its bulk, when in the state water, which shows what an astonishing lansive force it must exert when it is con- d; and hence its application to the steam- ine, of which it is the moving power. STEAM- Engine. WATER-Mill, an engine which is put in jon by the action of water, of which there four kinds: breast-mills, undershot-mills, ‘mills with horizontal wheels; of which, Weyer, the latter kind is by far the least com- ii, being very disadvantageous and defi- itm point of utility. Ina breast-mill the fer falls down upon the wheel at right an- to the float-boards or buckets placed all fad the wheel to receive it: if float-boards bused, the water acts only by its impulse ; if buckets, it acts also by the weight of ‘er in the buckets in the under quadrant ie wheel, which is considerable. In the ershot wheel fioat-boards only are used, the wheel is turned merely by the force he current running under it, and striking nthe boards. In the overshot wheel the eris poured over the top, and thus acts Kcipally, though not altogether, by its tht, for the fall upon the upper part can- be very considerable, lest it should dash Water out of the buckets. Hence it is Vent that an undershot wheel must require uch larger supply of water than the other; jbreast-inill is next, uxless the fall is yery bit; and an overshot-mill the least. 2 ; ] WAT a It was long believed that the float-boards of an overshot wheel ought to be so propor- tioned that when one of them was in a vertical position, or at the middle of its immersion, the next board should be just entering the water; but it is now well known that the more float-boards such a wheel has, the greater and more uniform, will be its effects. Accord- ing to the experiments made by M. Bossut on this subject, a wheel furnished with 48 float-boards produced a much greater effect than one furnished with 24; and the latter a greater effect than one with 12, their immer- sion in the water being equal. {f a stream of water impinge on the float- boards of an undershot wheel, and escape from it the very instant after it has made its impact, the quantity of water which actually impingés against the wheel, will be to the whole quantity which passes by it in a given time, as the difference between the velocities of the water and of the whecl to the absolute velocity of the water. Let WH bethe wheel, DA the stream of water, and let the float-board EF first receive the im- pact from the water at F,and quit it at C; also let DE’ be to IC as the absolute velocity of the water, to the velocity of ' the float-board. Then AC r EK D when I’ arrives at C, the particle at D will have passed at F; and taking DE=TI‘C, all the water in the space DE will pass by the wheel without impinging against it: for it cannot impinge on the float i, because that float emerges from the water at C; neither can it impinge on the subsequent float, be- cause it has already passed it. ‘Yherefore the whole quantity of water which passes by the wheel ia a given time, is to that which actually impinges against it, as DI’ to EF. Cor. 1. Hence we may correct the mistake of Mr. Waring, in his new doctrive of mills, who lays it down as a fundamental principle, that while the stream is invariable, whatever be the velocity of the wheel, the same number of particles or quantity of the fluid must strike it somewhere or other in a given time. See vol. i. of the Transactions of the American Society, p. 144. Cor. 2 'Vhe force of the impinging water is as the square of the difference between the velocities of the whee! and the water. For the force is as the relative velocity into the quantity of impinging matter, and the latter is manifestly as the relative velocity; therefore the force will be as the square of the relative velocity. : If W be a weight fastened to a line which is wound round the horizontal axis of an un- dershot water-wheel, A the altitude of a column? of water, equal to the foree of the impact of the water on the wheel, when the wheel is quiescent; V the velocity with which -~ nt” a Le WAT ihe water impinges on the float-boards, v the velocity of the circumference of the wheel, R the radius of the wheel, and r the radius of the axle: then will the velocity of the wheel wr’ bev = V—V 0/ 27. For the relative velocity with which the water strikes the wheel is V—v; whence, because the force of the stroke isas the square of the relative velocity, we have V* :(V —v)? ::A:F, the force of the water to tern the wheel when the velocity is v; and 2 4 PoA (= *’) Rr vivo But the acceleration of the wheel will cease when the force of the water to turn the wheel “is equal to the force of the weight which op- poses it ; that is, wr FR = wr, or F = Hence substituting this value of F for it in the preceding value of v, there results v = V EAL, ah? for the velocity of the wheel, when its acceleration ceases. This conclusion, it should be observed, is true only on the hypothesis, that the water escapes from the wheel as fast as it impinges. Cor. 1. Ifthe weight W vary, its momentum will be the greatest possible when the wheel has acquired its uniform velecity, if W — 4AR : or - Cor. 2. The greatest momentum generated in the ascending weight will be = = AV. Cor.3. When the momentum of the as- cending weight W is a maximum, that weight will be 4 of the weight, which would, if sus- pended from the axle, balance the force of the stream. Cor.4. When the momentum of the as- cending weight is a maximum, the velocity of the wheel will be 4 of the absolute velocity of the stream. Cor.5. When the uniform velocity of the ascending weight W is a maximum, the ra- 9wr 4A For the radius of the wheel being R, and that of the axle r, the uniform velocity of the ascending weight will be found by multiply- ing V—V eA ~"_ into the fraction a3 that dius of the wheel will be = AR is, the velocity of w will be = x V—V 23 } / ren which, supposing R variable, and Swr 4A¢ if the velocity of the stream be given, the greatest effect will be as the quantity of water expended. For by cor. 2, last prop. the great- making the fluxion = 0, gives R = WAT ¥ est effect is as = Av ; that is, since 4 constant, the effect is in a ratio compoun of the force of impact, and the velocity of | stream ; but the force of the impact is as, quantity of water expended and velocity ¢ jointly; consequently the effect is as quantity of water expended, and the squ! of the velocity ; or, if the velocity be giy, as the quantity of water expended. -~ | Cor. 1. When the expense of water is same, the greatest effect will be as the squ) of the velocity. | Cor. 2. The expense of water being } same, the effect will be as the height of | head of water. ; For V =/2g¢ h, whence ho V. Cor. 3. The aperture being the same, effect will be as the cube of the velocity, For the effect is as Q (the quantity of wa) into V* ; and when the aperture is given Q¢) whence, effect « V3. If all the water which passes by an unjj shot wheel be supposed to impinge agains} the force of the stream will be simply im} direct ratio of the relative velocity. : Because the number of particles wl} strike the wheel in a given time is gi} whatever be the velocity of the wheel. Cor. 1. According to this hypothesis, » velocity of the wheel will be equal to V= wr ‘? AK For in this case, we have V: V—v::A whence v= V — es ; which, by substitu for Fits value me becomes v = V — Ye Cor. 2. Retaining the same hypothesi AR & 2R78 it isa maximum. For sinee the uniform ¥ city of the circumference of the wheel i vV— er which, multiplied into ob . g AR’ R Vir ew rw R A R* cending weight w; hence its momentur Vite 2 Mine apes hich bej ‘ Ties a ar: > WhICD bDeIng a Maximum make its fluxion = 0, considering w as 4 able, whence results w = ee 7 Cor.3. On thesame supposition, the vel of the wheel will be half the velocity of stream, when the effect is a maximum. W vary its momentum will be = for the nnifom velocity of th For, by corl. vy = V — ane and in case of the last cor. w = ‘=. Putting r F for v in the former equation, it is transfo VARr i tov= V — _____ —1YV. ov—V SAR? 2 V | Cor. 4. Still retaining the same hypot “ale ‘ : WAT greatest momentum generated in the as- ng weight, will be = ¢ AY. haa Pie TF @iVX RoR the ascending weight; and the weight is the uniform velocity Zr 1 AV, the momentum. yed, wis by cor. 2 = Consequently x. ——— — n practice, the velocity of the wheel when machine is in its greatest perfection, will between one-third and one-half of the ve- ity of the stream. For the water does not escape the instant after it has made the yact, but is confined by the channel for 4e time; so that the succeeding water, ich would otherwise pass by the wheel in- caciously, drives the confined water against float-boards, and therefore acts in the same nner as if it actually impinged against the eel. Experiments show, however, that en the most work is done in a given time, yelocity of the wheel is much nearer the f than the third of the velocity. See in \. li. of the Philosophical Transactions an ‘ount of many interesting and important Seriments on this subject, of which also an itract-is given in vol. i. of Gregory’s Me- mics, to which work we are indebted for -present article, and to which the reader eferred for farther information. WATER-Poise. See HYDROMETER. WATER-Spout, a remarkable kind of teor, consisting of a moving column _ or lar of water, called by the French trompe, mits resemblance to the form of aspeaking mpet. ; {ts first appearance is in thé form of a deep ud, the upper part of which is white, and slower black. From the lower part of this ud there hangs, or rather falls down, what properly called the spout, resembling a iical tube, largest at the top. Under this tube is always observed a great tation of the water, flying up in the man- ‘ofa jet d’eau. For some yards above the face the water stands like a pillar or column, m the extremities of which it spreads, and as off in a kind of mist or smoke. This ne frequently descends as low as the middle the column, and continues for .sometime atiguous to it, though frequently it only ints to.it at some distance, either in a per- ndicular or in an oblique direction. Frequently it can be scarcely distinguished iether the cone or the column appears the it, both rising suddenly against each other. other times the water boils up from the sea a great height, without any appearance of pout pointing to itin any direction ; indeed, nerally speaking, the boiling or flying up of 2 water has the priority, this always pre- ling its being formed into a column. For >most part the cone does not appear hol- fr till towards the end, when the sea-water sami thrown up along its middle, as : WAT smoke up a chimney: soon after this, the spout or canal breaks and disappears; the boiling up of the water, and even the pillar, continuing to the last, and for some time af terwards ; sometimes till the spout form itself — again, and appear anew, which it will do se- veral times in a quarter of an hour., See a description of several water-spouts by Mr. Gordon, and by Dr. Stuart, in Phil 'Trans. Abr. vol, iv. p. 103, &e. M. de la Pryme, from a near observation of two or three spouts in Yorkshire, described in the Philosophical Transactions, num. 281, or Abr. vol. iv. p. 106, concludes, that the water-spout is nothing but a gyration of clouds caused by contrary winds meeting in a point, or centre; and there, where the greatest con- . densation and gravitation is, falling down into a pipe, or great tube, somewhat like Archi- medes’s spiral screw ; and in its working and whirling motion, absorbing and raising the. water in the same manner as the spiral screw does, and thus destroying ships, &c. Thus, June the 21st, he observed the clouds much agitated above and driven together, upon which they became very black, and were hurried round; whence proceeded a most audible whirling noise like that usually heard in a mill. Soon after there issued a long tube, or spout, from the centre of the congregated clouds, in which he observed a spiral motion, like that of a screw, by which the water was raised up. Again, August 15, 1687, the wind blowing at the same time out of the several quarters, created a great vortex and whirling among the clouds, the centre of which every now and then dropped down, in shape of a long thin black pipe, in which he could distinctly behold a motion like that of a screw, continually drawing upwards, and screwing up, as it were, wherever it touched. In its progress it moved slowly over a srove of trees, which bent under it like wands, — in a circular motion. Proceeding it tore off the thatch from a barn, bent a huge oak tree, broke one of its greatest limbs, and threw it toagreat distance. He adds, that whereas it is commonly said, the water works and rises in a column before the tube comes to touch it; this is doubtless a mistake, owing to the fineness and transparency of the tubes, which do most certainly touch the surface of the sea, before any considerable motion can be raised in it; but which do not become opaque and visible, till after they have imbibed a con- siderable quantity of water. The dissolution of water-spouts he ascribes to the great quantity of water they have glut- ted ; which, by its weight, impeding their mo- tion, upon which their force and even exist- ence depends, they break, and let go their contents, which so frequently prove fatal to whatever is found beneath. ; A notable instance of this may be seen in the Philosophical Transactions (num. 363, or Abr. vol. iv. p. 108) related by Dr. Richardson. WAT A spout, in 1718, breaking on Emmot-moor, near Coln, in Lancashire, the country was immediately overflowed; a brook, in a few minutes, rose six feet perpendicularly high; and the ground upon which the spout fell, which was 66 feet over, was torn up to the very rock, which was no less than 7 feet deep; and a’deep gulf was made for above half a mile, the earth being raised in vast heaps on each side. See a description and figure of a water-spout, with an attempt to account for it, in Franklin’s Exp. and Obs. p- 226, &e. Signor Beccaria has taken pains to show that water-spouts have an electrical origin, To make this more evident, he first describes the circumstances attending their appearance, which are the following: They generally appear in calm weather. The sea seems to boil, and to send up a smoke under them, rising in a hill towards the spout. At the same time, persons who have been near them have heard a rumbling noise. ‘The form of a water-spout is that of a ‘Speaking trumpet, the wider end being in the clouds, and the narrower end towards the sea. The size is various, even in the same spout. ‘The colour is sometimes inclining to white, and sometimes to black. Their position is sometimes perpendicular to the sea, some- times oblique ; and sometimes the spout itself is in the form of a curve. Their continuance is very various, some disappearing as soon as formed, and some continuing a considerable time. One that he had heard of continued a whole hour. But they often vanish, and pre- seutly appear again in the same place. ‘The very same things that water-spouts are at sea are some kinds of whirlwinds and hurricanes by land. They have been known to tear up trees, to throw down buildings, and make caverns in the earth ; and inall these cases, to scatter earths, bricks, stones, timber, &c. toa great distance in every direction. Great quantities of water have been left, or raised by them, so as io make a kind of deluge ; and they have always been attended by a prodi- gious rumbling noise. ‘That these phenomena depend upon elec- tricity cannot but appear very probable from the nature of several of them; but the con- jecture is made more probable from the fol- lowing additional circumstances. 'They gene- rally appear in months peculiarly subject to thunder-storms, and are commonly preceded, accompanied, or followed by lightning, rain, or hail, the previous state of the air being similar. Whitish or yellowish flashes of light have sometimes been seen moving with pro- digious swiftness about them. And lastly, the manner in which they terminate exactly re- sembles what might be expected from the prolongation of one of the uniform protube- rances of electrified clouds, mentioned before, towards the sea; the water and the cloud mutually attracting one another: for they suddenly contract themselves, and disperse cloud passed his zenith, and went gradu WAT * l almost at once; the cloud rising, and. water of the sea under it falling to its le But the most remarkable circumstance, | the most favourable to the supposition of tj depending on electricity, is, that they hy been dispersed by presenting to them sh pointed knives or-swords. ‘This, at least} the constant practice of mariners, in m parts of the world, where these water-ap abound, and he was assured by several them, that the method has often been doubtedly effectual. The analogy between the phenomens water-spouts and electricity, he says, may made visible by hanging a drop of water Wire communicating with the prime condug and placing a vessel of water under it, these circumstances, the drop assumes all\ various appearances of a water-spout, bot} its rise, form, and manner of disappearg Nothing is wanting but the smoke, wl) may require a great force of electricity to ¢ come visible. . Mr. Wilcke also considers the water-s as a kind of great electrical cone, raised \ tween the cloud strongly electrified and} sea or the earth, and he relates a very rem able appearance which occurred to him} and which strongly confirms his supposif| On the 20th of July 1758, at three o’cloe| the afternoon, he observed a great quantit} dust rising from the ground, and coveri field, and part of the town in which he was. ‘There was no wind, and the dust me gently towards the east, where appeare| great black cloud, which, when it was nea! zenith, electrified his apparatus positively, | to as great a degree as ever he had obset it to be done by natural electricity. 