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Bes Wy 
 
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 4 
 
DR webu ELONTS 
 PHILOSOPHICAL RECREATIONS, 
 
 By EDWARD RIDDLE. | 
 
PRINTED BY NUTTALL AND HODGSON, 
 GOUGH SQUARE, LONDON, ; 
 
2 
 
 boitd 
 
 Piet, Cirle HA Py TO2N-s 
 
 ad SCLENCE 
 
 AND 
 
 NATURAL PHILOSOPHY: 
 
 DR. HUTTON’S TRANSLATION OF MONTUCLA’S EDITION 
 OF OZANAM. - 
 
 THE PRESENT NEW EDITION OF THIS CELEBRATED WORK IS REVISED 
 
 By EDWARD RIDDLE, 
 
 MASTER OF THE MATHEMATICAL SCHOOL, ROYAL HOSPITAL, GREENWICH, 
 WHO HAS CORRECTED IT TO THE PRESENT ERA, AND MADE 
 
 NUMEROUS 
 ADDITIONS. 
 
 Hhis Bdition ts also Llustrated by upwards of Four Hundred Geoovcuts. 
 
 LONDON: 
 PRINTED FOR THOMAS TEGG, 
 No. 73, CHEAPSIDE. 
 1844, 
 
oe 
 e. 
 
 PREFATORY NOTICE. 
 
 Jacques Ozanam, the original composer of the ‘‘ Recreations in Mathe- 
 matics and Natural Philosophy,’ was born in 1640, at Bouligneu, in 
 what is now called the department of Ain, in France. His family was of 
 Jewish extraction, but had long been members of the Church of Rome ; 
 and having landed property to which some ecclesiastical patronage was 
 attached, Jacques, who was the younger of two sons, was intended for 
 the church ; and he accordingly entered on a course of study suitable for 
 his destination. | 
 
 He is said to have been of gay, lively, and expensive habits, and to have 
 shewn no inclination for theological pursuits. Chemistry, mechanics, &c. 
 
 which have a more obvious connection with the business of life, attracted 
 his attention; and after his father’s death, which took place about four 
 years after he began to read for the church, he abandoned theology, and 
 
 Y-attached himself to science. 3 
 o> His opportunities of receiving assistance in his scientific studies, were 
 ai So scanty, that he may be considered as having been self-taught; and 
 
 though he cannot be regarded as having attained great eminence as a 
 mathematician, even among his contemporaries, he was the author of a 
 «good many useful “works, whose popularity carried them through 
 «.. several editions. . ; 
 
 It would appear, however, that in attaching himself to science, he did 
 _¢not at first look to it as a means of living ; for soon after his father’s death 
 > he removed to Lyons, where he taught mathematics gratuitously, consider- 
 A ing it a degradation to receive pay for his instructions. 
 
 YW ‘It is probable, however, that he soon changed his opinion on the sub- 
 4. ject. He was addicted to gaming; his private pecuniary resources were 
 Y limited; and the stern realities of distress would speedily dissipate all 
 g_illusions about the dignity of teaching science for its own sake. 
 
 -- An act of striking and disinterested liberality, which he performed 
 
 _. towards two strangers, having been mentioned to the chancellor of France, 
 «& that distinguished personage invited the Lyonnese mathematician. to Paris ; 
 
 where, after some time spent in dissipation, he married a young woman 
 
 without fortune, but who proved to him a most excellent wife. After 
 bearing to him twelve children, all of whom died young, she died in 
 
 1701, deeply lamented by her husband. 
 
 * Oczanam subsisted in Paris by teaching mathematics, and met with con- 
 * siderable success, especially among. foreigners. But, upon the breaking 
 out of the war of the Spanish succession, most of his pupils quitting France, 
 his professiona] income became both small and precarious. 
 He lived for some years in comparative mdigence, but, towards the 
 close of his life, his difficulties were somewhat alleviated by his being 
 
 &28G 14 GRACE BARNES 
 
 s 
 
 a 
 z= 
 2 
 
vi PREFACE. 
 
 admitted an éléve of the Academy of Sciences. He died of apoplexy, at 
 Paris, April 3rd, 1717, aged 77 years. 
 
 Ye was of a mild and cheerful temper, generous to the full extent of 
 his means, and of an inventive genius; and his conduct after marriage 
 was irreproachable. He was devout, but averse to disputations about 
 points of faith. On this subject he used to say, ‘‘ It is the business of the 
 Sorbonne to discuss, of the Pope to decide, and of a mathematician to go 
 straight to heaven in a perpendicular line.” 
 
 Jean Erienne Monrvcua (who so greatly enlarged and improved the 
 “ Recreations”? of Ozanam, that he may be said to have made the work 
 his own,) was the son of a merchant at Lyons, where he was born on Sept. 
 5th, 1725. He was left an orphan at the age of sixteen, and was educated at 
 the Jesuit’s College in his native town. His attention was chiefly directed 
 to the ancient classics; but having a natural taste for philological studies, 
 and a powerful memory, he was enabled to acquire an accurate knowledge 
 of several modern languages ; among which Italian, German, Dutch, and 
 English are mentioned. Under Le Pere Béraud, who was subsequently the 
 tutor of Lalande, he made considerable proficiency in the study of mathe- 
 matics and physics. | " 
 
 Having completed his course of general education, he studied for the 
 legal profession, first at Toulouse, and afterwards in Paris; where at the 
 scientific soirées of M. Jambert, he became acquainted with Diderot, 
 D’Alembert, Lalande, and other scientific men of the highest character. 
 
 Having published several scientific works, by which he acquired much 
 reputation, he began to be employed by the government. He was sent as 
 Intendant Secretary to Grenoble, where he married the daughter of M. 
 Roland in 1763; and in the following year he was sent as secretary and 
 astronomer royal to the expedition for colonising Cayenne. 
 
 On his return to France, after a few years’ absence, he obtained the 
 situation of ‘‘ Premier Commis des Batiments,”’ and in addition the office 
 of ‘Censor Royal of mathematical works,” an appointment which was 
 merely honorary. 7 
 
 It would appear, that though the income which he derived from his offi- 
 cial appointment was not large, yet, from his prudent and economical 
 habits, it was sufficient for the immediate wants of himself and his family. 
 He employed his leisure in educating his children, and in scientific pur- 
 suits ; following the latter, it is said, in secrecy, lest he should be suspected 
 of neglecting his official duties. 
 
 It was at this time that he edited the new edition of the ‘‘ Recreations ;” 
 aud so carefully had he concealed his connection with the work, that, on 
 its completion, a copy of it was sent to him, in his capacity of censor, for 
 examination and approval. 
 
 Besides expunging from the work of Ozanam much that was absurd, 
 puerile, and obsolete, he enriched his edition with dissertations upon almost 
 every branch of practical science; and much of what he added is valu- 
 able even at the present day. 
 
 But the name of Montucla is best known from his “‘ History of the 
 Mathematics,’ which contains, besides what is strictly historical, treatises 
 upon all the leading departments of the pure and applied sciences; and 
 abounds with interesting details respecting the discoveries and improve- 
 ments which have contributed to their progress. 
 
PREFACE. Vil 
 
 The French Revolution put an end at once to his office and the little 
 savings which his regularity and economy had enabled him to make 
 from his income,—throwing him on the world in his old age, stripped of 
 every thing but his integrity, and the love and respect of his friends. He 
 died on the 18th of December, 1799. 
 
 In 1803, a translation into English of Montucla’s Edition of Ozanam’s 
 Recreations, by Dr. Charles Hutton, of Woolwich, was published in Lon- 
 don. In this Edition were incorporated many valuable additions and obser- 
 vations by the learned and judicious translator, who lived to superintend a 
 second edition, which, with still further improvements, was published in 
 1814. 
 
 Dr. Hutton was born in Percy Street, Newcastle-upon-Tyne, on August 
 14th, 1737. His. father, who- was employed in the coal’ works in the 
 
 ‘neighbourhood, was understood to be descended from a respectable family 
 in Westmoreland. He died when Charles was only five years old; and 
 his widow married a person named Fraim, whose employment was that of 
 a colliery over-man. 
 
 From an accident which happened to Charles at play, he was not sent 
 when a boy to work in the pits, as his brothers were; but kept at school 
 
 for some years, in the hope that he might be enabled to earn his bread by 
 his scholarship. He was taught to read by an old woman who conducted 
 a little school in the neighbourhood, and to write by a schoolmaster named 
 
 Robson, near Benwell, a village near Newcastle ; and he attended after- 
 
 _ wards a school at Jesmond, kept by the Rey. Mr. Ivison, a clergyman of 
 the English Church; and on Mr. Ivison’s removal to a curacy in the 
 
 county of Durham, Mr. Hutton succeeded him in his school at Jesmond. 
 
 It would appear that between his being the pupil and the successor of 
 Mr. Ivison, Hutton had worked for some time (probably not long) as a 
 miner at Old Long Benton colliery. 
 
 Mr. Hutton’s school at Jesmond soon increased so much that he was 
 obliged to remove toa larger room in the neighbourhood. 
 
 While conducting with such success his village seminary, he attended 
 in the evenings the school ofa Mr. James, at Newcastle, to prosecute his 
 studies in mathematics; and Mr. James some time after declining his 
 school, Mr. Hutton embraced the opportunity of settling in Newcastle as a 
 teacher. In that town, the metropolis of the northern counties, his suc- 
 cess was very great: and though his previous associates had been chiefly 
 among the humbler classes of society, his manners, as well as his talents, 
 rendered him acceptable as a private teacher in the families of the higher 
 classes. Among others, he had for his pupils the late Lord Chancellor 
 Eldon, and his lady, who was the daughter of a wealthy banker in New- 
 castle. 
 
 While in that town he published his Arithmetic, his Mensuration, and 
 his Tract on the principles of Bridges; and he made for the corpora- 
 tion a survey and plan of the town. He became also a leading writer in 
 the Ladies’ and Gentleman’s Diaries, and other scientific periodicals of 
 the day. - 
 
 On the death of Mr. Cowley, professor of Mathematics in the Royal 
 Military Academy at Woolwich, Mr. Hutton offered himself as a candidate 
 for the situation ; and after an examination, which lasted several days, the 
 
viii PREFACE. 
 
 examiners (Bishop Horsley, Dr. Makeslyne, and Col. Watson) unanimously 
 recommended him as preferable to all the other candidates, and peculiarly 
 well qualified to fill the situation, and he received his appointment accord- 
 ingly on May 24th, 1773. 
 
 Soon after his settlement at Woolwich he was elected a Fellow. of the 
 Royal Society ; and at a later period he received the degree of LL.D. 
 from the University of Edinburgh. 
 
 The Stationers’ Company appointed him general Editor of all their 
 Almanacs, except the Ladies’ Diary and Poor Robin, and he held the 
 appointment for forty-six years. : . 
 
 The editorship of the Ladies’ Diary afforded him an opportunity of 
 becoming acquainted with the talents and acquirements of many ingenious 
 individuals, who were improving themselves in science by endeavouring 
 to solve the mathematical questions proposed in the Diary; and as oppor- 
 tunity occurred, many of them were drawn by his kind discrimination from 
 obscurity, and placed in situations in which they have been eminently 
 useful to society. Indeed it has been justly said, that ‘ of this class of men 
 he was eminently the patron.” ; 
 
 After filling with distinguished ability the situation at Woolwich for 
 thirty-four years, he was permitted, at his own request, to retire; and the 
 Board of Ordnance assigned him a pension of £500 per annum, in testi- 
 mony of regard for his long and faithful services. 
 
 He settled in London, and enjoyed for the last sixteen years of his life the 
 society of all the leading men distinguished for science and worth in the 
 metropolis. 
 
 He died on January 27th, 1823, and was buried in the family vault, in 
 the Churchyard of Charlton, near Woolwich. 
 
 For a full account of the various scientific labours of Dr. Hutton, and of 
 the peculiarities by which he was distinguished as a teacher of science, we 
 must refer our readers to a memoir of him by his friend, and eventual suc- 
 cessor in the chair at Woolwich, Dr. Olinthus Gregory, published in th 
 Imperial Magazine for March 1823. 
 
 Both Editions having been long out of print, the present Editor was 
 induced to undertake the superintendance of a new one; in which, by omit- 
 ting what appeared trifling or of doubtful utility, and introducing in its 
 stead a popular account of the more interesting discoveries in modern 
 science, the work might continue to be to the present generation a useful 
 manual of Scientific Recreation, as its predecessors have been to the 
 generation which has passed. 
 
 Greenwich Hospital, 
 14th September, 1840. 
 
CONTENTS. 
 
 PART FIRST — ARITHMETIC. 
 
 CHAPTER I. 
 
 Or our Numerical System, and the different. kinds of Arithmetic . 
 
 CHAPTER II. 
 Of some Short Methods of performing Arithmetical Operations 
 
 CHAPTER III. 
 Of certain Properties of Numbers 
 
 CHAPTER IV. 
 Of Figurate Numbers Ae ae Se 
 
 CHAPTER V. 
 Of Right-angled Triangular Numbers... ae ; 
 
 CHAPTER VI. 
 Some curious problems respecting Square and Cube Numbers 
 
 CHAPTER VII. 
 Of Arithmetical and Geometrical Progressions, and of certain problems which 
 depend upon them cs se bn 
 
 CHAPTER VIII. 
 Of Combinations and Permutations x ee ve 
 
 CHAPTER IX. 
 
 Application of the doctrine of Combinations to games of chance and proba- 
 bilities ee ‘ eo aa ee 
 
 CHAPTER X. 
 _ Arithmetical Amusements in Divination and Combinations 
 
 CHAPTER XI. 
 Curious Arithmetical problems Acs we “ 
 
 CHAPTER XII, 
 Of Magic Squares... Le ap 
 
 CHAPTER XIII. 
 Political Arithmetic .. AS Ap a ve 
 
 Page 
 
 13 
 
 2] 
 
 24 
 
 28 
 
 31 
 
 41 
 
 48 
 
 62 
 
 82 
 
 O4 
 
x CONTENTS. 
 
 PART SECOND — GEOMETRY. 
 
 Containing a series of Geometrical problems and questions calculated for 
 
 exercise and amusement 
 
 Collection of useful tables of Lengths af Capacities vane in ancient an 
 
 modern times by: yh re < 
 
 PART THIRD — MECHANICS. 
 
 Containing v various problems «« °° 
 
 Of the Perpetual Motion 
 
 Historical account of some celebrated Reehaniels Works 
 Of the Steam Engine at ee 
 
 Balloons, Telegraphs, &c. : 
 
 Table of the Specific Gravities of nitiereat hades 
 
 Table of Weights, ancient and modern, as compared with ie Eugustan 
 
 Troy Pound ve oe : x 
 
 PART FOURTH — OPTICS. 
 
 Containing many curious probiers oe * 
 Camera Obscura 
 
 Camera Lucida i AD a oe 
 Plane Mirrors Ss a .s xc 
 Spherical Mirrors 
 
 Burning Mirrors es oe 
 Of Telescopes, both a ees aa Fedectian oa 
 Of Microscopes A ade oe ve 
 Of the Magic Lantern =e 
 Of Colours, and the Refrangbility of Light +e 
 Of the Rainbow oe oe ws 
 
 Of certain Optical Illusions 
 
 Of some curious phenomena in Colours and Vision! 
 Account of the most curious Microscopical Observations 
 On the Fixed Lines in the Prismatic Spectrum 
 
 Various properties of Solar Light “6 ce 
 Photogenic Drawing aie rh oe 
 Daguerréotype . vs 
 On the Infiexion and Diftrartion ae Lights ae 
 On the Polarization of Light .. oe ee 
 
 PART FIFTH — ACOUSTICS. 
 
 Containing every thing most curious in the science 
 Of Sound, with experiments fe 
 
 Of Echoes,—-how produced; account of most remarkable is 
 
 Experiments on the Vibration of Musical Strings 
 
 Method of adding, subtracting, multiplying, and dividing ‘Meet 
 
 Of the Resonance of Sonorous Bodies, with experiments 
 Of the different Systems of Music 
 
 Musical Paradoxes .. ay 
 
 On the cause of the pleasure arising fom ‘Music 
 
 ee 
 
— 
 
 CONTENTS. 
 On the properties of certain Instruments ee 
 Application of Music to a question in Mechanics 
 Method of improving Barrel Instruments e3 oe 
 On the Harmonica 2 
 On the Figures formed by light Pedies on noVvibrating Sarfacds oe 
 
 PART SIXTH — ASTRONOMY AND GEOGRAPHY. 
 
 Easy, curious, and useful problems ais is 4 a be 
 To find the Meridian Line of any place ee a rs oe 
 To find the Latitude of any place a re ‘2 e oe 
 
 Methods of finding the Longitude ; i 
 
 Table of Latitudes and Longitudes of Places oe 
 
 Various Astronomical problems oe : 
 
 To measure a Degree of a great circle .. 
 
 Of the Figure of the Earth 
 
 To find the distances between places on ihe Earth ots graphic aToiecuon 
 
 Itinerary Measures, ancient and modern ee ee 
 To represent the Terrestrial Globe ona plane... 
 On Refraction A ee Se oe oe 
 On Eclipses oh ee 
 To observe an Eclipse a the Moan “iy oe se 
 Face of the Moon, from the latest observations «-. 
 To observe an Eclipse of the Sun ee ee ais 
 Remarkable Phenomena of the Solar Eclipse of May 15th, 1836 oe oe 
 To measure the Height of Mountains .. oe : ats 
 - Table of Difference of Level .. wh vs 
 On the Constellations, and the method of Pao rine them oe RAT 
 View of the principal facts in Physical Astronomy on oe 
 Of the Sun, and the different Planets and Satellites ite Se 
 Of Comets : oe 
 Of the Fixed Stars, Double Stare, aiid Nehile ae i 
 Of Chronology VW OR Geis 
 To find the Bissextile ae. Golden Nahe tna Bpacts AY ae 
 To find the Moon’s Age a if oo) 
 Solar Cycle, Dominical Letter, &c. a9 P. a an 
 To find Easter-day, and the other Moveable Recs ue is ve 
 Lunar. Years, Calends, Nones, and Ides ee és ae 
 Julian and other Chronological Periods oe 
 ~ Lunar and other Cycles, corresponding to Ricee Shes af ay Christian era, 
 to find their places in the Julian Period Be 
 Epochs and Periods celebrated in history ee oe 
 Table of the most remarkable Epochs, Atras, and Hyents ee oh 
 Eminent British Philosophers = oe a de 
 
 Tables useful in Chronological Gemateione o* 
 
 PART SEVENTH — DIALLING. 
 
 Containing the most useful and interesting problems ee es Be 
 To find a Meridian Line ee “a ve 5 He 
 Methods of constructing an Equinoctial Dial i se we is 
 Methods of constructing Vertical Dials aS ee oe 
 
Xl CONTENTS. 
 
 Page 
 To trace out a Dial on any plane whatever . a A vat eES 
 Construction of various sorts of Dials .. ; 4 as «ee 
 Portable Dials, on a quadrant, a card, a ring, Re. Ay = .. 524 
 How the shadow of a Sun-dial might go backwards or oe 53 5828 
 Of the different kinds of Hours é ve os os ao 
 General method of describing Sun-dials ¢ on any plane “ 7 .. 534 
 
 PART EIGHTH — NAVIGATION. 
 
 Containing some of the most curious problems ale ae .. 538 
 On the Curve which a vessel describes while sailing on a given course .. 538 
 How a vessel may sail against the wind oe ws a Pees 
 Problems on the action of the Rudder *: ‘le ae pets «S| 
 On the most advantageous positions of the Sails a a a BAA 
 On the line of shortest distance between two given places = .. 545 
 On the most expeditious method of coming up with a vessel in chase ~. 547 
 
 On determining the Longitude at sea, and on the method~ of finding the , 
 Longitude on land, by observations of the transit of the moon’s limb and 
 a fixed star a1. ee e.€ ee 2.2 ee ee 548 
 
 PART NINTH — ARCHITECTURE. 
 
 Containing some curious particulars... - me oe SOD. 
 To cut a given tree into a beam of the erie) ABE ray mA os 552 . 
 Of the most perfect form ofan Arch «+> aie 7. ae creme” | 
 On diminishing the thrust of Arches .. .. 558 
 Curious problems on the construction and ° Rneeuirement of Ficors Roofs, . 
 Arches and Bridges ate ee a ve a .. 859 
 
 PART TENTH — PYROTECHNY. 
 
 Containing the most curious and amusing problems nie Ayr ont OES 
 Of Gunpowder, its composition, &c. ake ee es o« O72 
 Construction of the Cartridges of Rockets re Pe ee oh DLE 
 Brilliant Fire and Chinese Fire ee oa as wa BME tices) 
 Furniture of Rockets, of Serpents, Maroons, Stars, &c. a DOL 
 Globes, Fire-balls, &c. co Ie a ee *e3 ay eth 
 Fire of different colours A 50 ee oh t o«  D9S 
 Optical Pyrotechny .. 3 af i a Hs oa OUT 
 
 PART ELEVENTH — PHILOSOPHY. 
 
 Containing every thing most curious in Philosophy in general i ie eR OOD 
 Of Fire re 36 a 565 ap a ae -- 600 
 Of Air ny > Le an oe os - 602— 
 Of Water... ve we 36 on aie os «+ 604 
 Of Earths .. oe as a3 -» 606 
 Of the Air Pump, and Baperimehen orth it oy a? x oe OOg 
 Problems respecting Fountains, &c. —. : oy os 4 e012 
 Problems respecting Gravity, Falling Bodies: at ie as 2h 616 
 Curious problems respecting Fountains, &c. tee .. ae -- 619 
 
 On measuring the degree of Heat—Thermometers 
 
CONTENTS. 
 
 On Register Thermometers .. 
 
 Table of Boiling and Freezing points of different eoniéds 
 
 On the intense Cold at the tops of high mountains 
 
 On the Divisibility of Matter Ab 
 
 On the Velocity of a Cannon Ball, esta will Shable it is circulate 
 planet round the earth oe 4c “3 ee 
 
 Examination of a singular opinion vemnettie the Planets .. 
 
 On the probability and effects of the shock of a Comet 
 
 Of the Central Fire 
 
 Barometer — its construction ne use 
 
 The Marine and the Mountain EEO 
 
 The Sympiesometer .. oe 
 
 On the Expansibility of Air .. ee se te 
 Air Gun o ee me 
 To measure the pubatity of Rae which falls ee 
 
 Of the origin of Fountains .. a ve 
 
 On the circulation of Air in Mines 50 
 
 To measure the height of Mountains by the tavometer 
 Height of the principal Mountains on the earth 
 
 Dr. Hutton’s Rule for computing the heights of Mountains 
 Problems respecting Fountains and Sounds 
 
 On Capillary Tubes .. ie a ee ie 
 
 On Perpetual Motion oe ae . 
 -Prodigious force of Moisture to raise Foes oe “6 
 
 Papin’s Digester 5 of oe 
 
 Natural signs by which a change of Peinpeninre it in “the air may be predicted : 
 
 ‘Moisture and Dryness of the air — Hygrometer a8 
 
 On the loss of substance in the Sun by the emission of an 
 On Artificial Congelation a 
 Figures observed in Snow 
 
 On intermitting Fountains ae bc fe ev 
 Speaking and Ear Trumpets .. es Ac 
 Mechanism of Kites — Questions ht cares ae oe 
 
 The Divining Rod .. x oe ou 
 
 PART TWELFTH — MAGNETISM. 
 
 Of the Magnet and its various phenomena =e 
 Principal properties of the Magnet 
 Experiments on Magnetic Attraction 
 
 Poles of the Magnet .. 5h os 40 
 
 Direction of the Magnetic Current : 
 
 Methods of Magnetizing =. 2° 
 
 Direction of the Magnet—Declination and Variation : 
 
 Magnetism of non-ferruginous bodies .. ee De = 
 _ Diurnal Variation of Magnetic Needles .. ee <a oe 
 
 Dip of Magnetic Needles Ge oe oe 
 
 Of certain tricks performed by means Bf the Magnet oe -, 
 
 PART THIRTEENTH — ELECTRICITY. 
 Of Electricity a ote os on 3p 
 
 like a 
 
 734. 
 
X1V CONTENTS. 
 
 * 
 
 Electric Machines, deecnpeien of : 5 
 
 Electric Spark—Attraction, Repulsionasexperiments ee ae is 
 Difference between pointed and blunt bodies os Ae Sis “ 
 The Leyden Flask, and shock i ee = 7 F 
 Electric Battery ee ee ee ve a a 
 
 Magnetism produced by ibahien 
 
 Identity of Lightning and Electricity 
 
 Various experiments and speculations on Electric Biomella 
 
 To cause luminous characters to appear suddenly by Electricity .. 
 
 On the Cures said to have been performed by Electricity 
 
 Natural or Animal Electricity ~- ‘ F3 
 
 On the Modern discoveries in Wlectricity, Masncuean ani other lindrad 
 branches of Science : oP 
 
 Electrotype .. ae or oe ve sve he re. 
 
 e 
 
 PART FOURTEENTH — CHEMISTRY. 
 
 Of Salts and Acids .. we a oe 
 
 Vitriolic Acid ° a +e ee 
 
 Carbonic and Nitrous Acid... oe os 
 
 Muriatic, Fluoric, and Vegetable Acids oe 
 
 Of Alkalies — Fixed ve A ap 
 
 Volatile Alkali 30. ps oe Ae a6 
 
 Of Neutral Salts af ae ve ee ee 
 
 Of Oxygen a ee 7 ve ae 
 
 Of Affinities : . ve ak 
 
 Of Solution, Breeipitatent Effer vescence, &C. se ee ts we 
 Of Crystallization .. Be ee ate B #: 33 
 
 Various Chemical Experiments 
 Dissertation on the Philosopher's Sprlene on yeas Potabile, and on Palingenesy 
 
 Of Potable Gold se os : a rr 
 Of Palingenesy we o 
 Of the different vine of Phownarns: both Wntirak and artificial ah oi 
 Of some luminous Insects Le ae oy ie o* 
 Of some other Phosphoric Bodies : : se 4 
 Methods of making different sorts of onaal Pidsaherass as oe 
 Perpetual Lamps : ae oe Ap 
 On the physical possibility of ea ie a Pavoatank eae z a ve 
 Impossibility of procuring a perpetual Wick - .. ee ee 
 Impossibility of procuring indestructible aliment for Pe petuel Lamps— 
 pretended recipes for making indestructible oil ee oe 
 Impossibility of continually maintaining fire in a place absolutely ous oe 
 
 ERRATUM,.—P,. 550, line 7, for ‘ name,’ read ‘ value.’ 
 
MATHEMATICAL “AND PHILOSOPHICAL 
 
 RECREATIONS, 
 
 PART FIRST, 
 
 CONTAINING THE MOST CURIOUS PROBLEMS AND MOST INTERESTING 
 TRUTHS IN REGARD TO ARITHMETIC. 
 
 ARITHMETIC and geometry, according to Plato, are the two wings of the mathema- 
 tician. The object indeed of all mathematical questions, is to determine the ratios 
 of numbers, or of magnitudes; and it may even be said, to continue the comparison 
 of the ancient philosopher, that arithmetic is the mathematician’s right wing; for it 
 is an incontestible truth, that geometrical determinations would, for the most part, 
 present nothing satisfactory to the mind, if the ratios thus determined could not be 
 reduced to numerical ratios... This justifies the common practice, which we shall 
 here follow, of beginning with arithmetic. 
 
 This science affords a wide field for speculation and curious research : but in the 
 collection which we here present to the mathematical reader, we have confined our- 
 selves to what is best ealeulated to excite the curiosity of those who have a taste for 
 mathematical pursuits. 
 
 CHAPTER I. 
 
 OF OUR NUMERICAL SYSTEM, AND THE DIFFERENT KINDS OF ARITHMETIC. 
 
 Ir has been generally remarked, that all or most of the nations with which we are 
 acquainted, reckon by periods of ten-; that is to say, after having counted the units 
 from 1 to 10, they begin and add units to the ten; having attained to two tens, or 
 _ 20, they continue to add units as far as 30, or three tens; and so on, in succession, 
 _ till they come to ten tens, or a hundred; of ten times a hundred they form a thou- 
 
 sand, andsoon. Did this arise from necessity ; was it occasioned by any physical 
 
 - cause; or was it merely the effect of chance ? 
 No person, after the least reflection on this unanimous agreement, will entertain 
 _ any idea of its being the effect of chance. ‘There can be no doubt that this system 
 - derives its origin from our physical conformation. All men have ten fingers, a very 
 _few excepted, who, by some lusus nature, have twelve. The first men began 
 : B 
 
2 ARITHMETIC. 
 
 to reckon on their fingers. When they had exhausted them by reckoning the units, 
 it was necessary that they should form a first total, and again begin to reckon the 
 same figures till they had exhausted them a second time ; and so on in succession. 
 Hence the origin of tens, which being confined, like the units, to the number of the 
 fingers, could not be carried beyond it, without forming a new total, called a hun- 
 dred; then another called a thousand, and so on. 
 
 From these observations, a curious consequence may be drawn. If nature, instead 
 of ten fingers, had given us twelve, our system of numeration would have been 
 different. . After 10, instead of saying ten plus one or eleven, ten plus two or twelve, 
 we shonld have ascended by simple denominations to twelve, and should then have 
 counted twelve plus one, twelve plus two, &c., as far as two dozens; our hundred 
 would have been twelve dozens, &c. A six-fingered people would certainly have 
 had an arithmetic of this kind, which indeed would have sufficiently answered every 
 arithmetical purpose, and indeed would have been attended with some advantages, 
 which our numerical system does not possess. 
 
 In consequence of an idea of this kind, philosophers have been induced to examine 
 the properties of other numerical systems. The celebrated Leibnitz proposed one, 
 in which only two characters, 1 and 0, were to be employed. In this system of 
 arithmetic, the addition of an 0 multiplied every thing by two, as it does by ten in 
 common arithmetic, and the numbers were expressed as follow : 
 
 GRHG Th ck ete aines ¢ bain oe eso. 1 Fileven Vers suas Jalsintuc « chee 1011 
 MiG chats hide ta tee 10 Dwelvetu s tusa saoteaee 5 1100 
 Three ..... xn ab ee ons ae 11 TPhirteeh dois ies svecanen ts 1101 
 TOUTE EO ices vient «ues Aen Fourteen. 2c. ans ese 1110 
 RVC HN Vis, oh hu seed alee weet 101 Eifteen.. 0 .s/ia 5 ¥iw svat exer 1111l 
 OES ual a) gees $50 eae at epee 110 Bixteen: 96 ivediee bc aeies ser eke 
 Sevell.. seus. ae Ome ere ee! Thirty-tw0.vieesecccscesess 100000 
 Bob is, sic: os view aia we eae - 1000 Sixty-four ....++...+e00ee001,000,000 
 INTs ura os hs oe eee é owintalere weet Two thousand three huudred 
 
 epee ach div saiaia te erin te - 1010 and seventy-nine .. 100,101,001,011 
 
 As Leibnitz found in the above mode of expressing numbers some peculiar advan- 
 tages, he published, in the Memoirs of the Academy of Sciences at Berlin, rules 
 for performing, in this kind of arithmetic, the usual operations of common 
 arithmetic. But it may be readily perceived, that this new system, if introduced 
 into practice, would be attended with the inconvenience of requiring too many 
 figures: twenty would be necessary to express a number equal to about a million. 
 
 One curious circumstance in regard to this binary arithmetic must not be here 
 omitted. It serves to explain, as some pretend, a Chinese symbol, which has 
 occasioned great embarrassment to the ‘learned who have applied to the study 
 of the Chinese antiquities. This symbol, which is highly revered by the Chinese, 
 who ascribe it to their ancient emperor Fohi, consists of certain characters, formed 
 by the different combinations of a small whole line and a broken one. Father Bou- 
 set, a celebrated Jesuit, who resided some time in China as a missionary, having 
 heard of Leibnitz’ ideas, observed, that if the whole line were supposed to repre- 
 sent our 1, and the broken line our 0, these characters would be nothing else than a 
 series of numbers expressed by binary arithmetic. It is very singular, that a Chinese 
 enigma should find its (Xdipus only in Europe; but perhaps in this explanation there 
 is more of ingenuity than truth. 
 
 If the binary arithmetic of Leibnitz is entitled to no farther notice, than to be 
 classed among the curious arithmetical speculations, the case however is not the same 
 with duodenary arithmetic, or that kind which, as already observed, would have been 
 brought into use had men been born with twelve fingers. This arithmetic would 
 indeed have been as expeditious as the arithmetic now employed, and even somewhat 
 
 \ 
 
DIFFERENT KINDS OF ARITHMETIC. 3 
 
 more so; the number of the characters, which would have received an increase only 
 of two, to express ten and eleven, would have been as little burthensome to the 
 memory as the present characters, and would have been attended with advantages 
 which ought to make us regret that this system was not originally adopted. } 
 
 It is uot improbable, however, that the duodenary system would have been pre- 
 ferred, had philosophy presided at the invention; for it would have been readily seen 
 that twelve, of all the numbers from 1 to 20, is that which possesses the advantage 
 of being small, and of having the greatest number of divisors; for there are no less 
 than four divisors by which it can be divided without a ‘fraction, viz., 2, 3, 4, and 6. 
 The number 18 indeed has four divisors also ; but being larger than 12, the latter 
 deserves to be preferred for measuring the periods of numeration. The first of 
 these periods, from 1 to 12, would have had the advantage of being divisible by 2, 3 
 4,6; and the second, from 1 to 144, by 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72; 
 whereas, in our system, the first period, from one to ten, has only two divisors, 2 
 and 5; and the second, from one to a hundred, has only 2, 4, 5, 10, 20, 25, 50. It is 
 evident, therefore, that fractions would less frequently have occurred in the desig- 
 nation of numbers in that way, namely by twelves. 
 
 But what would have been most convenient in this mode of numeration, is that, 
 in the divisions and sub-divisions of measures, it might have introduced a duodecimal 
 progression, Thus, as the foot has by chance been divided into 12 inches, the inch 
 
 into 12 lines, and the line into 12 points; the pound might have been divided into 
 12 ounces, the ounce into 12 drams, and the dram into 12 grains, or parts of any 
 other denominations; the day might have been divided into 12 equal portions called 
 hours, the hour into 12 other parts, each equal to 10 minutes, each of these parts 
 12 others, and so on successively. ‘The case might have been the same in regard to 
 measures of capacity. 
 
 Should it be asked, what would be the advantages of such a division? we might 
 reply as follows. It is well known, by daily experience, that when it is necessary 
 to divide any measure into 3, 4, or 6 parts, an integer number in the measures of a 
 lower denomination cannot be found, or at least only by chance. Thus, the third or 
 the sixth of a pound averdupois does not give an exact number of ounces; and the 
 third of a pound sterling does not give an integer number of shillings. The case is 
 the same in regard to the bushel, and the greater part of the other measures of ca- 
 pacity. These inconveniences, which render calculations exceedingly complex, 
 would not take place if the duodecimal progression were every where followed. 
 
 There is still another advantage which would result from a combination of duode- 
 nary arithmetic, with this duodecimal progression. Any number of pounds, shillings, 
 and pence; of feet, inches, and lines; or of pounds, ounces, &c., being given, they 
 would be expressed as whole numbers of the same kind usually are in common arith- 
 metic. Thus, for example, supposing the fathom to consist of 12 feet, as must ne- 
 cessarily be the case in this system of numeration, if we had to express 9 fathoms 5 
 feet 3 inches and 8 lines, we should have no occasion to write 9f 5f 3i 81, but merely 
 9538 ; and whenever we had a similar number expressing any dimension in fathoms, 
 feet, inches, &c., the first figure on the right would express lines, the second inches, 
 the third feet, the fourth fathoms, and the fifth dozens of fathoms, which might be 
 expressed by asimple denomination, for example a perch, &c. In the last place, 
 
 when it might be necessary to add, or subtract, or multiply, or divide, similar quan- 
 tities, we might operate as with whole numbers, and the result would in like manner 
 express, according to the order of the figures, lines, inches, feet, &e. 
 
 It may easily be conceived how convenient this would be in practice. On this 
 account Stevin, a Dutch mathematician, proposed to adapt the subdivisions ot 
 weights and measures to our present system of numeration, by making them decrease 
 fn decimal progression. According to this plan, the fathom would have contained 
 
 B 2 
 
4 , ARIFHMETIC. 
 
 10 feet, the foot 10 inches, the inch 10 lines, &e. But he did not reflect on the in- 
 convenience of depriving himself of the advantage of being able to divide his mea- 
 sures, &c. by 3, 4, and 6, without a fraction, which is indeed of some importance. 
 On the suggestion of Borda, the centesimal division of the quadrant has been 
 adopted in France: Having divided the quadrant into 100 equal parts, called grades, 
 each grade is divided into 100 parts for minutes, and each minute into 100 parts for 
 seconds, &c. 
 Let S = any arc in the sexagesimal division, and D the same arc in the decimal 
 division ; then 
 S 90 9 9D 108 s 
 D= io 10? % po Pe Jo » and Db. 9 = s+ 9 
 Therefore, to reduce an arc expressed decimally to sexagesimals, multiply by 9, and 
 divide the product by 10. To reduce sexagesimals into decimals, to the are expressed 
 sexagesimally, add its ninth part. 
 Example.— What is the sexagesimal value of 348* 2896?. 
 34°2896 
 9 
 
 30°'86064 
 60 
 51°63840 
 60 
 38°30400 
 Answer.—-30° 51’ 38” 304. st 
 Conversely, it is required to express 30° 51’ 38” 304 decimally ? 
 
 60 | 38°304 
 60 | 51°6384 
 30°86064 
 Add 1-9th 3°42896 
 
 Answer — 3428960 
 
 On the basis of the decimal division the French have also constructed their system 
 of national measures. The distance from the pole to the equator being determined by 
 computations founded on an extensive series of trigonometrical operations; its ten — 
 millionth part, or the tenth part of a decimal angular second, equal to 39°371 English ~~ 
 inches, constitutes the metre, which is the French unit of length. From this metre ~ 
 the several measures of surface and capacity are derived, and a given measure of wa- 
 ter at its greatest concentration furnishes the standard of weights. 
 
 Whether, with its many apparent advantages, this method of division is ever likely — 
 to be generally adopted seems very doubtful. In the mean time its partial introduc- — 
 tion is productive of much inconvenience, as it not only deranges our habits, but — 
 lessens the utility of our instruments and tables, all of which have been adapted to 
 the old system. 1 
 
 It is evident that in the duodenary arithmetic, the nine first numbers might be ex- 
 pressed as usual, by the nine known characters, 1, 2, 3, &c. ; but as the period ought — 
 to terminate only at twelve, it would be necessary to express ten and eleven by sim- 
 ple characters. In this case we might choose ¢ to denote ten, and $ to denote eleven, 
 and then it is evident that, ‘ 
 
 10 would express twelve. 
 
 Wt yee ae bv ense thirteen, 
 TRE ets Li: obs . fifteen. 
 
 LBA hota tes seventeen. 
 AG oe ee ces eee twenty. 
 
 I® .ccceecorsee twenty-two. 
 
DIFFERENT KINDS OF ARITHMETIC. 
 
 5 
 Rothe ete ere ag + 5 twenty-three. 
 20 4ntver o.seeee twenty-four. 
 AD Cj: de Ba we Ss forty-eight. 
 OC) 1h eis On ee seventy-two. 
 
 100 ..........+. a hundred and forty-four. 
 
 300 ...++.+.-+-- four hundred and thirty-two. 
 1000 ........ ... one thousand seven hundred and twenty-eight. 
 2000 ...+.+....«. three thousand four hundred and fifty-six. 
 
 10,000 ............ twenty thousand seven hundred and thirty-six. 
 
 100,000, &c. ..,..... two hundred and forty-eight thousand eight hundred and .- 
 thirty-two. “ule 
 
 Thus the number denoted by the figures ¢943 would be eighteen thousand en 
 hundred and twenty-seven; for ¢000 is eighteen thousand two hundred and eighty- 
 900 is one thousand two hundred and ninety-six, 40 is forty-eight, and 3 is Tiebe 
 numbers, which if added will form the above sum. 
 
 It would be easy to form aset of rules for this new arithmetic, similar to those of 
 common arithmetic; but as it does not seem likely that this mode of calculation will 
 ever be brought into general use, we shall confine ourselves to what has been already 
 said on the subject, and only add,-that a book was printed in Germany, in which the 
 common rules of arithmetic were explained in all the systems, the binary, ternary, 
 quaternary, and so on, to the duodenary inclusively. 
 
 os 
 
 ON THE ARITHMETIC OF THE GREEKS. 
 
 The Greeks divided all their numbers, as we do ours, into periods of tens ; but for 
 the want of the happy idea of giving a local value to their numerical symbols, they 
 
 were obliged to employ thirty-six characters, most of which were derived from their 
 alphabet. 
 
 hus our digits’.....,..1, 2,.3)-4, 5, 6, 7, 8, 9; 
 Were represented by..«, B, y, 8, & s, % 1, 9, 
 And 10, 20, 30, 40, 50, 60, 70, 80, 90 
 Dye fyeuntee As fae OE nw ule eG 
 For hundreds they had p, 5, 7, v4, $, ¥% YW o, % 
 And for the thousands they had recourse again to the characters of the simple units 
 with the addition of a little dash below. Thus ,«== 1000, ,8 = 2000, &c. 
 With these characters they could express any number under 10,000, or a myriad. 
 Thus 991 was % Ga; 7382, ¢r28; and 4001, 3c. 
 In order to express myriads, they placed the letter M below the charactor repre- 
 
 senting the number of myriads they intended to indicate, as - for 10000, re for 
 
 - 43720000. This is the notation employed by Eutocius, in his commentaries on 
 Archimedes. 
 
 Diophantus and Pappus represented their myriads by the letters Mv, or more 
 simply still, by a point, placed after the number, thus 43728097 is expressed by 
 SroR. My ng % or rok. nGé. 
 
 The number 100,000,000 was the greatest extent of the Greek arithmetic; but 
 Archimedes indefinitely increased it, when he invented his system of Octates, or 
 periods of eight. He assumed 100,000,000 as a new unit, and called the numbers which 
 he formed with it numbers of the second order ; then assuming the square, cube, &c. 
 of 100,000,000 successively as a new unit, he obtained numbers of the third, fourth, 
 and higher orders. 
 
 This idea of Archimedes, we are informed by Pappus, Apollonius greatly improved; 
 
6 ARITHMETIC. 
 
 by reducing the octates to periods of fours, and dividing all numbers into orders of 
 myriads; thusthe number ........s-s.eeseeeecerses oes 1992.38 46. 2643; 
 written according to the notation of h ppollontus IB sssceess CHAK. ywrs. BY LY; 
 the first period of four to the right being units, the next myriads, the next double 
 myriads, or numbers of the second order. ‘The next would be numbers of the third 
 order, and so on indefinitely. 
 
 Having given a local value to his periods of fours, it was only necessary to 
 have done the same for single digits, to have arrived at the system in present use ; 
 and it is astonishing that he did not perceive the advantages of doing so; and the 
 ore singular, as the use of the cipher was not unknown to the Greeks, being always 
 
 Aoyed in their sexagesimal operations, when it was necessary, as in the division of 
 T circle, of which ours is still a representative, as is evident from the following 
 ¢kample : 
 
 y) Ors) iS yee 0%b9 Bo 174138" 
 
 Having given an idea of the Grecian notation for integer numbers, we next proceed 
 - to their method of representing fractions; which was by placing the denominator 
 above, and to the right of the numerator; thus 1669 represented 15. But when the 
 numerator was unity, a small dash was placed to the right of the denominator, as 7’ 
 for 4, “for }. And the fraction } had a particular character, as c or 7, Cor K’. 
 
 It now remains for us to explain the method employed by the ancients in perform- 
 ing the common rules of arithmetic with this complicated system of notation. 
 
 The examples below, of addition and subtraction, require no explanation, being 
 performed exactly as we do ours, proceeding from right to left; but to this method, 
 though so clearly the most simple, the Greeks did not constantly adhere, as there are 
 many instances which make it evident that they did both addition and subtraction 
 from left to right. 
 
 Example in Addition. Example in Subtraction. 
 wel. yD xa 847.3921 BK yyAs 93636 
 Emu 60.8400 B.yu 6 23409 
 Dr. Pree 908.2321 Clare x. 70227 
 
 In multiplication they usually proceeded from left to right, as we do in multiplica- 
 tion of algebra, and placed their successive products without much apparent order ; 
 but as each of their characters retained its own proper value wherever it was placed, 
 this want of order only rendered the addition a little more troublesome. 
 
 In the examples which follow we will mark the myriads by an m, the thousands 
 by ”, the hundreds by ”, &c., and so make the partial products in the Greek, and the 
 translation identical. With this arrangement the reader will find it extremely easy 
 to follow the work. 
 
 p y y 1” 5 3 
 p y y 1 5% 3 
 a. é Tr ] m Ue Bye 
 Y Re Gh p y sted Sle! 5” iliga 5 
 tTpyd 3” 1 59 
 B. yu yee ee: Ns = 23409 
 
 The division of the Greeks was still more intricate than their multiplication ; for 
 which reason, it seems, they generally preferred the sexagesimal division; and no ex- 
 ample is left at length by any of their writers, except in the latter form; but these 
 are sufficient to throw some light on the process they followed in the division of com- 
 mon numbers; and Delambre, in an essay subjoined to the French translation of 
 Archimedes, has accordingly supposed the following example : 
 
———— 
 
 SHORT METHODS OF COMPUTING. 7 
 
 TAB. yond Cawxy 3382™ 3” 3° 29(1” 8" 2 8 
 erBiy awry 1823 REG 
 py rx 9 150° 0.3 2 Gemiean a on. 
 PoE MU 145 84 
 8.,a 7%) x 9 4 1929 
 yu g 3 646 
 “eve 5469 
 zuvéd 5469 
 
 The example will be found, on a slight inspection, to resemble that sort of divi- 
 sion in which we divide feet, inches, and parts by similar denominations, which- 
 together with the number of characters they employed, must have rendered thistule 
 extremely laborious ; and that for the extraction of the square root must have been 
 equally difficult; the principle of which was the same as ours, except in the differ- 
 ence of the notation ; though it appears that they frequently, instead of making use cf 
 the rule, found the root by successive trials, and then squared it to prove the truth 
 of their assumption. | 
 
 CHAPTER II. 
 
 OF SOME SHORT METHODS OF PERFORMING ARITHMETICAL OPERATIONS. 
 
 SECTION I. 
 
 Method of Subtracting several Numbers from several other given Numbers, without 
 making partial Additions. 
 
 To give the reader an idea of this operation, one example will be sufficient. Let 
 it be proposed to subtract all the sums below the line at B, from all those above it 
 at A. Add, in the usual manner, all the lower figures of the first column on the 
 right, which will make 14, and subtract their sum from the 
 
 next highest number of tens, or 20. Add the remainder 6 to ee 
 the corresponding column above at A, and the sum total will be 3050 (4 
 23. Write down 8 at the bottom, and because there were here 26848 
 two tens, as before, there is nothing to be reserved or carried. 
 
 Add, in like manner, the figures of the second lower column, 2942 
 which will amount to 9, and this sum taken from 10 will anos ¢ B 
 leave 1; add 1 therefore to the second column of the upper Seer ang 
 numbers, the sum of which will be 20; write down O at the 162003 
 
 bottom, and because there were here two tens, while in the 
 
 lower column there was only one, reserve the difference, and substract it from the 
 next column of the numbers marked B before you begin to add. In the con- 
 trary case, that is to say when there are more tens in any one of the columns marked 
 B than in the corresponding column above it, the difference must be added. In the 
 last place, when it happens that this difference cannot be taken from the next column 
 below, for want of more significant figures, as is the case here in the fifth column, 
 we must add it to the upper one, and write down the whole sum below the line. By 
 proceeding in this manner, we shall have, in the present instance, 162003 for the 
 remainder of the subtraction required. 
 
 SECTION II. 
 
 Some Short Methods of performing Multiplication and Division. 
 I. Every one, in the least acquainted with arithmetic, knows, that to multiply and 
 
8 ARITHMETIC. 
 
 number by 10, nothing is necessary but to add to it a cipher; that to multiply by 
 
 100, two must be added, and so on. 
 Hence it follows, that to multiply by 5, we have only to suppose a cipher added to 
 
 the number, and then to divide it by 2. Thus, if it were required to multiply 127. 
 
 ‘by 5; suppose a cipher added to the former, which will give 1270, and then divide 
 by 2: the quotient 635 will be the product required. 
 In like manner, to multiply any number by 25, we must suppose it multiplied by 
 100, or increased by two ciphers, and then divide by 4. Thus 127 multiplied by 25, 
 will give 3175. For 127 when increased by two ciphers makes 12700, which being 
 
 divided by 4, produces 3175. 
 “ccording to the same principle, to multiply by 125, it will be sufficient to add 
 
 thyée ciphers to the multiplicand, or to suppose them added, and then to divide by 8. 
 ‘he reason of these operations may be so readily conceived, that it is not necessary 
 explain it. 
 
 II. The multiplication of any number by 11 may be reduced to simple addition, 
 | For it is evident that to multiply anumber by 11, is nothing else than to add the 
 
 number to its decuple, that is to say, to itself followed by a cipher. 
 
 Let the proposed number, for ex., be : : 5 ‘ “ 67583 
 To multiply this number by eleven, say 3 and 0 make 3; write down 3 743413 
 in the units place; then add 8 and 3, which makes 11; write down lin 
 the place of tens, and carry 1; then 5and 8 and 1 carried make 14; write down 4 in 
 the third place, or that of hundreds, and carry 1. Continue in this manner, adding 
 every figure to its next following one, till the operation is finished, and the product 
 will be 743413, as above. 
 
 The same number may be multiplied in like manner, by 111, if we first write down 
 
 the 3, then the sum of 8 and 3, then that of 5, 8, and 3, then that of 7, 5, and8, and ~ 
 
 so on, adding always three contiguous figures together 
 
 III. We shall only farther observe. that to multiply any number by 9, simple sub- 
 traction may be employed. Let us take, for example, the same number as 
 before, . ; : 4 ; : : ;. 2 ; . s : 67583 
 
 608247 
 To multiply this number by 9, nothing is necessary but to suppose a cipher added to 
 the end of it, and then to subtract each figure from that which precedes it, beginning 
 at the right. Thus 3 from 0 or 10, leaves 7; 8 from 2 or 12, leaves 4; and if we 
 continue in this manner, taking care to borrow 10 when the right-hand figure is ‘too 
 small to admit of the preceding one being subtracted from it, we shall find the pro- 
 duct to be 608247. : 
 
 The reason of these operations may be readily perceived. For itis evident, that 
 in the first we only add the number itself to its decuple ; and in the latter ae sub- 
 tract it from its decuple. But in order to form a clearer idea of the matin it ma 
 perhaps be worth while to perform the operation at full length. Y 
 
 Concise operations of a similar kind may be employed in certain cases of division: 
 as in dividing, for example, a given number by any power whatever of 5, Thus if 
 it were required to divide 128 by 5; we must double it, which will give 256; if a 
 then cut off the last figure, which will be a decimal, the quotient will be 25:6 or 
 25%; ‘To divide the same number by 25, we must quadruple it, which will give 512; 
 and if we then cut off the two last figures as decimals, we shall have for the atone 
 o and yyy. To divide by 125, we must multiply the dividend by 8, and cut off tues 
 figures. In like manner we may divide a given number by any other power of 5; 
 
 but it must be confessed that such short methods of calculation are attended with no 
 great advantage. 
 
\ SHORT METHODS OF COMPUTING, 9 
 
 SECTION III. 
 
 Short Method of performing Multiplication and Division by Napier’s Rods 
 
 or Bones. 
 
 When large numbers are to be multiplied, it is evident that the operation might be 
 performed much more readily, by having a table previously formed of each number of 
 the multiplicand, when doubled, tripled, quadrupled, and so on. Such a table indeed 
 might be procured by simple addition, since nothing would be necessary but to add 
 any number to itself, and we should have the double; then to add it to the double, 
 and we should have the triple, &c. But unless the same figure should frequcatly 
 recur in the multiplicand, this method would be more tedious than that which we 
 wished to avoid. 
 
 The celebrated Napier, the sole object of whose researches seems to have been to 
 shorten the operations of arithmetic and trigonometry, and to whom we are indebted 
 for the ingenious and ever-memorable invention of logarithms, devised a method of 
 forming a table of this kind in a moment, by means of certain rods, which he has 
 described in his work entitled Rabdologia, printed at Edinburgh in 1617, The con- 
 struction of them is as follows: | 
 Fig. 1. Provide several slips of card or ivory or metal rods, about nine times as 
 long as they are broad, and divide each of them into 9 equal squares. (Fig. 1.) 
 Inscribe at the top, that is to say in the first square of each slip or rod, one 
 of the numbers of the natural series 1, 2, 3, 4, &c., as far as 9 inclusively. 
 Then divide each of the lower squares into two parts by a diagonal, drawn 
 from the upper angle on the right hand to the lower one on the left, and in- 
 scribe in each of these triangular divisions, proceeding downwards, the double, 
 triple, quadruple, &c. of the number inscribed at the top; taking care, when 
 the multiple consists of only one figure, to place it in the lower triangle, and 
 when it consists of two to place the units in the lower triangle, and the tens 
 in the upper one, as seen in the figure. It will be necessary to have one of 
 these slips or rods, the squares of which are not divided by a diagonal, but in- 
 scribed with the natural numbers from 1 to 9. This one is called the index 
 rod. It will be proper also to have several of these slips or rods for each figure. 
 
 The rods being prepared as above, let us suppose 
 that it is required to multiply the number 6785399. 
 Arrange the seven rods inscribed at the top with the 
 figures 6785, &c., close to each other, and apply to 
 them on the left hand the index rod, or that inscribed 
 with the single figures (Fig. 2); by which means we 
 shall have a table of all the multiples of each figure 
 in the multiplicand ; and scarcely any thing more will 
 be necessary but to transcribe them. ‘Thus, for ex- 
 ample, to multiply the above number by 6; looking 
 for 6 on the index rod, and opposite to it in the first 
 square, on the right hand, we find 54; writing down 
 the 4 found in the lower triangle, and adding the 5 in the rae ine to the 4 in the 
 
 ; 6 the left, which makes 9; write down the 9, and 
 
 lower triangle of the next square on , 4 o.S fa the tower irate 
 : triangle of the same square to the 5 1n the low gt 
 
 a ei CE ean Baad in this manner, taking care to carry as in common addi- 
 Band we hall find the result to be 40712804, or the product of 6785399 multi- 
 
 plied by 6. 
 
10 ARITHMETIC. 
 
 Compound multiplication, or by several figures, may be performed in the same 
 manner, and with equal facility. Let us suppose, for example, that the same number 
 
 is to be multiplied by 839938. Write down the multiplicand, and 6785399 
 the multiplier below it in the usual manner; and as the first figure 839938 
 of the multiplier is 8, look for it in the index rod, and by adding 54283192 
 the different figures in the triangles of the horizontal column Pe. 
 
 opposite to it, the sum will be found to be 54283192, or the 61068591 
 product of the above number by 8, which must be written down. — 99356197 
 Then find the sum of the figures in the horizontal column opposite 54283192 
 
 to 3, and write the sum down as before, but carrying it one place “56993)4465262 
 farter to the left. Continue in this manner till you have gone ==> 
 through all the figures of the multiplier, and if the several partial products be then 
 added as usual, you will have the total product, as above expressed. 
 
 A similar artifice may be employed to shorten division, especially when large sums 
 are to be often divided by the same divisor. Thus, for example, if the number 
 1492992 is to be divided by 432, and if the same divisor must frequently occur, con- 
 struct, in the manner above described, a table of the multiples of 432, which will 
 scarcely require any farther trouble than that of transcribing the numbers, as may be 
 seen here on the left. 
 
 1... 489 1492992 ( 3456 
 2. 864 1296 
 
 ... 1296 969 
 4... 1728 1728 
 5 ... 2160 , A19 
 6 eos 9592 °2160 
 ” .. 8024 eset 
 8 ... 3456 2592 
 9 ... 3888 2592 
 
 0000 
 
 When this is done, it may be readily perceived, that since 432 is not contained in the 
 first three figures of the dividend, some multiple of it must be contained in the first 
 four figures, viz., 1492. To findthis multiple, you need only cast your eye on the table, 
 to observe that the next less multiple of 432 is 1296, which stands opposite to 3; write 
 down 3 therefore in the quotient, and 1296 under 1492, then subtract the former 
 from the latter, and there will remain 196, to which if you bring down the next 
 figure of the dividend, the result will be 1969. By casting your eye again on the 
 table, you will find that 1728, which stands opposite to 4, is the greatest multiple of 
 432 contained in 1969; write down 4 therefore in the quotient, and subtract as be- 
 fore. By continuing the operation in this manner, it will be found that the following 
 figures of the quotient are 5 and 6; and as the last multiple leaves no remainder, the 
 division is perfect and complete. 
 
 Remark.—Mathematicians have not confined themselves to endeavouring to simplify 
 the operations of arithmetic by such means: they have attempted something more, 
 and have tried to reduce them to mere mechanical operations. The celebrated Pas- 
 cal was the first who invented a machine for this purpose, a description of which 
 may be seen in the fourth volume of the Recueil des Machines preséntées 4 1’ Acadé- 
 mie. Sir Samuel Morland, without knowing perhaps what Pascal had done in this 
 respect, published, in 1673, an account of two arithmetical machines which he in- 
 vented, one of them for addition and subtraction, and the other for multiplication, 
 but without explaining their internal construction. The same object engaged the 
 attention of the celebrated Leibnitz about the same time; and afterwards that of 
 
PALPABLE ARITHMETIC. ll 
 
 the marquis Poleni. A description of their machines may be seen in the Theatrum 
 Arithmeticum of Leupold, printed in 1727, together with that of a machine invented 
 by Leupold himself, and also in the Miscell. Berol. for 1709. We have likewise the 
 Abaque rabdologique of Perrault, in the collection of his machines published in 1700. 
 It serves for addition, subtraction, and multiplication. The Recueil des Machines 
 présentées a l’ Académie Royale des Sciences contains also an arithmetical machine, 
 by Lespine, and three by Boistissandeau. Finally, Mr. Gersten, professor of mathe- 
 matics at Giessen, transmitted, in the year 1735, to the Royal Society of London, a 
 minute description of a machine of the same kind, invented by himself. We shall 
 not enlarge further on this subject, but proceed to give an account, which we hope 
 will be acceptable to the curious reader, of an ingenious method of performing +he 
 operations of arithmetic, invented by Dr. Saunderson, a celebrated mathematician, 
 who was blind from his infancy. 
 
 SECTION IV. 
 
 Palpable Arithmetic, or a method of performing arithmetical operations, which may be 
 practised by the blind, or in the dark. 
 
 What is here announced may, on the first view, appear to be a paradox; but it is 
 certain that this method of performing arithmetical operations was practised by the 
 celebrated Dr, Saunderson, who, though he had lost his sight when a child of a year 
 old, made so great progress in the mathematics, that he was elected to fill the professor’s 
 chair of that science, in the university of Cambridge. ‘The apparatus he employed, 
 to supply the deficiency of sight, was as follows: 
 
 Fig. 3 Let the square (Fig. 3.) be divided into four other squares, by two 
 * _ Jines parallel to the sides, and intersecting each other in the centre. 
 These two lines form with the sides of the square four points “of inter- 
 &—+;—4_ section, and these added to the four angles of the primitive square, 
 give altogether nine points. If a hole be made in each of these 
 points, into which a pin or peg can be fixed, it is evident that there 
 will be nine distinct places for the nine simple and significant figures of our arithme- 
 tical system, and nothing further will be necessary but to establish some order in 
 which these points or places, destined to receive a moveable peg, ought to be counted. 
 To mark 1, it may be placed in the centre; to express 2, it may be placed immedi- 
 ately above the centre; to express 3, at the upper angle on the right ; and so on in 
 succession, round the sides of the square, as marked by the numbers opposite to each 
 oint. 
 F But there is still another character to be expressed, viz., the 0, which in our arith- 
 metic is of very great importance. This character might be expressed in a manner 
 exceedingly simple, by leaving the holes empty ; but Saunderson preferred placing 
 inthe middle one a large-headed pin, unless when having unity to express, he was 
 obliged to substitute in its stead a small-headed pin. By these means he obtained the 
 advantage of being better able to direct his hands, and to distinguish with more 
 ease, by the relative position of the small-headed pins, in regard to the large one in 
 the centre, what the former expressed. This method therefore ought to be adopted ; 
 for Saunderson no doubt made choice of those means which were most significant to 
 his fingers. 
 
 As the reader has here seen with what ease a simple number may be expressed in this 
 manner, we shall now shew that a compound number may be expressed with equal 
 facility. If we suppose several squares to be constructed like the preceding, ranged 
 in a line, and separated from each other by small intervals, that they may be better 
 distinguished by the touch, any person acquainted with common arithmetic may 
 
 2S 
 
 6) Ss 
 
| 
 12 ARITHMETIC. f 
 { 
 
 Fig. 4. perceive, that the first square on the right will serve 
 
 ; - to express units; the next towards the left to ex- 
 
 HE {4 4. HH press tens ; the third to express hundreds, &c. Thus 
 
 3 8 0 Ss the four squares, with the pegs arranged as represented 
 (Fig. 4.) will express the number 3805. 
 
 If you therefore provide a board, or table, divided into several horizontal bands, 
 
 on each of which are placed seven or eight similar squares, according to circum- 
 
 Fig. 5. stances ; if these bands be separated by proper inter- 
 
 at vals, that they may be better distinguished; and if 
 
 | HE Hy co 271 all the squares of the same order, in each of the 
 
 2 bands, be so arranged, as to correspond with each 
 H+ HH LH 407 other ina perpendicular direction; you may perform, 
 
 by means of this machine, all the different opera- 
 
 4 LH = 953 tions of arithmetic. The reader will find (Fig. 5.) 
 
 ee —_— a representation of the method of adding three num- 
 ce LH LH HY 163] bers, and expressing their sum by a machine of this 
 
 - : —— kind. | 
 
 Saunderson employed this ingenious machine, not only for arithmetical operations, 
 but also for representing geometrical figures, by arranging his pins in a certain order, 
 and extending threads from the one to the other. But what has been said is suffi- 
 cient on this subject ; those persons who are desirous of farther information respecting 
 
 it, may consult Saunderson’s Algebra, or the French translation of Wolff’s Elements 
 Abridged, where this palpable arithmetic 1s explained at full length. 
 
 PROBLEM. 
 
 To multiply £11. 11s. 11d. by £11. 11s. 11d. 
 
 This problem was once proposed by a sworn accountant to a young man who had 
 been recommended to him as perfectly well acquainted with arithmetic. And indeed, 
 besides the difficulty which results from the multiplication of quantities of different 
 kinds, and from their reduction, it is well calculated to try the ingenuity of an arith- 
 metician. But it is not improbable that the proposer would have been embarrassed 
 by the following simple question: What is the nature of the product of pounds shillings 
 and pence multiplied by pounds shillings and pence? We know that the product of a 
 yard by a yard represents a square yard, because geometricians have agreed to give 
 that appellation to asquare surface one yard in length and one in breadth ; and 6 
 yards multiplied by 4 yards make 24 square yards; for a rectangular superficies 6 yards in 
 length and 4 in breadth, contains 24 square yards, in the same manner as the product 
 of 4 by 6 contains 24 units. But who can tell what the product of a penny by a 
 penny is, or of a penny by a pound ? 
 
 The question considered in this point of view, is therefore absurd, though ordinary 
 arithmeticians sometimes are not sensible of it. . 
 
PROPERTIES OF NUMBERS. _ 23 
 
 CHAPTER III. 
 
 OF CERTAIN PROPERTIES OF NUMBERS. 
 
 WE do not here mean to examine those properties of numbers which engaged so much 
 the attention of the ancients, andin which they pretended to find so many mysterious 
 virtues. Every one, whose mind is not tinctured with the spirit of credulity, must 
 laugh to think of the good canon of Cezene, Peter Bungus, collecting in a large 
 quarto volume, entitled De Mysteriis Numerorum, all the ridiculous ideas which 
 Nichomachus, Ptolemy, Porphyry, and several more of the ancients, childishly proga- 
 gated respecting numbers. How could it enter the minds of reasonable beings, to 
 ascribe physical energy to things entirely metaphysical? For numbers are mere 
 conceptions of the mind, and consequently can have no influence in nature. 
 
 None therefore but peopie of weak minds can believe in the virtues of num- 
 bers. Some imagine, that if thirteen persons sit down at the same table, one of them 
 will die in the course of the year; but there is a much greater probability that one 
 will die if the number be twenty-four. 
 
 I. The number 9 possesses this property, that the figures which compose its 
 multiples, if added together, are always a multiple of 9; so that by adding them, and 
 rejecting 9 as often as the sum exceeds that number, the remainder will always 
 be 0. This may be easily proved by trying different multiples of 9, such as 18, 27, 
 36, &c. © 
 
 This observation may be of utility, to enable us to discover whether a given num- 
 ber be divisible: by 9, for in all cases, when the figures which express any number, 
 on being added together, form 9, or one of its multiples, we may be assured that the 
 number is divisible by 9, and consequently by 8 also. 
 
 But this property does not exclusively belong to the number 9; for the number 3 
 has a similar property. Ifthe figures which express any multiple of 3 be added, we 
 shall find that their sum is always a multiple of 3; and when any proposed number is 
 not such a multiple, whatever the sum of the figures by which it is expressed exceed 
 a multiple of 3, will be the quantity to be deducted from the number, in order that it 
 may be divisible by 3 without a remainder. 
 
 We must not omit to take notice here, of a very ingenious observation of the 
 author of the History of the Academy of Sciences, for the year 1726, which is, that 
 if a system of numeration, different from that now in use, had been adopted, that for 
 example of duodecimal progression, the number eleven, or, in general, that preceding 
 the first period, would have possessed the same property as the number nine does in 
 our present system of numeration. By way of example,-let us take a multiple of 
 eleven, as nine hundred and fifty-seven, and let us express it according to that system 
 by the characters 7 6 5: it will here be seen that 7 and ¢ make seventeen, and 5 
 added makes twenty-two, which is a multiple of eleven. 
 
 This property of 9 and 3, in the decimal notation, admits of a very simple proof, 
 For let a be the digit in the unit» place, b, c, d, &c. those in the place of tens, hun- 
 dreds, &c.; then the number will be represented analytically by 1000 d+ 100 c+ 10 
 b+-a; orby 999+ 1.d+99+ 1. c+9-1.6-+a; or by 999 d+-99 c+-9 b+ 
 d+tc-+b+-a. But 999d+-99 c-+-9b is divisible both by 9 and 3; therefore, if the 
 whole number represented by 1000 d-++- 100 c +-10 5-+-a be divisible by 9 or 3, the 
 remaining part, d-++ c+ 5 -a, must also be divisible by 9 or 3. And a like proof 
 would apply to the digit and its factors preceding the last digit of the first period, in 
 any system of numeration. 
 
14 ARITHMETIC. 
 
 In addition to the foregoing observations of the French author, may be added the 
 following remarks on the same subject, lately made by an ingenious English gentle- 
 man. He first expresses all the products of 9 by the other figures, in the foltowing 
 manner, and then enumerates the curious properties. 
 
 9 -°-6+3=9 
 i 8 
 9..9 72...7+2=9 
 9 9 
 18 .21-+-8= 81..8+1=9 
 27..,24+-7 29 
 36-.8+6=—9 
 45..4+5=9 
 54..5+4=9 
 ae 
 63... 
 
 The component figures of the product, made by the multiplication of every digit 
 into the number 9, when added together, make NINE. 
 
 The order of those component figures is reversed, after the said number has been 
 multiplied by 5. 
 
 The component figures of the amount of the multipliers (viz. 45), when added 
 together make NINE. 
 
 The amount of the several products, or multiples of 9 (viz. 405), when divided 
 by 9, gives for a quotient 45 ; that is 4-+-5 = nine. 
 
 The amount of the first product (viz. 9), when added to the other products, whose 
 respective component figures make 9, is 81; which is the square of NINE. 
 
 The said number 81, when added to the aioypnenen amount of the several 
 products, or multiples of 9 (viz. 405), makes 486; which, if divided by 9, gives for 
 a quotient 54; that is, 5 --+-4= NINE. 
 
 It is also observable that the number of changes that may be rung on nine bells, is 
 362880 ; which figures, added together, make 27; that is, 2 +7 = NINE. | 
 
 And the quotient of 362880, divided by 9, is 40320 ; that is, 4+ 0+3+24+0= 
 NINE. 
 
 II. Every square number necessarily ends with one of these figures, 1, 4, 5, 6, 9; 
 or with an even number of ciphers preceded by one of these figures. This may be 
 easily proved, and is of great utility in enabling us to discover when a number is not 
 a square; for though anumber may end as above mentioned, it is not always however 
 a perfect square; but, at any rate, when it does not end in that manner, we are cer- 
 tain that it is not a square, which may prevent useless labour. In regard to cubic 
 numbers, they may end with any figure whatever; but if they terminate with ciphers, 
 they must be in number either three, or six, or nine, &c. 
 
 If a square number end with 4, the last figure but one will be even, as in 64, 144, 
 and 97344. 
 
 If a square number end with 5, it will end with 25; as 625, 1225. 
 
 If a square number end with an odd figure, the last figure but one will be even, 
 as 81529. But if it end with any even digit, except 4, the last figure but one will 
 be odd, as 36, 576, 13456. | 
 
 No square number can end with two even digits except two ciphers, or two oe 
 as 100, 144, 40000, 44944. 
 
PROPERTIES OF NUMBERS. 15 
 
 A square number cannot end in three equal digits, except they be three fours; nor 
 in more than three equal digits unless they be ciphers. 
 
 Il. Every square number is divisible by 8, or becomes so when diminished by 
 unity. This may be easily tried on any square number at pleasure. Thus 4 less ], 
 16 less 1, 25 less 1, 121 less 1, &c. are all divisible by 3; and the case is the same 
 with other square numbers. 
 
 Every square number is divisible also by 4, or becomes so when diminished by 
 unity. This may be proved with the same case as the former. 
 
 Every square number is divisible likewise by 5, or becomes so when increased, or 
 else diminished by unity. Thus, for example, 36 — 1, 49-+-1, 64-+1, 81—1, &c., 
 are all divisible by 5. 
 
 Every odd square number is a multiple of 8 increased by unity.. We have examples 
 of this property in the numbers 9, 25, 49, 81, &c.; from which if 1 be deducted the 
 remainders will be divisible by 8. 
 
 If a square number be either multiplied or Auded by a square, the product or the 
 quotient will be a square. 
 
 If a number be not a complete square, its square root cannot be represented either 
 by an integer, or by a rational fraction, either proper or improper. 
 
 IV. Every number is either a square, or divisible into two, or three, or four 
 squares. Thus 30 is equal to 25-+4-+1; 31=25+-4-+-1-+-1; 33=16+16-+1; 
 63 = 49 +-9+- 4-41, or 86= 25+ 1-+1. 
 
 - We shall here add, by anticipation, though we have not yet informed the reader 
 what triangular, or pentagonal, &c., numbers are, that 
 
 - Every number is either triangular, or composed of two or of three triangular num- 
 bers. And that 
 
 Every number is either pentagonal, or composed of two, or three, or four, or five 
 pentagonals, and so of the rest. 
 
 We shall add also, that every even square, after the first square 1, may be resolved 
 at least into four equal squares; and that every odd square may be resolved into 
 three, ifnotintotwo. Thus 81=36- 36+ 9; 121=81-+ 86+-4; 169= 144-++ 
 25; 625 = 400-144-481. 
 
 V. Every power of 5, or of 6, necessarily ends with 5 or with 6. 
 
 VI. If we take any two numbers whatever ; then either one of them, or their sum, 
 or their difference, is necessarily divisible by 3. Let the numbers assumed be 20 and 
 17; though neither of these numbers, nor their sum 387, is divisible by 3, yet their 
 difference is, for it is three. 
 
 It might easily be demonstrated, that this must mecepeatily be the case, whatever 
 be the numbers assumed. 
 
 _ VII. Iftwo numbers are of sucha nature, that their squares when added together 
 form a square, the product of these two numbers is divisible by 6. . 
 
 Of this kind, for example, are the numbers 3 and 4, the squares of which, 9 and 
 16, when added, make the square number 25: their product 12 is divisible hy 6. 
 
 From this property a method may be deduced, for finding two numbefs, the squares 
 of which, when added together, shall form a square number. For this purpose, mul- 
 tiply any two numbers together; the double of their product will be one of the 
 numbers sought, and the difference of their squares will be the other. 
 
 Thus if we multiply together 2 and 3, the squares of which are 4 and 9, their pro- 
 duct will be 6; if we then take 12 the double of this product, and 5 the differ- 
 ence of their squares, we shall have two numbers, the sum of whose squares is equal 
 to another square number ; for these squares are 144 and 25, which when added make 
 169, the square of 13. 
 
 VIII, When two numbers are such, that the difference of their squares is a square 
 
16 ARITHMETIC. fl 
 
 number; the sum and difference of these numbers are themselv es square numbers, 
 or the double of square numbers. 
 
 Thus, for example, the numbers 13 and 12, when squared, give 169 and 144, the 
 difference of which 25, is also a square number; then 25, the sum of gen numbers, 
 is a square number, and also their difference 1. 
 
 In like manner, 6 and 10, when squared produce 36 and 100, the difference of 
 which 64 is also a square number; then it will be found, that their sum 16 is a square 
 number, as well as their difference 4. 
 
 The numbers 8 and 10 give for the agerooce of their squares 36; and it may be 
 readily seen, that 18, the sum of these numbers, is the double of 9, which is a 
 square number, and that their difference 2 is the double of 1, which is also a square 
 number. 
 
 IX. Iftwo numbers, the difference of which is 2, be multiplied together, their pros 
 duct increased by unity will be the square of the intermediate number. 
 
 Thus, the product of 12 and 14 is 168, which being increased by 1, gives 169, the 
 square of 13, the mean number between 12 and 14. 
 
 Nothing is easier than to demonstrate, that this must always be the case; and it 
 will be found in general, that the product of two numbers increased by the square of 
 half their difference, will give the square of the mean number. : 
 
 X. A prime number is that which has no other divisor but unity. Numbers of 
 this kind, the number 2 excepted, can never be even, nor can any of them terminate 
 in 5, except 5 itself ; hence it follows, that except those contained in the first i 
 of ten, they must necessarily terminate in 1 or 3, or 7 or 9. 
 
 One curious property of prime numbers is, that every prime number, 2 and 3 ex- 
 cepted, if increased or diminished by unity, is divisible by 6. This may be readily 
 seen in any numbers taken at pleasure, as 5, 7, 11, 13, 17, 19, 23, 29, 31, &e.; but 
 I do not know, that any one has ever yet demonstrated this property 4 priort. But 
 the inverse of this is not true, that is, every number when increased or diminished 
 by unity is divisible by 6, is not, on that account, necessarily a prime number. ; 
 
 As it is often of utility to be able to know, without having recourse to calculation, 
 whether a number be prime or not, we have here subjoined a table of all the prime 
 numbers from 1 to 10,000. 
 
 Table of the Prime Numbers from 1 to 10,000. 
 
 Fe 163 263 373 | 479 601 719 | 853 977) 1093 | 1223 |= 
 
 2 
 3| 73| 167] 269| 379| 487| 607| 727] 857| 983 | 1097 | 12294. 
 5| 79| 173] 271] 383| 491| 613] 733] 859] 991 1231 | 
 7 83 | 179| 277) 389] 499| 617) 739] 863) 997 | 1103 | 1237 |@ 
 
 11| 89{ 181] 281| 397 619| 743| 877 1109 | 1249 | | 
 
 13| 97] 191] 283 503] 631] 751} 881 | 1009] 1117 | 1259 | 
 
 17 193| 293] 401] 509| 641] 757{ 883] 1013 | 1123 | 1277 | © 
 
 19] 101] 197 409 | 521| 643] 761] 887 | 1019 | 1129] 1279 | 
 
 23| 103] 199] 307] 419| 533] 647| 769 1021 | 1151 | 1283 
 
 29| 107 311} 421] 541] 653] 773]| 907 | 1031 | 1153 | 1289 | 
 
 31| 109] 211] 313] 431] 547] 659| 787| 911 | 1033 | 1163 | 1291 | © 
 
 37] 113| 223| 317| 433] 557| 661! 797} 919] 2039 | 1171 | 1297 | © 
 
 411) 127.1.. 997) B81 | 480.4 663)| 673 929 | 1049 | 1181 . 
 
 43 131 229 337 443 569 Oxi, 811 937 | 1051 | 1187 | 1301 
 47 137 233 347 449 571 683 821 941 | 1061 | 1193 | 1303 
 53 139 239 349 457 577 691 823 947 | 1063 1307 
 59 149 24] 353 461 587 827 953 | 1069 | 1201 | 1319 
 61 151 251 359 463 593 701 829| 967 | 1087 | 1213 | 1321 |@ 
 67 157 257 367 467 599 709 839 971 | 1091 | 1217 | 1327 
 
1361 
 1367 
 1373 
 1381 
 1399 
 
 1409 
 1423 
 1427 
 1429 
 1433 
 1439 
 1447 
 1451 
 1453 
 1459 
 1471 
 1481 
 1483 
 1487 
 1489 
 1493 
 1499 
 
 1511 
 1523 
 1531 
 1543 
 1549 
 1553 
 1559 
 1567 
 1571 
 1579 
 1583 
 1597 
 
 1601 
 1607 
 1609 
 1613 
 1619 
 1621 
 1627 
 1637 
 1657 
 1663 
 1667 
 1669 
 1693 
 1697 
 1699 
 
 1709 
 1721 
 1723 
 1733 
 1741 
 1747 
 1753 
 1759 
 1777 
 1783 
 
 1787 
 
 1789. 
 
 1801 
 1811 
 1823 
 1831 
 1847 
 1861 
 1867 
 1871 
 1873 
 1877 
 1879 
 1889 
 
 1901 
 1907 
 1913 
 1931 
 1933 
 1949 
 1951 
 1973 
 1979 
 1987 
 1993 
 1997 
 1999 
 
 2003 
 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 
 2477 
 
 2503 
 2521 
 2531 
 2539 
 2543 
 2549 
 2551 
 2557 
 2579 
 2591 
 2593 
 
 2609 
 2617 
 2621 
 2633 
 2647 
 2657 
 2659 
 2663 
 2671 
 2677 
 2683 
 2687 
 2689 
 2693 
 2699 
 
 “TABLE OF 
 
 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 
 3413 
 3433 
 3449 
 3457 
 3461 
 3463 
 3467 
 3469 
 3491 
 3499 
 
 3511 
 3517 
 3527 
 3529 
 3533 
 
 3535. 
 
 3541 
 3547 
 3557 
 3559 
 3571 
 3581 
 3583 
 3593 
 
 3607 
 3613 
 3617 
 3623 
 3631 
 
 3637 | 
 
 THE 
 
 3643 
 3659 
 3671 
 3673 
 3677 
 3691 
 3697 
 
 3701 
 3709 
 3719 
 3727 
 3733 
 3739 
 3761 
 3767 
 3769 
 3779 
 3793 
 3797 
 
 3803 
 3821 
 3823 
 3833 
 3847 
 3851 
 3853 
 3863 
 3877 
 3881 
 3889 
 
 3907 
 3911 
 3917 
 3919 
 3823 
 3929 
 3931 
 3943 
 3947 
 3967 
 3989 
 
 4001 
 4003 
 4007 
 4013 
 4019 
 4021 
 4027 
 4049 
 4051 
 4057 
 4073 
 4079 
 4091 
 4093 
 4099 
 
 4111 
 4127 
 
 PRIME NUMBERS. 
 