7 it towards the west, the dust then followin; and continuing to rise higher and higher it composed a thick pillar, in the form « sugar-loaf, and at length seemed to be in tact with the cloud. At some distance f this, there came, in the same path, ano’ great cloud, together with a long strean smaller clouds, moving faster than the | ceding. These clouds electrified his appari negatively, and when they came near | positive cloud, a flash of lightning was ¢ to dart through the cloud of dust, the posi cloud, the large negative cloud, and as fi the eye could distingnish, the whole trai smaller negative clouds which followed Upon this, the negative clouds spread 3 much, and dissolved in rain, and the air presently clear of all the dust. The w appearance lasted not abové half an h See Priestley’s Electr. vol. i. p. 438, &e. This theory of water-spouts has been ther confirmed by the account which Forster gives of one of them, in his “ Voy round the World,” vol. i. p. 121, &ce. On coast of New Zealand he had an opportu. of seeing several, one of which he has p cularly described. The water, he says, i gai a se of fifty or sixty fathoms, moved to- ds the centre, and there rising into vapour, he force of the whirling motion, ascended spiral form towards the clouds. Directly ¢the whirlpool, or agitated spot in the sea, oud gradually tapered into a long slender WAV. } ’ cht column of a cylindrical form. The er whirled upwards with the greatest vio- > which seemed to descend to meet the jig spiral, and soon united with it into a fein the spiral, and appeared to leave a (ow space in the centre; so that the water ned to form a hollow tube, instead of a { column; and that this was the case was lered still more probable by the colour, »ch was exactly like that of a hollow glass I, After some time, this last column was rvated, and broke like the others; and the anc of a flash of lightning, which at- ‘ed its disjunction, as well as the hail- ‘es which fell at the time, seemed plainly yidicate, that water-spouts either owe their jiation to the electric matter, or, at least, they have some connection with it. TATER-Wheel, the wheel of a water-mill, vhich the water acts asafirst mover. See rer-Mill and WHEEL. TAVE, in Physics, a volume of water ele- d by the action of the wind, &c. upon its Cl into a state of fluctuation, and accom- | ) ed by a cavity. ‘The extent from the bot- _ or lowest point of one cavity, and across | *leyation, to the bottom of the next cavity, ‘e breadth of the wave. aves are considered as of two kinds, hh may be distinguished from one another 1e names of natural and accidental waves. natural waves are those which are regu- proportioned in size to the strength of the il which produces them. The accidental ’s are those occasioned by the winds re- ig upon itself by repercussion from hills igh shores, and by the dashing of the *s themselves, otherwise of the natural i, against rocks and shoals; by which (as those waves acquire an elevation much e what tley can have in their natural /otton of the Waves. This.makes an article ne Newtonian philosophy, that author 4g explained their motions, and calculated velocity from mathematical principles, jar to the motion of a pendulum, and to eciprocation of water in the two legs of it and inverted cyphon or tube. $8 propositions concerning such canal, or FS the forty-fourth of the second book of rincipia, and is this: ‘“ If water ascend ‘lescend alternately in the erected legs of ‘nal or pipe; and a pendulum be con- ted whose length, between the point of ‘sion and the centre of oscillation, is \!to half the length of the water in the |; then the water will ascend and de- ‘lin the same time in which the pendu- iscillates.” 'The author hence infers, in ( 45, that the yelocity of wayes is in the WEA subduplicate ratio of their breadths; and ina prop. 46, he proceeds to tind the velocity of waves, as follows: “ Let a pendulum be ¢con- structed whose length between the point of suspension and the centre of oscillation is equal to the breadth of the waves, and in the time that the pendulum will perform one single oscillation the waves will advance forward nearly a space equal to their breadth. That which is called the breadth of the waves being the transverse measure lying between the deepest part of the hollows, or between the tops of the ridges.” Let ABCDEF represent a stagnant water ascending and descending in successive waves. Also let A, C, E, &c. be the tops of the waves; and B, D, F, &c. the intermediate hollows. Because the motion of the waves is carried on by the successive ascent and de- scent of the water; so that the parts of it, as A, C, E, &e. which are highest at one time, be- come lowest immediately after, and because the motive force, by which the highest parts de- scend and the lowest ascend, is the weight of the elevated water, that alternate ascent and descent will be analogous to the reciprocal inotion of the water in the canal, and observe the same laws as to the times ofits ascent and descent, and therefore (by prop. 44, above mentioned), if the distances between the high- est places of the waves, A, C, E, and the lowest, B, D, E, be equal to twice the length of any pendulum, the highest parts, A, C, E, will become the lowest in the time of one ogeil- lation, and in the time of another oscillation will ascend again. ‘Therefore between the passage of each wave, the time of two oscilla- tions will intervene ; thatis, the wave will de- scibe its breadth in the time that the pendu- lum will oscillate twice; but a pendulum of four times that length, and which therefore is equal to the breadth of the waves, will just oscillate once in that time, Q. E. I. “ Corol.1. Therefore waves, whose breadth is equal to 39% inches or 323 feet, will advance through a space equal to their breadth in one second of time; and therefore in one minute they will go over a space of 1953 feet; and in an hour a space of 11737 feet nearly, or two miles and almost a quarter. “ Corol. 2. And the velocity of greater or less waves will be augmented or diminished in the subduplicate ratio of their breadth. WAY-WISER. See PEDOMETER and PE- RAMBULATOR. WEATHER, denotes the state of the at- mosphere, with regard to heat and cold, dry- ness and moisture, wind, rain, hail, &c. Mr. Kirwan, in vol. v. of the Transactions of the Irish Academy, has laid down the fol- lowing rules, as the result of a careful exami- WEA nation of observations which had been made in England, during a period of 112 years, 1. When no storm has either preceded or foliowed the vernal equinox, the succeeding summer is generally dry, or at least so, five times out of six. 2. If a storm happen from an easterly point, on the 19th, 20th, or 21st day of May, the en- suing summer will, four times in five, be also dry. The same event generally takes place, if a storm arise on the 25th, 26th, or 27th days of March, in any point of the compass. 3. Should there be a storm, either at south- west, or at west-south-west, on the 19th, 20th, Qist, or 22d of March, the following summer is wet, five times out of six. In England, if the winters and springs be dry, they are mostly cold; but, if moist, they are generally warm: on the contrary, dry summers and autunins are usually hot; as moist summers are cold. ‘Thus, if the humi- dity or dryness of a particular season be de- termined, a tolerably correct idea may be formed respecting its temperature. ‘To these indications may be added the following max- ims; which, being the result of observations made by accurate inquirers, may so far be de- pended upon, as they will afford a criterion of the mildness or severity, and of the dryness or moisture, of future seasons. 1. A moist autumn, succeeded by a mild winter, is generally followed by a dry and cold spring; in consequence of which, vegetation is greatly retarded. 2. Should the summer be uncommonly wet, the succeeding winter will be severe ; because the heat or warmth ofthe earth will be carricd off by such unusual evaporation. Farther, wet summers are mostly attended with an in- creased quantity of frnit on the white-thorn and dog-rose; nay, the uncommon fruitful- ness of these shrubs is considered as the pre- sage of an intensely cold winter. 3. A severe winter is always indicated by the appearance of cranes and other birds of passage at an early period in autumn; because they never migrate southwards, till the cold season has commenced in the northern re- gions. 4, If frequent showers fall in the month of September, it seldom rains in May; and the reverse. 5. On the other hand, when the wind often blows from the south-west, during either sum- mer or autumn; when the air is unusually cold for those seasons, both to our sensations, and by the thermometer; at the same time, the mercury being low in the barometer ;— under these conditions, a profuse fall of rain may be expected. 6. Great storms, rains, or other violent commotions of the clouds, produce a kind of crisis in the atmosphere; so that they are attended with a regular succession, either of fine or of bad weather, for some months. Lastly, an unproductive year mostly suc- ceeds a rainy winter; as a rough and cold WED . | autumn prognosticates a severe Winter. also the article BAROMETER. WEATHER-Giass. HYGROMETER. ss WEDGE, one of the five mechanical po or simple engines, being a geometrical we or very acute triangular prism, applic the splitting of wood, rocks, or raising weights. The wedge is made of iron, / hard matter, and applied to the raisif vast weiyhts, or separating large or very) blocks of wood or stone, by introducin thin edge of the wedge, and driving it blows struck upon the back by hamme mallets. The wedge is the most powerful of a simple machines, having an almost unli advantage over all the other simple mech powers; both as it may be made vastly} in proportion to its height, in which consig§ own natural power; and as it is urged k force of percussion, or of smart blows, is a force incomparably greater in a time than any mere dead weight or pre such as is employed upon other mae And accordingly we find it produces « vastly superior to those of any other ] whatever; such as the splitting and the largest and hardest rocks; or eve raising and lifting the largest ship, by d a wedge below it; which a man can] the blow of a mallet. | To the wedge may be referred all | tools, and tools that have a sharp poi order to cut, cleave, slit, split; chop, ] bore, or the like; as knives, hatchets, sy bodkins, &e. 1] ! the See BAROMETER rt In the wedge, the friction again is very great, at least equal to the force} overcome ; because the wedge retains a sition to which it is driven; and therefo resistance is at least doubled by the fric Authors have been of various opinion cerning the principle from whence the’ derives its power. Aristotle considers two levers of the first kind, inclined te each other, and acting opposite ways. | Ubaldi, Mersenne, &c. will have them levers of the second kind. But De shows, that the wedge cannot be redu any lever at all. Others refer the we the inclined plane. And others again) De Stair, will hardly allow the wedge t any force at all in itself; ascribing mu greatest part to the mallet which drives) The doctrine of the force of the wed} cording to some writers, is contained 1) proposition: ‘If a power directly app! the head of a wedge, be to the resistal be overcome, as the breadth of the bac} the height; then the power will be ed the resistance ; and if increased, it wil come it.”’ | But Desaguliers has proved that, whi resistance acts perpendicularly agait] sides of the wedge, the power is to the) — wrI stance, as the thickness of the back is to | length of both the sides taken together. ithe same proportion is adopted by Wallis (. Math. vol. i. p. 1016); Keill ({ntr. ad |. Phys.); Gravesande (Elem. Math. lib. i. |, 14), and by almost all the modern mathe- icians. Gravesande, indeed, distinguishes mode in which the wedge acts, into two 42s, one in which the parts of a block of iid, &¢. are separated farther than the edge (penetrated to, and the other in which they ‘e not separated farther. See the work mentioned: see also the Treatises on ishanics, by Emerson and Gregory, and ‘Mlam’s Essay on the Wedge. \VEIGH, or Wey, an old English weight {56 pounds avoirdupois. 'VEIGHT, in Physics, a consequence of ity, being that quality in natural bodies, ‘which they tend downwards or towards fearth’s centre. See GRAVITY, and Spe- j GRAVITY. 7EIGHT, in Mechanics, denotes any thing He raised, sustained, or moved by a ma- fe, as distinguished from the power, or i by which the machine is put in motion. 7EIGHT, in Commerce, denotes a body of jown weight, appointed by law to be the ‘dard of comparison between different hitities of merchandise of certain descrip- (3; the weight itself being usually of lead, i, brass, or other metal. ‘ae great diversity of weights and mea- $, in all nations, for different kinds of jmoditics, has always been a. just subject egret and complaint; being the cause of jus disputes and deceptions, which it is st impossible to avoid under present cir- istances. And it is therefore much to be ied, though, perhaps, little to be expected, hone uniform system of weights should be Hted as applicable to all kinds of sub- ses; an attempt at which was made in nee during the revolution, but which it is has been lately laid aside by an imperial fee, in consequenee of the repeated repre- ations of people in trade ; so difficult is it )ercome prejudices and customs long esta- ied, however advantageous the change i be, when properly understood. See some }unt of this system in the subsequent part ‘iis article, and under the article Mega- ) the reign of King Richard I. it was (ned, that there should be only one weight one measure throughout England; and 'e Phil. Trans. No. 458, p. 457, we find an (ant of the analogy between English éhts and measures drawn up by Mr. Bar- ’ in which he states, that anciently the ': foot of water was assumed as a general dlard for all liquids. This cubic foot, of ». multiplied by 32, gives 2000lbs. the tht of a ton; and hence 8 cubic feet of ir made a hogshead, and 4 hogsheads a lor ton, in capacity and denomination, as as weight. W E'I Dry measures were raised on the same model. A bushel of wheat, assumed as a ge- neral standard for all sorts of grain, also weighed 622lbs. Eight of these bushels make a quarter, and 4 quarters, or 32 bushels, a ton. Coals were likewise sold by the chal- dron, supposed to weigh a ton, or 2000lbs. though in reality it probably weighs upwards of 3000Ibs. This principle, though not sufficiently ac- curate in some cases, was extremely obvious, and might have been improved so as to an- swer all the purposes of commerce ;, but un- fortunately instead of rendering it more sim- ple it has been made infinitely more compli- cated.by the different weights since intro- duced, Modern Europeon. Weights.—1. English weights. By the twenty-seventh chapter of Magna Charta, the weights all over England are to be the same; but for different commo- dities, there are two different sorts, viz. troy weight and avoirdupois weight. The origin from which they are both raised, is a grain of wheat gathered in the middle of the ear. in troy-weight, twenty-four of these grains make one pennyweightsterling; twenty penny- weights make one ounce; and twelve ounces one pound. By this weight we weigh gold, silver, jewels, and liquors. ‘The apothecaries also use the troy pound, ounce, and grain; but they differ from the rest in the intermediate divisions, They divide the ounce into eight drachms ; the drachm into three scruples, and the scru- ple into twenty grains. In avoirdupois weight the pound contains sixteen ounces, but, the ounce is less by near one-twelfth than the troy ounce; this latter containing 490 grains, and the former only 448. The ounce contains 16 drachms; 80 ounces avoirdupois are only equal to 73 ounces troy; and 17 pounds troy equal to 14 pounds avoirdupois. By avoirdupois weight are weighed meat, grocery wares, base metals, wool, tallow, hemp, drugs, bread, &c. Table of Troy Weight, as used by the Gold- smiths. Grains. 24 | Pennyweight. 480 | 20 | Ounce. 5760 | 240 | 12 | Pound. Apothecaries. Grains. 20 | Scruple. 60 | 3 {| Drachm. 480 | 24{ 8 | Ounce. 5760 | 288 | 96 | 12 | Pound. WEI Table of Avoirdupois Weight. Dram. i6 | Ounce. 256 | i6 | Pound. 7168 | 448 | 28 | Quarter. — 28672 | 1792} 112| 4 | Cwt. 573440'| 35840 | 2240 80 | 20 | 'Ton. Comparison between Troy and Avoirdupois Weight. 