 4129 
 4133 
 4139 
 4153 
 4157 
 4159 
 4177 
 
 4201 
 
 ' 4211 
 
 Q 
 
 42917 
 4219 
 4229 
 4231 
 4241 
 4243 
 4253 
 4259 
 4261 
 4271 
 4273 
 4283 
 4289 
 4297 
 
 4327 
 4337 
 4339 
 4349 
 4357 
 4363 
 4373 
 4391 
 4397 
 
 4409 
 4421 
 4423 
 444] 
 4447 
 4451 
 4457 
 4463 
 4481 
 4483 
 4493 
 
 4507 
 45135 
 4517 
 
 4519 | 
 
 4523 
 4547 
 4549 
 4561 
 4567 
 4583 
 4591 
 4597 
 
 4603 
 4621 
 4637 
 
 4639 
 4643 
 4649 
 4651 
 4657 
 46638 
 4673 
 4679 
 4691 
 
 4703 
 4721 
 4723 
 4729 
 4733 
 4751 
 4759 
 4773 
 4787 
 4789 
 4793 
 4799 
 
 4801 
 4813 
 4817 
 4831 
 4861 
 4871 
 4877 
 4889 
 
 4903 
 4909 
 4919 
 4931 
 4933 
 4937 
 4943 
 4951 
 4957 
 4967 
 4969 
 4973 
 4987 
 4993 
 4999 
 
 5003 
 5009 
 5011 
 5021 
 5023 
 5039 
 5051 
 5059 
 5077 
 5081 
 5087 
 5099 
 
 5101 
 5107 
 
 5113 
 5119 
 5147 
 5153 
 5167 
 5171 
 5179 
 5189 
 5197 
 
 5209 
 5227 
 5231 
 5233 
 5237 
 5261 
 5273 
 5279 
 5281 
 5297 
 
 5303 
 5309 
 5323 
 5333 
 5347 
 5351 
 5381 
 5387 
 5393 
 5399 
 
 5407 
 5413 
 5417 
 5419 
 
 | 6451 
 
 5437 
 5441 
 5443 
 5449 
 
 | 5471 
 
 5477 
 5479 
 5483 
 
 5501 
 5503 
 5507 
 5519 
 5521 
 5527 
 5531 
 5557 
 5563 
 5569 
 5573 
 5581 
 5591 
 
 5623 
 5639 
 5641 
 5647 
 
 5651 
 5653 
 5657 
 5659 
 5669 
 5683 
 5689 
 5693 
 
 5701 
 5711 
 5717 
 5737 
 5741 
 5743 
 5749 
 5779 
 5783 
 5791 
 
 5801 
 5807 
 5813 
 5821 
 5827 
 5839 
 5843 
 5849 
 5851 
 5857 
 5861 
 5867 
 5869 
 5879 
 5881 
 5897 
 
 5903 
 5923 
 5927 
 5939 
 5953 
 5981 
 5987 
 
 6007 
 6011 
 6029 
 6037 
 6043 
 6047 
 6053 
 6067 
 6073 
 6079 
 6089 
 6091 
 
 6101 
 6113 
 6121 
 6131 
 6133 
 
 6143 
 6151 
 6163 
 6173 
 6197 
 6199 
 
 6203 
 6211 
 6217 
 6221 
 6229 
 6247 
 6257 
 6263 
 6269 
 6271 
 6277 
 6287 
 6299 
 
 6301 
 6311 
 6317 
 6323 
 6329 
 6337 
 6343 
 6353 
 6359 
 6361 
 6367 
 6373 
 6379 
 6389 
 6397 
 
 6421 
 6427 
 
 | 6449 
 
 6451 
 6469 
 6473 
 6481 
 6491 
 
 6521 
 6529 
 6547 
 6551 
 6553 
 6563 
 6569 
 6571 
 6577 
 6581 
 6599 
 
 6607 
 6619 
 6637 
 6653 
 6659 
 
18 ARITHMETIC. : 
 7177 | 7451 | 7643) 7883 ) 8147 ) 8377 ) 8627 , 8831 | 9059 - {97491 | 
 7187 | 7457 | 7649 790) 8161 | 8387 | 8629 | 8837 | 9067 | 9311 | 9511 | 9767 
 7193 | 7459 | 7669| /o0. | $167 | 8389 | 8641 | 8839 | 9091 | 9319 | 9521 | 9769 
 
 oy 94ay7 | 7673 | 8171 8647 | 8849 9323 | 9533 | 9781 
 7207 | 1451 | y681| “917 | 8179 | 8419 | 8663 | 8861 | 9103 | 9337 | 9539 | 9787 
 7211 | J 450 | 7687 | 2927 | 8191 | 8423 | 8669 | 8863 | 9109 | 9341 | 9547 | 9791 
 7213 | a489 | 7691 1933 8429 | 8677 | 8867 | 9127 | 9343 |-9551 | ooo, 
 7219 | - 499 | 7699 | 2937 | s20g9 | 8431 | 8681 | 8887 | 9133 | 9349 | 9587 ibe 
 cane 7949 | so19 | 8443 | 8689 | 8893 | 9137 | 9371 9e11 
 
 , 7237 | 7507 a2 7951 | 504 8447 | 8693 9151 | 9374 | 9601 9817 
 7243 | 7517 | 7717 | 7963 | 2551 | g461 | 8699 | 8923 | 9157 | 9391 | 9613 orae 
 7247 | 7523 | 7723 | 7993 | cose | s467 e020 | oli 4 Gane tees 9833 
 7253 | 7529 | 7727 | goo9 | 8937 8707 | 8933 | 9173 9623 ogee 
 7283 | 7537 | 7741 | 2511 | ga43 | 8501 | 8713 | 8941 | 9181 | 9403 | 9629 | 9° 
 7297 | 7541 | 7753! soy | g963 | 8513 | 8719 | 8951 | 9187 | 9413 | 9631 ie: 
 307 | 2547 | 77571 go3q | sa6g | 8521 | 8731 | 8963 | 9199 | 9419 | 9643 ee 
 7309 | 2249 | 7799 | go53 | 8273 | 8527 | 8737 | 8969 9421 | 9649 mike 
 7391 | 2229 | 7789 | eosg | sas7 | 8537 | 8741 | 8971 | 9203 | 9431 | 9661 mie 
 7331 | 7561 | 7793 | gogo | s291 | 8539 | 8747 | 8999 | 9209 | 9433 | 9677 | 9 
 7333 | 2°73 | 7817 | sosi | 8293 | 8543 | 8753 9221 | 9437 | 9679 | 9901 
 7349 | 2577 | 7¢93| sos7 | 8297 | 8563 | 8761 | 9001 | 9227 | 9439 | 9689 | 9907 
 7351 | 7°83 | 7899 | 808g 8573 | 8779 | 9007 | 9239 | 9461 | 9697 | 9923 
 7369 | 2°89 | 7g41 | 8093 | 8311 | 8581 | 8783 | 9011 | 9241 | 9463 ra 
 7393 | 7°91 | 7853 8317 | 8597 9013 | 9257 | 9467 | 9719 | 9931 
 603 | 7867 | 8101 | 8329 | 8599 | 8803 | 9027 | 9277 | 9473 | 9721 | 9941 
 v4a11 | 7607 | 7873 | 8111 | 8353 8807 | 9041 | 9281 | 9479 | 9733 | 9949 
 
 7417 | 7621 | 7877 | 8117 | 8363 | 8609 | 8819 | 9043 | 9283 | 9491 | 9739 | 9967 
 7433 | 7639 | 7879 | 8123 | 8369 | 8623 | 8821 | 9049 | 9293 | 9497 | 9743 | 9973 | 
 ernie pees, (5 0979) IN AP 1 ee ee ee eee 
 
 Eratosthenes invented what he called a seive for excluding from a series of odd 
 numbers those which are not prime. The principle of the seive is this: 
 
 Having written down in consecutive order all odd numbers, from one to any re- 
 quired extent, as és es 7 
 
 1, 3, 5, 7, 9, 11, 18, 15, 17, 19, 21, 23, 25, 27, &c. 
 We begin with three, the first prime number, and over every subsequent third in 
 the series put a point, and from 5 a point is placed over every fifth number; from 
 7 over every seventh number, and so on. 
 
 Then the numbers which remain without points are prime numbers,—and adding 2, 
 the only even prime, we obtain all the prime numbers included in the series. 
 
 Every prime number greater than 3 is of one of the forms 6n-+-1, or6” —1. 
 
 For every number is either divisible by 6, or leaves, when divided by it, a remainder 
 of 1, 2, 3, 4, or 5; that is, every number is of one of the forms 6 n,6n-+ 1,6" +2, 
 6n+3, 6n+4, or6n-+5. But the first and fifth of these are divisible by 2, and 
 the fourth is divisible by 3, and are therefore not prime. Hence all prime numbers” 
 greater than 3 are of the form 6x-+-lor6n+5. But6n-5is of the same: 
 form, or would produce the same number, as 6 n—1. For takingn =2, 6n-+-5= 
 7, and taking n=3,6n+]=17. Therefore all prime numbers above 3 are of 
 the form 6x +1, or 62—1. 
 
 But though all prime numbers are Handa’ in these two forms, they include alah 
 many numbers which are not primes, For example, if n = 4, 62-1 = 25, which 
 is not prime, andif n = 6, 6n—1= 35, which is not prime. Indeed, it may be 
 demonstrated that no algebraic formula can contain prime numbers only. 
 
 With reference to the two forms under consideration, it has been proved that 
 whenever nis of the form 6 n’ n” + n’ + n’, 62-11 is not prime; and whenever nm 
 is not of that form 6 x +1 is prime. 
 
 Also that when n is of the form 6 n’ n” +n’ n pn’, 6n—l1is not prime, while it ial 
 always prime when n is not of that form. 5 
 
PERFECT AND AMICABLE NUMBERS. 19 
 
 XI. Another kind of numbers, which possess a singular and curious property, are 
 those called perfect numbers. This name is given to every number, the aliquot parts 
 of which, when added together, form exactly that number itself. Of this we have an 
 example in the number 6; for its aliquot parts are 1, 2, 8, which together make 6. 
 The number 28 possesses the same property; for its aliquot parts are 1, 2, 4, 7, 14, 
 the sum of which is 28. 
 
 To find all the perfect numbers of the numerical progression,take the double progres- 
 sion 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, &c.; and examine those 
 terms of it, which when diminished by unity, are prime numbers. , Those to which this 
 property belongs, will be found to be 4, 8, 32,128, 8192; for these terms when diminished 
 by unity, are 3, 7, 31, 127, 8191. Multiply therefore each of these numbers by that 
 number in the geometrical progression which preceded the one from which it is 
 deduced, for example 3 by 2, 7 by 4, 31 by 16, 127 by 64, 8191 by 4096, &c.; and 
 the result will be 6, 28, 496, 8128, 33550336, which are perfect numbers. 
 
 These numbers however are far from being so numerous as some authors have 
 believed.* The following is a series of numbers either perfect, or, for want of proper 
 attention, supposed to be so, taken from a memoir of Mr. Krafft, published in the 7th 
 volume of the Transactions of the Academy of Petersburgh. Those to which this 
 property really belongs are marked with an asterisk. 
 
 * 
 
 6 
 
 * 98 
 
 * 496 
 
 * 8128 
 130816 
 2096128 
 
 * 33550336 
 536854528 
 
 * 8589869056 
 * 137438691328 
 
 2199022206976 
 
 35184367894528 
 
 562949936644096 
 
 9007199187632128 
 
 144115187807420416 
 * 2305843008139952 128 
 36893488 143124135936 ~ 
 ' Thus we find that between 1 and 10 there is only one perfect number ; that there is 
 one between 10 and 100, one be*ween 100 and 1000, and one between 1000 and 
 10000; but those would be mistaken who should believe that there is one. also 
 between ten thousand and a hundred thousand, one between a hundred thousand and 
 -amillion, &c.; for there is only one between ten thousand and eight hundred mil- 
 lions. The rarity of perfect numbers, says a certain author, is a symbol of that of 
 perfection. 
 All the perfect numbers terminate with 6 or 28. 
 XII. There are some numbers called amicable numbers, on account of a certain pro- 
 _perty which gives them a kind of affinity or reciprocity, and which consists in their 
 being mutually equal to the sum of each other’s aliquot parts. Of this kind are the 
 numbers 220 and 284 ; for the first 220 is equal to the aliquot parts of 284, viz. 1, 2, 
 4, 71, 142; and, reciprocally, 284 is equal to the aliquot parts, 1, 2, 4,5, 10, 11, 20, 
 | 22, 44, 55, 110, of the other number 220. 
 
 * The rule given by Ozanam is incorrect, and produces a multitude of numbers, such as 130816, 
 2096128, &c., which are not perfect numbers. When Ozanam wrote his rule, he did not recollect 
 
 _ that one of the multipliers must be a prime number, But 511 and 2047 are not prime numbers. 
 
 | c 2 
 
 ——<—<—_$__-— 
 
- 
 
 20 ; ARITHMETIC. 
 
 Amicable numbers may be found by the following method. Write down, as in the 
 subjoined example, the terms of a double geometrical progression, or having the ratio 
 2, and beginning with 2; then triple each of these terms, and place these triple 
 numbers each under that from which it has been formed; these numbers diminished 
 by unity, 5,11, 23, &c. if placed each over its corresponding number in the geometrical 
 progression, will form a third series above the latter. Inthe last place, to obtain the 
 numbers of the lowest series, 71, 287, &c. multiply each of the terms of the series; 
 6, 12, 24, &c., by the one preceding it, and subtract unity from the product. 
 
 a 1] 23 47 95 19] 383 
 2 4 8 16 32 64 128 
 6 12 24 48 96 192 384 
 
 71 287 1151 4607 18431 73727 
 ‘Take any number of the lowest series, for example 71, of which its corresponding 
 number in the first series, viz. 11, and the one preceding the latter, viz. 5, as well as 
 71, are prime numbers: multiply 5 by 11, and the product 55 by 4, the correspond- 
 ing term of the geometrical series, and the last product 220, will be one of the num- 
 bers required. The second will be found by multiplying the number 71 by the same 
 number 4, which will give 284. 
 
 In like manner, with 1151, 47, and 23, which are prime numbers, we may find two 
 other amicable numbers, 17296 and 18416; but 4607 will not produce any amicable 
 numbers, because, of the two other corresponding numbers, 47 and 935, the latter is 
 nota prime number. The case is the same with the number 18431, because 95 is 
 among its corresponding numbers; but the following number 73727, with 383 and 
 191, will give two more amicable numbers, 9363584 and 9437056. 
 
 By these examples it may be seen, that if perfect numbers are rare, amicable 
 numbers are much more so, the reason of which may be easily conceived. 
 
 XIII. If we write down a series of the squares of the natural numbers, viz. 1, 4, 
 9, 16, 25, 36,49, &c.; and take the difference between each term and that which 
 follows it, and then the differences of these differences; the latter will each be equal 
 to 2,as may be seen in the following example. 
 
 1 4 9 16 25 36 49 
 Ist. Diff. 3 5 7 9 11 13 
 2d. Diff. 2 2 2 2 2 
 It hence appears, that the square numbers are formed by the continual addition of the 
 odd numbers 1, 3, 5, &c., which exceed each other by 2. . 
 
 In the series of the cubes of the natural numbers, viz. 1,8, 27, &c., the third, in- 
 stead of the second differences, are equal, and are always 6, as may be seen in the 
 following example. 
 
 Cubes ] 8 27 64 125 216 
 Ist. Diff. 7 19 37 61 91 
 
 2d. Diff. 12 18 24 ae 
 
 3d. Diff. 6 6 6 
 
 In regard to the series of the fourth powers, or biquadrates, of the natural numbers, 
 the fourth differences only are equal, and are always 24. In the fifth powers, the 
 fifth differences only are equal, and are invariably 120. 
 
 All this may be readily shewn, by taking the successive differences of the cxDa a 
 terms of the series x”, c-+1", xr--2", &e., giving to n the values 2, 3, 4, Ke, 
 succession. 
 
 These differences, 2, 6, 24, 120, &c. may be found by multiplying the series of the 
 numbers 1, 2, 3, 4, 5, 6, &c. For the second power, multiply the two first; for the 
 third power, the three first, and so on. 
 
 XIV. The progression of the cubes 1, 8, 27, 64, 125, &c. of the natural numbers, 
 1, 2, 3, 4, 5, 6, &c. possesses this coniantople property, that if any number of its 
 
 a a 
 
& 
 
 SQUARES AND CUBES.—FIGURATE NUMBERS. 91 
 
 terms whatever, from the beginning, be added together, their sum will always be a 
 square. Thus, 1 and8 make 9; if we add to this sum 27, we shall nave 36, which 
 ‘is still a square number; and if we add 64, we shall have 100, and so on. 
 
 The root of each square so formed isthe sum of the roots of all the component 
 cubes. Thus 13-23-33 = 36=1-+2-+3% 
 
 XV. The number 120 has the property of being equal to half the sum of its 
 aliquot parts, or divisors, viz, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 
 which together make 240. The number 672 is also equal to half the sum of its 
 aliquot parts, 1344. Several other numbers of the like kind may be found, and 
 some even which would form only a third, or fourth, of the sum of their aliquot 
 parts, or which would be the double, triple, or quadruple of that sum; but what 
 has been here said, will be sufficient to exercise those who are fond of such 
 researches. 
 
 CHAPTER IV. 
 
 OF FIGURATE NUMBERS. 
 
 Fig. 6. Ir there be taken any arithmetical progression, as for instance, the 
 
 A most simple of all, or that of the natural numbers 1], 2, 3, 4, 5, 6, 
 
 7, &c.; and if we take the first term, the sum of the two first, that of 
 
 EN the three first, and so on; the result will be a new series of numbers, 
 
 1, 3, 6, 10, 15, 21, 28, &c. called triangular numbers, because they. 
 
 ix can always be ranged in such a manner as to form an equilateral 
 triangle, as may be seen Fig 6. 
 
 : Fig. 7. 
 : The square numbers, as 1, 4, 9, 16, 25, 36, &c. arise froma like ia 
 addition of the first terms of the arithmetical progression, 1, 3, 5, 7, 9, 
 
 11, &c., the common difference of which is 2. These numbers, as is Ht 
 
 _ well known, may be arranged so as to form square figures. Fig. 7. 
 
 A similar addition of the terms of the arithmetical progression 1, 4, HH 
 7,10, 13, &c., the common difference of which is 3, will produce the i 
 
 Fig. 8. numbers 1, 5, 12, 22, &c., which are called pentagonal numbers, 
 
 na because they represent the number of points which may be arranged on 
 a the sides and in the interior part of a regular pentagon; as may be seen 
 
 Fig. 8; where there are three pentagons, having one common angle, 
 representing the number of points which increase arithmetically; the 
 first having two points on each side, the second three, and the third 
 four; and which progression, it is evident, might be continued ever 
 so far. / 
 
 It is in this sense, and in this manner, that we must conceive the figurate num- 
 bers to be arranged. 
 
 It is almost needless to say, that the progression 1, 5, 9, 13, 17, &c., the common 
 _ difference of which is 4, produces, by a similar addition, the hexagonal numbers, which 
 are 1, 6, 15, 28, 45, &e.; and that in like manner may be found the heptagonals, the 
  octagonals, &c, 
 There is another kind of polygonal numbers, which result from the number of 
 points that can be ranged in the middle, and on the sides, of one or more similar 
 | polygons, having a common centre. These are different from the preceding ; for the 
 _ series of the triangulars of this kind is 1, 4, 10, 19, 31, &c., which are formed by the 
 | successive addition of the numbers 1, 3, 6, 9, 12. 
 _ The central square numbers are 1, 5, 13, 25, 41, 61, &c. ; formed, in like manner, by 
 the successive addition of the numbers 1, 4, 8, 12, 16, 20, &e. 
 
1 
 
 ] 
 1 
 
 The central. pentagonal numbers are 1, 6, 16, 31, 51, 76, &c.; formed by the ad- 
 dition of the numbers 1, 5, 10, 15, 20, &c. 
 
 But we shall not enlarge farther on this kind of polygonal numbers, because they 
 are not those to which mathematicians usually give that name. Let us return there- 
 fore to the ordinary polygonal numbers. : 
 
 The radix of a polygonal number is the number of the terms of the progression 
 necessary tc be added in order to obtain that number. Thus the radix of the tri- 
 angular number 21, is 6, because that number results from the successive addition 
 of the six numbers 1, 2, 3,4, 5,6. In like manner, 4 is the radix of the square 
 ‘number 16, considered as a figurate number, because that number is produced by 
 adding the four terms 1, 3, 5, 7, of the progression of the odd numbers. 
 
 Having given this explanation of the nature of polygonal numbers, we shall now 
 present the reader with a few problems respecting them. 
 
 22 ARITHMETIC. 
 
 PROBLEM I. 
 To find whether any proposed Number is Triangular, or Square, or Pentangular, &c. 
 
 The method of finding whether a number be square, is well known, and serves as 
 a foundation for discovering the other figurate numbers. This being supposed ; then 
 to determine whether any gryen number is a polygonal number, the following general 
 rule may be employed. 
 
 Multiply by 8 the number of the angles of the polygon less 2; multiply this first 
 product by the proposed number, and to the new product add the square of a num- 
 ber equal to that of theangles of the polygon less 4: if the sum be a perfect square, 
 the given number is a polygon of the kind proposed. 
 
 It may easily be seen, that as the number of the angles in the triangle are 3, in the 
 square 4, in the pentagon 5, &c., we shall have, as the multiplier of the proposed 
 number, in the case of the triangular number, 8; in that of the quadrangular num- 
 ber, 16; in that of the pentagonal, 24; and in that of the hexagonal, 32. 
 
 In like manner, as the number of the angles less 4. gives for the triangle — 1; 
 for the square 0; for the pentagon, J] ; for the hexagon, 2, &c.; the numbers to be 
 added to the product, as before mentioned, will be, for the triangle, 1 (because the 
 square of —1is1); for the square, 0; for the pentagon, 1; for the hexagon,4; for 
 the heptagon, 9; &c. From these principles we may deduce the following rules, 
 which we shall illustrate by examples. 
 
 Suppose it were required to know whether 21 be a triangular number. 
 
 Multiply 21 by 8, to the product add 1, and the sum will be 169, which isa pene 
 square: consequently 21 is a triangular raimben 
 
 If we are desirous of knowing whether 35 be a pentagonal number, we must mould 
 tiply 85 by 24, and the product will be 840; to this product if 1 be added we shall 
 have 841, which is a square number: we may therefore rest assured that 35 is a pen- 
 tagonal number. 
 
 PROBLEM II. 
 
 A Triangular, or any Figurate Number whatever, being given; to find its Radix, or 
 the Number of the Terms of the Arithmetical Progression of which it is the Sum. 
 
 First perform the operation described in the preceding problem; and having found 
 the square root, the possibility of which will indicate whether the number be figurate 
 or not, add to this root a number equal to that of the angles of the proposed poly- 
 gon less 4, and divide the sum by the double of the same number of angles less 2: 
 the quotient will be the radix of the polygon. 
 
 The number to be added is, for the triangle — 1, that is to say 1 to be deducted ; 
 for the square it is 0; for the pentagon 1; for the hexagon-2; &c. 
 
FIGURATE NUMBERS. 93 
 
 As to the divisor, it may be easily seen that for the triangle it is 2 (because the 
 double of 3 less 2 is 2), for the square 4, for the pentagon 6, for the hexagon 8, &c. 
 
 Let it be required therefore to find the radix of the triangular number 36. 
 
 Having performed the operation explained in the preceding problem, and found the 
 product 289, the square root of which is 17, subtract unity from this number, and 
 divide the remainder by 2; the quotient 8 will be the radix or side of the triangular 
 number 36. 
 
 Let the radix of the pentagonal number 35 be required. 
 
 Having found, as before, the radix 29, add to it 1, which will give 30, and divide 
 by 6; the quotient 5 will be the radix of this pentagonal number, that is to say, of 
 the number formed by the addition of the 5 terms of the series 1, 4, 7, 10, 13. 
 
 PROBLEM III. 
 The Radix kg Polygonal Number being given; to find that Number. 
 
 The rule for this purpose is exceedingly simple. From the square of the given 
 radix, subtract the product of the same radix by a number equal to that of the angles 
 less 4; the half of the remainder will be the polygonal number required. 
 
 For example, what is the triangular number the radix of which is 12? 
 
 The square of 12 is 144; the number equal to that of the angles less 4is — I, 
 which being multiplied by 12 gives — 12: but according to the rule, — 12 ought to 
 be subtracted, which is the same thing as adding 12; in that case you will have 156, 
 which being divided by 2 gives 78. 
 
 What is the heptagonal number the radix of which is 20? 
 
 To find the number required, take the square of 20, which is 400; then multiply 
 20 by 3, which is the number of the angles less 4, and subtract 60, the product, from 
 400 ; if you then divide the remainder 340 by 2, the quotient 170 will be the number 
 sought, or the heptagon the radix of which is 20. 
 
 It may not be improper here to remark, that the same number may be a poly- 
 - gon or figurate number in different ways. Every number greater than 3 is a poly- 
 gon, of a number of sides or angles equal to that of its units. 
 
 Thus 36 is a polygon of 36 sides, the radix of which is 2; for the two first terms 
 _ of the progression are 1,35. The same number 386 is a square ; and lastly it is tri- 
 angular, having 8 for its radix. 
 
 In the like manner, 2] is a polygon of 21 sides; it is also triangular ; and lastly it 
 is octagonal, 
 
 PROBLEM IV. 
 
 To Ae the Sum of as many Triangular, or of as many Square, or of as many Pentagonal 
 Numbers, as we choose. 
 
 As by the successive addition of the terms of different pralriatrcs! progressions, 
 we obtain new progressions of numbers, called triangular numbers, square numbers, 
 pentagonals, &c. ; we can add also these last progressions, which will give rise to new 
 figurate numbers, of a higher order, called pyramidal numbers. Those which arise 
 from the progression of triangular numbers, are called pyramidals of the first order ; 
 those produced by the addition of the square numbers, pyramidals of the second 
 order; and those by the progression of the pentagonal numbers, pyramidals of the 
 third order. The same operation may be performed with the pyramidals ; which 
 gives rise to the pyramido-pyramidals. But as these numbers are of little utility, 
 and can answer no other purpose than that of exercising the genius of such as are 
 fond of analytical investigation, we shall not enlarge farther on the subject. We 
 shall therefore confine ourselves to giving a general rule for adding as many figurate 
 numbers as the reader may choose. 
 
e: 
 24 ARITHMETIC. 3 
 
 Multiply the cube of the number of terms to be added, by the number of the — 
 angles of the polygon less 2; to the sum add three times the square of the said — 
 number of terms, and subtract from it the product of the same number multiplied by 
 that of the angles less 5: if you divide the remainder by 6, you will have the sum of 
 the terms of the progression. : 
 
 For example, suppose it were required to find the sum of the first eight triangular 
 numbers. 
 
 The cube of 8 is 512; which being multiplied by the number of the angles of the 
 polygon less 2, or by 1, gives still 512; add to this number the triple of the square 
 of 8, or 192, which will make 704; then, as the number of the angles less 5, is — 
 2, multiply 8 by — 2, and you will have — 16; if you then add 16 to 704 you will 
 have 720, which being divided by 6, gives 120, for the sum of the eight first triangu- 
 lar numbers. 
 
 The same result may be obtained, with more ease, by multiplying the number of 
 the terms 8, by 9, and the product by 10, which gives also 720; which divided by 6, 
 the quotient is 120, as before. 
 
 In the case of a series of squares, the number of which we shall here suppose to 
 by 10, we have only to multiply the number of terms, viz. 10, by the same number 
 plus unity, or by 11, and then by the double of the same number plus unity, that is 
 to say by 21: the product of these three numbers, 2310, if divided by 6, gives 385, 
 for the sum of the first ten square numbers I, 4, 9, 16, &c. 
 
 CHAPTER V. 
 OF RIGHTANGLED TRIANGLES IN NUMBERS; OR RIGHTANGLED TRIANGULAR 
 NUMBERS. 
 
 RIGHTANGLED triangular numbers, are rational numbers so related to each other, 
 that the sum of the squares of two of them is equal to the square of the third. 
 
 The numbers 3, 4, and 5 have this property, 3? + 4? being equal to 5”. 
 Right-angled triangular numbers must be severally unequal; for, if the two less 
 ones could each be represented by a, and the third or greatest by 6, then 2 a? = 
 
 B2, b =a / 2, an irrational number, whatever is the value of a. 
 
 The area of arightangled triangle, whose sides are rational, eannot be equal to a 
 rational square. 
 
 If a, b, and c represent the sides of a triangle, and C be the angle opposite c; then 
 if C= 907, @ +e: if C= 1200 -+ab+h=-, and if C= 60° ae 
 —ab+tPac. 
 
 If » represent any number, and m any other number less than n, then n? + m?_ 
 will represent the hypothenuse of a rightangled plane triangle, of which the other 
 two sides are respectively n? — m7, and 2 n m. 
 
 For example, if » = 2, and m = 1, then n? + m? = 5, n*? — m? = 3, and2am= 
 4, which are rightangled triangular numbers. 
 
 If n = 7 and m = 2, the formule give 53, 45, and 28 for the numbers, and 53? 
 = 2809 = 45? + 287. 
 
 We shall now propose and solve a few of the most easy and curious problems 
 respecting right-angled triangular numbers. 
 
 PROBLEM I. 
 To find as many Rightangled Triangles in Numbers as we please. 
 This may be effected by the concluding formule which we have just given, but 
 we think it right to add the following methods. . 
 
RIGHTANGLED TRIANGLES IN NUMBERS. 25 
 
 Take any two numbers at pleasure, for example 1 and 2, which we shall call gene- 
 rating numbers; multiply them together ; then having doubled the product, we ob- 
 tain one of the sides of the triangle, which in this case will be 4. If we then square 
 each of the generating numbers, which in the present example will give 4 and 1, their 
 difference 3 will be the second side of the triangle, and their sum 5 will be the hypo- 
 thenuse. The sides of the triangle, therefore, having 1 and 2 for their generating 
 numbers, are 3, 4, 5. 
 
 If 2 and 3 had been assumed as generating numbers, we should have found the 
 sides to be 5, 12, and 13; and the numbers | and 3 would have given 6, 8, and 10. 
 
 Another Method.—Take a progression of whole and fractional numbers, as 1}, 22, 
 33, 44, &c., the properties of which are; lst, The whole numbers are those of the 
 common series, and have unity for their common difference. 2nd, The numerators 
 of the fractions, annexed to the whole numbers, are also the natural numbers. 3rd, 
 The denominators of these fractions are the odd numbers 3, 5, 7, &c. 
 
 Take any term of this progression, for example 3}, and reduce it to an improper 
 fraction, by multiplying the whole number 3 by 7, and adding to 21, the product, the 
 numerator 3, which will give 3. The numbers 7 and 24 will be the sides of a right- 
 angled triangle, the hypothenuse of which may be found by adding together the 
 squares of these two numbers, viz. 49 and 576, and extracting the square root of the 
 sum. The sum in this case being 625, the square root of which is 25, this number 
 will be the hypothenuse required. The sides therefore of the triangle produced by 
 the above term of the generating progression, are 7, 24, 25. 
 
 In like manner, the first term 1! will give the rightangled triangle 3, 4, 5. 
 
 The second term 22 will give 5, 12, 13. 
 
 The fourth 44 will give 9,40, 41. All these triangles have the ratio of their sides 
 different ; and they all possess this property, that the greatest side and the hypothe- 
 nuse differ only by unity. 
 
 The progression 1%, 241, 345, 419, &c., is of the same kind as the preceding. The 
 first term of it gives the rightangled triangle 8, 15,17; the second term gives the 
 triangle 12, 35, 37; the third the triangle 16, 63, 65,&c. All these triangles, it is 
 evident, are different in regard to the proportion of their sides; and they all have 
 this peculiar property, that the difference between the greater side and the hypothe- 
 nuse, is always the number 2. 
 
 PROBLEM II. | 
 
 To find any Number of Rightangled Triangles in Numbers, the sides of which shall 
 differ only by Unity. 
 
 _To resolve this problem, we must find out such numbers that the double of their 
 squares plus or minus unity shall also be square numbers. Of this kind are the num- 
 bers 1, 2, 5, 12, 29, 70, &c.; for twice the square of 1 is 2, which diminished by unity 
 leaves 1, a square number. In like manner, twice the square of 2 is 8, to which if 
 we add 1, the sum 9 will be a square number. And so on. 
 
 Having found these numbers, take any two of them which immediately follow each 
 other, as ] and 2, or 2.and 5, or 12 and 29, for generating numbers. The right- 
 angled triangles arising from them will be of such a nature, that their sides will differ 
 from each other only by unity. The following is a table of these-triangles, with 
 their generating numbers. 
 
 Gener. Numb. Sides. Hypoth. 
 1 2 3 4 5 
 2 5 20 21 29 
 5 12 119 120 169 
 12 29 696 697 985 
 29 70 4059 4060 5741 
 
 70 169 23660 23661 33461 e 
 
26 “ARITHMETIC. 
 
 But if the problem were, to finda series of triangles of such a nature, that the hypo- 
 _ thenuse of each should exceed one of the sides only by unity, the solution would be 
 
 much easier. Nothing in this case would be necessary but to assume, as the generat- 
 
 ing numbers of the required triangle, any two numbers having unity for their differ- 
 
 ence. The following isa table similar to the preceding, of the six first rightangled 
 
 triangles produced by the first numbers of the natural series. 
 
 Gener. Numb. Sides. Hypoth., 
 1 2 3 4, 5 
 2 3 5 12 13 
 3 4 7 24 25 
 4 5 9 40 4] 
 5 6 11 60 . 6) 
 6 7 13 84 85 
 
 If we assume, as generating numbers, the respective sides of the preceding series 
 of triangles, we shall have a new series of rightangled triangles, the hypothenuses of 
 which will always be square numbers ; as may be seen in the following table. 
 
 Gener. Numb. Sides, Hypoth. Roots. 
 3 4 7 24 25 5 
 5 12 119 120 169 13 
 7 24. 336 527 625 25 
 9 40 720 1519 1681 4] 
 ll 60 1320 3479 3721 61 
 13 84 2184 6887 1225 85 
 
 It may here be observed, that the roots of the hypothenuses are always equal to 
 the greater of the generating numbers increased by unity. 
 
 But if the second side and the hypothenuse of each triangle in the above table, 
 which differ only by unity, were assumed as the generating numbers, we should have 
 a series of rightangled triangles, the least sides of which would always be squares. 
 A few of these are as follow: | 
 
 Gener. Numb. Sides. Hypoth. 
 4, 5 9 40 ' 41 
 12 13 25 312 313 
 24 ao aan 49 1200 1201 
 40 Al 81 3280 3281 
 
 In the last place, if it were required to find a series of rightangled triangles, one 
 of the sides of which shall be always a cube, we have nothing to do but to take, as” 
 generating numbers, two following terms in the progression of triangular numbers, - 
 as 1, 3, 6, 10, 15, 21, &c. By way of example we shall here give the first four 
 of these triangles: 
 
 Gener. Numb. Sides. Hypoth. 
 1 3 6 8 10 
 3 6 36 27 45 
 6 10 120 64 136 
 10 15 300 125 326 
 
 PROBLEM III. 
 
 Lo find Three different Rightangled Triangles, the Areas of which shall be 
 all Equal. 
 
 The following are three rightangled triangles which possess this property. The 
 sides of the first are, 40, 42,48; those of the second 24, 70, 74; and those of the 
 third, 15, 112, 113. 
 
 a — 
 
RIGHTANGLED TRIANGLES IN NUMBERS. Pi 
 
 The method by which these triangles are found, is as follows: 
 
 Add the product of any two numbers to the sum of their squares, and that will . 
 give the first number; the difference of their squares will give a second; and 
 double the sum of their product and of the square of the least number, will give the 
 third. 
 
 If you then form arightangled triangle from the two first of the numbers thus 
 found, as generating numbers; asecond from the two extremes ; andathird from the 
 first and the sum of the other two; these three rightangled triangles will be equal 
 to each other. 
 
 No more than three rightangled triangles, equal to each other, can be found in 
 whole numbers ; but we may find as many as we choose in fractions or mixt numbers, 
 by means of the following formula: 
 
 With the hypothenuse of one of the above triangles, and the quadruple of its area, 
 form another rightangled triangle, and divide it by double the product which arises 
 from multiplying the hypothenuse of the triangle you made choice of by the difference 
 
 of the squares of the two other sides: the triangle thence produced will be the one 
 required. 
 
 PROBLEM IV. 
 
 To find a Rightangled Triangle, the Sides of which shall be in Arithmetical 
 Progression. 
 
 Take two generating numbers which have to each other the ratio of 1 to2; the 
 sides of the rectangled triangle thence produced will be in arithmetical pro- 
 gression. 
 
 The simplest of these triangles, is that which has for its sides 3, 4,and 5, arising 
 from the numbers 1 and 2 assumed as generating numbers. But it is to be observed, 
 that all the other triangles, which possess the same property, are similar to this one, | 
 and are only multiples of it. That there can be no other kind, might easily be 
 demonstrated in a great many different ways. 
 
 For let 2, x-+a, and +2 a, be the sides, then 22-+-2--a’=r-+2a’, and this 
 quadratic equation gives~== 3a. Therefore the sides are represented by 3a, 4a, 
 and 5a. 
 
 Remark.—If it were required to find a rightangled triangle, the three sides of 
 which should be in geometrical proportion, we must observe, that none such can be 
 found in whole numbers; for the two generating numbers ought to bein the ratio of 1 to 
 r/ /5—2, which is an irrational number. 
 
 _ PROBLEM V. 
 
 To find a Rightangled Triangle, the Area of which, expressed in Numbers, shall be 
 equal to the Perimeter, or in a given ratzo to it. 
 
 Of any square number, and the same square increased by 2, form a rightangled 
 triangle, and divide each of its sides by that square number: the quotients will give 
 the sides of a new rightangled triangle, the area of which, expressed numerically, will 
 be equal to the perimeter. 
 
 Thus, if we take, as generating numbers, i and 3, we shall have the triangle 6, 8, 
 
 10, the sides of which, if divided by unity, give the same 6, 8, 10, forming a triangle 
 having the property required ; for the area and the perimeter are each equak to 24. 
 Tn like manner, if we take 4 and 6 as generating numbers, we obtain for the required 
 triangle 5, 12, 13, which on trial will be found to possess the same property. 
 
 These triangles are the only two of the kind which can be found in whole num- 
 
— 28 _ ARITHMETIC. : 
 bers; but we may find abundance of them in fractional numbers, by means of the 
 squares 9, 16, &c.; such as the following: 49, 198, 202; or $8,575, 58, or in their 
 least terms, 17, 144, 145, 
 
 If it were required that the area of the proposed triangle should be only in a given 
 ratio to the perimeter, for example that of 3; take as generating numbers a square, 
 and the same square increased by 3, and form from them, as before directed, a right- 
 angled triangle : this triangle will possess the required property. Of this kind, in 
 whole numbers, are the two triangles 8, 15, 17,-and 7, 24, 25: and numberless others 
 may be found in fractional numbers. 
 
 CHAPTER VI. 
 
 SOME CURIOUS PROBLEMS RESPECTING SQUARE AND CUBE NUMBERS. 
 
 PROBLEM I. 
 Any Square Number being given, to divide it into Two other Squares. 
 
 INNUMERABLE solutions may be found to this problem, in the following manner. 
 Let 16, for example, whose root is 4, be the square to be divided into two other 
 squares, which, as may be easily seen, can be only fractions. 
 
 Take any two numbers, as 3 and.2; multiply them together; and by their product 
 multiply the double of 4, the root of the proposed square; the last product, which in 
 this case is 48, will be the numerator of a fraction, the denominator of which will be 
 13, the sum of the squares of the above numbers 3 and 2: the fraction 48 therefore 
 will give the side of the first square required, which square consequently will 
 be 323. 
 
 To obtain the second, multiply the given square by the above denominator 169, 
 and from the product 2704 subtract the numerator 2304: if we then take 20, the 
 root of 400 the remainder, (which will be always a square,) for a numerator, and 13 
 for a denominator, we shall have the fraction 39 for the side of the second square. 
 
 The two sides of the required squares therefore, are 48 and #3, the squares of which, 
 HH and 483, will be found equal to the square number 16. 
 
 If we had taken for the primitive numbers 2 and 1, we should have had the roots 
 8 and ¥, the squares of which are 386 and 44; the sum of which is 429 or 16. 
 
 The uumbers 4 and3 would have given the roots $$ and 38, the squares of which 
 SS and 784 still make up 199% or 16. 
 
 It may here be seen, that by varying the two first supposed numbers at pleasure, the 
 solutions also may be varied without end. 
 
 Remark.—Should it be here asked whether a given cube can, in like manner, be 
 divided into two other cubes? we shall reply, on the authority of an eminent analyst, 
 M. de Fermat, that it is not possible. It is equally impossible to divide any power 
 above the square into two parts, which shall be powers of the same kind; for ex- 
 ample, a biquadrate into two biquadrates. 
 
 PROBLEM It. 
 
 To divide a Number, which is the Sum of Two Squares, into Two other Squares. 
 
 Let the proposed number be 13, which is composed of the two squares 9 and 4: it 
 is required to divide it into two other squares. ; 
 
 Take any two numbers, for example, 4 and 3; and multiply the former, 4, by 6, the 
 double of 3 the root of one of the above squares; and the second 3 multiply by the 
 
SQUARE AND CUBE NUMBERS. 29 
 
 double of 2 the root of the other square; which will give as products 24 and 12. 
 Subtract the latter of these numbers from the former, and their difference 12 will be 
 the numerator ofa fraction, the denominator of which will be 25, the sum of the 
 squares of the numbers first assumed, Multiply this fraction 32, by each of the assumed 
 numbers, viz. 4 and 3, and you will have 48 and 38. If you then take the greater 
 of these numbers from the root of the greater square contained ia 13, viz. 3, the re- 
 mainder will be 32: and if you add the other to the side of the smaller square con- 
 tained in 13, viz. 2, you will have &. These two fractions then, 3% and §8, will be 
 the sides of the two squares sought, viz. 33 and 7328, which together are equal to 13. 
 
 By supposing other numbers, other squares may be obtained ; but these are suffi- 
 cient to shew the method of finding them. 
 
 Remark.—¥or a number to be divisible, in a variety of ways, into two squares, it 
 must be either a square, or composed of two squares. Of this kind, taking them 
 in order, are. the numbers 1, 2, 4,5, 8,9, 10, 138, 16, 17, 25, 26, 29, 32, 34, 36, 37, &c. 
 We do not know, nor do we think it possible to find, any method of dividing into 
 two squares any number which is not a square, or the sum of two squares; and we 
 
 are of opinion that it may be established as a rule, that every whole number, which. 
 
 is not a square, or composed of two squares, in whole numbers, cannot be divided, in 
 any manner, into two squares. A demonstration of this would be curious. 
 
 But every number is divisible, ina great variety of ways, into four squares ; for there 
 is no number which is not either a square, or the sum of two, or of three, or of 
 four squares. Bachet de Meziriac advanced this proposition,* the truth of which he 
 ascertained as far as possible by trying all the numbers from | to 325. It is added, 
 by M. de Fermat,} that he was able to demonstrate the following general and curious 
 properties of numbers, viz. : 
 
 That every number is either triangular, or composed of two or three triangular 
 numbers. 
 
 That every number is either square, or composed of two, or three, or four square 
 numbers. 
 
 And that every number is either pentagonal, or composed of two, or three, four, 
 or five pentagonal numbers. And so of the rest. 
 
 A demonstration of these properties of numbers, if they be real, would be truly © 
 
 curious. 
 
 PROBLEM IiIt. 
 
 To find Four Cubes, two of which taken together shall be equal to the Sum of the 
 other two. 
 
 This problem may be solved by the following simple method. Take any two 
 numbers of such a nature, that double the cube of the less shall exceed that of the 
 greater; then from double the greater cube subtract the less ; and multiply the re- 
 mainder, as well as the sum of the cubes, by the less of the assumed numbers: 
 the two products will be the sides of the two first cubes required. 
 
 In like manner, take the cube of the greater of the assumed numbers from double 
 the cube of the less; and multiply the remainder, as well as the sum of these two 
 cubes, by the greater of the assumed numbers: the two new products will be the 
 sides of the other two cubes. 
 
 For example, if we assume the numbers 4 and 5, which possess the above property, 
 we obtain, by following the rule, for the sides of the two first cubes, 744, 756; and 
 
 * Diophanti Alexandrini Arithmeticorum, lib. vi. cum Comm. C, G. Bacheti. Tolose. 1670. fol. p. 179. 
 + Ibid. p. 180. ; 
 
30 ARITHMETIC. 
 
 for those of the other two, 945 and 15, which being divided by 8, give for the two 
 first 248, 252; and for the two latter, 315, 5. 
 
 If the assumed numbers be 5 and 6, we shall have 1535 and 1705 for the sides of 
 the two first cubes ; and 2046 and 204 for those of the other two. 
 
 Remark.—A number composed of two cubes being given, it is possible to find two 
 other cubes, the sum of which shall be equal tothe former two. Vieta was of a con- 
 trary opinion; but M. de Fermat, in his Observations on the Arithmetical Questions 
 - of Diophantus, with a Commentary by Bachet de Meziriac, has pointed out a method 
 by which such cubes can be found. The calculation indeed extends to numbers which 
 are exceedingly complex, and sufficient to frighten the boldest arithmetician; as may 
 be seen by the following example, where it is required to divide the sum of the two 
 cubes 8 and 1 into two other cubes. By following the method of M. de Fermat, 
 Father de Billy found that the sides of the two new cubes were the following 
 
 numbers: > 
 12436177733990097836481, 
 
 60962383566137297449 
 and 487267171714352336560. 
 60962383566137297449 
 We must take these numbers on Father de Billy’s word ; for we do not know that 
 any one will ever venture to examine whether he has been deceived. 
 
 But it is possible to resolve, without much trouble, another question of a similar 
 kind, which is: To find three cubes which, taken together, shall be equal to a fourth. 
 By following the method pointed out in the above-mentioned work, it will be found 
 that the least whole numbers, which resolve the question, are 3, 4, and 5; for their 
 cubes added together make 216, which is the cube of 6. 
 
 We have confined ourselves to a few questions of this kind, but they might be 
 varied almost without end. ‘They are attended with a peculiar kind of difficulty 
 which renders them interesting, and on that account they have been an object of at- 
 tention to various analysts; such as Diophantus of Alexandria, among the ancients, who 
 wrote thirteen books on arithmetical questions, of which the first six only remain, 
 with another on polygonal numbers. Vieta too exercised his ingenuity on questions 
 of this kind; as did also Bachet de Meziriac, who wrote a commentary on the above 
 work of the Greek arithmetician. But this species of analysis was carried farther 
 than ever it had been before by the celebrated M. de Fermat. Father de Billy, about 
 the same time, gave proofs of the acuteness of his talents in this way, by his work 
 entitled Diophantus Redivivus, in which he far excelled the ancient analyst. M. 
 Ozanam likewise shewed great ability in this species of analysis, by the resolution 
 of several problems which had been considered as insoluble. He wrote a work on 
 this subject, but it was never published; and the manuscript, after his death, came 
 into the hands of the late M. Daguesseau, as we are informed by the historian of the 
 Academy of Sciences. 
 
 The Hindoos also were great adepts in such problems, as we learn from some alge- 
 braical works which have lately been found among them, an account of which may 
 be seen in the second volume of Tracts by the late Dr. Charles Hutton. 
 
» 
 
 ARITHMETICAL PROGRESSION. 31 
 
 CHAPTER VII. 
 
 'OF ARITHMETICAL AND GEOMETRICAL PROGRESSIONS, AND OF CERTAIN PROBLEMS 
 | WHICH DEPEND ON THEM, 5 
 
 SECTION I. 
 
 . 
 
 Explanation of the most remarkable properties of an Arithmetical Progression. 
 
 Ir there be a series of numbers, either increasing or decreasing, in such a manner, 
 that the difference between the first and the second shall be equal to that between 
 the second and third, and between the third and fourth, and so on successively; these 
 numbers will be in arithmetical progression. 
 
 The series of numbers I, 2, 3, 4, 5, 6, &c.; or 1,5, 9,13, &c. ; or 20, 18, 16, 14, 12, 
 &c. ; or 15, 12,9, 6,3, are therefore arithmetical progressions ; for in the first, the 
 difference between each term and the following one, which exceeds it, is always 1; 
 in the second it is 4: in like manner this difference is always 2 in the third series, 
 which goes on decreasing ; and in the fourth it is 3. 
 
 It may be readily seen, that an increasing arithmetical progression may be continued 
 ad infinitum ; but this cannot be the case, in one sense, with a decreasing series ; for 
 Wwe must always arrive at some term, from which if the common difference be taken, 
 the remainder will be 0, or else a negative quantity. Thus, the progression 19, 15, 
 11, 7,3, cannot be carried farther, at least in positive numbers; for it is impossible 
 to take 4 from 8, or if it be taken we shall have, according to analytical expression, 
 — 1;* and by continuing the subtraction we should have — 5, — 9, &c. 
 
 The chief properties of arithmetical progressions may be easily deduced from the 
 definitions which we have here given. For a little attention will shew, 
 
 lst. That each term is nothing else than the first, plus or minus the common dif- 
 ference multiplied by the number of intervals between that term and the first. 
 Thus, in the progression 2, 5, 8, 11, 14, 17, &c., the difference cf which is 3, there are 
 fiveintervals between the sixth term and the first; and for this reason the sixth 
 
 _term was equal to the first plus 15, the product of the common difference 3 
 by 5. But as the number of intervals is always less by unity than the number 
 of terms, it thence follows, that we may find any term, the place of which in 
 the series is known, if we multiply the common difference by the number expressing 
 that place less unity. According to this rule, the hundredth term of an increasing 
 progression will be equal to the first plus 99 times the common difference. If it be 
 decreasing, it will be equal to the first term minus that product. 
 
 In every arithmetical progression therefore, the common difference being given, to 
 find any term the place of which is known; multiply the common difference by the 
 number which indicates that place less unity, and add the product to the first term, 
 if the progression be increasing, but subtract it if it be decreasing: the sum or re- 
 mainder will be the term required. 
 
 2nd. In every arithmetical progression, the sum of the first and last terms, is equal 
 to that of the second and the last but one; and to that of the third and the last but 
 two, &c. ; in a word, to the sum of the middle terms if the number of the terms 
 be even, or to the double of the middle term if the number.of the terms be odd. 
 
 This may easily be demonstrated from what has been said: for let us call the first 
 term A, and let us suppose that there are twenty terms in the progression ; if it be 
 increasing, the twentieth term will be equal to A plus nineteen times the common 
 
 * As the quantities called negative are real quantities, taken in a sense contrary to that of the 
 quantities calied positive, it is evident that, according to mathematical and analytical strictness, an 
 arithmetical progression may be continued ad infinitum, decreasing as well as increasing; but we 
 here speak agreeably to the yulgar mode of expression. 
 
 . 
 
if 
 32 ARITHMETIC. 
 
 difference ; and their sum will be double the first term plus nineteen times that dif- 
 ference. But the second term is equal to the first plus the common difference, and 
 the nineteenth term, or last but one, according to our supposition, is equal to the first 
 plus eighteen times that difference. The sum therefore of the second and last but 
 one, is twice the first term plus nineteen times the common difference, the same as 
 before. And so of the third and last but two. . 
 
 3rd. By this last property we are enabled to shew in what manner the sum of all 
 the terms of an arithmetical progression may be readily found; for, as the first and 
 last terms make the same sum as the second and last but one, and as the third and 
 the last but two, &c.; in short as the two middle terms, if the number of terms be even ; 
 it thence follows, that the whole progression contains as many times the sum of the 
 first and the last terms, as there are pairs of such terms. But the number of pairs 
 is equal to half the number of terms ; consequently the sum of the whole progression 
 is equal to the product of the sum cf the first and last terms multiplied by half the 
 number of terms, or, what amounts to the same, to half the product of the sum of 
 the first and the last terms by the number of the terms of the progression. 
 
 If the number of the terms be odd, as 9 for example; it may be readily seen that 
 the middle term will be equal to half the sum of the two next to it, and consequently 
 of the sum of the first and the last. But the sum of all the terms, the middle term 
 excepted, is equal to the product of the sum of the first and last terms by the number 
 of terms less unity, for example 8 in the case here proposed, where there are 9 terms ; 
 consequently, by adding the middle term, which will complete the sum of the pro- 
 gression, and which is equal to half the sum of the first and the last terms, we shall 
 have, for the sum total of the progression, as many times the half sum above-men- 
 tioned, as there are terms in the progression; which is the same thing as the 
 product of half the sum of the first and last terms by the number of the terms, or 
 the product of the whole sum by half the number of terms. 
 
 When these rules are well understood, it will be easy to resolve the followin 
 questions. 
 
 PROBLEM I. 
 
 If a hundred stones are placed in a straight line, at the distance of a yard from each 
 other ; how many yards must the person walk, who undertakes to pick them up one 
 by one, and to put them into a basket a yard distant from the first sione ? 
 
 It is evident, that to pick up the first stone, and put it into the basket, the person 
 must walk 2 yards, one in going and another in returning; that for the second he 
 must walk 4 yards; and so on, increasing by two as far as the hundredth, which will 
 oblige him to walk two hundred yards, one hundred in going, and one hundred in 
 returning. It may easily be perceived also, that these numbers form an arithmetical 
 progression, in which the number of terms is 100, the first term 2, and the last 200. 
 The sum total therefore will be the product of 202 by 50, or 10100 yards, which 
 amount to more than five miles and a half. 
 
 PROBLEM It. 
 
 A gentleman employed a bricklayer to sink a well, “and agreed to give him at the rate of 
 three shillings for the first yard in depth, five for the second, seven for the third, and 
 so on increasing till the twentieth, where he expected to find water: how much was 
 due to the bricklayer when he had completed the work. 
 
 This question may be easily answered by the rules already given; for the diffe- 
 rence of the terms is 2, and the number of terms 20; consequently, to find the twen- 
 tieth term, we must multiply 2 by 19, and add 38, the product, to the first term 8, 
 which will give 41 for the twentieth term, 
 
ARITHMETICAL PROGRESSIONS, 33 
 
 If we then add the first and last terms, that is 3 and 41, which will make 44, and 
 multiply this sum by 10, or half the number of terms, the product 440 will be the 
 sum of all the terms of the progression, or the number of shillings due to the brick- 
 layer when he had completed the work. He would therefore have to receive £22. 
 
 PROBLEM III. 
 
 A gentleman employed a bricklayer to sink a well to the depth of 20 yards, and agreed to 
 give him £20 for the whole; but the bricklayer falling sick, when he had finished the 
 eighth yard, was unable to go on with the work: how much was then due to him? 
 
 Those who might imagine that two fifths of the whole sum were due to the work- 
 man, because 8 yards are two fifths of the depth agreed on, would certainly be mista- 
 ken ; for it may be easily seen that, in cases of this kind, the labour increases in pro- 
 portion to the depth. We shall here suppose, for it would be difficult to determine 
 it with any accuracy, that the labour increases arithmetically as the depth; conse- 
 quently the price ought to increase in the same manner. 
 
 To determine this problem, therefore, £20 cr 400 shillings must be divided into 
 20 terms in arithmetical progression, and the sum of the first eight of these will be 
 what was due to the bricklayer for his labour. 
 
 But 400 shillings may be divided into twenty terms, in arithmetical proportion, a 
 great many different ways, according to the value of the first term, which is here 
 undetermined: if we suppose it, for example, to be 1 shilling, the progression will be 
 
 1, 3,5, 7, &c., the last term of which will be 39; and consequently the sum of the 
 
 first eight terms willbe 64 shillings. On the other hand, if we suppose the first term 
 to be 103, the series of terms will be 103, 115, 123, 133, 143, which will give 112 
 shillings for the sum of the first eight terms. 
 
 But to resolve the problem in aproper manner, so as to give to the bricklayer his just 
 due for the commencement of the work, we must determine what is the fair value of a 
 
 - yard of work, similar to the first, and then assume that value as the first term of the 
 
 progression. We shall here suppose that this value is 5 shillings; and in that case 
 the required progression will be 5, 6}, 87, 944, 1174, 121%, &c., the common diffe- 
 rence of which is 39, and the last term 35. Now to find the eighth term, which is 
 necessary before we can find the sum of the first eight terms, multiply the common 
 difference 32 by 7, which will give 11,;, and add this product to 5 the first term, 
 which will give the eighth term 16},; if we then add 16), to the first term, and 
 multiply the sum, 21}, by 4, the product, 84 y4, will be the sum of the first eight 
 terms, or what was due to the bricklayer, for the part of the work he had compieted. 
 The bricklayer therefore had to receive 84, shillings, or £4, 4s, 2d. 
 
 PROBLEM IV. 
 
 A merchant being considerably in debt, one of his creditors, to whom he owed £1860, 
 offered to give him an acquittance tf he would agree to pay the whole sum in 12 month- 
 ly instalments ; that is to say, £100 the first month, and to increase the payment 
 by a certain sum each succeeding month, to the twelfth inclusive, when the whole debt 
 would be discharged: by what sum was the payment of each month increased ? 
 
 In this problem the payments to be made each month ought to increase in arith. 
 metical progression. We have given the sum of the terms, which is equal to the 
 sum total of the debt, and also the number of these terms, which is 12; but their 
 common difference is unknown, because it is that by which the payments ought to 
 increase each month. 
 
 Tofind this difference, we must take the first payment multiplied by the number 
 of terms, that is to say 1200 pounds, from the sum total, and the remainder will 
 
 D 
 
34 ARITHMETIC. 
 
 be 660; we must then multiply the number of terms less unity, or 11, by haif the 
 number of terms, or 6, and we shall have 66; by which, if the remainder 660 be 
 _ divided, the quotient 10 will be the difference required. The first payment, there- 
 fore, being 100, the second payment must have been 110, the third 120, and the 
 last 210. 
 
 SECTION II. 
 
 Of Geometrical Progressions, with an explanation of their Principal Properties. 
 
 If there be a series of terms, each of which is the product of the preceding by a 
 common multiplier; or what amounts to the same thing, each of which is in the 
 same ratio to the preceding; such a series forms what is called a geometrical pro- 
 gression. Thus 1, 2, 4, 8, 16, &c., form a geometrical progression; for the second 
 is the double of the first, the third the double of the second, and so on in succession. 
 The terms 1, 3, 9, 27, 81, &c. form also a geometrical progression, each term 
 being the triple of that which precedes it. 
 
 I. The principal property of geometrical progression is, that if we take any three 
 following terms, as 3, 9, 27, the product, 81, of the extremes will be equal to the 
 square of the middle term 9; in like manner, if we take four following terms, as 3, 
 9, 27, 81, the product of the extremes, 243, will be equal to the product of the two 
 means or middle terms, 9 and 27. 
 
 In the last place, if we take any successive number of terms, as 2, 4, 8, 16, 32, 64, 
 the product of the extremes, 2 and 64, will be equal to the product of any two which 
 are equally distant from them, viz. 4 and 32, or 8 and 16. If the number of the terms 
 -were odd, it is evident that there would be only one term equally distant from the 
 two extremes; and in that case, the square of this term would be equal to the pro- 
 duct of the extremes, or to that of any two equally distant from them, or from the 
 mean term. : ‘ 
 
 If. Between geometrical and arithmetical progression there is a certain analogy, 
 which deserves here to be mentioned, and which is, that the same results are obtained 
 in the former by employing multiplication and division, as are obtained in the latter 
 by addition and subtraction. When in the latter we take the half or the third, we 
 employ in the former extraction of the square, cube, &c. roots. 
 
 Thus, to find an arithmetical mean between any two numbers, for example 3 and 
 12, we add the two given extremes, and 73, the half of their sum 15, will be the 
 number required ; but to find a geometrical mean between two numbers, we must 
 multiply the two extremes, and extract the square root of their product. Thus, if 
 the given numbers were 3 and 12, by extracting the square root of their product 36, 
 we shall have 6 for the number required. | 
 
 If we take any geometrical progression whatever, as 1, 2, 4, 8, 16, 32, 64, &c., and 
 
 _write it down as in the subjoined example, with the terms of an arithmetical progres- 
 sion above it, in regular order, 
 OVE Seade a Ge uy 8 9 10 
 1. 2:24, 28.416 32. 640, 128,256 512.1024 
 the following properties will be remarked in this combination : 
 
 Ist. If any two terms whatever of the:geometrical progression, for example 4 
 and 64, be multiplied together, their product will be 256; if we then take the two 
 terms of the arithmetical progression corresponding to 4 and 64, which are 2 and 6, 
 and add them together, their sum 8 will be found over the above sum 256. 
 
 2d. If we take four terms of the lower series in geometrical proportion, for exam- 
 ple, 2, 16, 64, 512, the numbers of the upper series corresponding to them will be 
 1, 4, 6, 9, which are in arithmetical proportion ; for the difference between 4 and 1 
 is the same as that between 9 and 6. 
 
 3d. In the lower series, if we take any square number, for example 64, and in the 
 
GEOMETRICAL PROGRESSIONS, 35 
 
 upper series the term corresponding to it, viz. 6, the half of the latter will be found 
 to correspond to the square root of 64, the former, viz. 8. 
 
 By taking, in the lower series, a cube, for example 512, and in the upper series the 
 corresponding number 9, it will be found that the third of the latter, which is 3, 
 will correspond to the cube root of the former 512, which is 8. 
 
 Thus, it is evident, that what is multiplication in geometrical progression, is addi- 
 tion in arithmetical ; that what is division in the former, is subtraction in the latter ; 
 and, in the last place, that which is extraction of the square, cube, &c. roots, in 
 geometrical progression, is simple division by 2, 5, &c. in arithmetical. 
 
 This remarkable analogy is the foundation of the common theory of logarithms ; and 
 on that account seemed worthy of some illustration. 
 
 III. It is evident that all the powers of the same number, taken in regular order, 
 form a geometrical progression; as may be seen in the following example, which is a 
 series of the powers of the number 2, 
 
 24 8 16 32 64 128, &e. 
 The case is the same with the powers of the number 3, which form the series, 
 3 9 27 81 243 729, &ce. 
 
 The first of these series has this peculiar property, that if we take the first, bf 
 fourth, eighth, sixteenth, and thirty-second terms, and to them add unity, the result 
 will be prime numbers. 
 
 IV. The common ratio of a geometrical progression, is the number that results 
 from the division of any term by that which precedes it. Thus, in the geometrical 
 progression 2, 8, 32, 128, 512, the ratio is 4; for if we divide 128 by 32, 32 by 8, or 
 8 by 2, the quotient will be always 4. The ratio therefore acts an important part in 
 geometrical progression; the same that the common difference does in arithmetical, 
 that is to say, it is always constant. 
 
 To find any term then, for example the 8th, of a geometrical progression, the ratio 
 and first term of which are known, multiply the ratio by itself 7 times, or as many 
 times as there are units in the place of the required term less one ; or, what is the 
 same thing, raise the ratio to the 7th power; then multiply the first term by the 
 product, and the new product will be the eighth term required. For example, let the 
 first term of the progression be 3, and the ratio 2; to find the 8th term, raise 2 to the 
 7th power, which will be 128, and multipiy 128 by the first term 3, the product 384 
 will give the 8th term of the progression required. 
 
 We shall here observe, that had the 8th term of an arithmetical progression been 
 required, the first term and the common difference being given, we should have mul- 
 tiplied that difference by 7, and added the product to the first term ; which is a proof 
 of the analogy already mentioned in the second paragraph. 
 
 V. The sum of the terms of any given geometrical progression may be found in the 
 following manner : 
 
 Multiply the first term by itself, and the last by the second, and take the difference 
 of the two products. ‘Then divide this difference by that of the first two terms, and 
 the quotient will be the sum of all the terms. 
 
 Let us take, for example, the progression 3, 6, 12, 24, &c., the eighth term of aie 
 is 384, and let it be required to find the sum of these eight terms: the product of the 
 first term by itself is 9, and that of the last by the second is 2304; the difference of 
 these products is 2295; if this difference then be divided by 3, the difference of the 
 first and second terms, we shall have for quotient the number 765, which will be the — 
 suin of these eight terms. 
 
 VI. A geometrical progression may decrease in infinitum, without ever reaching 0 ; 
 for it is evident that any part of the quantity greater than 0 can never become 0, A 
 decreasing geometrical progression therefore may be extended without end; for by 
 
 “p22 
 
36 ARITHMETIC, 
 
 dividing the last term by the ratio of the progression, we stall have the touowing 
 term. q 
 We shall heze subjoin two of these decreasing progressions, by way of ex- 
 amples :— 
 lh b b ie ww ap &e. 
 ], 35 by a» dp als mp, &e 
 
 VII. The sum of an increasing geometrical progression is evidently infinite ; but 
 that of a decreasing geometrical progression, whatever be the number of terms as- 
 sumed, is ae finite. Thus the sum of all the terms, in infinitum, of the progres- 
 sion 1, 4, 41, 1,&c., is only 2; that of the progression 1, 3, J, oy, &c. in infinitum, is 
 only 14; &c. This necessarily follows from the method already given, for finding 
 the sum of any number of terms whatever of a geometrical progression; for if we 
 Suppose it prolongued in infinitum, and decreasing, the last term will be infinitely 
 small, or 0; the product of the second term by the last will therefore be 0; and 
 consequently, to find the sum, nothing will be necessary but to divide the square of 
 the first term by the mnaan ie ef the first and the second. In this manner it will be 
 igupd that the sum of 1,4, 4, 3, &c. continued in infinitum, is 2; and that of 1, 4, 
 
 $, &e. will be? or 14; for ies square of ] is 1, the difference of 14 and is 4, and 
 tee divided by 4 gives 2; in pee manner, 1 being divided by 3, Fin hs is the differ- 
 ence of | and d, gives 3. 
 
 Remark.—When we say that a progression continued in infinitum may be equal to 
 a finite quantity, we do not, like Fontenelle, pretend to assert that infinity can have 
 a realexistence. What is here meant, and what ought to be understood by all such 
 expressions, is that, whatever be the number of terms of a progression assumed, their 
 sum never can equal the determined finite quantity, though it may approach to it in 
 such a manner, that their difference will become smaller than any assignable quantity. 
 
 '.PROBLEM I. 
 
 If Achilles can walk ten times as fast as a tortoise, which is a furlong before him, can - 
 
 crawl ;x will the former overtake the latter, and how far must he walk before he 
 does so ? 
 
 This problem has been thought worthy of notice merely because Zeno, the founder — 
 
 of the sect of the Stoics, pretended to prove by asophism that Achilles could never 
 overtake the tortuise ; for while Achilles, said he, is walking a furlong, the tortoise 
 will have advanced the tenth of a furlong; and while the former is walking that 
 tenth, the tortoise will have advanced the hundredth part of a furlong, and so on zn 
 
 infinitum ; consequently an infinite number of instants must elapse before the hero can ° 
 
 come up with the reptile, and therefore he will never come up with it. 
 
 Any person however, possessed of common sense, may readily perceive that 
 Achilles will soon come up with the tortoise, since he will get before it. In what 
 then consists the sophism ? It may be explained as follows: 
 
 Achilles indeed would never overtake the tortoise, if the intervals of time during 
 which he is supposed to be walking the first furlong, and then the tenth, hundredth, 
 and thousandth parts of a furlong, which the tortoise has successively advanced before 
 him, were equal; but if we suppose that he has walked the first furlong in 10 
 minutes, he will require only one minute to walk the tenth of a furlong, and 
 6 
 hich Achilles will require to pass over the space gained by the tortoise, during 
 the preceding time, will go on decreasing in the following manner: 10, L, +5 a9, 
 td &C ; and this series forms a sub-decuple geometrical progression, the sum 
 of which is equal to 114, or the interval of time at the end of which Achilles will 
 
 have reached the tortoise. 
 
 1 of a minute to walk the hundredth, &c. The intervals of time therefore, — 
 
GEOMETRICAL PROGRESSIONS. 37 
 
 PROBLEM If. 
 
 Tf the hour and minute hands of a clock both begin to move exactly at noon, at what 
 points of the dial-plate will they be successively in conjunction, during a whole revo- 
 lution of the twelve hours 2 
 
 This problem, considered in a certain manner, is in nothing different from the pre- 
 ceeding. The minute hand acts here the part which Achilles did in the former, and 
 the hour hand, which moves twelve times slower, that of the tortoise. In the last 
 
 place, if we suppose the hour hand to be beginning a second revolution, and the 
 
 minute hand to be beginning a first, the distance which the one has gained over the 
 other will be a whole revolution of the dial-plate. When the minute hand has made 
 one revolution, the hour hand will have made but one twelfth ofa revolution, and so on 
 progressively. To resolve the problem therefore, we need only apply to these data, 
 the method employed in the former case, and we shall find that the interval from 
 noon to the point where the two hands come again into conjuction, will be x ofa 
 whole revolution, or, what amounts to the same thing, one hour and 4; of an hour. 
 They will afterwards be in conjunction at 2 hours and #,3 hours aud 4, 4 hours 
 and +, &c. ; and, lastly, at 11 hours 4}, that is to say at 12 hours. 
 
 PROBLEM IiIt. 
 
 A courtier having performed some very important service to his sovereign, the latter, 
 wishing to confer onhim a suitable reward, desired him to ask whatever he thought 
 proper, promising that it should be granted. The courtier, who was well acquainted 
 with the science of numbers, only requested that the monarch would give him a quan- 
 tity of wheat equal to that which would arise from one grain doubled sixty-three 
 times successively. What was the value of the reward ? 
 
 The origin of this problem is related in so curious a manner by Al-Sephadi, an 
 Arabian author, that it deserves to be mentioned. A mathematician named Sessa, 
 says he, the son of Daher, the subject of an Indian prince, having invented the game 
 
 _ of chess, his sovereign was highly pleased with the invention, and wishing to confer 
 
 on him some reward worthy of his magnificence, desired him to ask whatever he 
 thought proper, assuring him that it should be granted. The mathematician how- 
 ever only asked a grain of wheat for the first square of the chess-board, two for the 
 second, four for the third, and so on to the last or sixty-fourth. The prince at first 
 was almost incensed at this demand, conceiving that it was ill-suited to his liberality, 
 and ordered his vizier to comply with Sessa’s request; but the minister was much 
 
 _ astonished when, baying caused the quantity of corn necessary to fulfil the prince’s 
 order to be calculated, he found that all the grain in the royal granaries, and that 
 
 even of all. his subjects, and in all Asia, would not be sufficient. He therefore 
 informed the prince, who sent for the mathematician and candidly acknowledged 
 that he was not rich enough to be able to comply with his demand, the ingenuity of 
 which astonished him still more than the game he had invented. 
 
 Such is then the origin of the game of chess, at least according to the Arabian 
 historian Al-Sephadi. But itis not our business here to discuss the truth of this 
 story ; our*business being to calculate the number of grains demanded by the mathe- 
 matician Sessa. 
 
 It will be found by calculation, that the 64th term of the double progression, be- 
 ginning with unity, is 9223372036854775808. But the sum of all the terms of a 
 double progression, beginning with unity, may be obtained by doubling the last term 
 and subtracting from it unity. The number therefore of the grains of wheat equal 
 to Sessa’s demand, will be 18446744073709551615. Now if a standard pint contains 
 
38 ARITHMETIC. 
 
 9216 grains of wheat, a gallon will contain 73728, and, as eight gallons make 
 one bushel, if we divide the above result by eight times 73728, we shall have 
 312749974 12295 for the number of the bushels of wheat necessary to discharge the 
 promise of the Indian king; and if we suppose that one acre of land is capable of 
 producing, in one year, thirty bushels of wheat, to produce this quantity would require 
 1042499913743 acres, which make more than eight times the surface of the globe; 
 for the diameter of the earth being supposed equal to 7930 miles, its whole surface, 
 comprehending land and water, will amount to very little more than 126437889177 
 square acres. 
 
 Dr. Wallis considers the matter ina manner somewhat different, and says, in his 
 Arithmetic, that the quantity of wheat necessary to discharge the promise made to 
 Sessa, would form a pyramid nine miles English in length, breadth, and height; which 
 is equal to a parallelopiped mass having nine square leagues for its base, and of the 
 uniform height of one league. But as one league contains 15840 feet, this solid would 
 be equivalent to another one foot in height, and having a base equal to 142560 square 
 leagues. Hence it follows, that the above quantity of wheat would cover, to the 
 height of one foot, 142560 square leagues ; an extent of surface equal to eleven times 
 that of Britain, which, when every reduction is made, will be found to contain little 
 more than 12674 square leagues. 
 
 If the price of a bushel of wheat be estimated at ten shillings, the value of the 
 above quantity will amount to £15637498706147. 10s., a sum which, in all probability, 
 far surpasses all the riches on the earth. 
 
 Another problem of the same kind is proposed in the following manner : 
 
 A gentleman taking a fancy to a horse, which a horse-dealer wished to dispose of at as 
 high a price as he could, the latter, to induce the gentleman to become a purchaser, 
 offered to let him have the horse for the value of the twenty-fourth nail in his shoes, 
 reckoning one farthing for the first nail, two for the second, four for the third, and 
 so on to the twenty-fourth. The gentleman thinking he should have a good bargain 
 accepted the offer. What was the price of the horse ? 
 
 By calculating as before, the twenty-fourth term of the progression 1, 2, 4,8, &c., 
 will be found to be 8388608, equal to the number of farthings the purchaser ought to 
 give for the horse. The price therefore amounted to £8738. 2s. 8d., which is more 
 than any Arabian horse, even of the noblest breed, was ever sold for. 
 
 Had the price of the horse been the value of all the nails, at a farthing for the 
 first, two for the second, four for the third, and so on, the sum would have been 
 
 hae 
 
 double the above number, minus the first term, or 16777215 farthings, that is 
 
 £17476. 5s. 33d. 
 
 We shall conclude this chapter with some physico-mathematical observations on 
 the prodigious fecundity, and the progressive multiplication, of animals and vegeta- 
 bles, which would take place if the powers of nature were not continually meeting 
 with obstacles. 
 
 I. It is not astonishing that the race of Abraham, after sojourning 260 years in 
 Egypt, should have formed a nation capabie of giving uneasiness to the sovereigns of 
 
 that country. We are told in the sacred writings, that Jacob settled in Egypt with © 
 
 70 persons ; now if we suppose that among these 70 persons there were 20 too far 
 advanced in life, or too young, to have children; that of the remaining 50, 25 were 
 males and as many females, forming 25 married couples, and that each couple, in the 
 space of 25 years, produced, one with another, 8 children, which will not appear in- 
 credible in a country celebrated for the fecundity of its inhabitants, we shall find 
 that, at the end of 25 years, the above 70 persons may have increased to 270; from 
 which if we deduct those who died, there will perhaps be no exaggeration in making 
 
 them amount to 210. The race of Jacob therefore, after sojourning 25 years in 
 
 Egypt, may have been tripled. In like manner, these 210 persons, after 25 years 
 
- 
 
 HARMONICAL PROGRESSION, 39 
 
 more, may have increased to 630; and so on in triple geometrical progression : hence 
 it follows that, at the end of 225 years, the population may have amounted to 
 1377810 persous, among whom there might easily be five or six hundred thousand 
 adults fit to bear arms. 
 
 II. If we suppose that the race of the first man, making a proper deduction for 
 those who died, may have been doubled every twenty years, which certainly is not 
 inconsistent with the powers of nature, the number of men, at the end of five cen- 
 turies, may have amounted to 1048576. Now, as Adam lived about 900 years, he 
 may have seen therefore, when in the prime of life, that is to say about the five hun- 
 ‘dredth year of his age, a posterity of 1048576 persons. 
 
 _ III. How great would be the multiplication of many animals, did not the difficulty 
 of finding food, the continual war which they carry on against each other, or the 
 numbers of them consumed by man, set bounds to their propagation? It might easily 
 be proved, that the breed of a sow, which brings forth six young, two males and four 
 females, if we suppose that each female produces every year afterwards six young, 
 four of them females and two males, would in twelve years amount to 33554230. 
 
 Several other animals, such as rabbits and cats, which go with young only for a few 
 weeks, would multiply with still greater rapidity: in half a century the whole earth 
 would not be sufficient to supply them with food, nor even to contain them! 
 
 If all the ova of a herring were fecundated, a very few years would be sufficient 
 to make its posterity fill the whole ocean ; for every oviparous fish contains thousands 
 of ova which it deposits in spawning time. Let us suppose that the number of ova 
 amounts only to 2000, and that these produce as many fish, half males and half fe- 
 males ; in the second year there would be more than 200000; in the third, more than 
 200000000; and in the eighth year the number would exceed that expressed by 2 
 followed by twenty-four ciphers. As the earth contains scarcely so many cubic 
 inches, the ocean, if it occupied the whole globe, would not be sufficient to contain 
 all these fish, the produce of one herring in eight years ! 
 
 IV. Many vegetable productions, if all their seeds were put into the earth, would 
 in a few years cover the whcle surface of the globe. The hyosciamus, which of all 
 the known plants produces perhaps the greatest number of seeds, would for this pur- 
 pose require no more than four years. According to some experiments, it has been 
 found that one stem of the hyosciamus produces sometimes more than 50000 seeds: 
 now if we admit the number to be only 10000, at the fourth crop it would amount to 
 a 1 followed by sixteen ciphers. But as the whole surface of the earth contains uo 
 more than 5507634452576256 square feet ; if we allow to each plant only one square 
 foot, it will be seen that the whole surface of the earth would not be sufficient for 
 the plants produced from one hysociamus at the end of the fourth year ! 
 
 SECTION III. 
 Of some other Progressions, and particularly Harmonical Progression. 
 
 Three numbers are in harmonical proportion, when the first is to the last, as the 
 difference between the first and the second is to that between the second and the 
 third. - Thus, the numbers 6, 8,2, are in harmonical proportion ; for 6 is to 2, as 3, 
 the difference between the two first numbers, is to 1, the difference between the two 
 last. [his kind of relation is called harmonical, for a reason which will be seen 
 hereafter. 
 
 I. Two numbers being given, a third which shall form with them harmonical pro- 
 portion may be found, by multiplying these two numbers, and dividing their product 
 by the excess of the double of the first over the second. Thus, if 6 and 3 be given, 
 we must multiply 6 by 3,and divide the product 18 by 9, which is the excess of 12 
 the double of 6 over 3, the second of the numbers given. In this case the quotient 
 will be 2. . 
 
40 ‘ ARITHMETIC. 
 
 It may hence be readily seen that, in one sense, it is not always possible to find a 
 third number in harmonical proportion with two others; for when the first is less, if 
 its double be equal to or less than the second, the result will be an infinite or a 
 negative number. Thus, the third harmonic proportional to 2 and 4 is infinite; for 
 it will be found that the number sought is equal to 8 divided by 4—4, or 0. But 
 every person, in the least acquainted with arithmetic, knows that the more the deno- 
 minator of a fraction is inferior to unity, the greater the fraction; consequently, a 
 fraction which has 0 for its denominator is infinite. 
 
 If the double of the first number be less than the second, as would be the case 
 were it proposed to find athird harmonical to 2 and 6, the required divisor will bea 
 negative number. Thus, in the proposed example of 2 and 6, it will be —2; and 
 therefore the third harmonical required will be i2 divided by —2, that is —6.* 
 
 But this inconvenience, if it be one, is not to be apprehended when the greater 
 number is the first term of the proportion; for if the first exceeds the second, much 
 more will its double exceed it. In this case therefore, the third harmonical will 
 always be a finite and positive number. 
 
 II. When three numbers, in decreasing harmonical proportion, are given, for ex- 
 ample 6, 3, 2, it is easy to finda fourth: nothing is necessary but to find a third har- 
 monical to the two last, and this will be the fourth. The third and fourth may, in 
 like manner, be employed to find a fifth, and soc on; and this will form what is called 
 an harmonical progression, which may be always continued decreasing. In the 
 present example, this series will be found to be 
 
 6, By 25 85.08, Bas Ponds OC, 
 or 6, 3, 2, %, $, 1, %, 3, &e. 
 
 Had the two first numbers been 2 and 1, we should have had the harmonical pro- 
 gression 2, ], 4, 4, 4, 1, 4, 4, 4, 4, 4, &e. 
 
 {t is a remarkable property, therefore, of the series of fractions, having for their 
 numerators unity, and for their denominators the numbers of the natural progression, 
 that they are in harmonical progression. 
 
 Besides the numerical relation already mentioned, we find indeed, in the series of 
 these numbers, all the musical concords possible; for the ratio of 1 to 4 gives the 
 octave; that of } to 4, or of 3 to 2, the fifth; that of 4 to 4, or of 4 to 3, the 
 fourth ; that of } to, or of 5 to 4, the third major; that of } to-J, or of 6 to 5, the 
 third minor ; that of } to}, or of 9 to 8, the tone major, and that of } to 4, or of 
 10 to 9, the tone minor. But this will be explained at more length in that part of © 
 this work which treats of music. 
 
 PROBLEM. 
 
 What is the Sum of the Infinite Series of Numbers in Harmonical Progression, 
 1,3,9 44 4 he. ? 
 
 It has been already seen, that a series of numbers in geometrical progression, if 
 continued in infinitum, will always be equal to a finite number, which may easily be 
 cetermined. But is the case the same in the present problem ? 
 
 We will venture to reply in the negative, though an author, in the Journal de 
 Trevoux, has bestowed great labour in endeavouring to prove that the sum of 
 these fractions is finite. But his reasoning consists of mere paralogisms, which he 
 would not have employed had he been more of a geometrician ; for it can be demon- 
 strated that the series 1, 4, 4, 1, 4, &c., may be always continued in such a manner 
 as to exceed any finite number whatever. 
 
 * Sec what has been already said in regard to negative quantities, in the article on arithmetical 
 progression. 
 
2 
 _ 
 > = 
 ee 
 6 
 
 COMBINATIONS AND PERMUTATIONS. 41 % 
 
 SECTION IV. 
 
 Of various Progressions decreasing in infinitum, the Sums of which are known. 
 
 I. A variety of decreasing progressions, which have served to exercise the in- 
 genuity of mathematicians, may be formed according to different laws. Thus, for 
 example, the numerator being constantly unity, the denominators may increase in 
 the ratio of the triangular numbers 1, 3, 6, 10, 18; 21, &c. OF this kind is the 
 following progression : 
 
 4 3 3 4, To 15) wt &e. 
 
 Its sum is finite, and exactly equal to 2, or 1}. 
 In like manner, the sum ofa progression having unity constantly for its numerators, 
 and the pyramidal numbers for its denominators, as, 
 1, 45 to» Yo gs» de» &C. 
 is equal to ]}. 
 That where the denominators are the pyramidals of the second order, as 
 
 1, 3, ts) ab) ty he &e. 
 is equal to 12. ; 
 That where they are the pyramidals of the third order, as 
 : 1, 4, oy ahs The xa) &e., 
 is equal to 12. 
 
 The law therefore which these sums follow, is evident; and if the sum of a similar 
 progression, that, for example, where the denominators are the pyramidals of the 
 tenth order, were required, we might easily reply that it is equal to 14. 
 
 II. Let us now assume the following progression, 
 
 ay ladahoklideaser 
 in which the denominators are the squares of. the numbers of the natural pro- 
 gression, 
 
 If the reader is desirous to know its sum, we shall observe, with Mr. John Ber- 
 noulli, by whom it was first found, that it is finite, and equal to the square of the cir- 
 cumference of the circle divided by 6, or 4 of 3°14159°, 
 
 As to that in which the denominators are the cubes of the natural numbers, Mr. 
 Bernoulli acknowledges that he had not been able to discover it. 
 
 Those who are fond of researches of this kind, may consult a work of saives Ber- 
 noulli, entitled Tractatus de Seriebus Infinitis, whichis at the end of another pub- 
 lished at Bale in 1713, under the title of Ars Conjectandi, where they will find ample 
 satisfaction. They may also consult various other memoirs both of John Bernoulli, 
 to be found in the collection of his works, and of Euler, published in the Transac- 
 tions of the Imperial Academy of Sciences at Petersburgh. 
 
 CHAPTER VIII. 
 
 OF COMBINATIONS AND PERMUTATIONS, 
 
 Berore we enter on the present subject, it will be necessary to explain the method 
 of constructing a sort of table, invented by Pascal,* called the Arithmetical triangle, 
 which is of great utility for shortening calculations of this kind, 
 
 * This is a mistake in Montucla, as the triangle was invented some ages before Pascal: see Dr. 
 Hutton’s Tracts, 4to, p. 69. 
 
Cee " 
 
 42 ARITHMETIC. 
 
 First, form a band a 8 of ten equal squares, and below it another cp of the like kind, 
 but shorter by one square on the left, so that it shall contain only nine squares ; and 
 continue in this manner, 
 
 desl ds: 4h Gob whodeght ale dala] Geto an Gee B 
 Cab i2l 28 thee. 5s alae GS coe ee Sen ee 
 eer eee Cares ee reas: 
 
 Pik pil4dle 80.5 bl 20.046 85 sil oor aes 
 laligl Ged cull: |eteay 27 Ome 
 (ie dee a eel OL [geo Laban 
 
 ae ss TY 
 
 fg le RSs 
 
 meron rr et 
 
 sua 
 
 E 
 
 always making each successive band a square shorter. We shall thus have a series of 
 squares disposed in vertical and horizontal bands, and terminating at each end in a 
 single square so as to form a triangle, on which account this table has been called the 
 arithmetical triangle. The numbers with which it is to be filled up, must be dis- 
 posed in the following manner. 
 
 In each of the squares of the first band a B, inscribe unity, as well as in each of 
 those on the diagonal a E. 
 
 Then add the number in the first square of the band cp, which is unity, to that in the 
 square immediately above it, and write down the sum 2, in the following square. Add 
 this number, in the like manner, to that in the square above it, which will give 3, and 
 write it down in the next square. By these means we shall have the series of the 
 natural numbers, 1, 2, 3, 4, 5, &c. The same method must be followed to fill up 
 other horizontal bands; that is to say, each square ought always to contain the 
 sum of the number in the preceding square of the same row, and that which is imme- 
 diately above it. Thus, the number 15, which occupies the fifth square of the third 
 band, is equal to the sum of ten which stands in the preceding square, and of 5 which 
 isin the square aboveit. The case is the same with 21, which is the sum of 15 and 6; 
 with 35 in the fourth band, which is the sum of 15 and 20; &c. 
 
 The first property of this table is, that it contains, in its horizontal bands, the 
 natural, triangular, pyramidal, &c., numbers; for in the second, we have the natural 
 numbers 1, 2,3, 4, &c.; inthe third, the triangular numbers 1, 3, 6, 10, 15, &c.; in 
 the fourth, the pyramidals of the first order 1, 4, 10, 20, 35, &c.; in the fifth, the 
 
 pyramidals of the second order, I, 5,15, 35, 70, &c. This isa necessary consequence - 
 
 of the manner in which the table is formed ; for it may be readily perceived that the 
 number in each square isalways the sum of those which fill the preceding squares on 
 the left, in the band immediately above. 
 
 The same numbers willbe found in the bands parallel to the diagonal, or the hypo- 
 thenuse of the triangle. 
 
 But a property still more remarkable, which can be comprehended only by such of 
 our readers as are acquainted with algebra, is, that the perpendicular bands exhibit 
 
 the co-efficients belonging to the different members of any power to which a binomial, | 
 
 as a--3, can be raised. The third band contains those of the three members of the 
 square ; the fourth those of the four members of the cube; the fifth, those of the 
 five members of the biquadrate. But, without enlarging farther on this subject, we 
 shall proceed to explain what is meant by combinations. 
 
 By combinations are understood the various ways that different things, the number 
 
COMBINATIONS AND PERMUTATIONS. 43 
 
 of which is known, can be chosen or selected, taking them one by one, two by two, 
 three by three, &c., without regard to their order. Thus, for example, if it were 
 required to know in how many different ways the four letters a, 6, c, d, could be 
 arranged, taking them two and two, it may be readily seen that we can form with 
 them the following combinations ab, ac, ad, bc, bd, cd: four things, therefore, may 
 be combined, two and two, six different ways. ‘Three of these letters may be com- 
 bined four ways, abc, abd, acd, bed ; hence the combinations of four things, taken 
 three and three, are only four. 
 
 In combinations, properly so called, no attention is paid to the order of the things ; 
 and for this reason we have made no mention of the following combinations, ba, ca, 
 da, cb, db, dc. If, for example, four tickets, maiked a, 6, c, d, were put into a hat, 
 
 and any one should bet to draw out the tickets a and d, either by taking two at one 
 time, or taking one after another, it would be of no importance whether a should be 
 drawn first or last: the combinations ad or da ought therefore to be here considered 
 only as one. 
 
 But if any one should bet to draw out a the first time, and d the second, the case 
 would be very different; and it would be necessary to attend to the order in which 
 these four letters may be taken and arranged together, two and two: it may be 
 easily seen that the different ways are ub, ba, ac, ca, ad, da, bc, cb, bd, db, cd, dcw In 
 like manner, these four letters might be combined and arranged, three and three, 24 
 ways, as abc, acb, bac, bca, cab, cba, adb, abd, dba, dab, bad, bda, aed, adc, dac, dca, cad, 
 cda, bcd, dbc, cdb, bdc, cbd, dcb. This is what is called permutation and change 
 of order. 
 
 PROBLEM I. 
 
 Any number of things whatever being given ; to determine in how many ways they may 
 be combined two and two, three and three, &c., without regard to order. 
 
 This problem may be easily solved by making use of the arithmetical triangle. Thus, 
 for example, if there are eight things to be combined, three and three, we must take 
 ‘the ninth vertical band, or in all cases that band, the order of which is expressed by a 
 number exceeding by unity the number of things to be combined; then the fourth 
 horizontal band, or that the order of which is greater by unity than the number of 
 the things to be taken together, and in the common square of both will be found the 
 number of the combinations required, which in the present example will be 56. 
 
 But as an arithmetical triangle may not always be at hand, or as the number of 
 things to be combined may be too great to be found in such a table, the following 
 simple method may be employed. 
 
 The number of the things to be combined, and the manner in which they are to be 
 taken, viz. two and two, or three and three, &c., being given: 
 
 Ist. Form two arithmetical progressions, one in which the terms go on decreasing 
 by unity, beginning with the given number of things to be combined; and the other 
 consisting of the series of the natural numbers l, 2, 3, 4, &c. | 
 _ 2d. Then take from each as many terms as there are things to be arranged together 
 in the proposed combination. 
 
 3d. Multiply together the terms of the first progression, and do the same with 
 those of the second. 
 
 4th. In the last place, divide the first product by the second, and the quotient wil- 
 be the number of the combinations required. 
 
 I. In how many ways can 90 things be combined, taking them two and two ? 
 
 According to the above rule we must multiply 90 by 89, and divide the 
 
44 ARITHMETIC. 
 
 product 8010 by the product of 1 and 2, that is 2: the quotient 4005 will be the 
 number of the combinations resulting from 90 things, taken two and two. 
 
 Should it be required, in how many ways the same things can be combined three 
 and three, the problem may be answered with equal ease ; for we have only to mul- 
 tiply together 90, 89, 88, and to divide the product 704880 by that of the three num- 
 bers 1, 2,3; the quotient 117480 will be the number required. 
 
 In like manner, it will be found that 90 things may be combined by four and four, 
 2555190 ways ; for if the product of 90, 89, 88, and 87, be divided by 24, the product 
 of 1, 2, 3, 4, we shall have the above result. 
 
 In the last place, if it be required, what number of combinations the same $0 
 things, taken five and fiye, are susceptible of, it will be found, by following the rule, 
 that the answer is 43949268. 
 
 II].—Were it asked, how many conjunctions the seven planets could form with each 
 other, two and two, we might reply 21; for, according to the general rule, if we mul- 
 tiply 7 by 6, which will give 42, and divide that number by the product of 1 and 2, 
 that is 2, the quotient will be 21. ; 
 
 If we wished to know the number of all the conjunctions possible of these seven 
 planets, two and two, three and three, &c. ; by finding separately the number of the 
 conjunctions two and two, then those of three and three, &c., and adding them 
 together, it will be seen that they amount to 120. 
 
 The same result might be obtained by adding the seven terms of the double geo- 
 metrical progression 1, 2, #, 8, 16, 32, 64, which will give 127. But from this,num- 
 ber we must deduct 7, because when we speak of the conjunction of a planet, it is 
 evident that two of them, at least, must be united; and the number 127 compre-— 
 hends all the ways in which seven things can be taken one and one, two and two, 
 three and three, &c. In the present example, therefore, we must deduct the number 
 of the things taken one and one; for a single planet cannot form a conjunction. 
 
 PROBLEM II. . 
 Any number of things being given; to find in how many ways they can be arranged. 
 
 This problem may be easily solved by following the method of induction; for 
 
 Ist. One thing a can be arranged only in one way : in this case therefore the num- 
 ber of arrangements is = 1. 
 
 Qnd. ‘Two things may be arranged together two ways; for with the letters aand6 
 we can form the arrangements ab and da; the number of arrangements therefore is’ 
 equal to 2, or the product of 1 and 2. 
 
 3rd. The arrangements of three things a,b,c, are in number six; for ab can form 
 with c, the third, three different ones, bac, bca, cba, and there canbe no more. Hence 
 it is evident that the required number is equal to the preceding multiplied by 3, or 
 to the product of 1,2, 3. 
 
 4th. If we add a fourth thing, for instance d, it is evident that, as each of the pre- 
 ceding arrangements may te combined with this fourth thing four ways, the above 
 number 6 must be multipled by 4 to obtain that of the arrangements resulting from 
 four things ; that is to say, the number will be 24, or the product of 1, 2, 3, 4. 
 
 It is needless to enlarge farther on this subject; for it may be easily seen that, 
 whatever be the number of the things given, the number of the arrangements they 
 are susceptible of may be found by multiplying togetheras many terms of the natural 
 arithmetical progression as there are things proposed. 
 
/ 
 
 COMBINATIONS AND PERMUTATIONS. 45 
 
 Remark.—I\st. It may sometimes happen that, of the things proposed, one of them 
 is repeated, as a,a,6,c. In this case, where two of the four things préposed are the 
 same, it will be found that they are susceptible only of 12 arrangements instead of 
 24; and that five, where two are the same, can form only 60, instead of 120, 
 
 But if three of four things were the same, there would be only 4 combinations, in- 
 stead of 24; and five things, if three of them were the same, would give only 20, in- 
 stead of 120, or asixth part. But as the arrangements of which two things are sus- 
 ceptible amount to 2, and as those which can be formed with three en are 6, we 
 may thence deduce the following rule: 
 
 In any number of things, of which the different arrangements are requir edi if one 
 of them be several times repeated, divide the number of arrangements, found agcord- 
 ing to the general rule, by that of the arrangements which would be given by\the 
 things repeated, if they were different, and the quotient will be the number 
 required. 
 
 2nd. In the number of things, the different arrangements of which are required if. 
 
 there are several of them which occur several times, one twice for example, and 
 another three times ; nothing will be necessary but to find the number of the arrange- 
 ments according to the general rule, and then to divide it by the product of\the 
 numbers expressing the arrangements which each of the things repeated would) be 
 susceptible of, if instead of being the same, they were different. Thus, in the pre- 
 sent case, as the things which occur twice would be susceptible of two arrangements 
 if they were different; and as those which occur thrice would, under the like circum- 
 stances, give six; we must multiply 6 by 2, and the product 12 will be the 
 number by which that found according to the general rule ought to be divided. 
 Thus, for example, the five letters a, a,6,6, 6, can be arranged only 10 different ways: 
 for, if they were different, they would give 120 arrangements; but as one of them 
 occurs twice, and another thrice, 120 must be divided by the preduct of 2 and 6, or 
 12, which will give 10. 
 
 By observing the precepts given for the solution of this problem, the following 
 questions may be resolved. 
 
 1.—A club of seven persons agrced to dine together every day successively, as long as they 
 could sit down to table differently arranged. How many dinners would be necessary 
 - for that purpose ? 
 It may be easily found that the required number is 5040, which would require 13 
 years and more than 9 months. 
 
 IIJ.—The different anagrams which can be formed with any word, may be found in 
 dike manner. Thus, for example, if it be required, how many different words can be 
 formed with the four letters of the word amor, which will give all the possible 
 anagrams of it, we shall find that they amount to 24, or the continued product of 1, 
 2, 3,4. We shall here give them in their regular order. 
 
 AMOR MORA ORAM RAMO 
 AMRO MOAR ORMA RAOM 
 AOMR MROA OARM RMAO 
 AORM MRAO OAMR RMOA 
 7 ARMO MAOR OMRA ROAM 
 AROM MARO OMAR ROMA 
 
 Hence it appears that the Latin anagrams of the word amor, are in number seven, 
 viz., Roma, mora, maro, oram, ramo, armo, orma. But in the proposed word, if one 
 or more letters were repeated, it would be necessary to follow the precepts already 
 
46 
 
 ARITHMETIC. 
 
 given, Thus, the word Leopoldus, where the letter ¢ and the letter o both occu 
 twice, is susceptible of only 90720 different arrangements, or anagrams, instead o 
 362880, which it would form, if none of the letters were repeated; for, according t¢ 
 the before-mentioned rule, we must divide this number by the product of 2 by 2,0 
 4, which will give 90720. 
 The word studicsus, where the wu occurs twice, and the s thrice, is susceptible o 
 only 30210 arrangements ; for the arrangements of the 9 letters it contains, whicl 
 are in number 362880, must be divided by the product of 2 and 6, or twelve, and thy 
 quotient will be 30240. m 
 In this manner may be found the number of all the possible anagrams of any wort 
 whatever ; but it must be observed that however few be the letters of which ; 
 word is composed, the number of the arrangements thence resulting will be so grea 
 as (o require considerable labour to find them. 
 
 \Il.—How many ways can the following verse be varied, without destroying the 
 
 measure : 4 
 
 “ Tot tibi sunt dotes, Virgo, quot sidera czelo 2” 
 
 This verse, the production of a devout Jesuit of Louvain, named Father Bauhuys, 
 is celebrated on account of the great number of arrangements of which it is suscep: 
 tible, without the laws of quantity being violated ; and various mathematicians have 
 exercised or amused themselves with finding out the number. Erycius Puteanus 
 took the trouble to give an enumeration of them in forty-eight pages, making them 
 amount to 1022, or the number of the stars comprehended in the catalogues of the 
 ancient astronomers ; and he very devoutly observes, that the arrangements of these 
 ‘words as much exceed the above number as the perfections of the Virgin exceed that 
 of the stars.* 
 
 Father Prestet, in the first edition of his Elements of the Mathematics, says that 
 this verse is susceptible of 2196 variations; but in the second edition he extends the 
 number to 3276, 
 
 Dr. Wallis, in the edition of his Algebra, printed at Oxford, in 1693, makes them 
 amount to 3096. : 
 
 But none of them has exactly hit the truth, as has been remarked by James .Ber- 
 noulli, in his Ars Conjectandi. This author says, that the different combinations of 
 the above verse, leaving out the spondees, and admitting those which have no 
 cesura, amount exactly to 3312. The method by which the enumeration was made 
 may be seen in the above work. 
 
 The same question has been proposed respecting the following verse of Thomas 
 Lausius : 
 
 _ “ Mars, mors, sors, lis, vis, styx, pus, nox, fex, mala crux, fraus.’’ 
 
 It may be easily found, retaining the word mala in the antepenultima place, in 
 order to preserve the measure, that this verse is susceptible of 399168000 different 
 arrangements. 
 
 PROBLEM IIT. 
 
 Of the combinations which may be formed with squares divided by « diagonal into two 
 differently coloured triangles. 
 
 We are told by Father Sebastian Truchet, of the Royal Academy of Sciences, in 
 a memoir printed among those of the year 1704, that having seen, ; 
 
 during the course 
 of a tour which he made to the canal of Orleans, some square pore 
 
 elain tiles, divided 
 
 * See also Vossius de Scient. Math. Cap. Vii. 
 
COMBINATIONS AND PERMUTATIONS. \ 47 
 
 by a diagonal into two triangles of different colours, destined for pavng a chapel 
 
 and some apartments, he was induced to try in how many different ways they could — 
 
 be joined side by side, in order to form different figures. In the first place, it may 
 be readily seen that a single square, (Fig. 9.) accoriting to its 
 position can form four different figures, which however may 
 a = be reduced to two, as there is no difference between the 
 = S A first and the third, or between the second and fourth, than 
 what arises from the transposition of the shaded triangle 
 into the place of the white one. | 
 _ Now, if two of these squares be combined together, the result will be 64 different 
 ways of arrangement; for, in that of two squares, one of them may be made to 
 assume four different situations, in each of which the other may be changed 16 
 times. The result therefore will be 64 combinations. 
 We must however observe, with Father Sebastian, that one half of these combi- 
 nations are only a repetition of the other, in a contrary direction, which reduf{es 
 ‘them to 32; and if attention were not paid to situation, they might be reduced to 
 10. 
 In like manner, we might combine three, four, five, &c., squares together, and in 
 that case it would be found, that‘three squares are capable of forming 128 figures ; 
 
 * that four could form 256, &c. 
 
 The immense variety of compartments which arise, in this 
 manner, from so small a number of elements, is really astonish- 
 ing. Father Sebastian gives thirty different kinds, selected from 
 a hundred; and these even are only a very small part of those 
 which might be formed. The annexed figure (10.) exhibits one 
 of the most remarkable. 
 
 In consequence of Father Sebastian’s memoir, Father Douat, 
 one of his associates, was induced to pursue this subject still far- 
 ther, and to publish, in the year 1722, a large work, in which 
 it is considered ina different manner. It is entitled ‘‘ Méthode pour faire une 
 infinité de dessins différents, avec .des carreaux mi-partis de deux couleurs par 
 uné ligne diagonale ; ou, Observations du P. D. Donat, religieux Carme de la P. de 
 T. sur un Mémoire inseré dans l’Hist. de l’Acad. royale des Sciences de Paris, 
 
 année 1704, par le P. S: Truchet, religieux du méme ordre,” Paris 1722, in 4to. In 
 
 this work it may be seen that four squares, each divided into two triangles of different 
 colours, repeated and changed in every manner possible, are capable of forming 256 
 
 “different figures; and that these figures themselves, taken two and two, three and 
 
 three, and so on, will form a prodigious multitude of compartments, engravings of 
 
 which occupy the greater part of the book. 
 
 It is rather surprising that this idea should have been so little employed in archi- 
 tecture ; as it might furnish an inexhaustible source of variety in pavements, and other 
 works of the like kind. However this may be, it forms the object of a pastime, called by 
 the French Jeu du Parquet. The instrument employed for this pastime, consists of 
 a small ‘table, having a border round it, and capable of receiving 64 ora hundred 
 small squares, each divided into two triangles of different colours, with which people 
 amuse themselves in endeavouring to form agreeable combinations. 
 
48 Rea a ARITHMETIC, 
 
 CHAPTER IX. | 
 
 j | 
 
 Pale OF THE DOCTRINE OF COMBINATIONS TO GAMES OF CHANCE AND 
 
 j J 
 i | TO PROBABILITIES. 
 
 Tuov¢eH nothing, on the first view, seems more foreign to the province of the 
 mathematics than chance, the powers of analysis have, as we may say, enchained this 
 Pres and subjected it to calculation. It has found means to measure the differ- 
 ent degrees of probability ; and this has given birth toa curious branch of the ma- 
 thematics, the principles of which we shall here explain. 
 
 then an event can take place different ways, it is evident that the probability of 
 its/happening in a certain determinate manner, will be greater when, of the whole of 
 the ways in which it can happen, the greater number determine it to happen in that 
 
 yjanner. Ina lottery, for example, every one knows that the probability or hope of 
 ‘ebtaining a prize, is greater according as the number of prizes is greater, and as the 
 
 total number of the tickets is less. The probability therefore of an event is in the 
 compound ratio of the number of the cases which can produce it, taken directly, and 
 
 / of the total number of those according to which it may be varied, taken inversely ; 
 
 consequently it may be expressed by a fraction, having for its numerator the number 
 of the favourable cases, and for its denominator the whole of the cases. 
 
 Thus, in a lottery consisting of a thousand tickets, 25 of which only are prizes, the 
 chance of obtaining one of the latter will be represented by;33, or 4,; if the num- 
 ber of the prizes were 50, this probability would be double, for in that case it would 
 be equal to 3; but, on the other hand, if the whole number of tickets, instead of a 
 thousand, were two thousand, the probability would be only one half of the former, 
 that is {;. If the whole number of tickets were infinitely great, the number of 
 prizes still remaining the same, the provability would be infinitely small; and if the 
 whole number of tickets were prizes, it would become certainty, and in that case 
 would be expressed by unity. 
 
 Another principle of this theory, necessary to be here explained, the enunciation 
 of which will be sufficient to shew the truth of it, is as follows: 
 
 We play an equal game, when the money deposited is in direct proportion to the 
 probability of gaining the stake; for, to play an equal game, is nothing else than to 
 deposit a sum so proportioned to the probability of winning, that, after a great num- 
 ber of throws or games,the player may find himself nearly at par ; but for this purpose, 
 the sums deposited must be proportioned to the degree of probability, which each of 
 the players has in his favour. Let us suppose, for example, that A bets against B ona 
 throw of the dice,and that the chances are two to one in favour of A; the game will be 
 equal if, after a great number of throws, the parties separate nearly without any loss; 
 but as there are two chances in favour of A, and only one in favour of B, after three 
 hundred threws A will have gained nearly two hundred, and B one hundred; A 
 therefore ought to deposit two and B only one; for by these means, as A in winning’ 
 two hundred throws will gain 260, B in winning a hundred throws will gain 200 also. 
 In such cases therefore, it is said that two to one may be betted in favour of A. 
 
 PROBLEM I. 
 
 In tossing up, what probability is there of throwing a head several times successively, 
 or a tail; or, in playing with several pieces, what probability is there that they will 
 be all heads, or all tails ? 
 
 In this game it is evident, Ist, That as there is no reason why a head should come 
 up rather than a tail, or a tail rather than a head, the probability that one of the two 
 will be the case is equal to } , or an equal bet may be taken for or against. 
 
CHANCES AND PROBABILITIES. i 49 
 
 But ifthe game were for two throws, and any one should bet that a head will come 
 up twice, it must be observed, that all the combinations of head or tail, which can 
 take place in two successive throws with the same piece, are head, head; head, tail ; 
 tail, head ; tail, tail ; one of which only gives head, head. ‘There is therefore only 
 one case in four which can make the person win who bets to throw a head twice in 
 succession ; consequently the probability of this event is only; and he who 
 bets in favour of two heads, ought to deposit a crown, and the person who bets against 
 him ought to deposit three; for the latter has three chances of winning, while the 
 former has only one. To play an equal game then, the sums deposited by each 
 ought to be in this proportion. 
 
 It will be found also, that he who bets to throw a head three times in succession, 
 will have in his favour only one of the eight combinations of head and tail, which may 
 result from three throws of the same piece. The probability of this event there- 
 fore is 4, while that in favour of his adversary will be 7. Consequently, to play an 
 equal game, he ought to stake 1 against 7. 
 
 It is needless to go over all the other cases; for it may be easily seen, that the 
 probability of throwing a head four times successively is 7,; five times successively, 
 sy, &e. 
 
 It is unnecessary also to enumerate all the different combinations which may result 
 from head or tail; but in regard to probabilities, the following simple rule may be 
 employed. 
 
 The probabilities of two or more single events being known, the probability of 
 their taking place altogether may be found, by multiplying together the probabilities 
 of these events, considered singly. 
 
 Thus the probability of throwing a head, considered singly, being expressed at each 
 throw by }, that of throwing it twice in succession, will be 4 x } or 4; that 
 of throwing it three times, and three successive throws, will be 4 x 4 x 4, or 
 e &C. 
 
 2d. The problem, to determine the probability of throwing, with two, three, or 
 four pieces, all heads or all tails, may be resolved by the same means. When two 
 pieces are tossed up, there are four combinations of head and tail, one of which only 
 is all heads. When three pieces are tossed up together, there are 8, one of which . 
 only gives all heads, &c. The probabilities of these cases therefore are the same as 
 those of the cases similar to them, which we have already examined. 
 
 It may be easily seen indeed, without the help of analysis, that these two questions 
 are absolutely the same; and the following mode of reasoning may be employed to ° 
 prove it. To toss up the two pieces A and B together, or to toss them up in suc- 
 cession, giving time to A, the first, to settle before the other is tossed up, is certainly 
 the same thing. Let us suppose then, that when A, the first, has settled, instead of 
 - tossing up B, the second, A the first is taken from the ground, in order to be tossed 
 up a second time; this will be the same thing as if the piece B had been employed 
 for a second toss; for by the supposition they are both equal and similar, at least 
 in regard to the chance whether head or tail will come up. Consequently, to toss 
 up the two pieces A and B, or to toss up twice in succession the piece A, is the same 
 thing. Therefore, &c. 
 
 3d. We shall now propose the following question: What may a person bet, that 
 in two throws a head will come up at least once? By the above method it will be 
 found, that the chances are 3 to 1. In two throws, indeed, there are four combina- 
 tions, three of which give at least a head once in the two throws, and one only which 
 gives all tails; hence it follows, that there are three combinations in favour of the 
 person who bets to bring a head once in two throws, and only one against him. 
 
 1D) 
 
/ 
 
 50 ARITHMETIC, ? 
 
 PROBLEM II. 
 
 Any number of dice being given; to determine what probability there is of throwing an 
 assigned number of points. 
 
 We shall first suppose that the dice are of the ordinary kind, that is to say, having 
 six faces, marked with the numbers 1, 2, 3, 4, 5,6; and we shall analyse some of the 
 first cases of the problem, in order that we may proceed gradually to those that are 
 more complex. 
 
 Ist. tis proposed to throw a determinate point, 6 for example, with one die. 
 
 Here it is evident, that as the die has six faces, one of which only is marked 6, and 
 as any one of them may as readily come upas another, there are 5 chances against the 
 person who proposes to throw a six at one throw, and only one in his favour. 
 
 2d. Let it be proposed to throw the same point 6 with two dice. 
 
 To analyse this case, we must first observe that two dice give 36 different combi- 
 nations ; for each of the faces of the die A, for example, may be combined with each 
 of those of the die B, which will produce 36 combinations. But six may be thrown, 
 Ist, by 3and 3; 2d, by 2 with the die A, and 4 with the die B, which, as may be 
 readily seen, forms two distinct cases: 3d, by 1 with the die A, and 5 with the die B, 
 or 1 with B and 5 with A, which also gives two cases; and these are all that are 
 possible. Hence there are 5 favourable chances in 36; consequently the probability 
 
 of throwing 6 with two dies is », and that of not throwing it is 34. This therefore — 
 
 ought to be the ratio of the stakes or money deposited by the players. 
 By analysing the other cases, it will be found that, of throwing two with two dice, 
 there is one chance in 36; of throwing three, there are 2; of throwing four, 33 
 
 of throwing five, 4; of throwing six, 5; of throwing seven, 6; of throwing eight, 5;.— 
 
 of throwing nine, 4; of throwing ten, 3: of throwing eleven, 2; and of throwing 
 sixes, l. 
 Ifthree dice were proposed, with which it is evident the lowest point would be 
 three, and the highest eighteen, it will be found, by means of a similar analysis, that 
 in 216, the whole number of the throws possible with three dice, there is 1 chance of 
 
 throwing three ; 3 of throwing four ; 6 of throwing five, &c. : as may be seen in the — 
 
 annexed table, the use of which is as follows. 
 
 If it be required, for example, to find in how many ways 13 can be thrown with © 
 three dice, we must look in the first vertical column, on the left, for the number 13, ~ 
 
 and at the top of the table for 3, the number of the dice; and in the square below, 
 
 opposite to 13, will be found 21, the number of ways in which 13 may be thrown y 
 with three dice. In like manner, it will be found, that with 4 dice, it may be thrown ~ 
 
 140 ways; with five dice, 420; &c. 
 
CHANCES AND PROBABILITIES. 51 
 
 va 1 ie pa aay Aa vA 
 ‘ a » 
 Af : 
 a 
 
 Table of the different ways in which any point can be thrown with one, two, three, 
 or more dice. 
 
 Wumber of the Dice. 
 
 re ey LIT EVE oi Vico WL! 
 Pewee er | | 
 Ar Phar fen ties | | | 
 2h ks Sel Ma ale | | 
 SE TS: EB oak | : 
 Sola pk a Fen eae Fe 
 Ochs ole pelo. el [a oats | 
 7 POV Po las eoo felon eG 
 8 | RES wor ee a oo ole 
 ce (ee ROR er Sor 20 eee 
 -& | 10 | aot oy ie oo | oa eee 
 seal ae eo oy 104. P05 ie oh 
 Sol doe | 5 Ly fe eoa 25] 305" p4oe 
 mets | | | 21) 140" |°420 [756 
 E14 | | | 15 | 146 | 540 | 1161 
 A }15 | | | 10 | 140 | 651 | 1666 
 16 | | ror log hen oo 
 17 | | | 3 | 104 | 780 | 2856 
 18 | l Perk feed: ive0. poder 
 19 | | l | 56 | 735 | 3906 
 20 | | l | 35 | 651 | 4221 
 74 | | | 20 | 540 | 4339 
 27 | | | | 10 | 420 | 4221 
 23 | | | | 4 | 305 | 3906 
 “24 | | | fad | 20521 843] 
 25 | | | l | 126 | 2856 
 
 When it is once known how many ways a point can be thrown with a certain 
 number of dice, the probability of throwing it may be easily found: nothing is ne- 
 cessary but to form a fraction, having for its numerator the number of ways in which 
 the point can be thrown, and for denominator the number 6, raised to that power 
 indicated by the number of dice; as the cube of 6, or 216, for three dice; the bi- 
 quadrate, or 1296, for four dice; &c. 
 
 Thus, the probability of throwing 13 with three dice, is $4; of throwing it with 
 four, 449, ; &c. 
 
 Various other questions may be Siena concerning the throwing of dice, a few 
 of which we shall here examine. 
 
 Ist. When two players are engaged ; to determine the advantage or disadvantage of 
 
 the person who undertakes to throw a certain face, that for example marked 6, in. 
 a certain number of throws. 
 
 Let us suppose that he undertakes it at one throw: to find the probability of his 
 succeeding, it must be considered, that he who holds the die has only one chance of 
 winning, and five of losing ; consequently to undertake it at one throw, he ought to 
 stake no more than one to five. There is therefore a great disadvantage in under- 
 taking, on an even bet, to throw six at a single throw of one die. 
 
 To determine the probability of throwing the face marked 6 in two throws with a 
 single die, we must observe, as has been already said, in regard to tossing up, that 
 
 E2 
 MOMIVERSITY OF iLiine 
 | LIBRARY LIBRARY 
 cs UNIVERSITY OF ILLINOIS 
 
 4 AE URBANA-CHAMPAIGN 
 
52 ARITHMETIC. 
 
 this is the same thing as to undertake, in throwing two dice together, that one of them 
 shall have the side marked 6 uppermost. He then who holds the dice has only 11 
 chances, or combinations, by which he -can win; for he may throw 6 with the first 
 die, and 1, 2, 3, 4, or 5 with the second; or 6 with the second die, and 1, 2, 3, 4, or 5 
 with the first, or 6 with each die. But there are 25 combinations or chances unfa- 
 vourable to his winning, as may be seen in the following table: 
 
 Lee slecele era th ae 
 Le aoe se owl 40 le ae 
 [23 F003 | Os ied tae ae Chee 
 Wepie et Ie as eee Map ea eh 
 Tie ROP IS eS ee eee 
 
 Hence it may be concluded, that he who undertakes to throw a 6 with two dice, 
 ought to stake no more than 11 to 25; and consequently, that it would be disadvan- 
 tageous to do it on equal terms. 
 
 It must here be observed that 36, the number of all the chances or combinations 
 possible in two throws of the dice, is the square of 6, which is the number of the 
 faces of one die; and that 25, the number of the chances unfavourable to the person 
 who undertakes to throw a determinate face, is the square of 5, or of 1 less than the 
 same number 6. The number of the favourable chances therefore, in this case, is 
 equal to the difference of the squares 36 and 25, or of the square of the number of 
 the faces of one die, and of that of the faces of the same die less one. 
 
 In the case of undertaking to bring a6 in three throws with one die, we must 
 consider, in like manner, that this is the same thing as to undertake that, in throwing 
 three dice at once, one of them shall bring a6; but of the 216 combinations, which 
 result from three dice, there are 125 without a6, and 91 among which there is at 
 least one 6 ; consequently, he who engages to throw a 6, either in three throws with: 
 one die, or one throw with three dice, ought to bet no more than 91 to 125; and it 
 would be disadvantageous to undertake it on equal terms. 
 
 It is here to be observed, that the number 9] is the difference of the cube of the 
 number of the faces of one die, viz. 216, and of 125, the cube of the same number 
 less unity, or of 5. Hence it appears that, in general, to find the probability of 
 throwing a determinate face, ina certain number of throws, or at one throw with a — 
 certain number of dice, we must raise 6, the number of the faces of one die, to that — 
 power which is indicated by the number of throws agreed on, or by the number of 
 dice to be thrown at one time; we must then raise 6 less unity, or 5, to the same 
 power, and subtract it from the former; the remainder with this power of 5 will be — 
 the respective number of chances for winning or losing. 
 
 Thus, if a person should bet to throw at least one 3 with four dice, we must raise 
 6 to the 4th power, which is 1296, and subtract from it the fourth power of 5, or 
 625 ; the remainder 671 will be the number of chances for winning, and 625 that of 
 the chances of losing ; consequently there will be an advantage in laying an even 
 bet. ; 
 
 It is advantageous also to undertake, on an even bet, to throw any determinate 
 point, for example 3, in five throws, or with five dice; for if from the 5th power of 
 6, which is 7776, we deduct the 5th power of 5, or 3125, the remainder 4651 will 
 be the number of favourable chances, and 3125 that of the unfavourable. Conse- 
 quently, to play an equal game, he who bets on throwing the above point, ought to 
 deposit 4651 to 3125, or nearly 3 to 2. 
 
 2d. In how many throws may one bet, on equal terms, to throw a determinate doublet, 
 
 for example sixes, with two dice ? 
 
 It has been already shewn, that the probability of not throwing sixes with two 
 two dice, is 33; consequently the probability of their not coming up in two throws, 
 
CHANCES AND PROBABILITIES. 53 
 
 will be the square of that fraction; in three throws, the cube, &c. But as the. 
 powers of every number greater than unity, however small the excess, go on always 
 increasing, those of a number less than unity, however small the defect, go on always 
 decreasing: the. consecutive powers therefore of 33 will go on always decreasing. 
 
 Now let us conceive 32 to be raised to such a poe a as to be equal to 3; it will be 
 found that the 24th Bower of 33 is somewhat greater than 3; and that the 25th 
 power is somewhat less ;* renee it follows that one may lay an even bet with some 
 advantage, that another will not bring sixes in 24 throws with two dice, but that 
 there is some disadvantage in taking an even bet that they will not come up in 25 
 throws. Consequently, he who bets on throwing sixes in 24 throws, dees so with 
 disadvantage ; but if he lays an even bet that they will come up in 25 throws, the 
 advantage is in his favour. 
 
 3d. What probability is there of throwing any determinate doublet, for example two 
 
 threes, in one throw with two or more dice ? 
 
 To determine this question, we must first observe, that he who undertakes to 
 throw two threes with two dice, has only one favourable chance, in the 36 chances 
 or combinations given by two dice; and it thence follows that he ought to bet no 
 more than 1 to 35. 
 
 In the case of three dice, it will be found that he ought to bet no more than 16 to 
 900; for the number of chances or combinations possible with three dice is 216. 
 But when it is proposed to throw two threes with three dice, they may come up 16 
 different ways; for in the 36 combinations of the two dice A and B, all those in 
 which one 3 only is found, as 1, 3; 3,1, &c., being 10 in number, by combining with 
 the side marked 3 of the die C, give two threes. Besides the combination 3, 3 of 
 the dice A and B, by combining with one of the six faces of the third C, will give 
 two threes. Here then we have 16 ways of throwing two threes with three dice, 
 which give 16 favourable chances in 216. Consequently, the probability of throwing 
 two threes with three dice is }§; and no more ought to be betted on the success 
 of that event than 16 to 200, or 2 to 20. 
 
 If the probability of throwing two threes with four dice be required, we shall find 
 thatit is expressed by #21, ; for, of the 1296 combinations of the faces of four dice, there 
 are 150 which give two phrees: 20 that give 3, and one that gives 4, making altogether 
 171 throws; which give 2or38or4 threes. Consequently,no more than 17] to 1125, 
 or about 1 to 63, ought to be betted on throwing, at least, once threes with four dice. 
 
 In the last place, if the probability of throwing any doublet, at one throw, with 
 two or more dice, be required, it may be easily determined by the preceding method 
 of calculation; for if an indeterminate doublet be proposed, it is evident that the 
 probability is six times as great as when an assigned doublet is proposed; and there- 
 fore we have only to multiply the probabilities already found by 6. The probability 
 ‘therefore with two dice, will be $ or 4; with three dice, 2% or 4; with four dice, 
 1926 or 12, &c. So that there is an advantage in taking an even bet to throw at least 
 one doublet with four dice. 
 
 This property is not true when the number of dice exceeds three. The probability 
 of an assigned doublet with four dice is #44, which multiplied by 6, and added to the. 
 
 * Let n be the exponent of that power of 33 which is equal to 4; that is to say, let 
 _ ’ be equal to 3. As the unknown quantity m is in the exponent, it must be disen- 
 gaged from it, which may be done by means of logarithms. For 2 = 4, by taking 
 
 . the logarithms we shall have n log. 35 — n log. 36 = log. 3, or = — log. 2; for log. 
 = — log. 2. Hence n log. 35 — n log. 36 = — Jog. 2, or log. 2 = nm log. 36 — 
 ier 35. Therefore, n — —___!08- ?- Which gives n == 24°605, or 24¥, 
 
 log. 36 — log, 35 - 
 nearly. 
 
54 ARITHMETIC. 
 
 probability of a different face coming up with each die, viz. to &x& X4 xX Zor 282, 
 gives 1386, being 90 chances more than there is in all the four dice, rie is im- 
 possible. 
 
 The probability of an assigned doublet with four dice, viz. 4, includes the pro- 
 bability of some other doublet ; for aces, twos, or any other doublet may turn up at 
 the same throw, which cannot happen with two dice or with three; so that the 
 multiplier 6 will not answer to the probability of an indeterminate doublet, when 
 there are more than three dice. 
 
 In such cases it is the safest and easiest way to find the probability of the reverse 
 problem—of not throwing doublets—and then subtracting that probability from 
 unity or certainty, the remainder is the probability for doublets. 
 
 PROBLEM Iii. 
 
 Two persons sit down to play for a certain sum of money ; and agree that he who first 
 gets three games shall be the winner. One of them has got two games, and the other 
 one ; but being unwilling to continue their play, they resolve to divide the stake: how 
 much of it ought each person to receive ? 
 
 This problem is one of the first that engaged the attention of Pascal, when he 
 began to study the calculation of probabilities. It was resolved by Fermat, a cele- 
 brated geometrician, to whom he proposed it, by a different method, viz. that of com- 
 binations: we shall here give both. 
 
 It is evident that each of the players, in depositing his money, resigns all right to 
 it; but, in return, each has a right to what chance may give him ; consequently when 
 they give over playing, the stake ought to be divided in proportion to the probability 
 each had of winning the whole sum had they continued. 
 
 Case 1st. This proportion may be determined by the following reasoning. Since the 
 first player wants one game to be out, and the second two, it may be readily perceived, 
 that if they continue their play, and if the second should win one game, he would 
 want, in the same manner as the first, one game to be out ; and in that case, the two 
 players being equally advanced, their hopes or chances of winning would be equal. 
 This being supposed, they would have an equal right to the stake, and consequently 
 each ought to have an equal share of it. 
 
 It is evident therefore, that if the first should win the game about to be played, 
 the whole money deposited would belong to him; and that if he lost it he would 
 have a right only to the half. But the one case being as probable as the other, the 
 first hasa right to the half on both these sums taken together. But together they 
 make 3, the half of which i is 3; and this is the share of the stake belonging to the © 
 first players ; consequently that belonging to the second is only 3. 
 
 Case 2nd. The solution of the first case will enable us to solve the second, in 
 which we suppose that the first player wants one game to be out, and the second 
 three; for if the first should win one game, he would be entitled to the whole stake, 
 and if he lost one game, by which means the second would want only two games to 
 be out, 3 of the money would belong to the former, as the parties would then be in 
 the situation alluded to in the preceding case. But as both these events are equally 
 probable, the first ought to have the half of the two sums taken together, or the — 
 half of 3, that is 7: the remainder 3 will therefore be what belongs to the second. 
 
 Case 3rd. It will be found, by the like reasoning, if we suppose two games wanting 
 to the first player, and three to the second, that on ceasing to play, they ought to 
 divide the stake in such a manner that the former may have }j and the latter #,. 
 
 Case 4th. If four games were to be played, and if the first wanted only two 
 games, and the second four, the money ought oo be divided in such a manner that the 
 
 former should have }2 and the latter ¥. 
 
CHANCES AND PROBABILITIES, 55 
 
 But we may dispense with the above reasoning, and employ the following general 
 rule, deduced from it, which is to be applied by means of the arithmetical triangle. 
 Enter that diagonal of the arithmetical triangle, the order of which is equal to the 
 number of the games wanting to both players. As this number in the first case is 
 3, we must enter the third diagonal of the triangle; then because the first player 
 wants only one game, we must take the first number of that diagonal; but because 
 two are wanting to the second, we must take the sum of the two first numbers, 
 which will ioe 3. These two numbers therefore, 1 and 3, will indicate, that the 
 stake ought to be divided in the same proportion: consequently the first player 
 ought to have 3, and the second 4 
 
 As this rule may be easily apetiaa to every other case whatever, we shall ape 
 no farther on the subject. 
 
 The second method of resolving problems of this kind, which is that of combina- 
 tions, is as follows: 
 
 To resolve, for example, the fourth case, Hii oe: according to the supposition, the 
 first player wants two games to be out, and the second four, so that together they 
 want six games; take unity from that sum, and because 5 remain, we shall suppose 
 the five similar letters. a a a a a, favourable to the first player, and the five following, 
 b6bb 4, favourable to thesecond. These letters must be combined, as in the fol- 
 lowing table, where, of the 32 combinations which they form, the first 26, towards 
 the left, where a occurs at least twice, will indicate the number of chances which 
 the first has of winning; and the last 6, towards the right, in which @ never occurs 
 oftener than once, will indicate those favourable to the second. 
 
 aaaaa aaabb aabbb\abbbb 
 aaaab aabba abbba\bbbba 
 aaaba abbaa bbbaa|lbabbb 
 aabaa bbaaa ababb|bbabd 
 abaaa aabab. abbabjbbbab 
 buaaa abaab bbaab|bbbbb 
 
 baaab baabb 
 
 baaba babba 
 
 babaa bbaba 
 
 ababa eaebiaa bi 
 
 The expectation therefore of the first player, will be to that of the second, as 26 
 to 6, or as 13 to 3. 
 
 In like manner, to resolve the case where the first player is supposed to have won 
 three games, and the other none, as he must win who first gets four games, the num- 
 ber of the games wanting to both will be 5, which being diminished by unity, will 
 
 give 4. We must then examine in how many different st the letters a and 6 can 
 be combined four and four, which will be found to be 16, viz. 
 
 aaaa aabb abbb 
 aaab abab babb 
 aaba baab bbab ON 
 abaa abba bbba 
 baaa baba 
 bbaa 
 
 But, of these 16 combinations, it is evident there are 15 where a is found at least 
 once; which indicates that there are 15 combinations or chances favourable to the 
 first player, and one favourable to the second. Consequently they Cue to divide 
 the stake in the ratio of 15 to 1, or the former ought to have 4 of it, and the 
 latter 45. 
 
56 ARITHMETIC. 
 
 PROBLEM Iv. 
 Of the Genoese Lottery. 
 
 All persons are acquainted with the nature of lotteries, a kind of institution which 
 originated in Italy, and which was afterwards introduced into other countries of 
 Europe. It took its rise at Genoa, where it had long been customary to choose 
 annually by ballot five members of the senate, which was composed of 90 persons, 
 in order to form a particular council. Some idle persons took this opportunity of 
 laying bets, that the lot would fall on such’ or-such senators. The government then 
 seeing with what eagerness people interested themselves in these bets, conceived the 
 idea of establishing a lottery on the same principle; which was attended with so 
 great success, that all the cities of Italy wished to participate in it, and sent large 
 sums of money to Genoa for that purpose. The same motive, and that no doubt of 
 increasing the revenues of the church, induced the pope to establish one of the same 
 kind at Rome, the inhabitants of which became so fond of this species of gambling, 
 that they often deprived themselves and their families of the necessaries of life, that 
 they might have money to lay out in the lottery. Many of them also indulged in 
 every kind of foolery that credulity or superstition could inspire, in order to obtain 
 fortunate numbers. 
 
 The analysis of this kind of lottery is reduced to the solution of the following 
 problem. 
 
 Ninety numbers being given, five of which are to be drawn by chance; it is required 
 to determine what probability there is that among these five, there will be one, two, 
 three, four, or five numbers, which any one has chosen from among the 90 ? 
 
 It may be readily seen, that ifone determinate number only were proposed, and that 
 if no more than one number were to be drawn from the wheel, the adventurer would 
 have only one favourable chance in the 90; but as five numbers are drawn from the 
 wheel, this quintuples the chance favourable to the adventurer, so that he has five 
 favourable chances in the ninety. His probability therefore of winning, is 4; and, 
 to play an equal game, the stakes ought to be in the same ratio, or, what amounts to 
 the same thing, the proprietor of the lottery ought to reimburse the price of the ticket 
 18 times. 
 
 To determine what probability there is, that two numbers selected will both come 
 up, we must first find how many combinations may be produced by 90 numbers, 
 taken two and two. In treating on combinations we have already shewn, that in 
 this case they amount to 4005; but as five numbers are drawn from the wheel, 
 and as these five numbers, combined together two and two, give 10 twos, it thence — 
 results that, in these 4005 chances, there are only 10 favourable to the adventurer. 
 The probability therefore, that the two numbers selected may be among those drawn 
 
 : Hohe xO 1 : : 
 from the wheel, will be expressed by 4005 OF 4001: For this reason the proprietor 
 
 of the lottery ought to give the adventurer, in case he should win, 400} times the 
 price of the ticket. 
 
 To determine what probability there is, that three numbers selected will come up 
 among the five drawn from the wheel, we must find how many ways 90 numbers can 
 be combined three and three, or how many threes they make. These combinations 
 amount to 117480; but as the five numbers drawn from the wheel form 10 threes, the 
 adventurer has 10 favourable chances in 117480, and the probability in his favour is 
 m485 OF tag ‘To risk his money therefore on equal terms, the prize ought to be 
 11748 times the price of the ticket. - 
 
 In the last place, it will be found that in 511038 chances, there is only one favour- 
 able to the person who should bet that 4 determinate numbers will come up; and 1 
 
See 
 / 
 
 in 43949268 favourable to the person who should bet that five determinate numbers 
 will be the five drawn; consequently, in the last case, to risk his money on equal 
 terms, according to mathematical strictness, the adventurer, should he be successful, 
 ought to receive nearly 44 millions of times the money which he lays out. 
 
 CHANCES OF VARIOUS GAMES, 57 
 
 PROBLEM V. 
 
 A and B playing at piquet: A is first in hand, and has no ace: what probability is 
 there that he will get one, or two, or three, or four ? 
 
 It is well known that at this game -I2 cards are dealt to each of the players, 
 and that 8 remain in the pack, of which the first takes 5, and the last 3. This being 
 
 premised, it will be found that A’s chance to have any one ace is : : ve 
 to have two : ; : : 3 : ‘ : : : : - 859 
 to have three : : : : ‘ ; ‘ . : : : - 
 to have four : : : : “ . a 
 
 the sum of all these is 158, cased is equal to 3 952. 
 
 Hence it follows, that the probability of his having an ace among the five cards he 
 
 has to take in, is 332, the difference between which numbers is 71, so that one may 
 
 bet 252to71 that A will take in some of the aces. But let us suppose that A is last 
 in hand ; in that case it is required how much he may bet that he will have at least 
 one ace among his three cards ? 
 
 The probability of A pe an ace among his three cards is : - fy or 328 
 of having two it is : : : : : : re 
 ofhavingthree- . . 
 the sum of all which is 145, or 2? 
 
 Consequently, the ees that he will have either one, or two, or three in- 
 determinately, is 22. A may therefore take an equal bet with advantage, that he will 
 have one of the aces, for the ratio of the stakes would be 29 to 28. 
 
 PROBLEM VI. 
 
 At the game of whist, what probability is there, that the four honours will not be in the 
 hands of any two partners ? 
 
 De Moivre, in his Doctrine of Chances, shews that the chance is nearly 27 to 2 that 
 the partners, one of whom deals, will not have the four honours. 
 
 That it is about 23 to 1 that the other two partners will not have them. 
 
 That it is nearly 8 to 1 that they will not be found on any one side. 
 
 That one may bet about 13 to 7, without disadvantage, that the partners who are 
 first in hand will not count honours. 
 
 That about 20 to 7 may be betted, that the other two will not count them. 
 
 And, in the last place, that it is 25 to 16, that one of the two sides will count 
 honours; or that they will not be equally divided. 
 
 PROBLEM VII. 
 Of the game of the American Savages. 
 
 Weare told by Beron de la Hontan, in his Voyages en Canada, that the Indians 
 play at the following game: they have 8 nuts, black on the one side, and white 
 on the other: these they throw into the air, and if it happens, when they fall 
 to the ground, that the black are odd, the player wins the stake ; if they are all 
 black, or all white, he wins the double; but if there are an equal number of each, he 
 loses. 
 
 M. de Montmort, who analysed this game, finds, that he who tosses up the nuts, bas 
 
) 
 
 58 ARITHMETIC, 
 
 an advantage, which may be estimated at 3; ; and that to render the game equal, 
 he ought to deposit 22 when his adversary stakes 21. 
 | 
 
 PROBLEM VIII. 
 Of the game of Backgammon. 
 
 The game of backgammon is one of those where the spirit of combination is 
 displayed in a very striking manner, and where it is of great utility to know, at every 
 throw, what may be hoped or feared from the succeeding throws, whether your own 
 or those of your’adversary. The chances in this game, like those in others, may be 
 appreciated mathematically ; but we shall here confine ourselves to a small number 
 of examples, selected from those easiest to be comprehended. 
 
 I. A, being at play at backgammon, is obliged to make a blot ; now his throw is such, 
 that he can make it either where his adversary B may take it up with a single ace, or 
 where he can take it up by throwing seven in any manner : the question is, where should 
 he make the blot ? 
 
 As the number of chances for throwing one ace or more, is 11, and the number of 
 chances for throwing seven in any manner, are but 6, it will be safest to make the 
 blot where it may be taken up by throwing 7. 
 
 Il. Whether it is safer to make a blot, at backgammon, where it may be taken up ag an 
 ace, or where it may be taken up by a tré? 
 
 The number of chances for throwing one ace or more, and those for throwing one 
 tré or more, are each 11; but there are 2 chances for throwing deux ace, or 3; it 
 will therefore be safer to make the blot where it can be taken up only by anace. 
 
 The following table will shew the chances of taking up a single blot however 
 situated, 
 
 No. of No. of | 
 
 : Total Total 
 igen Chances. chances. hee Chances. chances. 
 Pail) fowl Tad 6.5 cite 
 
 Oe Vie Ce es Perr ene 8 | 5 [cares 
 Payal nls 9 | 4 [erie 
 a a ey ren 10 | 3 fr ee, 
 Ge ee Oe eee | 2 Saar 
 ee] te Saar lG ve eyo, 1 ees: 
 
 Hence, if a blot is liable sa be hit by any one face of the die, the mean pro- 
 
 bability of hitting it will be - Le Plea 27 = 3 nearly. 
 
 Ill. If two blots be made at backgammon, so as to be hit by two different faces of the 
 die, what is the probability of hitting one or both of them 2 
 _ By the first table it will appear, that the probability of throwing one or more, of any 
 two given faces, is 32. But besides this, one or both the blots may be at length 
 hit by the two dice, and the probability in this case will be different, according 
 to the number of points that will hit them, as in the following table: 
 
GAME WITH THE DICE. 59 
 
 =e AL | iar we ‘ otal | 
 aed Chances Petia eae Chances. ee 
 soo ake | a 2.6 [20-1F5 | 26 
 13) 202°" ] 3.4 | 20-213) 25 
 aed 20ris a BS? 120 402 a ae 
 15 | 20+4 | a m0 [20 8 A 5 7 
 Pi.8 fe £2032 57) 5 45°) 20t3 0 a oF 
 9.3 | 20+1+2] z 4.6 |20+3-+5] 28 
 2.4 | 20+1+ 3] 5.6 | 2+4+35]° 29 — 
 2.5 | 20+ 1+4| i ae 
 
 Hence the probability of hitting two such blots, will be at a medium 
 gi pee. — 2% 
 2x 367 3 
 
 IV. If there be three blots, so situated as to be hit by three different faces; the pro- 
 bability of hitting one or more of them is required ? 
 
 The first table will give the probability of hitting one or more of the blots with a 
 single face or faces; but besides this, there will be the probability of hitting one or 
 more ofthe blots with two dice, the least of which will be when the given faces are 1, 
 2, 3, which have 1 +- 2 =3 such chances, and the greatest when the given faces are 
 4, 5, 6, which have 3-+-4-+5= 12 such chances; the medium of these, viz. 
 
 : “aa on = 13, being added to 27, will make the whole probability about 27-8 
 = , which divided by the common denominator 36, becomes §} = 3. 
 
 Hence, if a player at backgammon makes 3 blots, which are severally within the 
 reach of being hit by a single face of the die, it is almost a certainty that one of them 
 
 at least will be hit. 
 
 PROBLEM IX. 
 
 A mountebank at a country fair amused the populace with the following game: he had 
 6 dice, each of which was marked only on one* face, the first with 1, the second with 
 2, and so on to the sixth, which was marked 6; the person who played gave him a 
 certain sum of money, and he engaged to return it a hundred-fold, if in throwing 
 these siz dice, the six marked faces should come up only once in 20 throws. If the 
 adventurer lost, the mountebank offered a new chance on the following conditions : to 
 deposit a sum equal to the former, and to receive both the stakes in case he should 
 bring all the blank faces in 3 successive throws. 
 
 -. Those unacquainted with the method to be pursued in order to resolve such pro- 
 blems, are liable to reason in an erroneous manner on dice of this kind; for observing 
 that there are five times as many blank as marked faces, they thence conclude that 
 it is 5 to 1 that the person who throws them will not bring any point. ‘They are 
 however. mistaken, as the probability, on the contrary, is near 2 to 1 that they will 
 not come up all blanks. 
 If we take only one die, it is evident that itis 5 to 1 that the person who holds it 
 will throw a blank ; but if we adda second die, it may be readily seen, that the 
 marked face of the first may comhine with each of the blank faces of the second, and 
 the marked face of the second with each of the blank faces of the first ; and, in the 
 last place, the marked face of the one with the marked face of the other: consequently, 
 of 36 combinations of the faces of these two dice, there are 11 in which there is at 
 least one marked face. But, as we have already observed, this number 11 is the 
 
60 ARITHMETIC. 
 
 difference of the square of 6, the number of the faces of one die, and of the square of 
 the same number diminished by unity, that is to say of 5. 
 
 If a third die be added, we shall find, by the like analysis, that of the 216 combina- 
 tions of three dice, there are 91 in which there is at least one marked face; and 91 
 is the difference of the cube of 6 or 216, and the cube of 5 or 125; the result will be 
 the same in regard to the more complex cases; and hence we may conclude that, of 
 _ the 46656 combinations of the faces of the 6 dice in question, there will be 31031 in 
 
 which there is at least one marked face, and- 15625 in which all the faces are blank ; 
 
 consequently the chance is 2 to 1 that some point at least will be thrown ; whereas, 
 by the above reasoning, it would appear that 5 to 1 might be betted on the contrary 
 being the case. 
 
 This example may serve to shew how diffident we ought to be in regard to the 
 ideas which occur on the first consideration of subjects of this kind ; and it may be 
 added that, in this case, our reasoning is confirmed by experience. But to return to 
 the problem; it is evident that, of the 46656 combinations of the faces of 6 dice, 
 there is only one which gives the 6 marked faces uppermost; the probability there- 
 fore of throwing them at one throw, is expressed by j,1;,; and as the adventurer was 
 allowed 20 throws, the probability of his succeeding was only 7224, which is nearly 
 equal to 545. ‘To play an equal game therefore, the mountebank should have en- 
 gaged to return 2332 times the money. But he offered only 100 times the stake, that 
 is, about the 23d part of what he ought to have offered, to give an equal chance, and 
 consequently he had an advantage of 22 to 1. 
 
 The chance offered to those who might lose was a mere deception; for the proposer 
 artfully availed himself of that propensity which every man, who had not sufficiently 
 examined the subject, would have to adopt the false reasoning above mentioned; and 
 the adventurer would have the less hesitation to accept the offer as it would seem 
 that he might bet 5 to 1 on bringing blanks every throw; whereas it is 2 to 1 that 
 the contrary will happen. But the chance of not bringing blanks in one throw, being 
 to that of bringing them, as 2 to 1; it thence follows, that the probability of not 
 bringing them three times successively, is to that of bringing them, as 8 to 1. To 
 play an equal game therefore, the mountebank ought to have staked 7 tol; conse- 
 quently, in the chance which he gave to the loser, in a game where he had an ad- 
 vantage of 22 to 1, he had still an advantage of 7 to 1. 
 
 PROBLEM X. 
 
 In how many throws with six dice, marked on all their faces, may a person engage, for 
 
 an even bet, to throw 1, 2, 3, 4, 5, 6. 
 
 We have just seen that there are 46655 chances to 1, that a person will not throw 
 
 these 6 points with dice marked only on one of their faces; but the case is very 
 
 different with 6 dice marked on all their faces; and to prove it, we need only observe 
 that the point 1, for example, may be thrown by each of the dice, as well as the 
 2, 3, &c.; which renders the probability of these six points, 1, 2, 3, &c. coming up, 
 much greater. . 
 
 But to analyse the problem more accurately, we shall observe, that there are 2 
 
 ways of throwing 1, 2, with two dice; viz. 1 with the die A, and 2 with the die B; 
 or 1 with the die B, and 2 with a. Ifit were proposed to throw 1, 2,3 with 3 dice ; 
 of the whole of the combinations of the faces of 3 dice, there are 6 which give the 
 points 1, 2,3; for 1 may be thrown with the die A, 2 with B, and3 with c; or 1 
 with a, 2 with c, and 3 with B; or 1 with B, 2 with A, and 3 with c; or 1 withs 
 2 with c, and 3 with a; or 1 with c, 2 with a, and 3 with B; or 1 withc 2 with 
 B, and 3 with a. 
 
GAMES WITH THE DICE. 61 
 
 It hence appears that, to find the number of ways in which 1, 2, 3 can be thrown 
 with 3 dice, 1, 2, 3 must be multiplied together. In like manner, to find the number 
 of ways in which 1, 2, 3, 4 can be thrown with 4 dice, we must multiply together 1, 
 2,3, 4, which will give 24; and, in the last place, to find in how many ways 1, 2, 3, 
 4, 5, 6 can be thrown with 6 dice, we must multiply together these six numbers, the 
 product of which will be 720. 
 
 If the number 46656, which is the combinations of the faces of 6 dice, be divided 
 by 720, we shall have 644 for the chances to 1, that these points will not come up at 
 one throw; consequently a person may undertake, for an even bet, to bring them in 
 65 throws; and one may bet more than 2 to | that they will come up in 130 throws. 
 In the last place, as the dice may be thrown 130 times and more, in a quarter of an 
 hour, a person may with advantage bet more than 2 to 1, that they will come up in 
 the course of that time. 
 
 He therefore who engages, for an even bet, to throw these points in a quarter ofan 
 hour, undertakes what is highiy advantageous to himself, and equally disadvantageous 
 to his adversary. 
 
 PROBLEM XI. 
 
 A certain person proposed to play with7 dice, marked on all their faces, on the following 
 conditions: he who held the dice was to gain as many crowns as he brought sixes ; 
 but if he brought none, he was to pay to his adversary as many crowns as there were 
 dice, that is 7. What was the ratio of their chances ? 
 
 To resolve this problem, we must analyze it in order. Let us suppose then, that 
 there is only one die; in this case it is evident, that as there is only 1 chance in 
 favour of him who holds the die, and 5 against him, the ratio of the stakes ought to 
 be that of 1 to 5. Ifthe first therefore gave a crown every time he did not throw 
 6, and received only the same sum when a6 came up, he would play a very un- 
 equal game. 
 
 Let us now suppose 2 dice. In the 36 combinations, of which the faces of 2 
 dice are susceptible, there are 25 which give no 6; 10 which give 1, and 1 which 
 gives 2. He therefore who holds the dice, has only 11 chances in his favour, 10 of 
 which may each make him gain a crown, and the remaining 1 make him gain two. 
 His chance then of winning, according to the general rule, will be 12 + 2; and be- 
 cause, if the 25 chances which do not give a 6 should take place, he would be 
 obliged to pay 2 crowns, the chance of his adversary will be $8. Consequently the 
 chance of winning will be to that of losing as 12 to 39, or 12 to 50, or less than 1 to 4. 
 
 To determine, in the more complex cases, the chances which give no 6, those 
 which give one, those which give two, &c. it must be observed, that they are al- 
 
 _ ways expressed by the different terms of the power of 5 +- 1, the exponent -of which 
 
 | 
 | 
 
 is equal to the number of the dice. Thus when there is only one die, the number 
 5 -+ | expresses, by its first term, that there are five chances without a 6, and one 
 which gives a 6; if there be two dice, as the product of 5-+-1 by 5-+-1, or the 
 square of 5+], is 25-++-10+1, the first term 25 indicates that there are 
 - chances, in the 36, which give no 6; 10 which give one, and 1 which gives 
 wo. 
 
 In like manner, as the cube of 5-++ 1 is 125 -+-75 + J5 +1, it denotes that, in 
 the 216 combinations of the faces of six dice, there are 125 in which there is 
 no 6; 75 in which there is one; 15 in which there are two, and 1 where there 
 are three. 
 
 The fourth power of 5 +- 1 being 625 + 500 -+- 150 + 20 + 1, it indicates, in the 
 same manner, that in the 1296 combinations of the faces of four dice, there are 
 
62 ARITHMETIC. 
 
 625 without a6; 500 which give 6; 150 which give two, 20 which give three, an 
 only 1 that gives four. 
 
 We shall pass over the intermediate cases, and proceed to that where 7 dice ar. 
 employed. In this case then it will be found, that the 7th power of 5-+ 1 is 7812. 
 + 109375 -+- 65625 + 21875 + 4375 + 525 + 35 + 1 = 279936. In the 279931 
 combinations of the faces of 7 dice, there are 78125 which give no 6; 10937: 
 where there is one; 65625 where there are two; 21875 where there are three, &e| 
 But as he who holds the dice would have to pay 7 crowns for each of the first 78122 
 chances, should they take place, we must consequently, according to the general rule, 
 multiply that number by 7, and divide the product by the sum of all the chances, 
 in order to obtain the chance against him, which is = 448873. To find the favourable 
 chance, we must multiply each of the other terms by the number of the sixes it 
 presents ; add together the different products, and divide the sum by the whole of 
 the chances, or 279936; in this manner we shall have, for the chance in favour of 
 the person who holds the dice, 335592. His chance of winning, therefore, is to that of 
 losing, as 325592 to 54687; that is to say, he plays a disadvantageous game, or it 
 is 54 to,32, or 27 to 16, or more than 3 to 2, that he will lose. 
 
 By a like process it may be found, in the case of eight dice, that the chance of the 
 person who holds them, is to that of his adversary, as 2259488 to 3125000, which 
 is nearly as 3 to 4. 
 
 If there were nine dice, the chance of the person who holds them, would be to 
 that of his adversary, nearly as 151 to 175, or nearly 25 to 29. 
 
 If there were ten dice, the chance of the former to that of the latter, would 
 be as 101176960 to 97656250, that is to say, nearly as 101 to 97%. The advantage 
 then begins to be in favour of the former, only when the number of the dice is 10; 
 and, to play an equal game, a less number ought not to be employed. 
 
 CHAPTER X. 
 
 ARITHMETICAL AMUSEMENTS IN DIVINATION AND COMBINATIONS. 
 
 PROBLEM I. : 
 To tell the Number thought of by a person. - 
 
 I. Desire the person, who has thought of a number, to triple it, and to take the : 
 exact half of that triple, if it be even, or the greater half if it be odd. Then desire 
 him to triple that half, and ask him how many times it contains 9; for the number 
 thought of will contain the double of that number of nines, and one more if it be 
 odd. 
 
 Thus, if 5 has been the number thought of; its triple will be 15, which cannot 
 be divided by 2 without a remainder. The greater half of 15 is 8; and if this’ 
 half be multiplied by 3, we shall have 24, which contains 9 twice: the number 
 thought of will therefore be 4 + 1, that is to say 5. 
 
 Proof.—\f the number be an even one, it may be represented by 2.x, and if an odd 
 one by 2x-+1. Then in the case of an even number ae X 3 X 3 represents the 
 operations which the person thinking of a number is requested to perform upon it. 
 The result is 9x, the ninth part of which doubled is 2 x, the number thought of. In 
 the case of the odd number as xX 3X 3=92-+ 41, which contains 9, x times, 
 and 2x-+-1 is the number thought of. = 
 
 In the same way may each of the following methods be shewn to be true. 
 
DIVINATION AND COMBINATIONS. 63 
 
 ; Il. Bid the person multiply the number thought of by itself; then desire him to 
 add unity to the number thought of, and to multiply that sum also by itself; in the 
 last place ask him to tell the difference of these two products, which will certainly 
 be an odd number, and the least half of it will be the number required, 
 
 | Let the number thought of, for example, be 10, which multiplied by itself gives 
 100 ; in the next place 10 increased by 1 is 11, which multiplied by itself makes 121, 
 ad the difference of these two squares is 21, the least half of which, being 10, is 
 the number thought of. 
 
 | This operation might be varied in the second step, by desiring the person to mul- 
 tiply the number by itself, after it has been diminished by unity, and then to tell the 
 ‘difference of the two squares; the greater half of which will be the number 
 thought of. 
 
 | Thus, in the preceding example, the square of the number thought of is 100, and 
 that of the same number less unity is 81; the difference of these is 19, the greater 
 half of which, or 10, is the number thoushs of. 
 
 IlJ.—Desire the person to take 1 from the number thought of, and to double the 
 ‘remainder ; then bid him take ] from this double, and add to it the number thought 
 of. Having asked the number arising from this addition, add 3 to it, and the third 
 ‘of the sum will be the number required. 
 
 Let the number thought of be 5; if one be taken from it there will remain 4, the 
 double of which, 8, being diminished by 1, and the remainder, 7, being increased by 
 5, the number thought of, the result will be 12: if to this we add 3, we shall have 
 ‘18, the third part of which, 5, will be the number required. 
 
 Remark.—This method may be varied a great many ways; for instead of doubling 
 
 the number thought of, after unity has been deducted from it, the person may be 
 desired to triple it; then after he has been desired to subtract a from that triple, 
 and to add the mower thought of, he must add 4 to it, and the + of the sum arising 
 from these operations will be the number required. - 
 _ Let the number required be w: if unity be subtracted from it the remainder will be 
 a—1; multiply this remainder by any number whatever, , and the product will be 
 nx —vn; again subtract unity, and we shall have for remainder nr —n — 1; if 2, the 
 number thought of, be then added, the sum will be (n-+ 1) « —n—1; and if to this 
 ‘sum we add the above multiplier increased by unity, that is to say 3, if the first 
 ‘remainder was doubled, 4 if it was tripled, &c., the result will be (n-+-1) x; which 
 ‘being divided by the same number, the Bo otiont will be x, the number required. 
 
 Unity, instead of being subtracted from the number thought of, might be added to 
 it; and then, instead of adding, at the end of the operation, the multiplier increased 
 by unity, it ought to be subtracted, after which the remainder may be divided as 
 
 above. 
 
 Let the number thought of, for example, be 7; if unity be added, the sum will he 
 8, and this sum tripled will give 24; if 1 be still added, we shall have 25, and this 
 sum increased by 7 will make 32; from which if 4 be deducted, because the number 
 | thought of was tripled after unity had been added, we shall have 28; one fourth of 
 which will be the number required. 
 
 | JIV.—Desire the person to add 1 to the triple of the number thought of, and to 
 ‘multiply the sum by 3; then bid him add to this product the number thought of, 
 ‘and the result will be a sum, from which if,3 be subtracted, the remainder will be 
 ten times the number required. If 3 therefore be taken from the last sum, and if 
 the cipher on the right be cut off from the remainder, the other figure will indicate 
 the number sought, 
 
64 ARITHMETIC. 
 
 Let the number thought of be 6; the triple of which is 18, and if unity be adde 
 it makes 19; the triple of this last number is 57, and if 6 be added it makes 63, fro! 
 which if 3 be subtracted the remainder will be 60: now if the cipher on the rigt 
 be cut off, the remaining figure 6 will be the number required. 
 
 Remark.—If 1 were subtracted from thrice the number thought of, the remainde 
 tripled, and the number thought of again added, it would be necessary, after the pei 
 son had told the result, which would always terminate with 7, to add 3 instead « 
 subtracting it, as in the above operation; and the sum would then be the decuple ¢ 
 the number thought of. 
 
 telling the number of which a person has thought, may 
 be added another ingenious one, by means of the 
 annexed columns of numbers, which are thus pre- 
 pared. 
 
 Having entered the geometrical series, 1, 2, 4, 8, 
 16, 32, as the top series of the six columns, the 
 other numbers in each column downward are pro- 
 duced by this rule. 
 
 To the first number add successively a unit as 
 often as is denoted by one less than the first num- 
 ber; and then, to the last of these, add a number 
 which is one more than the top number, and so on 
 till the columns are filled up. 
 
 Now having prepared the columns, it may be as 
 well, for the sake of secresy, to have them on dif. 
 ferent slips of paper. 
 
 Request a person to think of any number not 
 greater than the highest contained in the columns, 
 in the present case 63, and desire him to point out 
 all the columns in which it is contained, or shewing 
 each column separately, ask, Is the number there ? 
 Then, recollecting that the numbers at the top are 
 1, 2, 4, 8, 16, 32, add together in your mind the 
 figures of this series at the tops of all the columns 
 containing the number thought of, and the sum of 
 these numbers will be the number required. 
 
 Thus for example, if the person says his nnmberisin the 2nd, 5th, and 6th columns, 
 2 -+- 16 + 32, or 50, is the number: If he says it is in the Ist, 2nd, 4th, and 6th co- 
 lumns, 1+ 2-1-8 + 32, or 43 is the number. 
 
 The problem may be varied by requesting the person who thinks of a number to 
 give you those columns only which do not contain it, and you will then discover it by 
 subtracting the sum of the top numbers from the highest number, 63. ‘Thus, if the 
 number is not in the 2nd, 5th, nor 6th column, it must be 63 — 50, or 13. . 
 
 PROBLEM II, 
 Lo tell two or more numbers which a person has thought of. 
 
 I.—When each of the numbers thought of does not exceed 9, they may be easily 
 found in the following manner ; 
 
= 
 
 -f 
 
 DIVINATION AND COMBINATIONS. 65 
 
 Having made the person add 1 to the double of the first number thought of, desire 
 him to multiply the whole by 5, and to add to the product the second number. If 
 there be a third, make him double this first sum and add 1 to it; after which desire 
 him-to multiply the new sum by 5, and to add to it the third number. If there be 
 a fourth, you must proceed in the same manner, desiring him to double the preceding 
 sum; to add to it unity; to multiply by 5, and then to add the fourth number, and 
 so on, i 
 
 Then ask the number arising from the addition of the last number thought of, and 
 if there were two numbers, subtract 5 from it; if three, 55; if four, 555; and so 
 on; for the remainder will be composed of figures of which the first on the left will 
 be the first number thought of, the next the second, and so of the rest. 
 
 Suppose the numbers thought of to be 3,4,6: by adding 1 to 6, the double of the 
 first, we have 7, which being multiplied by 5, gives 35; if 4, the second number 
 thought of, be then added, we shall have 39, which doubled gives 78, and if we add 
 J], and multiply 79, the sum, by 5, the result will be 395. In the last place, if we 
 add 6, the third number thought of, the sum will be 401; and if 55 be deducted from 
 it, we shall have for remainder 346; the figures of which, 3, 4, 6, indicate in order 
 the three numbers thought ef, 
 
 One method we shall here omit, as we shall have o¢casion to emplay it in another 
 
 ~ amusement of the same kind, called the game of the ring, 
 
 IJ.—If one or more of the numbers thought of are greater than 9, two cases 
 must be distinguished: Ist, that where the number of the numbers thought of is 
 odd; 2d, that where it is even. 
 
 In the first case, desire the person to tell the sums of the first.and the second; of 
 the second and the third ; of the third and the fourth, &c., as far as the last, and then 
 
 _ the sum of the first and the last. Having written down these sums in order, add 
 
 together all those the places of which are odd, as the first, the third, the fifth, &c. ; 
 make another sum of ali those the places of which are even, as the second, the 
 fourth, the sixth, &c.; subtract this sum from the former, and the remainder will be 
 the double of the first number. 
 
 Let us suppose, for example, that the five following numbers are thought of, viz. : 
 
 _ 3,7, 13, 17, 20, which, when added two and two, as above, give 10, 20, 30, 37, 23: the 
 
 sum of the first and third and fifth is 63; and that of the second and fourth is 57: 
 if 57 be subtracted from 63, the remainder 6 will be the double of the first number 
 3. Nowif 3 be taken from 10, the first of the sums, the remainder 7 will be the 
 second number ; and, by proceeding in the same manner, we may find all the rest. 
 In the second case, that is to say, when the number of the numbers thought of is 
 even; ask, and write down as above, the sum of the first and the second; that of the 
 
 second and third; and so on as before; but instead of the sum of the first and the 
 _last, take that of the second and the last; then add together those which stand in 
 
 me 
 
 the even places, and form them into a new sum apart; add also those in the odd 
 places, the first excepted, and subtract this sum from the former; the remainder will 
 be the double of the second number; and if the second number thus found be sub- 
 tracted from the sum of the first and second, the remainder will be the first number ; 
 if it be taken from that of the second and third, it will give the third; and so of the 
 rest, 
 
 Let the numbers thought of be, for example, 3, 7, 13, 17: the sums formed as 
 above are 10, 20, 30, 24: the sum of the second and fourth is 44, from which if 30, 
 the third sum, be subtracted, the remainder will be 14, the double of 7, the second 
 number, ‘The first therefore is 3, the third 13, and the fourth 17. 
 
 Pr ‘ 
 
66 ARITHMETIG. 
 
 PROBLEM III. 
 
 A person having in one hand an even number of shillings, and in the other an odd, 
 to tell in which hand he has the even number. 
 
 Desire the person to multiply the number in the right hand by any even number 
 whatever, such as 2; and that in the left by an odd number, as 3; then bid him add — 
 together the two products, and if the whole sum be odd, the even number of 
 shillings will be in the right hand, and the odd number in the left; if the sum be 
 even, the contrary will be the case. 
 
 Let us suppose, for example, that the person has8 shillings in his right hand, and 
 7 inhis left; 8 multiplied by 2 gives 16, and 7 multiplied by 3 gives 21; the sum of 
 which, 37, is an odd number. 
 
 If the number in the right hand were 9, and that in the left 8, we should have 
 9 x 2 = 18, and 8 X 3 = 24; the sum of which two products is 42, an even 
 number. 
 
 Investigation.—Let x represent the number in the left, and y that in the right 
 hand, and let 2 » and 2m -— 1 represent any even and odd numbers. Then 
 Qnyt+tA2m+1l.¢x=2.ny+ma-+z is the sum of the products directed to 
 be taken. Nows2.ny-—+ mx is necessarily even, Therefore when the whole pro- 
 duct is even, 2, the remaining term, is also even; and when odd, x is odd, which is 
 the rule. 
 
 PROBLEM IV. 
 
 A person having in one hand a piece of gold, and in the other a piece of silver, 
 to tell in which hand he has the gold, and in which the silver. 
 
 For this purpose, some value, represented by an even number, such as 8, must be 
 assigned to the gold, and a value represented by an odd number, such as 3, must 
 be assigned to the silver: after which the operation is exactly the same as in the 
 preceding example. 
 
 Remarks.—\. To conceal the artifice better, it will be sufficient to ask whether 
 the sum of the two products can be halved without a remainder ; for, in that case, a 
 total will be even, and in the contrary case odd. 
 
 Il. It may be readily seen that the pieces, instead of being i in the two hands of 
 the same person, may be supposed to be in the hands of two persons, one of whom 
 has the even number, or piece of gold, and the other the odd number, or piece of ; 
 silver. he same operations may then be performed in regard to these two persons 
 as are performed in regard to the two hands of the same person, calling the one pri- 
 vately the right, and the other the left. 
 
 PROBLEM V. 
 The Game of the Ring, 
 
 This game is nothing else than an application of one of the methods employed — 
 to tell several numbers thought of, and should be performed in a company not ex-— 
 ceeding 9, in order that it may be less complex. Desire any one of the company to 
 take a ring, and to put it on any joint of whatever finger he may think proper. The 
 question then is to tell what person has the ring, and on what hand, what finger, 
 and what joint. 
 
 For this purpose, call the first person 1, the second 2, the third 3, and so on; also 
 
ot 
 
 DIVINATION AND COMBINATIONS. 67. 
 
 call the right hand J, and the left 2: the first finger of the hand, that is to say the 
 thumb, must be denoted by 1, the second by 2, and so on to the little finger ; 
 and the first joint of each finger, or that next the extremity, must be called 1, the 
 second 2, and the third 3, 
 
 Let is now suppose that the fifth person has taken the ring, and put it on the first 
 joint of the fourth finger of his left hand. To resolve the problem, nothing is 
 necessary but to discover these numbers 5, 2, 4, 1, which may be done in the 
 following manner. 
 