175 troy pounds are equal to 144 ayoirdupois pounds. 175 troy ounces are equal to 192 avoirdupois ounces. 1 troy pound contains 5760 grains. _ 1 avoirdupois pound contains 7000 grains. 1 avoirdupois ounce contains 4374 grains. L avoirdupois dram contains 27.84375 ers. 1 troy pound contains 13 .0z. 2.651428576 drams avoirdupvis. 1 avoirdupois Ib. contains 11b. 20z. 11 dwts. 16 gr. troy. Therefore the avoirdupois lb. is to the Ib. troy as 175 to 144, and the avoirdupois oz. is to the troy oz. as 437% is to 480. Goldsmiths and jewellers, &c. have a par- ticular class of weights for gold and precious stones, viz. carat and grain; and for silver, the pennyweight and grain. In the mint they have also a peculiar subdivision of the troy grain: thus, dividing the grain into 20 mites, the mite into 24 droits, the droit into 20 periots, the periot into 24 blanks. The dealers in wool have likewise a parti- cular set of weights, viz. the sack, weigh, tod, stone, and clove; the proportions of which are as below; viz. the sack containing 2 weighs, the weigh....... ie saunth 61 tods, thE TAGS. Geccees siees oo. 2 Stones, thé; stones! s.iival..i ses 2 cloves, the clove................ 7 pounds. Also 12 sacks make a last, or 4868 pounds. Farther, 56 Ib. of old hay, or 60 1b. new hay, make a truss. 40 lb. of straw, make a truss. 36 trusses make a load of hay or straw. 14lb. make a stone. Slb. of glass a stone. Other nations have also certain weights pe- euliar to themselves: thus, Spain has its ar- robas, containing 25 Spanish pounds, or one- fourth of the common quintal: its quintal macho, containing 150 pounds, or one-half of the common quintal, or 6 arrobas ; its adarme, containing one-sixteenth of its ounce. And for gold, it has its castillan, or one-hundredth ofa pound; and its tomin, containing 12 grains, or one-eighth of a castillan. ‘Che same are in use in the Spanish West Indies. WEI Portugal has its arroba, containing 39 bon arratals, or pounds. Savary also me its faratelle, containing 2 Lisbon pounds its rottolis, containing about 12 pounds, for gold, its chego, containing four ¢, The same are used in the Portuguese Indies. . Italy, and particularly Venice, have migliaro, containing four mirres; the containing 30 Venice pounds; the gz containing a sixth part of an ounce. € has five kinds of weights, viz. large we} whereby all merchandises are weighed ; custom-house ; cash weights for piastres) other specie ; the cantara, or quintal, f coarsest commodities; the large balance raw silks, and the small balance for the commoidities. Sicily has its rottolo, 32 \ half pounds of Messina. ; Germany, Flanders, Holland, the towns, Sweden, Denmark, Poland, &e.; their schippondt, which, at Antwerp) Hamburgh, is 300 pounds; at Lubeck,} and at Koningsberg, 400 pounds. In Sy¥ the shippondt for copper is 320 pounds :; the shippondt for provisions 400 pounds Riga and Revel, the schippondtis 400 por at Dantzic, 340 pounds; in Norway pounds; at Amsterdam, 300; containi lyspondis, each weighing 15 pounds. in Moscow, they weigh their large} modities hy the bercheroct, or berkewits) taining 400 of their pounds. hey havi the poet or poede, containing 40 pounc one-tenth of the bercheroct. In order to show the proportion of tl veial weights used throughout Europ} shall add a reduction of them to one sta viz. the London pound. 1. Proportion of the weights of the| cipal places of Europe. The J001b. of England, Scotland, an| land, are equal to r i. Ib. oz. 91 8 of Amsterdam, Paris, &c. 96 8 of Antwerp or Brabant. 88 0 of Rouen, the viscounty weig 106 0 of Lyons, the city weight. 90 9 of Rochelle. 107 11 of Toulouse and Upper Langt 113 0 of Marseilles or Provence, 81 7 of Geneva. 93 5 of Hamburgh. 89 7 of Frankfort, &c. 96 1 of Leipsic, &e. 137 4 of Genoa. 132 11 of Leghorn. 153 11 of Milan. 152 0 of Venice. 154 10 of Naples. 97 0 of Seville, Cadiz, &e. 104 13 of Portugal. 96 5of Liege. 112 2of Russia. 107 4, of Sweden. 89 of Denmark, leak as Me) WEI 5 of Antwerp. 9) of Bayonne and Paris. ‘54 of Bergen in Norway. of Bologna. of Bremen. of Breslaw. - of Cadiz. of Copenhagen. of Dublin and Edinburgh. of London. of Madrid. of Naples. of Riga. of Rome. of Stettin. of Venice. wWCOWODUOFKRUONNUU Ww = ENGLISH WEIGHTS. Troy Weight. ‘oz, #drdms. scruples. grains. grammes. 112 = 96 = 283 = 5760 = 372.96 fis 8= 24=— 480= 31.08 7 1= 3= 60= 3.885 De bees hy 20 S279 1.395 ee 1= 0.06475 ii + Avoirdupois Weight. oz, drams, grains. grammes, = 16 = 256 — 7000 = 453.25 Pe 16 = 437.5. . =. 28,32 Bie ee oO ae Sl OL GERMAN. ‘Tbs. or grs. English troy = 74]bs. or grs. | German apothecaries’ weight. ‘oz. Nuremberg, medic. weight=7 dr. 2sc. mark Cologne =7 oz. 2 dwt. 4gr. English | 9 grains English. t troy. Dutcu. lish Troy. | wedish Weights, used by Bergman and Scheele. ie Swedish pound, which is divided like \inglish apothecary, or troy, pound, weighs | grains troy. te kanne of pure water, according to man, weighs 42250 Swedish grains, and pies 100 Swedish cubic inches. Hence ‘anne of pure water weighs 48088.719444 lish troy grains, or is equal to 189.9413 lish cubic inches; and the Swedish lon- tinal inch is equal to 1.238435 English jitudinal inches. wrtion of the Weights of several Places in rope compared with those of Amsterdam. ie 100 lb. of Amsterdam, are equal to parison of English and Foreign Weights. | Ib. Dutch = 1 lb. 3 oz. 16 dwt. 7 gr. Eng- : Ibs. Dutch = 1038 lbs. English troy. ) WEI From these data, the following rules are educed: 1. To reduce Swedish longitudinal inches to English, multiply by 1.2384, or divide by 0.80747. 2. To reduce Swedish to English cubic inches, multiply by 1.9, or divide by 0.5265. 3. To reduce the Swedish pound, ounce, dram, scruple, or grain, to the corresponding English troy denomination, multiply by 1.1382, or divide by °8786. 4. To reduce the Swedish kannes to Eng- lish wine pints, multiply by .1520207, or di- vide by 6.57804. 5. The lod, a weight sometimes used by Bergman, is the 32d part of the Swedish pound: therefore, to reduce it to the English troy pound, multiply by .03557, or divide by 28.1156. Correspondence of English Weights with those used in France before the Revolution. The Paris pound, poids de marc of Charle- magne, contains 9216 Paris grains: itis di- vided into 16 ounces, each ounce into 8 gros, and each gros into 72 grains. It is equal to 7561 English troy grains. The English troy pound of 12 ounces con- tains 5760 English troy grains, and is equal to 7021 Paris grains. The English avoirdupois pound, of 16 ounces, contains 7000 English troy grains, and is equal to 8538 Paris grains. Toreduce Paris grains to English troy grains, divide by............ 1.2189" ~ To reduce English troy grains to Paris grains, multiply by........ To reduce Paris ounces to Eng-> lish troy, divide DY..........sce0-0 aA To reduce English troy ounces 1.015704 to Paris, multiply by ............. Or the conversion may be made by means of the following tables : I. To reduce French to English Troy Weight. The Paris pound = 7561 The ounce:.:..,... = 472.5625(€ English Pheri gros... vase = 59.0703 ¢ troy grains. "Phe grain ....s.c. = 8204 Il. To reduce English Troy to Paris Weight. The English troy poundof2 _ "021 LO Ounees . Liat nGaesas , re y A The troy ounce........-.....00«0« = 585.0893 3 The dram of 60 grains........ = -73.1354 U& The pennyweight, or de-. _ a nier, of 24 graims.........§. 29.2541 3 The scruple of 20 grains..... = 24.8784 | “4 THE QTain .....cccccseesoesens bidemkeee: Sp Lee Til. Lo reduce English Avoirdupois to Paris Weight. The avoirdupois pound Pa of 16 ounces, or 7000 + — 8538. of trOY QTAINSsv..e.ceeceeeeees ‘ 5 TRE OUNCE; ..,..d6:vseerssentd caso = 533.6250 7 ¢ 3C WEI TABLE Shewing the Comparison between French and English Grains. ( Poid de Marc.) Fr. grs. = Eng. grs. ©Cono Ou GO = Ore 09 5 Soooe fp) So 70 © @ oo 100 200 300 400 500 600. 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Milligramme ... Centigramme... Decigramme.... ‘Gramme.... Decagramme... Eng. gers. = French grs. New French Weights. HWW aE Hecatogramme = Kilogramme.... = Myriogramme.. = 154440.2344 = 22 12 We have made a slight mention in the pre- ecding part of this article of the French sys- English grains. 0164 1544 1.5444 15.4440 154.4402 = 0 0 5.65 1544.4023 = 15444.0234 = 0.8203 1 1.2189 1.6407 2 2.4578 2.461 1 3 3.6568 3.2815 4 4.8757 4.1019 9) 6.0947 4.9223 6 7.3136 5.7427 7 8.5325 6.5631 8 9.7515 7.3835 9 10.9704 | 8.205 10 12.189 | 16.407 20 24.378 24.611 30 36.568 32.815 40 48.757 41.019 50 60.947 49.223 60 73.156 07.427 70 85.525 65.631 80 97.515 73.835 90 109.704 82.03 100 121.89 164.07 200 243.78 246.11 300 365.68 328.15 400 487.57 410.19 300 609.47 492.23 600 731.36 574.27 700 853.25 656.31 800 975.15 738.35 900 1097.04 820.3 1000 1218.9 1640.7 2000 2437.8 2461.1 3000 3656.8 3281.5 4000 4875.7 4101.9 5000 6094.7 4922.3 6000 7313.6 5742.7 7000 8532.5 6563.1 8000 9751.5 7383.5 9000 10970.4 $203.0 10000 12189.0 Avoirdupois Ib. oz. dr 0 3 8.5 235 WET by tem of weights, as deduced from the me system of measures; and although, — have observed above, this system is repo to have been abandoned by the Freneh vernment; yet the ingenuity employed - tablishing it, and the simplicity of it, y once accurately determined, render it ne sary for us to give the reader an idea, original construction. Pe On comparing the magnitude of the te trial arc, extending from Barcelona to 1 kirk, as it was given by the operations lambert and Mechain, with that of the measured in Peru about the year 1740, it concluded that the quadrant of the meric situated towards the north pole, was 513 French toises ; and therefore that the m or ten millioneth part of this, was 0518 toises, answering to about 39°37023 En inches. if ‘The unit of weight, which is called grai is the absolute weight of the cube of the. dredth part of the metre of distilled w. taken at its maximum of density, answe to the temperature of 42° of Fahrenh thermometer. at Lefebure Gineau was charged with ¢ thing that related to this observation, @ ther to this union of operations all ex re delicate. The precision which he prop to attain, excluded at once a method W at first glance, appeared very simple, which consisted in taking a cubic ¥ whose side had a given ratio with the | dredth part of the metre, weighing it alone, and then weighing it again after it filled with distilled water. The differene tween the weights would give the weig) the volume of water employed; but it be conceived, without entering into detail, the result would be affected with va errors, which it would be impossible e€ to avoid or to appreciate. . Another method was therefore ado susceptible of a much greater exacines consisted in weighing specifically in they a hollow cylinder of copper, whose vo had before been compared with that o cube, whose side is the hundredth part metre. The operation made known the w of the volume of distilled water, equal to of the cylinder, and thence was inferres weight of the cube of the same water ¥ represented the required unit. We tre will be gratifying to the reader if we here a detailed account of the process follow order to obtain that result. ‘ The machine déstined to measure tl linder had been constructed, with as 1 care as judgment and ingenuity, by Fe one of the most distinguished artists in] Without tarrying to give its descript will suffice to say, that it would render ¢ | ciable a difference equal to a two thous or even a four thousandth of a line: this ee Sil luation was made by means of a lever, iy whose arms is ten times longer thar other; the whole is so disposed that whil WEI differences, which it is the question to mine, occasion in the smaller arm moye- s equal to those differences, the motions » larger arm which are decuple, and which e become perceptible, by means of a no- applied to the extremity of that arm, the two thousandth of a line measured he play of the shorter arm. itwithstanding all the attention of the artist in the fabrication of the cylinder, orm of that solid was necessarily found to fected by a multitude of little inequalities, h might have sensibly influenced the re- had they been neglected; for here an committed on either of the two dimen- of the cylinder; that is, the height and jiameter of the base, is, if we may so <, a cubic error, and not merely a linear , as in the determination of a simple dis- », The surface of the body must there- be followed or traced in all its parts from yoint to another, and a sufficient number tights and of diameters must be mea- fat different places, in the bases and of urve surface, to reduce the capacity of cylinder, which was the object of the ition, to that of a perfeetly regular cy- rof equal volume. This operation ter- ‘ted, the cylinder was weighed in the air, oying a process as simple as ingenious, h made the inconvenience disappear that casioned by the almost inevitable in- lity between the arms of even the best uted balances. The body to be weighed laced in one of the basins or scales, the » basin being charged with any weights ever, till the beam of the balance became ontal. Afterwards the body was taken the first basin, and known weights sub- ed for it until the beam had again as- d the horizontal position. It is evident the weight of this body is exactly repre- sd by the sum of the weights which have substituted for it, though it may happen ‘this sum differs from that of the weights ae other side, as a necessary consequence e faulty construction of the balance. ssides, the weight of the cylinder in the iscertained by means of this procedure, the advantage of furnishing precisely the result as if it had been determined in a um. For first, the weights are substi- 1 for the cylinder, being of the same mat- s that body, their volume would be equal iat of the solid part of the cylinder; and or this relation the loss in the air will be ion both sides. But farther, there was ein one of the bases of the cylinder a + orifice, which established a communi- bn between the interior-air, and that of atmosphere. It hence results, that at the aent of the experiment the interior air was he same density as that which had been ced by the cylinder; the surrounding would therefore be in equilibrio with it, thus the loss of weight was nothing in that t. WET The cylinder was then weighed in the water, and as the weight which kept it in equilibrio: was then solely sustained by the air, it was requisite to estimate the small loss it expe- rienced in that fluid, as no longer common to the cylinder immersed in the water. Regard was also had to the little augmentation of weight which was eccasioned with respect to the cylinder, by the air comprised within it. Finally, the result was reduced to what would have obtained in water taken at its maximum of density, andit was found that the new unit of weight, or the gramme, answered to 18°82715 grains of the old French weight, or 15°444 grains of English troy weight. Haiiy’s Phil. Universal Standard of Weight and Measure.