 Desire some one to double the first number 5, which will give 10, and to subtract 
 1 from it ; desire him to multiply 9, the remainder, by 5 which will give 45; to this 
 product bid him add the second number 2, which will make 47, and then 5 which will 
 make 52: desire him to double this number, and the result will be 104, and to sub- 
 tract 1, which willleave 103. Desire him to multiply this remainder by 5, which 
 will give 515, and to add to the product the third number 4, or that expressing the 
 finger, which will give 519: then bid him add 5, which will make 524, and from 1048, 
 the double of this sum, let him subtract 1, which will leave 1047: then desire him 
 to multiply this remainder by 5, which will give 5235, and to add to this product 1, 
 the fourth number, or that expressing the joint, which will make 5236; in the last 
 place bid him again add 5, and the sum will be 5241, the figures of which will indi- 
 cate, in order, the person who has the ring, and the hand, finger, and joint, on which 
 it was put. 
 
 It is evident, that all these operations amount, in reality, to nothing else than 
 multiplying by 10, the number which expresses the person ; then adding that which 
 expresses the hand; multiplying again by 10,and so on.* But as this artifice is too 
 easily detected, it might be better to employ the method taught in Prob. II. 
 No. 1, to discover any number of numbers thought of at pleasure ; for, on account of 
 the number which must be subtracted, the operation will be more difficult to be com- 
 prehended. 
 
 The problem might be proposed in the follawing manner, and be resolved by 
 the same process. 
 
 Three or more persons having each selected a card, the number of the spots of 
 which does not exceed 9, to tell the number of the spots of each. 
 
 Desire the first person to add 1 to double the number of the spots of his card; to 
 multiply the sum by 3, and to add to the product the spots of the card of the second 
 person: then desire him to double that sum; to add unity to it, to multiply the 
 whole by 5, and to add to this product the spots of the card of the third person: by 
 subtracting from the last result 55, if the number of the persons be 3; 535, if it be 
 4; 5555, if it be 5, the figures which compose the remainder will indicate, in order, 
 the spots of the cards selected by each person. 
 
 _ This process may be demonstrated with as much ease as the former; let the num- 
 bers to be guessed, less than 10, bez, y, z: we confine ourselves to three, for the sake 
 of brevity. If 1 be added to the double of the first number, we shall have 27-1 1, 
 and multiplying by 5, the product will be 10 2-+-5; if the second number y be 
 
 * For the satisfaction and information of the reader, we shall here give the following demonstra- 
 tion. Let the four numbers to be guessed be z, y, 2; w: according to the above method, we must 
 double 2, which will give 22; if 1 be then subtracted we shall have 2 2 — J, and multiplying by 5, 
 the result will be 10 2 — 5. If y, the second number, be added, we shall have 102 — 5+ y, and 5 
 added to this sum will make 10 z + y, which being doubled will give 20a + 2y; if 1 be subtracted, 
 there will remain 20a + 2y — 1, which multiplied by 5 will give 1002+ 10y—5,; to this product 
 if the third number z, and 5 be added, the sum will be 100 a2 + 10y + %; and, if unity be taken 
 from the double of this sum. the result will be 200 2 + 20y + 2z—1; if we then multipiy by 5, 
 we shall have for product 1000 2 + 100 y + 10 x —5; and by adding 5 and the last number, wv, the 
 sum will be 10007 +100 y+10z+u. Ifa, y, z, uw represent numbers, below 10, as 5, 2, 4, 1, 
 the sum will be 5000 + 200 + 40 + 1, or 5241. If the numbers were 9, 6, 5, 4, the sum for the 
 same reason, would be 9654; which is a demonstration of the process abuve indicated. 
 
 FQ 
 
68 ARITHMETIC. 
 
 added, the sum will be 10 ++ 5 + y, and 1 added to the double will make 20 2 + 
 10 +2 y +1, which multiplied by 5 gives 100 «+ 50-++ 10 y-++ 5; if we then add 
 the third number z, we shallhave 100 x + 50+ 10 y a 5-+2z, or 100x+ 10y-+ 
 z+ 55: ifa, y, z are, for example, 5, 6, 7, this expression will be 567 ++ 55, or 612. 
 hae this last sum therefore, if we deduct 55, the remainder will be 567, which in- 
 dicates in order the three numbers to be guessed. 
 
 For the sake of brevity, we shall not give any. other example, as the ‘reader may 
 recur to that before given in Prob. II. 
 
 PROBLEM VI. 
 
 To guess the number of spots on any card, which a person has drawn froma 
 whole pack. 
 
 Take a whole pack, consisting of 52 cards, and desire some person in company to 
 draw out any one at pleasure, without shewing it. Having assigned to the different 
 cards their usual value, according to their spots, call the knave 11, the queen 12, 
 and the king 13. Then add the spots of the first card to those of the second; the 
 last sum to the spots of the third, and so on, always rejecting 13, and keeping the 
 remainder to add to the following card. It may be readily seen that it is needless to 
 reckon the kings, which are counted 13. If any spots remain at the last card, sub-— 
 tract them from 13, and the remainder will indicate the spots of the card that has 
 been drawn; if the remainder be 1], it has been a knave; if 12 it has been a 
 queen; but if nothing remains, it has been aking. The colour of the king may be 
 known by examining which one among the cards is wanting. 
 
 If you are desirous of employing only 32 cards, the number used at present for 
 piquet, when the cards are added as above directed, reject all the tens; then add 4 
 to the spots of the last card, and a sum will be obtained, which taken from 10, if it 
 be less, or from 20 if it exceeds 10, the remainder will be the number of the card that 
 has been drawn; so that if 2 remains, it has been a knave, if 3a queen, if 4 a king, 
 and so on. - 
 
 If the pack be incomplete, attention must be paid to those deficient, in order that 
 the number of the spots of all the cards wanting may be added to the last sum, after 
 as many tens as possible have been subtracted from it; and the sum arising from this | 
 addition must, as before, be taken from 10 or 20, according as it is greater or less than — 
 10. It is evident that by again looking at the cards, the one which has been drawn 
 may be discovered. 
 
 The demonstration of this rule is as follows: since, in a complete pack of canted 
 there are 13 of each suit, the values of which are l, 2, 3, &c., to 13, the sum of all” 
 the spots of each suit, calling the knave 11, the queen 12, and the king 13, is seven 
 times 13. or 91, whichis a multiple of 13; consequently the quadruple of this sum is 
 a multiple of 13 also: if the spots then of all the cards be added together, always 
 rejecting 13, we must at last find the remainder equal to nothing. It is therefore 
 evident that if a card, the spots of which are less than 13, has been drawn from 
 the pack, the difference between these spots and 13 will be what is wanting 
 to complete that number: if at the end then, instead of reaching 13, we reach 
 only 10, for example, it is evident that the card wanting is a three; and if we reach 
 13, it is also evident that the card wanting is one of those equivalent to 18, or 
 a king. 
 
 If two cards*have been drawn from the pack, we may tell, in like manner, the 
 number of spots which they contain both together: that is, how much is want- 
 ing to reach 13, or that deficiency increased by 13; and to know which two, — 
 nothing is necessary but to count privately how many times 13 has been com- 
 
DIVINATION AND COMBINATIONS, 69 
 
 pleted, for with the whole of the cards it ought to be counted 28 times: if it 
 be counted therefore only 27 times, with a remainder, as 7 for example, the spots 
 of the two cards drawn amount together to 6: if 13 be counted only 26 times, 
 with the same remainder, it may be concluded that the two cards formed to- 
 gether 13-+-6, or 19. 
 
 The demonstration of the rule given when the same number of cards is used, 
 as that employed for the game of piquet, viz. 32, calling the ace 1, the knave 2, 
 the queen 3, the king 4, and assigning to the other cards the value of their spots, 
 is attended with as little difficulty; for in each suit there are 44 spots, making 
 altogether 176, which, as well as 44, is a multiple of 11; we may therefore always — 
 count to 11, rejecting 11, and the number wanting to reach 11, will be the 
 value of the card which has been drawn. 
 
 But the same number 176, if 4 were added to it, would be a multiple of 10 
 or of 20; and hence a demonstration also of the method which has been taught, 
 
 PROBLEM VII. 
 
 A person having an equal number of counters, or pieces of money, in each hand, 
 to find how many he has altogether. 
 
 Desire the person to convey any number, as 4, for example, from the one hand 
 to the other, and then ask him how many times the less number is contained in 
 the greater. Let us suppose that he says the one is triple of the other; and in 
 this case multiply 4, the number of the counters conveyed from one hand into 
 the other, by 3, and add to the product the same number 4, which will make 
 16. In the-last place, from the number 3 subtract unity, and if 16 be divided by 
 2, the remainder, the quotient 8 will be the number contained in each hand, and 
 consequently the whole number is 16. 
 
 Let us now suppose that when 4 counters are conveyed from one hand to the 
 other, the less number is contained in the greater 23 times: in this case we must, 
 as before, multiply 4 by 24, which will give 94; to which if four be added, we 
 shall have 134, or 4; if unity be then taken from 24, the remainder will be 14, 
 
 r 4, by which if 4° be divided, the quotient, 10, will be the number of counters 
 in each hand, as aie be easily proved on trial. 
 
 Proof.—Let x be the number in each hand, a the number transferred from the 
 one hand to the other, and » =the multiple which the sum is of the remainder. 
 1 
 
 il 
 
 —_——. ee ——— n 
 The z+-a=2n.2 —a, orn—1.e=n-+1.a; whencer=- -a@; a gene- 
 
 ral rule. 
 PROBLEM VIII. 
 
 Several cards being presented, in succession, to several persons, that they may each 
 choose one at pleasure; to guess that which each has thought of. 
 
 Shew as many cards to each person as there are persons to choose; that is to 
 say, 3 to each if there are 3 persons. When the first has thought of one, lay 
 aside the three cards in which he has made his choice. Present the same num- 
 ber to the second person, to think of one, and lay aside the three cards in the 
 like manner. Having done the same in regard to the third person, spread out 
 the three first cards with their faces upwards, and place above them the next 
 three cards, and above these the last three, that all the cards may thus be dis- 
 posed in three heaps, each consisting of three cards. Then ask each person in 
 which heap the card is which he thought of, and when this is known it will be 
 easy to tell these cards, for that of the first person will be the first in the heap 
 
70 ARITHMETIC. 
 
 to which it belongs; that of the second will be the second of the next heap, | and 
 that of the third will be the third of the last heap. : 
 
 PROBLEM Ix. 
 Three cards being presented to three persons, to guess that which each has chosen_ 
 
 As it is necessary that the cards presented to the three persons should be dis- 
 tinguished, we shall call the first a, the second s, and the third c; but the 
 three persons may be at liberty to choose any of them at pleasure. This choice, 
 which is susceptible of six different varieties, having been made, give to the first 
 person 12 counters, to the second 24, and to the third 36: then desire the first 
 person to add together the half of the counters of the person who has chosen the 
 eard a, the third of those of the person who has chosen B, and the fourth part 
 of those of the person who has chosen c, aud ask the sum, which fe be either 
 23 or 24; 25 or 27; 28 or 29, as in the following table: 
 
 First. Second. Third. Sums. 
 12 24 36 
 A B c 23 
 A Cc B Q4 
 B A Cc 25 
 c A B OT 
 B Cc A 28 
 c B A 29 
 
 This table shews, that if the sum be 25, for example, the first person must 
 have chosen the card g, the second the card a, and the third the card c; and 
 that if it be 28, the first person must have chosen the card B, the second the 
 card c, and the third the card a; and so of the rest. 
 
 PROBLEM X. 
 
 A person having drawn, from a complete pack of fifty-two cards, one, two, three, 
 four, or more cards, to guess the whole number of the spots which they contain. 
 
 -Agssume any number whatever, such as 15, for example, greater than the num- 
 ber of the spots of the highest card, counting the knave 11, the queen 12, and 
 the king 13, and desire the person to add as many cards from the pack, to the 
 first card he has chosen, as will make up 15, counting the spots of that card; 
 let him do the same thing in regard to the second, the third, the fourth, &c. ; 
 and then desire him to tell how many cards remain in the pack. When this is 
 done, proceed as follows: ; 
 
 Multiply the above number 15, or any other that may have been assumed, by 
 the number of cards drawn from the pack, which we shall here suppose to be 3; 
 to the product, 45, add the number of these cards, which will give 48; subtract 
 the 48 from 52, and take the remainder 4 from the cards left in the pack: the 
 result will be the number of spots required. 
 
 Let us suppose, for example, that the person has drawn from the pack a 7, a 
 10, and a knave, which is equal to 11: to make up the number 15 with a 7, 
 eight cards will be required; to make up the same number with a 10, will re- 
 quire five; and with the knave, which is equal to 11, four will be necessary. 
 The sum of these three numbers, with the 3 cards, makes 20, and consequently 
 32 cards remain in the pack. To find the sum of the numbers 7, 10, 11, mul- 
 tiply 15 by 3, which will give 45; and if the number of the cards drawn from 
 the pack be added, the sum will be 48, which taken from 52, leaves 4, If 4 
 then be subtracted from 32, the remainder, 28, will be the sum of the spots 
 
DIVINATION AND COMBINATIONS. 71 
 
 contained on the three cards drawn from the pack, as may be easily proved by 
 trial, 
 
 Another Example.—Let us suppose two cards only drawn from the pack, a 4 
 and a king, equal to 13; if cards be added to these to make up 15, there will 
 ‘remain in the pack 37 cards. 
 
 If 15 be multiplied by 2, the product will be 30, to which if 2, the number of 
 the cards drawn from the pack, be added, we shall have 32; and if 32 be taken 
 from 52, the remainder will be 20. In the last place, if 20 be subtracted from 
 37, the number of the cards left in the pack, the remainder, 17, will be the 
 number of the spots of the 2 cards drawn from the pack. ; 
 
 Remarks.—I. If 4 or 5 cards are drawn from the pack, it may sometimes hap- 
 pen that a sufficient number will not be left to make up the number 15; but 
 even in this case the operation may be still performed. For example, if 5 cards, 
 the spots contained on which are 1, 2, 38, 4, 5, have been drawn; to complete. 
 with each of these cards the number 15 would require, together with the 5 cards, 
 at least 65; but as there are only 52, there are consequently 13 too few. He 
 who counts the pack must therefore say that 13 are wanting. 
 
 On the other hand, he who undertakes to tell the number of the spots, must 
 multiply 15 by 5, which makes 75; and to this if 5, the number of the cards, be 
 added, it will give 80; that is to say, 28 more than 52: if 13 then be subtracted from 
 28, the remainder J5 will be the number of the spots contained ov these 5 cards. 
 
 But if we suppose that the cards left in the pack are, for example 22, which would 
 be the case if the five cards drawn were the 8, 9, 10, knave = 11, and queen = 12, 
 it would be necessary to add these 22 to the excess of 5 times 15+ 5, over 52, that 
 is to say to 28, and we should have 50 for the spots of these 5 cards, which is indeed 
 the exact number of them. ; 
 
 II. If the pack consists not of 52 cards, but'of 40, for example, there will still be 
 no difference in the operation: the number of the cards, which remain of these 40, 
 must be taken from the sum produced by multiplying the made up number by that 
 of the cards drawn, and addiug to the product the number of these cards. 
 
 Let us suppose, for example, that the cards drawn are 9, 10, 11, that the number 
 to be made up is 12, and that the cards left in the pack are 31. Theu 12 x 3= 
 36, and 3 added for the 3 cards, makes 39, which subtracted from 40 leaves 1. If] 
 then be taken from 31, the remainder 30 will be the number of the spots re- 
 quired. 
 
 Ill. Different numbers to be made up with the spots of each card chosen might be 
 assumed ; but the case would still be the same, only that it would be necessary to 
 
 add these three numbers to that of the cards, instead of multiplying the same num- 
 ber by the number of cards drawn, and then adding the number of the cards. In 
 this there is so little difficulty, that an example is not necessary. 
 
 IV. The demonstration of this method, which some of our readers perhaps may 
 be desirous of seeing, is exceedingly simple, and is as follows. Let a be the num- 
 ber of cards in the pack, ¢ the number to be made up by adding cards to the spots 
 of each card drawn, and 6 the cards left in the pack; let 2, y, z express the spots 
 of the cards, which we shall here suppose to be 3, and we shall then have, for the 
 number of the cards drawn, ce—2-+c—y-—+c—z-++3; which with the cards left 
 in the pack b, must be equal to the whole pack. Then 3c+3—2—y-—z+5) 
 =a, or rty+2=3c-43-+5)—a, or =b—(a—3e—3). Butr+y+z is 
 the whole number of the spots; is the number of cards left in the pack, and a— 
 3 c¢—3 is the whole number of cards in the pack, less the product of the number 
 
72) ARITHMETIC. 
 
 to be completed by the number of the cards drawn, minus that number. There- 
 fore, &c. 
 
 PROBLEM XI. 
 
 Three things being privately distributed to three persons ; to guess that which each 
 has got. 
 
 Let the three things be a ring, a shilling, and a glove. Call the ring A, the shilling 
 E,and the glove 1; and in your own mind distinguish the persons by calling them 
 first, second, and third. Then take twenty-four counters, and give one of them to 
 the first person, two to the second, and three to the third. Place the remaining 18 
 on the table, and then retire, that the three persons may distribute among themselves 
 the three things proposed, without your observing them. When the distribution has 
 been made, desire the person who has the ring to take from the 18 remaining counters 
 as many as he has already ; the one who has the shilling to take twice as many as he 
 has already, and the person who has the glove to take four times as many; according 
 to the above supposition then, the first person has taken 1, the second 4, and the 
 third 12; consequently one counter only remains on the table. When this is 
 done, you may return, and by the number left can discover what thing each has got, 
 by employing the following words : 
 
 1 2 3 5 6 7 
 Par fer César jadis devint si grand prince. 
 
 To make use of these words, you must recollect that in all cases there can remain 
 
 only 1 counter, or 2, 3, 5, 6, or 7, and never 4: it must be likewise observed that 
 each syllable contains one of the vowels which we have made to represent the three 
 things proposed, and that the above line must be considered as consisting only of six 
 words: the first syllable of each word must also be supposed to represent the first © 
 person, and the second syllable the second. This being comprehended, if there 
 remains only 1 counter you must employ the first word, or rather the two first 
 syllables, par fer, the first of which, that containing a, shews that the first person 
 has the ring, represented by a; and the second syllable, that containing E, shews 
 that the second person has the shilling, represented by =; from which you may easily 
 conclude that the third person has the glove. If two counters remain, you must 
 take the second word César, the first syllable of which, containing ©, will shew 
 that the first person has the shilling, represented by £, and the second syllable, 
 containing A, will indicate that the second person has the ring, represented by a: 
 you may then easily conclude that the third person has the glove. In general, 
 whatever number of counters remain, that word of the verse which is pointed out 
 by the same number must be employed. 
 
 Remarks.—Instead of the above French verse, the following Latin one might be 
 used. 
 Losi? aes ben Opt 
 
 Salve certa anime semita vita quies. 
 This problem might be proposed in a manner somewhat different, and might be 
 applied to more than three persons: those who are desirous of farther information 
 
 on the subject, may consult Bachet in the 25th of his ‘‘ Problémes plaisants et — 
 delectables.” 
 
DIVINATION AND COMBINATIONS, 73. 
 
 PROBLEM XII. 
 
 Several numbers being disposed in a circular form, according to their natural series, 
 to tell that which any one has thought of. 
 
 The first ten cards of any suit, disposed ina circular form, as seen in the figure 
 below, may be employed with great convenience for performing what is announced 
 in this problem. The ace is here represented by the letter a annexed to 1, and 
 the ten by the letter x joined to 10. 
 
 2 3 4 
 B Cc D 
 TA ES i 
 10 «x FE 6 
 I H G 
 9 8 7 
 
 Having desired the person who has thought of a number or card to touch also 
 any other number or card, bid him add to the number of the card touched the 
 number of the cards employed, which in this caseis 10. Then desire him to count that 
 sum in an order contrary to that of the natural numbers, beginning at the card he 
 touched, and assigning to that card the number of the one which he thought of; 
 for by counting in this manner, he will end at the number or card which he thought 
 of, and consequently you will easily know it. 
 
 Thus, for example, if the person has thought of the number 3, marked c, and 
 has touched 6, marked F; if 10 be added to 6, it will make 16; if 16 be then 
 counted * from F, the number touched, towards E, D, c, B, A, and so on in the 
 retrograde order, counting 3, the number thought of, on r, 4on E, 5 on D, 6 on c, 
 and so round to 16, the number 16 will terminate on c, and shew that the person 
 thought of 3, which corresponds to c. 
 
 | 
 | 
 
 Remarks.—1. A greater or less number of cards may be employed at pleasure. 
 If there are 15 or 8 cards, 15 or 8 must be added to the number of the card 
 touched. 
 
 2. To conceal the artifice better, you may invert the cards, so as to prevent the 
 spots from being seen; but you must remember the natura! series of the cards, and 
 the place of the first number, or the ace, that you may know the number of the card 
 touched, in order to find the one to which the person ought to count. 
 
 PROBLEM XIII. 
 
 Two persons agree to take alternately numbers less than a given number, for example 
 11, and to add them together till one of them has reached a certain sum, such as 100; 
 by what means can one of them infallibly attain to that number before the other ? 
 
 The whole artifice of this problem consists in immediately making choice of 
 certain numbers, which we shall here point out. Subtract 11, for example, from 
 100, the number to be reached, as many times as possible, and the remainders will 
 be 89, 78, 67, 56, 45, 34, 23, 12, and 1, which must be remembered; for he who 
 by adding his number less than 11, to the sum of the preceding, shall count one 
 of these numbers before his adversary, will infallibly win, without the other being 
 able to prevent him. 
 
 _ These numbers may be found also, with still greater ease, by dividing 100 
 
 * It isto be observed that the person must not count this sum aloud, but privately in his own 
 mind. 
 
 er 
 
74 - ARITHMETIC. 
 
 by 11, and adding 11 continually to 1, the remainder, which will give 1, 12, 23, 
 34, &c. 
 
 Let us suppose, for example, that the first person, who knows the game, takes 
 1 -for his number: it is evident that his adversary, as he must count less than 
 1], can at most reach 11 by adding 10 to it. The first will then take 1, which 
 will make 12; if the second takes 8, which will make 20, the first will take 3, 
 which will make 23; and proceeding in this manner successively he will first reach’ 
 34, 45, 56, 67, 78, 89. When he attains to the last number it will be impossible 
 for the second to prevent him from getting first to 100; for whatever number the_ 
 second takes, he can attain only to 99, after which the first may say, ‘‘ and 1 makes 
 100.” If the second takes | after 89, it will make 90; and his adversary may 
 finish by saying, ‘and 10 make 100.” 
 
 It is evident that when two persons are equally well acquainted with the game, 
 he who begins must necessarily win. 
 
 But if the one knows the game and the other does not, the latter, though first, 
 may not win; for he will think it highly advantageous to take the greatest number 
 possible, that is to say 10; and in that case the other, acquainted with the nicety 
 of the game, will take 2, which with 10 will make 12; one of the numbers he 
 ought to secure. But he may even neglect this advantage, and take only 1 to 
 make 11; for the first will probably still take 10, which will make 21, and the 
 second may then take 2, which will make 23; he may then wait a little longer to 
 get hold of some of the following numbers 34, 45, 56, &c. 
 
 If the first is desirous to win, the least number proposed must not be a measure 
 of the greater; for in that case the first would have no infallible rule to direct 
 him in his operations. For example, if 10, which measures 100, were assumed, 
 instead of 11, by subtracting 10 from 100 as many times as possible, we should 
 have the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90, the first of which, 10, could. 
 not be taken by the first; for being obliged to employ a number less than 10, 
 if the second were as well acquainted with the game, he might take the complement 
 to 10; and would thus have an infallible rule for winning. 
 
 PROBLEM XIV. 
 Sixteen counters being disposed in two rows, to find that which a person has thought of. 
 
 The counters being arranged in two rows, as a and B, desire the person to think 
 of one, and to observe well in which row it is. 
 
 0920 £0000 
 ©0906 010-9 O'9 
 Omekic Oa 
 O1rese OCT © Oo 
 eoooos 
 66 420-4 
 OO, 0 O80-0,0 Oe 
 jay ©} fo) oo 
 oo 5.6 f 
 econo oOCcC Oooo 
 oo O & 
 
 Let us suppose that the counter thought of is in the row a; take up that whole 
 row, in the order in which it stands, aiid dispose it in two rows c and pb, on the 
 right and left of the row B; but in arranging them, take care that the first of the 
 row A may be the first of the row c; the second of the row a the first of the row 
 pD; the third of the row a, the second of the row c, and so on; then ask again in 
 which of the vertical rows, c or D, the counter thought of is. Suppose it to be in” 
 c: take up that row as well as the row D, putting the last at the end of the 
 first, without deranging the order of the counters, and, observing the rule already 
 
DIVINATION AND COMBINATIONS, 79 
 
 given, form them into two other rows, as seen at © and r; then ask, as before, in 
 which row the counter thought of is. Let us suppose it to bein E: take up this 
 tow, and the row F, as above directed, and form them into two new rows, on the 
 tight and left of sn. After these operations, the counter thought of must be the 
 first of one of the perpendicular rows, H and 1; if you therefore ask in which 
 row it is, you may easily point it out; and as it is here supposed that each of the 
 counters has some distinguishing mark, you may desire them to be mixed together, 
 and still be able to tell the number thought of, by observing the mark. 
 
 It may be readily seen that, instead of counters, cards may be employed; and 
 when you have discovered, by the above means, the one thought of, you may cause 
 them to be mixed, which will better conceal the artifice. 
 
  Remark.—If a greater number of counters or cards, arranged in two vertical 
 rows, be supposed, the counter or card thought of will not necessarily be the first 
 in the row to which it belongs, after the third transposition: if there be 32 counters 
 or cards, four transpositions will be necessary; if there are 64, five; and so on, 
 before it can be said, with confidence, that the counter or card thought of occupies 
 the first place in its row; for if this counter or card were at the bottom of the 
 perpendicular row A, supposing 16 counters in each row, or 32 altogether, it would 
 not arrive at the first place till after four transpositions: if there were 64, or 32 
 in each row, it would require five; and so on, as may be easily proved by trial. 
 
 PROBLEM XV. 
 
 A certain number of cards being shewn to a person, to guess that which he has 
 thought of. 
 
 To perform this trick, the number of the cards must be divisible by 3; and to do 
 it with more convenience, the number must be odd. 
 
 The first condition, at least, being supposed, desire the person to think of a card; 
 then place the cards on the table with their faces downward; and, taking them up 
 in order, arrange them in three heaps, with their faces upward, and in such a manner 
 that the first card of the packet shall be the first of the first heap; the second the first 
 of the second, and the third the first of the third; the fourth the second of the first, 
 and so on. When the heaps are completed, ask the person in which heap is the card 
 thought of, and when told, place that containing the card thought of in the middle; 
 then turning up the packet, form three heaps, as before, and again ask in which is 
 the card thought of. Place the heap containing the card thought of still in the 
 ‘middle, and, having formed three new heaps, ask which.of them contains the card 
 thought of. When this is known, place it as before between the other two; and 
 again form three heaps, asking the same question. Then take up the heaps for the 
 last time; put that containing the card thought of in the middle, and placing the 
 packet.on the table, with the faces of the cards downward, turn up the cards till you 
 count half the number of those contained in the packet; 12 for example, if there 
 be 24, in which case the 12th card will be the one the person thought of. 
 
 If the number of the cards be, at the same time, odd, and divisible by 3, as 15, 21, 
 by &e., the trick will become much easier ; for the card thought of will always te 
 that in the middle of the heap in which it is found the third time ; so that it may be 
 easily distinguished without counting the cards; nothing will be necessary for this 
 purpose, but to remember, while you are forming the heaps the third time, the card 
 which is the middle one of each. Let us suppose, for example, that the middle card 
 of the first heap is the ace of hearts; that of the second the king of hearts, and that 
 of the third the knave of spades; it is evident, if you are told that the heap con- 
 taining the required card is the third, that this card must be the knave of spades. 
 You may therefore cause the cards to be shuffled, without touching them any more, 
 
mi 
 
 76> . ARITHMETIC. 
 
 and then, looking them over for the sake of form, may name the knave of spades 
 when it occurs. 
 
 PROBLEM XVI. 
 
 Fifteen Christians and fifteen Turks being at sea in the same vessel, a dreadful storm 
 came on, which obliged them to throw all their merchandise overboard ; this however 
 not being sufficient to lighten the ship, the captain infor med them that there was no pos- 
 sibility of its being saved, unless half the passengers were thrown overboard also, 
 Having therefore caused them all to arrange themselves in a row, by counting from 9 
 to 9, and throwing every ninth person into the sea, beginning again at the first of the 
 row when it had been counted to the end, it was found that after fifteen persons had 
 been thrown overboard, the fifteen Christians remained. How did the captain arrange 
 these thirty persons so as to save the Christians ? 
 
 The method of arranging the thirty persons may be deduced from these two 
 
 French verses : Rs 
 Mort, tu ne failliras pas 
 
 En me livrant le trepas. 
 Or from the following Latin one, which is not so bad of its kind: 
 Populeam virgam mater regina ferebat. 
 
 Attention must be paid to the vowels A, E, I, 0, U, contained in the syllables of these 
 - verses; observing that a is equal to 1, E to 2,1 to 3, 0 to 4, and u to 5. You must 
 begin then by arranging 4 Christians together, because the vowel in the first syllable 
 is o; then five Turks, because the vowel in the second syllable is vu; and so on to 
 the end. By proceeding in this manner, it will be found, taking every ninth person 
 circularly, that is to say, beginning at the first of the row, after it is ended, that the 
 lot will fall entirely on the Turks. 
 
 . The solution of this problem may be easily extended still farther. Let it be 
 required, for example, to make the lot fall upon 10 persons in 40, counting from: 
 12 to 12: Arrange 40 ciphers in a circular form, as below; 
 
 rte 
 0000000000000000 
 
 00O00000000000000 
 ttt t 
 
 Then, beginning at the first, mark every twelfth one with a cross; continue in this ~ 
 manner, taking care to pass over those already crossed, still proceeding circularly, 
 till the required number of places has been marked ; if you then count the places 
 of the marked ciphers, those on which the lot falls will be easily known: in the 
 present case they are the 7th, the 8th, the 10th, the 12th, the 21st, the 22d, the 24th, 
 the 34th, the 35th, and the 36th. : 
 
 A captain, obliged to decimate his company, might employ this expedient, to make 
 the lot fall upon those most culpable. 
 
 It is related that Josephus, the historian, saved his life by means of this expedient. 
 Having fied for shelter to a cavern, with forty other Jews, after Jotapat had been 
 taken by the Romans, his companions resolved to kill each other rather than sur- 
 render. Josephus tried to dissuade them from their horrid purpose, but not being. 
 able to succeed, he pretended to coincide with their wishes, and retaining the authority _ 
 he had over them as their chief, to avoid the disorder which would necessarily be 
 
AMUSING PROBLEMS. rif: 
 
 the consequence of this cruel execution, if they should kill each other at random, 
 he prevailed on them to arrange themselves in order, and, beginning to count from 
 one end to a certain number, to put to death the person on whom that number 
 should fall, until there remained only one, who should kill himself. Having all 
 agreed to this proposal, Josephus arranged them in such a manner, and placed himself 
 in such a position, that when the slaughter had been continued to the end, he re- 
 mained with only one more person, whom he persuaded to live. 
 
 Such is the story related of Josephus by Hegesippus; but we are far from war- 
 ranting the truth of it. However, by applying to this case the method above indi- 
 cated, and supposing that every third person was to be killed, it will be found that 
 the two last places on which the lot fell were the 16th and 31st; so that Josephus 
 must have placed himself in one of these, and the person he was desirous of saving 
 in the other. 
 
 PROBLEM XVII. 
 
 A man has a wolf, a goat, and a cabbage, to carry over a river; but being obliged to 
 transport them one by one on account of the smallness of the boat, in what manner 
 is this to be done, that the wolf may not be left with the goat, nor the goat with the 
 cabbage ? 
 
 He must first carry over the goat, and then return for the wolf; when he carries 
 over the wolf, he must take back with him the goat, and leave it, in order to earry 
 over the cabbage; he may then return and carry over the goat. By these means, 
 the wolf will never be left with the goat, nor the goat with the cabbage, but when 
 ‘the boatman is present. 
 
 PROBLEM XVIII. 
 
 Three jealous husbands, with their wives, having to cross a river at a ferry, find a boat 
 without a boatman ; but the boat is so small that it can contain no more than two of them 
 at once. How can these six persons cross the river, two and two, so that none of the 
 women shall be left in company with any of the men, unless when her husband is 
 present ? 
 
 - The solution of this problem is contained in the two following Latin disticks : 
 
 It duplex mulier, redit una, vehitque manentem, 
 Itque una; utuntur tunc duo puppe viri. 
 Par vadit et redeunt bini, mulierque sororem 
 Advehit ; ad propriam fine maritus abit. 
 That is: “ Two women cross first, and oue of them, rowing back the boat, carries 
 over the third woman. One of thé three women then returns with the boat, and 
 remaining, suffers the two men, whose wives have crossed, to go over in the boat. 
 One of the men then carries back his wife, and leaving her on the bank, rows over 
 the third man. In the last place, the woman who had crossed enters the boat, and 
 returning twice, carries over the other two women.” 
 
 This question is proposed also under the title of the three masters and the three 
 valets. The masters agree very well, and the valets also; but none of the masters 
 ‘can endure the valets of the other two; so that if any one of them were left with any 
 of the other two valets, in the absence of his master, he would infallibly cane him. 
 
 PROBLEM XIx. 
 
 In what manner can counters be disposed in the eight external cells of a square, so 
 that there may be always 9 in each row, and yet the whole number shall vary from 20 
 to 82? 
 Ozanam proposed this problem in the following manner, with a view no doubt to 
 excite the curiosity of his readers; 
 
73 . ARITHMETIC. 
 
 A certain convent eonsisted of nine cells, one of which in the middle was occupied 
 by a blind abbess, and the rest by her nuns. The good abbess, to assure herself that 
 the nuns did not violate their vows, visited all the cells, and finding 3 nuns in each, 
 which made 9 in every row, retired to rest. Four nuns however went out, and the 
 abbess returning at midnight to count them, still found 9 in each row, and therefore 
 retired as before. The 4 nuns then came back, each with a gallant, and the abbess 
 on paying them another visit, having again counted 9 persons in each row, entertained 
 no suspicion of what had taken place. But-4 more men were introduced, and the 
 abbess again counting 9 persons in each row, retired in the full persuasion that no 
 one had either gone out or come in, How was all this possible ? 
 
 This problem may be easily solved by inspecting the four following figures: the 
 first of which represents the original disposition of the counters in the cells of the 
 square; the second that of the same counters when 4 are taken away; the third the 
 manner in which they must be disposed when these 4 are brought back with 4 others; 
 and the fourth that of the same counters with the addition of 4 more. It is here 
 evident that there are always 9 in each external row; and yet, in the first case, the 
 whole number is 24, while in the second it is 20, in the third 28, and in the fourth 
 32, 
 
 Sty 3 nl is 7 Ete Sa he a 
 IL] 3) 3 |} yf] 1 | 
 PTR a EES 40d steal iad 
 
 1 7 1 
 1 7 ] 
 
 It would seem that Ozanam had not observed that these variations might have 
 deen carried still farther: that four men more might have been introduced into the 
 convent, without the abbess perceiving it; and that all the men might have after- 
 wards gone out with six nuns, so as to leave only 18, instead of the 24 who were in 
 
 the cells at first. The possibility of this will appear by inspecting the two following 
 figures, 
 
 0 9 0 5 0 4 
 
 V. 9 9 VI 0 0 
 ee eee a ee « 
 
 0 9 0 4 0 ) 
 
 It is almost needless to explain in what manner the illusion of the good abbess 
 arose, It is because the numbers in the angular cells of the square were counted 
 twice; these cells being common to two rows. The more therefore the angular cells 
 are filled, by emptying those in the middle of each band, these double enumerations 
 become greater; on which account the number, though diminished, appears always 
 to be the same; and the contrary is the case in proportion as the middle cells are 
 
 filled by emptying the angular ones, which renders it necessary to add some units to 
 have 9 in each band. 
 
AMUSING PROBLEMS. 79 
 
 PROBLEM XxX, 
 
 . 
 y 
 
 A gentleman has a bottle, containing 8 pints, of choice wine, and wishes to make a 
 
 present of one half of it to a friend ; but as he has nothing to measure it, except two 
 other bottles, one capable of containing 5 and the other 3 pints, how must he manage, 
 so as to put exactly 4 pints into the bottle gapable of containing 5? 
 
 To enable us to resolve this problem we shall call the bottle containing the 8 
 
 pints, A; that of 5 pints, B; and that of 3 pints, c; supposing that there are 8 pints 
 of wine in the bottle a, and that the other two are empty, as seen at D. 8 5 3 
 Having filled the bottle B with wine from the bottle a, in which there A BC 
 will remain no more than 3 pints, as seen at £, fill the bottle c from s, pvp 8 0 O 
 and consequently there will remain only 2 pints in the latter, as seenat ge 3 5 O 
 -¥F: then pour the wine of c into a, which will thus contain 6 pints,as F 38 2 8 
 ‘seen at G, and pour the two pints of Binto c,as seenat H. Inthe last Gc 6 2 O 
 place, having filled the bottle B from the bottle a, in which there will no 6 O 2 
 ‘remain only 1 pint, as seen at 1, fill up c from 8, in which there willre- 1 1 5 2 
 main 4 pints, as seen at K; and thus the problem is solved. K 1 4 8 
 Remark.—If you are desirous of making the four pints of wine remain in the 
 
 bottle a, which we have supposed to be filled with 8 pints, instead of 
 remaining in the bottle ps, fill the bottle c with wine from the bottle a, 
 in which there will remain only 5 pints, as seen at D; and pour 3 pints 
 of c into B, which will consequently contain 3 pints, as seen at E: 
 having then filled c from a, in which there will remain no more than 2 
 pints, as seen at F; fill up B from c, which will thus contain only 1 
 pint, as seen at a. In the last place, having poured the wine of the 
 bottle B into the bottle a, which will thus have 7 pints, as seen at H; 
 pour the pint of wine which is inc into B, consequently the latter 
 will contain ] pint, as seen at 1; and then fill up c from A, in which there 
 will remain only 4 pints, as was proposed, and as seen at kK. 
 
 Rae DOs wey 
 
 RIT Woes Ow 
 RH ONnrwnwoorpm & 
 BO et a 09 S69 SG 109 
 
 PROBLEM XXI. 
 
 A gentleman has a bottle containing 12 pints of wine, 6 of which he is desirous of 
 giving to a friend ; but as he has nothing to measure it, except two other bottles, one 
 of 7 pints, and the other of 5, how must he manage, to have the 6 pints in the bottle 
 
 capable of containing 7 pints ? 
 
 This problem is of the same nature as the preceding, and may be solved in the 
 like manner. Let T represent the twelve-pint, s the seven-pint, and Fr the five-pint 
 bottle. The bottle rT is full, and the other two, s and F, are empty, tong 
 as seen ata. Fill the bottle r with wine from 7, so that T shall con- 
 tain only 7 pints, as seen at H; then pour into s the wine contained in 
 F, which will remain empty, and the bottle s will contain 5 pints, as 
 seen at 1; having filled F from 1, the latter will contain only 2 pints, 
 the bottle s will contain 5, and the bottle F will be full, as seen at K; 
 in the next place fill s from Fr, and will still contain only 2 pints, 1 
 while s contains 7, and F 3, as seen atx; then empty sintoT, and M 
 F into s, by which means T will contain 9 pints, and s 3, F remaining N 
 empty, as seen at m: fill F from T, and pour from F into s as muchas o 11 
 will fill it, so that there will then be 4 pints in 7, 7 pintsins,and1l Pp 6 
 pint in F, as seen at N: pour the 7 pints from s into T, and the pint contained in F 
 into s, after which T will contain 11 pints, s 1, and F will be empty, as seen at o, 
 In the last place, having filled the five-pint bottle F from the bettle r, and poured 
 
 Ae © 
 —d mel 
 Ronnwpwyyp w 
 AeNIwwEernoon 
 SOF CwMMNOuUNos 
 
80 ARITHMETIC. 
 
 6 
 a 
 
 these 5 pints from F into s, which already contains 1, it will be found that Tr contains 
 6 pints, and that s contains 6 also; so that the desired result has been obtained. 
 
 PROBLEM XXile 
 
 To make the knight move into all the squares of the chess board, in succession, without 
 passing twice over the same. ® 
 
 As the reader perhaps may be unacquainted with the 
 movement of the knight at the game of chess, we shall 
 here explain it. The knight being placed in the square | hp rere 
 A, cannot move into any of those immediately surround- 
 ing it, as 1, 2, 3, 4, 5, 6, 7, 8; nor into the squares 9, OeluSut? a aldie ae 
 
 10, 11, 12, which are directly above or below, and on |——|——|——|——|—= 
 each side of it; nor into the squares 13, 14, 15, 16, which Ce OA Ties 
 are in the diagonals; but only into one of those which, 16 Tinos "Sa 
 
 in the annexed figure, are empty. 
 
 Several eminent men have amused themselves with this problem, such as Mont- 
 mort, Demoivre, and Mairan, and each of these has given a solution of it. In those 
 of the two former, the knight is supposed to be placed at first in one of the angular 
 squares of the chess board; in that of the third, he is supposed to begin to move 
 from one of the four central squares; but in our opinion it was not known, till 
 within these few years, that placing the knight in any square whatever, he may be 
 made to traverse the whole chess board, and even in such a manner that, without 
 returning the same way, he shall pass a: second time over the board under the like 
 conditions. For this last solution we are indebted to M. W , a captain in the 
 Kinski regiment of dragoons, in the imperial service. . 
 
 We shall here give four tables, representing these four solutions, with an explana- 
 tion and some remarks. 
 
 I, M. Montmort. II. Demoivre. 
 
AMUSING PROBLEMS, gi 
 
 Ill. M. Mairan. IV. M. W——~. 
 
 62 \51 | 56 | 53 °)'18 
 
 124 55 | 58 |-49 | 60 
 
 Of these four ways of resolving the problem, that of Demoivre is doubtless the 
 easiest to be remembered ; for the principle of his method consists in filling up, as 
 much as possible, the two exterior bands, which form as it were a border, and not 
 entering the third, till there is no other method of moving the knight from the place 
 where he is, to one of the two first, a rule which in the clearest manner subjects the 
 movement of the knight to a certain necessary progress, from his first step, to the 
 50th, and beyond it; for from the cell marked 50 there is no choice in placing him, 
 except on those marked 51 and 63; but the cell 51 being nearer the band, ought to 
 be preferred, and then the movement must necessarily be through 52, 53, 54, 55, 56, 
 57, 58, 59, 60, 61. When he arrives at 61, it is a matter of indifference whether he 
 be placed in the cell marked 64, for he may thence proceed to the last but one 63, 
 and end at 62; or be placed in 62, to proceed to 63, and end at 64. It may therefore 
 be said that the movement of the knight in this solution is almost constrained. 
 
 The case is not the same with the fourth, which it is difficult to practise in any 
 other manner than from memory; but it is attended with one very great advantage, 
 which is, that you may begin, as already said, at any cell at pleasure ; because the 
 author took the trouble to bring the knight at the conclusion to a place from which 
 he can pass into the first. His movement therefore is in some measure circular, and 
 interminable, by adhering to the condition of not passing twice over the same cell, 
 till after 64 steps. 
 
 It may be readily seen, that to make the knight perform this movement without 
 
 confusion, the cell he has quitted must be marked at each step. For this purpose a 
 counter may be placed in each-cell, and removed as the knight passes over it; or, 
 what will be still better, a counter may be placed in each cell when he has passed it. 
 
 PROBLEM XXIII. 
 
 To distribute among 8 persons, 21 casks of wine, 7 of them full, 7 of them empty, and 
 7 half full, so that each shall have the same quantity of wine, and the same number 
 of casks. 
 
 This problem admits of two solutions, which may be clearly comprehended by 
 means of the two following tables. 
 
 Persons. full casks, empty. half full. 
 Ist 2 2 3 
 i 2d 2 2 3 
 3d 3 3 ] 
 G 
 
82 
 
 Persons. 
 
 Ist 
 Il. 2d 
 3d 
 
 It is evident that, in these two combinations, each person will have 7 casks, and 
 
 34 casks of wine. 
 
 But it may be easily seen that the whole number of the casks must be divisible by 
 
 empty. 
 
 ARITHMETIC. 
 full casks. 
 3 3 
 3 3 
 ] ] 
 
 half full. 
 1 
 1 
 5 
 
 the number of persons, otherwise the thing required would be impossible. 
 
 It will be found, in like manner, that if 24 casks were to be divided among 3 per- 
 sons, under the same conditions, we should have three different solutions, as follow: 
 full casks. 
 
 Persons. 
 
 Ist 
 III. 2d 
 
 If there should be 27 casks to be divided, there would be three solutions also : 
 empty... 
 
 Persons. 
 Ist 
 
 I. }24 
 3d 
 
 Ist 
 
 Il. 2d 
 3d 
 
 Ist 
 Til. 2d 
 
 36 
 
 full casks. 
 
 C9 DD DD bO 69 69 
 
 A CODD BAe 09 09 09 
 
 CHAPTER XI. 
 
 empty. 
 
 We GO PbO DD bo 69 09 
 
 Hm OO RD ee OO 09 OO 
 
 half full. 
 
 COND ChE PNW 
 
 half full. 
 
 m9 OY ee eT 09 69 C9 
 
 CONTAINING SOME CURIOUS ARITHMETICAL PROBLEMS, 
 
 A gentleman, in his will, gave orders that his property should be divided among his 
 children in the following manner :—The eldest to take from the whole £1000, and the . 
 7th part of what remained ; the second £2000, and the Tth part of the remainder ; 
 the third £3000, and the 7th part of what was left; and so on to the last, always 
 
 The children having followed the disposition of the testator, tt. 
 
 was found that they had each got an equal portion: how many children were there, 
 
 what was the father’s property, and to how much did the share of each child amount ? — 
 
 increasing by £1000. 
 
 It will be found by analysis, that the father’s property was £36000; that there 
 were 6 children, and that the share of each was £6000. 
 
 Thus, if the first takes £1000, the remainder of the property will be £35000, the 
 7th part of which £5000, together with £1000, makes £6000. The remainder, after 
 deducting the first child’s portion, is £30000, from whichif the second takes £2000, 
 the remainder will be £28000, but the 7th part of this sum is £4000, which if added 
 
 PROBLEM I. 
 
 to the above £2000, will make £6000, and so on. 
 
AMUSING PROBLEMS. 83 
 
 PROBLEM It. 
 
 A gentleman meeting a certain number of beggars, and being desirous to distribute 
 among them all the money he had about him, finds that if he gave sixpence to each he 
 would have 2s. too little ; but that by giving each a groat, he would have 2s. 8d. over : 
 how many beggars were there, and what sum had the gentleman in his pocket ? 
 
 There were 28 beggars, and the gentleman had in his pocket 12 shillings ; for if 28 be 
 multiplied by 6, the product will be 168, from which if 2 shillings or 24 pence be sub- 
 tracted, as he wanted 24 pence to be able to give each sixpence, the remainder will 
 be 144 pence = 12 shillings; but by giving each of the beggars 4 pence, he had 
 occasion only for 112 pence, or 4 times 28; consequently he had 32 pence, or 2s. 8d. 
 remaining. 
 
 PROBLEM III. 
 
 A gentleman purchased for £110. a lot of wine, consisting of 100 bottles of Burgundy, 
 and 80 of Champagne ; and another purchased at the same price, for the sum of 
 £95, 85 bottles of the former, and 70 of the latter: whatwas the price of each kind 
 of wine ? 
 
 It will be found that the Burgundy cost 10s. per bottle, and the Champagne 153. 
 as may be easily proved. 
 | PROBLEM IY. 
 
 A gentleman, on his death-bed, gave orders in his will, that if his lady, who was then 
 pregnant, brought forth a son, he should inherit two thirds of his property, and the 
 widow the other third ; but that if she brought forth a daughter, the mother should 
 inherit two thirds, and the daughter one third ; the lady however was delivered of two 
 children, a boy and a girl, what was the portion of each ? 
 
 The only difficulty in this problem is to discover, in what manner the testator 
 would have disposed of his property, had he foreseen that his lady would have been 
 
 delivered of two children. It has generally been explained in the following manner : 
 
 As the testator desired that if his wife brought forth a boy, the latter should have 
 two thirds of his property, and the mother the other third, it hence follows that his 
 intention was to give the son a portion double to that of the mother; and as he gave 
 orders that in case she brought forth a daughter, the mother was to have two thirds 
 of the property, and the daughter the other, there is reason to conclude that he 
 intended the portion of the mother to be double that of the daughter. Consequently, 
 to combine these two conditions, the property must be divided in such a manner, 
 that the son may have twice as much as the mother, and the mother twice as much 
 as the daughter. If we therefore suppose the property to be £30000, the share of 
 the son will be £17142, that of the mother £8571]#, and that of the daughter 
 £42853. 
 
 Asa supplement to this problem, another is differently proposed. In case the 
 mother should be delivered of two sons and a daughter, in what manner must the pro- 
 perty be divided ? 
 
 _ In our opinion, no other answer can be given to this question, than what would be 
 given by the gentlemen of the bar; that the will, in such a case, would be void; for 
 
 ~achild having been omitted in the will, all the laws with which we are acquainted 
 
 would pronounce its nullity: Ist, because the law is precise; and 2d, because it is 
 impossible to determine what would have been the disposition of the testator, had he 
 had two sons, or had he foreseen that his wife would be delivered of two. 
 
 Gc 2 
 
84 ARITHMETIC. 
 
 PROBLEM V. 
 
 A brazen lion, placed in the middle of a reservoir, throws out water from its mouth, its 
 eyes, and tts right foot. When the water flows from its mouth alone, it fills the reservoir 
 in 6 hours ;: from the right eye it fills it in 2 days ; from the left eye in 3, and from the 
 
 footin 4. In what time will the bason be filled by the water flowing from all these aper- 
 tures at once ? 
 
 To solve this problem it must be observed, that as the lion, when it throws the 
 water from its mouth, fills the bason in 6 hours, it can fill } of it inan hour; and 
 thas as it fills it in 2 days when it throws the water from its right eye, it can fill 4 of 
 it in an hour. It will be found, in like manner, that it can fill 4, of it in an hour 
 when the water flows from its left eye, and 4, when it flows from its foot. By 
 throwing the waters from all these apertures at once, it furnishes in an hour } + 4 + 
 dy + dy, and these fractions added together are equal to $4. We must therefore make 
 the following proportion: If}, are filled in one hour or 60 minutes, how many 
 minutes will the whole bason, or 383, require: or as 61 is to 288, so is 1 hour, to the 
 answer, which will be 4h. 43m. 1644 seconds. 
 
 PROBLEM VI. 
 
 A mule and an ass travelling together, the ass began to complain that her burthen was 
 too heavy. ‘‘ Lazy animal,” said the mule, “‘ you have little reason to complain ; for 
 if I take one of your bags, I shall have twice as many as you, and if £ give you one of 
 mine we shall then have only an equal number.” With how many bags was each loaded ? 
 
 This problem, which is one of those commonly proposed to beginnersin Algebra, is 
 taken from a collection of Greek epigrams, known under the name of the Anthology ; 
 and has been translated, almost literally, into Latin as follows: 
 
 Una cum mulo vinum portabat asella, 
 
 Atque suo graviter sub pondere pressa gemebat. 
 Talibus at dictis mox increpat ipse gementem : 
 Mater, quid luges, tenerze de more puelle? 
 Dupla tuis, si des mensuram, pondera gesto: 
 
 At si mensuram accipias, zequalia porto. 
 
 Dic mibi mensuras, sapiens geometer, istas? 
 
 The analysis of this problem has also been expressed in indifferent Latin. verses, 
 which we shall here give, to gratify the reader’s curiosity. 
 
 Unam asina accipiens, amittens mulam et unam 
 
 Si fiant equi, certe utrique ante duobus 
 
 Distabant a se. Accipiatsi mulus at unam, 
 
 Amittatque asina unam, tunc distantia fiet 
 
 Inter e03 quatuor, Muli at cum pondera dupla 
 
 Sint asinz, huic simplex, mulo est distantia dupla. 
 Ergo habet hc quatuor tantum, mulusque habet octo, 
 Unam asiue si addas, si reddat mulus et unam, 
 Mensuras quinque hac et septem mulus habebunt. 
 
 That is: “ As the mule and the ass will both have equal burthens when the for- 
 
 mer gives one of his measures to the latter, itis evident that the difference between 
 the measures which they carry is equal to 2, Now if the mule receives one from the 
 ass, the difference will be 4; but in that case the mule will have double the number 
 of measures that the ass has; consequently the mule will have 8, and the ass 4. If 
 the mule then gives one to the ass, the latter will have 5and the former 7:’’ these 
 were the number of the measures with which each was loaded, and which solve the 
 problem. 
 
 This problem, which might be expressed in a great variety of forms, is not the only — 
 
 eS 
 
AMUSING PROBLEMS. 85 
 
 one furnished by the Greek Anthology. The following are a few more, translated 
 into the Latin verse by Bachet de Meziriac, who inserted them in a note to one of the 
 problems of Diophantus. ; 
 
 I. 
 
 Aurea mala ferunt Charites, equalia cuique 
 
 Mala insunt calatho; Musarum his obvia turba 
 Mala petunt, Charites cunctis zqualia donant ; 
 Tune equalia tres contingit habere, novemque : 
 
 Dic quantum dederint numerus sit ut omnibus idem ? 
 
 That is: ‘“‘ The three Graces, carrying each an equal number of oranges, were met 
 by the nine Muses, who asked for some of them; and each Grace having given to 
 each Muse the same number, it was then found that they had all equal shares: How 
 many had the Graces at first ?” 
 
 The least number which will answer this problem is 12; for if we suppose that 
 each Grace gave one to each Muse, the latter would each have three; and there 
 would remain 3 to each Grace. 
 
 The numbers 24, 36, 48, &c., will also answer the question; and after the distri- 
 bution is made, each of the Graces and Muses will have 6, or 9, or 12, &c. 
 
 idl 
 
 Dic, Heliconiadum decus O sublime Sororum, 
 Pythagora, tua quot tyrones tecta frequentent, 
 
 i Qui, sub te, sophie sudant in agone magistro? 
 
 : Dicam ; tuque animo mea dicta, Polycrates, hauri; 
 
 * Dimidia horum pars preclara mathemata discit, 
 Quarta immortalem naturam nosse laborat ; 
 Septima, sed tacite, sedet atque audita revolvit ; 
 Tres sunt foeminei sexus. 
 
 “ Tell me, illustrious Pythagoras, how many pupils frequent thy school? One 
 half, replied the philosopher, study mathematics, one fourth natural philosophy, one 
 seventh observe silence, and there are three females besides.” 
 
 The question here is, to find a number, the 3, 4, and 4 of which ++ 8, shall be 
 equal to that number. It may be easily replied that this number is 28. 
 
 II. 
 
 Dic quota nunc hora est? Superest tantum ecce diei 
 Quantum bis gemini exacta de luce trientes. 
 
 ** A person being asked what o’clock it was, replied, the hours of the day which 
 remain, are equal to 2 of those elapsed.” 
 
 If we divide the day, as-the ancients did, into 12 equal portions, the question will 
 be to divide that number into two such parts, that 3 of the first may be equal to the 
 second ; in this case the result will be 54 for the number of the hours elapsed; and 
 consequently for the remainder of the day 6§ hours. 
 
 IV. 
 
 Hic Diophantus habet tumulaum, qui tempora vitea 
 Ilius mira denotat arte tibi. 
 
 Egit sextantem juvenis, lanugine malas 
 Vestire hinc coepit parte duodecima. 
 
 Septante uxori post hec sociatur et anno 
 Formosus quinto nascitur inde puer. 
 
 Semissem etatis postquam attigit ille paterne, 
 Infelix subita morte peremptus obit. 
 
 Quatuor estates genitor lugere superstes 
 Cogitur, hinc annos illius assequere, 
 
 “This is the epitaph of the celebrated mathematician Diophantus. It tells us that 
 -Diophantus passed the sixth part of his life in childhood, and the twelfth part in the 
 
86 ARITHMETIC. 
 
 state of youth; that after a seventh part of his life and five years more were elapsed, 
 
 he had a son, who died when he had attained to half the age of his father, and that’ 
 
 the latter survived him only four years.” 
 To resolve this problem, we must find a number, the 4, 7, 4, and 4 of which +5 
 + 4, shall be equal to the number itself. This number is 84. 
 
 PROBLEM VII. 
 
 The sum of £500 having been divided among four persons, it was found that the 
 shares of the first two amounted to £285; those of the second and third to 
 
 £220; those of the third and fourth to £215; and that the share of the first 
 
 was to that of the last as 4 to 3. What was the share of each? 
 
 The solution of this problem is exceedingly easy. The first had £160, the second 
 £125, the third £95, and the fourth £120. 
 
 It is to be observed, that without the last-mentioned condition, or a fourth one ot 
 
 some. kind or other, the problem would be indeterminate ; that is to say, would be 
 susceptible of a great many answers: the last condition however limits it to one 
 only. 
 
 PROBLEM VIII. 
 
 A labourer hired himself to a gentleman on the following conditions: for every day 
 
 | he worked he was to receive 2s. 6d.; but for every day he remained idle he 
 was to forfeit 1s. 3d.: after 40 days’ service he had to receive £2. 15s. How 
 many days did he work, and how wany remain idle ? 
 
 He worked 28 days of the 40, and remained idle 12. 
 
 PROBLEM IX. 
 
 A bill of exchange, of £2000 was paid with half-guineas and crowns; and the 
 number of the pieces of money amounted to 4700. How many of each sort were 
 employed ? 
 
 There were 3000 half guineas and 1700 crowns. 
 The solution of this and that of the preceding problem are left as exercises for the 
 young student. 
 
 PROBLEM X. 
 
 A gentleman, having lost his purse, could not tell the exact sum it contained, but 
 
 recollected that when he counted the pieces two by two, or three by three, or five 
 
 by five, there always remained one; and that when he counted them seven by 
 seven, there remained nothing. What was the number of pieces in his purse ? 
 
 It may be readily seen that, to solve this problem, nothing is necessary but to find 
 a uumber which when divided by 7 shall leave no remainder; and which when 
 
 divided by 2, 8, 5, shall always leave 1. Several methods may be employed for this. 
 
 purpose ; but the simplest is as follows: 
 
 Since nothing remains when the pieces are counted seven by seven, the number of 
 them is evidently some multiple of 7; and since 1 remains when they are counted 
 two by two, the number must be an odd multiple: it must therefore be some of the 
 series 7, 21, 35, 49, 63, 77, 91, 105, &c. 
 
 This number also, when divided by 3, must leave unity ; but in the above series, 7, 
 49, and 91, which increase arithmetically, their difference being 42, are the only 
 numbers that have the above property. It appears likewise, that if 91 be divided 
 by 5, there will remain I; and we may thence conclude that the first number which 
 
 aioe, 
 
AMUSING PROBLEMS. 87 
 
 answers the question is 91: for it is a multiple of 7, and being divided by 2, 3, or 5, 
 the remainder is alwys 1. 
 
 Several more numbers, which answer this question, may be found by the following 
 means: continue the above progression, in this manner: 7, 49, 91, 183, 175, 217, 
 259, 301, until you find another term divisible by 5, that leaves unity; this term. 
 will be 301, and wil! also answer the conditions of the problem; but the difference 
 
 between it and 9] is 210, from which it may be concluded, that if we form the pro- 
 gression : 
 
 91, 301,511, 721, 931, 1141, &e: 
 all these numbers will answer the conditions of the problem also. 
 
 It would therefore be still uncertain what money was in the purse, unless the 
 owner could tell nearly the sum it contained. Thus, for example, if he should say 
 that there were about 500 pieces in it, we might easily tell him that the number was 
 51]. 
 
 Let us now suppose that the owner had said, that when he counted the pieces two 
 by two there remained 1 ; that when he counted them three by three there remained 
 2; four by four, 3; five by five, 4; six by six, 5; and, in the last place, that when 
 he counted them seven by seven, nothing remained. 
 
 It is here evident that the number, as before, must be an odd multiple of 7, and 
 consequently one of the series 7, 21, 35, 49, 63, 77, 91, 105, &c. But the numbers 
 35 and 77, of this series, answer the condition of leaving 2 as a remainder when 
 divided by 3, and their difference is 42. For this reason we must form a new arith- 
 metical progression, the difference of which is 42, viz. 
 
 35, 77, 119, 161, 203, 245, 287, &c. 
 
 We must then seek for two numbers in it, which when divided by 4 shall leave 
 3asremainder. Of this kind are the numbers 35, 119, 203, 287; and therefore we 
 must form a new progression, the difference of the terms of which is 84. 
 
 35, 119, 203, 287, 371, 455, 539, 623, &c. 
 Ia this new progression we must seek for two terms, which when divided by 5, 
 shall leave 4; and it will be readily seen that these numbers are 119 and 539, the 
 difference of which is 420. A series of terms therefore which answer all the con- 
 ditions of the problem except 1, is 
 
 119, 539, 959, 1379, 1799, 2219, 2639, &c. 
 But the last condition of the problem is, that the required number, when divided 
 by 6, leaves 5 as remainder. This property belongs to 119, 959, 1799, &c., always 
 adding 840; consequently the number sought is one of those in that progression. 
 For this reason, as soon as we know nearly within what limits it is contained, we 
 shall be able to determine it. 
 
 If the owner therefore of the purse had said, that it contained about 100 pieces, 
 
 _ the number required would be 119; if he had said there were nearly 1000, it would 
 be 959, &c. 
 
 Remark.— The solution of this problem, according to the method taught by Oza- 
 nam, would be imperfect. For after finding the smallest number which answers the 
 conditions of the problem, viz. 119, he would merely say, that to obtain the other 
 numbers which answer them, the numbers 2, 3, 4, 5, 6, 7, ought to be successively 
 multiplied together, and their product 5040 added to 119, the first number found: 
 this would give the number 5159, which would answer the proposed conditions aiso. 
 But it may be readily seen, that there are several other numbers, between 119 and 
 5159, which answer these conditions, viz. 959, 1799, 2639, 3479, 4319. 
 
 In treating of chronology, we shall give the solution of another problem of the 
 same kind; viz., To find any year of the Julian period, the golden number, cycle of 
 the sun, and cycle of indiction, for that year, being given. 
 
88 ; ARITHMETIC. 
 
 PROBLEM XI. 
 
 A sum of money, placed out at a certain interest, increased in 8 months to £3616. — 
 13s. 4d.; and in two years and a half it amounted to £3937. 10s. What was 
 the original capital, and at what rate of interest was it placed out ? 
 
 That young algebraists may have an opportunity of exercising their own ingenuity, 
 we shall here give the answer only of this problem. By employing the proper 
 means of analysis, they will find that if r =the interest of one pound for a year, a 
 6.b6—a 
 lsa—46 
 a aud b, gives r = th of a pound, or the rate £5. per cent. per annum; and hence 
 the capital is easily found to be £3500. 
 
 the first amount and b the second, that r= , which, with the given value of 
 
 PROBLEM XII. 
 
 Three women went to market to sell eggs; the first of whom sold 10, the second 
 25, and the third 30, all at the same price. As they were returning, they began 
 to reckon how much money they carried back, and it was found that each had 
 the same sum. How many eggs did they sell, and at what price? 
 
 It is evident that, to make what is announced in this problem possible, these women 
 must have sold their eggs at two different times, and at different prices; for if the 
 one who had the least number of eggs sold a very small number at the lowest price, 
 and the remainder at the highest, while the one who had the greatest number sold 
 the greater part at the lowest price, and could sell only a very small number at the 
 highest, it may be easily seen that they might have got equal sums of money. 
 
 The question then is to divide each of the numbers 10, 25, 30, into two such parts, 
 that if the first part of each be multiplied by the first price, and the second by the 
 second, the sum of the two products shall be equal. 
 
 This problem is indeterminate, and susceptible of ten different solutions, It is, in — 
 the first place, necessary that the difference of the prices of the first and the second 
 sale shall be an exact divisor of 15, 20, 5, the differences of the three numbers given ; 
 but the least divisor of these three numbers is 5, and for this reason the prices must 
 be 6 and 1, or 7 and 2, or 8 and 3, &c. 
 
 If we suppose the two prices to be 6 and 1, we shall have seven different solu- 
 tions, as seen in the following table : 
 
 Women. Ist sale. 2d sale. Total amount. 
 Ist. 4 eggs at 6d. 6 at ld. 30 
 I 2d. 1 24. 30 
 3d. 0 30 30 
 Ist. 5 5 35 
 ef 2d. 2 23 35 
 3d. 1 29 35 
 Ist. 6 4 40 
 Til. 2d. 3 22 40 
 3d. 2 28 40 
 Ist. 7 3 45 
 IV, 2d. 4 21 45 
 3d. 3 27 45 
 Ist. 8 2 50 
 V. 2d. 5 20 50 
 3d. 4 26 50 
 Ist. 9 ] 55 
 VI. } 2d. 6 19 55 
 3d. 5 25 55 
 Ist. 10 0 60 
 Vil. } 2a vi 18 60 
 3d. 6 24 60 
 
Me 
 
 | AMUSING PROBLEMS. 89 
 
 f we suppose the two prices to be 7 and 2, we shall have also the three following 
 solutions : 
 
 _ Women. Ist sale. 2d sale. Total amount. 
 Ist. 8 eggs at 7d. 2 at 2d. 60 
 AK } 24 2 93 60 
 od. 0 30 60 
 Ist. 9 1 65 
 II. } 24 3 22 65 
 3d. 1 29 65 
 Ist. 10 0 70 
 III. } 2 4 21 70 
 3d. 2 28 70 
 
 It would be needless to try 8 and 3, or any other number, as no solution could be 
 obtained from them, for reasons which will be seen hereafter. 
 
 Remarks.— We are told by M. de Lagny, in the second part of his ‘* Arithmetique 
 Universelle,” p. 456, that this question is susceptible of no more than six solutions ; 
 but the author is here mistaken, for we have pointed out ten. As it may afford plea- 
 sure to those who are studying algebra, to be made acquainted with the method em- 
 ployed for obtaining them, we think it our duty here to give it. 
 
 We shall call the price at which the women sold the first time u; and that at 
 which they sold the second time p. 
 
 If x then be the number of the eggs sold by the first woman, at the price u, the 
 number of those sold at the price p will be 10 — x; the money arising from the first 
 sale will be x u, that of the second will be 10 p — pz, and the sum total will be zu 
 +10p — pz. If z be the number of eggs sold by the second woman, at the first sale, 
 we shall have uz for the money arising from the first sale, and 25 p — pz for that 
 arising from the second, making together z u-+- 25 p — pz: 
 
 In like manner, if y represent the number of eggs sold, the first time, by the third 
 woman, we shall have uy for the money arising from the first sale, 30 p — py for 
 that of the second, and for the total of the two sales uy-+30p— py. But, by 
 the supposition, these three sums are equal; consequently ru-+-l0 p—px=zu 
 + 25 p—pz=uy-+ 30 p — py, from which we obtain the three following new 
 equations : 
 ru—pr=zu—pz+ 5p 
 ru—pe—=wuy— py 0p 
 z2u—pz=uy—py+i5p 
 And dividing the whole by u — p, we have these three others, viz., 
 
 ves 15 p 
 z Aghios 
 20 
 ee ree 
 5 
 Wh area 
 
 from which it may be concluded, that « — p must be a divisor of 15, 20, and 5, 
 otherwise 15 p 20 p o'p , would not be integral numbers, which it is ne- 
 Uu 
 
 Saar ie i = : 
 cessary they Peal be. tT But fe only number which is a divisor of 15, 20, and 5, is 
 5, which shews that the prices of the two sales could be only 5 and 0, 6 and 1, 7 and 
 2, 5 and 3, &c. 
 
 It may he easily seen, that the supposition of 5 and O will not answer the con- 
 ditions, since in that case there would have been only one sale. 
 
 We must therefore try the second supposition, 6 and 1, viz. «= 6 and p=1, which 
 -gives for the two last equations, x= y-+-4,z=y-+l. 
 
90 ARITHMETIC. 
 
 But we have here three unknown quantities, and only two equations; for which) 
 reason one of these unknown quantities must be assumed at pleasure. Let us take 
 y, and first suppose it = 0. | 
 
 This will give = 4, and z= I, and we shall have the first solution, which shews 
 that the first woman sold the first time 4 eggs at 6 pence each, and consequently, 
 the second time 6 at 1 penny each; while the second sold 1 the first time at 6 pence, 
 and the other 24 at 1 penny each, and the third sold all her eggs at the second price: 
 they would then all have 30 pence each. 
 
 By making y = 1, we shall have the second solution. 
 
 By making y = 2, we shall have the third. 
 
 By making y = 3, we shall have the fourth. 
 
 By making y = 4, we shall have the fifth. 
 
 By making y = 5, we shall have the sixth. 
 
 By making y = 6, we shall have the seventh, 
 
 We cannot suppose y to be greater than 6, because then we should have z = 10; 
 which is impossible, as the first woman has only 10 eggs to sell. 
 
 We must therefore proceed to the following supposition, viz. u = 7, and p = 2, 
 which gives two equations, 7 = y-+ 8; z=y-+-2. 
 
 If y here be first made = 0, we shall-have z = 8, and z = 2, which gives the 
 eighth solution, 
 
 By making y = 1], we shall have the ninth. 
 
 By making y = 2, we shall have the tenth. } 
 
 But y cannot be made greater, for then would be greater than 10, which is im- 
 possible. 
 
 It would be useless also to try for the values of u and p, 8 and 3; for these would 
 necessarily give to z a value greater than 10, which cannot be the case. __ 
 
 We may therefore rest assured that the problem is susceptible of no more solu- 
 tions than the ten above-mentioned. 
 
 PROBLEM XIII. 
 
 To find the number and the ratio of the weights with which any number of pounds, 
 Jrom unity to a given number, can be weighed in the simplest manner. 
 
 Though this problem on the first view seems to belong to mechanics, it may be 
 readily seen that it is only an arithmetical question: for, to solve it, nothing is ne- 
 cessary but to find a series of numbers beginning with unity, which, added or sub- 
 tracted from each other in every way possible, shall form all the numbers from unity 
 to the greatest proposed. ; 
 
 It may be solved two ways; either by addition alone, or by addition combined with 
 subtraction. In the first case, the series of weights which answers the problem, 
 is that of the numbers increasing in double progression; in the second, it is that of 
 those in the triple progression. 
 
 Thus, for example, with weights of 1 pound, 2 pounds, 4 pounds, 8 pounds, and 
 16 pounds, we can weigh any number of pounds up to 31: for, with 2 and 1 we ean 
 form 3 pounds; with 4 and 1, 5 pounds; with 4 and 2, 6 pounds; with 4, 2, and 1, 
 7 pounds, &e With the addition of a weight of 32 pounds, we can weigh as far as 
 63 pounds; and so on, doubling the last weight, and deducting from that double 
 unity. ‘ 
 
 But by employing weights in the triple progression, 1, 3, 9, 27, 81, all weights from 
 ] pound to 121 can be weighed with them: for, with the second less the first, that 
 is to say, putting the first into one scale and the second into the other, we can make 
 2 pounds ; by putting both in the same scale, we can form 4 pounds; by putting 9 
 on the one side and 3.and 1 on the other, 5 pounds; by 9 on the one side and 
 
AMUSING PROBLEMS. 91 
 
 3 on the other, 6 pounds; by 9 and 1 on the one side and 3 on the other, 7 pounds ; 
 nd so on, 
 
 It is here evident that this last method is the simplest, being that which requires 
 he least number of different weights. 
 
 Both these progressions are more advantageous than any of the arithmetical ones ; 
 as will appear on trial; for if the increasing arithmetical weights, 1, 2, 3, 4, &c. were 
 employed, 15 would be necessary to weigh 120 pounds; to weigh 121 with weights 
 in the increasing progression 1, 3, 5, 7, &c., 11 would be required. No other pro- 
 gression would make up all the weights possible, from 1 pound to the greatest re- 
 sulting from the whole of the weights. The triple proportion therefore is the most 
 convenient of all. 
 
 _ The solution of this problem may be of the greatest utility in commerce, and the 
 ordinary concerns of life, as it affords the means of weighing any weight whatever 
 with the least possible number of different weights. 
 
 PROBLEM XIV. 
 
 A country woman carrying eggs to a garrison, where she had three guards to pass, 
 sold at the first, half the number she had and half an egg more; at the second, 
 the half of what remained and half an egg more; and at the third, the half of 
 the remainder and half an egg more: when she arrived at the market place, she 
 had three dozen still to sell. How was this possible without breaking any of the eggs ? 
 
 It would appear, on the first view, that this problem is impossible; for how can 
 half an egg be sold without breaking any? The possibility of it however will be 
 evident when it is considered, that by taking the greater half of an odd number, we 
 take the exact half + 4. It will be found therefore that the woman, before she passed 
 the last guard, had 73 eggs remaining, for by selling 37 of them at that guard, which 
 is the half +- 4, she would have 36 remaining. In like manner, before she came to 
 the second guard she had 147 ; and before she came to the first, 295. 
 
 ' This problem might be proposed also in a different manner, as follows : 
 
 / PROBLEM XV. 
 
 A gentleman went out with a certain number of guineas, in order to purchase 
 necessaries at different shops. At the first he expended half his guineas and 
 half a guinea more ; at the second, half the remainder and half a guinea more ; 
 and so at the third. When he returned he found that he had laid out. all his 
 money, without having received any change. How was this possible ? 
 
 He had 7 guineas, and at the first shop expended 4, at the second 2, and at the 
 third 1; for 4 is the half of 7 and 4 more; the remainder being 3, its half is 14, and 
 4 more makes 2; but 2 taken from 8 leaves 1, the half of which is 3, and 3 makes 
 1; consequently nothing more remains. 
 
 _ Remark.—If the number of places at which the gentleman ietied all his money 
 were greater, nothing would be necessary but to raise 2 to such a power, that the 
 exponent should be equal to the number of places, and to diminish it by unity. 
 Thus, if there were 4, as the fourth power of 2 is 16, the required number would be 
 15; if there were 5, the fifth power of 2 being 32, the required number would be 31. 
 
 PROBLEM XVI. 
 
 Three persons have each such a number of crowns, that if the first gives to the 
 other two as many as they each have ; and if the second and third do the same ; 
 they will then all have an equal number, namely 8. How many has each ? 
 
 The first has 13, the second 7, and the third 4; as may be easily proved, by dis- 
 tributing the crowns of each as announced in the problem. 
 
92 ARITHMETIC. 
 
 PROBLEM XVII. | 
 
 A wine merchant who has only two sorts of wine, one of which he sells at 10s.,_ 
 
 and the other at 5s. per bottle, being asked for some at 8s. per bottle, wishes to 
 
 know how many bottles of each kind he must mix together, to form wine worth 
 8s. per bottle ? 
 
 The difference between the highest price, 10s., and the mean price required, is Qs 
 and that between the mean price and the lowest is3; which shews that he must tall 
 3 bottles of the wine at the highest price, and two of that at the lowest. ‘This mix 
 ture will form 5 bottles, worth 8s. each. 
 
 In problems of this kind, in general, as the difference between the highest price 
 and the mean price, is to the difference between the mean price and the lowest, so is 
 the number of measures at the lowest price, to that of those at the highest, which 
 must be mixed together to have a similar measure at the mean price. 
 
 PROBLEM XVIII. 
 
 A gentleman is desirous of sinking £100,000, which together with the interest is to 
 
 become extinct at the end of 20 years, on condition of receiving a certain annuity 
 
 during that time. What sum must the gentleman receive annually, supposing ims 
 
 terest to be at the rate of five per cent.?— 
 
 The sum which the gentleman ought to receive annnally i is £8014. 19s. 2d. 1°7f. 
 
 If the number of years were small, for example 5, this problem might be resolved, 
 without algebra, by the retrograde method, and false position; for if we suppose the 
 sum, which at the last year exhausts the capital and interest, to be £10,000, we shall 
 find that the capital alone at the commencement of that year was £95231}; and if 
 we add the £10,000, which were paid at the end of the year preceding the last, the 
 sum, £1952314, will be the capital increased with the interest of the 4th year; con- 
 sequently the capital at the beginning of that year was only £18594,6; whence it 
 follows, that before the payment, at the end of the third year, the sum was £28594 
 which represented the capital increased with the interest of the third year. By 
 thus going back to the commencement of the first year, the original capital will be 
 found to be £43294. 15s. 4d. We must then make the following proportion: As 
 this capital is to the supposed sum of £10000, so is the sum to be sunk, on the 
 above conditions, to the annuity, or sum to be received every year. 
 
 But it may be readily perceived, that in the case of 20 or 30 years, this method 
 would require very long calculations, which are greatly shortened by algebra.* 
 
 PROBLEM XIX. 
 
 What is the interest with which any capital whatever would be increased, at the 
 end of a year, if the interest due at every instant of the year were itself to be- 
 come capital and to bear interest? 
 
 This problem, to be well understood, has need of explanation. A person might 
 place out his money under this condition, that the interest due at the end of a month, 
 which at the interest of 5 per cent. would make a 60th of the capital, should be added 
 to the capital, and bear interest the following month at the same rate; that at the 
 
 * If a be the capital, m the interest, a m the number of years; the annuity or sum to be re- 
 
 ax(1lt+ 1)” 
 ceived every year, will be —————T n>? which in the case of 20 ae 
 mx (Lt fot ayn 20 years, and allowing in 
 i & es : 2°6584 
 terest to be at 5 per cent., (sm being then == 20) will be found = a x Saleas 
 - oO eo . 
 
AMUSING PROBLEMS. 93 
 
 piration of this month, the interest of the above sum, which would be a 60th -+ 
 4, of the original capital, should be still added to the capital, increased by the in- 
 erest of the first month, and bear interest the following month, and so on to the end 
 f the year. 
 
 ! What is done here in regard toa month, might be done in regard to a day, an hour, 
 -minute, or even a second, which may be considered as a part of. the day infinitely 
 mall: the question then is to know, what in this case would be the interest pro- 
 luced by the capital at the end of the year, the interest of the first second being at 
 he rate of five per cent. or sth. 
 
 It might be supposed, on the first view, that this compound and super-compound 
 nterest would greatly increase the 5 per cent., and yet it will be found that it pro- 
 luces an increase scarcely sensible ; for if the capital be 1, the same capital increased 
 with simple interest, at five per cent., will be 1 -+ 3, or 1 + 4§§, and when in- 
 sreased with the interest accumulated every second, it will be 1,335, or rather, when 
 
 05127 
 expressed more exactly, 15t37., 
 
 PROBLEM XX. 
 
 A dishonest butler, every time he went into his master’s cellar, stole a pint from a 
 particular cask, which contained 100 pints, and supplied its place by an equal 
 quantity of water. At the end of 30 days, the theft being discovered, the butler 
 was discharged. Of what quantity of wine did he rob his master, and how much 
 remained in the cask ? 
 
 It may be readily seen that the quantity of wine which the butler stole did not 
 amount to 30 pints; for the second time that he drew a pint from the cask, taking 
 the hundredth part of what it contained, it had already in it a pint of water, and as 
 he each day substituted for the liquor he stole a pint of water, he every day took 
 less than a pint of wine. To resolve, therefore, the problem, nothing is necessary 
 but to determine in what progression the wine which he every day stole decreased. 
 
 For this purpose, we must first observe, that after the first pint of wine was drawn, 
 there remained in the cask no more than 99 pints, and the pint of water which had. 
 been added. When a pint therefore was drawn from the mixture, it was only #2, of 
 a pint of wine}; but before the pint was drawn, the cask contained 99 pints of wine ; 
 consequently, after it was drawn, there remained 99 pints -- #9, that is to say $Y, or 
 98 pints + 745. When the third pint was drawn, the wine contained in it would 
 
 be only $8 + shoo Which being taken from the quantity of wine in the cask, viz, 
 98,1; pints, would leave 2299, or 97 pints ++ saogy° 
 
 It must here be remarked, that 98% is the square of 99 divided by 100; and that, 
 ce is the cube of 99 divided by the square of 100, and so cn; consequently, when 
 the second pint is drawn, the wine remaining will be the square of 99 divided by the- 
 
 first power of 100; after the third, it will be the cube of 99 divided by the square 
 of 100, &c. Whence it follows, that after the 30th pint is drawn, the quantity of 
 wine remaining will be the 30th power of 99 divided by the 29th power of 100. 
 But it may be found, by logarithms, that this quantity is 73%,, consequently the 
 quantity of wine stolen is 26. 3.* 
 
 * If the usual method of calculation were employed, it would be necessary to find the 30th power 
 99, which would contain not less than 59 figures, and to divide it by unity followed by 58 ciphers; 
 whereas if logarithms be used, nothing is necessary but to multiply the logarithm of 99 by 30, which 
 will give 598690560, and to subtract the product of the logarithm of 100 multiplied by 29, which 
 
 is 580000000. ‘The remainder, 18690560, is the logarithm of the required quantity ; which, in the 
 tables, will be found to be nearly 723i 
 
94 ARITHMETIC. 
 
 PROBLEM XXI. 
 
 A and B can perform a certain piece of work in 8 days, A and C can do it inQ 
 days, and B and C in 10 days; how many days will each of them require to per- 
 form the same work, when they labour separately 2 
 
 A will perform it in 1434 days; B in 1733 days; and C in 23%, days. 
 
 PROBLEM XXII. 
 
 An Englishman owes a Frenchman £1. 11s. ; but has no other money to pay his 
 debt than seven shillings pieces, and the Reopen has only French crowns, valued 
 at five shillings. How many seven shillings pieces must the Englishman give to 
 the Frenchman, and how many crowns must the latter give to the former, that 
 the difference shall be equal to 31 shillings in favour of the Frenchman, so that 
 
 the debt may be paid? 
 
 The simplest numbers that answer this question, are 8 seven shillings pieces, and 5 
 crowns; for 8 seven shillings pieces make 56 shillings, and 5 crowns make 25; con- 
 sequently their difference, of which the Frenchman has the advantage in this kind of 
 exchange, is 3i shillings. 
 
 This problem is susceptible of an infinite number of solutions; for it will be found 
 that the same result may be obtained with 13 seven shillings pieces and 12 crowns; 
 18 seven shillings pieces and 19 crowns; always increasing the number of seven shil- 
 lings pieces by 5, and that of the crowns by 7. 
 
 Remark.—For the sake of young algebraists, we shall here give the analytical 
 solution of this problem. Let x represent the number of the seven shillings pieces, 
 and y that of the crowns; 7x then will be the sum given by the Englishman, and 
 that given by the Frenchman willbe = 5y. But as the difference of these two sums 
 must be equal to 31, we shall have 7x —5y = 31 shillings ; consequently 7 x = 81 
 
 + 5y, and «= 31 whe. Y—4 + 21" 3 4 oy shillings. But «is a whole number, and 4 
 
 being one also, 4 5y must be a whole number, and three times that quantity, 
 
 7 
 which is ocr ey =1-+42y+ aay must also be a whole number. Consequently 
 ay must be a whole number. Put it equal to u, then y = 7 u — 2, and 2, which 
 is equal to a =5u+3. Ifu=1,theny=5,ande=8. Ifu= 2, then. 
 y=12,andr=13. If u=3, theny = 19, and « = 18, &e. 
 
 CHAPTER XII. 
 
 OF MAGIC SQUARES. 
 
 THE name Magic Square, is given toa square divided into several other small 
 equal squares or cells, filled up with the terms of any progression of numbers, but. 
 generally an arithmetical one, in such a manner, that those in each band, whether 
 horizontal, or vertical, or diagonal, shall always form the same sum. 
 
MAGIC SQUARES. 95 
 
 There are also squares in which the product of all the terms in each horizontal, or 
 vertical, or diagonal band, is always the same. We shall speak of these also, but 
 in a cursory manner, because they are attended with as little difficulty as the 
 former. 
 
 These squares have been called magic squares, because the ancients ascribed to 
 them great virtues; and because this disposition of numbers formed the bases and 
 principle of many of their talismans. 
 
 According to this idea, a square of one cell, filled up with unity, was the symbol 
 of the deity, on account of the unity and immutability of God; for they remarked 
 that this square was by its nature unique and immutable; the product of unity by 
 itself being always unity. 
 
 _ The square of the root 2 was the symbol of imperfect matter, both on account of 
 the four elements, and of the impossibility of arranging this square magically, as will 
 be shewn hereafter. 
 
 A square of 9 cells was assigned or consecrated to Saturn; that of 16 to Jupiter ; 
 that of 25 to Mars; that of 36 tothe Sun; that of 49 to Venus; that of 64 to Mer- 
 cury ; and that of 81, or nine on each side, to the Moon. 
 
 Those who can find any relation between the planets and such an arrangement of 
 
 numbers, must no doubt have minds strongly tinctured with superstition; but such 
 was the tone of the mysterious philosophy of Jamblichus, Porphyry, and their dis- 
 ciples. Modern mathematicians, while they amuse themselves with these arrange- 
 ments, which require a pretty extensive knowledge of combination, attach to them 
 mo more importance than they really deserve. 
 _ Magic squares are divided into even and odd. The former are those the roots of 
 which are even numbers, as 2, 4, 6, 8, &c. ; the latter of those the roots of which 
 are odd, and which, by a necessary consequence, have an odd number of cells; such 
 are the squares ot 3, 5, 7,9, &c. As the arrangement of the latter is much easier 
 than that of the former, we shall first treat of them. 
 
 SECTION I. 
 Of Odd Magie Squares. 
 
 There are several rules for the construction of these squares; but, in our opinion, 
 the simplest and most convenient, is that which, according to M. de la Loubere, is 
 employed by the Indians of Surat, among whom magic squares seem to be held in as 
 much estimation as they were formerly among the ancient visionaries before men- 
 tioned. 
 
 We shall here suppose an odd square, the root of which is 5, and that it is 
 required to fill it up with the first 25 of the natural numbers. In this case, begin 
 by placing unity in the middle cell of the horizontal 
 band at-the top; then proceed from left to right, as- 
 cending diagonally, and when you go beyond the square, 
 transport the next number 2 to the lowest cell of that 
 vertical band to which it belongs; set 3 in the next cell, 
 ascending diagonally from. left to right, and as 4 would 4 | 6 | 13 | 20 | 22 
 go beyond the square, transport it to the most distant |——~-|——|——|——|——_ 
 cell of the horizontal band to which it belongs; set Sin | 19 | 12] 19 | 21) 3 
 the next cell, ascending diagonally from left to right, |—_|_,1|,-|o | ¢ | 
 and as the following cell, where 6 would fall, is already 
 occupied by 1, place 6 immediately below 5; place 7 and 
 8 in the two next cells, ascending diagonally, as seen in the figure; and then, in 
 consequence of the first rule of transposition, set 9 at the bottom of the last vertical 
 band ; then 10, in consequence of the second, in the last cell on the left of the second 
 horizontal band; then 11 below it, according to the third rule: after which continue 
 
 SS |e eee ee -_——~— 
 
96%) >- ARITHMETIC, 
 
 to fill up the diagonal with the numbers 12, 13, 14, 15; and as you can ascend no 
 farther, place the following number 16 below 15; if you then proceed in the same 
 manner, the remaining cells of the square may be filled up without any difficulty, as) 
 seen in the above figure. 
 
 The following are the squares of 3 and7 filled up by the same method ; and as hed 
 examples will be sufficient to exercise such of our readers as have a taste for amuse- 
 ments of this kind, we shall proceed to a few general remarks on the properties of a 
 square arranged according to this principle. ' 
 
 8 1 6 46 6 8 17 | 26 | 35 37 
 Tig © fig “8 [14 | 16 [25 | 34 | 36 | 45 
 “4 | 9 2{ [is |15| 24] 33 | 42 | 4a] 4 
 
 “a1 | 23 | 32 | 41 | 48] 3 | 12 
 “22 | 31| 40| 49| 2 | 11 | 20 
 
 22 | 31} 40 | 49 
 
 Ist. According to this disposition, the most regular of all, the middle number of 
 the progression occupies the centre, as 5 in the square of 9 cells, 13 in that of 25, 
 and 25 in that of 49; but this is not necessary in the arrangement of all magic 
 squares. 
 
 2d. In each of the diagonals, the numbers which occupy the cells equally distant 
 from the centre, are double that in the centre: thus 30 + 20 = 47 +3 = 28+ 22 
 = 24 + 26, &c., are double the central number 25. 
 
 3d. The case is the same with the cells centrally opposite, that is to say, those 
 similarly situated in regard to the centre, but in opposite directions both laterally 
 and perpendicularly: thus31 and 19 are cells centrally opposite, and the case is the 
 same in regard to 48 and 2, 13 and 37, 14 and 36, 32 and 18. But it happens 
 that, according to this magic arrangement, those cells opposite in this manner, are 
 always double the central number, being equal to 50, as may be easily proved. 
 
 _ 4th. It may be readily seen, that it is not necessary that the progression to be 
 arranged magically, should be that of the natural numbers J, 2, 3, 4, &c.: any 
 arithmetical progression whatever, 3, 6, 9, 12, &c., or 4, 7, 10, 13, 16, &c., may be 
 arranged in the same manner. | 
 5th. Nor is it necessary that the progression should be continued: it may be 
 disjunct, and the rule isas follows. If the numbers of 
 the progression, arranged according to their natural 
 order in the cells of the square, exhibit in every direc- 1 ro Be ee 
 tion, vertical and horizontal, an arithmetical progres- |_| |. |. |._|..+¢ 
 sion, they are susceptible of being arranged magically 
 in the same square, and by the same peace Let us 13 |.14.J 150) decane 
 take, for example, the series of numbers 1, 2, 3, 4,5; {——)——|——|——);—— 
 4; 8, 9, 10,115 13, 14, glOs\LG; elie; 20) 2h 22, 23 ; 19 | 20 | 21 | 23 | 23 | 
 25, 26, 27, 28, 29: as these, when arranged in the cells §(——|——|———|— | 
 of a square, every where exhibit an arithmetical pro- 260) PON eT) eee 
 gression, they may be arranged magically; and indeed, 
 according to the above rule, they may be formed into the annexed magic square. 
 
 Moscopulus, a modern Greek author, and Bachet have also invented magic 
 
 squares. But their methods are inferior to one contrived by M. Poignard, and im- 
 
: 
 io. 
 4 
 
 =a 
 
 MAGIC SQUARES. 97 
 
 proved by M. de la Hire. Of this method we now proceed to give a short 
 account. 
 
 Let it be required to fill up a square having an odd root, such as 5. Having con- 
 structed the square of cells, place in the first horizontal row at the top, the five 
 first numbers of the natural progression, in any order, at 
 pleasure, which we shall here suppose to be 1, 8, 5, 2, 
 
 4; then make choice of a number, which is prime to the - 
 root 5, and which when diminished by unity does not |—— 
 measure it: let this number be 3; and for that reason | | 7 | 
 
 begin with the third figure of the series, and count | 4 | 1 | 3 | 5 | 9. 
 from it to fill up the second horizontal band 5, 2, 4, | — 
 J, 3; then begin again by the next third figure, includ- 3 
 ing the 5, that is to say by 4, which will give for the |~— 
 third band 4,.1, 3, 5,2; by following the same process, 
 we shall then have the series of numbers 3, 35, 2, 4, 1, 
 
 to fill up the fourth band: continue in this manner, always beginning at the third 
 figure, the preceding included, until the whole square is filled up. This square will 
 be one of the components of the required square, and will be magic; for the sum of 
 each band, whether horizontal, or vertical, or diagonal, is the same, as the five num- 
 bers of the progression are contained in each without the same figure being ever 
 ‘repeated. 
 
 Now construct a second geometrical square of 25 
 cells, in the first band of which inscribe the multiples S07 OM eno 
 of the root 5, beginning with a cipher, viz. 0, 5, 10, (——;—~—|—— 
 15, 20, and in any order at pleasure, such for example 
 as 5, 0, 15, 10, 20: then fill up the square according to 
 the same principle as before, taking care not to assume 
 the same number in the series always to begin with. | 20 
 Thus, for example, as in the former square, the third |—— 
 figure in the series was taken, in the present one the 
 fourth must be assumed; and thus we shall have a 
 square of the multiples, as seen in the annexed figure. This is the second eompo- 
 nent of the required magic square, and is itself magic, since the sum of each band in 
 every direction is the same. 
 
 Now to obtain the magic square required, nothing is 6 |} 3 | 20 | 12.) 24 | 
 necessary but to inscribe, in a third square of 25 cells, .———;———-—-|_—_|_— 
 the sum of the numbers found in the corresponding | 15 | 22) 9 | 1 | 18 
 cells of the preceding two; for example 5-+ 1, or 6, in 
 
 _ the first on the left, at the top of the required square; 
 0 + 3, or 3in the second, andso on; by these means we 23. }.10:) 2) erg eee 
 shall have the annexed square of 25 cells, which will |——|——.——~——-|—— 
 necessarily be magic. 47) 1S 2 
 
 By these means, any of the numbers may be made to 
 fall in any cells at pleasure; for example 1 in the central 
 cell; nothing is necessary for this purpose, but to fill 
 up the middle band with the series of numbers in such 
 a manner that 1 may be in the centre, as seen in the 
 annexed figure ; and then to fill up the rest of the square 
 according to the above principles, begiuning at the 
 highest band, when the lowest has been filled up. 
 
 H 
 
ie (ef rhe \ahe ys: Buy 
 labe ae Cp aes -P 
 
 ¢ is i « 
 is i fet se 
 ad aay 
 
 i ia G 
 “ale ARITHMETIC. 
 Aa 
 Oe a blG 
 . A 
 
 bea 2 ino form the second square, place a cipher in the 
 
 | centre, as seen in the annexed figure, and fill up the 
 fi remaining cells in the same manner as before, taking 
 | care not to assume the same quantities as in the former 
 . for beginning the bands. 
 
 ‘ A (4 bea In the last place, form a third square by adding to- |—~~|—~)_—.—_|_—— 
 aw : Ab cgether the numbers in the similar cells, and you will 0 
 oe Lap, ‘aa the annexed square, where 1 will necessarily | 39 | 19 | 1 | 18! 24 
 cae 
 
 EQSTY CRN YS ge 
 SHIT ES 
 
 ‘eh a goccupy the centre. Ln te pete 
 
 24 5 17 | 21 
 
 Remarks.—I. We must here observe, that when the number of the root is not 
 prime; that is, if it be 9, 15, 21, &c., itis impossible to avoid a repetition of.some 
 of the numbers, at least in one of the diagonals; but in that case it must be ar- 
 ranged in such a manner, that the number repeated in that diagonal shall be the 
 middle one of the progression; for example 5, if the root of the square be 9; & 
 if it be 15; and as the square formed of the mujtiples will be liable to the same 
 accident, care must be taken, in filling them up, that the opposite diagonal shall 
 contain the mean multiple between 0 and the greatest; for example 36 if the root— 
 be 9; 105 if it be 15. 
 
 II. The same thing may be done also in squares main have a prime number 
 for their root. By way of example we shall here form a magic square of the first 
 two of the following ones: 
 
 1 2 3 
 a a) aS ee ee ae 104.8 as toe 11 | 2 | 10119 | 23 
 Buttes Mahe ts (1 By 904 10 PO Ss Abas 99 1/15 4°) gm) 496 
 ala ae Nk a {5k QO Obi Ont AS 90 | 24113143 47 
 CARRS Coe Gas Bot 5h 20cr 30710 9'| 18 | 31 | 1945 
 Slay) 99 1s | 4 | ovl'5 ‘b15 Kae }-2e 3 126 1 Ay (ogee 
 
 in the first of which 3 is repeated in the diagonal descending from right to left, 
 and in the second 10 is repeated in the diagonal descending from left to right. 
 This however does not prevent the third square, formed by their addition, from 
 being magic. 
 
 SECTION II. 
 Of Even Magic Squares. 
 
 The construction of these squares is attended with more difficulty than that of 
 the odd squares, and the degree of difficulty is different, according as they are 
 evenly even, or oddly’ even: for this. reason we must divide them into two 
 classes. 
 
D aig eres ciate 999 
 
 Squares evenly even, are those the root of which when Halved is even, or can 
 be divided by 4 without remainder ; of this kind are the squares of 4, 8, 12, &c. 
 The oddly even are those the root of which when halved gives an odd number ; 
 as those of 6, 10, 14, &c. 
 
 As the ancients have left us no general rule on this subject, but only some 
 examples of even squares magically arranged, we shall here give the best methods 
 invented by the moderns, and shall begin with squares evenly even. 
 
 Let us suppose then that the annexed square ABCD is 
 
 to be filled up magically, with the first 16 of the natural ee eae 
 numbers: fill up first the two diagonals; and for that 1 4 
 purpose begin to count the natural numbers, in order, ——| ——|——}—— 
 1, 2, 3, 4, &e., on the cells of the first horizontal band _ Ota 
 
 from left to right ; then proceed to the second band, and oT eo Lae 
 
 when you come to the cells belonging to the diagonals, in- 
 scribe the numbers counted as you fall upon them; by 13 16 
 which means you will have the arrangement repre- — 
 
 sented in the annexed figure. 
 
 When the diagonals have been thus filled, to fill up the cells which remain 
 vacant, begin to count the same numbers, proceeding from the angle p in the 
 cells of the lower band, going from right to left, and ee 
 then in the next above it; and when any cells are found 
 empty, fill them up with the numbers that belong to 
 them; in this manner you will have the square 16 
 filled up magically, as ~seen in the annexed figure, 
 and the sum of each band and each diagonal will be 
 34. 
 
 Rule for Squares evenly even. 
 
 Having given, according to M. de la Hire, a very general rule for odd squares, 
 which is capable of producing a great number of variations, we shall do the same in 
 regard to even squares; especially as it will equally serve for evenly even and oddly 
 even magic squares. It is as follows: 
 
 Let it be required, for example, to fill up magically a square of 8 cells on each 
 side, . 
 
 For this purpose, arrange, in the first 
 horizontal band in a square of that kind, 
 the first eight numbers of the arithmetical 8 
 progression, but in such a manner, that |—— 
 those equally distant from the middle 
 - shall form the same sum; viz. that of the 
 root augmented by unity, which in this 
 case is 9; the second band must be the 8 
 inverse of the first; the third must be |——-/——-|—— 
 like the first ; the fourth like the second, 1 
 and so on alternately, till the half of the |—— 
 square is filled up; after which the other 
 half may be formed by merely reversing 1 
 the first, as may be seen in the above 
 figure. This will be the first primitive square. 
 
 To form the second, fill it up according to the same principle with the multiples 
 of the root, beginning with 0, that is to say, 0, 8, 16, 24, 32, 40, 48,56; taking care 
 that the extremes shall always make 56, but instead of arranging these numbers in a 
 horizontal direction, they must he arranged vertically, as in the following figure. 
 
 H 2 
 
 w os 
 Ae Oe ae Se" 
 
ARITHMETIC. 
 
 48 8 48 8 8 
 ERE DUE LEY 
 
 Pian seen 
 We eclcna cy cal oa ee 
 may Sap ge ee 
 Aa sacs (an ard ss. 
 ecenraue se 
 
 3 Wen es Orn ee eR 
 
 When this is done, add together the similar cells of the two squares, and you will 
 have a square of 8 on each side, as in the last figure above. 
 Without enlarging farther on squares evenly even, we shall give the simplest me- 
 
 thod of constructing squares oddly even. 
 
 Method for Squares oddly even. 
 
 We shall take, by way of example, the square 
 of the root 6. ‘To fill it up, inscribe in it the first 
 six numbers of the arithmetical progression, 1, 2, 
 3, &c., according to the above method; which 
 will give the first primitive square, as in the an- 
 nexed figure. 
 
 The second must be formed by filling up the 
 cells in a vertical direction, according to the same 
 principle, with the multiples of the root, begin- 
 ning at 0, viz. 0, 6, 12, 18, 24, 30. 
 
 The similar cells of the two squares if then 
 added, will form a third square, which will re- 
 quire only a few corrections to be magic. This 
 third square is as here annexed. 
 
 ne | ee ff | | 
 
 24 6 24 | 24 6 24 
 ~o | 30| 0 | 0 | 30| o- 
 a2} is | 12] a2 | is | 12 
 “ye | 12/18 | 1s | 12 | 18. 
 “30 | 0 | 30| 30] 0 | 30 
 
 aus A 
 
 29 | 12 | 27 | 28 7 26 
 2|si| 4] 3 |36| 5 
 17 [24] 15 | 16 | 19 | 14 
 23 | 18 “a1 | 22 | 13 | 20. 
 “a2 | | 34|33| 6 | 35 
 “11 |30| 9 | 10| 25 | 6 
 
MAGIC SQUARES. 101 
 
 To render the square magic, leaving the corners fixed, transpose the other numbers - 
 of the upper horizontal band, and of the first vertical one on the left, by reversing 
 all the remainder of the band; writing 7, 28, 27, 12, instead of 12, 27, &c., and in 
 the vertical one, 32, 23, 17, and 2, from the top downwards, instead of 2, 17, &c. 
 
 It will be necessary alsc to exchange the num- 
 bers in the two cells of the middle of the second 
 -horizontal band at the top, of the lowest of the 32 | 31 | 3 
 
 4 36 5 
 second vertical band on the left, and of the last |__|. J. ~~) jd 
 on the right. The numbers in the cells a and B 230.260 ini S | 16 | 19 | 20 
 must also be exchanged, as well as those inc and) |——~|———|—_~|——_ + —— 
 D; by which means we shall have the square M4 ne rete 
 corrected and magically arranged. ose hirsah sg | 6 Veet 
 
 | 
 11 | 25 | 10] 27 | 30) 8 | 
 i 
 
 SECTION III. 
 Of Magic Squares with Borders. 
 
 Modern arithmeticians have added a new difficulty to the subject of magic squares, 
 by proposing not only to arrange magically in a square a progression of num- 
 bers, but by requiring that this square, when lessened by a band on each side, or two 
 or three bands, &c., shall still remain magic; or a magic square being given, to add 
 to it a border of one or more bands, in such a manner, that the enlarged square thence 
 resulting shall be still magic. 
 
 To give an example of this construction, let it be required to form a magie square 
 of the root 6, and to fill it up with the natural numbers, from 1 to 36. The first 
 even magic square possible being that of 4 on each side, we shall first arrange it 
 magically, filling it up with the mean terms of the progression, to the number 16, 
 and reserving the first and the last 10 for the border. For the interior square there- 
 fore we shall take the numbers 1], 12, &c., as far as 26 inclusively, and shall give 
 them any magic disposition whatever: there will then remain the numbers 1], 2, &c., 
 as far as 10, and 27 as far as 36, for the border. 
 
 To dispose these numbers in the border, first 
 place the numbers 1, 6, 31, 36, in the four corners, 
 and in such a manner that diagonally they shall 
 make 37. As each band must make 111, it will be 
 necessary to place in the first band four such num- 
 bers,that their sum shall be 104; and as their com- 
 plements to 37 must be found in the lowest, where 
 there is already 67, it will be necessary that they 
 should together make 44: there are several com- 
 binations of these numbers, four and four, which 
 can make 104, and their complements 44; but it 
 is necessary at the same time that four of those re- 
 maining should make 79, to fill up the first vertical band, while their complements 
 make 69 to complete the last. This double condition limits the combination to 
 35, 34, 30, 5, which may be placed in the first band in any order whatever, provided 
 their complements be placed below each of them in the last band; and the four 
 numbers requisite to fill up the first vertical band will be 33, 28, 10, 8, which may 
 be arranged any how at pleasure, provided the complement of each be placed oppo- 
 site to it in the corresponding cell on the other side. 
 
 — 
 
102 ARITHMETIC. 
 
 It is not absolutely necessary that 1, 6, 31, 36 
 should be placed in the four corners of the square: 
 if we suppose them to be filled up, in the same 
 order, with 2, 7, 30, 35, it would be then neces- 
 sary that the four first numbers should make 102, 
 and their complements 46, while the four last 
 make 79, and their complements 69: but it is 
 found that the first four numbers are 36, 31; 27, 
 8, and the second 34, 32, 9, 4. The first being 
 _ arranged any how in the four empty cells of the 30 | j 
 
 first band,and their complements below, the second 
 must be arranged in the cells of the first vertical 
 band, and their complements each at the extremity of the same horizontal band; 
 by which means we shall have the new square with a border, as seen above. 
 
 If it were required to form a bordered square of the root 8; it would be neces- 
 sary to reserve for the interior square of 36 cells, the 36 mean numbers of the 
 progression ; and they might be formed into a bordered square around the magic 
 square of 16 cells; with the 28 remaining numbers, we might then form a border 
 to the square of 36 cells, &c. 
 
 Hence it appears, in what.manner we might form a magic square, which when 
 successively lessened by one, two, or three bands, shall still remain magic. 
 
 6 | 10 | 29 | 35 | 
 
 SECTION IV. 
 Of another kind of Magic Square in Compartments. 
 
 Another property, of which most magic squares are susceptible, is, that they 
 are not only magic when entire, but that when divided into those squares into 
 which they can be resolved, these portions of the original square are themselves 
 magic. A square of 8 cells on a side, for example, formed of four squares, each 
 having 4 for its root, being proposed, it is required that not only the square of 
 64 shall be disposed magically, but each of those of 16, and that the latter even, 
 however arranged, shall still compose a magic square. 
 
 What is here required, is easy; and this is even the simplest method of all 
 for constructing squares that are evenly even, as will appear from what follows. 
 
 To construct a square of 64, in this manner, take the first 8 numbers of the 
 natural progression, from 1 to 64, and the 8 last, and arrange them magically in a 
 square of 16 cells; do the same thing with the 8 terms which follow, the first 8 and — 
 the 8 which precede the last 8, and by 
 these means you will have a_ second tes) ba Ta oo ee 
 
 magic square; form a similar square of iodo Paha ae 
 the 8 following numbers with their cor- | 60 | 6 | 7 | 57} 521141] 15 | 49 
 responding ones, and another with the |——|——|——j——j——|—_|____|___ 
 
 16 means: the result will be four squares | 8 | 58 | 52} 5 | 16 | 50] 51 | 13 
 
 of 16 cells, the numbers in which will Pata ts CPE PY 5 r 
 - be equal when added together, either in : 1 he ae 
 bands or diagonally; for they will every | ,, | 49 461 20 | os \'se eae 
 
 where. be 130.. It is therefore evident, 02 of 3) iL ee et 
 that if these squares be arranged side by | 44 | 22 | 23 | 41 | 36 | 30 / 31 | 33 
 side, in any order whatever, the square |——|——|-—|——}——|——/_—_|__ 
 
 24 | 42 | 43 | 21 § 32 | 34] 35 |.29 
 
 resulting from them will be magic, and 
 the sum in every direction will be 260. | 4, 19 | 18 [a8 Po? | 9 dope ee 
 
iain ae 
 
 MAGIC SQUARES. 103 
 
 SECTION V. 
 Of the Variations of Magic Squares. 
 
 The square having 3 for its root is suceptible of no variation: whatever method 
 may be employed, or whatever arrangement may be given to the numbers of the 
 progression from 1 to 9, the same square will always arise, except that it will be 
 inverted, or turned from left to right, which is not a variation. But this is not 
 the case with the square having 4 for its root, or that of 16 cells: this being sus- 
 ceptible of at least 880 variations, which M. Frenicle has given in his Treatise on 
 Magic Squares. 
 
 The square of 5 is susceptible of, at least, 57600 different combinations; for 
 according to the process of M. de la Hire, the 5 first numbers may be arranged 
 120 different ways inthe first band of the first primitive square; and as they may 
 be afterwards arranged in the lower bands, beginning again by two different quan- 
 tities, this will make 240 variations, at least, in the primitive square; which, com- 
 bined with the 240 of the second, form 57600 variations in the square of 5. But 
 there are doubtless a great many more, for a bordered square of 5 cannot be 
 reduced to the method of M. de la Hire; but one bordered square of 5, the 
 corners remaining fixed, as well as the interior square of 3, may experience 36 
 variations. What a number therefore of other variations must be produced by 
 changing the interior square and the angles! 
 
 A bordered square of 6, when ence constructed, the corners remaining fixed, and 
 the interior square being composed of the same numbers, may be varied 4055040 
 different ways ; for the interior square may be varied and differently transposed in 
 the centre 7040 ways: each of the horizontal bands at top and at bottom, the ex- 
 tremities remaining fixed, may be varied 24 ways; for there are four pairs of num- 
 bers susceptible of changing their place, which may be combined 24 ways; and 
 there are also four pairs in the vertical bands between the corners. ‘The number 
 of the combinations, therefore, is the product of 7040 by 576, the square of 24, which 
 gives 4055040 variations. But the corners may be varied, as well as the numbers 
 assumed to form the interior square; and it hence follows, that the wnole number 
 of the variations of a square of 6, while it still remains bordered, is equal to several 
 millions of times the former. 
 
 The square of 7, by M. de la Hire’s method alone, may be varied 406425600 dif- 
 ferent ways. 
 
 These variations, however numerous, ought to excite no surprise; for the number 
 of dispositions, magic or not magic, of 49 numbers, for example, forms one of 62 
 figures, of which the preceding is, as we may say, but a part infinitely small, 
 
 SECTION VI. 
 
 Of Geometrical Magic Squares. : 
 
 We have already observed, in the beginning of this chapter, that numbers in geo- 
 
 metrical progression might be arranged in the cells of a square, and in such a man- 
 ner, that the product of these numbers, in each band, whether vertical, or horizon- 
 
 tal, or diagonal, shall always be the same. 
 
 To construct a square of this kind, the same principles must be followed as 
 in the construction of other magic squares; and this may be 
 easily demonstrated from the property of logarithms. Without 128| 1 | 32 
 enlarging further therefore on this subject, we shall confine our- a) 
 
 selves to giving one example; it is that of the 9 first terms 4°1..16 5] 64 
 of the double geometric progression, 1, 2, 4, 8, &c. arranged Gaol obel a. 
 in a square of 3 cells on each side. The product is evi- 
 
 dently the same in every direction, viz, 4096. 
 
104 ee . ARITHMETIC. 
 
 Remarks.—The ingenious Dr. Franklin, it seems, carried this curious speculation — 
 further than any of his predecessors in the same way. He constructed both a 
 magic square of squares, and a magic circle of circles. The magie square of — 
 
 squares is formed by dividing a great square into 256 little squares, in which all 
 the numbers from 1 to 256, or the square of 16, are placed in 16 columns, which 
 may be taken either horizontally or vertically. Their chief properties are as 
 follow: 
 
 1. The sum of the 16 numbers in each column or row, vertical or horizontal, 
 is ais 
 
 2, Every half column, vertical and horizontal, makes 1028, or just one half of the 
 
 same sum 2056. 
 
 3. Half a diagonal ascending, mided to half a diagonald descending, makes also 
 the same sum 2056; taking these half diagonals-from the ends of any side of the 
 square to the middle of it; and so reckoning them either upward or downward ; or 
 sideways from right to left, or from left to right. 
 
 4, The same with all the parallels to the half diagonals, as many as can be drawn 
 in the great square: for any two of them being directed upward and downward, 
 from the place where they begin, to that where they end, their sums still make the 
 same 2056. Also the same holds true downward and upward; as well as if taken 
 sideways to the middle, and back to the same side again. 
 
 5. The four corner numbers in the great square added to the four central num- 
 bers in it, make 1028, the half sum of any vertical or horizontal column which 
 contains 16 numbers; and also equal to half a diagonal or its parallel. 
 
 6. If asquare hole, equal in breadth to four of the little squares or cells, be cut 
 in a paper, through which any of the 16 little cells in the great square may be 
 seen, and the paper be laid upon the great square; the sum of all the 16 numbers 
 
 seen: through the hole, is always equal to 2056, the sum of the 16 numbers in any 
 
 horizontal or vertical column. 
 _ The magic circle of circles is composed of a series of stridtore foi 12 to 75 in- 
 clusive, divided into 8 concentric circular spaces, and ranged in 8 radii of numbers, 
 
 with the number 12 in the centre; which number, like the centre, is common to all — 
 
 these circular spaces, and to all the radii. 
 
 The numbers are so placed, that Ist, the sum of all those in either of the con- 
 
 centric circular spaces above mentioned, together with the central number 12, amount 
 to 360, the same as the number of degrees in a circle. 
 
 2. ‘The numbers in each radius also, together with the central number 12, make 
 just 360. 
 
 3. The numbers in half of any of the above circular spaces, taken either above or — 
 
 below the double horizontal line, with half the central number 12, make just 180, or 
 half the degrees in a circle. 
 
 4. If any four adjoining numbers be taken, as if in a square, in the radial divisions 
 of these circular spaces, the sum of these, with half the central number, make also 
 is same 180. 
 
 . There are also included four sets of other bane spaces, bounded by circles 
 ee excentric with regard to the common centre; each of these sets containing 
 five spaces. For distinction, these circles: are drawn with different marks, some 
 dotted, others by short unconnected lines, &c.; or still better with inks of divers 
 colours, as blue, red, green, yellow. 
 
 These sets of excentric circular spaces intersect those of the concentric, and each 
 other; and yet, the numbers contained in each of the excentric_ spaces, taken all 
 around through any of the 20, which are excentric, make the same sum as those in 
 the concentric, namely 360, when the central number 12 is added. 
 
POLITICAL ARITHMETIC. _ 105 
 
 Their halves also, taken above or below the double horizontal line, with half the 
 entral number, make up 180. 
 
 It is observable, that there is not one of the numbers, but what belongs at least 
 to two of the circular spaces; some to three,some to four, some to five; and yet 
 they are all so placed, as never to break the required number 360, in any of the 28 cir- 
 cular spaces within the primative circle. 
 
 CHAPTER XIII. 
 
 POLITICAL ARITHMETIC, 
 
 Since politicians have acquired juster ideas respecting what constitutes the real 
 strength of states, various researches have been made in regard to the number of the 
 inhabitants in different countries, in order to ascertain their population. Besides, as 
 almost all governments have been under the necessity of making loans for the most 
 part on annuities, they have naturally been induced to examine according to what 
 progression mankind die, that the interest of these loans may be proportioned to 
 the probability of the annuities becoming extinct. These calculations have been 
 distinguished by the name of political arithmetic, and as it exhibits several curious 
 facts, whether considered in a political or a philosophical point of view, we have 
 thought it our duty to give it a place here. 
 
 SECTION I. 
 Of the Proportion between the Males and the Females. 
 
 Many people imagine that the number of the females born exceeds that of the 
 males; but it has long since been proved that the contrary is the case. More boys 
 than girls are born every year ; and since the year 163], a. small interval excepted, 
 we have a register of births, in regard to sex, and it has never been observed that 
 ‘the number of the females born ever equalled that of the males. It is found, by 
 taking a mean or average term ina greater number of years, that the number of the 
 ‘males born is to that of the females, as 18 to 17. This proportion is nearly that 
 which prevails throughout all France ; but, to whatever reason owing, it seems at 
 Paris to be as 27 to 26. 
 
 This kind of phenomenon is observed, not only in England and in France, but in 
 every other country. We may be convinced of the truth of it by inspecting the 
 
 ‘monthly and other periodical publications, which at*the commencement of 
 every year give a table of the births that have taken place in most of the capital 
 cities of Europe: it may there be seen that the number of the males born, always 
 exceeds that of the females; and consequently it may be considered as a general 
 law of nature. 
 
 We may here observe a striking instance of the wisdom of Providence, which has 
 thus provided for the preservation of the human race. Men, in consequence of the 
 active life for which they are naturally destined by their strength and their courage, 
 are exposed to more dangers than the female sex ; war, long sea voyages, employments 
 laborious or prejudicial to health, and dissipation, carry off great numbers of the 
 males; and it therice results, that if the number born of the latter did not exceed 
 that of the females, the males would rapidly decrease, and soon become extinct. 
 
 SECTION II. 
 
 Of the Mortality of the Human Race, according to the Different Ages. 
 In this respect, there is apparently a considerable difference between large towns 
 
106 ARITHMETIC. 
 
 : 
 and the country; but this arises from the women in town rarely suckling their own 
 children; and consequently the greater part of their children being put out to nurse 
 in the country, as it is in the period of childhood that the greatest mortality prevails, 
 it becomes most apparent in the country. To make an exact calculation, it ought 
 to be founded on the deaths which happen in the towns, as well as in the country; 
 and this M. Dupré de St. Maur has endeavoured to do, by comparing the registers 
 of three parishes in Paris, and twelve in the country. 
 
 According to the observations of this author, in 23994 deaths, 6454 of them were 
 those of children not a year old; and carrying his researches on this subject as far as 
 possible, he concludes, that of 24000 children born, the numbers who attain to dife 
 ferent ages are as follow: 
 
 Ages. Number, } Ages. Number. | Ages. Number, 
 
 2 17540 BU ec Pasi ile oy pelo ee eae QD y's oh acon Nore ae SO 
 3 15162 BO lamtessd ad lah EO Ol vir giee cia eabtasks 71 
 4 .. 14177 AO ie ay eed Bue olislteient Die. esti icon aes 63 
 5 ~» 1o4th BEV Se ok o's ett an Meg OU G58 b Ms wa retaen 47 
 6 12968 SO Bia dant ree BLY, QE bee thie wits sate 40 
 7 -- 12562 DD in top swities sham od as 2D ieekch’ tig, cela 33 
 oe! sop SUR tary ts bla) BOs “alsa Woe hag wl MeO Gr. via deeb 23 
 MER od ieee ne al DOLD BD ie «ent low i eeoe Oictimeis bene 18 
 1Ote eine Tweets SOL TOOL all 6 os\noe en goes OB ab te fear, ee 16 
 ROE ves ee 7 LIADS TOL. Seat ee OUT D9 Dees ies esis 8 
 ae ss pens sete ge 10909 9 | (BO .2? nel, ae hg OT - | LOO ee 
 TOL bs «- 10259 eo ey) NE I OSD 
 
 Such then is the condition of the human species, that of 24000 children born, 
 scarcely one half attain to the age of 9; and that two thirds are in their grave before 
 the age of 40; about a sixth only remain at the expiration of 62 years ; a tenth after 
 70; a hundredth part after 86; about a thousandth part attain to the age of 96; and 
 six or seven individuals to that of 100. 
 
 We must however observe, that the authors who have treated on this subject, 
 differ from each other. According to the table of M. de Parcieux, for example, the 
 half of the children born do not die before 31 years are completed ; but according to 
 M. Dupré de St. Maur they are cut off before the commencement of the ninth year. 
 This difference arises from the table of M. de Parcieux having been formed from lists 
 of annuitants, who are always select subjects: for a father never thinks of pur- 
 chasing an annuity on the life of a child who is sickly, or has a bad: constitution. 
 The laws of mortality in these cases therefore are different; and if the one is a 
 general and common law, the other is that which public administrators, who grant 
 annuities, ought to consult with great care, that they may not make loans too 
 burthensome. = 
 
 SECTION III. 
 
 Of the Vitality of the Human Species, according to the Different Ages, or Medium 
 of Life. . | 
 When a child is born, to what age may a person bet, on equal terms, that it will 
 attain? Or if the child has already attained to a certain age, how many years is it 
 probable it will still live? These are two questions, the solution of which is not 
 only curious, but important. 
 We shall here give two. tables on this subject; one by M. Dupré de St. Maur, and 
 the other by M. Parcieux; and add to them a few general observations. 
 
POLITICAL ARITHMETIC. 107 
 
 TIME to LIVE, 
 
 M. de St. Maur. M. de Parcieux. | 
 
 AGE. | YEARS. | MONTHS. 
 
 — — a 
 
 YEARS. | MONTHS. 
 
 0 8 
 
 1 33 41 9 
 2 38 42 8 
 3 40 43 6 
 4 41 44 2 
 5 41 6 44 5 
 6 42 44 3 
 
 7 42 3 44 
 8 41 6 43 9 
 9 40 | 10 43 3 
 10 40 2 42 8 
 20 33 5 36 3 
 6 
 40 22 l 25 6 
 50 16 7 19 5 
 60 11 1 14> tay 
 70 6 2 9 2 
 75 4 6 6a =10 
 
 80 3 7 5 
 
 3 3 
 
 2 2 
 
 mnwwrann - 
 
 Two observations here occur, in regard to these tables. The first is respecting the 
 difference between them. In that of M. Parcieux, the time assigned to each age to 
 live, is more considerable, and the reason has been already mentioned. We have 
 even suppressed the first year in the table of M. Parcieux, because the difference was 
 by far too great, and in our opinion it arose from two causes. Ist. No one ever 
 thinks of purchasing an annuity for a child in its first year, until the goodness of its 
 constitution has been fully ascertained. 2d. It is not at the birth of a child, but in 
 the course of the first year, towards the middle or end, that such a measure is 
 hazarded ; for as annuities remain sometimes several months, and even a whole year, 
 to be filled up, people are not under the necessity of sinking money on so young a 
 life, and have time during the course of several months to acquire some certainty 
 respecting the constitution of the subject. In our opinion, therefore, the 34 years of 
 vitality, assigned by M. de Percieux to a child just born, ought to be considered as ap- 
 plicable to a child from 6 to 9 months old, and more ; but it is during the first months 
 of the first year that the life ofachildis most uncertain, and that the greatest number die. 
 
 The second observation, which is common to both tables, is, that vitality, exceed- 
 
 ingly weak at the moment of birth, goes on increasing after that period, till it comes 
 to another, at which it is the greatest ; for the chance is less than 3 to 1 that a new 
 born child will attain to the end of its first year,* and one may take an even bet that 
 
 * According to the principles explained in treating of probabilities, the probability of a child newly 
 born being alive at the end of a year, is to that ofits dying before that period, as the number of the 
 children alive at the end of a year, is to the number of those dead; that is to say, as 1754) to 6460; 
 which is somewhat less than the ratio of 3to 1. In the other cases the calculation is the same. ‘lake 
 the number of those who have died in the course of the year, and divide by it the number cf those 
 alive ; this will express what may be betted to 1, that the person who has completed. that year will 
 complete another. 
 
108 ARITHMBTIC. 
 
 it has only 8 years to live; but when it has attained to the commencement of the 
 second year, one may bet 6 to 1 that it will attain to the third; and it is an even 
 chance that it will live 33 years. Ina word, it is-seen by the table of M. Dupré de 
 St. Maur, that it is towards the age of 10 years, or betwen 10 and 15, that life is most 
 secure. At that period, one may take an even bet that the child will live 43 years ; 
 and it is 125 to 1 that it will live a year, or 25 to 1 that it will live five years. 
 Beyond that period the probability of living a year longer decreases. At the age of 
 20, for example, it is somewhat less than+16 to 1, that the person will not die within 
 the five following years. Whena person has reached his sixtieth year, it is no more 
 than 3} to 1 that he will attain to the beginning of his sixty-fifth year. 
 
 SECTION IV. 
 Of the number of Men of different ages in a given number. 
 
 It may be deduced from the preceding observations, that when the inhabitants of a 
 country amount toa million, the number of those of the different ages will be as 
 follows : 
 
 Between 0 and l yearcomplete 38740 55 60 os p 37110 © 
 d2and 5 es he 119460 60 65 wd he 28690 
 5 10 vai -» — 99230 65 70 Ae ist 21305 
 
 10 15 me 5 94530 70 75 re ys 13195. 
 15 PN PR 88674 15 80 re pallees 7065 
 20 OD ME sei Mew 82380 80 BO Buse h os 2880 
 25 30 ee oe 77650 85 90 ee Se 1025 — 
 30 35 oe oe 71665 90 95 ap a 335 
 385 AO) ita « alins « 64205 95 100 «eee 82 
 40 45 a pa) 57230 Above 100 years ... “ 3 or 4 
 4 Somers Pea 
 
 Thus ina country peopled with a million of inhabitants, there are about 573460 
 between the age of 15 and 60; and as nearly one half of them are men, this number 
 of inhabitants could, on any emergency, furnish 250,000 men capable of bearing 
 arms, even if an allowance be made for the sick, the lame, &c., who may be supposed 
 to be among that number. 
 
 SECTION V. 
 Of the Proportion of the Births and Deaths to the whole Number of the Inhabitants 
 
 of a Country.—The consequences thence resulting. 
 
 As it would be difficult to number the inhabitants of a country, and much more to 
 repeat the enumeration as often as it might be necessary to ascertain the population, 
 means have been devised for accomplishing the same object, by determining the pro- 
 portion which the births and deaths bear to the whole number of the inhabitants ; 
 for as registers of births and deaths are regularly kept in all the civilized countries of 
 Europe, we may judge, by comparing them, whether the population has increased of 
 decreased ; and in the latter case can examine the causes which have produced the 
 diminution. 
 
 It is deduced, for example, from Dr. Halley’s tables of the state of the populations 
 of Breslaw, about the year 1690, that among 34000 inhabitants, there took place, every 
 year on anaverage, 1238 births; which gives the proportion of the former to the lat- 
 ter as 273 tol. In regard to cities, such as Breslaw, where there is no great influx 
 of strangers, we may therefore adopt it as a rule, to multiply the births by 27} in 
 order to find the number of the inhabitants. 
 
 i 
 
POLITICAL ARITHMETIC. 109 
 
 There appeared in 1766, a very interesting work on this subject, entitled, ‘* Re- 
 herches sur la Population des la Généralités d’Auvergne, de Lyon, de Rouen, et 
 e quelques Provinces et Villes du Royaume,” &c., by M. Messance. By an enumera- 
 ion of the inhabitants of seventeen small towns or villages in the generality of 
 Luvergne, compared with the average number of births in the same places, the author 
 hews that the number of births is to that of the inhabitants, as 1 to 244, 4, 4: a 
 imilar enumeration, in twenty-eight small towns or villages of the generality of Lyons, 
 ave the ratio of 1 to 283; and by another made in five small towns or villages of 
 he generality of Rouen, it appeared that the ratio was as 1 to 274 and 3. But as 
 hese three generalities comprehend a very mountainous district, such as Auvergne, 
 mother which is moderately so, as the generality of Lyons, and a third which con- 
 ists almost entirely of plains or cultivated hills, as the generality of Rouen, there is 
 ‘eason to conclude that these three united afford a good representation of the aver- 
 we state of the kingdom; combining therefore the above proportions, which gives 
 shat of 1 to 254, this will give the proportion of births to the number of the inha- 
 yitants, for the whole kingdom, without including the great cities: so that for two 
 girths in the year we shall have 51 inhabitants. 
 
 But as, in towns of any magnitude, there are several classes of citizens who spend 
 their lives in celibacy, and who contribute either nothing or very little to the popu- 
 lation, it is evident that this proportion, between the births and effective inhabitants, 
 must be greater. M. Messance says, he ascertained, by several comparisons, that the 
 ratio nearest the truth, in this case, is 1 to 28, and that this is the proportion 
 which ought to be employed in deducing, from the number of births, the number of 
 the inhabitants of a town of the second order, such as Lyons, Rouen, &c. ; which 
 agrees pretty well with what Dr. Halley: found in regard to the city of Breslaw. 
 
 In the last place, for cities of the first class, or the capitals of states, such as Paris, 
 London, Amsterdam, &¢., where a great many strangers, attracted either by pleasure 
 or business, are mixed with the inhabitants, and where great luxury prevails, which 
 increases the number of those who live in voluntary celibacy, it is very probable that 
 the above ratio must be raised, and that it ought to be carried to 30 or 31. 
 : M. Kerseboom, in his book entitled, ‘‘ Essai de Calcul politique, concernant la 
 quantité des habitans des Provinces de Hollande et de Westfriesland,” &c., printed 
 ‘at the Hague in 1748, has endeavoured to shew that to obtain the number of the inha- 
 pitants in Holland, the number of the births ought to be multiplied by 35. 
 Tfthis be the case, there is reason to conclude that marriages are less fruitful or less 
 ‘numerous in Holland than in France; and this difference may be founded on physical 
 causes. 
 
 If these calculations be applied to determining the population of great cities, it will 
 be seen that the opinions entertained in general on this subject, are erroneous ; for 
 it is commonly said that Paris contains a million of inhabitants; but the number ot 
 births there, taking one year with another, never exceeds 19500, which, multiplied by 
 30, gives 585000 inhabitants ; if we employ as multiplier the number 31, we shall 
 have 604500, and this is certainly the utmost extent of the population of Paris. 
 
 \ 
 { 
 
 SECTION VI. 
 
 Of some other Proportions between the Inhabitants of a Country. 
 
 We shall present to the reader a few more short observations in regard to popu- 
 ation. The book, which we quoted in the preceding paragraph, shall still serve us 
 as a guide. 
 
 By combining together the three generalities above mentioned, it is found, 
 
 Ist. That the number of the inhabitants of a country, is to that of the families, 
 
 as 1000 to 2224; so that 2000 inhabitants give in common 445 families, and con- 
 
 ‘s 
 
 x ) | 
 
110 ARITHMETIC. 
 
 seqnently 44 heads on an average for each, or 9 persons for two families. In th 
 respect, those of Auvergne are the most numerous ; those of the Lyonnois, are next 
 and those of the generality of Rouen are the least numerous. By taking a mean, | 
 is found also, that in 25 families, there is one where there are six or more children, 
 
 2d. The number of male children born exceeds, as has been said, that of females 
 and this excess continues till a certain age; for example, the number of boys ¢ 
 the age of fourteen, or below, is greater than that of the females of the same agi 
 and in the ratio of 80 to 29. The whole number of the females, however, exceeé 
 that of the males, in the ratio of about 18 to 19. We here see the effect of th 
 great consumption of men, occasioned by war, navigation, laborious employments 
 and debauchery. 
 
 3d. It is found that there are three marriages annually among 337 inhabitants; s 
 that 112 inhabitants produce one marriage. 
 
 4th. The proportion of married men or widowers, to married women or widows 
 is nearly as 125 to 140; and the whole number of this class of society, is to th 
 whole of the inhabitants, as 265 to 631, or as 53 to 126. 
 
 oth. According to King and Kerseboom, the number of widowers is to that o 
 widows, as 1 to 3 nearly ; so that there are three widows for one widower. Thi 
 at least is deduced from the enumerations made in Holland and in England. Bu 
 is the case the same in France? It is to be regretted, that the above-mentione 
 author did not make researches on this subject. In our opinion, however, this pro 
 portion is pretty near the truth, and it will excite no astonishment when it i 
 considered that the greater part of the men marry late, in comparison of th 
 women. 
 
 6th. If the above proportion between widowers and widows be admitted, it thence 
 follows, that among 631 inhabitants there are 118 married couples, 7 or 8 widowers 
 and 21 or 22 widows; the remainder are composed of children, people in a state 
 of celibacy, servants, or passengers. . 
 
 7th. It thence results also, that 1870 married couples give annually 357 chil. 
 dren; for a city of 10000 inhabitants would contain that number of married 
 couples, and give 357 annual births. Five married couples therefore, of all ages, 
 produce annually one birth. 
 
 8th. The number of servants is* to the whole number of the inhabitants, as 136 
 to 1535 nearly ; which is somewhat more than the eleventh part, and_less than the 
 tenth. 
 
 The number of male servants is nearly equal to that of the female, being in the 
 ratio of 67 to 69; but it is very probable that. in large cities, where agreat deal 
 of luxury prevails, the proportion is different. 
 
 9th. The number of ecclesiastics of both sexes, that is to say, secular as well as 
 tegular, comprehending the nuns, is to the inhabitants of the above three gene- 
 ralities, as 1 to 112 nearly: this is contrary to the common opinion, which sup- 
 poses the proportion to be much greater. 
 
 10th. By dividing the territory of these three generalities among their inhabi- 
 tants, it is found, that the square league would contain 864; but the square league 
 contains 6400 acres; each man therefore, on an average, would have 7+, acres, and 
 each family being composed, one with another, of 4} heads, 331 acres would fall to 
 the share of each family. But it is to be observed that the generality of Rouen, 
 considered alone, is much more populous, since it contains 1264 inhabitants for each 
 square league, which gives to each head no more than 5 acres. 
 
 lth. It appears by the same enumerations, that a very sensible increase in the 
 population has taken place since the beginning of the last century. It is indeed 
 found, that the annual number of the births has been augmented ; and by comparing 
 the present period with the commencement of the last century, there is reason to 
 
eee TE Pes 
 
 POLITICAL ARITHMETIC. Ill 
 
 Biclude, that the number of the Shatiiaetet is now greater than what it was at the 
 yeginning of the century, in the ratio of 1456 to 1350; which makes less than a 
 wwelfth, and more than a thirteenth, of increase. This is doubtless owing to 
 she great extent to which agriculture and commerce have been carried, and to the 
 cessation of those wars which so long exhausted the interior of France. The wound 
 ziven to the nation by the revocation of the edict of Nantes seems healed, and even 
 nore; but had it not been for that event, France, in all probability, would contain a 
 sixth more of inhabitants than it did at the commencement of the 18th century; for 
 the number who expatriated in consequence of that revocation, amounted perhaps to 
 a preelith part of the whole people. 
 
 SECTION VII. 
 Some Questions which depend on the preceding Observations. 
 
 The following are some of those questions, in the solution of which the preceding 
 observations may be employed: we shall not explain the principles on which each 
 isresolved; but shall merely confine ourselves to referring to them sometimes, that 
 we may leave to the reader the pleasure of exercising his own ingenuity. 
 
 Ist. The age of a man being given, that of 30 for example, what probability is 
 there that he will be living at the end of a determinate number of years, such 
 as 15. 
 
 Seek in the table of the second section for the given age of the person, viz. 30 
 years, and write down the number opposite to it, which is 11405; then take from 
 the same table the number opposite to 45, which is 7008, and form these two 
 numbers into a fraction, having the latter for its numerator, and the former as its 
 denominator; this fraction will express the probability of a person of 30 years of 
 age living 15 years, or attaining to the age of 45. 
 
 _ The demonstration of this rule is obvious to every one who understands the 
 theory of probabilities. 
 
 _ 2d. A young man 20 years of age borrows £1000, to be paid, capital and interest, 
 when he attains to the age of 25; but in case he dies before that period, the debt to 
 become extinct. What sum ought he to engage to pay, on attaining to the age 
 bof 25? 
 
 It is evident that if it were certain he would not die before the age of 25, the sum 
 to be then paid would be the capital increased by five years’ interest, which we here 
 ‘suppose to be simple interest: the sum therefore which in that case he ought to en- 
 gage to pay, on attaining to the age of 25, would be £1250. But this sum must be 
 increased, in proportion to the danger of the debtor dying in the course of these five 
 years, or in the inverse ratio of the probability of his being alive when they are ex- 
 pired. As this probability is expressed by the fraction Toso we must multiply the 
 above sum by this fraction inverted, or by 49998, which will give £1329. 3s. lld., that 
 is to say, £79. 3s. lld. for the risk of losing the debt, which certainly cannot be 
 considered as usury. 
 
 _ 3d. A state or an individual having occasion to raise money on annuities, what 
 interest ought to be given for the different ages, legal interest being at the rate opts 5 
 “per cent. ? 
 
 The vulgar, who are accustomed to burthensome loans, entertain no doubt that 10 
 per cent. is a great deal for any age below 50, and that such a method of borrowing 
 cannot be advantageous to the state. But thisis a great mistake; for it will be 
 found by calculation, employing the before-mentioned data, according to the table of 
 _M. de Parcieux, that 10 per cent. cannot be allowed before the age of 56. According 
 | tothe same table, no more than 6! can be given at the age of 20; 6h at 25; 64 at 30; 
 Tat 40; Stat 50; 10 at 56; 114, at 60; 162 at 70; 272 at 80; 394, at 85. 
 
 M 
 
112 ARITHMETIC, : bi] 
 
 Ai 
 
 It is also a very great error to believe, that on account of the number of person 
 who lay out money on these loans made by governments, they are soon freed fron 
 a part of the annuities by the death of a part of the annuitants. The slowness of th 
 increase of annuities in tontines, is a sufficient proof of the falsity of this idea; be 
 sides, the great number of the persons is the very cause why the extinction of th 
 annuities takes place according to the laws of probability above explained. A fortu 
 nate circumstance, at the end of some years, may free an individual from an annuity 
 established on the life of a man aged 30; but if this annuity were divided among 30( 
 persons of nearly the same age, it is certain that he would not be freed from this 
 burthen before the expiration of about 65 years ; and at the end of 32 or 33 years one 
 half of the annuitants would still be living. This M. de Parcieux has shewn, in the 
 clearest manner, by examining the lists of different tontines. 
 
 4th. Legal interest being at 5 per cent. ; at what rate of interest may an annuity be 
 granted on the lives of two persons, whose ages are given, and payable till the death o} 
 the last survivor ? 
 
 5th. What interest may be allowed on a capital, sunk for an annuity on the lives of 
 two persons, whose ages are given, and payable only while both the annuitants are 
 living ? | 
 
 6th. A certain person, whose age is given, has an annuity, secured on the public 
 Junds, of £1000; but being in want of money, he is desirous to sell it. How much is 
 zt worth ? ) 
 
 7th. A, aged 20, and B, aged 50, agree to purchase, on their Joint lives, an annuity 
 of £1000, to be equally divided between them, during their lives, with a reversion to 
 the survivor. How much ought each of them to contribute towards the purchase 
 money ? 
 
 Sth. How much ought each to contribute, supposing it stipulated between them, that 
 B, the eldest, should enjoy the whole till his death 2 ee 
 
 9th. Legal interest being at 5 per cent., what is the worth of an annuity of £100, on 
 the lives of three persons, whose ages are given, and payable till the death of the last 
 survivor ? 
 
 10th. An annuity ts purchased for the life of a child, of 3 years of age, on this con- 
 dition, that the annuity at the end of each year is to be added to the purchase money, 
 till the annuity equals the capital sunk. At what age will the annuity be due, legal 
 interest being at 5 per cent.? 
 
 Many people imagine that a capital can be deposited in the bank of Venice on this 
 condition, that nothing is received for the first 10 years, but after that period the 
 
 _ annuitant receives an annuity equal to the capital. This however is entirely ground- 
 less, as has been shewn by M. de Parcieux in his “ Addition a l’Essai sur les Probabie 
 lités de la durée de la Vie Humaine,” published in 1760; for it is there shewn, by a 
 calculation, the demonstration of which is evident, that if £100, for example, were 
 sunk on the life of a child 3 years of age, it could not begin to enjoy an annuity of 
 £100 till it had attained to the age of 45 or 46. 
 
 The table of M. de Parcieux presents, on this subject, two things very curious. 
 For example, on the above supposition, if the increase of the annuity were not stopped 
 till the end of 54 years, the person ought to receive £205 per annum during the re- 
 mainder of his life ; if it were not stopped till 58 years, he ought to receive till the 
 time of his death £300; and by stopping it only at 75 years, he would be entitled to 
 £2900 per annum: inthe last place, if the arrears due each year were left, on the 
 like conditions, to accumulate till the 94th year, the annuity for the remainder of 
 the person’s life ought to be £134069. 19s. 2d.,a sum which must appear prodi- 
 gious. 
 
 But it may seem astonishing that M. de Parcieux should begin his calculations 
 only at the age of 3 years. It is very true that people do not venture capitals in the 
 
 : 
 
POLITICAL ARITHMETIC. : 113 
 
 purchase of annuities on the lives of new-born children ; but if ever such an establish- 
 ment existed at Venice, it is evident that it must have been only on the supposition 
 of the money being risked on the life of a child just born, because great mortality 
 takes place during the first year. For this reason we have examined what would be 
 the result of such a supposition; and we have found that, if the sum of £100 were 
 sunk, on the above conditions, on the life of a child just born, it ought, according to 
 the table of M. Dupré de St. Maur, to procure it an annuity of £10. 15s.; that this 
 ‘sum sunk in like manner, at 8 per cent., at the end of the first year, by adding the 
 annuity, would give at the end of the second year £11. lls. 7d. These £11. 11s. 7d. 
 sunk at 6,8, per cent., which is the interest that might be allowed at the commence- 
 ment of the third year, would, at the end of the third, or the commencement of the first 
 fourth, amount to £12. 5s. 1d., and by a calculation similar to that of M. de Parcieux, 
 ‘it will be found, that the annuity would be increased to £100 at about theage of 36; 
 which is still very far distant from what is commonly believed. 
 
 If legal interest be supposed to be 10 per cent., as it was in the 16th century, it 
 
 will be found, that it would be only about the 26th year that a person could receive 
 -an annuity equal to the capital sunk at the time of his birth. 
 Those who are desirous of farther information on this subject, may consult 
 ‘ Demoivre’s Essay upon Annuities on Lives, and M. de Parcieux’s ‘‘ Essai sur les Pro- 
 ' babilités de la durée de la Vie Humaine,” and.Dr. Price on Reversionary Payments. 
 The other authors who have treated mathematically on these matters, are Dr. Hal- 
 ley, Sir William Petty, Major Graunt, King, Davenant, Simpson, Maseres; and 
 - among the Dutch, the celebrated John de Wit, grand pensionary of Holland, M. 
 : Kerseboom, Struyk, &c. 
 
114 GEOMETRY. 
 
 PART SECOND. 
 
 CONTAINING A SERIES OF GEOMETRICAL PROBLEMS AND QUESTIONS, 
 CALCULATED FOR EXERCISE AND AMUSEMENT. 
 
 PROBLEM I. 
 
 From the extremity of a given right line to raise a perpendicular, without continuing the 
 line, and even without changing the opening of the compass if necessary. 
 
 I. Let a B (Fig. 1.) be the given straight line, and a the ex- 
 tremity, from which it is required to raise a perpendicular, without 
 prolonging it. 
 
 From a towards B assume 5 equal parts at pleasure; and extending 
 the compasses from a, so as to include 3 of these parts, describe the 
 are of a circle; then from 8, the extremity of the fourth part, with 
 an opening equal to the 5 parts, describe another; these two arcs will 
 necessarily cut each other in a certain point c, from which if a straight line, asc a, be 
 drawn, it will be perpendicular to a B. 
 
 For the square of c A, which is 9, added to the square of a b, which is 16, are 
 together equal to 25, the square c b: the triangle c a 6 is therefore rightangled at a. 
 
 We might assume also, for the radius of the arc to be described from the point a, 
 a line equal to 5 parts; for the base 12, and for the other radius 13; because 5, 12, 
 and 13, forma rightangled triangle. Indeed, all the rightangled triangles in numbers, 
 of which there are a great variety, may beggmployed in the solution of this problem. 
 Fig. 2. II. On any part whatever of the given line a B (Fig. 2.), describe 
 
 an isosceles triangle a cB, that is, so that the sides ac, cB shall be 
 ae equal; and continue acto Db, so that cp shall be equal to cs; if a 
 s line be then drawn from D to B, it will be perpendicular to ap. The 
 
 demonstration of this is so easy that it requires no illustration. 
 
 Fig. 1. 
 
 Bb A 
 
 PROBLEM Ii. 
 
 To divide a given straight line into any number of equal parts, at pleasure, without 
 repeated trials. 
 
 Fig. 3. Let it be proposed, for example; to divide the line a B (Fig. 
 
 ¢ 3.) into 5 equal parts. Make this given line the base of an 
 
 ! equilateral triangle 4 Bc; and from the point c, in the side cB, 
 continued if necessary, set off 5 equal parts, which we shall 
 suppose to terminate at D, and make cr equal to cp; then 
 make pF, for example, equal to one of the five parts of cD; 
 and draw cF, which will interseet aB ina: it is evident that 
 BG will be the fifth part of a B. 
 
 If pf were equal to 3 of cp, by drawing c f we should have 
 g, as the point of intersection of c f and AB, which would give Bg equal to 2 of 
 AaB. And so on. 
 
 PROBLEM IIT. 
 
 Without any other wnstrument than a few pegs and a rod, to perform on the ground the 
 greater part of the operations of geometry. 
 
 It is well known that most geometrical operations may. be performed in the fields, 
 
 ag 
 
GEOMETRICAL PROBLEMS, 115 
 
 by means of the graphometer: and it would even seem that this instrument is abso- 
 lutely necessary in practical geometry. 
 
 | A geometrician however may happen to be unprovided with such an instrument, 
 and even destitute of the means of procuring one. We shall suppose him in the 
 woods of America, with nothing but a knife to cut a few pegs, anda long stick to 
 serve him as a measure: he has several geometrical operations to perform, and even 
 inaccessible heights to measure; how must he proceed to accomplish what is 
 here proposed ? 
 
 We shall suppose also, that the reader is acquainted with the method of tracing 
 out a straight line on the ground, between two given points; and in what manner it 
 may be indefinitely continued on either side, ce. This being premised, we shall now 
 proceed to give a few of those elementary problems of geometry, required to be 
 
 performed, without employing any other line than a straight one, and even excluding 
 the use of a cord, with which the are of a circle may be described. 
 Ast. Through a given point to draw a straight line, parallel to a given straight line. 
 Fig. 4: Let AaB (Fig. 4.) be the straight line, and c the point 
 a ‘ through which it is required to draw a straight line pa- 
 a ee rallel.to 4B. From the point ¢ draw the line cB, to any 
 ik point in a B, and divide cB into two equal parts in p; in 
 this point D fix a peg, and from any point a in the given 
 straight line, draw, through p, an indefinite line ap £, and make pe equal to ap: 
 if a straight line be then drawn through the points c and £, it will be parallel to aB. 
 2d. From a given point in a given straight line, to raise a perpendicular. 
 Fig. 5. Divide the given line a B (Fig. 5.) into two equal parts, ac 
 and cB; and from the point c draw, any how at pleasure, the 
 , linecd; makecp equal toca; drawpaA, and make aE equal 
 to ac, and a¥F equal to ap; through the points £ and F draw 
 the line FE G; and if EG be made equal to EF, we shall have 
 r the point c, which with the point a will determine the position 
 
 of the perpendicular a a. 
 
 \e For the sides a p and ac of the triangle cab, being re- 
 
 e spectively equal to the sides aF and aE of the triangle uaF, 
 these two triangles are equal; and, in the triangle p.c a, the sides cp and c a being 
 equal, the sides £ a and E F of the other will be equal also: the angle EF A therefore 
 will be equal to EB AF, and consequently tocap. But in the triangle ra a, the side 
 FG is equal to aB, for FG by construction is the double of FE, and FE or aE is 
 equal to Ac, which is the half of as: the triangles Fac and A DB then are equal; 
 since the sides F G, F A, are equal to the sides a B, A D, and the included angles equal; 
 the angle F a G will therefore be equal to aps; but the latter isa right angle, because 
 the lines cB, c D, Cc a, being equal, the point p is in the circumference of a semicircle, 
 described on the diameter as, The angle FAG then is a right angle, and ca is 
 perpendicular to A B. 
 
 3d. From a given point a, to draw a straight line perpendicular to a given straight 
 line. 
 
 Assume any point B (Fig. 6.) in the indefinite line Bc; and, 
 having measured the distance 8 A, make Bc equal toB A; draw 
 c A, which must be measured also, and then form this propor- 
 tion: as Bc is to cD, the half of ac, so is Ac toa fourth pro- 
 portional, which will be cE; if cr be then made equal to this 
 fourth proportional, we shall have the point £, from which if 
 the line aE be drawn through a, it will be the perpendicular 
 required, 
 
 Fig. 6. 
 
 12 
 
116 GEOMETRY. 
 
 4th. To measure a distance aB, accessible only at one of tts extremities, as the 
 breadth of a river or ditch, &c. | 
 
 First fix a peg at a (Fig. 7.); then another in any point ¢, | 
 assumed at pleasure, and a third at D, in the straight line be-_ 
 tween the points Band c; continue the lines c A and D A inde- 
 finitely beyond a, and make the lines a x and a F respectively 
 equal to ac and aD; in the last place, fix a peg at G, in such 
 a manner as to be in a straight line with a and B, ard also with 
 rand es: the distance ac will then be equal to a B. 
 
 If it be found impossible to proceed far enough from the line 
 AB towards E or F, we may take in aE or AF, only the half or 
 the third of ac and aD, for example ae, af; if a peg be then fixed in g, so as to 
 fall in the continuation of both the lines Ba and ef, we shall have a g equal to the 
 half or the third of a B respectively. 
 
 Fig. 8. Now let the distance a B (Fig. 8.) be inaccessible through- 
 out. The solution of this case may be easily deduced from 
 that of the former: for having fixed a peg inc, and having 
 continued by a series of pegs the lines Bc and ac, if the 
 parts cE and cF be, by the above means, made respectively 
 equal to Bc and cA, or the half or the third of these lines, 
 it may be readily seen that the line which joins the points & 
 and F, will be equal to the line required, or to the half or third of it; and that in 
 either case it will be parallel to it; which resolves the problem, to draw a line paral- 
 lel to an inaccessible line. 
 
 These examples are sufficient to shew in what manner a person, who has only 
 a slight knowledge of geometry, may execute the greater part of geometrical opera- 
 tions, without any other instruments than those which might be procured in a wood 
 by means of a knife. It must indeed be allowed that one can never be in such 
 circumstances, unless on some very extraordinary occasion; but, however, it may 
 afford satisfaction to those who have a turn for geometry, to know in what 
 mauner they might proceed, if ever such a case should happen. 
 
 It is remarkable, that it is not perhaps possible to resolve in this manner, that is 
 to say, without employing the arc of a circle, the very simple problem, and one of the 
 first in the elements of geometry, viz., to describe an equilateral triangle. We have 
 often attempted it, but without success, while trying how far we covld proceed in 
 geometry by the means of straight lines only. 
 
 PROBLEM IV. 
 
 To describe a circle, or any determinate arc of a circle, without knowing the centre, and 
 without compasses. 
 
 To those who are little acquainted with geometry, this will appear to be a sort 
 of paradox; but it may be easily explained by that proposition, in which it is de- 
 monstrated, that the angles whose summits touch the circumference, and whose sides 
 pass through the extremity of the chord, are equal. 
 
 Fig. 9 Let A, c, B, (Fig. 9.) be three points in the required 
 
 ae circle or arc; having drawn the lines a cand cB, make 
 
 an angle equal to 4 cB of any solid substauee, and fix 
 
 two pegs in a and B; if the sides of the determinate 
 
 angle be then made to slide between these pegs, the 
 
 vertex or summit will describe the circumference of 
 
 the circle. So that if the summit or vertex be fur- 
 
 nished with a spike or pencil, it will trace out, as it 
 revolves between A and B, the required are. 
 
 su 
 
GEOMETRICAL PROBLEMS. 117 
 
 | If another angle of the like kind were constructed, forming the supplement of acs 
 to two right angles, and if it were made to revolve with its sides always touching 
 the points a and B, but with its summit in a direction opposite toc, it would describe 
 the other segment of the circle, which with the are a cB would make up the whole 
 circle. 
 
 It may sometimes happen that it is necessary to describe, through two given points, 
 the are of a determinate circle, the centre of which is at a great distance, or inac- 
 cessible on account of some particular causes. Should it be required, for example, 
 to describe on the ground a circle, or the arc of a circle, with a radius equal to 2 or 
 8 or 4 hundred yards; it may be readily seen that it would be impracticable to do it 
 by means of a cord; the mode of operation therefore must be as follows. In a and 
 ‘ B, (Fig. 10.) the extremities of that line which we here suppose 
 Fig. 10. to be the chord of the required arc, the amplitude or subtending 
 . angle of which is known, fix two pegs, and then find out, by 
 means of a graphometer or plane table, any point c, in such a 
 position, that ac and Be shail form an angle, ac B, equal to the 
 given angle, and in that point tix a peg; then find out another 
 point d, so situated that a d and Bd shall form an angle, adz, 
 equal to the former; if the points f and e be found in like 
 manner, it is evident that the points c, d, e, and f will be in the 
 are of a circle capable of containing the given angle. If the points g,h, i,k, be then 
 found, on the other side of a B, so situated, that the angle agB, cr ahB, &c. shall 
 ‘be the supplement of the former, the points c¢, d, e.f,g, 4,7, k, will evidently be all in 
 a circle, 
 
 PROBLEM Vv. 
 
 Three points, not in the same straight line, being given, to describe a circle which 
 shall pass throuah them. 
 
 Fig. 11. Let the three points be those marked 1, 2, 3, (Fiz. 11.): from 
 one of them as a centre, that for example marked 2, and with any 
 radius at pleasure, describe a circle ; and from one of the other two 
 points, 1 for example, assumed as a centre, make with the same 
 radius two intersections in the circumference of the first circle, 
 as at Aand B; draw the line a B, and assuming the third point 3 as 
 a centre, make with the same radius two more intersections in 
 the circumference of the first circle, as D and ©: if DE be then 
 
 drawn, it will cut the former line a‘B in the point c, which will be the centre of 
 the circle required. If a circle therefore be described from this point as a centre, 
 through one of the given points, its circumference will pass through the other two. 
 It may be readily seen that this construction is the same, in principle, as the com- 
 mon one, taught by Euclid and all other elementary writers; for it is evident that the 
 the lines 1 a, 2 A, 1B, 2B, are equal to each other; consequently the line a B is per- 
 _pendicular to that which would join the points 1 and 2, or to the chord 1 2 of the 
 required circle; hence it follows, that the centre of the circle is in the line a B: for 
 the same:rcason this centre is in the line px, and therefore it is in the point waere 
 they intersect each other. 
 
 If the three given points were in a straight line, the lines a8 and p E would be- 
 
 come parallel, and consequently there would be no intersection. 
 
 SS 
 
 PROBLEM VI. 
 
 An engineer, employed in a survey, observed from a certain point the three angles 
 formed by three objects, the positions of which he had before determined : it is re- 
 quired to determine the position of that point, without any farther operation. 
 
 This problem, reduced to an enunciation purely geometrical, might be proposed in 
 
118° i GEOMETRY. 
 
 the following manner: a triangle, the sides and angles of which are known, being 
 given, to determine a point from which, if three lines be drawn to the three angles, 
 they shall form with each other given angles. 
 
 In this problem there are a great number of cases; for either the three angles, 
 under which the distances of the three given points are perceived, occupy the 
 whole extent of the horizon, that is to say are equal to four right angles, or occupy 
 only the half, or less than the half. In the first case, it is evident that the required 
 point is situated within the given triangle; in the second, it is situated in one of the 
 sides; and in the third, it is without. But for the sake of brevity we shall here 
 confine ourselves to the first case. 
 
 Let it be required then to determine, between the points 
 Fig. 12. A, B, c, (Fig. 12 ) the distances of which are given, a point D so 
 situated, that the angle ap B shall be equal to 160 degrees, 
 cps to 180°, and cpa to 70°. On the side aB describe an 
 are of a circle capable of containing an angle of 160°; and on 
 the side Bc another capable of containing an angle of 130°; 
 the point where they intersect each other will be the point 
 required. ; 
 
 For it is evident that this point is in the circumference of the are described on the 
 side a B, and capable of containing an angle of 160°; because from all the points of 
 that are, and of no other, the distance a B is seen under the angle of 160°. In like 
 manner the point D must be found in the are described on the side pc, and capable 
 of containing an angle of 130°; consequently it must be in the place where they 
 intersect each other, and no where else. . 
 
 Remark.—On this construction, a trigonometrical solution may be founded, to de- 
 termine in numbers the distance between p and the points a, B, and c; but we shal 
 leave this to the ingenuity of the reader. : 
 
 PROBLEM VII. 
 
 If two lines meet in an inaccessible point, or a point which cannot be observed, it is 
 proposed to draw, from a given point, a line tending to the inaccessible point. 
 
 Fig. 13. Let the unknown and inaccessible point be o (Fig. 13.), 
 the lines tending to it Ao and Bo; and let & be the point 
 from which it is required to draw a straight line tending 
 A towards o. 
 
 Through the point = draw any straight line rc, inter- 
 secting Ao and Bo in the points p and c; and through any 
 point F, assumed at pleasure, draw Fa parallel to it; then 
 make this proportion: ascp is to DE, so is ra to cH; if 
 the indefinite line HE be then drawn through the points © and u, it will be the line 
 required, 
 
 - Or if the given point be e, make this proportion, as cp is to ce, so is FG to F h; 
 the line eh will be that required. 
 
 The demonstration of this problenr will be easy to those who know, that, in any 
 triangle, if lines be drawn parallel to the base, all those drawn from the vertex of the 
 triangle, will divide them proportionally. 
 
 Cg 
 
 PROBLEM VIII. 
 The same supposition being made; to cut off two equal portions from the lines Bo 
 and Ao (Fig. 14.) 2 
 From the point a, draw ac perpendicular to po, and ap perpendicular to ao; 
 if the angle ca be then divided into two equal parts by the line ax, meeting BO 
 in , this line will eut off from Boand ao the two equal parts, Ao and Eo. 
 
GEOMETRICAL PROBLEMS. 119 
 
 This may be easily cemonstrated, by shewing that, in 
 consequence of this construction, the angle oA E becomes 
 equal tooEA. But the angle oA is equal to the angle 
 oac plus CAE; and the angle oF A is equal to oDA or 
 oac plus EAD, or Eac, which is equal to it; the angle 
 
 o AE then is equal too A, and the triangle 0 4 & is isosceles, 
 therefore, &c. 
 
 PROBLEM IX. 
 The same supposition still made ; to divide the angle A 0 & into two equal parts, (See 
 
 last figure.) 
 
 Construct the same figure as in the preceding problem; then between the two. 
 given lines draw any line FG, parallel tothe line Ae; and divide the lines ar and FG 
 into two equal parts in H and 1: the line H1 will divide the angle a o Einto two equal 
 parts. ‘The demonstration of this is so easy that it requires no illustration. 
 
 These problems, as may be readily seen, contain operations of practical geometry 
 of great utility in certain cases; such, for example, as when it is necessary to cut 
 
 roads through a forest, or when it is required to make them tend to, or end at, a 
 common centre. } 
 
 PROBLEM X. 
 Two sides of a triangle, and the included angle, being given ; to find its area. 
 
 Multiply one of the sides by half the other, and the product by the sine of the 
 included angle: this new product will be the area. 
 
 It may be easily demonstrated, that the area of every triangle is equal to half the 
 rectangle of any two of its sides, multiplied by the sine of the included angle. 
 
 Let anc (Fig. 15.) be a triangle, having an acute 
 
 Fig. 15. angle at a; produce ac towards d, and from a as a 
 
 ie ae Cea centre, with the distance a B, describe the semicircle 
 
 -BF Ob; then from the point a, draw F a perpendicular to 
 
 Ac; and from the point B, draw BD also perpendicular 
 to ac. 
 
 It is here evident that the two triangles Fac and 
 BAC are respectively to each other as AF is to BD; that is to say, as radius is 
 to the sine of the angle B a c, or as unity is to the number which expresses that sine ; 
 the triangle F a c then being equal to half the rectangle of F a by ac, the other will 
 be equal to that half rectangle multiplied by the sine of the angle B ac. 
 
 This property enables us to avoid that tedious process, necessary to be employed 
 in order to find out the measure of the perpendicular let fall from the extremity of 
 oue of the known sides on the other, that the latter side may be then multiplied by 
 the half of this perpendicular. 
 
 Thus for example, let the two sides az and ac be respectively equal to 24 and 63 
 yards; and let the included angle be 45°. The product of 63 by 12 is 756, and the 
 sine of 45° is 0°70710; if 756 therefore be multiplied by 0°70710, according to the 
 method of decimal fractions, the product. will be 534%. 
 
 eee Ae T) 
 
 PROBLEM XI. 
 
 To find the superficial content of any trapezium or quadrilateral figure, without know- 
 ing its sides. 
 
 Fig. 16. The solution of this problem is a consequence of the pre- 
 <<" ceding. Let the given trapezium be aBepn (Tig. 16.) ; measure 
 ae the diagonals ac and BD, as well as the angle which they make 
 Saas at the point where they intersect each other in ©; if these diag- 
 onals be then multiplied together, and half their product by the 
 sine of the above angle, the last product will be the area. This method is far shorter 
 
120. . GEOMETRY. ; 
 
 than if we should reduce the trapezium to triangles, in order to find the area of each 
 of them. 
 
 Corollaries.—A very curious theorem, which no author has before remarked, may 
 be deduced from this problem, It is as follows: If two quadrilateral figures have 
 their diagonals equal, and intersecting each other at the same angle, whatever may 
 be their difference in other respects, these quadrilateral figures will be equal as to 
 their area. 
 
 Fig. 17. No.1. Ist. Thus, the quadrilateral ancp (see last figure) is equal 
 
 to the parallelogram abcd (Fig. 17. No. 1.), which has its diags 
 e onals equal to those of aBcD, and inclined toward each other 
 
 Ud gp at the same angle. 
 d 
 Fig. 17. No. 2. 2d. The same quadrilateral a Bc, is equal to the triangle 
 B BA c (Fig. 17. No, 2.), formed by the two lines ac and a B, equal 
 to the diagonals ac, pz, and inclined at the same angle. 
 A C | 
 Fig. 17. No.3. 3d. The same quadrilateral will be equal also to the triangle 
 
 ABC (Fig. 17. No. 3.), if the lines ac and pz of that triangle, 
 
 are equal to the diagonals of the quadrilateral, and equally 
 inclined. 
 