— Tt has been shown in the preceding article that the French mathematicians have em- ployed for their standard of measure, the ten millioneth part of a quadrant of the meridian, contained between the equator and north pole; and thence have deduced their unit or standard of weight. This, however, is only one of many different standards that have been at different times proposed for this purpose ; nor is it, we conceive, the best that might have been adopted, being subject to this im~ portant objection; that it depends upon an accurate measure of a quarter of a meridian of the earth, which is obviously a very difli- cult, not to say an impossible, task to per- form, particularly as there is every reason to suppose that meridians differ essentially amongst themselves. What therefore has been said in approbation of this system in the preceding article, must be understood as only applying to the division according to the de- cimal scale, and not to the standard itself. Amongst the various projects for a universal measure (on which that of weight obviously depends), there are indeed but few that deserve — any particular notice, except those that are founded on experiments with pendulous bo- dies in some assumed position on the surface of the earth, or on the mensuration of some of the lines or circles measured on its surface after the manner adopted by the French, and which has been already explained. We have therefore, in this place, only to attend to those depending on the former principle, and of these we shall only mention one or two of the most plausible. Huygens, in his “ Horologium Oscillato- rium,” was, as far as we can ascertain, the first who proposed the distance from the point of suspension to the centre of oscillation, of a pendulum vibrating seconds, as the length of a universal standard yard, the third part of which was to be denominated @ horary foot, to which all measures were to be referred, This standard is liable to two objections ; for, Ist, it is not invariable, the length ofa second’s pendulum, like the space descended in the first second from quiescence, being different at different distances from the equator. And - 2dly, the difficulty of exactly measuring the distance from the point of suspension to the 3C2 a mee a WEI centre of oscillation is such, that it is probable no two measures would come to the same result. Several expedients were proposed after this of M. Huygens, but none that deserved any attention, until the year 1779, when a Mr. Hatton, in consequence of a premium offered by a society of Arts and Manufactures, ** for obtaining invariable standards for weights and measures communicable at all times and to all nations,” proposed a plan which con- sisted in the application of a moveable point of suspension to one and the same pendulum, in order to produce the full and absolute ef- fect of two pendulums, the difference of whose lengths was the intended measure. Here also the ratio of their lengths was easily determined, from observing the number of vibrations performed in a given time at each point of suspension. Whence there being two equations and two unknown quan- tities, the actual length of the pendulums themselves, might be easily deduced by simple algebraic rules. The late ingenious Mr. Whitehurst much improved upon Mr. Hat- ton’s original notion, in his essay published in 1787, under the title of “ An Attempt towards obtaining invariable Measures of Length, Ca- pacity, and Weight, from the Mensuration of Time,” &c. Mr. Whitehurst’s proposal is to obtain a measure of the greatest length that conve- nience will permit, from two pendulums whose vibrations are in the ratio of 2 to 1, and whose lengths coincide with the English standard in whole numbers. His numbers were chosen with considerable judgment and skill. Ona supposition that the length of the second’s pendulum in the latitude of London is 39:2 inches, the length of one vibrating 42 times in a minute must be 80 inches; and that of another vibrating 84 times in a minute must be 20 inches; their difference 60 inches, or 5 fect, in his standard measure. The differ- ence in the lengths of the two pendulums, however, resulting from his experiments was 59:892 inches, instead of 60, the discrepancy being occasioned by an error in the original assuinption of 39°2 inches instead of 39°128, or 393 inches, as it is very nearly. Still Mr. Whitehurst has accomplished a principal part of his grand design, by showing how an in- variable standard may always be found for the same latitude. But this is by ne means all that is wanted. Perhaps the least objectionable way would be to take for the length of the metre the length ofasimple pendulum vibrating seconds at the equator, at a certain height above the surface of the sea, when the thermometer is at a fixed medium temperature; the length of the metre would then be about 39:027 English inches, instead of 39'37023, the metre of the new French system. The magnitude of the are, the stere, the gramme, &e. (or any other terms we thought proper to introduce lor similar purposes), might have the same relations to the metre as in the French system. WHE Thus should we possess a standard from the gravitating force of the globe inhabit, and which might be safely consid as invariable, so long as the constitution of earth and its time of rotation remains the ge The material standard itseif might chosen of some shape that should possess double advantage of being little affecte changes of temperature, and being a 4 dulum whose distance between the poin suspension and centre of oscillation, sh be exactly equal to a fixed dimension of pendulum that might readily be meas with exactness. Sucha body we have inari angled cone, or one, the diameter of w base is equal to its altitude; for when cone is suspended by its vertex as a cent) motion, the centre of oscillation is in centre of its base; and when it is susper by its base, the centre of oscillation coine with the vertex of the solid: the lengt the isochronous simple pendulum bein, both cases equal to the altitude of the or to the diameter of its base. It wil easy for any person who takes an intere; these speculations to pursue the present Haiiy’s Philos. vol. i. WERST, a Russian measure of ler equal to 3500 English feet. WEST, one of the four cardinal being that opposite to the east. WEY. See WEIGH. WHALE, a constellation. See Ceru: WHEEL, in Mechanics, a simple mae consisting of a circular piece of wood, m or other matter, which revolves on an ; This is otherwise called wheel and axl axis in peritrochio, as a mechanical p being one of the most frequent and use any. In this capacity of it, the wheel kind of perpetual lever, and the axle anc lesser one; or the radius of the wheel that of its axle may be considered as longer and shorter arms of a lever, the ce¢ of the wheel being the fulcrum or poi suspension. Whence it is, that the pow this machine is estimated by this rule, a: radius of the axle is to the radius of the wi or as the circumference to the circumfere so is any given power to the weight it sustain. Wheels, as wellas their axles, are frequt dented, or cut into teeth, and are then ol upon innumerable occasions; as in jé clocks, mill-work, &c.; by which means’ are capable of moving and acting on another, and of being combined togeth any extent; the teeth either of the a circumference working in those of ( wheels or axles; and thus, by multiplyin; power to any extent, an amazing effe produced. To compute the power of a combinati wheels; the teeth of the axle of every acting on those in the circumference 0 next following. Multiply continually ther the radii of all the axles, as also the’ of all the wheels ; then it will be, as the fol Pi WHI ict is to the latter product, so is a given r applied to the circumference, to the tit can sustain. ‘Thus, for example, in inbination of five wheels and axles, to he weight a man can sustain, or raise, » force is equal to 150 pounds, the radii » wheels being 30 inches, and those of xles 3 inches. we 3X 3X 3x 3X 3=— 248, | 30 x 30 x 30 x 30 x 30 = 24300000, fore as 243 : 24300000 :: 150: 15000000Ib. eight he can sustain, which is more than tons weight. So prodigious is the in- e of power in a combination of wheels! tit is to be observed, that in this, as well very other mechanical engine, whatever ‘ned in power, is lost in time; that is, the ht will move as much slower than the r, as the force is increased or multiplied, h, in the example above, is 100000 times mee, haying given any power, and the ht to be raised, with the proportion be- a the wheels and axles necessary to that :; to find the number of the wheels and Or, having the number of the wheels xles given, to find the ratio of the radii wheels and axles. Here, putting = the power acting on the last wheel, = the weight to be raised, = the radius of the axles, = the radius of the wheels, =the number of the wheels and axles; ‘by the general proportion, as 1: R® :: ; therefore pR* = wr” is a general theo- whence may be found any one of these sitters or quantities, when the other four riven. Thus, to find » the number of Is: we have first _ — ” then n = ee Oe eet p log. R. — log. r to find ©, the ratio of the wheel to the pe n it is i as a/ toa r £ er-WHEEL of a mill, is that which re- 3 the impulse of the stream by means of lat boards. dershot Water-WHEELS, are those in : the motion of the stream exceeds that 2 wheel, so that the fluid impels them rd by its action against the float boards, {are immersed in it to a certain depth. VatTer-Mill. shot Water- WHEEL, is that which con- of a frame of open buckets, so disposed _ the rim of the wheel as to receive the from its fall, in such a manner that one .s loaded with water while the other is 7, in consequence of which the loaded lescends, and empties its buckets, while on the other side revolve round and are ‘in their turn; and by means of which a ‘m circular motion is produced. Fyreus’s WHEEL. See ORFFYREUS. HIRLING Table, a machine intended WIN to represent the several phenomena in phi- losophy and nature, as the principal laws of gravitation, and of the planetary motions; for a particular description of which see Fergu- son’s Lectures on Mechanics, lect. 2; and Desagulier’s Ex. Phil. vol. i. lect. 5. WHISTON (WILLIAM), an ingenious Eng- lish mathematician and divine, was born in 1667, and died in 1752, upwards of 84 years of age. He was author of numerous works on philosophy and religion; of the former, his Theory of the Earth, and his Astronomical Lectures, are the only ones which it is neces- sary to enumerate in this place; but a com- plete list of all his works may be seen in Dr. Hutton’s Math. Dict. WHITEHURST (Joun), an ingenious English mechanic and philosopher, was born in the county of Chester in 1713, and died in 1788, in the 75th year of his age. He published, in 1778, An Enquiry into the Original State and Formation of the Earth ; second edition in 1786; and a third in 1792. In 1787, he published “ An Attempt toward obtaining Invariable Measures of Length, Ca- pacity, and Weight, from the Mensuration of Time.” His plan is to obtain a measure of the greatest length that convenience will permit, from two pendulums whose vibrations are in the ratio of 2 to 1, and whose lengths coincide nearly with the English standard in whole numbers. The numbers which he has chosen show much ingenuity. On a supposition that the length of a second’s pendulum, in the la- titude of London, is 392 inches, the length of one vibrating 42 times in a minute must be 80 inches; and of another vibrating 84.times in a minute must be 20 inches; and their dif- ference 60 inches, or five feet, is his standard measure. By the experiments, however, the difference between the lengths of the two pendulum rods was found to be only 59892 inches, instead of 60, owing to the error in the assumed length of the second’s pendulum, 392 inches being greater than the truth, which ought to be 394 very nearly. By this experi- ment, Mr. Whitehurst obtained a fact, as ac- curately as may be in a thing of this nature, viz. the difference between the lengths of two pendulum rods whose vibrations are known ; a datum, whence may be obtained, by calculation, the true lengths of pendulums, the spaces through which heavy bodies fall in a given time, and many other particulars re- lating to the doctrine of gravitation, the figure of the earth, &c. WILKINS (Dr. Joun), an ingenious and learned English divine, was born in 1614. He was one amongst the early members and promoters of the Royal Society, and author of several works on politics, religion, and phi- losophy ; the latter of which were, the “‘ Ma- thematical and Philosophical Works” of the Right Rev. John Wilkins, 8vo. 1708; and “* An Essay towards a real Character and a Philosophical Language,” folio, 1688. Dr, Wilkins died in 1672, in his 59th year. WINCH, the bent handle employed for WIN turning round wheels, and producing circular motion. WIND, a sensible current in the atmos- phere. The motions of the atmosphere are subject in some degree to the same laws as those of the denser fluids; if we remove a portion of water in a large reservoir, we see the surrounding water flow in to restore the equilibrium ; and if we impel in any direction, a certain portion, an equal quantity moves in a contrary direction from the same cause; or if a portion being rarefied by heat, or con- densed by cold, ascends in the one instance and descends in the other, a counter-cyrrent is the visible and natural result; and similar effects are found to follow the same causes in the atmospheric fluid. Thus, no wind can blow, without a counter or opposite current, nor can any wind arise, without a previous derangement of the general equilibrium ; the general causes of which may be stated as follows: 1. The ascent of the air over certain tracts, heated by the sun. 2. Evaporation causing an actual increase in the volume of the at- mosphere. 3. Rain, snow, &c. causing an actual decrease in its volume by the destruc- tion of the vapour. Currents thus produced may be permanent and general, extending over a large portion of the globe; periodical, as in the Indian ocean; or variable, and as it were occasional, or at least uncertain, as the winds in temperate climates. General or permanent winds blow always nearly in the same direction. In the Atlantic and Pacific oceans, under the equator, the wind is almost always easterly; it blows, in- deed, in this direction, on both sides of the equator to the latitude of 28°. More to the northward of the equator, the wind generally biows between the north and east; and the farther north we proceed, we find the wind to blow to a more northern direction; more to the southward of the equator it blows between the south and east; and the farther to the south, the more it comes in that direction. Between the parallels of 28° and 40° south iatitude, in that tract which extends from 30° west to 100° east longitude from London, the wind is variable, but it most frequently blows from between the N.W. and S.W. so that the outward-bound East-India ships generall y run down their easting on the parallel of 36° south. Navigators have given the appellation of trade-winds to these general winds. Periodical Winds.—Those winds which blow ina certain direction for a time, and at certain stated seasons change, and blow for an equal space of time from the opposite point of the compass, are called monsoons. During the months of April, May, June, J uly, August, and September, the wind blows from south- ward over the whole length of the Indian ocean, vz. between the parallels of 28° N, and 28° S. latitude, and between the eastern coast of Africa and the meridian which passes through the western part of J apan; but in the WIN other months, October, November, Dec January, February, and March, the all the northern parts of the Indian ¢ shift round, and blow directly contrary ° course they held in the former six mo} For some days before and after the cha there are calins, variable winds, and tren dous storms, with thunder, &c. % Causes of the Wind.—Philosophers diffe their opinions respecting the cause of t periodical winds; but a more probable th of the general trade-winds is, that they occasioned by the heat of the sun in regions about the equator, where the ¢ heated to a greater degree, and conseque rarefied more than in the more northern } of the globe. From this expansion of th in these tropical regions, the denser aj higher latitudes, rushes violently toward, equator from both sides of the globe. this conflux of the denser air, withou other circumstances intervening, a d northerly wind would be produced in northern tropic, and a southern oné in other tropic; but as the earth’s diurna tion varies the direct influence ofthe sun the surface of the earth, and as by thatm this influence is communicated from ea west, an easterly wind would be produc this influence alone prevailed. On accon the co-operation of these two causes a same time, the trade-winds blow natr from the N.E. on the north, and from S. E. on the south of the line, throughou whole year ; but as the sun approaches n the tropic of Cancer in our summer se the point towards which these winds ar! rected will not be invariably the same they will incline more towards the nor| that season, and more towards the soul our winter, | ; The land and sea breezes in the t climates may be considered as partial it ruptions of the general trade-winds; an| cause of these it is not very difficult to ex) From water being a better conducter ol than earth, the water is always of a more temperature. During the day, therefor land becomes considerably heated, th rarefied, and consequently in the aftern breeze sets in from the sea, which i heated at that time than the land. O other hand, during the night the earth its surplus heat, while the sea continues even in its temperature. Towards moi therefore, a breeze regularly proceeds the land towards the ocean, where the | warmer, and consequently more rarefied} on shore. | The cause of the monsoons is not sq understood as that of the general trade-¥i but what has been just remarked sugges least, a probable theory on the subject. } well known, that at the equator the chi of heat and cold are occasioned by the di motion of the earth, and that the diffet between the heat of the day and the ni! almost all that is perceived in those tri i WIN ns; whereas in the polar regions the it vicissitudes of heat and cold are occa- ed by the annual motion of the globe, ch produces the sensible changes of winter ‘summer; consequently, if the heat of the ) was the only cause of the variation of winds, the changes, if any, that would be juced by those means in equatorial re- 1s, ought to be diurnal only, but the inges about the pole should be experienced yonce in six months. As the effects aris- from the heat of the sun upon the air fst be greater at the equator than at the ) 28, the changes of the wind arising from ] expansion of the air by the sun’s rays st be more steady in equatorial than in arregions. The incontrovertible evidence havigators proves this truth, that winds are te variable towards the poles, and more ‘stant towards the equator. But in sum- ir, the continual heat, even in high lati- es, comes to be sensibly felt, and produces nges on the wind, which are distinctly ‘ceptible. In our own cold region, the pets of the sun on the wind are felt during ‘ summer months; for