 4th. In the last place; this same quadrilateral ancp (Fig. 
 16.), will be equal to the quadrilateral abcd (Fig. 17. No. 4.) 
 the diagonals of which do not intersect each other, if ac and 
 db are equal to ac and pB, and if the angle bec is equal to the 
 angle BEC. 
 
 PROBLEM XII. 
 
 Two circles, not entirely comprehended one within the other, being given; to find a 
 point from which, if a tangent be drawn to the one, it shall be a tangent also to the 
 other. 
 
 : Through the centres a and g (Fig. 18. No. 1.), of 
 Hig 38, avo. the two eee draw the ne sear line a BI: 
 then from the centre a draw any radius ac, and 
 
 : through the centre B draw the radius Bp paraliel to 
 
 it. If the points c and p be joined by the line cp, 
 
 ; tS it will meet ax in x, which will be the point re- 
 
 g 4, quired ; that is to say, if 1m be drawn from the point. 
 I, a tangent to one of the circles, it will bea tangent 
 also to the other, 
 
 Fig. 18. No. 2. When the circles do not cut each other, the point 
 
 s = 1 (Fig. 18. No. 2.) may happen to fall between them. 
 
 ma VO To find it, in that case, nothing is necessary but to 
 
 nn No ai draw the radius BD parallel to ac, and in a direction 
 i 
 
 es x opposite to that of Fig.18. No.1. axsandcp will 
 (ea intersect each other in the point 1, which will have 
 ce the same property as the former. 
 
 Remark.—We cannot here help observing, that if any secant whatever, as 1p H or 
 1dh (Fig. 18. No. 1.), be drawn from the point 1, through the two circles, the rect« 
 angle of 1p and 14, or of 1d and th, will be al ways the same, that is, equal to the 
 rectangle of the two tangents1nandrF. In like manner, the rectangle of 1c and 
 IG, or of r¢ and rg, will be equal to the rectangle of the same tangents. This isa 
 very remarkable extension of the well known property of the circle, by which the 
 rectangle of the two segments rp and 1G is equal to the square of the tangent IE. 
 
GEOMETRICAL PROBLEMS. 121 
 
 PROBLEM XIII. 
 
 A gentleman, at his death, left two children, to whom he bequeathed a triangular field, 
 to be divided equally between them; in the field is a well, which serves for watering 
 it; and asit is necessary that the line of division should pass through this well, in 
 what manner must it be drawn, so as to intersect the well, and divide the field, at the 
 same time, into two equal parts. 
 
 Fig. 19. Let the given triangle be a Bc (Fig. 19.), and the given point 
 
 B, beg. From the point « draw the lines ep and &R, parallel to the 
 base and the side B crespectively, and meeting them inDandR; 
 let the base ac be divided into two equal partsin m; and having 
 drawn the line pm from the point D, draw BN parallel to it, and 
 <i eee divide c N into two equal parts,-in 1; on 1R describe the semi- 
 
 K circle IK R, in which apply RK = RC; and, having drawn 1K, 
 
 if 1F be made equal to it, the points F and & will determine the line FEG. ; 
 
 Remark.—It is evident that cr must be at least double of cr; otherwise c R could 
 not be applied in the semicircle described on rn, which would render the problem 
 impossible. 
 
 In numbers. —Let a B= 48 fathoms, sc= 42, ac= 380, cp = 18, and DE orcR 
 = 6; conséquently cm will be=15. Butcp:cm::cB: cn, that is to say 18; 15 
 +: 42: 35; hence it follows that cn= 35, and c1= 174; and as cr is equal to 6, 
 we shall have 1R = 11}. But the triangle 1k R being right-angled, 1K = 
 r/ IR? — RK? = / 1321 — 36 = J/ 963, or #1, fathoms, which gives cr = 27yy 
 fathoms. 
 
 Fig. 20. The demonstration of this constructidn is too prolix to be 
 given in this work; and there are even a variety of cases which 
 it would be tedious to explain. We shall therefore confine 
 ourselves to one of the simplest; that is, where the point 5 is 
 in one of the sides (Fig. 20.) 
 
 The construction in this case is exceedingly easy ; for having 
 
 A © divided ac into two equal parts in M, and drawn EM, and BN 
 parallel to it; if the point N falls within the triangle, by drawing the line EN, the 
 problem will be solved; but if the point wn falls without the triangle, it will be neces- 
 sary to draw the line a£; then No parallel to it, through the point n, and o£ through 
 the point o: the last line, oF, will solve the problem. 
 
 For because Em is parallel to BN, the triangle MBE=MNE; and if the triangle 
 cme be added’ to each, we shall have the triangles cpm and cEN equal to each 
 other. But the triangle cpm is the half of the triangle apc, because AM=MC; 
 consequently cEN is the half of AaBcalso. In like manner, because § A is parallel 
 
 ‘to No, the triangles ANE and AoE are equal; and therefore if the triangle AGE, 
 
 which is common to both, be taken away, the triangle anc will be equal to GoE; 
 hence it follows, that if we add to the space ca Ge the triangle GOE, we shall have 
 the space caoE=the triangle ce N, which we have already shewn to be equal to 
 the half of abe. 
 
 But if the gentleman had left the field to be divided equally 
 
 Fig. 21. among three children, by lines proceeding from the given point 
 3 © (Fig. 21.); if we suppose one line of division £ B already 
 drawn, it would be necessary to proceed as follows: 
 Divide the base ac into three equal parts; and let the points 
 of division be p anda; draw the line ED, and BF parallel to it ; 
 i\ then draw the line EF from the point £, and if the point F does 
 LC ot fall without the triangle, the trapezium BEFAB will 
 
 be one of the thirds required. 
 
 a 
 
122 GEOMETRY. 
 
 But if the point F falls without the triangle, we must proceed as above directed ; 
 that is to say, the line EA must be drawn towards the angle a, and Fo parallel to if 
 from the point Fr, as far as the side a B, which it meets, we shall suppose in 0; the 
 line Eo will give the triangle Bor, equal to the third of the triangle proposed. 
 BEICB, the other third, may be found in like manner; consequently the remainder 
 of the figure will be a third also. The three lines therefore, £ 0, £1, and £ B, which 
 proceed from the point £, will divide the proposed triangie into three equal parts. 
 
 By the same method a triangle might be divided into 4, or 5, or 6, &c. equal parts, 
 by lines all proceeding from a given point; and this point may be assumed even 
 without the triangle. 
 
 PROBLEM XIV. 
 Two points and a straight line, not passing through them, being given; to describe a 
 circle which shall touch the straight line, and pass through the two given points. 
 
 Fig. 22. Let the given line be a B (Fig. 22.) and the given 
 
 Es points c and p. Join these two points, and on the middle 
 
 of the line c D raise the perpendicular & r, meeting the 
 given straight line in F; and on the same line let fall the 
 perpendicular © H; draw F c, and from the point £, with 
 
 nued, int; draw 18, and through the point c draw c k 
 parallel to it: the point x will be the centre, and x c the radius of the circle 
 required. 
 
 For if the perpendicular x u be let fall from the point k, on the line a B, it will be 
 equal to K c, which is equaltox p. But F Eis toF K, as EH istoKL,andasEr 
 to kK c; therefore E His toK LaseEItoxKc; and consequently, as E I is equal to 
 E H, K L will be equal to K c: therefore, &c. 
 
 It may be readily seen, that if the given line passed through 
 Fig. 28. one of the given points, the centre of the required circle would 
 be in the point K (Fig. 23.) where c K, drawn perpendicular to 
 
 it into two equal parts in E. 
 In the first case, the problem might be resolved in a different 
 
 manner, viz., by continuing the line c p (Fig. 22.) till it meets 
 
 the radius E H, describe a circle intersecting F c, conti- _ 
 
 A B, intersects E K, which is perpendicular to c p, and divides © 
 
 A Bin ™; then taking a mean proportional between mc and — 
 
 M D, and making mM L equal to it; if a circle were then described through the points — 
 
 Cc, D, L, it would be the one required. But this solution would be attended with 
 
 difficulty, if the point m were at a great distance, whereas in the former case this is — 
 
 a matter of indifference. 
 
 PROBLEM XV. 
 
 Two lines azand cD (Fig. 24.) with a point © between them, being given; to describe 
 
 a circle which shall pase through this point, and touch the 
 Fig 24. B two lines. 
 
 If the two lines meet, as at F, draw the line F u dividing 
 the angle’3 F D into two equal parts; or, if they are parallel, 
 draw one, such as F H (Fig. 25.) equally distant from both; 
 D then from the point © draw £ G14, perpendicular to F H, and 
 
 Fig. 25. make G 1 equal to GE; the points 1 £ will be so situated that 
 
 if a circle, touching one of the given lines, be described through 
 ng them, it will touch the other given line also; which reduces 
 er f this problem to the preceding one. 
 
 A_D ’ 
 
ee es a ee 
 ae os ee a 
 
GEOMETRICAL PROBLEMS. 123 
 
 Various demonstrations of the forty-seventh proposition of the first book of Euclid, by 
 the mere transposition of parts. 
 
 - The beauty of this elementary proposition, and the difficulty beginners often find 
 
 to comprehend the demonstration, have induced some geometricians to invent others 
 
 _ofasimpler nature. Some of these are very ingenious, and worthy of notice, be- 
 
 - eause it can be seen on the first view, that the square of the hypothenuse is com- 
 
 posed of the same parts as the squares of the two sides; with this difference only, 
 that they are differently arranged. Some of these demonstrations are as follow: 
 
 Fig 26. Ist. Describe the right angled triangle a Be (Fig. 26.), 
 
 ‘rc . and on the two sides of it, a c and B c, construct the two 
 
 un \x 
 
 | THEOREM I. 
 
 I squares cGandcp, On the base a B raise the two per- 
 pendiculars a 1 and 8 , the former meeting c F, continued, 
 ak in 1, and the latter meeting E Din H; and then draw 1 H. 
 
 It may, in the first place, be easily demonstrated that a I 
 
 \| ¢ and Bare equal toa xB; so that aru B is the square of 
 
 the base a 8B; for it may be readily seen that the triangle 
 B HD is equal and similar to the triangle B a c, as wellas 
 the triangle 1G a; so that 5 Hand a 1 are each equal to aA B. 
 
 It may be shewn, with equal ease, that the small triangle kK E His equal tor FO: 
 and lastly, that the triangle 1 K L is equal to aoc. 
 
 But the constituent parts of the two squares are, the quadrilateral c B H K, the 
 triangle B D H, the triangle K HE, the quadrilateral a a o F, and the triangle 4 c 0, 
 which we shall shew to be the same that compose the square A B 11; for the quad- 
 rilateral c 8 H K is common, and the triangle BH D is equal toBca, and may be sub- 
 stituted for it, and transposed intoits place. In like manner, we may conceive the 
 triangle aco transposed intor KL; there will then. remain, in the square of the 
 hypothenuse, the vacuity 1a, and we shall have, to fill it up, the quadrilateral 
 ¥0AG, with the triangle KEH: let the triangle k EH be transposed into o F1, which 
 is equal to it, and it will complete the triangle I a G, which is equal and similar to 
 1A 1; hence it follows that the square of the hypothenuse is composed of the same 
 parts as the squares of the other two sides. 
 
 We may therefore cut these parts from a piece of card, and first compose the two 
 squares of the two sides, and then that of the hypothenuse, which will form a sort of 
 amusement in combination. 
 
 2d. The second method, which is nearly the same as the pre- 
 ceding, will appear perhaps a little more evident. Let cp and 
 cr (Fig. 27) be the squares of the two sides, which contain the 
 right angle of the triangle a c B: having continued F a until a 
 is equal cB, on the side rH construct the square FHDG: and 
 on A B, the hypothenuse, the square Ax. It may be easily 
 proved that the angles © and n will be in the sides of the former, 
 and that AH, BD, EG, NF will be all equal, as well as FA, BH, 
 DE, GN. . 
 
 But it may be readily seen that, by drawing the line n1 parallel to F 8, the 
 two squares c D and CF will be composed of the parts 1, 2, 3, 4, 5; and the 
 square AE is composed of the parts 1, 5, 6, 7, 8 But the parts.l and 5 are com- 
 mon, and the parts 6 and 2 are evidently equal: it remains then that the parts 4 
 and 3 should be equal to the parts 7 and 8. But this is also evident; for the part 3 
 is equal to 9, and the part 8 to 5, consequently the parts 4 and 3, or 4, and 9, are 
 equal to the parts 7 and 8, or 7 and 5, since the rectangle F I is divided into two 
 
124 GEOMETRY. 
 
 equal parts by the diagonal. The squares of the sides then are composed of the 
 same parts as the square of the hypothenuse, and consequently they are equal. 
 
 3d. Retaining the same construction, it is evident that the square F D is equal | 
 
 to the squares of the two sides a c and cB of the right-angled triangle a c B, plus 
 
 the two equal rectangles c Gandc Hu. But the square a E, of the hypothenuse, 
 
 is equal to the same square less the four equal triangles ABH, BED, EGN, NFA, 
 which taken together are equal to the two rectangles above mentioned, since each of 
 the triangles is the half of one of the rectangles. The quantity by which the square 
 
 F D exceeds the squares of the sides of the right-angled triangle a cB, is the same as 
 
 that by which it exceeds the square of the hypothenuse: these squares and that of 
 
 the hypothenuse are therefore equal; for quantities which are less than a third by an 
 equal quantity, are themselves equal. 
 
 We shall now give a few propositions which are only generalizations of the forty- 
 seventh of the first book of Euclid, and from which that celebrated proposition is 
 deduced as a simple corollary. 
 
 THEOREM II. 
 
 If a square be described on each of the sides of any triangle a B c (Fig. 28 and 29.) ; 
 and if a perpendicular B D be let fall from one of the angles, as B, on the opposite 
 side a c; if the lines B E and BF be drawn in such a manner that the angles AEB 
 and c F B shal! be equal to the angle 8B; and lastly iff 1 and F 1 be drawn parallel to 
 c G, the side of the square, the square of a B will be equal to the rectangle a 1, and 
 the square of B c tothe rectangle cL; consequently the sum of the squares on A Band 
 BC will be equal to the square of the base less the rectangle E L, if the angle B be 
 obtuse, and plus the same rectangle if the angle B be acute.* 
 
 The demonstration of this theorem is as follows: the 
 triangle AEB is similar to the triangle a Bc, because the 
 
 angle ABC; consequently Ac: AB’: AB: AE, whence 
 it follows that the rectangle of Ac X AF, or of AE X 
 AH, which is the same since AH = AC, is equal to the 
 square of A B. 
 
 In like manner it may be proved that the square of 
 Bc is equal to the rectangle cL. 
 
 But it may be readily seen, that if the angle B be 
 obtuse, the line B = will fall between the poiuts a and 
 p, and the line BF between c and p; the contrary of 
 which is the case if the angle B be acute; and that 
 
 the perpendicular B D, when the angle B is a right 
 one. ; 
 
 In the first case then it is evident, that the sum of 
 the squares of the sides, is less than the square of the 
 base by the rectangle EL. 
 
 rectangle EL. 
 
 Lastly, that if the triangle be right-angled at 3B; as 
 the rectangle EL vanishes, the sum of the squares of 
 the sides is equal to the square of the base; which is a very ingenious generali- 
 zation of the celebrated theorem of Pythagoras. 
 
 * For this ingenious theorem, from which is deduced the famous problem of the right-angled 
 
 triangle, we are indebted to Clairault, junior, who published it at the age of sixteen, in a small work © 
 
 printed in 1731. ‘This young man would certainly have trodden in the steps of his brother, had he 
 not been cut off by a premature death. 
 
 angle Ais common, and the angle a E B equal to the 
 
 these two lines are confounded with, or coincide with, 
 
 And in the second case, that they exceed it by the 
 
GEOMETRICAL PROBLEMS. 125 
 
 THEOREM III. 
 
 Let a 3c (Fig. 30.) be a triangle, and let any parallelogram c & be described on the 
 side a c, and any parallelogram 8B ¥ on the side AD; continue the sides D £ and K F 
 till they meet in the point u, from which draw the straight line UH a L, and make LM 
 equal ton A; tf the parallelogram c 0 be then completed on the base B c, by drawing 
 BO or CN parallel to u M, this parallelogram will be equal to the two c B and BF. 
 
 Continue oBand Nc till they meet the sides of the paral- 
 30. lelograms BF and cer, in Pp and R, and draw pr. 
 
 Then since cr and HA are parallel, and comprehended be- 
 tween the same parallels, viz. c a and DH, they are equal; 
 consequently c Ris equal toL M. In like manner it may be 
 demonstrated that 8 P is equalto LM. crand bP therefore 
 are equal, and the figure B P R Cis a parallelogram equal 
 to BN. 
 
 Now it is evident that the parallelogram r L, on the base 
 RC, is equal to the parallelogram Rc A H, because it is on the same base and be- 
 tween the same parallels; and for the same reason the parallelogram acDE= 
 ACRH; consequently the parallelogram A CDE=RCLG. 
 
 It may be demonstrated, in like manner, that the parallelogram sk FA=BPGL; 
 consequently the two parallelograms cE, BY, are together equal to BPRC, or to 
 B CN O, which is equal to it. 
 
 Corollary.—The reader, if in the least acquainted with geometry, may readily 
 see that this very ingenious proposition is only a generalization of the celebrated 
 proposition by which it is proved, that in every right-angled triangle, the squares 
 of the two sides, containing the rignt angle, are equal tothat of the hypothenuse. 
 For.if we suppose that the triangle B ac is right-angled at a, and that the two 
 parallelograms c £ and BF are the two squares, it may be easily conceived that the 
 third parallelogram BN will be also a square, viz. that of the hypothenuse ; in con- 
 sequence of the preceding demonstration then, these two first squares will be equal 
 to the third. This theorem is extracted from Pappus Alexandrinus. 
 
 THEOREM Iv. 
 
 If the base of a triangle be bisected, the squares of the other two sides are equal to twice 
 the square of half the base, and twice the square of the line joining the middle point 
 of the base to the vertical angle. 
 
 Fig. 31. For let p (Fig. 31.) be the middle of the base, andc rE a 
 perpendicular from the vertex c on the base a 8. Then a c? 
 =avn*+npnce?+2an.peandnc?=sBn?-+pc? — 2 Bp. 
 DE or BC? =a D?-+ Dc? 2ap. ne. Whence by adding we 
 
 Dp E B have ac?-+pc?=2a v?+- 2d c%, 
 
 THEOREM V. 
 
 In every quadrilateral figure whatever, the sum of the squares of the four sides, is equal 
 to the sum of the squares of the two diagonals, plus four times the square of the line 
 which joins the middle of these diagonals. 
 
 Fig. 32. Leta scp (Fig. 32,) be a quadrilateral figure, the two diago- 
 
 Q nals of which are Ac and B D; and let us suppose them 
 
 divided each into two equal parts in © and F, and that the 
 
 straight line » F has been drawn. It may be demonstrated, that 
 
 the squares of the four sides, taken together, are equal to the 
 
 squares of the two diagonals, plus four times the square 
 of E ¥. 
 
: 
 126 GEOMETRY. 
 
 It is said that weare indebted to Euler for this elegant and very curious pr oblem, 
 which may be easily demonstrated by the aid of the preceding theorem. 
 
 For join re tos and p. Then as?+Bpe?= 2a £?-+ 2r B, anda v?+pe=s 
 2a40?+2 5 pv’, Therefore a B?-+3c?+av?-+pei =—4ar?+2E B+ Qe ye 
 But B r?-+-Ep?=28F?-+-2F 2’; therefore 2 pe? -+2eD?=4BF?-+4FrE?, There-' 
 fore aB+ BC?-+cD?-+avP—45BH?+4BrP+4 rR =~A +B D?+4 Fe, 
 
 Corollary.—-If the quadrilateral is a parallelogram, then & F coincide, and the pro- 
 position shews that the square of the sides of a parallelogram are together equal to 
 the squares of the diagonals. 
 
 The preceding theorem, therefore, is only a particular case of the present one. 
 
 PROBLEM XVIP 
 The three sides of a rectilineal triangle being given; to determine its superficial content, 
 without measuring the perpendicular let fall from one of the angles on the opposite 
 side. 
 
 From half the sum of the three sides subtract each of the three sides separately ; 
 multiply the three remainders together, and the product by the half sum of the sides; 
 the square root of the last product is the area required. 
 
 Let the three sides, for example, be 50, 120, and 150 yards; the half sum of which 
 is 160; the first difference is 110, the second 40, and the third 10: the product of 
 these four numbers is 7040000, the square root of which is 2653 and #, nearly, which 
 is the area, 
 
 It may be easily shewn, that the usual method, that is to say, by finding the per- 
 pendicular let fall from one of the angles on the opposite side, would require a much 
 more tedious calculation. 
 
 Remark.—By this method we have a very easy rule for finding the radius of the 
 circle inscribed in a triangle, the three sides of which are given: nothing is necessary 
 but to multiply together the difference between each side and the half sum; to 
 divide the product by this half sum, and to extract the square root of the quotient: 
 the result will be the radius required. 
 
 Thus, in the above example, the product of the differences is 44000; which 
 divided by 160, gives 275 ; the square root of this quotient 167%, is the radius of the 
 eircle inscribed in the given triangle. 
 
 PROBLEM XVII. 
 
 In surveying the side of a hill, ought its real surface to be measured, or only the space 
 occupied by its horizontal projection ? 
 
 It may be easily proved that, in this case, the horizontal projection or base only 
 ought to be measured; for the object of surveying is nothing else than to determine 
 the quantity of any kind of production that land is capable of producing, or the num- 
 ber of the buildings that can be erected on it. But it is evident that as trees and 
 plants always rise in a direction perpendicular to the horizon, an inclined plane can 
 contain no more than the horizontal one which corresponds to itas its base. In like 
 manner, no more buildings can be raised on inclined ground, than on its horizontal 
 projection; because the walls of an edifice must always be vertical: a little more 
 care only is required in building on such ground than on horizontal. . 
 
 Another reason is, that inclined ground, compared with the horizontal ground in 
 the neighbourhood, contains less vegetable earth or mould, as part of it is always 
 carried away by the rains, and deposited on the lower grounds; consequently it is 
 not capable of supplying nourishment to such a quantity of productions as the 
 other. 
 
7 
 
 re 
 * 
 
 GEOMETRICAL PROBLEMS. 127 
 
 ' It is therefore evident that the horizontal surface only, and not the real or in- 
 clined surface, ought to be measured, unless these considerations are thought to be of 
 little value in adjusting the price. 
 
 _ Remark.—It is in topographical descriptions of mountainous countries chiefly, that 
 care should be taken to reduce the whole to a horizontal plane; for if we suppose 
 that a country has been surveyed, and that, in measuring the sides of pretty steep 
 mountains, the real and not the horizontal distances of places have been taken, it 
 will be impossible, in constructing a map, to make the measures agree. This indeed 
 would be the same thing as if one should attempt to transfer to the plane or base of 
 a pyramid, the triangles which form its inclined sides: for if one of the triangles were 
 Jaid down on it, all the rest would be falsely represented. 
 
 PROBLEM XVIII. 
 To form one square of five equal squares. 
 
 Fig. 33. No. 1. Divide one side of each of four of the squares, as A, B, C, D, 
 
 -" asl (Fig. 33. No. 1.) into two equal parts, and from one of the 
 \ A | N B I ¢ | angles adjacent to the opposite side draw a straight line to 
 — ~ the point of division; then cut these four squares in the 
 
 \ D | | Ez | direction of that line, by which means each of them will be 
 fas divided into a trapezium and a triangle. 
 
 Lastly, arrange these four trapeziums and these four 
 triangles around the whole square £, as seen Fig. 33. No. 2; 
 and you will have a square evidently equal to the five squares 
 given. 
 
 Remark.—By means of the solution to the following pro- 
 blem, one square may be formed of any number of squares at 
 pleasure ; for any number of squares may be transformed into 
 an oblong, and we shall shew, in the next problem, how an 
 oblong may be resolved into several parts, susceptible of being arranged in such a 
 manner as to form a square. 
 
 PROBLEM XIX. 
 
 Any rectangle whatever being given; to convert it, by a simple transposition of parts, 
 into a square. : 
 
 Fig. 34, No. 1. Let the given rectangle be ancy (Fig.34. No.1.) To cut.it into 
 several parts susceptible of being arranged in a square, first find 
 the geometric mean proportional between the sides Ba and a D; 
 make ax equal to that mean proportional, and draw EF per- 
 pendicular to AE. EF will cut ab in the point F, which will 
 either fall beyond p, in regard to the point A, or on the point 
 D itself, or between p and a: this forms three cases, the last 
 of which subdivides itself into two, but if one of them be well 
 understood, there will be no difficulty in the rest. 
 
 Case \st. In the first place then, let the point F be beyond 
 p, as seen (Fig. 34. No.1.) As the line eF will intersect cp 
 in the point 1, make a c equal to DL, and draw GH perpendicular to aE, by which 
 means GH will cut off from the triangle A Bg, the small triangle a GH. 
 
 Fig.34. No.2. : Then cut the given rectangle ac into four parts, according to the 
 
 ines AE, EL, and Gu, and the result will be the trapezium a ELD, 
 the triangle Ec, the trapezium G Br H, and the small triangle a GH, 
 which we shall respectively denote by the letters a, b,c, d; lastly, 
 arrange these four parts as seen Fig. 34, No, 2, and you will havea 
 perfect square. 
 
128 GEOMETRY. 
 
 The demonstration may be easily found, by considering, in Fig. 34,:No. 1, the 
 square constructed on AE, Viz. AEKI; but it is first necessary to shew, that’ 
 if at be drawn parallel to Er, and x1, through the point v, parallel to a £, the’ 
 rectangle A E K1, thence resulting, will be a square. Now this is easy; for if 1K 
 be continued till it meet Bc produced in Pp, we shall evidently have the rectangle | 
 AEKI equal to the parallelogram AE PD, which is equal to the rectangle ABCD, or 
 that of apand a.p; hence it follows that az into alis equalto AaB xX AD. But 
 the square of a x is equal to AB into AD, consequently AE into a1 is the same thing” 
 as the square of an. | 
 
 This being demonstrated, draw Lc parallel to ap, and LM parallel to ar; then, | 
 from the points m and c, to a p and ar, draw the perpendiculars MN andau. It is. 
 here evident that the triangle am N is equal and similar to ELC: in like manner the 
 triangle A GH is equal and similar to pL K; and the trapezium BEHG is equal and 
 similar to ND 1M, for BE is equal and parallel toDN, BG toMN, DI to EH, and MI 
 toGcu. The four parts AELD, ECL, BEHG, AGH, which compose the rectangle 
 Ac, are therefore equal to the four AELD, AMN, NDIM, and DLK, which compose | 
 the square AEK I, or its equal, that of the same figure, No. 2, &c. 
 
 Fig. 34. No.3. Case 2d. If the point F falls on the point p, the solution of the) 
 problem will be exceedingly easy ; for in that case the triangle d| 
 vanishes, since DL vanishes; the square equal to the rectangle, 
 therefore, will be composed of the rightangled isosceles triangle | 
 AE D (Fig. 34, No. 3.), and the other two right-angled and isosceles 
 triangles ABE and CDE, equal to each other, and to the half of the 
 former; consequently these parts may be arranged in a square) 
 without any difficulty. This case indeed can never exist but when 
 the side AB is exactly the half of ap: the rectangle a c is then composed of two. 
 
 equal squares. But the manner in which two equal squares may be fomed into one 
 is well known. | 
 
 c D 
 
 | 
 
 Fig. 35. No. 1. Case 3d.. Let us now suppose, that the point F falls be-| 
 tween a and p (Fig. 85. No. 1.), but in such a manner that’ 
 FD is less thanks. In this case make EG equal to FD, and | 
 
 . draw GH perpendicular to aE; by which means the rectangle | 
 
 oe f ac will be divided into four parts, viz., the ‘tri- Fig. 35. No.2. 
 
 A I angle AEF, the trapezium cDFE, the trapezium 
 
 hee ABGH, and the triangle EGu; which we shall aN. | 
 
 , distinguish by the letters a,b,c, d. If these four | 
 
 parts be arranged as seen Fig. 35. No. 2, we shall a | 
 havea perfect square,asmay beeasily demonstrated. 
 
 If ¥ p be exactly equal to BE, it is evident, that instead of the trae. 
 pezium A BGH, we should have a triangle a Bh; so that the square to. 
 be formed would consist of three triangles, and a trapezium EC DF, as) 
 
 seen Fig. 35. No. 3. 
 If rp exceeds EB, and is exactly equal to ar, draw D P parallel to. 
 
 Fie.38.No.4, 2 and if the rectangle be cut according to the lines a 5, & F, and PD, 
 
 3 , there will be formed three triangles and a parallelogram Ep, which if | 
 
 K 
 [| arranged as seen Fig. 35. No. 4, will compose the square AIK E. 
 Lastly, we may suppose the height a p of the given rectangle to be | 
 see such, that having the general construction described in the first part of | 
 
 4 this problem, the line F p exceeds the line a F, or is any multiple of it, | 
 with or without a remainder. In that case, to resolve the problem, set off the line 
 A Fas many times as possible on Fp. For the sake of simplification, we shall here 
 suppose that the former is contained in the latter only once, with the remainder uD. - 
 Draw 1 M parallel to e F, and by these means we shall have the parallelogram LM EF, 
 
 | 
 
GEOMETRICAL PROBLEMS. 129 
 
 which may be placed in Fa No}; then make EG equal to pt, and draw cH perpen- 
 dicular to AE; cut the rectangle a BcD according to the lines Af, EF, ML, and au 
 into five parts, viz., the triangle arr, the parallelogram ri M8, the trapeziums 
 LDCM, AHGB, rat the triangle GHE; which we shall distinguish by the letters 
 a,b,c, d,e; these five parts can be arranged into a perfect square, as AIK E, which is 
 composed of the triangle a, the parallelogram b, the trapeziums ¢ and d, and the 
 small triangle e. 
 If A F were contained twice in FD, six parts: would be requisite ; two of them pa- 
 rallelograms as b. 
 By a sort of retrograde progress, the following problem may be resolved. 
 
 PROBLEM XX. 
 
 To cut a given square into 4, or 5, or 6, &c. dissimilar parts, which can be arranged so 
 as to form a rectangle. 
 
 Let it be required, for example, to divide the square Ar Kr (Fig. 35. No. 1. ), into 
 four parts susceptible of such an arrangement. On the side EK assume EF greater 
 than the half of it, and draw ar; make ao equal to ny, and draw om parallel to 
 _AF; lastly, from the point m, where om meets rk, draw MN per pendicular to aF; 
 the four parts required will be the triangles AEF, o m1, and the two trapeziums 
 AOMN, MNFK, which may be arranged in such a manner, as to form the rectangle 
 ABCD. ‘To those who have comprehended the solution of the preceding problem, 
 this will appear evident, 
 
 Fig. 36. If five parts be required, assume EF (Fig. 36. ), of such a 
 length, that it may be contained in £ K twice, with a remainder ; 
 let these parts of the line EK be EF and Fo, and let the re- 
 mainder be ok; draw AF, and, making a N and NP each equal 
 to Er, draw No and PQ parallel to ar, the latter of. which 
 will meet the side Kr in Q; from this point draw QR perpen- 
 dicular to No; and we shall have two triangles, a parallelo- 
 gram, and two trapeziums, which are evidently susceptible of 
 being formed into an oblong such as ABCD; since they are 
 the same parts into which that oblong might be divided, in 
 order to. form, by their transposition, the square aEKI; 
 therefore, &c, 
 
 PROBLEM XXI. 
 To divide a line in extreme and mean ratio. 
 
 A line is divided in extreme and mean ratio, when the whole line is to one of the 
 segments, as that segment is to the other. As a great many geometrical problems 
 ‘are reduced to this division, some of the geometricians of the sixteenth century gave 
 it the name of the divine section. But without adopting so emphatical a denomina- 
 
 ' tion, we shall proceed to the solution of the problem. é 
 Let the line, to be divided in extreme and mean ratio, be a B 
 Fig. 37. (Fig 37.) From its extremity B raise the perpendicular Bc, 
 ; and make it equal to the half of a B; draw ac, and make cp 
 equal toc B; if A E be then made equal to the remainder a p, 
 the line a B will be divided as required, and we shall have this 
 
 ratio; ABisto AEas A EIstoE B, 
 
 Fig. 88. No. 1. Remarks.— The line a 6 (Fig. 38. No. 1.) being divided in 
 , extreme and mean ratio, if its greater segment be added to 
 b eo © it, we shall have the line dc, also divided in extreme and 
 mean ratio, in the point a; so that b c will be to ba as b ais toae. 
 K 
 
130 GEOMETRY. 
 
 2d. The line b a (Fig. 38. No. 2.), being divided, in the same. 
 Fig. 38. No.2. manner in c, if ¢ d be made equal to the small segment b c, ca will! 
 6 cia then be divided in the same manner; that is to say, ca will be to 
 . edascdtoda. | 
 
 PROBLEM XXII. 
 
 | 
 
 On a given base to describe a right angled triangle, the three sides of which shall ad 
 in continued proportion. 
 
 Fig. 39. On the given base a B (Fig. 39.) describe a semicircle ;| 
 divide a B in extreme and mean ratioinc, and raise the perpen-| 
 dicular cp till it meet the semicircle in p; then draw the lines. 
 
 LN Ap and DB: the triangle asp will be the one required, and, 
 AB will have the same ratio to a D as A D has to D B, as might! 
 be easily demonstrated. 
 
 ote |W) 
 
 PROBLEM XXIII. 
 
 Pwo men, who run equally well, propose for a bet to start from a, and to try who shall 
 first reach B, after touching the wallc pv, (Fig. 40.) What course must be pursued in’ 
 order to win ? 
 
 Fig. 40. It may be readily seen that, to determine the course to be 
 pursued in order to win, it will be necessary to determine the) 
 position of the lines a E and E B, of such a nature, that their | 
 sum shall be less than that of all the others, as ae, eB, &e. | 
 But it may be demonstrated that this sum is the least possible, 
 when the angle aE c is equal to the angleBED. For let us | 
 suppose A c drawn perpendicular to cp, and continued till 
 c F.be equal to a c,and that EF and EB have been drawn ; in | 
 this case the angles aEc and cuF will be equal. But anc is equal tops p by the | 
 supposition, consequently the angles ce F and BED will be equal also; and it thence 
 follows, that, as cp isa straight line, Fr B will likewise be one. But BEF is equal | 
 to b E and £a taken together, as Be anderare to Be ande a; the course BEA | 
 therefcre will be shorter than any other Be a, for the same reason that B F is shorter | 
 than the lines B e and e F. - # 
 To find then the point E, we must draw a c and B D perpendicular to the line cp, | 
 and thén divide c p in £, in such a manner, that c E shall be to EDasca toDB, | 
 
 PROBLEM XXIV. 
 A point, a circle, and a straight line, being given in position, to describe a circle which 
 shall pass through the given point, and touch the circle and 
 Fig. 41. straight line. 
 
 Through the centre of the given circle draw B & (Fig. 41 ) 
 perpendicular to the given straight line, and let it cut the 
 circle in B and F; draw also B a to the given point a, and | 
 take 8 G a fourth proportional to B a, BE, BF; if a circle be | 
 then described through the points a and a, touching the line 
 cD, it will touch also the given circle. | 
 
 If the point a be within the circle, the construction will be 
 the same: in this case itis evident that the line which ought 
 to be touched by the required circle, must enter the given circle 
 also; and there are even two circles which will resolve the pro- 
 
 blem, as may be seen in Fig, 42. ° | 
 
GEOMETRICAL PROBLEMS. 131 
 
 PROBLEM XXV. 
 
 Two circles and a straight line being given; to describe a circle which shall touch 
 
 them all. 
 This problem is evidently susceptible of several cases; for the circle which touches 
 
 the straight line may inclose both the other circles, or only one of them, or may leave 
 them both without it; but, for the sake of brevity, we shall confine ourselves to the 
 last case, and leave the rest to the sagacity of our readers, who, when they compre- 
 hend this solution, will find no difficulty to resolve the rest. 
 
 Let there be given two circles, whose radii are ca andca 
 (Fig. 43.), and let the line p = be given in position. In the 
 present case, on the radius c a make a o equal c a, and with the 
 radius co describe a new circle; draw also beyond pE the 
 line de parallel to p &, and distant from it by a quantity equal 
 to ca; then, by the preceding problem, describe a circle 
 through c, which shall touch the circle having for its radius co, 
 and also the straight line de; let the centre of this circle be B; 
 if its radius be diminished by the quantity ao or ca, the circle 
 described with this new radius will evidently be a tangent to 
 the two given circles, as well as to the straight line Dz, 
 
 PROBLEM XXVI. 
 Of inscribing regular polygons in the circle. 
 ; The following general method of inscribing regular polygons - 
 Fig. 44. : : ig: : : é 
 in the circle, is given in various books of practical geometry. 
 On the diameter a B (Fig. 44.) of the given circle, describe an 
 equilateral triangle ; and divide this diameter into as many equal 
 parts as the required polygon is intended to have sides; then 
 from x, the summit of the triangle, draw through c, the extre- 
 mity of the second division, the line Ec; and continue it till it 
 meet the circumference of the circle in D: the chord Ap, they 
 say, will be the side of the required polygon to be inscribed. 
 We have noticed this method merely to say that it is erroneous, and could be in- 
 
 vented only by a person ignorant of geometry, or else intended only as near the 
 
 truth. For it may be easily demonstrated that it is false, even when employed for 
 
 finding the simplest polygons, such for example as the octagon. lt will be found in- 
 deed, by trigonometrical calculation, that the angle Dca, which ought to be 45°, is 
 | 48° 14’; whence it follows, that the chord apis not the side of the inscribed 
 octagon. 
 
 None of the regular polygons can be inscribed geometrically and without trial, by 
 
 _meanis of a rule and compasses, except the triangle, and those polygons deduced from 
 it, by doubling the number of sides, as the hexagon, the dodecagon, &c. 
 
 The square and those polygons deduced from it in like manner, as the octagon, the 
 
 | sedecagon, &c. 
 
 The pentagon and those deduced from it, as the decagon, and the eikosiagon, &c. 
 The pentedecagon and its derivatives, as the polygon of thirty sides, &c. 
 The rest, such as the heptagon, enneagon, endecagon, &c., cannot be described by 
 
 means of the rule and compasses alone, without trial; and all those who have 
 attempted this method, have failed or have produced ridiculous paralogisms. 
 
 The following in a few words, is the method of describing geometrically in a 
 
 “circle, the five primitive polygons, which may be inscribed with the rule and 
 _ compasses. 
 
 Kee 
 
 iP a 
 #3 
 
eri 
 
 132 GEOMETRY. 
 Fig. 45. Divide the circle a DB & (Fig. 45.) into four equal parts, 
 B by the two diameters a B and DE, intersecting each other at} 
 
 right angles; then divide the radius cp into two equal parts, 
 in F, and draw oF G parallel to A B: the line Eq will be the 
 side of the inscribed equilateral triangle, as well asG o 
 and o£. | 
 The line — B, as every one knows, will be the side ol| 
 the square. 
 If uw be made equal to the radius, it is in like manner) 
 evident, that it will be the side of the hexagon. | 
 Divide the radius a c into two equal parts in 1, and draw E1; make 1K equal ta! 
 1c, and the chord £E L equal to the remainder EK: EL will be the side of the decagon;, 
 and by making the arc LM equal to the arc EL, we shall have the chord © m for the| 
 side of the pentagon, 
 Then divide the arc om, which is the difference between the are of the pentagon. 
 and that of the triangle, into two equal parts in n, and draw the straight line o Ny 
 which will be the side of the pentadecagon, or polygon of 15 sides. | 
 Remark.—T he heptagon is susceptible of a construction, not geometrical, but. 
 approximated, which is pretty near the truth, and which on that account deserves to 
 be known; it is as follows: First describe an equilateral triangle, or at least deter-) 
 mine the side of one, the half of which will be the side nearly of the insusceptible| 
 heptagon. It will be found indeed by calculation, that the side of the triangle, radius | 
 being unity, will be equal to 0:86602, the half of which is 0-43301, and the side) 
 of the heptagon is 0°43387 ; the difference therefore between it and half the side of 
 triangle, isless than a thousandth part. Whenever then the thousandth part of the 
 radius of the given circle is an insensible quantity, the above construction will ap- 
 proach very near to the truth. — 
 It is much to be wished that methods of construction equally simple, and as 
 near the truth, could be discovered for all other polygons; which indeed is not 
 possible 
 
 | 
 
 PROBLEM XXVII. 
 
 The side of a polygon of a given number of sides being known; to find the centre of 
 the circumscriptible circle. . : 
 This problem is, in some measure, the reverse of the former, and may be easily, 
 solved for the same polygons. ! 
 We shall say nothing of the triangle, the square, and the hexagon, because those 
 who are acquainted with the first elements of geometry, know how to find the 
 centre of an equilateral triang!e and a square, and that the side of the hexagon is 
 equal to the radius of the circumscriptible circle. 
 We shall begin therefore with the pentagon.—Let a : 
 Fig. 46. (Fig. 46.) be the side of the pentagon: at the extremity 0) 
 which raise the perpendicular a c, equal to 4B; draw BC, anc 
 cut off from it ce =ac, and make sr=BeE; then with the 
 centre a, and the radius ar, describe an are of a circle, anc 
 from the point 8, with the. radius Ba, describe another inter- 
 secting the former in G: the line BG will be the position of the 
 second side of the pentagon, and the two perpendiculars on the 
 middle of the sides a B and Bq will give, by their intersée: 
 tiou, he position of the centre u. 
 For the octagon.—Let a B (Fig. 47.) be the given side; or 
 this line describe a semicircle, and raise the radius c @ perpendi. 
 
 | 
 
GEOMETRICAL PROBLEMS, 133 
 
 cular, and indefinitely continued; draw the side of the square 
 BG, and make c F equal to the halfof Ba; drawre perpendi- 
 cular to the diameter, and through the point ©, where it cuts 
 the semicircle, draw an, which will meet ca continued in D: 
 this point p will be the centre of the circle required. 
 
 For the decagon.—If a 8 (Fig. 46.) be’the given side, find, as 
 
 if a pentagon were to be constructed, the line Br, and from the 
 points a and B with the radius a Fr, describe the isosceles triangle 
 AhB: the point will be the centre of the decagon, 
 
 For the dodecagon, and any other polygons what- 
 ever.—Let the line given for the side of the polygon 
 be aB (Fig. 48.) With any radius whatever cp 
 describe a circle, and inscribe in it the required 
 dodecagon or polygon, the side of which we shall 
 
 “suppose to be DE: continue De to F, ifap exceeds 
 Dx, so that D F shall be equal to a B, and then draw 
 c £, and its parallel F G: the point where the latter 
 
 meets the diameter DH continued, will evidently be 
 
 ‘the centre of the circle, in which the required polygon is inscriptible. 
 
 Though we have given particular methods for the pentagon, octagon, and decagon, 
 
 ‘itis evident that the last method may be applied equally to them all. ; 
 
 | ‘We shall conclude this article, on polygons, with the two useful tables, one of 
 ‘which contains the sides of the polygons, the radius of the circle being given, and the 
 
 ‘other the length of the radius, the side of the polygon being known. If the radius of 
 the circle then be expressed by 100000, the side of the inscribed triangle will be 
 
 Meron an unit of .......6.. 173205 that of the decagon ..... - 61803 
 that of the square........ 14142] that of the endecagon .... 56347 
 that of the pentagon .... 117557 ‘ that of the dodecagon .... 51763 
 that of the hexagon..... - 100000 that of the tredecagon.... 47844 
 that of the heptagon .... 86777 that of the tesseradecagon. 44503 
 that of the octagon ..se0e50 76536 that of the quindecagon .. 41582 
 
 that of the enneagon ...- 68404 
 On the other hand, if the side of the polygon be 100000, the radius of the circle wilt 
 
 be, in the case of the triangle 577385 of the decagon .......-.5 161804 
 Mr the square.....s...... 70710 of the endecagon ........ 177470. 
 
 | ofthe pentagon ....... - 85065 of the dodecagon ........ 193188. 
 _ of the hexagon ....-5ee°* 100000 of the tredecagon........ 209012 
 of the heptagon......ee00 115237 of the tesseradecagon .... 224703. 
 of the octagon ...ece.... 130657 of the quindecagon ...... 240488. 
 
 of the enneagon .eee.e.. 146190 
 
 PROBLEM XXVIII. 
 | Method of forming the different regular bodies. 
 
 _ It was long ago demonstrated in geometry, that there can be only five bodies: 
 terminated by regular figures, all equal to each other, and forming with one another 
 equal angles. These bodies are: 
 
 The tetraedron, which is formed by four equilateral triangles. 
 
 The cube, or hexaedron, formed of six equal squares. 
 _ The octaedron, formed of eight equal equilateral triangles. 
 _ The dodecaedron, formed of twelve equal pentagons. 
 _ The icosaedron, formed of twenty equilateral triangles. 
 
 Two methods may be employed to form any one of these regular bodies. The 
 first is, to construct a sphere, and then to cut off the excess, so that the remainder 
 
s v" ; —— CC 
 % 
 
 134 GEOMETRY. 
 
 shall form the regular body required ; the other, which resembles the process used | 
 in stone-cutting, consists in first tracing out on a plane, made at hazard, one of the 
 faces of the body to be formed, and then cutting out the adjacent faces, under the 
 
 determinate angles. 
 To resolve then the problem in question, we shall first answer the following’ 
 
 . 
 
 f 
 
 questions. 
 Ist. The diameter of a sphere being given, to find the sides of the faces of each of 
 
 the regular bodies. | 
 2d. To find the diameters of the less circles of that sphere, in which the faces of 
 each of these bodies are inscriptible. | 
 3d. To determine the opening of the compasses, with which each of these circles: 
 may be described on the surface of the same sphere. . | 
 4th. To determine the angles which the contiguous faces form with each other, in 
 their common intersection. | 
 
 | 
 . 
 ‘ 
 
 Ist. A sphere being given; to find the sides of the faces of each of the five | 
 regular bodies. | 
 
 Fig. 49. Let apc (Fig. 49.) be the half of a great circle of the given 
 sphere, and ac one of its diameters. Divide a c into three 
 equal parts, and let ar be two thirds; draw x1 perpendicular. 
 to the diameter, cutting the circle in x, and join aE: this 
 line will be one of the faces of the tetraedron, and cx will be 
 that of the cube or hexaedron. | 
 
 Then, through the centre r, draw the radius F B, perpendi- 
 cular to ac, cutting the circle in B, and join a B: this linea B 
 will be the side of the octaedron inscribed in the same sphere. 
 
 The side of the dodecaedron will be found, by dividing rc, tbe side the hexaedron, 
 in mean and extreme ratio, and taking for the side of the dodecaedron the larger 
 segment C kK. ; ! 
 
 Lastly, from a, the extremity of the diameter, draw the perpendicular a a, equal! 
 to ac, and from the centre F drawn the line FG, intersecting the circle in H; AH 
 will be the side of the icosaedron. . 
 The radius ofthe circle being 10000, the side of the tetraedron will be found, by 
 calculation, to be equal to 16329; that of the hexaedron or cube, 11546;. that of 
 the octaedron, 14142; that of the dodecaedron, 77136; and that of the icosaedron,, 
 10514. 
 
 24. To find the radius of the lesser circle of the sphere, in which the face of the 
 proposed regular body ts inscriptible. | 
 
 ' 
 ' 
 
 The method of determining the radius of the circle circumscriptible to the triangie 
 the square, and the pentagon, which are the only faces of the regular bodies, has| 
 been shewn already, and consequently the problem is thus solved. | 
 
 To express them in numbers, as we know that when the side of the equilateral) 
 triangle is 10000, the radius of the circumscribing circle is 5773, therefore, as the side! 
 of the tetraedron is 16329, nothing is necessary but to say, As 10000 is to 5773, so is. 
 16329 to a fourth proportional, which will be 9426. | 
 
 It will be found, in like manner, that the radius of the lesser circle, in which the) 
 octaedron can be inscribed, is 8164. | 
 
 And it will be found also, that the radius of the circle in which the face of the 
 icosaedron can be inscribed, is 6070. 
 
 The side of the square being 10000, the radius for the circumscribing circle, as is 
 well known, is 7071; which will give for the radius of the face of the hexaedron, 
 8164. . 
 
 ~ 
 
 a 
 a 
 4 f 
 
ae 
 
 - 
 , 
 
 GEOMETRICAL PROBLEMS. 135 
 
 Lastly, the side of the pentagon being 10000, we shall have for the radius of the 
 circumscribing circle 8506, which will give for the radius of the face of the dodecae- 
 - dron, 6070. 
 
 $d. To determine the opening of the compasses, with which the circle, capable of re- 
 ceiving the face of the regular body, ought to be described on the sphere. 
 
 Fig. 50. This is very easy; for if pF (Fig. 50.) be the radius of the 
 lesser circle of the sphere, capable of receiving the given face, 
 it is evident that FD is the opening of the compasses proper for 
 describing this circle on the surface of the sphere. But EF is 
 the sine of the angle rc D, which will consequently be given; 
 and F pis the double of the sine of half this first angle: rp 
 therefore may be found by seeking in the tables for the angle 
 FCD, then halving it, afterwards seeking for the sine of that 
 half, and then doubling this sine. This operation will give the 
 _yalue of rp, which in the case of the tatraedron will be 11742; in those of the 
 _hexaedron and octaedron, 9192; and in those of the dodecaedron and icosaedron, 
 — 6408. 
 
 KF 
 > 
 
 4th. Zo find the angle formed by the faces of the different regular bodies. 
 
 Fig. 51. Describe a circle (Fig. 51.) as large as possible, and determine 
 C in it the side of the regular body required ; if a perpendicular 
 be then let fall from the centre on this side, it will be the dia- 
 meter of a second circle, which must also be described. We 
 A B shall here suppose that this diameter is a B. 
 
 Describe then, on the side of the regular body found, the 
 proper polygon, or at least find the centre of the circumscribing 
 circle, and from this centre let fall a perpendicular on the side 
 _ which has been found; in the second circle already mentioned, make the lines a p 
 
 and ac equal to this perpendicular, and the angle Dac will be equal to the angle 
 | required. 
 
 It will be found, by calculation, that this angle, for the tetraedron, is 70° 32’; for 
 the hexaedron, 90°; this is evident because the faces of the cube are percendicular 
 ' to each other; for the octaedron, 109° 28’; for the dodecaedron, 116° 34’; and for 
 | the icosaedron, 138° 12’. 
 _ We shall here collect all these dimensions in the following table, where we sup- 
 pose the radius of the sphere to be ]0000 parts. 
 
 D 
 
 Names of the Sides Radii of the Distances Angles of 
 
 : of the sircum scribing from the the contiguous 
 esegwiar bodies. faces. ‘ circles. poles. faces. 
 Tetraedron 16329 9426 11742 70° 32’ 
 Hexaedron 11540 8164 9192 90 00 
 Octaedron 14142 8164 9192 109 28 
 Dodecaedron 77336 6070 6408 116 34 
 
 Iccsaedron 10514 6070 6408 138 10 
 
 | 
 
 It will now be easy to trace out, by either of the above methods, any required 
 
 _ regular body whatever. 
 _ First method.—Let it be required, for example, to form a dodecaedron from a 
 sphere, First describe a circle of a diameter equal to that of the sphere, and deter- 
 _ mine in it the side of the dodecaedron, or the side of the pentagon, which is one of 
 its faces; also the radius of the circle in which this pentagon can be inscribed, and the 
 
= 
 Te 
 wes 
 
 136. GEOMETRY. 
 
 opening of the compasses proper for describing it on the sphere; which may be easily | 
 done by the precepts before given. | 
 
 Or, if we suppose the radius of the proposed sphere to be 10000 parts, take upon | 
 a scale 6408 of these parts, and-with this opening of the compasses describe, on the | 
 surface of the sphere, a circle on the circumference of which the five angles of the | 
 inscriptible pentagon may be determined ; from two neighbouring points describe, | 
 with the same opening of the compasses, two arcs, the intersection of which will be 
 the pole of a new circle, equal to the former: continue in this manner, from every | 
 two points, and you will have the five poles of the five faces, which rest on the 
 first. In like manner, you may easily determine the other poles, the last of which, 
 if the operation be exact, ought to be diametrically opposite to the first. Lastly, 
 from these twelve poles, describe two equal circles, which will both be cut into five | 
 equal parts, and these will determine twelve segments of a sphere, which being cut 
 off, will give the twelve faces of the dodecaedron required. | 
 
 Second method.—Having first found out, on the proposed block, a plane face, des | 
 scribe on it the polygon belonging to the regular body required ; then cut out, on 
 each side of this polygon, a new plane, inclined according to the proper angle, as | 
 determined in the above table, or which has been traced out by means of the 
 geometrical construction before given, and you will thus obtain so many plane faces, | 
 on which new polygons, having one side common with the first polygons, must be | 
 described. If the same thing be done on these polygons, you will at length arrive | 
 at the last, which, if the operation has been exactly performed, must be perfectly | 
 equal to the first. 
 
 } 
 
 5th. To form the same bodies of a piece of card. | 
 
 If you are desirous of forming these bodies of a piece of card or stiff paper, the: 
 following method will be the most convenient. 
 
 First trace out on the card all the faces of the. 
 
 required body, viz. four triangles for the tetrae-| 
 
 Wet dron, as seen Fig. 52, No. 1, six squares for the 
 
 Oy Ne2 | Weg cube, as No. 2, eight sylaineral triangles for the 
 
 octaedron, No. 3, twelve pentagons for the dode- 
 
 caedron, No. 4, and twenty equilateral triangles 
 
 x A for the icosaedron, No. 5. If you then cut the 
 
 Fig. 52. 
 
 edges, it will be easy to fold up the faces so as to 
 
 join, and if they be then glued together, you will 
 
 have the regular body complete. 4 
 
 The ancient geometricians made a great many geometrical speculations respecting’ 
 these bodies ; and they form almost the whole subject of the last books of Euclid’s 
 Elements. A modern commentator on Euclid, M. de Foix Candalle, has even ex, 
 tended those speculations, by inscribing these bodies within each other, and com- 
 paring them under different points of view; but, at present, such researches are. 
 considered as entirely useless. 'They were suggested to the ancient, by their be-. 
 lieving that these bodies were endowed with mysterious properties, on which the) 
 '|\explanation of the most secret phenonomena of nature depended. With these 
 bodies they compared the celestial orbs, &e. But since rational philosophy has) 
 begun to prevail among mankind, the pretended energy of numbers, and that of the! 
 regular bodies of nature, have been consigned to oblivion, along with the other. 
 visions, which were in vogue during the infancy of philosophy, and the reign of 
 Platonism. For this reason, we shall say nothing farther of these speculations, and. 
 confine ourselves to a very curious problem, respecting the cube or hexaedron, | 
 
A 
 
 GEOMETRICAL PROBLEMS. 13z- 
 
 PROBLEM XXIX. 
 
 To cut a hole in a cube, through which another cube of the same size shall be able to 
 pass. 
 
 If we conceive a cube raised on one of its angles, in such a manner, that the 
 diagonal passing through that angle shall be perpendicular to the plane which it 
 touches ; and if we suppose a perpendicular let fall on that plane from each of the 
 elevated angles, the projection thence resulting will be a regular hexagon, each side 
 and each radius of which may be found in the following manner. 
 
 Fig. 53 On the vertical line a B (Fig. 53.) equal to the diagonal of the 
 
 a2 cube, the square of which is triple to that of the cube, describe a 
 semicircle, and make ac equal to the side of the cube; from the 
 point c let fall, on the horizontal tangent of the circle in B, the per- 
 -€ pendicular cE, BE will be the side and the radius of the required 
 hexagou abcd, Fig. 54. 
 
 When this operation is finished, describe on its hexagonal projec- 
 tion, and around the same centre, the square which forms the pro- 
 jection of the given cube placed on one of its bases, so that one of 
 
 Fig. 54. sides shall be parallel, and the other perpendicular to the diameter 
 a ac: it may be demonstrated, that this square can be contained within 
 the hexagon, in such a manner, as not to touch with its angles any 
 b of the sides: a square hole therefore, equal to one of the bases 
 of the cube, may be made in it, in a direction parallel to one of its 
 d diagonals, without’ destroying the continuity of any side; and con- 
 sequently another cube of equal size may pass through it, provided 
 c it be made to move in the direction of the diagonal of the former. 
 
 A 
 
 PROBLEM XXX. 
 
 With one sweep of the compasses, and without altering the opening, or changing the 
 
 centre, to describe on oval. 
 
 This problem, as is the case with others of a similar kind, is a mere deception ; 
 for it is not specified on what’ kind of surface the required curve ought to be de- 
 scribed. Those to whom this problem is proposed, will think of a plane surface, 
 and therefore will consider it impossible, as it really is; while indeed the surface 
 meant is a curved one, on which it may be easily performed. 
 
 If a sheet of paper be spread round on a cylindric surface, and if a circle be de- 
 scribed upon it with a pair of compasses, assuming any point whatever as a centre, 
 it is evident that, when the sheet of paper is extended ona plane surface, we shall 
 
 _ havean oval figure, the shortest diameter of which will be in the direction correspond- 
 _ ing to that of the axis of the cylinder. 
 
 We should however be deceived, were we take this curve for the real ellipsis, so 
 
 well known to geometricians. The method of describing the latter is as follows. 
 
 | PROBLEM XXXI. 
 
 | To describe a true oval or ellipsis geometrically. 
 
 . The geometrical oval is a curve with two unequal axes, and having in its greater 
 axis two points so situated, that if lines be drawn to these two 
 points, from each point of the circumference, the sum of these 
 two lines will always be the same. 
 
 Let aB (Fig. 55.) then be the greater axis of the ellipsis to 
 be described ; and let DE, intersecting it at right angles, and 
 dividing it into two equal parts, be the lesser axis, which is also 
 divided into two equal parts in c; from the point D asa 
 
138 GEOMETRY. 
 
 centre, with a radius equal to ac, describe an are of a circle, cutting the 
 greater axis in F and f: these two points are what are called the foci: fix in 
 each of these a pin, or, if you operate on the ground, a very straight peg; then 
 take a thread, or a chord if you mean to describe the figure on the ground, having 
 its two ends tied together, and in length equal to the line a 8, plus the distance F f; 
 place it round the pins or pegs rf; then stretch it as seen at FG f, and witha 
 pencil, or sharp pointed instrument, make it move round from B, through D, a, and 
 FE, till it return again to B: the curve described by the pencil on paper, or on the 
 ground by any sharp instrument, during a whole revolution, will be the curve 
 required. . 
 
 This ellipsis is called the Gardener’s Oval; because when gardeners describe that 
 figure, they use this method. 
 
 It is here seen that the geometric ellipsis, or oval, is, as we may say, a circle with 
 two centres; for in the circle the distance from the centre to any point of the cir- 
 cumference, and from that point back to the centre, is always equal to the same sum, 
 viz. the diameter. In the ellipsis, where there are two centres, the distance from 
 one of them to any point of the circumference, and from that point to the other 
 centre, is always equal to the same sum, or to the greater diameter. 
 
 A circle therefore is nothing else than an ellipsis, the two foci of which, by con- 
 tinually approaching, have at length been united and confounded with each other. 
 
 Another method of describing an ellipsis, which may be also used sometimes, is as 
 follows. 
 
 Let apc (Fig. 56.) be a square, and Bu and BI the two 
 * semi-axes of the ellipsis to be described. Provide a rule, such | 
 
 as E D, equal to the sum of these two lines, and having taken | 
 E F equal to Bi, fix in the point F, by some mechanism which | 
 may be easily invented, a pencil or piece of chalk, capable of 
 tracing out a line upon paper; then make this rule turn in the | 
 given right angle, in such a manner, that its two extremities | 
 shall always touch the sides of that angle, and during this | 
 movement the pencil fixed in F will describe a real geometrical | 
 ellipsis. 
 
 It may be readily seen, that if the pencil or chalk were fixed in the point @, which 
 divides D E into two equal parts, the curve described would be a circle. 
 
 Remark.—Another sort of oval, very much used by architects and engineers, when, | 
 they intend to form a fiat or an acute arch, is called by the French workmen anses | 
 de paniers. It consists of several arcs of circles having different radii, which mu- 
 tually touch each other, and which represent pretty nearly a geometrical ellipsis. — 
 But it has one fault, which is, that however well these arcs touch each other, a nice 
 eye will always observe at the place of junction an inequality, which is the effect of | 
 the sudden transition of one curve to another that is larger. For this reason, any | 
 arch which rises on its pier without an impost, seems to form an inequality, though 
 the arch at its junction with the pier may touch it exactly. 
 
 This inconvenience however is compensated by one advantage, which is, that for | . 
 the voussoirs of the arch, there is no need but of two panneaux, or model boards, if | 
 the quarter of the oval be formed of two ares, or of three if it be formed of threé; | 
 whereas, if it were a real ellipsis, it would have occasion for as many panneaux as I 
 voussoirs. If any one however should have the courage, and it would require no i 
 small degree of it, to surmount this difficulty, we entertain no doubt that the real | 
 ellipsis would have more beauty than this bastard kind of it. 
 
 r 
 
 a 
 
GEOMETRICAL PROBLEMS, 139 
 
 | : PROBLEM XXXII. 
 On a given base, to describe an infinite number of triangles, in which the sum of the 
 : two sides, standing on the base, shall be always the same. 
 
 This is only a corollary to the preceding problem. For on a given base let there 
 be described an ellipsis, having the two extremities of that base as its foci: all the 
 points of the ellipsis will be the summits of as many triangles on the given base FG f, 
 Fg /f, (Fig. 55.), and the sum of their sides will be the same; consequently they will 
 all have the same perimeter, and the greatest triangle will be that which has its two 
 sides equal; for it is that which has the summit at the most elevated part of the 
 ellipsis. 
 
 THEOREM VI. 
 
 Of all the isoperimetric figures, or figures having the same perimeter, and a determi- 
 nate number of sides, the greatest is that which has all its sides and all its angles 
 equal. 
 
 Fig. 57. We shall first demonstrate this theorem in regard to triangles. 
 Let acs (Fig. 57.) then be a triangle on the base a B, the sides 
 of which ac and cB are unequal. We have already shewn, 
 that if there be constructed a triangle a FB, the equal sides of 
 which ar and FB are together equal to a c and cB, the triangle 
 AFB will be greater than acB. 
 
 For the same reason, if there be constructed, on A F asa base, 
 the triangle abr, the sides of which, a b and /F, are equal to 
 each other, and together ‘equal to a B and BF, the triangle a br will be greater than 
 AFB. In like manner, if we suppose Fa and ab equal, and their sum equal toFa 
 and a 8, the latter triangle ra 6 will be still greater than AFB, which has the same 
 perimeter, &c. But it may be readily seen by this operation, that the three sides of 
 a triangle always approximate towards equality, and that, by conceiving it continued 
 ad infinitum, the triangle would at length become equilateral, and consequently the 
 equilateral triangle will be the greatest of all. 
 
 For example, if the three sides of the first triangle be 12, 18, 5, the sides of the 
 second will be 12, 9, 9; those of the third 9, 10}, 104; those of the fourth 10}, 93, 
 93 ; those of the fifth 93, 101, 101; those of the sixth 10}, 9}%, 9}§; those of the 
 ‘seventh 945, 104, 104,, and so on; by which it is seen that the difference always de- 
 creases ; so that at last the three sides become 10, 10, 10, and the triangle will then 
 be the greatest of all. 
 
 If we now take a rectilineal polygon, such as A BCD E F (Fig. 
 58.), all the sides of which are unequal : draw the lines a c, CE, 
 and & A. By what has been already shewn it will be seen, that 
 if an isosceles triangle a 5c be described on A c, in such a 
 manner, that a band 6c shall be together equal to a B and 
 Bc, the polygon, though of the same perimeter, will become 
 greater by the excess of the triangle a b c above aBc. Ifthe 
 same thing be done all around, the surface of thepolygon will be 
 continually augmented; all its sides and its angles will more 
 and more approach to equality ; and consequently the greatest ofall will be that which 
 has all its sides and angles equal. 
 
 We shall now demonstrate, that, of two regular polygons, having the same peri- 
 meter, the greater is that which has the greatest number of sides, For this purpose 
 let any polygon, an equilateral triangle for example, be circumscribed round a circle, 
 and let x ru1(Fig. 59.), be an hexagon circumscribed about the same circle: it is evi- 
 
 dent that the perimeter of the latter will be less than that of the triangle; for the 
 
“a 
 
 140 GEOMETRY. 
 
 Fig. 59. parts F Er, G H, and IK, are common, and the side g F is 
 B less than FB plus B G, &c. ; a hexagon, concentric to the’ 
 former, and equal in perimeter to the triangle, which we 
 
 here suppose to be MN 0, will therefore be without the 
 hexagon K F H; consequently the perpendicular c 7 will | 
 
 be greater thanc Lt. But asthe triangle has the same 
 
 at perimeter as the hexagon m N 0, their-areas will be as 
 
 the perpendiculars c 1, ¢ J, let fall from the centre of the 
 
 fe D circle; and therefore the hexagon, having the same peri- 
 
 eed meter as the triangle, will be the greater. 
 
 What has been demonstrated in regard to a triangle and hexagon of the same peri- 
 meter, is evidently applicable to any other two polygons, one of which: has a number | 
 of sides double to that of the other; consequently the more sides a polygon of a 
 determinate perimeter has, the greater is its area. 
 
 Remarks.— 1st. This leads us to a consequence much celebrated in geometry, 
 which is: that ofall the figures, having the same perimeter, the circle is of the greatest 
 capacity ; fora circle is only a polygon of an infinite number of sides, or, to use a more 
 geometrical expression, is the last of the polygons resulting from their sides being con- 
 tinually doubled ; consequently it is the greatest of all. 
 
 2d. We shail here remark also, that if upon any determinate base, and witha de- 
 terminate perimeter, there be described several figures, the greatest will be that 
 which has the greatest number of sides, beside the base, and which approaches nearest 
 to regularity ; hence it follows, that if it be required tu describe, with a determinate 
 length, ona given base, the greatest figure, that figure will be the segment of a circle, 
 viz. a segment having that base for its chord, and for its arc the given length. 
 
 All these things may be demonstrated by a mechanical consideration. For let us 
 suppose a vessel, the sides of which are flexible, and that any liquor is poured into 
 it ; the sides it is certain will arrange themselves in such a manner as to contain the 
 greatest quantity possible. On the other hand, it is well known that the vessel will 
 assume the cylindric form ; that is to say its Dae and the sections parallel to the base 
 will be circular; hence it follows that, of all figures having the same perimeter, the 
 circle is that which comprehends the greatest area. 
 
 By means of the above observations it will be easy to solve the following questions. 
 
 I,—a has a field 500 poles in circumference, which is square; B has one of the 
 same circumference which is an oblong, and proposes to A an exchange. Ought — 
 the latter to accept the offer ? + | 
 
 It is easy to answer that he ought not ; and a would sustain more loss by the ex- 
 change the greater the inequality is between the sides of the field belonging tos, This 
 inequality might even be such, that the latter field would be only the half, or the 
 fourth, or the tenth part of that of a. For let us suppose the field of a to be 100 
 poles on each side; and that the field of B is a rectangle, one side of which is 190 
 poles, and the other 10, by which means it will have the same perimeter as the other; 
 it will however be found that the surface of the latter will be only 1900 square poles, 
 while that of the former will be 10000. If one side of the field belonging to B were 195 
 poles, and the other 5, which would still make the perimeter 400 poles; its surface. 
 would be only 975 poles, which is not even a tenth part of that of the field belong- 
 ing to a. 
 
 Il.—A farmer borrowed a sack of wheat, measuring 4 feet in length, and 6 feet in cir 
 cumference ; for which he returned two sacks of the same length, and each 3 feet in 
 circumference: did he return the same quantity of wheat? E | 
 
GEOMETRICAL PROBLEMS, 141 
 
 * He returned only half the quantity ; for two equal circles, having the same perimeter, 
 _ taken together, as a third, do not contain the same area; the area of both is only 
 the half of the third, each of them being but a fourth of it. 
 
 IlIl.—4 green-grocer purchased for a certain sum, as many heads of asparagus as 
 could be contained in a string a foot in length; being desirous to purchase double 
 that quantity, he returned next day to the market, with a string of twice the length, 
 and offered to double the price of the former quantity, for as many as it would con- 
 tain. Was his offer reasonable ? 
 
 _ No—the man was in an error to imagine that a string of twice the length would con- 
 en only double the quantity of what he purchased the preceding day; for a circle 
 which has its circumference double to that of another, has its diameter double also. 
 But the area of a circle, the diameter of whichis double to that of another, is equal 
 to four times the area of the other. 
 
 Remark.—It remains for us to observe here that as the circle of all the figures 
 having an ecjual perimeter, is the greatest ; the sphere among the solids is that which 
 contains the greatest volume. Thus, if it were required to make a vessel of a deter- 
 minate capacity, but in such a manner as to save the materials as much as possible, 
 it ought to be in the form ofa sphere. But this will be better illustrated by the fol- 
 lowing problem. 
 
 PROBLEM XXXIII. 
 
 A gentleman wishes to have a silver vessel of a cylindric form, open at the top, capable 
 of containing a cubie foot of liquor ; but being desirous to save the material as much as 
 possible, requests to know the proper dimensions of the vessel. 
 
 If we suppose that the vessel ought to bea line in thickness, for example, it is 
 evident that the quantity of the matter will be proportional to the surface. The 
 question then is: Of all the cylinders, capable of containing a cubie foot; to deter- 
 mine that which shall have the least surface, exclusive of the top. 
 
 It will be found that the diameter of the base ought to be 16 inches 4 lines; 
 and the height 8 inches 2, lines, which is the ratio of nearly 2 to 1 between the 
 diameter and the height. 
 
 If it were required to have the vessel in the form of a cask, close at both ends, the 
 question would be: To find a cylinder which shall have its whole surface, compre- 
 hending the two bases, greater than that of any other of the same capacity. In this 
 case the diameter of the base ought to be 13 inches, and the height 12 inches 58 line. 
 
 PROBLEM XXXIV. 
 On the form in which the Bees construct their Combs. 
 
 The ancients admired bees on account of the hexagonal form of their combs. They 
 
 observed that, of all the regular figures which can be united, without leaving any 
 
 vacuum, the hexagon approaches nearest to the circle, and with the same capacity 
 
 has the least perimeter ; whence they inferred that this animal was endowed with a 
 
 sort of instinct, which made it choose this figure as that which, containing the 
 
 same quantity of honey, would require the least wax to construct the comb; for it 
 
 appears that bees do not prepare wax on its own account, but in order to construct 
 
 their combs destined to be the repositories of their honey, and receptacles for their 
 young. 
 
 This however is far from being the principal wonder in regard to the labour .of 
 bees, if we can give the name of wonder to alabour blindly determined by a peculiar 
 organization; for it may be remarked, in the first place, that it is not absolutely 
 wonderful that small animals, all endowed with the same activity and the same 
 force, pressing outwards, from within, small cells all arranged close to each other, 
 
142 GEOMETRY. 
 
 and all equally flexible, should give them, by a sort of mechanical necessity, a hexa- | 
 gonal form. If we suppose indeed a multitude of circles, or small cylinders, highly 
 flexible and somewhat extensible, close to each other, and that forces acting inter. 
 nally, and all equal, tend to make heir sides approach each other, by filling up. 
 the vacuities left between them, the first form they will assume will be the hexa-' 
 gonal; after which all these forces remaining in equilibrium, nothing will tend oF 
 change that form. 
 
 However, not to deprive the bees of the admiration which they have excited in 
 the above respect, we shall remark that this is not the manner in which they labour. 
 They do not first make circular cells, and then transform them into hexagons by 
 extending them in concert: The cells, which terminate an imperfect comb, are 
 composed of equal planes inclined to each other, nearly in that angle which the hexa-_ 
 gonal form requires. But let us proceed to another singularity, still more wonder- 
 ful, in regard to the labour of bees. 
 
 This singularity consists in the manner in which the bottom of their cells is formed. 
 We must not indeed imagine that they are all uniformly terminated by a plane perpen- 
 dicular to their axes; there is a method of terminating them which employs less wax, 
 and even the least possible, still leaving to the cells the same capacity ; and it is this 
 method which these insects adopt, and which they execute with great precision. 
 
 To execute this disposition, itis necessary, in the first place,that the tworows ofcells, | 
 of which it is well known acomb consists, and which stand back to back, should not be | 
 
 Fig. 60. arranged so as to make their axes correspond, but in such a manner 
 beat’ that the axis of the one may be ina line with the common juncture 
 
 of three posterior. As is seen Fig. 60, where the hexagon de- 
 scribed with black lines corresponds with the three formed of | 
 dotted lines, which represent the plane of the posterior cells; 
 aud it is thus that the cells of bees are arranged, to suit the dis- | 
 position of their common bottoms. | 
 
 In the second place, to give an idea of this disposition, let us | 
 suppose an hexagonal prism, the upper base of which is the héxa- | 
 gona B CDE F (Fig. 61.) with a triangle ax c inscribed in it. | 
 
 Let the axis Pp o be continued to s, and through the point s and | 
 the side a c let a plane pass, which shall cut off from the prism | 
 the angles, so as to forma rhomboidal face a sc 1; such is 
 one of the bottoms of the cel) of a comb: if two other similar | 
 planes be made to pass through s and the sides az and xc, they 
 will form the othertwo; so that the bottom is terminated by a 
 
 triangular pyramid. | 
 
 It may be readily seen, that wherever the point s may be situated, as the pyramid 
 AcosS is always equal to Ac BT, and as the case is the same with the rest, the capa- | 
 city of the cell will not vary, whatever be the inclination of that part of the bottom | 
 turned towards ac. But the case is different with the surface where there is such | 
 an inclination, that the whole surface of the prism and of its bottom will be less | 
 than with any other inclination. It has been found, by the researches of geometri- | 
 cians, that, for this purpose, the angle formed by the bottom with the axis ought to | 
 he 54° 44’; from which there results the smaller angle of the rhombus a Tc or ASG, | 
 equal to 70° 32’, and the other sa T or sc of 109° 28’, 
 
 But this is exactly the inclination of the sides of the parallelogram, formed we 
 each of the three inclined planes of the bottom of the cells of a comb, as appears 
 by the measurement of a great many of these cells. Hence there is reason to con- | 
 clude, that bees construct the bottom of their cells in the most advantageous form, | 
 so as to have the least surface possible, and in such a manner indeed, as can be de- 
 
-GEOQMETRICAL PROBLEMS, 143 
 
 termined only by modern geometry.* Who can have given to these insects, so con- 
 temptible, not in the eyes of the philosopher, who never despises the least of the 
 works of the Deity, but in the eyes of the vulgar, that wonderful instinct, which 
 directs them to perform so perfect a work, but the supreme Geometrician, of whom 
 Plato said, what is verified more and more as we become acquainted with the works 
 of nature, that he does every thing numero, pondere, et mensura. 
 
 PROBLEM XXXV. 
 What is the greatest polygon that can be formed of given lines ? 
 
 It may be demonstrated that the greatest polygon that can be formed with given 
 lines, is that about which a circle can be circumscribed. 
 But it may be still asked, whether there. be any particular order, in regard to 
 
 the sides, capable of giving a greater polygon than any other arrangement. We can 
 
 answer that there is not; and that, whatever be the arrangement, if the polygon can 
 be inscribed in a circle, it will be always the same; for it may be easily demon- 
 
 strated, that whatever be this order, the size of the circle will not vary; the poly- 
 gon will always be composed of the same triangles, having their summits at its 
 centre: the’only difference will be, that they will be differently arranged. 
 
 PROBLEM XXXVI. 
 
 What is the largest triangle that can be inscribed ina circle; and what is the least 
 
 that can be circumscribed about tt ? 
 
 The triangle required in both these cases is the equilateral. 
 The case is the same with the other polygons. The greatest quadrilateral figure 
 
 that can be inscribed in the circle, is the square; this figure also is the least of all 
 those that can be circumscribed about a circle. 
 
 The regular pentagon is likewise the greatest of all the five-sided figures that can 
 be inscribed in the circle; and the same figure is the least of all the pentagons that 
 
 ean be circumscribed about the circle. And so on. 
 
 PROBLEM XXXVII. 
 
 AB (Fig. 62.) is the line of separation between two plains; one of which Ac1B con- 
 sists of soft sand, in which a vigorous horse can scarcely advance at the rate of a 
 league per hour ; the other ABDX ts covered with fine turf, where the same horse, 
 without much fatigue, can proceed at the rate of a league in half an hour ; the two 
 places c and D are given in position, that is to say the distance c A and DB of each 
 from the line of boundary a 8, as well as the position and length of AB, are known ; 
 now tf a traveller has to go from D to c, what route must he pursue, so as to employ 
 the least time possible on his journey ? 
 
 Most people, judging of this question according to common ideas, would imagine 
 that the route to be pursued by the traveller, ought to be the straight line. In this 
 however they would be deceived, as may be easily shewn; for if the straight line 
 
 Fig. 62. cED be drawn, it may be readily conceived that it will be gaining 
 an advantage to perform, in the first plain, where it is difficult to 
 travel, the part of the journey c F, which is somewhat shorter than 
 cE; and to perform in the second, where it is much easier to 
 travel, the part FD, longer than D x, that is to say, than the space 
 which would be passed over by going directly from c to D; so that 
 less time would really be employed to go from c to d, byc rand Fp, 
 than by cz and ED, though the road by the latter is shorter. 
 
 * The Abbe Delisle says improperly, in the notes to the fourth book of his Translation of the 
 Georgics, that Reaumer, having proposed this problem to Koenig, the latter, after a great many 
 calculations, at length found the angie of the inclination of the planes which form the bottom cf 
 these cells. Nothing however is easier than the solution of this problem by means ef fluxions; 
 two lives of calculation are sufficient, and a solution may even be given without that assistance. 
 
144 GEOMETRY. 
 
 This indeed may be demonstrated by calculation. For if ua be drawn per 
 pendicular to AB, through the point F, it will be found that one can go fron) 
 c to D, in the least time possible, when the sines of the angles crc and pF H an 
 to each other respectively in the inverse ratio of the velocity with which the tra! 
 veller can pass over the planes ACIBand A BDK, that is to say, in the present case| 
 as 1 to 2; and therefore the sine of the angle crc, ought to be half only of that o| 
 
 the angle DF H. | 
 
 PROBLEM XXXVIII. 
 
 Ona given base to describe an infinite number of triangles, in such a manner, that the sun 
 of the squares of the sides shall be constantly the same, and equal to a given square. 
 
 Let aB (Fig. 63 and 64.) be thi 
 given base, which must be divide 
 into two equal parts inc; ther 
 from the points a and B, with | 
 radius equal to half the diagona| 
 of the given square, describe a1| 
 isosceles triangle, of which F is thi 
 vertex; draw Cr, and from the point c, with the radius c F, describe a semicircle o1| 
 A B, produced if necessary : all the triangles having a B for their base, and whose ver. 
 tices are at F, f, ¢, in the circumference of the circle, will be of such a nature, tha 
 the sum of the squares of their sides will be equal to the square given. | 
 
 Remark.—Every one knows that when the sum of the squares of the sides is equa. 
 to the square of the base, the triangle is right-angled, and has its vertex in the cir 
 cumference of the circle described on that base. Here it is seen, that if the sum o 
 the squares of the sides is greater or less than the square of the base, the vertices 0 
 the triangles, which in this first case are acute-angled, and in the second obtuanl| 
 angled, are always in a semicircle also, having the same centre, but on a diametei 
 greater or less than the base of the triangle; whichis a very ingenious generalizatior 
 of the well known property of the right angled triangle. | 
 
 PROBLEM XXXIX. 
 
 On a given base, to describe an infinite number of triangles, in such a manner, that the 
 ratio of the two sides, on that base, shall be constantly the same. 
 
 Divide the given base a 3 (Fig. 65.) in such a mannei 
 Fig. 65. in p, that a D may be to p B, in the given ratio, which we 
 shall here suppose to be as 2 to 1. Then say, as the 
 difference between ap and pBis toDB, so is A Bto BB; 
 z andif aD exceeds pz, BE must be taken in the directior 
 A Db BEC Ea BE; then divide p & into two equal parts in c, and from 
 c.as a centre, with the radius cp or cE, describe a semi- 
 circle on the diameter D E: all the triangles, as AFB, Af B, A $B, &c., having the 
 same base 4 B, aud their verticesr, f, $, in the circumference of this semicircle, will 
 be of such a nature, that their sides aF, FB; Af, fB; A, $B, will be in the same 
 ratio, viz., that of ap to DB, orof AEto EB, which is the same thing. 
 
 But the centre c will be found much easier by the following construction: on A D 
 describe the equilateral triangle aq@p, and on DB, the equilateral triangle DHB; 
 through their summits, G and u, draw a straight line, which being continued will 
 cut the continuation of a B in the point c, and this point will be the centre required, 
 
 THEOREM VII. 
 