while the weather that season of the year is fine, the wind ferally becomes stronger as the time of the 'r advances, and dies away towards the Ming, and assumes that pleasing serenity i delightful to our feelings. Such are the thal changes of the wind in northern cli- tes. The annual revolution of the sun duces still more sensible effects. The bvalence of the western winds during sum- ir we may attribute to this cause, which is ‘Imore perceptible in France and Spain ; [sause the continent of land to the eastward, ng heated more than the waters of the lantic ocean, the air is drawn, during that ‘ison, towards the east, and consequently yduces a western wind. But these effects are much more perceptible i countries near the tropics than with us. r when the sun approaches the tropic of meer, the soil of Persia, Bengal, China, and 3 adjoining countries, becomes so much wre heated than the sea to the southward of dse countries, that the current of the general \de-wind is interrupted, so as to blow, at at season, from the south to the north, con- to what it would do if no Jand was there. it as the high mountains of Africa, during ‘the year, are extremely cold, the low untries of India, to the eastward of it, be- ime hotter than Africa in summer, and the tis naturally drawn thence to the eastward. om the same cause it follows, that the trade- ‘nd in the Indian ocean, from April till Oc- iber, blows in a north-east direction, contrary that of the general trade-wind in open seas the same latitude ; but when the sun retires wards the tropic of Capricorn, these north- n parts become cooler, and_ the general jade-wind assumes its natural direction. Having given the most obvious causes of ,@ periodical monsoons in the Indian seas, it | is necessary to observe, that no monsoon takes place to the southward of the equator, except in that part of the ocean adjoining to New Holland. There the same causes concur to produce a monsoon as in the northern tropic, and similar appearances take place. From October till April the monsoon sets in from the N. W. to S.E. opposite to the general course of the trade-wind on the other side of the line; and here also the general trade-wind resumes its usual course during the other months, which constitute the winter season in these regions. It may not be improper to conclude this account of the tropical winds, by enumerating some of the principal inflec- tions of the monsoons. Between the months of April and October the winds blow constantly from W.S.W. in all that part of the Indian ocean which lies between Madagascar and Cape Comorin, and in the contrary direction from October till April, with some small variation in different places; but in the Bay of Bengal these winds are neither so strong nor so constant as in the Indian ocean. It must also be remarked, that the S.W. winds in those seas are more southerly on the African side, and more west- erly on the side of India ; but these variations are not so great as to be repugnant to the general theory. The cause of this variation is, as was before intimated, that the moun- tainous lands of Africa are colder than the flatter regions of Arabia and India; conse- quently the wind naturally blows from these cold mountains, in the summer season, to- wards the warmer lands of Asia, which oc- -easions those inflections of the wind to the eastward during the summer months. The peninsula of India lying so much farther to the south than the kingdoms of Arabia and Persia, adds greatly to this effect; because the wind naturally draws towards them, and produces that easterly variation of the mon- soon which takes place in this part of the ocean, while the sandy deserts of Arabia draw the winds more directly northward, near the African coast. A similar chain of reasoning will serve to explain any other inflections or variations that may occur in the perusal of books of travels, &c. Variable Winps.—In the temperate zones the direction of the winds is by no means so recular as between the tropics. Even in the same degree of latitude we find them often blowing in different directions at the same time; while their changes are frequently so sudden and so capricious, that to account for them has hitherto been found impossible. When winds are violent, and continue long, they generally extend over a large tract of country; and this is more certainly the case when they blow from the north or east than from any other points. By the multiplica- tion and comparison of meteorological tables, some regular connection between the changes of the atmosphere in different places may, in time, be observed, which may at last lead to a WIN satisfactory theory of the winds. It is from such tables chiefly, that the following facts have been collected: In Virginia, the prevailing winds are be- tween the south-west, west, north, and north- west; the most frequent is the south-west, which blows more constantly in June, July, and August, than at any other season. The north-west winds blow most constantly in November, January, and February. At Ips- wich, in New England, the prevailing winds are also between the south-west, west, north, and north-east ; the most frequent is the north- west. But at Cambridge, in the same pro- vince, the most frequent wind is the south-east. The predominant winds at New York are the north and west; and in Nova Scotia north- west winds blow for three-fourths of the year. The same wind blows most frequently at Montreal in Canada; but at Quebec the wind generally follows the direction of the river St. Lawrence, blowing either from the north- east or south-west. At Hudson’s Bay west- erly winds blow for three-fourths of the year ; the north-west wind occasions the greatest cold, but the north and north-east are the vehicles of snow. It appears from these facts, that westerly winds are most frequent over the whole east- ern coast of North America; that in the south- ern provinces, south-west winds predominate ; and that the north-west become gradually more frequent as we approach the frigid zone. In Egypt, during part of May, and during June, July, August, and September, the wind blows almost constantly from the north, vary- ing sometimes in June to the west, and in July to the west and the east; during part of September, and in October and November, the winds are variable, but blow more re- gularly from the east than any other quarter; in December, January, and February, they blow from the north, north-west, and west; towards the end of February they change to the south, in which quarter they continue till near the end of March; during the last days in March and in April, they blow from the south-east, south, and south-west, and at last from the east ; and in this direction they con- tinue during a part of May. In the Mediterranean the wind blows nearly three-fourths of the year from the north; about the equinoxes there is always an east- erly wind in that sea, which is generally more constant in spring than in autumn. ‘These observations do not apply to the gut of Gib- raltar, where there are seldom any winds except the east and the west. At Bastia, in the island of Corsica, the prevailing wind is the south-west. In Syria the north wind blows from the au- tumnal equinox to November; during De- cember, January, and February, the winds blow from the west and south-west; in March they blow from the south, in May from the east, and in June from the north. From this month to the autumnal equinox, the wind WIN changes gradually as the sun approaches equator ; first to the east, then to the so and Jastly to the west. At Bagdad, then frequent winds are the south-west and no west; at Pekin, the north and the south, Kamtschatka, on the north-east coast of A the prevailing winds blow from the west. In Italy, the prevailing winds differ ¢} siderably according to the situation of , places where the observations have been me at Rome and Padua, they are northerly, Milan easterly. All that we have been ak to learn concerning Spain and Portugal! that on the west coast of these countries, ¢ west is by far the most common wind, pa; cularly in summer; and that at Madrid k wind is north-east for the greatest part of ¢ summer, blowing almost constantly from | Pyrenean mountains. At Berne in Sw, erland, the prevailing winds are the north ¢| west; at St. Gothard, the north-east; at Li sanne, the north-west and south-west. Father Cotte has given us the result observations made at 86 different places} France; from which it appears, that along % whole south coast of that kingdom the wil blows most frequently from the north, nor, west, and north-east; on the west coast, fri the west, south-west, and north-west; and) the north coast from the south-west. Tha’ the interior parts of France, the south-w wind blows most frequently in 18 places; 4 west wind in 14; the north in 13; the so in 6; the north-east in 4; the south-east in the east and north-west each of them 0) On the west coast of the Netherlands, as as Rotterdam, the prevailing winds are pi bably the south-west, at least, this is the ¢ at Dunkirk and Rotterdam. It is rou also, that along the rest of this coast, from 4 Hague to Hamburgh, the prevailing win are the north-west, at least these winds ¢ most frequent at the Hague and at Franek The prevailing wind at Delft is the south-ea and at Breda, the north and the east. i In Germany, the east wind is most freque at Gottingen, Munich, Weissenfels, Duss dorf, Saganum, Erford, and at Buda in Hu gary; the south-east at Prague and Wur burg; the north-east at Ratisbon; and t west at Manheim and Berlin. ; From an average of ten years of the regist kept by order of the Royal Society, it appez that at London the winds blow in the follo ing order: Winds. Days. Winds, Days South-west...... 112 South-east...... 32 North-east ...... 58 East .........0.0. . 2S North-west...... 50 South............ 18 West...... Boy aeate 63° North wsc..)..c3et 16 It appears, from the same register, that t south-west wind blows at an average me frequently than any other wind during eve month of the year, and that it blows longs in July and August; that the north-east blo most constantly during January, March, Ap WIN Nay, and June, and most seldom during F'e- ary, July, September, and December; and fat the north-west wind blows oftener from avember to March, and more seldom during »ptember and October than any other months. the south-west winds are also most frequent » Bristol, and next to them are the north- st. The following table of the winds at Lan- ster has been drawn up from a register kept + seven years at that place: Winds Days. Winds. Days, South-west...... 92 South-east..... 35 North-east .....- 7. JNGOFEE ond dincs 30 ee RR 51 North-west.... 26 Dee estnes ed CT Se a RD he 4 e following table is an abstract of nine ars observations made at Dumfries by Mr. pland: Winds Days. Winds. Days. 821 North. ..... vanes, (GS est .,..... sessee 69 North-west.... 252 EEC Ct. ts cae 2 68 South-east..... 282 South-west...... 503 North-east... 142 Che following table is an abstract of seven ws observations made by Dr. Meek at mbuslang, near Glasgow: , Winds. Days. Winds. Days. jouth-west...... 174 North-east..... 104 Vorth-west....... 40 South-east.... 47 tappears from the register from which this le was extracted, that the north-east wind ws much more frequently in April, May, (June, and the south-west in July, August, | September, than at any other period. 2 south-west is by far the most frequent d all over Scotland, especially on the west st. At Saltcoats, in Ayrshire, for instance, Jows three-fourths of the year; and along ‘whole coast of Murray, on the north-east + of Scotland, it blows for two-thirds of the r. East winds are common over all Great fam during April and May; but their in- ce is felt most severely on the eastern st. he following table exhibits a view of the ber of days during which the westerly ‘easterly winds blow in a year at different Sof the island. Under the term westerly imcluded the north-west, west, south-west, ‘south; the term easterly is taken in the e latitude. oad Places, Wester. eastett y. 0 {London...... Lenard Beet, ac 132 @ jLancaster........... peut 216 149 MW {Liverpool .............6 190 | 175 ® iDumfries................ 227.5| 137.5 0 |Branxholm, fifty- four miles south 232 133 west of Berwick ‘7 |Cambuslang............ 214 | 151 8 |Hawkhill, near Edinburgh ....... Pome Ld Mean | 220.3| 144.7 WIN In Ireland the south-west and west are the grand trade-winds, blowing most in summer, autumn, and winter, and least in spring. The north-east blows most in spring, and nearly double to what it does in autumn and winter. The south-east and north-west are nearly equal, and are most frequent after the south- west and west. At Copenhagen the prevailing winds are the east and south-east; at Stockholm, the west and north. In Russia, from an average of a.register of 16 years, the winds blow from November to April in the following order: W.N.W. E. S.W. S. N.E. N. SAE. Days 45° 26 23 . 22: 20.19 14 /12 And during the other six months, W.N.W. E. S.W. S. N.E. N. SE. Days 27. 27.19. 24) :22 1553218 The west wind blows during the whole year 72 days; the north-west 58; the south-west and north 46 days each. During summer it is calm for 41 days, and during winter for 21. In Norway, the most frequent winds are the south, the south-west, and south-east. The wind at Bergen is seldom directiy west, but generally south-west or south-east; a north- west, and especially a north-east wind, are but little known there. From the whole of these facts, it appears that the most frequent winds on-the south coasts of Europe are the north, the north-east, and north-west; and on the western coast, the south-west: that in the interior parts which lie most contiguous to the Atlantic ocean, south-west winds are also most fre- quent; but that easterly winds prevail in Germany. Westerly winds are also most frequent on the north-east coast of Asia. It is probable that the winds are more con- stant in the south temperate zone, which is in a great measure covered with water, than in the north temperate zone, Where their di- rection must be frequently interrupted and altered by mountains and other causes. M. De la Caille, who was sent thither by the French king to make astronomical ob- servations, informs us, that at the Cape of Good Hope the principal winds are the south- east and north-west; that other winds seldom Jast longer than afew days; and that the east and north-east winds blow very seldom. 'The south-east wind blows in most months of the year, but chiefly from October to April; the north-west prevails during the other six months, bringing along with it rain, and tem- pests, and hurricanes. Between the Cape of Good Hope and New Holland, the winds are commonly westerly, and blow in the following order: north-west, south-west, west, north. In the Great South Sea, from latitude 30° to 40° south, the south-east trade-wind blows most frequently, especially when the sun ap- proaches the tropic of Capricorn; the wind next to it in frequency is the north-west, and next to that is the south-west. From south latitude 40° to 50°, the prevailing wind is the north-west, and next the south-west. From WIN 50° to 60°, the most frequent wind is also the north-west, and next to it is the west. Thus it appears that the trade-winds some- times extend farther into the south temperate zone than their usual limits, particularly during summer; that beyond their influence the winds are commonly westerly, and that they blow in the following order: north-west, south-west, west. Such is the present state of the history of the direction of the winds. In the torrid zone they blow constantly from the north-east on the north side of the equator, and from the south-east on the south side of it. In the north temperate zone they blow most fre- quently from the south-west; in the south temperate zone from. the north-west, chang- ing, however, frequently to all points of the compass; and in the north temperate zone blowing, particularly during spring, from the north-east. Foree and Velocity of the Winp.—As to the velocity of the wind, its variations are almost infinite ; from the gentlest breeze, to the hur- ricane which tears up trees and blows down houses. It has been remarked, that our most violent winds take place when neither the heat nor the cold is greatest; that violent winds generally extend over a great tract of country, and that they are accompanied by sudden and great falls in the mercury of the barometer. The reason appears to be, that violent winds succeed the precipitation in rain of a large quantity of vapour, which previously constituted a part of the bulk of the atmo- sphere; and this precipitation cannot take place when the general temperature ap- proaches to either extreme. The wind is sometimes very violent at a distance from the earth, while it is quite calm at its surface. On one occasion Lunardi went at the rate of 70 miles an hour in his balloon, though it was quite calm at Edinburgh when he ascended, and continued so during his whole voyage. The same thing happened to Garnerin and his companion in their aerostatic voyage to Colchester; they having been carried from London to Colchester, a distance of at least 60 miles, in three quarters of an hour, making the velocity of the wind, at.that time, 80 miles per hour, or 14 miles per minute. This again may be illustrated by the motions of dense fluids, which are always impeded in the parts contiguous to the sides and bottom of the vessels; and the same thing happens in tide-rivers, where the boatman, when he wishes to proceed with the tide, commits himself to the middle of the stream: but when he has to strive against it, he keeps close to the shore. It is, therefore, not the upper parts of the atmosphere which are accelerated, but the lower are retarded by friction against the surface of the earth. The following table, drawn up by Mr. Smeaton, will give the reader a pretty precise idea of the velocity of the wind in different circumstances : WIN Miles | Fect per |Perpendicular Force on one s a ie’ Second. Avoirdnpois Pounds an uare Foot Parts. 4 d .005 Hardly perceptible, .020 ; } Just perceptible. gis t Gently pleasant. Boe t Pleasant, brisk. 1.968 ) 3.075 § 4.429 ) 6.027 § 7,873 9.963 § 73.35)12.300 Storm or tempest. $8.02)17.715 Great storm. 117.36/31.490 Hurricane. F Hurricane that tear; up trees, and carrie buildings before it. é Very brisk. i : High wind. ‘ : Very high wind. i > - 146.7 ss200 Winp-Gage. See ANEMOMETER. Winp-Gun. See Arr-Gun. © ie WINDLASS, or Windlace, a particu machine used for raising heavy weights, | guns, stones, anchors, &c. : This is a very simple machine, consist) only of an axis or roller, supported horizont| at the two ends by two pieces of wood at pulley: the two pieces of wood meet at being placed diagonally so as to prop é other; and the’ axis or roller goes through ¥ two pieces, and turnsinthem. The pulle fastened at top, where the pieces join. Las} there are two staves or hand-spikes which} through the roller, to turn it by; and the re which comes over the pulley, is wound off } on.the same. WINDLAss, in a ship, is an instrument} small ships placed upon the deck, just all the foremast. It is made of a piece of tim six or eight feet square, in form of an 4, tree, whose length is placed horizontally wy two pieces of wood at the ends, and u which it is turned about by the help of he spikes put into holes made for that purp This instrument serves for weighing anch or hoisting of any weight in or out of} ship, and will purchase much more than capstan, and that without any danger tot who heave: for if in heaving the wind about, any of the handspikes should happe break, the windlass would stop of itsell means of a catch for that purpose. | WINDMILL, a well known engine w, receives its motion from the impulse ob wind; the machinery of which does not within the nature of this work to describe shall therefore merely state here a few pring relating to the position, &e. of the sails, W, admits of pure mathematical investiga though it must be acknowledged nofy standing, that the results thus obtainet not in all cases, indeed but in very few, a! with what have been determined from }f tical experiments. . . \ : a WIN ‘First, we may observe, that were the sails »t square upon their arms or yards, and per- endicular to the axletree, or to the wind, no jotion would ensue, because the direct wind ould keep them in an exact balance. But y setting them obliquely to the common sis, like sails of a smokejack, or inclined ke the rudder of a ship ; the wind, by strik- ig the surface of them obliquely, turns them yout. Now this angle which the sails are to iake with their common axis, or the degree * weathering, as the mill-wrights call it, so as iat the wind may have the greatest effect, is matter of nice inquiry, and has occupied ie thoughts of the mathematician and artist. In examining the compound motions of the idder of a ship, we find that the more it yproaches to the direction of the keel, or to ie course of the water, the more weakly this rikes it; but on the other hand, the greater the power of the lever to turn the vessel yout. The obliquity of the rudder has, at \e same time, both an advantage and a dis- \lvantage. It has been a point of inquiry, erefore, to find the position of the rudder hen the ratio of the advantage over the dis- lvantage is the greatest. And M. Renau, his theory of the working of ships, has und, that the best situation of the rudder is en it makes an angle of about 55 degrees th the keel. ‘The obliquity of the sails, with regard to leir axis, has precisely the same advantage id disadvantage, with the obliquity of the dder to the keel. And M. Parent, seeking ‘the new analysis the most advantageous ation of the sails on the axis, finds it the me angle of about 55 degrees. This ob- juity has been determined by many other athematicians, and found to be more ac- irately 54° 44’. See Maclaurin’s Fluxions, 733 ; Simpson’s Fluxions, prob. 17, p. 521. his angle, however, is only that which ves the wind the greatest force to put the il in motion, but not the angle which gives e force of the wind a maximum upon the il when in motion: for when the sail has a in degree of velocity, it yields to the ‘nd; and then that angle must be increased, ‘give the wind its full effect. Maclaurin, his Fluxions, p. 734, has shown how to termine this angle. It may be observed, that the increase of is angle should be different according to e different velocities from the axletree to € farther extremity of the sail. At the ‘ginning, or axis, it should be 54° 44’; and ence continually increasing, giving the vane ist, and so causing all the ribs of the vane fie in different planes. It is farther observed, that the ribs of the e, or sail, ought to decrease in length m the axis to the extremity, giving the ne a curvilinear form; so that no part of @ force of any one rib be spent upon the st, but all move on independent of each her. The twist above-mentioned, and the WIN diminution of the ribs, are exemplified in the wings of birds. As the ends of the sail nearest the axis cannot move with the same velocity which the tips or farthest ends have, although the wind acts equally strong upon them both. Ferguson’s Lecture on Mechanics, p. 54, by Brewster, suggests, that perhaps a better po- sition than that of stretching them along the arms directly from the centre of motion, might be to have them set perpendicularly across. the farther ends of the arms, and there ad- justed lengthwise to the proper angle: for in that case both ends of the sails would move with the same velocity ; and being farther from the centre of motion they would have so much the more power, and then there would be no occasion for having them so large as they are generally made; which would render them lighter, and consequently there would be so much the less friction on the thick neck of the axle, when it turns in the wall. Mr. Smeaton (Phil. Trans. 1759), from his expe- riments with windmill sails, deduces several practical maxims: as, 1. That when the wind falls upon a con- cave surface, it is an advantage to the power of the whole, though every part, taken sepa- rately, should not be disposed to the best advantage. By several trials he has found that the curved form and position of the sails will be best regulated by the numbers in the following table. Sixth Parts of the Angle with the Radius or Sails. Axis Angle with the Plane of Motion, OF RN ss aimless CLE lan dale oleh tk 19 Ms ar ab 2 ROS OR OL AS pare 18 middle FW A Aenea | Vp: 2 FERN DOES 16 Ly pe ER) TENOR Fd E15 caus tebtelnacte 124 Ee EAE onary. tS Ba eS eo Te 7 end 2. That a broader sail requires a greater angle; and that when the sail is broader at the extremity, than near the centre, this shape is more advantageous than that of a paral- lelogram. 3. When the sails, made like sectors of circles joining at the centre or axis, filled up about seven-eighths of the whole circular space, the effect was the greatest. 4. The velocity of windmill sails, whether unloaded or loaded, so as to produce a max- imum of effect, is nearly as the velocity of the wind ; their shape and position being the same. 5. The load at the maximum is nearly, but somewhat less than as the square of the ve- locity of the wind. 6. The effects of the same sails at a max- imum, are nearly, but somewhat less than, as the cubes of the velocity of the wind. 7. In sails of a similar figure and position, the number of turns in a given time, are re- ciprocally as the radius or length of the sail. 8. The effects of sails of similar figure and position are as the square of their length. WIN 9. The velocity of the extremity of Dutch mills as well as of the enlarged sails, in all their usual positions, is considerably greater than the velocity of the wind. M. Parent, in considering what figure the sails of a wind-mill should have, to receive the greatest impulse from the wind, finds it to be a sector of an ellipsis, whose centre is that of the axletree of the mill; and the less semi-axis the height of 32 feet; as for the greater, it follows necessarily from the rule that directs the sail to be inclined to the axis in the angle of 55 degrees. On this foundation he assumes four such sails, each being a quarter of an ellipse ; which he shows will receive all the wind and lose none, as the common ones do. ‘These four surfaces, multiplied by the lever, with which the wind acts on one of them, express the ‘whole power the wind has to move the ma- chine, or the whole power the machine has when in motion. A wind-mill with six elliptical sails, he shews, would still have more power than one with only four. It would only have the same surface with the four; since the four contain the whole space of the ellipses, as well as the six. But the force of the six would be greater than that of the four in the ratio of 245 to 231. Ifit were desired to have only two sails each, being a semi-ellipsis, the surface would be still the same: but the power would be diminished by near one-third of that with six sails ; because the greatness of the sectors would much shorten the lever, with which the wind acts. The same author has also considered which form, among the rectangular sails, will be most advantageous ; 2. e. that which shall have the product of the surface by the lever of the wind the greatest. The result of this inquiry is, that the width of the rectangular sail should be nearly double its length; whereas, usually, the length is made almost five times the width. The power of the mill, with four of these new rectangular sails, M. Parent shows, will be to the power of four elliptic sails, nearly as 18 to 28; which leaves a considerable advan- tage on the side of the elliptic ones; and yet the force of the new rectangular sails will still be considerably greater than that of the com- mon ones. M. Parent also considers what number of the new sails will be most advantageous ; and finds that the fewer the sails, the more surface there will be, but the power the less. Farther, the power of a wind-mill with six sails is de- noted by 14; that of another with four will be as 13; and another with two sails will be de- noted by 9 That, as to the common windmill, its power still diminishes as the breadth of the sails is smaller, in proportion to the length; and there- fore the usual proportion of 5 to 1 is exceed- ingly disadvantageous. WINTER, one of the four seasons or quar- ters of the year. This quarter commences on ’ WOL ° the day when the sun’s distance from the ze nith of the place is the greatest, or when th sun has the greatest declination on the con trary side of the equator, vzz. in northern la titudes, when he has the: greatest south declination; and in southern latitudes wher he has the greatest northern declination; an it ends when he next crosses the equinoctia Winter, however, is sometimes ape to mean one of the two principal divisions the year, viz. into summer and winter; andi which case it begins at the autumnal equino and continues to the spring equinox, incl ing an interval of 178 days, 18 hours, 29 nutes. WIT, or Witt (JouHN DE), the celebrate pensioner of Holland, and one of the greate politicians of his time, was the son of Jacob d Wit, burgomaster of Dort; he was born in 162€ and became well skilled in civil law, polities mathematics, and other sciences ; and wrote} treatise on the Elements of curved Line published by Francis Schooten. Havin; taken his degree of doctor of law, he travelle into foreign courts, where he became esteent ed for his genius and prudence. At his re turn to his native country, in 1650, he becam pensionary of Dort, then counsellor-pensio ary of Holland and West Friesland, inten dant and register of the fiefs, and keeper ¢ the great seal. He was thus at the head@ affairs in Holland ; but his opposition to th re-establishment of the office of stadtholdey which he thought a violation of the freedo and independence of the republic, cost hir his life, when the Prince of Orange’s part prevailed. He and his brother Cornelius wel assassinated by the populace at the Haguei A 1674, aged 47. | WITCH, in the Higher Geometry, a cury defined by the equation _ a (ax—x”) y— 24 . P A point in the curve may be thus obtained! the semicircle ADC on the diameter AC bein given, the point M in the curve is such thé drawing MB perpendicular to the diamete AC (B being in AC) to cut the circle in I itmay be AB: BD:: AC: BM. This cury has two opposite, equal, and infinite branche) to which a line drawn through A, perpend! cular to the diameter AC of the circle, is a) asymptote. WOLFF, or Wo trFius, (CHRISTIAN), a eminent philosopher, was born at Breslaij 1679. At Jena university he made a mo extraordinary progress, and in 1702 he r paired to Leipsic, where he opened his le tures by a famous dissertation, called Phik sephia practica universalis methodo math matica conscripta. He was invited by universities of Giessen and of Halle to acce the professional chair of mathematics. F went to Halle, 1707, and to his academic honours was soon after added the title_ counsellor to the king of Prussia. In h XEN atin oration on the morality of the Chinese, | 1721, he spoke with such applause of their ailosophy and virtues, that the university as offended, and not only his tenets were probated, but he was, by the representation ‘the body of divines, ordered to leave the yuntry in twenty-four hours. He retired to assel, and became professor of mathematics id philosophy at Marbourg, and counsellor to ‘elandgrave. ‘The favourable opinion of the arned atoned for the persecution of Halle tiversity, he was declared honorary professor ‘the Petersburg academy of sciences, and omitted a member of that of Paris, &c. ejudices at last passed away, and in 1741, th some reluctance, he assumed the office of ivy counsellor, of vice-chancellor, and of ofessor of the law of nature and of nations the university of Halle. He afterwards is raised to the dignity of chancellor of the iversity, and created a baron of the Roman apire. This great man, whose whole life is devoted to advance science and virtue, sd at Halle, of the gout in his stomach, , aged 76. His works in Latin and Ger- in'are more than sixty in number, the best which known area Course of Mathematics, ‘ols. 4to. ; Philosophia Rationalis, sive Lo- a, 4to.; a System of Metaphysics, 4to.; s Nature, 8 vols. 4to.; Horze Subsecivee agdeburg; Dictionary on the Mathematics, WREN (Sir CHRISTOPHER), an eminent iglish philosopher and mathematician, and © of the most learned and celebrated archi- ts of his age, was born at Knoyle, in Wilt- e, in 1632, When very young he discovered a surpis- ‘genius for the mathematics, in which sci- *e he made great advances before he was teen years of age. In 1657 he was made fessor of astronomy in Gresham college, ndon ; and his lectures, which were much yuented, tended greatly to the promotion ‘eal knowledge. He proposed several me- s by which to account for the shadows X ENOCRATES, a celebrated ancient Gre- 1 philosopher, was born at Chalcedon in 95th Olympiad. At first he attached him- to Aischines, but afterwards became a ‘iple of Plato, who took much pains in ivating his genius, which was naturally vy. As long as Plato lived, Xenocrates one of his most esteemed disciples ; after death he closely adhered to his doctrine; yin the second year of the 110th Olympiad, 00k the chair in the academy, as the suc- or of Speusippus. XEN returning backward ten degrees on the dial of king Ahaz, by the laws of nature, One sub- ject of his lectures was upon telescopes, to the improvement of which he had greatly con- tributed: another was on certain properties of the air, and the barometer. In the year 1658 he read a description of the body and different phases of the planet Saturn; which subject he proposed to investigate, while his colleague, Mr. Rooke, then professor of geo- metry, was carrying on his observations upon the satellites of Jupiter. The same year he communicated some demonstrations concern- ing cycloids to Dr. Wallis, which were after- wards published by the doctor at the 'end of his treatise upon that subject. About that time also, he resolved the problem proposed by Pascal, under the feigned name of John de Montford, to all the English mathematicians ; and returned another to the mathematicians in France, formerly proposed by Kepler, and then resolved likewise by