 In a circle, if two chords, as ap and cp (Fig. 66. ) intersect each other at right 
 angles ; the sum of the squares of their segments, CE, AE, ED, and&E B, will always 
 be equal to the square of the diameter. 
 
GEOMETRICAL PROBLEMS. 145 
 
 The demonstration of this curious and elegant theorem, is exceedingly easy ; for 
 
 it may be readily seen, if the lines BD and ac be drawn, that their two squares are 
 
 together equal to the squares of the four segments in question. 
 
 Fig. 66. Moreover, by making the arc Fc equal to aD, we shall have the 
 
 arc FD equal to ac, and consequently the angle F p c equal to 
 
 ACE, which is itself equal to aBD; the angle rpB therefore 
 
 will be a right angle, since it is equal to & p B and DBE, which 
 
 together make aright angle; hence the squares of rp and DB 
 
 are equal to the square of the hypothenuse r B, which is the 
 diameter. 
 
 It must here be remarked, that the result would be the same, 
 if we suppose the point e, where the chords meet, to be without the circle; in that 
 case the four squares, viz. those of e a, e b, ec, and e d, would still be together equal 
 to the square of the diameter. 
 
 Remark.—Circles being to each other as the squares of their diameters; it is evi- 
 dent that if on EA, EB, EC, and ED, as diameters, four circles be described, these 
 circles will be together equal to the circleacsp. And they will also be propor- 
 tional; for we know that BE istoECasEDistoEA. But if four magnitudes are 
 proportional, their squares are so also. Moreover, it is evident that whatever be the 
 position of these two chords, their sum will always be equal, at the most, to two 
 diameters if they both pass through the centre; or at least to one, if one of them 
 passes through the centre, and the other almost at the distance of a radius. By 
 means of this theorem, therefore, it will be easy to solve the following problem. 
 
 PROBLEM XL. 
 
 To find four proportional circles, which taken together shall be equal to a given circle, 
 and which shall be of such a nature, that the sum of their diameters shall be equal 
 to a given line. 
 
 It is evident, for the above reasons, that the given line must be less than twice the 
 diameter of the given circle, and greater than once that diameter, or, which is the 
 same thing, that the half of this line must be less than the diameter of the given 
 circle, and greater than its radius. 
 
 ° This being premised; let the given line, the sum 
 of the diameters of the required circles, be a6 
 (Fig. 67.) the half of whichis ac; let ap BE be the 
 given circle, the two diameters of which are aB and 
 DE, perpendicular to each other. On the radiica 
 and cE continued, make the lines cr and ca equal 
 to ac, and draw Fa, which will necessarily intersect 
 cH, the square of the radius of the circle. In the 
 part 1K of that line comprehended within the square, 
 assume any point L, from which draw the lines q, 
 anduw 7, the one parallel and the other perpendicular 
 | to the diameter a B; through the points mand Nn, where 
 they intersect the circumference of the circle, draw MR and N Q, the one parallel and 
 the other perpendicular to a B: the chords Ns and mT will be the two chords required. 
 
 For it is evident that N@ and mR are equal to Lq, and ur, which are together equal 
 to cc or cr, or to the half ofa; the whole chords then are together equal to ab; 
 consequently, by the preceding theorem, they solve the problem, and the four 
 circles described on the diameters NO, OM, 08, and oT, will be equal to the circle 
 ADBE. 
 
 Remark.—The line rc may happen only to touch the circle; in which case any 
 point, except the point of contact, will equally solve the problem 
 
 L 
 
146 GEOMETRY. 
 
 But ifre intersect the circle, as seen Fig. 68, the point 1 
 must be assumed in that part of the line 1K, which is with. 
 out the circle, as seen in the same figure. 
 
 This solution is much better than that given by M. Oza, 
 nam ; for he tells us to take on ae (Fig. 67.) a portion less thar 
 the radius, and to set it off from cto q: then to draw the 
 lines gm and wR, and to set off the remainder of ac from ¢ 
 tor; but it is necessary that the point 7 should fall beyond R 
 otherwise the two semi-chords would not intersect each | 
 In the last place, according to the magnitude of ac, in regard to the radius 
 there is a certain magnitude which must not be exceeded, and which M. Ozanan: 
 does not determine: this therefore renders the solution defective. 
 
 | 
 : 
 
 PROBLEM XLI. 
 Of the Trisection and Multisection of an Angle. 
 
 This problem is celebrated on account of the fruitless attempts made, from tim) 
 to time, to resolve it geometrically, by the help of arule and compasses, and of thi! 
 paralogisms and false constructions given by pretended geometricians. But it 1) 
 now demonstrated, that the solution of it depends on a geomeiry superior to th 
 elementary, and that it cannot be effected by any construction in which a rule ani 
 compasses only, or the circle and straight line, are employed, except in a very fey, 
 cases ; such as those where the are which measures the proposed angle is a whol| 
 circle, or a half, a fourth, or a fifth part of one. None therefore but people ignoran| 
 of the mathematics attempt at present to solve this problem by the common geometry | 
 
 But though it cannot be solved by the rule and compasses alone, without repeate | 
 trials, there are some mechanical constructions or methods, which, on account 0) 
 their simplicity, deserve to be known. They areas follow: 
 
 Let it be ‘proposed, for example, to divide a 
 
 Fig. 69. angle ABC (Fig. 69.) into three equal parts. Fror 
 
 the point a let fall, on the other side of the angle) 
 the perpendicular ac, and through the same point ,| 
 draw the indefinite straight line az parallel to BC. 
 if from the point B you then draw to AE aline BE! 
 in such a manner, that the part FE, intercepted be| 
 tween the lines acand ak, shall be equal to twice the line a B, which may b! 
 done very easily by repeated trial, you will have the angle rBc equal to the thi 
 part of aBc. 
 
 IffE indeed be divided into two equal parts in p, and if Ap be drawn; as th. 
 triangle Fax is right-angled, p will be the centre ef the circle passing throug’ 
 the points Fr, a, B; consequently pa, p£, and pF willbe equal to each other, and t, 
 the line a 3B; the triangle A p& then will be isosceles, and the angles pa¥ and DE } 
 will be equal; the external angle ADF, which is equal to the two interior one} 
 DAE and DEA, will therefore be the double of each of them. But as the triang) 
 BAD is isosceles, the angle a BDis equal to ApB, and the angle arp, or its equi! 
 FBC, is half of the angle aBD; consequently the angle a Bc is divided by Bg, } 
 such a manner, that the anglers c is the third part of it. 
 
 Another method.—Let the. given angle be Ac) 
 
 Fig. 70. (Fig. 70.): from the vertex of it as a centre, dé 
 
 A scribe a circle, and continue the radius B c indefinite) 
 ee to x; then draw theline arin such a manner, thi’ 
 
 i ~ the part px, intercepted between BE and the circun 
 B é Wee fe ference of the circle, shall be equal to the radius BC 
 
 ifou be then drawn through the centre c, varallel 4 
 A 5, the angle 8 cu will be the third part of the given angle acB. 
 
GEOMETRICAL PROBLEMS. 5 147 
 
 If the radius cp be drawn, it may be readily seen that the angle Hoa is equal, 
 on account of the parallel lines, toca Dorcpa. But the latter is equal to the 
 angles DCE andpDEC, or to the double of one of them; since cp and DE are equal 
 by construction; and as the angle HcBis equal to Dcx& or DEC, the angle ACH is 
 the double of H cB, and consequently a cBis the triple of HcB. 
 
 PROBLEM XLII. 
 The Duplication of the Cube. 
 
 To double a rectilineal surface, or any curve whatever, as the circle, square, 
 triangle, &c., is easy; that is to say, one of these figures being given, it is easy to~ 
 construct a similar one, which shall be the double or any multiple of it whatever, or 
 which shall be in any given ratio to it at pleasure: nothing is necessary for this pur- * 
 pose, but to find the mean geometrical proportional between one of the sides of the 
 given figure, and the line which is to that side in the given ratio, this mean will be 
 the side homologous to that of the given figure. Thus, to describe a circle double 
 of another, a mean proportional must be found between the diameter of the former 
 and the double of that diameter; this proportional will be the diameter of the double 
 circle, &c. The case is the same with every other ratio. 
 
 All this belongs to the elements of geometry. But to construct a double solid 
 figure, or a figure in a given ratio to another similar figure, isa much more difficult 
 problem, which cannot be solved by means of the circle and straight line, or of the 
 rule and compasses, unless a method of repeated trial, which geometry rejects, be 
 employed. This at present is clearly demonstrated; but the demonstration is not 
 susceptible of being comprehended by every one. 
 
 Respecting the origin of this problem, a very curious circumstance is related. 
 During the plague at Athens, which made a dreadful havoc in that city, some persons 
 being sent to Delphos to consult Apollo, the deity promised to put an end to the 
 destructive scourge, when an altar, double to that which had been erected to him, 
 should be constructed. The artists who were immediately dispatched to double the 
 altar, thought they had nothing to do, in order to comply with the demand of the 
 oiacle, but to double its dimensions. By these means it was made octuple; but the 
 god, being a better geometrician, wanted it only double. As the plague still con- 
 tinued, the Athenians dispatched new deputies, who received for answer, that the 
 altar was more than double. It was then thought proper to have recourse to the 
 geometricians, who endeavoured to find out a solution of the problem. There is 
 reason to think that the god was satisfied with an approximation, or mechanical solu- 
 tion; had he required more, the situation of the people of Athens would have 
 deserved pity indeed. 
 
 There was no necessity for introducing a deity into this business. ~What is more 
 natural-to geometricians than to try to double a solid, and the cube in particular, 
 after having found the method of doubling the square and other surfaces? This is 
 the progress of the human mind in geometry. 
 
 Geometricians soon observed that, as the duplication of any surface consists: in 
 finding a geometrical mean between two lines, one of which is the double of the 
 other, the duplication of the cube, or of any solid whatever, consists in finding the 
 first of two continued mean proportionals between the same lines. We are indebted 
 for this remark to Hippocrates of Chios, who from being a wine merchant, ruined by 
 shipwreck or the officers of the excise at Athens, became a geometrician. Since that 
 time, all the efforts of geometricians have been confined merely to the finding of two 
 continued geometrical mean proportionals between two given lines, and these two 
 problems, viz., that of the duplication of the cube, or, more generally, of the con- 
 struction of a cube in a given ratio to another, and that of the two continued mean 
 proportionals, have become synonymous. 
 
 Lie 
 
 ie 
 “3 
 
148 
 
 GEOMETRY. 
 
 The different methods of solving this problem, some of which require repeated 
 trial, and some no other instruments than a rule and compasses, are as follow: 
 
 Fig. 71. 
 
 s 
 
 ‘Be. 
 “See evee™ 
 
 > 
 ! 
 is 
 a 
 
 Ist. Let the two lines, between which it is required to find 
 two mean proportionals, be aB and ac (Fig. 71.) Form a 
 them the rectangle B ac pb, and continue the sides 4B and ac’ 
 indefinitely; draw the two diagonals of the rectangle inter. 
 secting each other in E; and we shall then have the solution: 
 of the problem, if the line rpG terminated by the sides of 
 the right angle F Ac, be drawn through the point D, in such a 
 manner, that the points G and F shall be equally distant from 
 the point £; for in that case the lines aB, cG, BF, and AC, 
 
 will be continued in proportion. | 
 
 Or, with £ as a centre, describe an arc of a circle, as FIG, in such a manner, that 
 by drawing the line Fa, it shall pass through the point p: we shall then havea 
 solution of the problem. , | 
 
 Another method is as follows: Circumscribe a circle about the rectangle BAcD; 
 
 then through the point p, draw the line F a, in such a manner, that the segments FD. 
 and GH shall be equal: the lines ca and pF will be continued mean vroportionals'| 
 
 between AB and ac. 
 
 Cc B 
 
 2d. Form aright angle of the two given lines aB and BO 
 (Fig. 72.); and having continued gc and 4B indefinitely, 
 from the point B asa centre, describe the semicircle DEA; 
 draw also the line a c, and in the continuation of it find a point! 
 Gc of such a nature, that by drawing the line peut, the 
 segments GH and HX shall be equal to each other: the line 
 BH will be the first of the two means. ; 
 _ 3d. Let ca (Fig, 73.) be the first of the given lines: 
 from the point c, with the radius cB, equal to the half 
 
 of c A, describe a circle, and in this circle make the chord 
 BD equal to the second of the given lines, which must 
 
 be continued indefinitely ; draw the indefinite line a DE, 
 and from the point c draw the line c EF, in such a man 
 ner, that the part EF, intercepted with the angle DF, 
 shall be equal to cB; the line pF will then be the first! 
 
 of the required mean proportionals, and cE will be the’ 
 
 A 
 
 -gecond. This construction is that of Sir Isaac Newton. : | 
 
 PROBLEM XLIII. 
 
 An angle, which is not an exact portion of the circumference, being given, to find us 
 value with great accuracy, by neans of a pair of compasses only. | 
 
 From the vertex of the given angle, with as great a radius as possible, describe € 
 circle, and mark its principal points of division, as the half, third, fourth, fifth, sixth 
 eighth, twelfth, and fifteenth parts of the circumference; then by means of the | 
 compasses take the chord of the given arc, and set it off along the circumference 
 from a determinate point, going round it once, twice, thrice, &c.; and counting the 
 number of times that the chord is applied to the circumference, until you fall exactly, 
 on one of the points of division, which cannot fail to be the case after a certail 
 number of revolutions, unless the given arc be incommensurable to the circumfer: 
 ence; then examine what the point of division is, or how many and what aliquo’ 
 parts of the circumference it is distant from the first point ; add the number of de. 
 grees which it gives to the product of 360 degrees multiplied by the complete num: 
 ber of turns made with the compasses, and divide the sum by the number of time: 
 
 - 
 
GEOMETRICAL PROBLEMS. \ 149 
 
 ‘that the compasses were applied to the circumference: the quotient will be the 
 number of degrees, minutes, and seconds, required. 
 Let us suppose, for example, that the compasses, with an opening equal to the 
 chord of the given arc, have been applied to the circumference seventeen times, and 
 that after four complete revolutions they have coincided exactly on the second divi- 
 sion of the circle divided into five equal parts. The fifth part of the circumference 
 is 72°, and two fifths are 144°; if 144 then be added to the product of 360° by 4, 
 which is the number of the complete revolutions, and if the sum 1584° be divided 
 ; by 17, the quotient 93° 10’ 35” will be the value of the required are. 
 
 PROBLEM XLIV. 
 
 A straight line being given; to find, by an easy operation, and without a scale, toa 
 thousandth, ten thousandth, hundred thousandth, §c. part, nearly, its proportion to 
 another. 
 
 Let the first or least of these lines be called a, and second B. 
 
 Take with a pair of compasses the extent of the line a, and set it off as many 
 times as possible on B: we shall here suppose that a is contained in the latter three 
 times, with a remainder. 
 
 Take this remainder in the compasses, and set it off, in like manner, on the line z, 
 as often as possible: we shall suppose that it is contained in it seven times, with a 
 remainder. 
 
 Take the second remainder, and perform the same operation on the line B, in which 
 we shall suppose it¢o be contained 13 times, with a remainder ; and, in the last place, 
 let us suppose that this third remainder is contained in B exactly 24 times. 
 
 Then form the following series of fractions ; 3, 314, 3-7!tg. g-n-Pgaq. and reduce them 
 to decimal fractions, which will be 0°333333, 0:047619, 0°003663, 0°000152. The 
 given line is in decimals equal to the first of these fractions, minus the second, plus 
 the third, minus the fourth, which gives 0°289225, without the error of one of these 
 parts entirely, that is to say of a millionth part. 
 
 It may be easily seen that no scale, however small the divisions, could give so ap- 
 proximate a ratio; and even if we suppose such a scale to exist, there would still 
 remain an uncertainty in regard to the division on which the extremity of the given 
 line would fall ; whereas, a line applied with the compasses along a greater one, can 
 never leave any uncertainty in regard to the number of times it is contained in it, with 
 or without a remainder. 
 
 If the above fractions be added in the usual manner, we shall find that the given 
 line is equal to 4895 of the second. 
 
 PROBLEM XLV. 
 
 To make the same body pass through a square hole, a round hole, and an 
 elliptical hole. 
 
 We give a place to this pretended problem, merely because it is found in all the 
 Mathematical Recreations hitherto published; for nothing is easier to those who are 
 in the least acquainted with the simplest geometrical bodies. 
 
 Provide a right cylinder, and suppose it to be cut through its axis; this section 
 will be a square or a rectangle; if cut through a plane perpendicular to the axis, the 
 section will be a circle; and if cut obliquely to that axis, the section will be an 
 ellipsis. Consequently, if three holes, the first equal to this rectangle, the second 
 to the circle, and the third to the ellipsis, be cut in a piece of wood or pasteboard, 
 it is evident that the cylinder may be made to pass through the first of these holes, 
 by moving it in a direction perpendicular toits axis ; it will also pass through the cir- 
 cular hole when moved in the direction of its axis; and through the elliptical, hole, 
 when held with the proper degree of obliquity ; in all these cases it will exactly touch 
 
fr 7 4 ‘ rae tat iar ie a = cal , | 
 j “ : ~ ae safle po vag) 
 ‘ , ball Bl 
 . : 4 
 ; : fi & 
 ba? | 
 
 150 ii "GEOMETRY. 
 
 the edges of the hole, so that if the hole were smaller it would be impossible to mak 
 
 it pass through it. 
 This problem might be solved $F means of other bodies; but it is so simple the 
 
 nothing farther needs be said on the subject. 
 
 PROBLEM XLVI. 
 
 To measure the circle, that is to say, to find a rectilineal space equal to the circle, ¢ 
 more generally, to find a straight line equal to the circumference of the circle, ori 
 a given arc of that circumference. 
 
 We are far from pretending to give an exact and perfect solution of this problem)! 
 it is more than probable that it will ever baffie the efforts of the human mind; bu} 
 it is allowed in geometry, that when a problem cannot be completely solved, it} 
 some merit to approach near to it, and the more so when the unknown quantity 1) 
 circumscribed within the nearest limits. But though geometricians despair of eve. 
 being able to find the exact measure of the circle, they have accomplished thing | 
 highly worthy of notice; for they have found means to approach so near to it tha| 
 even if the radius of a circle were equal to the distance between the sun and the firs| 
 of the fixed stars, it is certain that its circumference might be found from the radius) 
 without the error of a hair’s breadth. This is doubtless more than sufficient t| 
 answer the nicest purposes in the arts; but it must be allowed that it would giv| 
 great pleasure to a geometrical genius, to be able to tell exactly the measure of th’ 
 circle; thatis to say, toknowit with the same precision that we know, for example 
 that a parabolic segment is equal to two thirds ofa parallelogram baving the same bast 
 and the same altitude. 
 
 1.—The diameter of a circle being given; to find, in approximate numbers, the crew 
 ference ; or vice versa. 
 
 When moderate exactness only is required, we may employ the proportion o 
 Archimedes, who has demonstrated that the diameter is to the circumference nearly} 
 as | to 3}, or as 7 to 22, | 
 
 If we therefore make this proportion: as 7 is to 22, so isa given diameter to ¢| 
 fourth term; or if we triple the diameter and add to it a seventh, we shall have the 
 Seencnferente very nearly. 
 
 The circumference of acircle, the diameter of whichis equal to 100 feet, will ‘be foun: 
 therefore to be 314 feet 3 inches 5} lines: the error in this case isabout 1 inch 6 lines | 
 
 If we are desirous of approaching still nearer to the truth, we must employ the. 
 proportion of Metius, which is that of 113 to 355: we must therefore say as 113 te | 
 355, so is the given diameter to the required circumference. The same diameter ag 
 before being supposed, we shall find the circumference to be 314 feet, 1 inch, Loy 
 lines; the difference between which and the real circumference is less than a line. | 
 
 If still greater exactness be required, we have only to employ the proportion oj) 
 10000000000 to 31415926535 ; the error in this case, if the circumference were a great| 
 circle, such as the equator of the darth, would be, at most, half a line. 
 
 To find the diameter, the circumference being given, it is evident that the in- 
 verse proportion must be employed. We must therefore say as 22 is to 7, or as 35 
 to 113, or as 314159 is to 100000, or as 31415926535 to 10000000000, so is the given 
 circumference to a fourth term, which will be the diameter required. 
 
 Il.—The diameter of a circle being given; to find the area. 
 
 Archimedes has demonstrated that a circle is equal to the rectangle of half the 
 radius by the circumference. Find therefore the circumference, by the preceding 
 paragraph, and multiply it by half the radius, or the fourth part of the diameter: the 
 product will be the area of the circle, and the more exact the nearer to the truth the 
 circumference has been found. 
 
GEOMETRICAL PROBLEMS. 161 
 
 _ By employing the proportion of Archimedes, the error, in a circle of 100 feet diame- 
 Ey, will be about 3} square feet. 
 
 That of Metius : ae give an error less than of square inches, or about a sixth of 
 a square foot. As the circle in question would contain about 7854 square feet, the 
 error, at most, would be only one 47124th part of the whole area. 
 
 But the area of a circle may be found, without determining the circumference ; 
 for it follows, from the proportion of Archimedes, that the square of the diameter 13s 
 to the area, as 14 to 11; from that of Metius, that it is as 452 to 355; from the pro- 
 portion of 100000 to 314159, thatis as 100000 to 78539, or with still greater exact- 
 ness as 1000000 to 785398. 
 
 The area of the circle therefore will be found by making this proportion, as 14 is 
 to 11, or as452 to 355, oras 1000000 is to 785398, so is the square of the given diame. 
 
 ter, to a fourth proportional, which, if the Jast proportion has been employed, will be 
 very near the truth. 
 
 Ill — Geometrical constructions for making a square very nearly equal to a given circle, 
 or a straight line equal toa given circular circumference. 
 
 Having shewn some methods for finding numerically, and very near the truth, the 
 proportion between a circle and the square of its diameter, we shall now give some 
 geometrical constructions, exceedingly simple and ingenious, for aceomplishing the 
 game object. 
 
 Fig. 74. Ist. Let 3 anc (Fig. 74.) be a circle, of which ac is the dia- 
 - meter, and ABa quadrant; let Az,ED, and pe be chords equal 
 
 to the radius ; from the point B, draw to the points & and p, the 
 i lines BE and BD, intersecting the diameter in F and q@; the sum 
 c a of the lines Br and re will be equal to the quadrant of the 
 
 circle, within a five thousandth part. 
 
 2d. Let ap (Fig. 75.) be the diameter 
 of the circle, c thecentre, and cB the radius 
 perpendicular to that diameter. In ap 
 continued, make DE equal to the radius; 
 then draw BE, and in AE continued make 
 EF equal to it; if to this line EF, its fifth 
 part FG be added, the whole line a g& will 
 be equal nearly within a 17000th part to the 
 circumference described withthe radius c A, 
 For if p A be supposed equal to 100000, a c will be found equal to 314158, 
 
 with less than an unit of error: but the circumference corresponding to this diameter 
 
 is, with the difference of nearly an unit, 314159; the error therefore at most is qyyq5q Of 
 the diameter, or about the 17000th part. 
 3d. If the semicircle a Brc (Fig. 76.) be given; from the 
 
 Fig, 76. extremities a and c of its diameter, raise two perpendiculars, 
 CG one of them c x equal to the tangent of 30°, and the other ae 
 equal to three times the radius; if the line GE be then drawn, 
 it will be equal to the semi-circumference of the circle, within 
 ‘a hundred thousandth part nearly. 
 
 For it will be found by this construction, the radius being 
 supposed to be 100000, that the line eG, within a unit nearly, 
 is equal to 314162, and tne semi-circumference would be, with 
 the difference of nearly an unit, 314159; the error therefore is 
 about y2;55 of the radius, .or less than a hundred thousandth 
 part of the circumference. 
 
i >) oe 4 
 ‘ny { 
 ” 
 
 3 | a | 
 152 GEOMETRY. 
 
 4th. Let a (Fig. 77.) be the centre ot the giver 
 circle, and DE and cB-its two diameters, perpendicula; 
 to each other. On any radius, such as aD, make aj 
 equal to half the side Ec of the inscribed square; dray 
 BFI indefinitely, and to the point u, draw Fu dividing 
 Ac in extreme and mean ratio, AH being the lesse; 
 segment; if cx be drawn parallel to ru through th 
 point c, the square BLKI, constructed on B14, will b 
 nearly equal to the circle of which Bc is the dia 
 meter. 
 
 For it will be found by calculation, that Bu and BF are respectively equal t¢ 
 69098 and 61237, the radius being 100000; 21 therefore will be found equal te 
 88623, the square of which is 78540, &c., the square of the diameter being 100000 
 &c., while the circle is 78539, &c. 
 
 5th. Inscribe in the given circle a square, and to three times the diameter add: 
 fifth part of the side of the square; the result will be a line which will differ fro 
 the circumference by about a 17000th part only. } 
 
 IV.—Several methods for making, either numerically or geometrically, and very near the 
 truth, a straight line equal to the given are of a circle. | 
 
 Ist. Let the given arc, which ought never to exceet| 
 ay Fig. 78. 30°, be na (Fig.78). To obtain the length of it very| 
 nearly in a straight line, draw Bu perpendicular to the 
 ‘diameter aB, and continue the diameter to p, so that! 
 AD shall be equal to the radius; if DE be then drawn, 
 it will cut off from Bu the line BE somewhat less, but| 
 very nearly equal to the arc Ba. “ST 
 But if the line d fg e, be drawn in such a manner, that the segment d f, intercepte¢ 
 between the circle and the diameter continued, shall be equal to the radius, the 
 straight line Be will then be somewhat greater than the arc Bg; but very near it 
 if the are does not exceed 30 degrees. 
 _For this theorem we are indebted to Snellius; but it was first demonstrated by| 
 Huygens. We shall shew hereafter, that it is very useful in trigonometry. 
 2d. It has been demonstrated also by Huygens, that twice the chord of half an 
 are, plus the third of the difference between that sum and the chord of the whole 
 arc, is nearly equal to the arc itself, when it does not exceed 30°. 
 
 For if we suppose the arc to be 30°, the chord will be 25882 parts, the diameter! 
 being 100000; that of half the same arc, or of 15°, will 13053, the double of whick 
 is 26106; if from this we subtract 25882, the difference will be 224, the third o}| 
 which, 742, added to 26106, will give 261802 for the arc of 30°. Twelve times this | 
 arc ought to give the whole circumference ; but 261802 multiplied by 12, is equal tc. 
 314168, and the circumference is 314159, the difference therefore is only the nine 
 hundred thousandth part of the radius. | 
 
 It being remembered that a circle is a polygon, whose sides are indefinitely small 
 and infinite in number, the following is a simple method of arriving approximately 
 
 at the ratio of the circumference to the diameter. 
 Fig. 79. Let aB be the semi-side of a regular polygon, c its centre; along Ac 
 m produced take cD=cpB; then the isosceles triangle Bc pv will have the 
 angle D = 3 8CA, and the perpendiculars cl and rr on BD and ap re. 
 I spectively, will give the middle points 1 and F of those lines. Since ther 
 IF =}3BA, and p= toi B8C4, IF is the semi-side of a regular polygor 
 B of the same perimeter as the first, but having twice the number o! 
 sides. 
 
: GEOMETRICAL PROBLEMS. 153 
 ; x 
 
 ' Calling the radii of the circumscribed and inscribed circles of the first polygon, 
 iz. Be and CA, R, 7, and those of the’second, pt and DF, R’,7”, we have 
 prF=4(pe-+ca) = 3 (wc+ca),or’ =} (R+7); 
 
 md by the rectangular triangle crp, we have 
 
 DI=a/ DF X DC, OTR = / RY 
 
 £ the four radii, r, 7, B’,7’, therefore, the first two being known, the last {wo may 
 ye deduced from them. 
 
 r, B3 r’, R’; r”, R’; ol Ble: “fl 
 in which cach term r is half the sum, and each term R the square root of the product 
 of the two terms which precede it. 
 Now in a hexagon the side is equal to the radius of the circumscribing circle; if 
 then we call this radius 1, the perimeter of the figure will be 6, andc a, the radius 
 
 of the inscribed circle, = ,/ BC? — BA? =»/ 2 = 0°866025. And from these 
 values of R and r we deduce successively the following results. 
 
 r = 0'866025 | 0°949469| 0°954588 | 0:954908 
 R = 1000000 | 0°957662 | 0°955100 | 0°954940 
 7 = 0'933013 | 0°963566 | 0°954844* | 0°954924 
 R = 0°965925 | 0°95561 . | 0°954972 | 0°954932, &c. 
 
 Here r, 7’, r”.... are readily found, and the calculations of R, R’, R”.... are very 
 rapid by logarithms; and moreover when r and R agree in the first half of the 
 figures, as at the place we have marked with an asterisk, R as well as 7 may be found 
 by taking half the sum of the two preceding terms. 
 
 We arrive at last to R = 7 = ‘954929; or the radius of the circumscribing = ra- 
 dius of the inscribed circle, therefore these circles coincide with each other, and with 
 the polygon which lies between them. Then “954929 is the radius of a circle whose 
 perimeter is 6, and the proportion 
 
 . 2 x °954929 or 1:909858 ; 1:36 ; 3°14159 
 gives 3'14159 for the circumference of a circle whose diameter is1. By this methed 
 we can carry the approximation to any point we wish. 
 
 In the Mathematical Tracts of Dr. Charles Hutton, several series are investigated 
 for computing the circumference of a circle from its diameter. 
 
 The following is better adapted to computation than any other that has yet been 
 discovered. 
 
 Let a be an arc of 45°, to radius unity. Then 
 
 f 4 4 8a 126 16 y 
 oO Ie Broek alia e707 4 
 
 9°10 
 Bal nck 4 Sree minl 2 Asc 1G y 
 |-50 * [1+ 3-400 + 5-100 + 7-100 + 9100 + &* J 
 
 Where a, @, y, &c. denote the preceding terms in each series. 
 
 Remark.—As we promised to give a short account of the different attempts made 
 respecting the quadrature of the circle, we shall here discharge our promise. What 
 we are going to say on the subject, is only an abstract from a very curious work, 
 published by Jombert in 1754.* ; 
 
 It will first be proper to divide those who have employed themselves on this 
 problem, into two classes. The first, consisting of able geometricians, were not led 
 away by illusions. Being aware of the difficulty or impossibility of the problem, 
 
 * The author of that curious little work was Mcntucla himself.—Note by Dr. Hutton. 
 
154 GEOMETRY. 
 
 they confined themselves merely to the finding out methods of approximation mot 
 and more exact ; and their researches have often terminated in discoveries in almo; 
 every part of geometry. , 
 The other class consists of those who, though scarcely acquainted with the el¢ 
 ments of geometry, and scarcely knowing on what principles the problem depend 
 have made every effort to solveit, by accumulating paralogismson paralogisms. Lik} 
 the unfortunate Ixion, condemned to roll up a heavy burden eternally without bein 
 able to bring it to the place of its destination, we find them twisting and turning th 
 circle in every direction, without advancing one step further. When a geometricia| 
 has convinced them of an error in their pretended demonstrations, we see thei 
 returning a few days after, with the same demonstration in a new form, but equall 
 contemptible. Very often they do not hesitate to contest the best established truth | 
 in the elements of geometry; and, in general, sensible of the weakness of the) 
 knowledge in this department of science, they consider themselves as speciall 
 illuminated by Heaven to reveal truths to mankind, the discovery of which it he| 
 withheld from the learned, in order to confer the honour of it on idiots. Such: 
 the ridiculous but real picture of this sort of men. It may be readily conceived thé 
 in the shert history we are about to give of the quadrature of the circle, we sha) 
 _not be so unjust towards the eminent geometricians, as to couple them with suc) 
 visionaries. The singular flights of the latter will only furnish us, towards the en 
 of this article, with matter for an amusing addition to it. 
 Geometry had scarcely been introduced among the Greeks, when the quadratur 
 or measure of the circle began to give employment to all gers who possessed | 
 mathematical genius. Anaxagoras, it is said, exercised himself upon it while i] 
 prison; but with what success we are not informed. 
 The question had been already become celebrated in the time of Aristophane}: 
 and perhaps had made some geometrician lose his senses; for in order to ridicul 
 the celebrated Meto, that comic writer introduces him on the stage, promising t| 
 square the circle. | 
 Hippocrates of Chios certainly made it an object of his research: for it could }| 
 only by endeavouring to square the circle that he discovered his famous lunuley’ 
 Some even ascribe to him a certain combination of lunules, from which, as the 
 pretend, he deduced the quadrature of the circle; but in our opinion without an | 
 foundation ; for as he held a distinguished place among the geometricians of h) 
 time, he égald not be a dupe to the paralogism of a school-boy: his object was on), 
 to shew, that if the lunule described on the side of an inscribed hexagon, could }) 
 made equal to a rectilineal space, the quadrature of es circle could be thence di 
 duced; and in this he was perfectly right. 
 It is very probable that geometricians were not long ignorant that the circle | 
 equal to the rectangle of half the circumference by the radius. Before the time: 
 Plato, geometry had been enriched with more difficult discoveries, yet this truth 
 first found in the writings of Archimedes. Something more however was necessary’ 
 the proportion between the circumference and the diameter, or the radius, remain¢: 
 to be determined ; and this discovery occasioned, no doubt, many a sleepless night * 
 that profound geometrician. Not being able to succeed with geometrical precision, 1’ 
 had recourse to approximation, and found, by calculating the length of an inscribe 
 ' polygon of 96 sides, and that of a similar one circumscribed, that the diameter beir | 
 1, the circumference would be more than 3}¢, and less than 319, or 3}. For he shewe’ 
 that the inscribed polygon is somewhat less than 34, and that the chraumner ne | 
 somewhat greater than 34} 
 Since that time, if at rea ncuneKs be not required, to find the ratio of the diameti 
 to the circumference, the proportion of 1 to 31, or of 7 to 22, is employed; that is 1) 
 
GEOMETRICAL PROBLEMS. 155 
 
 ‘say, the diameter is tripled, and one seventh of it is added: this seventh 1s never 
 neglected, but by the most ignorant workmen. 
 _ This object, we know, engaged the attention of several more of the ancient geo- 
 metricians; among whom were Apollonius, and one Philo of Gadara; but the 
  exactest approximations which they found have not reached us. 
 The first of the modern geometricians, who made any additions to what the ancients 
 had transmitted to us, respecting the measure of the circle, was Peter Metius, a 
 geometrician of the Netherlands, who lived about the end of the sixteenth century. 
 _ Being employed in refuting the pretended quadrature of oue Simon a Quercu, he 
 _ found this very remarkable proportion, which approaches exceedingly near to the 
 truth between the diameter and the circumference, viz. as 113 to 355. The error 
 | is scarcely the ten millionth part of the circumference. 
 | ‘After him, or about the same time, Vieta, a celebrated French analyst and geome- 
 _ trician, expressed the ratio of the circumference to the radius by the proportion of 
 10000000000 to 31415926535, and shewed that the latter number was too small, but 
 that if its last figure were augmented by only one unit, it would be too great. About 
 | the same period also Adrian Romanus, a geometrician of the Netherlands, carried 
 this approximation to 16 figures; but all these were far exceeded by Ludolph van 
 Ceulen, a native of the Netherlands likewise, who carried this proportion to 35 
 figures, and shewed that, if the diameter be unity followed by 35 ciphers, the cir- 
 cumference will be greater than 314159265358979323846264338327950288, and less 
 than 314159265358979323846264338327950289. He was so proud of this labour, 
 which however required less sagacity than patience, that, like Archimedes, he 
 requested it might be inscribed on his tomb-stone: his desire was complied with, 
 and this singular monument is still to be seen, it is said, in one of the towns of 
 Flanders. 
 
 Willebrord Snell, another countryman of Metius, made several important ad- 
 ditions to what had been done on this subject, in his book entitled ‘*Cyclometria.” 
 He discovered the method of expressing, by a very approximate proportion and an 
 exceedingly simple calculation, the magnitude of any are whatever; and he made 
 
 use of it to verify the calculation of van Ceulen, which he found to be correct. He 
 then calculated a series of polygons, both inscribed and circumscribed, always 
 doubling the number of sides, from the decagon to that of 5242880 sides ; so that 
 when a proportion between the diameter and circumference of the circle pretended 
 to be exact is proposed, one may refute it by means of this table, and shew which 
 is the circumscribed polygon greater than the supposed value of the circumference, 
 aud what circumscribed polygon it surpasses; in either case this will serve to prove 
 the falsity of the pretended rectification of the circular circumference. 
 _ The celebrated Huygens, when very young, enriched the theory of the measure of 
 the circle with a great many new theorems. He combated also the pretended 
 quadrature of the circle, which Father Gregory St. Vincent, a jesuit of the Nether- 
 Jands, announced as discovered, and requiring only a few calculations, which he 
 dexterously forgot to make. Gregory St. Vincent, however, was an able geome- 
 -trician; he wrote an answer to Huygens, and the latter replied ; some of Gregory’s 
 pupils entered the lists also; and another jesuit, a geometrician, combated on 
 the same side, But it is certain, whatever Father Castel may have said, that 
 Gregory was mistaken, and that his large work, which contains some very inge- 
 nious things, ended with an error, or something unintelligible. As he pretended 
 to have found the quadrature of the circle, why did he not perform those calcu- 
 lations which are necessary to express it numerically? But this was never done, 
 either by him, or by any of his pupils, who carried on the dispute with a great 
 ‘deal of asperity. 
 James Gregory, a celebrated geometrician’ in Scotland, undertook, in 1668, to 
 
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 4 
 
 demonstrate the absolute impossibility of the quadrature of the circle. This he 
 did by a very ingenious method of reasoning, which deserves perhaps to be better 
 examined. However it did not meet with the approbation of Huygens, and this 
 produced a very warm dispute between these two geometricians. But, it must be: 
 confessed, that Gregory gave several very ingenious methods for approaching nearer 
 to the measure of the circle, and even to that of the hyperbola. 
 
 The higher geometry supplies us witha great number of different methods for find- 
 ing, by approximation, the measure of the circle, and the greater part of them} 
 are easier than the preceding: but this is not a proper place for entering into an! 
 explanation of them. We shall content ourselves with observing, that by means} 
 of these methods the approximation of Ludolph van Ceulen has been carried as’ 
 far as 127 figures or decimals. Sharp, an English geometrician, first carried it to} 
 74 figures; Mr. Machin extended it to a hundred, and M. de Lagny, continued it 
 to 127: it is as follows. If the diameter be unity, followed by 127 ciphers, the| 
 circumference will be greater than 314159265358979323846264338327 9502884197 1.| 
 6939937510582097494459230781 74062962089986280348253421170679821480865132- j 
 723066470938446, and less than the same number, when the last figure is increased 
 only by unity. The error therefore is less than a portion of the diameter ex.’ 
 pressed by unity, divided by unity followed by 127 ciphers. If we suppose a 
 circle, the diameter of which is a thousand millions of times greater than the 
 distance of.the sun from the earth, the error in the circumference would be a 
 thousand millions of times less than the thickness of a hair. 
 
 It is even possible to go still further; and Euler has pointed out the method, 
 in the Transactions of the Imperial Academy of Sciences at Petersburgh; but | 
 if must be confessed that it would be superfluous labour. f 
 
 We cannot conclude better this short history of the quadrature of the circle, | 
 than by an account, which will no doubt amnse some of our readers, of those’ 
 who have miscarried in their attempts to solve this problem, or who have faller 
 into ridiculous errors on the subject } 
 
 The first, among the moderns, who pretended to have found the quadrature | 
 the circle, was Cardinal de Cusa. One of his methods was, to roll a circle oi| 
 cylinder over a plane, till the point which first touched it should touch it again | 
 and he then endeavoured, by a train of reasoning, which displayed nothing geome’ 
 trical, to determine the length of the line thus described. He was refuted by 
 Regiomanrtanus, in 1464 or 1465 } 
 
 After him, that is to say, about the middle of the sixteenth century, Orontiu; 
 Fineus, though professor royal of the mathematics, rendered himself famous by. 
 his paralogisms, not only in regard to the quadrature of the circle, but also in re.| 
 gard to the trisection of an angle and the duplication of the cube. Peter Nonius’ 
 however, a Portuguese geometrician, and J. Borelli his former pupil, clearly exposec | 
 the fallacy of his reasoning. The same Orontius Fineus published also a work or! 
 Gnomonics, which is nothing but a series of paralogisms. 
 
 We are astonished to.find the celebrated Joseph Scealiger fall, soon after, into the 
 same error. As he had no great esteem for geometricians, he was desirous tc 
 shew them the superiority of a man of letters, in solving, by way of amusement, 
 what had so long puzzled them: he attempted the quadrature of the circle, and| 
 seriously imagined he had discovered it, by giving, as the measure of it, a quantity! 
 which was only a little less than the inscribed dodecagon, It was therefore ne. 
 great difficulty for Vieta, Clavius, and others, to refute his reasoning :- this threw 
 him into a violent passion; and according to the practice of that period, exposed: 
 the latter in particular to a great many epithets not very decent, while it confirmeé 
 
 Scaliger more and more in his opinion, that geometricians were destitute of commor. 
 sense. 
 
 156 GEOMETRY. 
 
GROMETRICAL PROBLEMS, 157 
 
 We are sorry to include, among this class, the celebrated Danish astronomer, 
 Longomontanus, who pretended to prove that the diameter of a circle is to the 
 circumference, exactly as 100000 is to 314185. Soon after, the famous Hobbes 
 imagined also that he had found the quadrature cf the circle; and being refuted by 
 Dr. Wallis, he undertook to prove that the whole system of geometry before taught 
 was nothing buta series of paralogisms, This forms the subject of a work entitled, 
 ‘*De Ratiociniis et Fastu Geometrarum.” 
 
 Olivier de Serres, the agriculturist, by weighing a circle and a triangle, equal to 
 the equilateral triangle inscribed, believed he had found that the circle was exactly 
 the double of it. This weak man did not see that this double is exactly the 
 hexagon inscribed in the same circle. 
 
 A. M. Dethlef Cluver pretended, in 1695, to have squared the circle; and he reduced 
 the problem to one much easier, which he announced in the following manner: ‘‘ In- 
 venire mundum Menti Divine analogum.” He unsquared the parabola, and endea- 
 voured to prove that Archimedes had been gecared in regard to the measure of that 
 figure. 
 
 Leibnitz endeavoured to engage him ina dispute with M. Nieuwentyt, who then 
 started a great many. difficulties against these new calculations ; but the attempt did 
 not succeed. 
 
 Though these ridiculous attempts, as appears, ought to have prevented others, 
 men were seen, and are still seen daily, falling into errors of the like kind. About 
 30 years ago, a M. Liger pretended that he had found out the quadrature of the 
 circle, by demonstrating that the square root of 24 was the same as that of 25; and 
 that of 50 the same as that of 49: this he demonstrated, according to his own 
 terms, not by geometrical reasoning, which he abhorred, but by mechanism combined 
 with figures. 
 
 A certain Sieur T. de N—— found out something not less curious, viz., that 
 curves ought not to be measured by comparing them with straight lines, but by 
 comparing them with curves. This being once demonstrated, the quadrature of the 
 circle is merely children’s amusement. 
 
 M. Clerget made another discovery no less interesting, viz., that the circle is a 
 polygon ot a determinate number of sides; and he thence deduced, which is very 
 curious, the magnitude of the point where two unequal spheres touch each other. 
 He demonstrated also the impossibility of the motion of the earth. No one before 
 him had been able to suspect the least affinity between these questions. 
 
 But what shall we say of the complex calculations of the late M. Basselin, 
 professor in a university, who, after as much labour almost as Van Ceulen, Teak a 
 proportion between the diameter and circumference beyond the limits even of Ar- 
 chimedes? This weak man, who had so happily discovered the quadrature of the 
 circle, was ignorant, till some days before his death, that Archimedes had squared 
 the parabola. He proposed also, had he recovered from his malady, to examine the 
 process of Archimedes, being fully convinced that the geometrician of Syracuse had 
 been deceived. 
 
 But if these men incurred only ridicule, and ridicule confined to the circle of a 
 small number of geometricians, we are now going to introduce one to whom the 
 ambition of squaring the circle cost much dearer. We allude to the Sieur Mathulon, 
 who, from being a manufacturer of stuffs at Lyons, commenced geometrician and 
 mechanist ; but with less success than Hippocrates of Chios, who, from being a wine 
 merchant at Athens, became an illustrious geometrician. Sieur Mathulon, about 
 forty years ago, deposited the sum of 1000 crowns at Lyons, and having announced 
 to geometricians and mechanists the discovery of the quadrature of the circle and 
 perpetual motion, declared he would give the above sum to the person who should 
 prove that he was in an error. M. Nicole, of the Academy of Sciences, proved that 
 
158 GEOMETRY. 
 
 his knowledge of geometry was very limited; that his pretended quadrature was 
 
 mere paralogism; and demanded the 1000 crowns, which were adjudged to hin 
 The Sieur Mathulon demurred, and maintained that he ought to prove also the falsii| 
 of his perpetual motion; but he lost his suit, and M. Nicole gave up the 1000 crow) 
 to the general hospital at Lyons, to which they were delivered. 
 
 Had the Chatelet of Paris been equally severe, a similar folly would have co 
 much more to a man of some property, who, about thirty years ago, announced tl 
 quadrature of the circle; defied the whole world to refute him; and at last, by we 
 of challenge, deposited 10000 livres to be adjudged to the person who should prov 
 that he was mistaken, It is impossible, without lamenting the weakness of tk 
 human mind, to see this grand discovery reduced to dividing a circle into four equi 
 parts, by perpendicular diameters, turning these quadrants with their four angle 
 outwards, so as to form asquare, and then pretending that this square is equal to th 
 circle. According to the principles of this pretended mathematician, for two figure 
 to be equal it is not necessary that they should touch each other throughout thei. 
 whole extent ; itis sufficient that they touch, or can touch. Thus the square is not onl | 
 equal to the inscribed circle, but even to any figure included in the circle, the salien | 
 angles of which touch the circumference. 
 
 It would not have been difficult to shew to any other person than the author, tha 
 this was absolute nonsense. Three persons appeared as claimants of the 1000 
 livres; the cause was tried at the Chatelet, but this tribunal was of opinion that 
 man’s fortune ought not to suffer for the errors of his judgment, when these error. 
 are not prejudicial to society. On the other hand, the king decreed that the be 
 should be considered as void; and that both parties should take back their money 
 The author extorted from the Academy of Sciences a sentence, by which he wa 
 desired to study the elements of geometry; but he was still convinced that futur) 
 ages would blush for the injustice done to him by that in which he lived. Befor 
 we conclude this article, we must say a few words, respecting M. le Rohberger d 
 Vausenville, who in a work called ‘‘ Consultation sur la Quadrature du Circle,” ask | 
 geometricians, whether the quadrature of the circle would not be found, if mean 
 could be devised for determining the centre of gravity of a sector of a circle, i | 
 common parts of the radius and the circumference of the same circle. We do no. 
 rightly understand what the author means by common parts of the radius and thi 
 circumference. If he means those parts of the radius in which it is usual to expres}! 
 the circumference—as when it is said that if the radius be 100, the circumferenci d 
 will be 314we can answer, in the name of all geometricians, that the quadrature o 
 the circle would, in that case, be found. We will even not hesitate to tell him, tha’ j 
 in whatever manner he determines, in the axis of a sector or arc of a circle, its centre’ 
 of gravity, provided that in this determination the arc itself is not employed as one 
 of the data, he will have solved this famous problem; for who does not know thai 
 the distance of the centre of gravity of the semi-circumference, for example, from | 
 the centre, forms a third proportional to the fourth part of the circle and the radius i 
 But this determination of the centre of gravity of the sector, or arc of a circle, is ¢. 
 discovery rather to be wished than hoped for. ! 
 
 M. de Vausenville had no need to challenge, either individually or in general, al)’ 
 the geometricians of Europe, and even those of Turkey and Africa, where the | 
 meaning of the words centre of gravity is certainly not known; and he had stil), 
 
 less occasion to inform them that if they did not refute him he would consider thei! 
 silence as a sign of their defeat, and that his quadrature was acknowledged as resting| 
 on a solid foundation. This bravado will certainly excite neither the Eulers,, 
 the d’'Alemberts, nor the Bernouillis, &c., to attack his quadrature. Either M. de 
 Vausenville is right, and in that case mathematicians will] acknowledge his discovery, | 
 and bestow on him every just praise; or his pretended quadrature is a mere paralo« 
 
 \ 
 
 ————— 
 
 ql 
 
 i 
 
 | 
 | 
 
GEOMETRICAL PROBLEMS. : 159 
 
 _gism, and of course it will meet with as little attention as that of Henry Sullamar, 
 areal Bedlamite, who found it in the number 666, inscribed on the forehead of the 
 beast in the Revelations ; or those of many others which deserve, in like manner, to 
 be consigned to oblivion. 
 
 PROBLEM XLVI. 
 Of the length of the Elliptical Circumference. 
 
 We have spoken in a pretty full manner of the circular circumference, the exact 
 determination of which in length would give the quadrature of the circle. But no 
 author, as far as we know, has said any thing satisfactory, or useful in a practical 
 view, respecting the circumference of the ellipse. It is however necessary, in many 
 cases, and even in practical geometry, to know the length of that curve; in the 
 higher geometry there are also a great many problems the solution of which depends 
 on the same knowledge: a few observations therefore on this subject may be of 
 utility. . 
 
 Some authors, who have written on practical geometry, are of opinion that the 
 circumference of an ellipse is an arithmetical mean between the circumference of the 
 circles described on the two axes as diameters; but this is a mistake; and had they 
 possessed a little more of the spirit of geometry, they would have readily perceived 
 it; for it may be easily demonstrated that this is false in an ellipse much elongated, 
 as in that which has the greater axis 20, and the lesser 2. ‘The circumference-of 
 this ellipse indeed will certainly be greater than 40, while the mean proportional 
 between the circumferences of the circles described on its axes, as diameters, will be 
 only 343. 
 
 The rectification of the elliptical circumference is a problem which is almost the 
 same, in regard to the quadrature of the circle, as the latter is to a common problem 
 in geometry. John Bernouilli is the only person who has given a method susceptible 
 of being reduced to practice for measuring the length of the elliptic line. He shews 
 indeed, in an excellent memoir, published in his works, how to determine the circular 
 circumferences which are limits alternately less and greater than the circumference of 
 a given ellipse: and by this method we have calculated the following table. We 
 have supposed a series of ellipses, one half of the common greater axis of which is 
 10 parts, while the half of the less axis becomes successively 1, 2, 3, &c., as far as 
 
 10, the last value given by a circle; and we have found that the length of the cir- 
 cumferences of the ellipse is as here expressed. 
 
 —_——— TT Oct 
 
 Common length of the greater axis, 20.* 
 
 Length of the mean cir- 
 
 Lesser Length of the cumterence of the circles 
 
 axis. elliptic circumference. described on the greater 
 and less axis. 
 
 2 40°63245 34°5579 
 4 42°01968 37°6990 
 6 43°68526 40°8406 
 8 46°02506 43°9822 
 10 48°44215 47°1238 
 12 ; 51°05407 50°2654 
 14 53°82377 53°4070 
 16 56'72739 56°5486 
 18 5981022 596902 
 20 62°83185 62°83185 
 
 eee 
 
 ® Sir Jonas Moore also calculated a like table for the elliptic circumferences, extending ten 
 
160 ) GEOMETRY. 
 
 _It here appears, that the circumference of the circle, which forms a mean betwe 
 those described on the greater and less axis, is always less than the elliptic line, ar 
 the more sensibly so, the more the ellipsis differs from a circle; in the first of tl 
 above ellipsis the error is the 7th part. 
 
 By the help of this table all the mean lengths of the ellipsis between the precedii 
 may be calculated: nothing is necessary but to take the proportional parts. 
 
 Let us suppose, for example, that the greater axis of a semi-ellipse is 20 feet, at 
 that the half of its less axis is 74 feet; it is evident that, in this case, the whole) 
 the less axis will be 15 feet. This ellipsis then will hold a mean place between th: 
 in which half the less axis is 14 of the greater, and another in which the less axis is Hl 
 But by dividing the difference between the lengths of these two ellipses into tw 
 equal parts, it will be found, without any considerable error, that the length of tk 
 circumference of the mean ellipse will be 55:27558 parts, the axis being 20; cons«| 
 quently the half of the proposed ellipsis having its transverse diameter equal to 2) 
 feet and its conjugate to 74, will be 27 feet 6 inches and 8 lines: the error bein 
 scarcely a line, ; | 
 
 PROBLEM XLVIII. | 
 To describe geometrically a circle, the circumference of which shall approach very nea . 
 ty to that of a given ellipse. 
 
 It is to Mr, John Bernouilli also that we are indebted for this simple and ingeniou’ 
 method of describing a circle isoperimetrous to a given ellipse. As it may serve as | 
 supplement to what we have said of the rectification of the ellipse, we shall her 
 give it a place. | 
 
 Form the two axes of the given ellipsis into on 
 straight line, as ap (Fig. 80.), in which aB is equal ti 
 the greater axis, and 8p to the less: let this line az b 
 the diameter of a semicircle AED, which must be di 
 vided into 4, 8, 16, or 32 parts, &c. at pleasure, and ac 
 cording as greater precision may be required. We shal 
 here suppose the number of equal parts to be 16. Fron 
 
 4 the point 8 draw to each point of division straight lines 
 then take the 16th part of the sum of all these lines zg A, Bl, B2, B38, &c., as far a 
 BD inclusively; and if with the line hence arising as radius, a circle be described 
 you will have a circular circumference so nearly equal to that of the given ellipse 
 that it will not differ from it one hundred thousandth part, even in the most unfa 
 vourable case, such as that, for example, where the ratio of the axes of the ellipse 
 is as 10 to l. he 
 
 It may be readily seen, that if the semicircle had been divided into 8 parts, i 
 would have been necessary to take only the 8th part of the sum of all the lines 
 drawn to the points of division, including the points p and a. 
 
 If this operation were performed with a circle of a foot radius, the precision 0} 
 the result would approach very near the truth; and by means of a geometric scale 
 with exceedingly minute divisions, a very satisfactory numerical approximation might 
 be found without caleulation.* 
 
 times as far, thatis, to a hundred different ellipses,the conjugate axes gradually increasing from | 
 to 100, which is the constant transverse. The numbers indeed are set down to four decimals, but 
 they are not commonly true to more than two.—Aote by Dr. Hutton, 
 
 * It is noticed above, by Montucla, that an arithmetical mean between the two axes of an ellipse, 
 has been often taken as the diameter of a circle of equal circumference with the ellipse. It may be 
 added that this rule always gives the perimeter in defect, or less than just. 
 
 Another rule, almost as easy, which gives the perimeter always in excess, or more than just, is 
 this: Square each axis, and take the arithmetical mean between these squares ; that is, add the 
 squares together, and take half the sum; then extract the square root of this mean, and it will be 
 nearly the diameter of a circle of equal circumference, 
 
GEOMETRICAL PROBLEMS, 161 
 
 PROBLEM XLIX. 
 To determine a straight line nearly equal to the arc of any curve whatever. 
 
 _ We shall suppose that the amplitude of the given arc is not very considerable, as 
 not more than 20 degrees; that is to say, if tangents be drawn at the extremities 
 ofthe arc, and then perpendiculars to these tangents, the angle included between 
 these perpendiculars shall be at most 20 degrees. 
 
 This supposition being made; draw the chord of the arc; and then find, either 
 
 ‘by calculation, or by means-of the compasses, the third of the tangents compre- 
 hended between the place where they meet and the points of contact; if we then add 
 ‘to this third two thirds of the chord, we shall have a straight line so nearly equal 
 to the arc, that in the present case the difference will be but a ten thousandth part. 
 ‘But if the amplitude be only about 5 degrees, the error will not be a millionth part, 
 as has been shewn by M. Lambert, member of the Academy of Sciences at Berlin, in 
 a very interesting work, published in German, which is highly worthy of being 
 ‘translated. 
 _ Ifthe amplitude of the given are be greater, as about 50 degrees for example, 
 nothing will be necessary but to divide it into three parts nearly equal, and to draw 
 ‘tangents to the extremities of the are and to the points of section, which will give a 
 portion of the polygon circumscribed about the curve; if the three chords of the 
 three parts of the are be then drawn, and if two thirds of these three chords be added 
 tothe third of the tangents, forming the circumscribed polygon, the result will be a 
 line equal, within a hundredth thousandth part, to the length of the given are. — 
 
 | PROBLEM L. 
 A circle, having @ square inscribed in it, being given ; to find the diameter of a cirele in 
 which an octagon of a perimeter equal to the square can be inscribed. 
 
 Fig.81. Let a B (Fig. 81.) be the diameter of the given circle, and 
 AD theside of the inscribed square. Divide 4D into two 
 equal parts in ©, and raise  F perpendicular to aD, meeting 
 the given circle in F; if 4 F be then drawn, it will be the 
 diameter of a circle, in which if an octagon be inscribed it 
 will be equal in perimeter to the given square. 
 
 For it is evident that the circle described on the diameter 
 AF, will pass through the point E, since the angle a EF is 
 . aright angle. It is also evident that the line drawn from I, 
 the centre of the second circle, to the point u, will be parallel top r, because the sides 
 apand AF of the triangle p A F are bisected in the points BE and I. But the angle ar D 
 is half a right angle, being halfof pc A, which is a right angle, since the chord of wee 
 inscribed square subtends an are of 90°: consequently the angle ALE is equal to 45 ; 
 whence it follows that Az isthe side of the octagon inscribed in the circle having 
 And it is evident that eight times a zis equal to four 
 
 ‘a F for its diameter. 
 
 times A D. 
 
 | Remark.—If ax be, in like manner, divided into two equal parts in G, and if GH be: 
 
 drawn perpendicular from the point «, till it meet the second circle ; by drawing AH, 
 
 that line will be the diameter of a third circle, in which, if a polygon of 16 sides be 
 
 inscribed, it will be isoperimetrous to the above square or octagon. 
 Hence it follows, that if this operation were infinitely continued, we should obtain 
 
 ess as the former is in defect, if an arithmetical mean 
 
 it will be the diameter of a circle of equal Cy ae 
 i , SOE ; 
 
 ference, within the fifty thousandth part of the whole, and is the nearest approximating rule ye 
 
 ‘given. These rules are Taker from my treatise on Mensuration, where several others may also be 
 
 ‘seen, both for the whole circumference and also for any part of it.— Note by Dr. Hutton. 
 
 M 
 
 _ As this latter rule is nearly as much in exc 
 between them be taken, that is half their sum, 
 
162 GEOMETRY. 
 
 a circle or polygon of an infinite number of sides, isoperimetrous to a given squat 
 The circumference of this circle therefore would be equal to the perimeter of } 
 square, and we should thus have the quadrature of the circle. 
 
 We have seen a very ingenious attempt to discover the quadrature of the circlall 
 this principle. The author, M.Janot, professor of mathematics in the Royal Milita’ 
 School, reduced the problem to a very exact equation, but complex, by the solutin 
 of which be expected to obtain this last diameter; but when he seriously tried to redu, 
 it, he found the two members of his equation to be composed of the same term) 
 which of course gave him no solution. 
 
 1 
 | 
 | 
 
 PROBLEM LI. 
 
 The three sides of a rightangled triangle being given; to find the value of its angi 
 without trigonometrical tables. 
 
 We shall first suppose that the ratio of the hypothenuse to the least side is greater | 
 not less than 2to], in order that the angle opposite to that side may be at most abo| 
 30°; for the error will be less the more that angle is below 30°. | 
 
 This being premised ; let us suppose, for example, that the hypothenuse of thet| : 
 angle is equal to 13, the greater side comprehending the right angle 12, and the le! 
 5. We must then make this proporticn: as twice the hypothenuse, plus the great. 
 side, or 38, is to the less side or 5, so is three times unity or 3, to a fourth prope 
 tional, which will be 13. But 38 reduced toa decimal fraction is 0°39473: if | 
 number be divided by 0: 1745, the quotient will be the number of the degrees a. 
 parts of a degree contained in the angle opposite to the less side. This quotient 
 22,621. which makes 22° 37’ 15”. By the tables it will be found to be 22° 37’ 28”. | 
 
 If the sides of the triangle are nearly equal, such for example as 3, 4, 5, we mu 
 suppose in the triangle a line c p (Fig. 82.) dividing the angle opposite to t. 
 
 * side A B, or that represented by 3, into two equal par) 
 
 Eng Bae But it is known that in this case the opposite side a B will | 
 
 - divided in the same ratio as the adjacent sides; consequeni. 
 Iw the segment B p may be fonnd by the following analogy: 
 F As the sum of the two other sides or 9, is to 3, the third sic 
 
 so is cB or 4, to B D, which will be ¥ or 4; if the squares of. 
 and 4, or of cBandBD, be then added together, by eters the square root of t 
 sum, which in decimals is 17°777, we shall have for the square root 4°21637, whi 
 will be the value of cp. Inthe last place, by applying the above rule to the triang 
 BCD, we shall find the anglegpc p to be 18° 26’7”, and consequently its double, ort’ 
 angle a c B, 36° 52’ 14”. By trigonometrical tables the latter will be found to | 
 36° 52’ 15”; so that the difference is only one second. 
 
 PROBLEM LII. 
 
 An arc of a circle being given, in degrees, minutes, and seconds ; to find the correspon 
 ing sine, without the help of trigonometrical tables. 
 
 the preceding; but it appears to be the best hitherto proposed, especially as it is eas, 
 and may be readily remembered by means of an observation we shall make at the er 
 and which will shew its source as wellas the demonstration of it. | 
 
 In this problem there are three cases, which require three different methods | 
 operation. The given arc may exceed 60°, or it may be less, or at most not me. 
 than 30°; and in the last place it may be greater than 30°, but less than 60° 
 
 Ist. We shall suppose that the are exceeds 60°, and that its sine is require 
 Take its complement to 90°, and reduce that arc into parts of the radius, which *| 
 shall suppose to be 100000; for this purpose, nothing is necessary but to multip, 
 the degrees it contains by 1745/,, and the minutes by 29°09, and then to add t 
 
 : 
 q 
 The solution we are going to give of this problem, is not so simple and short 
 
a 
 . ad 
 — 
 \ oe ~ . 
 _ 
 
 GEOMETRICAL PROBLEMS, 163 
 
 products. Square this arc thus reduced, and raise it also to the fourth power; 
 divide the square of it by 2, and from the quotient subtract unity or the radius; 
 divide the fourth power of it by 24, and add the quotient to the above remainder ; 
 the number thence resulting will be nearly the sine of the given arc. 
 
 Let the given arc, for example, be 70° 30’; its complement to 90° is 19° 30’, which 
 reduced to parts of the radius, as beforesaid, will give 34025. The square of this 
 number, suppressing the five last figures, which are useless, because we have no occa- 
 sion for more than 100000 parts of the radius, is 11583, and its half 5792, which 
 taken from 100000 leaves 94208. Square 11583, which will give the fourth power of 
 34035; and if five figures be suppressed, as useless for the reason before mentioned, 
 we shall have 1341, which must be divided by 24. The quotient, whichis somewhat 
 less than 56, being added to 94208, will make 94264, which will be the sine of 70° 30’. 
 And this is exactly what it will be found to bein the tables of sines. 
 
 2d. Let us nowsuppose that the given arc is at most 30°. Find the cube and fifth 
 power of that arc reduced to parts of the radius; then divide the cube by 6, and the 
 fifth power by 120; if the first quotient be subtracted from the are, and the second 
 be added to the remainder, we shall have the value of the sine, a very small error 
 excepted. 
 
 Let the given arc, for example, be 30°. When reduced to 100000th part of the 
 radius it will give 52362, the cube of which, suppressing the last ten figures, will be 
 14354. The sixth part of this number is 2392, which taken from the are 52362, 
 leaves 49970. The fifth power of the same number 52362, suppressing the last 
 twenty figures, is 3935, which divided by 120 gives 32; if 32 be added to the above 
 remainder, we shall have 50002 for the sine of 30°; whichit is well known is exactly 
 50000; consequently the error is only two units in the last figure. 
 
 3d. If the arc be between 30° and 60°, for example 45°; take the difference be- 
 tween that are and 60°, which is 15°, and add to it 60°; the sum will be 75°, the sine 
 of which must be found by the first rule. 
 
 Then find that of 15° by the second; and subtract it from that of 75°; the re- 
 wainder will be the sine of 45°; for according toa theorem in trigonometry, that 
 the sines of two arcs, equally distant from 60°, have for difference the sine of that are 
 by which each of these two arcs differs from that of 60°. 
 
 If, instead of the sine of an arc, that of its complement be required, the same rules 
 may be employed: the sine of the complement of 20°, for example, is the right sine 
 of 70°; and on the other hand,’ the sine complement of 70°, is the right sine of 
 20°; by which it may be readily seen, that to find the sine complement of an arc, 
 nothing is necessary but to find the right sine of the complement of the are. 
 
 When the right sine and the sine complement of an are are known, it will be 
 easy to find the tangent by the following proportion: As the sine complement, or 
 
 cosine, is to the sine, so is radius to the tangent: nothing therefore is neces- 
 sary but to divide the sine, increased with any number of ciphers at pleasure, by the 
 cosine, ; 
 
 Remark.—We have here givena method for supplying the place of tables, so neces- 
 sary in practical trigonometry, or of forming them very expeditiously, in cases when 
 they are not at hand, or cannot be procured. I was once myself in such a situation, 
 having lost my baggage, which was taken from me by a party of the Iroquois Indians, 
 when posted at Oswego in Canada. In that dreary abode, I endeavoured to amuse 
 myself by the study of geometry. An opportunity of performing some trigonometrical 
 operations occurred: but, being destitute of books, I fortunately remembered the 
 theorem of Snellius, which serves as a basis for the solution of the preceding problem : 
 in short, I recollected two expressions, in infinite series, which give the value of the 
 
 sine and cosine, the are being given. The first, as is well known, a being made to 
 M 2 
 
164 GEOMETRY. 
 
 represent th i ah 3 a an a? at 
 P e are, isa ~—— 120 5040 “Cs and the second 1 — ey + oa a 
 
 a® , 
 0. &c. But when the are a is very much below the value of the radius or unity, — 
 
 it is evident that the three first terms of each will be sufficient, because all the 
 following terms become excessively small. After what has been said, the demonstra= 
 tion of these rules may be easily discovered. 
 
 PROBLEM LIII. 
 A circle and two pcints heing given; to describe another circle, which shall pass through if 
 these points, and touch the former circle. P 
 
 It is here evident, that these two points must be both within, or both without 
 the given circle. . 
 
 a eel ry 
 
 } Let the two given points then be A _ 
 Fig. 84, and B, as in the two Figures 83. and 84, 
 = Join these points by a straight line aB;_ 
 and through one of them, for example 
 A, and the centre of the given circle, — 
 draw the straight line a1rH, intersecting 
 i oN it in the two points H andr; then take © 
 2 sas av a fourth proportional to-a B, a H, 
 
 ! Sa ae aa AI, and from the point p draw the two 
 tangents D E, D e; lastly, from the point 
 A, through the two points of contact draw the two lines Ea F, e Af, intersecting the — 
 circle in F and f: the circle described through the two points a and B, and through F, — 
 will touch the given circle in; andif one be described through the points a, 3B, and 
 J, it will touch the given circle in f. E 
 
 PROBLEM LIV. 
 Two circles and a point being given: to describe a third circle, which shall pass 
 through the given point, and touch the other two circles. } 
 Fig. 85. Let the centres of the two given circles be the points A and 
 ES c, (Fig. 85.) and their radii aB, cp. In the line which joins ~ 
 
 their centres continued, find the point Fr, the tangent from 
 which to one of the circles shall be the tangent to the other, 
 (by Prob. xii.), and join the point F to the given point E: then 
 make FG a fourth proportional to FE, FB, FD; and by the pre- 
 
 ceeding problem, through the points c and & describe a circle - 
 which shall touch one of the two circles AB or cD::.-this third 
 eircle will touch also the other circle. 
 
 PROBLEM LV. 
 Three circles being given ; to describe a fourth, which shall touch them all. 
 
 It may be readily seen that this problem is susceptible of a great number of differ- 
 ent cases and solutions; for the required circle may contain the three given circles, 
 or only two of them, or even one; or the given circles may all be without it. But, 
 for the sake of brevity, we shall confine ourselves to one of these cases, that where 
 the circle to be described must leave the other three without it. : 
 
 Let the three given circles then be denoted by a, B, c, (Fig. 86.) and let their 
 radii be Aa, Bb, cc; let a also be the greatest, 3 the mean one, and c the 
 least. In the radius a a make ad equal to ce, the radius of the least circle, and 
 from A asa centre, with the radius a d describe a new circle. In the radius 8d. 
 
 i 
 
GEOMETRICAL PROBLEMS. 165 
 
 Fig. 86. make be equal to cc also, and from B asa 
 centre, with the radius B e, describe another 
 eircle: then, by the preceding proposition, 
 through the centre c describe a circle, 
 which shall touch the two new circles; 
 let its centre be EB, and its radius E G; di- 
 minish this radius by the radius cc, and 
 from the same centre E describe another 
 circle, which will evidently touch the three 
 first circles given 
 
 For since the circle described from the 
 centre A with the radius a d, is within the 
 proposed circle a, by the quantity ad or ce, it is evident that if the radius E G@ be 
 diminished by that quantity, the circle described with this new radius, instead of 
 touching the interior cirele, having ad for its radius, will touch the proposed 
 circle, the radius of whichis aa. 
 
 It may be seen also that the same circle described with the radius EG, less ce, 
 will touch externally the circle which has for its radius Bb. Lastly, it will touch 
 externally the circle having ce for its radius; consequently it will touch them all 
 three externally. 
 
 Remark.—This problem had some celebrity among the ancients; and indeed it is 
 
 attended with a certain degree of difficulty. It terminated a treatise of Apollonius, 
 entitled ‘‘ De Contactibus,’’ which has been lost, but which Vieta, a celebrated geo- 
 metrician who lived about the end of the sixteenth century, restored, and which may 
 be found in his works, printed in Latin, at Leyden, in 1646, in folio, with the 
 title of ‘‘ Apollonius Gallus, seu exsuscitata Apollonii Pergei de ‘Tactionibus 
 Geometria.” 
 _ Newton has given a beautiful and ingenious solution of this problem; but that of 
 Vieta appeared to us preferable for the present work, being founded on easier prin- 
 ciples. We cannot omit this opportunity of observing, that the above work of Victa 
 is a most elegant piece of geometry, treated in the manner of the ancients. 
 
 PROBLEM LVI. 
 
 What bodies are those, the surfaces of which have the same ratio to each other as 
 their solidities ? 
 
 _ This problem was proposed, in the form of an enigma, in one of the French Jour- 
 
 mals, entitled the Mercury, of the year 1773. 
 
 ‘* Reponds-moi, d'Alembert, qui découvre les traces 
 Des plus sublimes vérités ; 
 Quels sont les corps dont les surfaces 
 Sont en méme rapport que leurs solidités ? ” 
 
 We do not find that d’Alembert condescended to answer this problem, for it may 
 be readily seen, by those in the least acquainted with geometry, that two bodies well 
 Known, the sphere and the circumscribed cylinder, will solve it. Archimedes de- 
 Mmonstrated long ago, that the sphere is equal to two thirds of that cylinder, both in 
 surface and solidity, provided the two bases of the cylinder are comprehended in 
 the former ; and this is the answer which was given to the enigma in the following 
 Mercury. 
 
 _ But we may go a little further, and say, that there are a great number of bodies 
 which, when compared with each other, and with the sphere, will answer the problem 
 also: such are all solids formed by the circumvolution of a plane figure circumscribed 
 about the same sphere, and even all plane-faced solids, regular or irregular, that can 
 
' 1665)” . GEOMETRY. 
 
 be circumscribed about the same sphere ; for the solidity of all these bodies is the - 
 product of their surfaces by the third of the radius of the inscribed sphere, — 
 while the solidity of the sphere is the product of its surface by the third of its 
 radius. . : 
 
 Thus, the equilateral cone is to the inscribed sphere, both in surface and solidity, 
 as 9 to 4. , 
 
 The case is the same in regard to the sphere and the circumscribed isosceles cone; _ 
 except that the ratio, instead of 4 to9, will be different according to the elongation — 
 or oblate form of the cone. 
 
 “If the sphere and the circumscribed cylinder possess this property, it is because the _ 
 latter is a body produced by the circumvolution of the square circumscribed about the 
 great circle of the sphere, on an axis perpendicular to two of the parallel sides. 
 
 If the square and inscribed circle revolved around the diagonal of the square, the 
 surface and solidity of the body, thus produced, would be to each other as ,/ 2 
 1S5tO Ls 
 
 We shall here propose a similar problem : 
 
 < eat 
 
 What are those figures, the surfaces and perimeters of which are to each other in the 
 same ratio ? 
 
 The answer is easy: the circle and all polygons, regular or irregular, circumscripte 
 ible of it. | 
 
 THEOREM VIII. 
 
 The dodecagon inscribed in the circle, is 2 of the square of the diameter, or equal te 
 the square of the side of the inscribed triangle. 
 This theorem, which is exceedingly curious, was first remarked by Snellius, a — 
 Dutch geometrician. 
 . Fig.87. Let ac (Fig.87.) be the radius of a circle, in which is in-— 
 scribed the side aB of the hexagon; and if a p and pB, be the 
 sides of the regular dodecagon, it thence follows that, by draw- 
 ing the radius pc, it will cut the side a 8B perpendicularly, and 
 divide it into two equal parts. But it is evident, that the 
 area of the dodecagon is equal to 12 times one of the triangles” 
 ADC, or DcB; and as the triangle a Dc is equal to the product 
 of the radius by the half of a r, or by the fourth part of the radius, that is to say, is 
 equal to a fourth of the square of the radius, the twelve will be equal to three times — 
 the square of the radius, or to three fourths of the square of the diameter. 
 On the other hand, the side of an equilateral triangle inscribed in a circle, the © 
 diameter being unity, is equal to ,/2; consequently its square is also equal to 3 of the 
 square of the diameter, or to the dodecagon, 
 
 IB 
 
 Remark.—Two of the inscribed polygons, viz. the square and the dodecagon, pos- 
 sess the property of having a numerical ratio to the square of the diameter; for the 
 inscribed square is exactly the half; but of the regular polygons circumscribed, this 
 property belongs only to the square. 
 
 Irregular polygons however, and even a great variety of them, commensurable to 
 the square of the radius, might be inscribed in a given circle. 
 
 Let the diameter of the circle, for example, be 1; and let the four sides of the 
 inscribed quadrilateral be 5, %, 43, %: its surface will be rational, and equal to 329 
 of the square of the diameter. 
 
 PROBLEM LVII. 
 \ If the diameter as (Fig. 88.) of a semicircle acB, be divided into any two parts 
 
GEOMETRICAL PROBLEMS. 167 
 
 whatever, aD and DB; and if on these parts as diameters there be described two 
 
 semicircles AED and DF B, a circle is required equal to the remainder of the first 
 semicircle. 
 
 Fig. 88. From the point D raise pc perpendicular to a B, till it 
 meet the semicircle acB: if a circle be then described 
 having Dc for its diameter, it will be that required. 
 
 The demonstration of this problem, so well kiown, is 
 deduced from a theorem in the second book of the Elements 
 ‘a. of Euclid, viz., that the square of aB is equal to the 
 
 squares of aD and DB, and twice the rectangle of ap and 
 >B; a Fectatitte to which the square of Dc is equal by the property of the circle. 
 mstead of these squares if’ we substitute semicircles, which are in the same ratio, 
 he problem will be demonstrated. 
 
 PROBLEM LVIII. 
 
 A square being given; to cut off its angles in such a manner, that it shall be trans- 
 JSormed into a regular octagon. 
 
 Fig. 89. Let the given square be aBcD (Fig. 89.) In the two sides 
 Dc and pA, which meet in D, take any two equal segments 
 whatever, DI and Dx, and draw the diagonal1K; make DL 
 equal to twice Dk, plus the diagonal rx, and draw Li; if cm 
 be then drawn parallel to Lr, through the point c, it will cut 
 off from the side of the square the quantity pM, to which if 
 DN be made equal, by drawing the line NM, we shall have the 
 MI E A _ side of the octagon required. If AF, AF, BG,BH,cP, and 
 co be made equal to the line p a; by drawing EF, GH, and oP, 
 the required octagon will be completed. 
 
 i > 
 
 Remark.—The solution above given, is an example of what often happens in em- 
 ploying the algebraic calculus in the solution of geometrical problems ; for there is 
 a solution much more simple, and that of a nature to be self- 
 
 Fig. 90. evident even to a beginner. It is as follows: 
 
 Draw the diagonal a c (Fig. 90.) of the square, and alsoznF 
 bisecting the opposite sides in £ and F, and the diagonal in a. 
 Draw GH so as to bisect the angle cGE; so shall EH be half 
 the side of the octagon. Therefore make c1, BK, BL, AM, AN, 
 DO, DP, each equal to cH, and the angles of the octagon will 
 be found. 
 
 PROBLEM LIX. 
 
 A triangle awc being given; to inscribe in it a rectangle, in such a manner, that FH 
 or GI shall be equal to a given square. 
 
 On the base B c (Fig. 91.) describe the rectangle zp, 
 
 equal to the given square, and let F& be the point 
 
 os where ac is intersected by the side of this rectangle 
 
 \ parallel to Bc. On acc describe a semicircle, and 
 
 je having raised the perpendicular £1, till it meet the 
 
 ft circumference, draw cL; on CK, the half of ac, de. 
 
 Jr scribe also a semicircle, in which make em equal to 
 
 ; cL: if KF and K@ be then made equal to KM, we 
 
 shall have the points F and c, through which if two lines be drawn parallel to the 
 
 base till they meet a xB, and also two other lines perpendicular to the base, they will 
 
» 
 168 GEO METRY. 
 
 form the rectangles rH and G1, equal to each other, as well as to the rectangle Dz, 
 which was equal to the given square: therefore, &c. 
 
 PROBLEM LX. 
 
 the given angle, and make the rhombus LEGA, equal to the 
 apply pF equal to pL, and draw EF; lastly, if eu be made 
 
 H DI will be the one required. 
 
 PROBLEM LXI. 
 Of the Lunule of Hippocrates of Chios. 
 
 Though the quadrature of the circle be in all probability impossible, means have 
 been devised to find certain portions of the circle which are demonstrated to be 
 equal to rectilineal spaces. The oldest instance of a circular portion, which may be 
 thus squared, is that of the lunules of Hippocrates of Chios; the construction of 
 which is as follows: 
 
 Let aBc (Fig. 93.) be a right-angled triangle, on the hy- 
 pothenuse of which describe the semicircle a Bc, touching 
 the right angle B: if semicircles be then described on the 
 sides. AB and Bc, the spaces in the form of a crescent, 
 
 Fig. 93. 
 
 va 
 
 a ABC. 
 
 Through a given point D (Fig. 92.) within an angle BA c,to draw a line HI, in such 4 
 
 a manner, that the triangle 1H A shall be equal to a given square, 
 Fig. 92. Through the point p draw LE parallel to one of the sides of — 
 
 given square. On the line DE describe a semicircle, in which ~ 
 
 ' 
 
 equal to EF, aud HDI be drawn through the point g, the line — 
 
 To nds 
 
 AEBHA and BDCGB, will together be equal to the triangle — 
 
 For it is well known that the semicircle on the base ac, — 
 
 is equal to the two semicircles AEB and BDC, because circles are to each other as 
 
 the squares of their diameters: if the segments AB and BGC, which are common 
 
 to both, be taken away, there will remain, on the one hand, the triangle a Bc, and: 
 on the other the two spaces in the form of a crescent, AEBH and BDCc«G, and these 
 remainders will be equal: therefore, &c. 
 
 If the sides a b, bc, are equal, as in Fig. 94, the two lunules — 
 
 aed will evidently be equal, and each will be half of the triangle 
 abc, thatis to say, equal to the triangle b fa or b fe. 
 Hence we obtain a simpler construction of the lunule of 
 
 from the point F asa centre, there be described through 4 
 and c, the arc of acircle AD cou the base A c, the lunule A BCD 
 will be equal to the triangle c a F. 
 
 Since the square of F c is double the square of & c, or of EF, 
 the circle described with the radius rc will be double that de- 
 scribed ‘with the radius EC: consequently a fourth part of 
 
 of the second, or to the semicircle age. If the common sege 
 ment ADCEA therefore be taken away, the remainders, that 
 
 Hippocrates. Let apc (Fig. 95.) be a semicircle on the — 
 diameter A c, and a Fc an isosceles right-angled triangle. If — 
 
 the former, or the quadrant r Apc, will be equal to the half — 
 
 is to say, the triangle a Fc, on the one hand, and the lunule asep 4, on the ~ 
 
 other, will be equal. 
 