himself, to which they never gave any solution. In 1660, he invented a method for the construction of solar eclipses; and in the latter part of the same year, he, with ten other gentlemen, formed themselves into a society, to meet weekly, for the improvement of natural and experimental philosophy, being the founda- tion of the Royal Society; of which learned body he was chosen president, in 1680. He died in 1723, at 91 years of age. WRIGHT (Epwarp), an English mathe- matician, was born in Norfolk about the mid- dle of the 16th century, and died in 1615. He was one of those who laboured to bring the logarithmic tables of Napier and Briggs to perfection, and was the inventor of that division of the meridian on which the Mer- cator sailing is founded. He was also the author of two works of navigation, the one entitled, ‘‘ The Correction of certain Errors in Navigation,” published in 1599; and the other entitled “ The Haven-finding Art,” of which the date is not mentioned. Xenocrates was celebrated among the Athe- nians, not only for his wisdom, but for his vir- tues. He was an admirer of the mathema- tical sciences ; and was so fully convinced of - their utility, that when a young man, who was unacquainted with geometry and _ astro- nomy, desired admission into the academy, he refused his request, saying, that he was not yet possessed of the handles of philoso- phy. In fine, Xenocrates was eminent both for the purity of his morals and for his ac- quaintance with science, and supported the XEN credit of the Platonic school by his lectures, his writings, and his conduct. He lived to the first year of the 116th Olympiad, or the 82d of his age, when he Jost his life by acci- dentally falling, in the dark, into a reservoir of water. XENOPHANES, the founder of the Ele- atic sect of philosophy among the Greeks, was born at Colophon, probably about the 65th Olympiad. From some cause or other he left his country early, and took refuge in Sicily, where he supported himself by recit- ing, in the court of Hiero, elegiac and iambic verses, which he had written in reprehension of the theogonies of Hesiod and Homer. From Sicily he passed over into Magna Greecia, where he took up the profession of philosophy, and became a celebrated preceptor in the Pythagorean school. Indulging, however, a greater freedom of thought than was usual among the disciples of Pythagoras, he ven- tured to introduce new opinions of his own, and in. many particulars to oppose the doc- trines of Epimenides, Thales, and Pythagoras. Xenophanes possessed the Pythagorean chair of philosophy about seventy years, and lived to the extreme age of a hundred years; that is, according to Eusebius, till the 81st Olym- piad. The doctrine of Xenophanes concern- ing nature is so imperfectly preserved, and ob- scurely expressed, that it is no wonder that it has been differently represented by different writers. Perhaps the truth is, that he held the universe tobe one in nature and substance, but distinguished in his conception between the matter of which all things consist, and that latent divine force which, though not a dis- tinct substance, but an attribute, is necessarily inherent in the universe, and is the cause of all its perfection. Xenophanes was the au- thor of several poetical works, among which are mentioned a poem on the foundation of Colophon, some of the elegies above referred to, and a treatise concerning nature ; all of which, however, with the exception of a few fragments, are lost. XENOPHON, an Athenian, son of Gryl- lus, celebrated as a general, an historian, and a philosopher. In the school of Socrates he received those precepts which afterwards so eminently distinguished him. Being invited by Proxenus, one of his intimate friends, to accompany Cyrus the younger in an expedi- tion against his brother Artaxerxes, king of Persia, he previously consulted Socrates, who XIP strongly opposed it. Xenophon, howevye ambitious of glory, hastened with precipit tion to Sardis, where he was introduced to tl young prince. In the army of Cyrus, Xen phon showed that he was a true disciple | Socrates. After the decisive battle in tl plains of Cunaxa, and the fall of Cyrus, t] prudence and vigour of his mind were calk into action. The ten thousand Greeks | had followed the prince were now at the di tance of above 600 leagues from home, su rounded on every side by a victorious enem without money, without provisions, and wit out a leader. Xenophon was selected fre among the officers to superintend the retre This celebrated retreat was at last happ effected by the Greeks, who returned hot after a march of 1155 parasangs, or leagu: which was performed in 215 days, after | absence of 15 months. He had no sooner } turned from Cunaxa, than he sought n honours in following the fortune of Agesile in Asia, where he conquered with him int} Asiatic provinces, as well as at the battle Corona. His fame, however, did not ese# the aspersions of jealousy: he was publij banished from Athens for accompanying Cyt avainst his brother; and being now with a home, he retired to Scillus, a small town the neighbourhood of Olympia. In this tary retreat, he dedicated his time to liter pursuits ; but his peaceful occupations w soon disturbed by a war which arose betwt the Lacedzmonians and Elis. From the) ter place he retired to the city of Cori where he diedin the 90th year of his age, years before the Christian era. The work# Xenophon are the Anabasis, the Cyropat his Hellenica, Memorabilia, besides of} tracts. The simplicity and the elegancé Xenophon’s diction have procured ‘him ‘ name of the Athenian Muse, and the Be Greece. His sentiments, as to the divi and religion, were the same as those of | venerable Socrates. The best editions of the works of Xe phon are those of Frankfort in 1674, an¢ Oxford, in Greek and Latin, 5 vols. 8vo. 14 His Cyropedia, Anabasis, and Memorak Socrates, have been published separately) Oxford; and there is a good translation of Cyropzdia by Spelman. . XIPHIAS, in Astronomy, the Dorado, Sword-Fish. See ConsTELLATION. ¥ YEA ye ig | Y ARD, an English measure of length, and ed also by several other European nations. The English yard contains 3 feet, and is to 4-5th of the English ell, to 7-9th of the Paris ell, to 4-3d of the Flemish ell, to 56-5th of the Spanish vasa or yard. WEAR, in Astronomy and Chronology, the ‘tion of time occupied by the sun in pass- ‘over the twelve signs of the zodiac, and in ich is comprehended the several changes he seasons. e mean solar year, according to the ob- tions of the best modern astronomers, ains 365 days, 5 hours, 48 minutes, 48 ds; the quantity assumed by the authors e Gregorian calendar, is 365 days, 5 hours, minutes ; but in the civil or popular account year contains 365 days, 6 hours, or ra- 865 days for three years in succession, . every fourth year 366 days.. See Bis- PEdLE. ‘he vicissitude of seasons seems to have m occasion to the first institution of the r. Man, naturally curious to know the ise of their diversity, soon conjectured that ‘pended upon the motion of the sun, and ‘efore gave the name year to the space of 2in which that luminary seemed to per- ahis whole course, by returning again to Same point of its orbit. ‘ccording to the accuracy of their obser- ons, the year of some nations was more fect than that of others, but none of them e€exact, nor whose parts did not shift with d to the parts of the sun’s course. ecording to Herodotus, it was the Egyp- 8 who first formed the year, making it to sist of 360 days, which they subdivided into months, of 30 days each. Mercury Tris- ‘istus added five days more to the account ; which form of the year Thales is said to > instituted amongst the Greeks; and 2e, with successive improvements, it has 1 handed down to the moderns. he Solar Year is cither astronomical or he Astronomical Solar YEAR, is that which recisely determined by astronomical ob- ations, and is of two kinds, tropical and eal, or astral. ropical or Natural Y £AR, is the time which jun, or rather the earth, employs in pass- through the 12 signs df the zodiac, and th, as stated above, contains 365 days, 5 hours, 48 minutes, 48 seconds, which is the only natural year, because it always keeps the same seasons in the same months. Sidereal Year, or Astral Year, is the space of time the sun takes in passing from any fixed star, till his return to it again. ~This consists of 365 days, 6 hours, 9 minutes, 11 seconds, being 20 minutes, 29 seconds longer than the true solar year. Anomalhistic YEAR, is the interval which is occupied by the sunin passing from apogee to apogee, or from perigee to perigee: it is greater than the sidereal year by the time re- quired to describe the annual progression of the apogee. The length of the anomalistic year is 365 days, 6 hours, 14 minutes, 1 se-- cond. Lunar Y ean, is the space of 12 lunar months. Hence, from the two kinds of synodical lunar months, there arise two kinds of lunar years; the one astronomical, the other civil. Lunar Astronomical YEAR, consists of 12° lunar synodical months; and therefore con- tains 354 days, 8 hours, 48 minutes, 38 se- conds, and is therefore 10 days, 21 hours, 0 minutes, 10 seconds shorter than the solar year. A difference which is the foundation of the epact. Lunar Cwil YEAR, is either common or em- bolismic. The Common Lunar YEAR, consists of 12- lunar civil months; and therefore contains 354 days. And The E’mbolismic, or Intercalary Lunar YEAR, consists of 13 lunar civil months, and there- fore contains 384 days. Thus far we have considered years and months, with regard to astronomical princi- ples, upon which the division is founded. By this, the various forms of civil years that have formerly obtained, or that do still obtain, in divers nations, are to be examined. ; Civil Year, is that form of the year which every nation has contrived or adopted for com- puting their time by. Or the civil is the tro- pical year, considered as only consisting of a certain number of whole days: the odd hours and minutes being set aside, to render the computation of time, in the common occa- sions of life, more easy. As the tropical year is 365 days, 5 hours, 49 minutes, or almost- 365 days, 6 hours, which is 365 days and a quarter; therefore, if the civil year be made 365 days, every 4th year it must be 366 days, to keep nearly to the course of thesun. And YEA hence the civil year is either common or bis- sextile. The Common Civil YEAR, is that consisting of 365 days; having seven months of 31 days each, four of 30 days, and one of 28 days ; as indicated by the following well-known me- morial verses : Thirty days hath September, April, June, and November ; February twenty-eight alone, And all the rest have thirty-one. Bissextile, or Leap YEAR, contains 366 days, having one day extraordinary, called the in- tercalary, or bissextile day, and takes place every 4th year. This additional day to every 4th year was first introduced by Julius Ceesar ; who, to make the civil years keeps pace with the tropical ones, contrived that the six hours which the latter exceeded the former should make one day in four years, and be added be- tween the 24th and 23d of February, which was their 6th of the calends of March; and as they then counted this day twice over, or had bis sexto calendas, hence the year itself came to be'ealled bis sextus, and bissextile. However, among us, the intercalary day is not introduced by counting the 23d of Febru- ary twice over, but by adding a day at the end of that month, which therefore in that year contains 29 days. A farther reformation was made in the civil year by pope Gregory. . The civil or legal year, in England, for- merly commenced on the day of the Annun- ciation, or 25th of March ; though the histo- rical year began on the day of the circumci- sion, or 1st of January; on which day the German and Italian year also begins. The part of the year between these two terms was usually expressed both ways; as 1740-6, or 1743, But by the act for altering the style, the civil year now commences with the Ist of Ja- nuary. Ancient Roman YEAR. This was the lunar year, which, as first settled by Romulus, con- tained only ten months, of unequal numbers of days, in the following order: viz. March 31; April 30; May 31; June 30; Quintilis 31; Sextilis 30; September 30; October 31; November 30; December 30; in all 304 days; which came short of the true lunar year by 50 days; and of the solar by 61 days. Hence, the beginning of Romulus’s year was vague, and unfixed to any precise season; to remove which inconvenience, that prince ordered so many days to be added yearly as would make the state of the heavens correspond to the first month, without calling them by the name of any month. Numa Pompilius corrected this irregular constitution of the year, composing two new months, January and February, of the days that were used to be added to the former year. Thus Numa’s year consisted of 12 months, of different days, as follow ; viz. | YEA January 29; February 28; March ~ April 29; May 31; June 2 Quintilis 31; Sextilis 29; September 2 October 31; November 29; December 2 in all 355 days ; therefore exceeding the qui tity of a lunar civil year by one day ; that o Junar astronomical year by 15 hours, I1n nutes, 22 seconds; but falling short of t common solar year by 10 days; so that its ginning was still vague and unfixed. Numa, however, desiring to have it beg at the winter solstice, ordered 22 days to intercalated in February every 2d year, | every fourth, 22 every 6th, and 23 every § year. 4 But this rule failing to keep matters ey recourse was had to a new way of inters lating ; and instead of 23 days every 8th ye only 15 were to be added. The care of 1 whole was committed to the pontifex ma) mus; who, however, neglecting the trust, things run to great confusion. And thus Roman year stood till Julius Caesar reforr it. See CALENDAR. é The Ancient Egyptian YEAR, called a the year of Nabonassar, on account of epocha of Nabonassar, is the solar year 365 days, divided into 12 months of 30d each, besides five intercalary days adde the end. The names, &c. of the months as follows: 1. Troth. 2. Paophi. 3. Atl 4. Chojac. 5.Tybi. 6. Mecheir. 7. Phai noth. 8. Pharmuthi. 9. Pachon. 10. Pai 11. Epiphi. 12. Mesori ; beside the vp ETOLYOPLEVOLL. . The Ancient Greek YEAR, was lunar; ¢ sisting of 12 months, which at first had 30 a-piece, then alternately 30 and 29 days, ¢ puted from the first appearance of the 1 moon; .with the addition of an embolis month of 30 days every 3d, 5th, 8th, 11th, 1 16, and 19th year of a cycle of 19 years order to keep the new and full moons to same terms or seasons of the year. Their} commenced with that new moon, the moon of which comes next after the sum solstice. The order, &c. of their months thus: 1. ‘ExarouCawy, containing 29 days. Marayaurywy, 30. 3. Bondpomiwy, 29. 4. peextnpiwr, 30. 5. Tluave}iwv, 29. 6. Tlooes 30. '7.Tapundswy, 29. 8. Arbecngswy, 30. 9.1 OnCorswy, 30. 10. Mavuxiwy, 30. 11. Oweyn 29. 12. LxspoPopswy, 30. The Ancient Jewish YEAR, is a lunar } consisting commonly of 11 months, alternately contain 80 and 29 days. It made to agree with the solar year, eith the adding of 11, and sometimes 12 days the end of the year, or by an emboli month. The names and quantities of months stand thus: 1. Nisan, or Abib, 30 2. Jiar, or Zius, 29. 3. Siban, or Siwan 4. Thammuz, or Tammuz, 29. 5. Ab 6. Elul, 29. 7. Tisri, or Ethanim, 30. 8.. chesvam, or Bull, 29. 9. Cisleu, 30. 10. beth, 29. 11. Sabat, or Schebeth, 30. Adar, in the embdolismic year, 30. Ada ZEN ommon year, was but 29. Note, in the tive year, Cisleu was only 29 days; and i » redundant year, Marchesvam was 30, he Persian YEAR, is a solar year of about lays ; consisting of 12 months of 30 days ,, with five intercalary days added at the | Mahometan, and Turkish Y BARS, id also the year of the Hegira, is a lunar | ITH, an Arabic word, used in astro- fy to denote the vertical point of the hea- |, or that point directly over our heads. zenith is called the pole of the horizon, x 90° distant from every point of that Ge NITH Distance, the arc intercepted be- n any celestial object and the zenith, x the same as the co-altitude of an ob- ENO (ELEATES), an eminent Grecian sopher, was born about 500 years before st. He was a zealous friend of civil li- , and is celebrated for his courageous successful opposition to tyranny ; but the ‘asistency of the stories related by different ars concerning him in a great measure ‘oys their credit. He chose to reside in mall native city of Elea rather than at ms, because it afforded freer scope to his pendent and generous spirit, which could easily submit to the restraints of autho- It is related, that he vindicated the ath with which he resented reproach, by ig, “ If I were indifferent to censure, I Id also be indifferent to praise.” The in- ion of the dialectic art has been impro- 7 ascribed to Zeno; but'there can be no it that this philosopher, and other meta- ical disputants in the Eleatic sect, em- ed much ingenuity and subtlety in exhi- ‘g¢ examples of most of the logical arts, th were afterwards reduced to rule by totle and others. If Seneca’s account of /philosopher deserves credit, he reached aighest point of scepticism, and denied the | existence of external objects. ENO, another Greek philosopher of consi- ible eminence, was born at Citium in the of Cyprus. He was founder of the Stoics, »et which had its name from that of a por- at Athens, where Zeno was accustomed leliver his discourses. The father of our osopher was a merchant, but readily se- ded his son’s inclinations, and devoted | to the pursuits of literature. In the way housiness he had frequent occasion to visit ZOD , year, equal to 354 days, 8 hours, and 48 mi-’ nutes, and consists of 12 months, which con- tain alternately 30 and 29 days. The Hindoo YEAR differs from all these, and is indeed different in different provinces of India. The best account that we have of it is by Mr. Cavendish, in the Phil. ‘Trans. of the Royal Society of London for the year 1792. ~ ee Athens, where he purchased for his son several of the most renowned works of the celebrated Socratic philosophers. ‘These Zeno read with avidity, and determined to visit the city where so much wisdom was found. Upon his first arrival in Athens, going accidentally into the shop of a bookseller, he took up the commen- taries of Xenophon, with the perusal of which he was so much delighted, that he asked the bookseller where he might meet with such men. Crates, the cynic philosopher, was at that moment passing by ; the bookseller point- ed to him, and said, foliow that man. Heim- mediately became his disciple, but was soon dissatisfied with his doctrine, and joined him- self to other philosophers, whose instructions were more accordant to his way of thinking. Zeno staid long with no master; he studied under all the most celebrated teachers, with a view of collecting materials from various quarters for a new system of his own. ‘To this [Polemo| alluded when he saw Zeno com- ing into his school ; “ I am no stranger,” said he, “ to your Phenician arts, I perceive that your design is to creep slily into my garden, and steal away the fruit.” From this period Zeno avowed his intention of becoming the founder of anew sect. The place which he chose for his school was the painted porch, the most famous in Athens. Zeno excelled in that kind of subtle reasoning which was in his time very popular. Hence his followers were very numerous, and from the highest ranks in society. Among these, was Anti- gonus Genates, king of Macedon, ‘who ear- nestly solicited him to go to his court. He possessed so large a share of esteem among the Athenians, that on account of his integrity, they deposited the keys of their citadel in his hands; they also honoured him with a golden crown and a statue of brass. He lived to the age of 98, and at last, in consequence of an accident, voluntarily put an end to his life. ZETETIC Method, an old term for what we now call analytic method. See ANALYSIS and SYTHESIS. ZODIAC, in Astronomy, an imaginary ring 3D ZOD er broad cirele, in the heavens, in form of a belt or girdle, within which the planets all make their excursions. In the very middle of it runs the ecliptic, or path of the sun in his annual course; and its breadth, compre- hending the deviations or latitudes of the earlier known planets, is by some authors ac- counted 16, some 18, and others 20 degrees. The zodiac, cutting the equator obliquely, makes with it the same angle as the ccliptic, which is its middle line, which angle, conti- nually varying, is now nearly equal to 23° 28’; which is called the obliquity of the ecliptic, and constantly varies between certain limits which it can never exceed. . See Ecuiptic and OBLIQUITY. The zodiac is divided into 12 equal parts, of 30 degrees each, called the signs of the zodiac, being so named from the constella- tions which anciently occupied them. But the stars having a motion from west to east, those constellations do not now correspond to their proper signs; from whence arises what is call- _ed the precession of the equinoxes. And there- fore when a star is said to be in such a sign of the zodiac, it is not to be understood of that constellation, but only of that dodecatemory or 12th part of it. Cassini has also observed a tract in the heavens, within whose bounds most of the comets, though not all of them, are observed to keep, and which he therefore calls the zo- diac of the comets. This he makes as broad as the other zodiac, and marks it with signs or constellations, like that; as Antinous, Pe- gasus, Andromeda, Taurus, Orion, the Lesser Dog, Hydra, the Centaur, Scorpion, and Sa- gittary. It is a curious fact, that the solar division of the Indian zodiac is the same in substance with that of the Greeks, and yet that it bas not been borrowed either from the Grecks or the Arabians. The identity, or at least strik- ing similarity of the division is universally known; and Montucla has endeavoured to prove that the Bramins received it from the Arabs. His opinion, we believe, has been very generally admitted; but in the second volume of the Asiatic Researches, the accom- plished president,Sir William Jones, has proved unanswerably, that neither of those nations borrowed that division from the other; that it has been known among the Hindoos from time immemorial; and that it was probably invented by the first progenitors of that race, whom he considers as the most ancient of mankind, before their dispersion. The Greek zodiac originally contained only eleven signs: the Scorpion in their zodiac oc- cupying the place of two. See Montucla’s History of Mathematics, vol. i. p. 79. ZODIACAL Light, a brightness some- times observed in the zodiac, resembling that of the galaxy or milky way. It appears at ceriain seasons, viz. towards the end of winter and in spring, after sunset, or before his ris- ing, in autumn and beginning of winter, re- sembling the form of a pyramid, lying length- ZON ways with its axis along the zodiac, its being placed obliquely with respeet te horizon. This phenomenon was first dese and named by the elder Cassini, in 168; was afterwards observed by Fatio, in | 1685, and 1686 ; also by Kirch and Eim in 1688, 1689, 1691, 1693, and 1694, | Mairan, Suite des Mem. de l’Acad. Rq des Sciences, 1781,-p: 3. vi The zodiacal light, according to Maira the solar atmosphere, a rare and subtile 4 either luminous by itself, or made so rays of the sun surrounding its globe; by a greater quantity, and more extensp about his equator, than any other part. | Mairin says, it may be proved from ni observations, that the sun’s atmosphere s ‘times reaches as far as the earth’s orbit, there meeting with our atmosphere, prod the appearance of an aurora borealis. ‘| The length of the zodiacal light varies s¢ times in reality, and sometimes in appear: only, from various causes. oa Cassini often mentions the great res blance between the zodiacal light and the: of comets. The same observation has | made by Fatio; and Euler endeavoure prove that they were owing to similar eau See “‘ Decouverte de la Lumiere Celeste paroit dans le Zodiaque,” art. 41. Lett M. Cassini, printed at Amsterdam, in It Euler, in Mem. de I’Acad. de Berlin, tom This light seems to have no other mo than that of the sun itself; and its ém from the sun to its point is seldom less t 50 or GO degrees in length, and more that degrees in breadth; but it has been knowl -extend to 100 or 103°, and from 8° to broad. . It is now generally acknowledged, that : electric fluid is the cause of the ee alis, ascribed by Mairan to the solar @ sphere, which produces the zodiacal light: which is thrown off chiefly and to the great distance from the equatorial parts of thes by means of the rotation on his axis, and: tending visibly as far as the orbit of the i where it falls into the upper regions of) atmosphere, and is collected chiefly tows the polar parts of the earth, in consequel of the diurnal revolution, where it forms | aurora borealis. And hence it has been sl gested, as a probable conjecture, that the s may be the fountain of the electrical fluid, a that the zodiacal light, and the tails of come as well as the aurora borealis, the lightm and artificial electricity, are its various a not very dissimilar modifications. See B D’Astronomie Physique, art. 254; and G gory’s Translation of Haiiy’s Philosophy, ¥ il. art. 628. y ZONE, in Geography and Astronomy, ¢ vision of the earth’s surface, by means off rallel circles, chiefly with respect to the degr of heat in the different parts of that surface The ancient astronomers used the fel zone, to explain the different appearances the sun and other heayenly bodies, with” ZON seth of the days and nights; and the geo- phers, as they used the climates, to mark e sitnation of places; using the term cli- ate when they were able to be more exact, id the term zone when less so. The zones were commonly accounted five ‘number; one a broad belt round the mid- » of the earth, having the equator in the very iddle of it, and bonnded towards the north id south by parallel circles passing through e tropics of Cancer and Capricorn. ‘This rey called the torrid zone, which they sup- sed not habitable, on account ofits extreme iat. Though sometimes they divided this to two equal torrid zones, by the equator, he to the north, and the other south; and jen the whole number of zones was ac- unted six. Next, from the tropics of Cancer and Ca- icorn, to the two polar circies, were two her spaces called temperate zones, as being sderately warm; and these they supposed be the only habitable parts of the earth. ‘Lastly, the two spaces beyond the tempe- tle zones, about either pole, bounded within je polar circles, and having the poles in the iddle of them, are the two frigid or frozen wes, and. which they supposed not habi- gle on account of the extreme cold there. ence, the breadth of the torrid zone is {ual to twice the greatest declination of the n, or obliquity of the ecliptic, equal to 46°56’, (twice 23° 28’ Hach frigid zone is also of ¥same breadth, the distance from the pole j the polar circle being equal to the same ob- tye ZON liquity 23° 28’. And the breadth of each tem- perate zone is equal to 43° 4’, the complement of twice the same obliquity. The difference of zones is attended with a great diversity of phenomena. 1. In the tor- rid zone, the sun passes through the zenith of every place in it twice’ a year; making as it were two summers in the year; and the in- habitants of this zone are called amphiscians, because they have their noon-day shadows projected different ways in different times of the year, northward at one season, and south- ward at the other. 2. In the temperate and frigid zones, the sun rises and sets every natural day of 24 hours. Yetevery where, but under the equa- tor, the artificial days are of unequal lengths, and the inequality is the greater, as the place is farther from the equator. The inhabitants of the temperate zones are called heteroscians, - because their noon-day shadow is cast the same way all the year round, viz. those in the north zone toward the north pole, and those in the south zone toward the south pole. 3. Within the frigid zones, the inhabitants have their artificial days and nights extended out to a great length; the sun sometimes skirting round a little above the horizon for many days together; and at another season never rising above the horizon at all, but mak- ing continual night for a considerable space of time. The inhabitants of these zones are called periscians, because sometimes they have their shadows going quite round them im the space. of 24 hours. 7 _ FINIS. ie : | 3 | ) WHITTINGHAM and ROWLAND, Printers, Goswell Street, London, ERRATA. MD , Sheet. B. col. 4, line 31, Maupertius (Maupertuis); c. 7, 1. 67, hyperbole (hyperbola); c. 10,1. 44, Dol- land ‘(Dollond); ce. t0, 1.57, Pollus (Pollux). ' ©. col. 16, line 40, AGNESTA (Acuest); c. 31,1. 38, Pantoloiga ( Pantologia); c. 32,1. 30, Dynam- que (Dynamique); 1. 33, Movement (Mouve- men). D. col. 12, line 19, Montulca (Montucla); c. 11,1. 4, Wilden (Wilder); 1. 7, Nicoles (Nicole); c. 19, 1. 4, dele (= 8 feet); c. 30,1. 27, 234 (284); c. 27, 1. 15, fr. bot. Horseley (Horsley). FE. col. 1, line 18. Rationes (Rationis); c. 6, bottom line, DAC (DAB); c. 20, 1. 19, edtvina (Redivivus); c. 23,1. 10, fr. bot. vols. (vol. 3). F. col. 4, line 48, Alter (Altar); c. 13,1. 4, aeornprog (cprdpiec) $ c. 17, 1. 11, Arinarius (frendrius)’; c. 31, I. 20, Mornin (Asrerism) $ c. 32, 1. 14, Phisical (Ph ysical). G. col. 10, line 29 and 34, Vester (Vesta); c. 28, 1, 49, aurora (aurore); c. 29,1. 42, automatas (automata). I. col. 3, line 25, Disquisiones (Disquisitiones) ; c. 22,1. 16, invitantus (invitantur); 1. 2, fr. bot. unque (ungue). K. col. 6. line 36, year of Julian (Julian year); col. 8, }. 46, demonstrations (denominations) ; c. 12,1. 57, Capatna (CAPELLA); c 15,1. 20, Infinitorem (Infinitorum); c. 24, 1. 23, Al- hazon (Alhazen); c. 26,1. 1, fr. bot. curve- lineal (curvilineal). L. col. 5, 1. 33, of (or) — 3n (3x); c. 6,1. 44, if (of); c. 14, 1. 34, plant (planet); col. 16,1. 50, Schoolen (Schooten) ; 1.1, fr. bot. Witto ‘Craig (Witt. Craig); c. 17,1. 38, 1750 (1756). M. col. 2,1. 30, Mathematique (Mathematiques) ; 1. 31, determinations (determination); Meri- dian (Meridien) ; c. 10,1. 50 and 54, product (products); c. 14,1. 31, Rhronius (Rhonius); ¢. 26 731,,4 210. bot. comets (comet); 1. 11, movemens (mouveimens). N. col, 1, line 1 fr. bot. Montuclar (Montucla) ; c. 15,1. 3 fr. bot. dele (Robertson) ; c. 18,1.25, Acturus (Arcturus); c. 19,1. 38, Capalla (Ca- pella); the same mistake occurs next page. O. col. 28, line 6 fr. bot. cupsis (cuspis). P. col. 5, line 12, dele (oblique) ; the rule does not apply to the oblique surface; c. 15, |. 14, Myer’ (Myers’); c. 17, 1. 46, 1750 (1756). Q. col. 4, line 9 fr. bot. Sterling (Stirling); c. 5, 1. a7, h highest (the highest); c. 10,1. 32, Bouger (Bouguer) ; c. 19,1. 42, Barly (Baily). $. col. 5, line 29, Cavello (Cavallo); 1. 30, Mairne (Nairne) ; c. 9, in the equations to the curve, for P(e); c. 12,1. 2 fr. bot. Memoires (Me- moirs); c. 12, 1. 9 fr. bot. perameter (para- meter); c. 19, 1. 6, equiescent (quiescent); c. 17, 29, and 17 fr. bot. different (defferent); c. 99° 1, 1 fr. bot. reduced a (reduced to a); c. 26, 1. 38, Lecroix (Lacroix). JT. col. 3, line 9 fr. bet. aquus (equus); c. 20,1. 15, 1 l a™ (a™); 1.17, a” (a”); c. 20,1. 4, explosion (expansion); c. 21,1]. 7 fr. bot. x7 (x7); c. 28, 1, 23, invested (inverted); 1. 16 fr. bot. object is (objects are); 1. 11 fr. bot. Ore. (Horn); c. 29,1. 7, factors (factors of); 1. 22, dele (such) $ c. 31, 1.35 pegs wanes). UV. col. 7, line 13, Mobili (Mobile); 1. 21, corro- Sheet. sive, add (menstruum); c. 16, I. 20 fr. | expression (expressions); c. 18, I. 10 fr, |; substitutes (substitute); Table of Fluents, || ani = “tes (a0? ee x = =) form & 2) c. 31, art. Focus, the line CPi omitted in the fig. G (a). X. col. 14, line 39 and 46, Columb (Conlon, c. ST, 1. 14 fr. bot. contract (contact). Y. col. 2, line 11, Alter (Altar); c. 7,1. 35, azoe (azote); c. 14, 1. 16 fr. bot. Pilkington (Pi. erton); c. 21, 1. 34, Leisle (Leslie); c. SHht fr. bot. dele (the velocity of). Z. col. 16, line 37, art (act); .c. 24, 1. i (sasoc)$ ce. 29, |. 15, they (there ). AA. col. 17,line 35 and 36, 2 (2); c. 18, 1h dele (ef); c. 26,1. 14, hy pothesis (hypoth BB. col. 7, line 9 fr. bot. Emmerson (Emerso, c. 13, 1. 18, 19, &c. dele (A); c. 11, WY), bot. Incipiorum (Principiorum). CC, col. 4, line 28, dele (Morgan); c. 5, 1. 10, bot. 17 (77); 1. 36, dele (of) c. 9, 1. 22.) summationes (de summationes) ; the same . curs c. 11, }. 23 fr. bot.; c. 16,1. 21, ems) for p and r read (d and gq); 1. 17, tune (tim: c. 26, 1.2, his (her); c. 29, 1. 43, ‘latitute & tude); c. "98, 1. 6 fr. bot. PW (PN). 4 DD.col. 29, line 5, finding the roots (finding | limits of the roots); c. 32,1. 22, ADE (AD) EEF. col. 4, line 1, Acyop (nayecds easune (cepsBpxo} c. 10,1. 1, fr. bot. or (a); c. 19, 1. 26, Mar) et Avignon” (Marins” Syne So c. 20,1), Catlet’s (Callet’ s)3c.27,1.11 ,ADEF (ADB) * c. 29,1. 13, Bockles (Bockler). FF. col. 26, line 20 fr. bot. dele (the); c. 30, I 7 fr. bot. of (or). GG. col. 1, line 18, BC (BG); 1. 24, OB (AT 1. 26,2 DB = BC (2 DB x BC); c, 3h, MontMonr (Montmort). HH.col. 29, |. 5, fr. bot.; this work is not oil | Newton’ S, though commonly given in the ; | ) of his works, IT. col. 28, line 21, OzaNnnam (OZANAM); 1. 4, nobulosity (nebulosity); c. 32, 1. I bot DG= DF (AG= AF). KK.col. 1, line 6, G (F). | LL. col. 1, line 23 fr. bot. Creswell (Cressi c./ 10; 1. 12, same (contrary); c. 16,1. 27 38, Vester (Vesta); c. 2%, 1. 38, PLEn (PLENUM). sires ~ ——— é - i MM. col. 28, line 7'fr. bot. Vaga (Vega). N N. col. 9, line 2, 3, 4, for a (A); for m (h). OO. col. 8, ‘line 21, quadratrixes of Dinostrates the (are the quadratrixes of Dinostrates &) QQ. col. 16, 1.17, 1632 (1682). | RR. col. 30, line 33, Sectionen Conicorum (Se tionum Conicarum). TT. col. 3. line 3, dele (or); ¢. 18, 1. 38, 5 c. 30, Ex. 1. 26 (20). XX. col. 1, line 19, Be? to D e? (BC? to cet). 3 B. col. 1, line 38, Horner (Horn). The words i in the parenthesis are the correctio of those which precede them. The above method of arranging the errata h been found necessary in consequence of the not being paged. 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