 Remarks.— We shall take this opportunity of making the reader acquainted with 
 several curious observations, added by modern geometricians to the discovery of 
 
 Hippocrates. a 
 
 ‘ 
 4 
 if 
 
. 
 . 
 
 é 
 J 
 4 
 
 | 
 | 
 
 GEOMETRICAL PROBLEMS. 169 
 
 Ist. From the centre F (Fig. 96.) if there be drawn 
 any straight line whatever F x, cutting off a portion of the 
 lunule AE GA; that portion will still be squarable, and 
 equal to the rectilineal triangle a HE right-angled at H. 
 
 For it may be easily demonstrated, that the segment 
 A E will be equal to the semi-segment a GH, 
 
 2d. From the point £, if £ 1 be let fall perpendicular on 
 ac, andif rr and F E be drawn, the same portion of the 
 lunule AE GA, will be equal to the triangle a F 1. 
 
 For it may be easily demonstrated that the triangle 
 A FL is equal to the triangle AH E. 
 3d. The lunule therefore may be divided in a given ratio, by aline drawn from 
 the centre r: nothing more is necessary than to divide the diameter a c in such a man- 
 ner, that ax shall be tocr in that ratio; to raise EI perpendicular to a c, and to 
 draw the line FE; the two segments of the lunule a GEand «ec will be in the 
 ratio of Ar tore. 
 
 All these remarks were first made by M. Artus de Lionne, bishop of Gap, who 
 published them in a work entitled ‘‘Curvilineorum Ameenior Contemplatio,” 1654, 
 4to, and afterwards by other geometricians. 
 
 4th. If the two circles, forming the lunule of Hippocrates, be completed, the result 
 will be another lunule, which may be called Conjugate, and in which mixtilineal 
 spaces, absolutely squarable, may be found. 
 
 From the point F, if there be drawn any radius F M, intersecting the two circles 
 in R.and mM; we shall have the mixtilineal space R AMR, equal to the rectilineal tri- 
 angle L a R: which can be easily demonstrated; for it may be readily seen that the 
 segment aR of the small circle, is equal to the semi-segment L a m of the greater. 
 
 Hence it follows, that if the diameter m o touch the small circle in F, the mixt tri- 
 
 angular space ARF mA, will be equal to the triangle as r, right-angled ins, or to half 
 
 the lunule aGcBA. } 
 5th. There are also some other portions of the lunule of Hippocrates that are abso- 
 lutely squarable; which, as far as we know, were never before remarked. 
 
 Fig. 97. Letapera (Fig. 97.) be a lunule, and let aB be a 
 tangent to the interior are. Draw the lines EA andea 
 making with a Bequalangles ; if from the point B there 
 be then drawn the chords B E, Be, which will be equal, we 
 shall have the mixtilineal space, terminated by the two 
 circular arcs, EBe, AGY, and the straight lines ae and 
 F E, equal to the rectilineal figure ea E Be. 
 
 This would be true, even if the figure a Bc F A were not absolutely squarable ; 
 that is to say, though a Bc should not be a semicircle, provided the two circles were 
 always in the ratio of 2 to 1. 
 
 PROBLEM LXII. 
 To construct other lunules, besides that of Hippocrates, which are absolutely squarable. 
 
 The lunule of Hippocrates is absolutely squarable, because the chords a B, Bc (Fig. 
 95.) and a c are such, that the square of the last is equal to the squares of the other © 
 
 _ two; so that by describing on the last an arc of a circle, similar to those subtended 
 
 by a Band Bc, the twosegments A Band BC are equal to ADC. 
 
 This method of considering the lunule of Hippocrates conducts us to more general 
 views ; for we may conceive in acircle any equal number of chords at pleasure ; for 
 example four, as A B, BC, cD, and DE, (Fig. 98.) of such a nature, that by drawing 
 the chord ar, the square of it shall be quadruple ef one of them; or more generally, 
 the number of these chords being n, the square of A E may be to that of a B, as n 
 
170 GEOMETRY. 
 
 : tol. Thus, if we describe on A & an arc similar to 3 
 Fig. 98. those subtended by the chords a B, Bc, &c., the seg- is 
 B c ment AE will be equal to the segments aB, Bo, &., 
 together: if from the rectilineal figure a BcDE, there- 
 fore, we take away the segment a F, and add it to the - 
 segments A B, BC, &c., the result will be alunule formed 
 of the ares Ac £, and a £, which will be equal to the ree- 
 = tilineal polygon ABCDE. 
 
 The question thenis to resolve the following geome- 
 trical problem: 
 
 In a given circle to inscribe a series of equal chords, a B, BC, CD, &c., in sucha 
 manner, that the square of the chord’a , by which they are all subtended, shall be to 
 the square of one of them, as the whole number of them is to unity ; triple, if there 
 
 _ are three ; quadruple, if there are four, &c. 
 
 But we shall confine ourselves to cases that can be constructed by the elements 
 of geometry, which will still give us two lunules similar to those of Hippocrates, the 
 one formed by circles in the ratio of 1 to 3, and the other by two circles in that of 
 1 to 5, besides two other lunules, formed by circles in the ratio of 2 to 3 and of 3 to5. 
 
 Construction of the first Lunule. 
 
 Fig. 99. Let az (Fig. 99.) be the diameter of the lesser circle, 
 
 £ with which the lunule is to be constructed. Continue aB 
 to p, so that BD shall be equal to the radius, and on ap asa 
 diameter describe the semicircle a ED, cutting Br, drawn 
 perpendicular to AD,in E; draw DE, and make pD F equal to 
 it: on aF describe also a semicircle AHF, intersecting the - 
 radius c G, perpendicular to A B,,in H; draw A 4, and in the 
 given circle make the chords A1, 1K, and KL, equal to aH; 
 then draw aL, and on that chord, with a radius equal to Dz, describe an arc of a 
 circle aL; by these means we shall have the lunule AGBL A, equal to the rectilineal 
 figure AIKLA. 
 
 Construction of the second Lunule, where the circles are as 1 to 5. 
 
 Fig. 100. Continue the diameter of the given or less circle, till 
 _~ the part pp (Fig. 100.) be equal to half the radius; and 
 ‘2 draw the indefinite line p= perpendicular to ap; then 
 from the point s, which divides the radius ac into two 
 equal parts, with a radius equal to 3 times ac, describe 
 an arc of a circle, cutting the before-mentioned perpen- 
 dicular in E: make EF equal toi of ac, and pu equal 
 to the radius; divide H F into two equal parts in G; and 
 from G as a centre, with a radius equal to cH, describe 
 an are of a circle cutting the straight line a p in 1: then 
 make pK equal to HI, and draw K R perpendicular to 
 the diameter, intersecting the semicircle described on 
 Acin1; lastly draw aL, and let the chords am, MN, 
 WO, OP, PQ, be made equal to it: if an arc of a circle be then described on AQ, 
 with a radius equal to pE, the lunule ANPQA will be equal to the rectilineal figure — 
 AMNOPQA. 
 Lunules absolutely squarable may therefore be formed with circles, which are to 
 each other in the ratios of 1 to 2, 1 to 3, and of 1 to 5. But there are no others 
 formed by circles in simple multiple or sub-multiple ratio, which can be constructed 
 
 i‘ 
 
GEOMETRICAL PROBLEMS. 171 
 
 erely by the rule and compasses. Those which might be formed with circles in 
 1e ratio of J to 4,1 to 6,or1 to 7, &c., would require the assistance of the higher 
 eometry. ‘The trisection of an angle, or the finding of two mean proportionals, is 
 problem of the same nature, and of the same degree, and to be solved only by the 
 ume means. But there are still two which may be constructed by the help of sim- 
 le geometry, and which are formed by circles in the ratio of 2 to 3 and of 3 to 5. 
 ‘or the sake of brevity, we shall confine ourselves to shewing the method of con- 
 truction. 
 Fig. 101. For the first.—Let there be any circle, the radius of which 
 ; is supposed to be 1; inscribe in it a chord a 8 (Fig. 101.), equal 
 to / $ LPP yTs #2, and let it be twice repeated, from B to c, and 
 from c to p: draw the chord ap, and on ap describe an arc 
 similar to the are axc; if the two equal chords aE and ED 
 be then drawn, the lunule ABcDE£EA will be equal to the rec- 
 tilineal polygon ABCDEA. 
 
 For the second. —-In a circle a circle, the radius of which is 1, inscribe a chord equal to 
 V§—24/3—/ 8-4/7 % Phy % and carry it round five times: draw the chord of the 
 Bictuple are, and duscuilie on it an are with a radius = ,/ $; in this arc inscribe the 
 three chords of its three equal parts, which may be dare by common geometry, 
 because each of these thirds is similar to a fifth of the first are already given. We 
 shall then have a lunule equal to a retilineal figure, formed by the five chords of the 
 small circle and the three chords of the greater. 
 
 PROBLEM LXIItI. 
 
 A lunule being givens to find in it portions absolutely squarable, provided the circles 
 
 by which it is formed are to each other in a certain numerical ratio. 
 Fig. 102. 
 
 Let apcpaA (Fig. 102, 103, 104.) be a lunule, formed by 
 two circles in any of the above ratios, a Bc being a portion 
 of the lesser circle, and apc of the greater. Draw az the 
 tangent of the arc anc, and then draw the line a¥F in such 
 a manner, that the angle Fac shall be to the angle Fac in 
 the same ratio as the less circle is to the greater; then one 
 of the three following things will take place; aF will bea 
 tangent to the circle a Bc, (Fig. 102.); or it will cut it, as 
 in f, (Fig. 103.); or as in ¢, (Fig. 104.) 
 
 In the first case, the lunule will be absolutely squarable, 
 and equal to the rectilineal figure k aA Lc (Fig. 102.) 
 
 In the second, this lunule, minus the circular segment’a f, 
 will be equal to the rectilineal figure a fk CLA, or the space 
 AKCL, plus the triangle ax f, (Fig. 103.) 
 
 In the third, the same lunule, plus the circular segment 
 aq, will be equal to the rectilineal space ag K cla, or the 
 space ak el minus the triangle ak 6, (Fig. 104.) 
 
 We omit the demonstration, both for the sake of brevity, 
 and because it may be easily conceived from what has been 
 
 “ie already said. 
 
 Hence it may be readily seen, that if the given circles have to each other a 
 certain ratio, which will admit of the angle F ac (Fig. 102, 103.) being constructed 
 with the rule and compasses, in such a manner as to be to the angle EAc, in the 
 reciprocal ratio of these circles, we may draw the line F A, which will cut off from 
 
GEOMETRY. 
 
 the lunule the portion aDcB//f A, equal to an assignable ree 
 tilineal space. Now this will always be the case whe 
 the less circle is to the greater in the ratio of 1 to 2, or 1 to 
 8, or 1 to 4, or 1 to 5, &c.; for the angle Fr ac must then be 
 double, triple, quadruple, or quintuple of ac: in this there 
 is no difficulty. The case will be the same if the less cirele 
 is to the greater in the ratio of 2 to 3, or 2 to 5, or 2 to 7, 
 &c.; or the arc ADC, being susceptible of geometrical tric 
 section, as is often the case, if the greater circle be to the — 
 less as 3 to 4, or 3 to 3, or 3 to 7, &c. 
 
 te 
 
 Another method.—Let aF (Fig.105.) be a tangent to — 
 Fig. 105. the circle ABC in A, and Ara tangent to the arcane } 
 in the same point. Draw the line a c in such a manner, — 
 that the angle FAG may be to the angle EaG, as the™ 
 greater circle is to the less; that is, that the angle FAE 
 shall be to za G, as the greater circle minus the less ig 
 to the latter: the line a will then fall either on ac, 
 or above it, as in a G, or below it as in ag. 
 
 Now, in the first case, it may be easily demonstrated, 
 that the lunule is absolutely squarable. 
 
 In the second, it may be shewn that the same lunule, minus the mixtilineal tri- — 
 angle MGcM, is equal to an assignable rectilineal space. 
 
 In the third, it mey be proved that the same lunule, if the mixtilineal triangle 
 cmg be added, will be equal to the same rectilineal space. 
 
 Lastly, let there be drawn, in any of the preceding figures, between ac and aw 
 as Fig. 105, any line whatever aN, forming with the tangent aE any angle Nax; then, 
 in the angle Fa £, draw another line an, in such a manner, that the angle n a £ shall 
 be to EANasFakE tocar. It may still be demonstrated, that the mixtilineal 
 figure formed of the two arcs Nn, AP, and the two lines an, PN, will be equal toa 
 rectilineal space; which may be found, by dividing the are Nn into as many parts, 
 similar to the are a P, as the number of times that the less circle is contained in the 
 greater: this may be performed geometrically, if the ratio of the one circle to the 
 other be as 1 to 2, or 1 to 3, or 1 to 4, &c. If we here suppose it, for example, to 
 be as 1 to 3, we shall have three equal chords, no, o£, and EN, and the portion of — 
 the lunule in question will be equal to the rectilineal figure anoEN a, since the 
 three segments, 0, 0 E, &c., are together equal to the segment a P. 
 
 Remark 1.—We have also proposed and solved the following problem. 
 
 A lunule, not squarable, but formed by two circles in the ratio of 1 to 2, being given; 
 to intersect it by a line parallel to its base, which shall cut off from it a portion abso- 
 lutely squarable. 
 
 Remark 2.—The following method for dividing circles, &c. is so curious, that it 
 is well deserving of a place here, in addition to the foregoing ways of dividing them 
 into certain portions. 
 
 To divide geometrically Circles and Ellipses into any number of parts at pleasure, and 
 in any proposed ratios. 
 
 Although the learned labours of all ages have failed in their attempts at the geo- 
 metrical quadrature of the circle, and even of the division of the circumference into 
 any number of equal parts at pleasure; yet our own time has furnished the solution 
 of a problem but little less curious, and heretofore esteemed almost, if not altogether, 
 as difficult as it; namely the division of the circle into any proposed number of parts 
 
»y 
 
 GEOMETRICAL PROBLEMS. 173 
 
 hatever, of equal perimeter, and the areas either equal or in any proportion to each 
 other. The solution of this seeming paradox was first published by Dr. Hutton, in 
 his quarto volume of Tracts, in 1786. That curious solution was, in substance, ‘as 
 follows: 
 Fig. 106. Divide the diameter a 3B (Fig. 106.) of the given circle into 
 st as many equal parts as the circle itself is to be divided into, at 
 the points, D, E, &c. Then on the lines ac, aD, az, &c., as 
 diameters, describe semicircles on one side of the diameter a B, 
 and also on the lines BE, BD, BC, &c., on the other side of that 
 diameter; then will these semicircles divide the whole given 
 circle in the manner proposed, viz. into parts which are all equal 
 to each other, both in area and in perimeter. 
 For, the several diameters of the dividing semicircle being in arithmetical pro- 
 gression, and the diameters of circles being in the same proportion as their circume 
 ferences, these also will be in arithmetical progression. But, in such a’progression, 
 the sum of the extremes being equal to the sum of each pair of terms that are 
 equially distant from them; therefore the sum of the circumferences on ac and cB, 
 is equal to the sum of those on ap and vB, and to the sum of those on ak and EB, 
 &e.; and each sum equal to the semicircumference of the given circle on the 
 whole diameter az. Therefore all the parts have equal perimeters; and each pe- 
 rimeter is equal to the whole circumference of the first given circle: which satisfies 
 one of the conditions in the problem. 
 Again, the same diameters being in meg pectiak to each other as the numbers ], 2, 
 3, 4, &c., and the areas of circles being as the squares of their diameters, the semi- 
 circles will be as the square numbers 1, 4, 9, 16, &c., and consequently the differ- 
 ences between all the adjacent semicircles are as the terms of the arithmetical pro- 
 gression, 1, 3, 5, 7, &c.: and here again the sums of the extremes and of every two 
 equidistant means, make up the several equal parts of the circle: which is the other 
 condition of the problem. 
 
 But this subject admits of a still more geometrical form, and is capable of being 
 rendered very general and extensive, and is moreover very fruitful in curious con- 
 sequences. For first, in whatever ratio the whole diameter is divided, whether into 
 equal or unequal parts, and whatever be the number of the parts, the perimeters 
 
 Fig. 107. of the parts will always be equal. For since the cireum- 
 ferences of circles are in the same proportion as their dia- 
 meters, and because a B (Fig. 107.) and ap-++-pB, and ac—+- 
 cB are all equal, therefore the semicircumferences c, aud 6+ 
 d,and a-t-e, are all equal; and constantly the same, whatever 
 pe the ratio of the parts ab, Dc, cB, of the diameter. We 
 shall presently find too that the spaces T v, Rs, and PQ, will 
 be universally as the same parts, AD, Dc, cB, of the dia- 
 meter. 
 
 The semicircles having been described as before mentioned, erect c E perpendicular to 
 ‘AB, and join pe. Then will the circle on the diameter BE, be equal to the space 
 Pa. For, join ak. Now the space Pp is=semicircle on aB — semicircle on ac: 
 ‘but the semicir. on aB==semicir. on aE-+semicir. on BE, and the semicir. on ac 
 = semicir. on AE — semicir. on cre; therefore semicir. a B — semicir. ac =semicir. 
 BE-+semicir. CE, that is, the space P is =semicir. BE + semicir. CE; to each of 
 ‘these add the space Q, or salts semicir. on Bc; then p+ Q@=semicir, BE-+ semicir, 
 cE+ semicir. BC, that is, Pp + @== double the semicir. BE, or = the whole circle 
 “On BE. 
 | In like manner, the two spaces Pp gand ks together, or the whole space P QRS, is 
 
174 GEOMETRY. 
 
 equal to the circle on the diameter Br. And therefore the space R s alone, is equal 
 to the difference, or the circle on BF minus the circle on BE. 
 
 But, circles being as the squares of their diameters, BE*, BF, and these again 
 being as the parts or lines Bc, BD, therefore the spaces PQ, PQRS, RS, TV, are 
 respectively as the lines Bc, BD, cp, aD. And if Bc be equal to cD, then will rq 
 be equal to RS, as in the first or simplest case. 
 
 Hence to find a circle equal to the space R s, where the points p and c are taken 
 
 at random: From either end of the diameter, as A, take a G equal to Dc, erect — 
 
 GH perpendicular to a B, and join au; then the circle on 4 H will be equal to the 
 
 “spacer 8s. For, the space P Q is to the space R Ss, as BC is toc D or A G, that is as 
 BE? to A BH’, the squares of the diameters, or as the circle on B E to the circle on aH. 
 But the circle on B £ is equal to the space PQ; therefore the circle on a H is equal 
 to the space R 8. 
 
 Hence, to divide a circle in this manner, into any proposed number of parts, that 
 shall be in any ratio to one another: Divide the diameter into as many parts, at the 
 points p,c, &c., and in the same ratios as those proposed; then on the several 
 distances of these points from the two ends a and B as diameters, describe the alter. 
 nate semicircles on the different sides of the whole diameter azB; and they will 
 divide the whole circle in the manner proposed. ‘That is, the spaces T Vv, R S, PQ, 
 will be as the lines A D, Dc, c 3. 
 
 But these properties are not confined to the circle alone. ‘They are to be found 
 also in the ellipse, as the genus of which the circle is only a species. Forif the 
 
 Fig. 108. annexed figure be an ellipse described on the axis A B (Fig. 108.) 
 ee On the area of which is, in like manner, divided by similar semi- 
 ellipses, described on A D, A C, B C, BD, as axes, all the semi- 
 perimeters f, a e, bd, c, will be equal to one another, for the 
 same reason as before in the circle, namely, because the periphe- 
 ries of similar ellipses are in the same proportion as their diame- 
 ters. Andthe same property would still hold good, if a B were 
 any other diameter of the ellipse, instead of the axis; describ- 
 ing on the parts of its semi-ellipses which shall be similar to 
 those into which the diameter a B divides the given ellipse. 
 
 And farther, if a circle be described about the ellipse, on the diameter a B, and 
 lines be drawn similar to those in the second figure ; then by a process the very same 
 as before ‘in the circle, substituting only semi-ellipse for semicircle, it is found that 
 the space 
 
 ms 
 
 PQ == the similar ellipse on the diameter zz, 
 PQRSs =the similar ellipse on the diameter B F, 
 RSs == the similar ellipse on the diameter a u, 
 
 or to the difference of the ellipses on BF and B E; also the elliptic spaces P Qa, P QR8, 
 RS, TV, are respectively as the lines B c, BD, DC, A D; being the same ratios as the 
 the circular spaces. And hence an ellipse is divided into any number of parts, in any 
 assigned ratios, after the same manner as the circle is divided, namely, dividing the 
 axis, or any diameter, in the same manner, and on the parts of it describing similar 
 semi-ellipses. 
 
 With respect to the above method of dividing acircle, Dr. Hutton gives the follow- 
 ing anecdote, which we think will be interesting to our readers :— 
 
 ‘* About the year 1770, Mr. James Ferguson, the ingenious lecturer on astronomy 
 and mechanics, in his peregrinations, came to Newcastle, where I then resided, to give 
 his usual course of public lectures; on which occasion, with the assistance of my 
 friends, I not only procured him a numerous and respectable audience, but also ac- 
 commodated him with the free use of the new school rooms, which I had lately built, 
 to deliver his lectures in. As Mr. F. commonly amused my family and friends at 
 
GEOMETRICAL PROBLEMS. 175 
 
 evenings, with shewing his ingenious mechanical contrivances and drawings, on one 
 of these occasions he produced a very neatand correct drawing on a large scale, being 
 a construction of this problem in the prolix way in which it had been given by Mr. 
 Hawney ; but which he exhibited as a great curiosity. I ventured to state to him, 
 that I thought a much simpler construction might be found out for this problem, 
 which was then new to me. As Mr. F. expressed a wish to see such a thing as a 
 simpler construction, which however he seemed to have his doubts of procuring, I 
 was induced to consider it that evening, before going torest, and discovered the con- 
 struction above given. 
 
 ‘The next morning I shewed it to him, and he seemed much pleased with its 
 apparent simplicity, but doubted whether it might be exactly true. On referring him 
 to the avcompanying demonstration, 1 was much surprized by his reply that he could 
 not understand that, but he would make the drawing ona large scale, which was 
 always his way to try if such things were true. In my surprise, I asked where he 
 had learned geometry,—by what Euclid or other book, —wher he frankly stated that 
 he had never learned any geometry, nor could ever understand the demonstration of any 
 one of Euclid’s propositions, _ Accordiugly, next morning, with a joyful countenance, 
 he brought me the construction neatly drawn out on a large sheet of pasteboard, 
 saying he esteemed ita treasure, having found it quite right, as every point and line 
 agreed to a hair's breath, by measurement on the scale.” 
 
 PROBLEM LXIV. 
 Of various other Circular Spaces absolutely Squarable. 
 Ist. Let there be two concentric circles, through which is drawn the line 6B (Fig. 
 109.) a tangent or secant to the interior circle. Drawc a and c B, forming an angle 
 
 ; A cD, and make the are DF in proportion to the are Da, as the 
 Fig. 109. square of c v is to the difference of the squares of c B and cp: 
 
 AY be equal to the rectilineal triangle a c B. 
 
 It is evident that, to render the position of cz determinable 
 by common geometry, the ratio between the ares a p and DF: 
 must be that of certain numbers as 1 to 1,J to 2, 1 to 3, &e., or 
 2 tol, 2to3,&c. Consequently, the difference of the squares 
 of the radii of the two circles, must be to the square of the 
 less, as 1 to 1, or 2 tol, or3tol, &c. Thesectors of the different circles being then 
 in the compound ratio of the squares of their radii and of their amplitudes, we shall 
 have the sector Bc Eequal to acr¥; if the common sector pcF therefore be taken 
 away, and the space A DB be added to both, the rectilineal triangle a c B will be equal 
 to the space AF EB. 
 
 2d. Let there be any sector, as acBGA (Fig. 110.) of which a Bis the chord. 
 In a double, or quadruple, or octuple circle, take a sector ac bga, the angle of which 
 Fig. 110 shall be the half or the fourth, or the eighth part of the angle 
 6 . ACB, which it is possible to do with the rule and compasses ; let 
 this second sector be disposed as seen in the figure, that is to 
 say, in such a manner, that the arcag bshall stand on the chord 
 A xB. We shall then have the space aagb Bea, equal to the 
 rectilineal figure E c F c, minus the two triangles aa © and 
 ¥ OBYF. 
 This is almost evident; for, by the above construction, the sector AcBG is equal 
 to acbg; if the part therefore whichis common be taken away, there will be an 
 equality between what remains on the one hand, viz. the kind of lunule acs bga, 
 plus the two triangles Aaxr, and B 6 Fr, and what remains on the other, or the recti- 
 
 ae Ai if c x be then drawn, we shall have the mixtilineal space ABEF 
 is a 
 
176 GEOMETRY. 
 
 lineal figurezcrc: this kind of lunule therefore is equal to the above rectilineal 
 figure, diminished by these two triangles. Y 
 Fig. 111. 3d. If two equal circles cut each otherin a and B (Fig. 111.) ~ 
 and if any line a c be drawn intersecting the interior arc in xz, — 
 and the exterior in c, it is evident that the are E B will be equal 
 to the are Bc; and consequently the segment & B will be equal — 
 to the segment Bc. Hence it follows, that the triangle formed — 
 by the two arcs E B and 8 6, and the straight line Ec, will be 
 equal to the rectilineal trianglez Bc. Lastly, that if a p bea 
 tangent in a, to the arcA EB; the mixtilineal figure azBep, 
 will be equal to the rectilineal triangle a D B. 
 Fig. 112. 4th. If two equal circles touch each other inc (Fig. 112.) and 
 ifa third equal circle be described through the point of contact; — 
 the curvilineal space AFC EDBA will be equal to the rectilineal — 
 
 hs 
 sf (22s quadrilateral ABDC. 
 
 a 3 For ifc B be drawn a tangent to the first two circles, the space — 
 
 comprehended by the arcs cF a and a Band the straight linecs, 
 bp: is equal to the rectilineal triangle c a B, as has been shewn 
 already. The case is the same with the mixtilineal space cEDB 
 in regard to the triangle cp B: therefore, &c. 
 
 5th. Theabove remark was made by M. Lambert, in the ‘‘ Acta Helvetica,” vol. 
 ii. But other spaces of the same form may be found equal to rectilineal figures, 
 though bounded by circular arcs, two of which only are equal. 
 
 Let aBcp (Fig. 113.) bea circle, 
 Fig. 113. from which it is required to cut off, 
 aT ee by two other circular arcs, a space 
 of the above kind absolutely squar- © 
 able. On an indefinite right line 
 make the parts c E, EF, FH, each 
 equal to the side of the square in- 
 : ‘ scribed in the given circle: and let 
 = a Peo Grit the third part rH be divided into © 
 . two equal parts in @: on the ex- 
 tremity of cE raise the perpendicular £1, and let it be intersected in 1, by a circle 
 described from G as a centre with the radius cc. Draw c1, and make c k equal to 
 it; lastly, on F @ describe a semicircle, cutting, in L, the line KL perpendicular to 
 FG; draw HL, and in the given circle make the chords A B and a D equal to it. 
 If with a radius equal to cx, there be then described ares, passing through the points 
 A and B, A and D, with their convexity turned towards c; we shall have the space 
 bounded by the arcs AB, AD, and BCD, equal to the rectilineal space formed by the 
 chords AB, AD, and the four chords D M, Mc, CN, and nB, of the four equal portions 
 of the arespcn. 
 
 But, as enough has been said on this subject, we shall only add one reflection, 
 which is, that these quadratures ought not to be considered as real quadratures of a 
 curvilineal space. All the marvellous in these operations, as M. de Fontenelle has 
 very properly remarked, consists in a kind of geometric legerdemain, by means of 
 which as much is dexterously added on the one hand, to a rectilineal space, as is 
 taken from it on the other. It was not in this manner that Archimedes first 
 squared the parabola, and in which modern geometricians have given the quadrature 
 of so many other curves. All these things however appeared to us sufficiently curious 
 to entitle them toa place in a work of this nature. 
 
 N 
 D 
 
 ae 
 
GEOMETRICAL PROBLEMS. | 97) 
 
 PROBLEM LXxy. 
 Of the measure of the Ellipse or Geometrical Oval, and of its parts. 
 
 It may be easily demonstrated, that the ellipse (Fig. 114.) 
 is to the rectangle of its axes ABand DE, as the circle is to 
 the rectangle of its axes, or to the square of its diameter a B, 
 since each axis is equal to the diameter. 
 
 Thus, as the circle is 4! nearly of the square of its diame- 
 ter, the ellipse is also {4 of the rectangle of its axes. 
 
 Nothing then is necessary, but to multiply the rectangle of 
 
 : the axes of the given ellipse by 11, and to divide the product 
 by 14; the quotient will give the area. 
 
 We shall here add, that each segment or sector of the ellipsis is always in a given 
 ratio to the sector or segment of a circle, as is easy to be determined. Let the 
 
 ; elliptical sector Fc G, (Fig. 115.) for example, or the segment 
 
 Fig. 115. FBG, be given: on the axis AB describe a circle from the 
 
 centre c; and if G¥ be continued to pand &, we shall have 
 
 the elliptical sector Fc GB to the circular sectorD c EB, asFG 
 
 to DE, or as the less axis of the ellipsis is to the greater: the 
 
 elliptical segment BFG will also be to the circular segment 
 
 DBE, as FG toD&, or as the less axis of the ellipsis to the 
 greater. 
 
 Let there be likewise, in an ellipsis, any segment whatever, 
 asnop. On the axes let fall two perpendiculars from n and p, and continue them 
 till they meet the circlein N and P; if NP be then drawn, we shall have the segment 
 nop to the circular segment Nop, in the same ratio as the less axis is to the 
 greater. From thisis deduced the solution of the following problem. 
 
 PROBLEM LXVI. 
 To divide ihe sector of an ellipsis into two equal parts. 
 Fig, 116. Let it be required, for example, to divide the elliptical sector 
 | pcs (Fig. 116.) into two equal parts, by a line asca, 
 On the diameter 4B describe a circle; and having drawn pr 
 perpendicular to A B, continue it to E, and draw Ec, which will 
 - give the circular sector ECB; divide the arc EB into two equal 
 partsinr, and draw Fu perpendicular to the axis aB; then 
 from the centre c, to the point a, where that perpendicular cuts the ellipsis, draw 
 ‘the line cc, the elliptical sector Bc will be equal to Gc D, as the circular Bc F is 
 tOFCE. 
 
 The case would be the same if the sector were equal to the 4th part of an ellipsis, 
 or any higher part ; and also if the sector were comprehended between any two semi- 
 diameters of the ellipsis, as pc and d c. 
  Inthis case, from the points p and d, let fall on the axis the perpendiculars p1 
 and di, which when continued will cut the semicircle ArBin E and e; divide the 
 ‘arc Ee into two equal parts inf, and draw fh perpendicular to a x, cutting the 
 ellipsis ing: the line cg will divide the sector pc d into two equal parts. 
 
 PROBLEM LXVII. 
 
 A carpenter has a triangular piece of timber ; and, wishing to make the most of tt, ts 
 desirous to know by what means he can cut fromit the greatest right-angled quadran- 
 gular table possible. In what manner must he proceed ? ; 
 
 . N 
 
 | 
 } 
 
 -— 
 
178 : GEOMETRY. 
 
 Fig. 117. Let the given triangular piece of timber be a Bc (Fig, 
 zB 117.) Divide the two sides a B, Bc, into two equal parts, 
 in F and Gc, and draw Fa: then from the points F and @, 
 
 } N draw FH and G1 perpendicular to the base: the rectangle © 
 A c FI, will be the greatest possible that can be inscribed in the © 
 + * triangle, and will be exactly the half of it. 
 
 If the triangle be right-angled at a (Fig. 118.) the question may be 
 Fig. 118. solved in two different ways, by which there may be obtained the two 
 rectangular tables ¥ ¢ and F1, which will each be the greatest inscript-— 
 ible in the given triangle, and both equal. 3 
 When the triangle has all its angles acute, the solution will be 
 ¥ different according to the side assumed as base. There will conses— 
 /J\/\ quently be three, and each will give a table more or less elongated, 
 and always of the same area, otherwise the greatest would exclusively 
 solve the problem: such are the rectangles F1, GL, and Kk m (Fig.119,) 
 
 But the carpenter having consulted a geometrician, the latter 
 observed that it would be most advantageous to convert this 
 piece of timber into an oval table: and in what manner then 
 must he proceed to trace out on it the greatest oval pos. 
 sible. 
 
 Let the given triangular piece of wood, as before, be ABe 
 (Fig. 120.) First divide each side into two equal parts in F, D, 
 and ©; these three points will be the points of contact where 
 the ellipsis touches the sides of the triangle; if the lines a E, 
 cF,and Bp be then drawn, intersecting each other in «, the 
 point G will be the centre of the ellipsis. 
 
 Then, make G L equal to G £, and through @ draw & 0, 
 parallel to Be, and through the point p draw pq parallel to 
 AE; then take ap, a mean proportional between a@ and co: if 
 the triangle B ac be isosceles, the lines eu and cr will be the 
 semi-axis of the ellipsis; and we have already shewn in what manner an ellipsis may 
 be described when the two axes are given. 
 
 But if the angle LGP be acute or obtuse, the ellipsis may be traged out at once 
 by means of an instrument, described in Prob. xxii.; for it is of little importance 
 whether the angle of the two given diameters be a right angle or not. This method 
 will always be equally successful ; with this only difference, that when the above 
 angle is not aright angle, the portions of the ellipsis, described in the two adjacent — 
 angles, LG P and uGR, will not be equal and similar. | 
 
 The two axes may be determined also directly: the method may be found in books 
 on conic sections, and to these we must refer, as the nature of this work will not 
 admit of entering deeply into the subject. 
 
 PROBLEM LXVIII. 
 
 The points sw and c (Fig. 121.) are the adjutors of two basons in a garden, and A is 
 the point where a conduit is introduced, and to be divided into two parts, in order to 
 supply B and c with water. Where must the point of separation be, that the sum of 
 the three conduits, aD, DB, and DC, and consequently the expense in pipes, shall be 
 the least possible 2 
 This problem, which belongs to that branch of civil engineering that relates to 
 
 the conveyance of water, when reduced to geometrical language, may be enounced — 
 
 as follows: Ina triangle a Bc, to find a point, from which if three lines be drawn 
 to the three angles, the sum of these lines shall be the least possible. Now it is 
 
a 
 4 
 ty 
 + 
 
 fy GROMETRICAL PROBLEMS, 179 
 
 evident that there must be such a point, and that its position being found, the 
 expense in pipes will be less than if the point of separation were assumed in any 
 other place. ' 
 
 It would be tedious to explain the reasoning by means of which this problem is 
 solved; and it would be impossible to employ calculation without great prolixity. 
 ‘We shall therefore only observe, that it may be demonstrated, that the required 
 point D must be so situated, that the angles apc, aDB, 
 Fig. 121. and BDC shall be equal to each other, and consequently 
 each equal to 120°. 
 
 To construct this problem, on the side ac as a chord 
 
 describe an arc of a circle apc, capable of coutaining an 
 angle of 120°, or equal to one third of the circle of which 
 it forms a part; if the same thing be done on another of 
 the sides, as Bc, the intersection of these two circular 
 arcs will determine the required point p; and it is from 
 this point that the conduit must be divided, in order 
 : to be conveyed thence to B and to c. 
 : Such, at least, would be the solution of the problem if the three pipes ap, pc, and 
 DB, were all to be of the same bore. But an intelligent engineer will not make the 
 : pipes equal in size; he will be sensible that to give greater height to the jet, it will 
 be proper that the pipes ps and pc should not together admit a greater quantity 
 of water than the pipe ap, otherwise the water in these pipes, after coming from 
 the pipe D, would be in a state of stagnation, and would not receive the impulse 
 necessary to make it rise to its greatest height. 
 
 The solution of the problem, in this new case, is as follows: We shall suppose 
 that the bore of the pipe aD, or its capacity, is exactly double that of the other two; 
 that is to say, that the diameters are in the ratio of 10 to 7 nearly; for by these 
 means the water will always sustain an equal pressure in the former and in the two 
 latter. We shall suppose also, that the price of the foot of each kind of these pipes 
 is in the same ratio, because, in economical problems of this sort, it is the ratio of 
 the prices that ought chiefly to be considered. 
 
 These things being premised, we shall find that the point of separation of prt 
 
 pipes ought to be in d, so situated, that the angles cd a and Bd a shall be equal, and 
 of such a nature, that the sine of each shall be to radius as 10 is to 14; or more 
 _ generally, as the price of the foot of the larger pipe is to double that of the smaller. 
 Hence it will be easy, according to this hypothesis, to determine the angle, which 
 _ will be found to be 132° 56 or near 133°. 
 If on the sides aB and ac then, of the triangle apo, there be described two cir- 
 cular arcs, each containing an angle of 133°, their point of section will be in d, where 
 the main pipe ought to be divided, to convey water to Band c, so as to incur the least 
 possible expense in pipes. 
 
 Fig. 122. Remark.—By extending this problem, we may suppose that the 
 -. main pipe is to convey water to three given points, B, c, E, (Fig. 122.) 
 1 In that case it may be demonstrated, that if the four pipes were 
 j equal, the point of separation could not be placed more advantage. 
 Ps ously, at least for diminishing the quantity of the pipes, than in 
 the place where the lines a E and Bc intersect each other; but this 
 £ perhaps would not be the most advantageous disposition for making 
 
 the water to be thrown up with the greatest force. 
 _ The same observation, made in regard to the first solution of the problem, may be 
 | made here also. To give greater force to the jet, the main pipe ought to be nearly 
 triple in size to each of the rest. Let us suppose then that the price of a 
 
 wn 2 
 
 { 
 
 —————— 
 
’ 
 
 1890 GEOMETRY. 
 
 foot of the former, is to that of a foot of the others, as mis ton; and in the last 
 place, to simplify the problem, the solution of which would be otherwise exceed< 
 
 ingly complex, we shall suppose that the lines at and Bec cut each other at right 
 
 angles: this being the case, it will be found, that the angle ere ought to be such, 
 that, radius being unity, the cosine of it-shall be $n./ 4nn —(m — 1), or, what 
 
 amounts to the same thing, the sine of the angle pc ¥F must be equal to the above © 
 
 expression. 
 If we suppose then, for example, that m is to n as 5 to 3, we shall have the above 
 
 expression equal to 0°71496, which is the sine of an angle of 45° 38’. If the angle 
 pcF therefore be made equal to from 45° to 46°, the point F will be that where the ~ 
 .principal pipe ought to be divided. 
 
 If m were ton as 2 to 1, the above expression would become equal to 0°86600, 
 which is the sine of an angle of 60°; in this case therefore the angle pc F ought to 
 be made equal to 60°, or each of the angles prc and DF B equal to 30°. 
 
 It is here evident that, to render the problem susceptible of a solution, m and 2 
 
 ' must be such, that the above expression shall not be imaginary, nor greater. than 
 
 unity. In either of these cases there could be no solution; and this would indicate, 
 at most, that the division ought to be made at the point a, or at as great a distance 
 
 as possible from the line pc. This expression also must not be = 0: in that case © 
 
 we ought to conclude that the division should be made at tle point b. 
 
 PROBLEM LXIX. ‘ bz: 
 
 Geometrical paradoz of lines which always approach each other, without ever being 
 able to meet or to coincide. 
 
 Every person, in the least acquainted with geometry, knows, that if two straight 
 lines, in the same plane, approach each other, they will necessarily meet in a common 
 
 point of intersection. We say in the same plane, for it they were in different planes, 
 
 ea 
 
 it is evident that they might approach till a certain term,,without cutting each other, — 
 
 and that they would then diverge from each other more and more. If we suppose, 
 for example, two parallel and vertical planes, on one of which is drawn a horizontal 
 line, and on the other one inclined to the horizon, it may be readily conceived that 
 they will not be parallel, and yet they can never intersect each other, their least dis- 
 
 tance being necessarily that of the two planes. Here then we have two lines not — 
 parallel, which never meet: but this is not the sense in which the problem is 
 
 understood. 
 It may be demonstrated that there are many lines, and in the same plane, which 
 continually approach each other, and which however can never meet. ‘They are 
 
 indeed not straight lines, but a curve combined with a straight line, or two curved - 
 
 lines together. We shall here give a few examples of these lines, which are very 
 familiar to those who are versed in the higher geometry. 
 Fig. 123. In the indefinite straight line a e, (Fig. 123.) take equal parts 
 
 pendiculars Bb, cc, Dd, Ee, &c., which decrease, ‘acookien toa 
 progression, no term of Ane can become 0, though it may 
 become indefinitely small; let these terms decrease, for exam- 
 ple, according to the progression 1, 3,4,3,4,2, &c.: it is evi- 
 dent that the curve passing through the summits of the lines, 
 decreasing according to this progression, can never meet the 
 line G, however far continued, since its distance from that line can never become 0; 
 it will however approach it more and more, and in such a manner, as to be nearer it 
 than_any quantity, however small. This curve, in the present case, is that so well 
 known to geometricians under the name of the hyperbola; which has the property 
 
 AB, BC, CD, &c.; and from the points 8, c, p, &c., raise the per- 
 
 of being contained between the branches of two rectilineal angles, having their — 
 
GEOMETRICAL PROBLEMS. — 18l 
 
 vertices opposed to each other, towards which it approaches more and more, without 
 ever touching them. , 
 
 If the progression, according to which these lines B 8, ce, pd, &c. decrease, were 
 
 1, 3,4) a> te &c., the line passing through the points 4, c,d, e, &c., would still ap- 
 proach more and more to the straight line a.c, without ever meeting it; for whatever 
 might be the distance of any term of this: progression, it could never become = 0. 
 _ Another example.—Without the indefinite line ar, 
 (Fig. 124.) assume any point p, from which draw Pa 
 perpendicular to ag, and any other lines at pleasure, PB, 
 PC, PD, &c., more and more inclined; in the continuation 
 of which make the lines aa, Bb, ce, &c., always equal: 
 it is evident that the line passing through the points a, 
 b,c, d, &e., never can meet the line ar, though it may 
 approach it more and more, and nearer than any deter- 
 minate quantity; because F f becomes more and more inclined. This curve is that 
 known to geometricians by the name of the conchoid, and was invented by Nico- 
 medes, a Greek geometrician, to serve for the solution of the problem respecting two 
 mean proportionals, 
 
 A great many other examples might be found in the higher geometry ; but these. 
 will be sufficient for our purpose. 
 
 PROBLEM LXX. 
 
 In the island of Delos, a temple consecrated to Geometry was erected, on a circular 
 basis (Fig. 125.), and covered hy a hemispherical dome, having four windows in its 
 circumference, with a circular aperture at the top, so combined, that the remainder 
 of the hemispherical surface of the dome was equal toa rectilineal figure ; and in 
 the cylindric part of the temple was a door, absolutely squaradle, or equal to a rec- 
 tilineal space. What geometrical means did the architect employ in the construction 
 of this monument 2 
 
 Fig. 125. ' Every person, acquainted with the principles of geometry, knows 
 that the measure of a hemispherical surface depends on that of 
 
 ' the circle, which is equal to the surface of a cylinder having the 
 samé base and the same altitude, The ingenuity of this construc- 
 tion then was, Ist. ‘To have cut from the dome, by the apertures 
 above mentioned, spherical portions of such a nature, that the 
 remainder should be equal to a figure purely rectilineal. 2d. To 
 have described in the cylindric part, or circular wall of the temple, 
 another figure which was squarable. The method that might 
 have been employed is as follows : 
 
 Let us first suppose a fourth part of the hemispherical dome, 
 having for its base the quadrant aczg (Fig. 126.) Take the are 
 BD, equal to one fourth of the are aB, as the breadth of the are 
 that ought to separate the windows; and draw ap the chord of 
 the remainder. Now let sce be any section whatever, through 
 the axis of the dome sc, and let its intersection with a p be F; 
 make CE, CF, CG, continually proportional; in the axis cs make. 
 the line cu equal to EG, and draw 1 parallel to cB, which will 
 intersect the quadrant su in 1: then will r be one of the points 
 of the window required; and the series of points 1, determined 
 in this manner, will give the contour of that window, the surface 
 
 of which will-be equal to double the segment a £ D, while the spherical portion s ar ps - 
 
 Will be equal to double the rectilineal triangle c aD. 
 
182 GEOMETRY. 
 
 this triangle, plus the spherical sector spB, which is equal to double the circular 
 sector cpB, or to the fourth of the spherical sector saEB; if from this sector, 
 therefore, there be cut off the fourth part suM, by a plane parallel to its base, and 
 distant from the vertex s by the fourth part-of the radius sc, the remainder of this 
 hemispherical quadrant, that is to say the surface arp B MLA, will be equal to double 
 the rectilineal triangle cab. If the other quadrants of the hemispherical dome be 
 then made similar to the present one, the whole dome, the apertures deducted, will 
 be equal to eight times the triangle acv. 
 In regard to the aperture to be made in the circular wall of the temple, and which 
 must be equal to a rectilineal space, nothing is easier; though it be a part of a cylin- 
 Fig. 127. dric surface. Let anprF (Fig. 127.) represent one half of this 
 surface; assume, as the breadth cf the door to be formed, the 
 chord Gu, parallel to the diameter ap; make Gi and HK, 
 
 may have that proportion which good taste and the character of 
 
 the work require ; if through the points 1, K, and the line a p, 
 PA a plane be then made to pass, which by its intersection with the 
 Ceti cylindric surface will determine the curve rL kK, we shall have the 
 
 cylindric aperture G B H K1, a little arched at the top, which will be to the rectangle — 
 
 of cB by Gu, as the sine of the angle Lc Bis to the sine of half the right angle. 
 The problem of the Greek geometrician therefore is solved. 
 
 This problem might be varied a great many ways, During my dreary residence, in 
 1758, at a post in Canada, I amused myself with these variations, and I resolved the 
 problem by making the whole of the surface of the temple absolutely squarable. I 
 left only one aperture in the dome, viz.a hole at the top, like that ofthe Pantheon at 
 
 Rome, and I made the four windows in the cylindric part of the temple, &c. All this . 
 
 however will be easy to any one versed in geometry. 
 
 Remarks.—1. This problem is nearly the same as that proposed by Viviani, in 
 1692, under the title of ‘‘ Enigma Geometricum,” which was easily solved by Leib- 
 nitz, Bernouilli, and the Marquis de ’Hépital. An account of it may be seen in my 
 ‘** History of the Mathematics,” vol. ii. book i. Viviani’s solution is ingenious and 
 elegant ; but as the dome, according to this solution, would not be susceptible of con- 
 struction, because it would bear upon four points, which in architecture is absurd, we 
 have made some changes in the enunciation, by adding the circular aperture at the top. 
 By these means the dome will bear upon parts that have some solidity, each window 
 being separated from the other by an are which forms a sixth part of the whole cir- 
 ference. 
 
 2. Father Guido-Grandi has remarked, that if a polygon, for example the triangle 
 ABC (Fig. 128.) be inscribed in the circular base of acone, and if on each side of 
 
 this polygon a plane be raised perpendicular to the base, the por- 
 
 Fig. 128. tion of the conical surface, cut off towards the axis, is equal to a 
 
 rectilineal space. For it may be easily demonstrated that this sur- 
 
 face is to that of the rectilineal polygon a Bc, which correspends to 
 
 it perpendicularly below, as the surface of the cone is to the circle 
 
 of its base; that is to say, as the inclined side of the cone s D, 
 is to £ DT the radius of that base. 
 
 The portions also of the cone cut off by the above planes, 
 towards the base, are evidently in the same ratio with the seg- 
 ments of the circle on which they rest. In fact, whatever figure 
 be inscribed in the base, if we conceive a right cylindric surface 
 
 which are perpendicular to the base, of such asize, that the door - 
 
 ' 
 
 The whole surface of this fourth part of the dome will be equal then to double 
 
OS 
 t on 
 
 A COLLECTION OF PROBLEMS. 183 
 
 raised from the circumference of the figure, it will cut off from the conical sur- 
 face a portion which will be to it in the same ratio. 
 
 This Italian geometrician, who was of the order of the Camaldules, thought proper 
 to give to this conical portion absolutely squarable, the name of ‘ Velum Camal- 
 dulense.”’ In like manner, a Franciscan took it into his head to construct a sun-dial 
 
 on a body which resembled a sandal, and to print a description of it, under the title of 
 *¢ Sandalion Gnomonicum.” 
 
 PROBLEM LXXI. 
 
 Tf each of the sides of any irregular polygon whatever, as ABCDEA (Fig. 129.) 
 be divided into two equal parts, as ina, b,c, d, e; and if the points of division 
 in the contiguous sides be joined; the result will be a new polygon abcdea: 
 if the same operation be performed on this polygon; then on the one resulting 
 
 Jrom it; and so on ad infinitum; it is required to find the point where these 
 divisions will terminate. 
 
 Fig. 129. This problem, impossible to be resolved perhaps by considera- 
 Aas a tions purely geometrical, is susceptible of a very simple solution, 
 ¢ deduced from another consideration, and which shall be given in 
 z A a subsequent page. 
 eos In the mean time our readers may exercise their ingenuity 
 
 upon it, as we shall only add, that. it was proposed in 1750, by M. D 
 he had it from M. Buffon. 
 
 » who said 
 
 A COLLECTION 
 
 OF VARIOUS PROBLEMS, BOTH ARITHMETICAL AND GEOMETRICAL; 
 THE SOLUTION OF WHICH IS PROPOSED BY WAY OF EXERCISE TO 
 MATHEMATICAL READERS. 
 
 Tuosr who study mathematics cannot begin too early to exercise their talents with 
 the solution of the problems presented by that science; for it is by such exercise that 
 the inventive faculty is called forth and strengthened. We have therefore thought 
 it our duty to subjoin to this part of the Mathematical Recreations, a selection of 
 problems proper for exercising and amusing young mathematicians, They are of 
 different degrees of difficulty, that they may be suited to the different capacities of 
 those who read this work. Some curious theorems have been inserted among them; 
 
 and, as the demonstration of these is required, they may serve also to exercise their 
 ‘ingenuity. 
 
 It may be here observed, that, as the most of these problems are far ftom being 
 difficult if the resources of algebraic calculation be employed, it is therefore proposed 
 that the solutions of themshould be found by means of pure geometry ; as it is well 
 known that algebraic analysis gives, for the most part, complex solutions, while those 
 which arise from analysis purely geometrical are far more simple and elegant. 
 
 ARITHMETICAL AND GEOMETRICAL PROBLEMS AND THEOREMS. 
 
 Prosiem 1.—In aright-angled triangle, given the base, the sum or difference of 
 the other two sides, and the area, to determine the triangle ? 
 
 Pros. 11.—Given the base, the ratio of a) other two sides, and the area, to deter- 
 mine the triangle. 
 
 Pros, 111.—The base, the angle comprehended by the two other sides, and the area 
 being given, to determine the triangle. 
 
184 . - * GEOMETRY. 
 
 Pros. tv.— Three lines being given in position, on a plane, to draw another line 
 through them, which shall be cut by them into two parts, in a given ratio. a. 
 
 Pros. v.—Four lines being given in position on a plane, to draw another line 
 through them, which shall be cut into three parts, in a given ratio. 
 
 Prog. vi.— What is the probability of throwing anace, or any one of the faces of@ 
 die, in three throws; that is, either at the first, second, or third throw ? 
 
 Pros. vir.—Atthe game of Piquet, A is first in hand, and has no ace ; what pro- 
 bability is there that he will take in from the pack either one, or two, or three, 
 or four aces, 
 
 Pros. vii1r.—What is the probability of throwing one ace, and no more, in four 
 successive throws ? 
 
 Prog. 1x.—In alottery, where the number of blanks is to that of the prizes as 39 
 to .1,as was the case in the year 1720, how many tickets must be purchased that the 
 buyer may have an equal chance for one or more prizes ? 
 
 Pros. x.—Ifa man has in his hand a certain number of pieces of money, as for 
 example 12, how much may be betted to 1 that in tossing them all up at once or 
 separately, there shall be as many heads as tails. 
 
 Prog. x1.—Four lines being given of such a nature, that any three of them are 
 together greater than the fourth, to construct of them a quadrilateral figure inscrip- 
 tible in a circle, or which can be circumscribed about it ? a 
 
 Theorem 1.—If from the three angles of any right-angled triangle, three lines be 
 drawn perpendicular to the opposite sides, they will all cut each other in the same 
 point. | 
 
 Theor. 2.—-If lines be drawn from these angles, dividing each of them into two 
 equal parts, or cutting the opposite sides into two equal parts, these three lines will 
 all pass through the same point. y 
 
 Pros. x11.—A trapezium being given, to divide it into two‘equal parts, or in any 
 given ratio, by a line passing through a given point, éither in one of the sides, or 
 within the trapezium, or without it. 
 
 Pros. x111.—Ina given circle to inscribe an isosceles triangle of a given magnitude. 
 —Itis evident that this triangle must be less than the equilateral triangle, inscribed 
 in the givencircle; for the latter is the greatest of all those that can be inscribed 
 in it. 
 
 Pros. xiIv.—To circumscribe about a given circle an isosceles triangle off a 
 given magnitude.—This triangle must be greater than the circumscribed equila- 
 teral triangle ; since the latter is the least of all those that can be circumscribed. 
 
 Pros. xv.—In an isosceles triangle to describe three circles, each of which shall 
 touch two sides of the triangle, and which all three shall touch each other. 
 
 Pros. xvi1.—To do the same thing in a scalene triangle. 
 
 Prog. xvit.— What is the value of this analytical expression, rJ/ 2 / 2am &C., 
 in infinitum ?—The answer is 2; but a demonstration is required. In like manner 
 the value of YW 2,/ 3 3, &c., in infinitum, is 3; and so of any other number. 
 
 Pros. xvrire—In a pyramid, * four triangular faces, if the sides of these four tri- 
 angles be given; required the angles formed by the faces of this pyramid, the perpen- 
 dicular let fall from any of the angles.on the base, and the solidity of the pyramid. 
 
 Pros. x1x.—To cuta given trapezium into four equal parts, by lines intersecting 
 each other at right angles. 
 
 Pros. xx.—A gentleman has an irregular quadrangular piece of ground, from which 
 he is desirous, for the purpose of making a parterre, to cut the largest oblong pos- 
 sible, with its angles touching the sides of the quadrilateral: how is this to be 
 done ? 
 
 _ Pros. xxt.—Given the area of a nent dagied triangle, and the sum of the ute 
 sides, to determine the triangle. 
 
 . 
 x 
 
A COLLECTION OF PROBLEMS. 185 
 
 Prox. xx11.—If froma pack, consisting of 52 cards, 13 of each suit, 5 cards be dealt 
 to one person; what is the chance that two of them shall be trumps, or of any suit 
 that is proposed. 
 
 Pros. xx111.—About a given circle to circumscribe a triangle, of a given perime- 
 ter ; provided this perimeter be greater than that of the equilateral triangle circum- 
 scribed. 
 
 Pros. xx1v.—In a triangle, not equilateral, to find a point which, if three per- 
 pendiculars be drawn to the three sides, they shall be together equal to a given line. 
 —wWe have excluded the equilateral triangle, because it may be easily demonstrated, 
 that from whatever point, within such a triangle, perpendiculars are let fall on the 
 
 sides, their sum will be always the same. 
 
 The case is the same in regard to every regular polygon; and even those that 
 are irregular, provided the sides are equal. 
 
 Pros. xxv.—In a given circle toinscribe an isosceles triangle, or to circumscribe 
 about it a triangle of a given perimeter.—This problem not being always possible, 
 as may be easily seen, it is required to assign its limitations. 
 
 Pros. xxvi.—In a given circle to inscribe, or to circumscribe about it, any 
 triangle whatever, of a determinate perimeter. 
 
 Pros. xxvit.—In a given quadrilateral to inscribe an ellipsis; that is to say, 
 to describe in it an ellipsis which shall touch its four sides. 
 
 Pros. xxvit.—A jeweller has a valuable plate of agate, in the form of an irre- 
 gular trapezium, andis desirous to cut from it the largest oval possible for the lid 
 of a snuff-box: in what manner must he proceed ?—It is evident that this problem 
 expressed geometrically is as follows: In a given quadrilateral to inscribe the 
 largest ellipse possible: a problem which is certainly not easy. It is proper to in- - 
 form those who may be disposed to try it, that it requires a profound knowledge of 
 
 analysis. The foilowing also might be proposed: 
 
 About a given quadrilateral to circumscribe the least ellipsis possible. 
 
 Pros. xx1x.—A point and a straight line being given, in what line will be 
 found the centres of all the circles passing through the given point, and touching 
 the given line. 
 
 Prog. xxx.—Required the same thing in regard to all the circles that touch a 
 given circle and a given straight line.— This straight line may be without the given 
 circle; or it may touch it, or intersect it. 
 
 Pros. xxx1.—Any two circles being given, in what line will be found the 
 centres of all the circles that touch the given circles: whether the touching circle 
 comprehends them both within it, or touches the one without and the other 
 within ? 
 
 Pros. xxx11.—The base of a triangle, the sum of the two other sides, and the 
 line drawn from the vertex to the middle of the base, being given; to determine 
 the triangle. 
 
 Pros. xxxu1.—Given the three lines, drawn from the angles ofa triangle to the 
 middle of each of the opposite sides; to determine the triangle. 
 
 Pros. xxx1tv.—Given the base of a triangle, and the sum and the difference of 
 the squares of the sides, to determine the triangle—This problem is susceptible of a 
 very simple and very elegant construction; for the vertex of this triangle is in the 
 circumference of a certain circle, and is also in a certain straight line. 
 
 Pros. xxxv.—Given the three lines drawn from the angles of a triangle to the 
 opposite sides, dividing each of these angles into two equal parts; to determine the 
 triangle. 
 
 Pros. xxxvi.—Any number of points being given, to draw a shrateht line among 
 them in such a manner, that if a perpendicular be let fall on it from each of these 
 
 4 
 
186 er GEOMETRY. 
 
 points, the sum of the perpendiculars on the one side, shall be equal to the sum of 
 those on the other side. | 
 
 Pros, xxxvit.—The same suppositicn being made, it is required that the sum of 
 the squares of the perpendiculars drawn on the one side, shall be equal to the sum of 
 the squares of those on the other; or that the sum of these perpendiculars, raised 
 to any power whatever n, shall be on both sides equal. 
 
 Pros. Xxxvur.—In any trapezium, given the four sides and the area, to deter- 
 mine the trapezium. 
 
 Pros. Xxx1x.—An angle being given, to find a point from which if two perpendi- 
 culars be let fall on its sides, the quadrilateral formed by them and the sides of the 
 angle, shall be equal to a given square. 
 
 Pros. xu.—As there are an infinite number of points which will answer the pro- 
 blem, it is proposed to find the line traced out by them, or the curve which they 
 form. 
 
 Pros. xL1.—To find four numbers in arithmetical progression, to which if four 
 given numbers, such as 2, 4, 8, 17, be added, their sums shall be in geometrical pro- 
 gression. 
 
 Pros. xLi1.—Two couriers, a and z, set out at the same time; a from Paris 
 for Orleans, the distance between whichis 60 miles, and n from Orleans for Paris ; 
 and they travel at such a rate that a reaches Orleans 4 hours after meeting B, 
 and B reaches Paris 6 hours after meeting a: how many miles per hour did each 
 travel ? : ; , 
 
 Pros. xtu1.—A certain sum, placed out at interest, amounted at the end of a 
 year to £1100, and at the end of eighteen months to £1120; what was the sum, and 
 at what rate of interest was it lent ? 
 
 Pros. xi1v.—Two bills of exchange, one of £1200, payable in 6 months, and 
 the other of £2000, payable in 9 months, were discounted at the same time, and 
 at the same rate of interest, for £120, at what rate of interest were they dis- 
 counted ? 
 
 Pros. xLv.—How many ways can £100. be paid by guineas, at 21 shillings, and 
 pistoles at.17 shillings, each ? 
 
 Pros. xtv1—An angle and a point within it being given, to draw through that 
 point a straight line intersecting the two sides of the angle, in such a manner, that 
 the rectangle of their segments towards the vertex, shall be equal to a given square. 
 This given square must not be less than a certain square, which gives rise to the 
 following problem. 
 
 Pros. xivi1r.—The same supposition being made as in the preceding case, re-. 
 quired the position of the line passing through the given point, when the rectangle of 
 the sides of the angle cut off towards the vertex is the least possible. 
 
 Pos. xivit.—Three lines being given in position, to find a point from whieh 
 the three perpendiculars drawn to these lines shall be in a given ratio. We shall 
 here observe that this problem is susceptible of a very simple and very elegant 
 solution, without calculation. 
 
 Pros. XL1x.—Given two circles in a given ratio, as of 1 to 2, for example, and 
 which cut each other, but in such a. manner as not to form a squarable lunule; it is 
 proposed to draw through these circles a line parallel to that which joins the 
 peints of intersection, so that the part of the lunule cut off above may be eqnal to 4 
 rectilineal space. . 
 
 Pros. L.—The same supposition being made, it is proposed to cut the two 
 circular arcs by a third, which shall be of such a nature that the concavo-con- 
 vex triangle, formed by these three arcs, shall be equal to a rectilineal space, if 
 - possible, 
 
A COLLECTION OF PROBLEMS. 187 
 
 Pros. 11.—Three persons have together £100; and it is known that 9 times the 
 soney of the first, plus 15 times that of the second, plus 20 times that of the third, 
 equal to £1500. How much money has each ?—It may be here proper to ob- 
 prve, that this problem, as well as the 45th, 52d, 57th, and 58th, is susceptible of 
 syeral solutions; andto solve them completely it will be necessary to find all the 
 ifferent answers, and toshew that there can be no more; for by repeated trials it 
 
 ‘ould not be difficult to find some of them. 
 
 Pros. i11.—A farmer bought 100 calves, sheep, and pigs, for the sum of £100., at 
 ae rate of £3. 10s. for the calves, £1. 6s. 8d. for the sheep, and 10s. for the pigs: 
 low many of each kind did he purchase? 
 
 Pros. tu1.—Three merchants enter into partnership, and agree to advance each 
 £10000. towards a certain adventure; two of them paid down the money, but the 
 aird advanced only the half of his share, that is £5000; the adventure having 
 iiled, they lost not only their capital but 50 per cent. more: What must each contri- 
 ute to make good the loss? 
 
 Pros. tiv.—In a rectilineal triangle, given the base, the rectangle of the other 
 wo sides, and the included angle, to determine and construct the triangle. 
 
 Pros. tv.—An are of a circle being given, to divide itinto two parts, the sines 
 f which shall be in a given ratio. 
 
 Pros. ivi.—If a person draws 4 cards from a pack, containing 32, what probabi- 
 ‘ty is there, or how much may be betted to 1, that among these four cards there 
 rill be one of each colour? 
 
 Pros. rvu.—It is required to divide 24 into three such parts, that if the first be 
 qultiplied by 36, the second by 24, and the third by 8, the sum of those products 
 nay be 516? 
 
 Pros. tvit.—How many ways may four sorts of wine, the prices of which are 16d. 
 Od., Sd. and 6d. per quart, be mixed, so as to make 100 quarts in all, worth 12d. 
 yer quart ? 
 
 Pros. trx.—To find a number of such a nature, that if 12 and 25 be successively 
 \dded to it, the sums shall be square numbers. 
 
 Pros. Lx.—To find three numbers, the squares of which shall be in arithmetical. 
 wrogression. 
 
 Pros. rx1.—Any number of points being given, to find another, from which if 
 straight lines be drawn to all the rest, the sum of these lines shall be equal to a given 
 ine. 
 
 _ Pros. ixu.—The same supposition being made as before, the sum of the squares 
 of the lines drawn from the required point must be equal to a given square. 
 
 It is very singular that the last problem is susceptible of a construction much 
 sasier than the preceding. We shall here observe, merely for the purpose of excit- 
 ng the curiosity of the geometrical reader, that in the latter the required point, and 
 ill those that solve the problem, for there are a great many which do so, are situated 
 in the circumference of a certain circle; and it is very remarkable that the centre of 
 this circle is the centre of gravity of the given points, supposing each of them to 
 be charged with the same weight. — 
 
 It may be observed also, thatif it were required that the square of one of the 
 lines drawn, plus the double of the second, plus the triple of the third, &c., should 
 make the same sum, it would be necessary to suppose the first point loaded with a 
 single weight, the second with a double weight, the third with a triple one, &c., and 
 their centre of gravity would still be the centre of the required circle. 
 
 The solution of this problem was not unknown to the ancient geometricians. It 
 was one of those of the Loca plane of Apollonius ; and this may serve to give us & 
 more favourable idea of their analysis than is generally entertained. 
 
188 
 
 COLLECTION OF USEFUL TABLES. 
 
 TABLE 
 
 OF THE LENGTH OF THE FOOT, OR OTHER LONGITUDINAL MEASURI 
 USED IN ITS STEAD, AMONG THE DIFFERENT NATIONS AND IN THI 
 PRINCIPAL CITIES OF EUROPE. 
 
 Havine frequently experienced great embarrassment, while engaged in cer’ 
 tain researches, from not being able to obtain accurate information respecting thi 
 measures of different countries, whenever an opportunity occurred we colleetec 
 with great care the proportions of these foreign measures, both ancient an¢ 
 modern, as compared with our own, and it is hoped our readers will conside; 
 themselves indebted to us for the following table on this subject, which there i 
 reason to think is the fullest and most complete ever given. All the different mea. 
 sures are compared with the English foot, which is here supposed to be divided inte 
 12 inches, each inch into 12 lines, and each line into 10 parts: which makes the foo) 
 to consist of 1440 of these parts. The first column in the table shews the number oj 
 these parts which each measure contains; and the second the value of it in Englisk 
 feet, inches, lines, and tenths of a line. 
 
 ANCIENT FEET. 
 
 parts. ft. in. lin. pts 
 
 Ancient Roman foot 1.2.6... .cecees ile ghia cep eiatenbet ale ts 1392 Oll 7 2 
 Greek and Ptolemaic .......es+esees Sn aide ak eee es evess | 1458 1 O leg 
 Greek *PhyleterianW «ses ss.dcns see Sonia se tae ae salen eeeOSk 1 2 OF 
 Foot of Archimedes, or probably of Sicily and Syracuse .. 1051 0 8 9mm 
 MDWUSIAN so sag seid Taki Geauihe oe de 2 ahte Wlaeere cele odisie ep hie sh LOT 1 1 i 
 Macedoniat | ./..cgeec ses eee eue oA Aiea ee Salve at <-+) 1670 1 1 Tie 
 Egyptian ...... Date aie pions epee Site woth Shoe Setiay Bion e @ 2046 158 ¢ 
 Hebrew ee aise g sacuetan ese Gini eebewrck ies au sce sees ohne Sladen 1 2 6 
 The natural (hominis vestigium)...ccc++cceceescee voce Faery ey 0 9 oa 
 ‘Arabiengens : 1h ees ok RTA MAS Sah RRO TELS GN 1577 1 1 i 
 Babylonian pak soos pegs y <yi8is als vasa te K€ leis’ ep Givin nis sie 'gitlaer le EOS 1 1, 833 
 lata et nhtiet e's 5 uipusiedesats 5 ee Be SEA -. 1635 1 1 43 
 
 MODERN FEET. 
 
 Bpglish yond ce eet INA ee Se ON ee he oan, 1440 1 0 Om 
 A lbord Ghee Se aneeene wales kee ee bate igen eae he sire, 1116 0.9. ome 
 Arasterdam’ yivgy sees se wees ciiledclsus tyes Smid fas Fares we toca 011 1 5 
 Ancona and the Ecclesiastical States........... een botele 1846 1 3 46 
 Antwerp ..... EER ar Reape Sa Sache ceaet acy voeal ERG etch 1353 O11 3 3 
 Aquileia; >. oe aetinae se eee pissed \al's Seoiethclapwecird ks nate etek totois 1624 1 1, 6m 
 Aree Herd. apa. ok 5 A eORe Tlelt ee eee Siielea ts Pei -» 1279 010 7 9 
 PRCA DUTE 1s :55 plese sinh ae spa wide oycicliets’s ae wee ee ks & 1399 O11 ta 
 SE) ee le spk-eiag sto! dusinheuha SW Reis tile abil. dick: 1279 010 7 9 
 Barcelona. ..... RK oe ee ke ees See Bile an tee 1428 01110 8 
 
LONGITUDINAL MEASURES. 
 
 ee seeoee @eoeneaereee ese esi eceteoeserte & eee 
 earereereneaeoe oeoeer esc eomaeeeweeeeebr see ee eesesees 
 ®Ceoeeesr se oOresses ee eeereenes ceoaereeoeseseosd e 
 @eeesesee eeeoeen e@oeeserrsen e@eetoneeaseeene 
 
 eeeoveveeee0 e@ecseeeveevegeoeet ev aseseevnesre 
 
 jourg en Bresse and Bugey .-+eeeees- 
 
 eneeogceeueeevegcneaese 
 
 jremen e@eceoeeeeeeeeaeeeeeoeGeoeoseeoevse02 G28 
 
 jrescia . eooevueeveveeeorese 
 
 eecereeceeere 
 eeoesee cee es eeeeeaaeeOGnveeeree 
 
 jreslaw COCO OHP SoH RET HES HSH SHOTS HOCH OHESH HHT EHHOHOETE 
 
 iruges coves eeoveds cevece 
 
 eer>rsevoeeeeeetceeoseSeeveese see eben 
 srussels teewessceoeeseeeOs2eF 6CeeeeeFeseoeerteeere sores ever s 
 
 Dhambery and Savoy s.cceesseeee 
 
 Dhina— Tribunal of mathematics....ccccoccccersveccsece 
 | Bie PAPOGEMES oreie since doleierescisisicte’s-asaierera tart s-ale-a'ae 
 OTIG. -6)e csp ewsccenccetcscatsenrnnscvansnnsees , 
 Jonstantinople POs e scy ee! peeks Hots both oe of : 
 
 Jopenhagen Sera eNCa tacts Maa kk ees oe Reiss a 6:0 . 
 
 Cracow ce cece erecta avec tesecaseeseeeceneers ve 
 Dantzic erieevecestoseeneues enews es/e aries ce's'ee's Woche’ 
 Delft Meese vawgie adie oe 8 elelc'c Nios a's siee's waie'e a ae 6 ae Sere ate 
 Denmark .eeeeeesceceeeces rere as ain ativere sre mn 8 slateiets . 
 Dijon dc Welecihs Mele sta eonuncnreceseneepacned vices . 
 Dordrecht ...... Baits as Se ene 62 Vee ss Seale Weleiwie wit vin e's = 
 Ferrara shacss Sivianio hee yon ie0e Sa SiSupisiaie 55 8m 8 G16 .6.bis 4) a0 m2 
 MIOrence scccccccecccece pr asiaarelsie otis e ee eb e a aleis hse ° 
 Franche-Comté... «.... Fone eis s-0'es'e 930s 09.0 0x00 oe 
 Frankfort on the Main °.5,2ssccaccheverecion erry oe 
 Genoa (the palm) ...esscese cecvenesnccerns seersees 
 
 Geneva Some ty ar ee eae Seals mw eia\ein/Saeie'* ters damcurh ies 
 Grenoble and Dauphigny ......0+.. PRA A hy ER 20 oe 
 Haerlem Sb sen ehiniete > «<a MP aan Ary cree ek AAO oe 
 Halle in Saxony 1. ess eeeeseeeee eee eeees eielets edna Sete ti 
 Hamburgh ......-...sesecereencees pense lajelane te wbreia lene ts 
 Heidelberg (Palatinate) ...2.+e.s-06. Scaeereeeeceve ans 
 Tnspruck 2,06... see ees ceeeeeeeeeees aiaie ss tials ate Sis te . 
 Maeehorm ..... w asatitete Lee, Bek h e BW estiumianote 24 ars 
 BEEMSIC hos ‘sees AE ee cinveile Marches ate eie se wow vy 
 BPP VOCT | oe. o eve ceees ME PIRI. PR ms hererapiirel tne 
 _ ASS noes ice aA Ried TIP Ta ae ene 
 BUMEGSIER la iac is  e'6i Ga ons oie > Bais pla wie ture stoke ei scete ats 
 Lombardy, foot of Luitprand or Aliprand .............. 
 MMMPETAINIG <1. o 602 solar cess Sige ness Spa ewisess Oni sere . 
 TMIDECHs oy scene Cc EDR ORE eWacgess aaneeiere eraleipralare sfatete 
 MMEECALS «occas swaces SSRI seis oie ea pietcieinlegsiee 
 Lyons and the Lyonnese, Fores and Baujalois .......... 
 SUEC UL cig Vccstitih«cpseepts preeriere fe Bistotatete oi 
 MRC neue recess cc tenses ccuass POs ee aie ss 
 Malta (the palm) ........cceccsscecsevarsecreecs nao 
 Mantua (the brasso) ...... eee atoe ses tie: cece ee 
 
 BEES “nee err aie dks ous. ote CL SAC Sdn were 
 
 189 
 
 . in. lin. pts. 
 
 ll 4 
 5 2 
 11 10 
 0 2 
 211 
 0 4 
 Lid 
 6 8 
 1 6 
 8 11 
 10 9 
 bes 
 Lab 
 LLy 
 10 10 
 2 4 
 111 
 0 7 
 2 0 
 1l Oo 
 6 6 
 0 6 
 0 4 
 9 3 
 3 9 
 11 11 
 2-0 
 ll 2 
 9 9 
 iG heey 
 1075 
 lk-3 
 ll 8 
 ll 2 
 10 10 
 Ling 
 11 10 
 0 4 
 Cus 
 Lied 
 jiers 
 Bod 
 lies 
 1e 
 hep. 
 Les 
 10 11 
 10 11 
 10 11 
 6 3 
 
 Rees ep co Chae at ea ar coh et ici eo! cnn os st Coc co fm core’ Oo =F 6g OF St Go) OO a tO yt 7 i Gocco tO se I 
 
190. LONGITUDINAL MEASURES. 
 Mechlin ..0..¢003 Shee se Vig Weare cee oe eed Sone meas teas 
 Dl entZ~ ke. valeic a ee Sel ews welled «wee de fokta Cane aecéeds 
 Milan—Decimal foot. .cccecacessvesosesessaees sieve 
 —Aliprand ditto ...... cae ev ectedvenses des veccee 
 MOdeMA TN, she Galatis s.oaieie 6s/0's a tereisueldlatouale tote: slot e aigtetoraiad we 
 MONRCO*), cae vadinscee deta Cade cies soe rhvedeservudiace 
 Montpellier (the pany). . 5. .0se<seeseteceseseveversnyve 
 NGSCOWU. psiaris ess. see Riie eek teens a ens eis Senet eth 
 NEUNICH Ss sf ctaie ss ele BS No see TREND oes sacla ik De mindy tesa 
 Naples (the palm) ....... Pere e: Tr eee eee Saito 
 Netherlands, see Maestricht. 
 NUT OM ele = CON TLOOU sreiaisie « s-si0 #isloie 4S os spats aioloicleral yee 
 Country f006.. i. 200.0 <.0 sists tempt yatta o- Sethe dats 
 Bat taen cave or cat ie one oe ees wets a ln'te fore i6 Bp ise ta covetavetcears 
 Palermo ...... Bre eis tale let ehebalaiaceeias 34 mac eve ee si isieker a A amekr arti 
 Hari LOOUa re su case's ciaistaiatiasipissiere rete BYalaretglers) here at tots 
 —metre..... sco odo éecas eee cere et ceserece erence 
 MATING Wes dis\sa.0 p'caeeas cote reeaees stare cuislscacaia ET ten 
 PAVED ts avec css Ls Shel os area A suai Wales & ool oaieiw ere eter ie lela 
 TATORMIC Febc ales o 45's Wie ctl oc"o siete gsaheia Gide inet ert detente Seti 
 Provence, see Marseilles. 
 Febiglandish 10067 ciices «cece ss «c pee © Si aslare/ elas a ote ke of store 
 TRIP OREGE) ves cia e nig ee ce oh co's $4 060.0 a celdiea H eG atieinm ese ose 
 BHOMS ENG IMAIIN) Weeeesee cp usb cnsses gas 6 ¢ eeecetecn ee 
 POULT SAS HUE ALIS gra't s'e sais oe 5 vie eeig ened a ys 6 Stiga deal ov9 
 
 Savoy, see Chambery. 
 POVILIEMNO A NGAUS A ede e ae eb dee oa obese 06d ease eee tae 
 
 PENDS ICOMIMON LOO Nes c'c'v acts Buca e eee SER eT ue 
 Stettin in Pomerania ..... oles pias stores Wid erases eee “ora 
 Stockholmi +... «sos ess « Sais aie ake Apis iui tis ated yay Mabe ae ey 43 Co 
 Strasburgh— Town font hes Cee naiceh eters RCs Gs ceeky 6 ait 
 COUNUEY MLO v9 sin scares ewig owes hee Gee alee 
 TOUGH Gs ocak 9 a ES Ae vy elekesres oerice paca 
 Turin (Piedmont) ~%...>. Rea ehe> wma heh eres cay eve etre 
 TENE. GP aicinc s oie sleet RIA ee re eo ee ee 
 PV BHAUOLIC Ces weir atay veces eels s Foce am nia ocak yc cree oh 
 V CCG Bike hs csc 0 edits os ce tucte ates Ph wc Sea teewe athe a. 
 Wy COT eres sce's.c sisie ee o'e oe ke hk Sielene vec coe eens ou 6 eis se NE 
 WAGE Tare ee e's TREES bee oo nae « oe vache mone pees se Ree 
 Wiennarta ei es ceSeeks cs Nah Damas e's Sie s bak Keb Sik a eee TE 
 Vienne in Dauphipny ). 22... 00-00 AYA CH, Seve oe eae 
 Ulm aitte a'siletsiits as baa a4 3 aie ole siecle cece Stare a vt ath ete 
 UITDINOSE eee tees Cae Nes cle we eSeet «ey ts ree 
 Utrecht Hih.,..0 esses @ siecepereisies ame ST 
 Warsaw Pica: ESA roe a siea e Newws ace SAME 
 Weesel 2.01. s06selcare. ae Reese's ei stepiiiae mi we cee oe 
 BUTTE Ge? Ne vralsnts Cr Geass neces as Cs Cons rahe ete eioie 
 
 ie 
 > 
 
 mh = oe 
 
 —OoO- 
 
 SOF OF OF RF ew eB BH ORF Rr OD OWE EBO 
 
 — 
 
 Comm WF OF OO 
 
 — 
 
 8 
 
 — 
 
 KON MOCO OCOK Be Ow 
 
 1 
 
 4 
 0 
 7 
 b 
 7 
 5 
 8 
 1 
 ] 
 0 
 0 
 3 
 9 
 
 — 
 
 ee 
 
 SCORND CORNY ROONOROEN SH oR OD 
 
 fr Gr co Co 
 
MEASURES. 7 191 
 
 TABLE 
 
 29F SOME OTHER MEASURES, BOTH ANCIENT AND MODERN, COMPARED WITH THE 
 ENGLISH STANDARD. 
 _ The ancient cubit in general was a foot and a half. The Hebrews however had 
 three cubits. 
 
 Ist. The common cubit, which was a foot and a half Hebrew measure, or 2617 of 
 those parts of which the English foot contains 1440. 
 
 9d. The sacred and modern cubit, which was one Babylonic foot and three quarters, 
 or 2883 or 2861 parts of the English foot. 
 _ 8d. The great geometric cubit, which was 9 Hebrew feet, or 6 lesser cubits. 
 ) Grecian feet. 
 The orgya of the Greeks was s+eeecssseecsevessvecscoreses 6 
 Thearura °+cccesscee a plavsryiple se Cob er cen nes ecco eresces rece 50 
 The plethron ++esecsccacccsreecverencverercracncecensrece 100 
 { The diplethron ++++scscsceceosccecoseccvcessveevevaresecs 200 
 
 Roman feet. 
 
 The hexapeda of the Romans Was *eeccccesecscscsesvesovcos 6 
 | The decempeda erereccsersccserecereccescerocsseccareres 10 
 
 MEASURES OF PARIS. 
 French feet. English feet. 
 
 Toise of Paris -sscesrcousacccccocevvecresccces 6 6°3959 
 Metre, OF NEW MEASUPEseeerserssreeseaccerrece Oss 3:2854 
 
 The royal perch Aine e ole clevatarnlele! eiseie.6\6 Sooo. “VR 93°4515 
 f The mean perch+ressossecewsscccesccssvccecare 20 91°3195 
 . The lesser perch used at Paris secceersececsee-ee 18 19-1876 
 
 The acre is 100 square perches. 
 The are is 100 square metres. 
 
 } 
 
 MEASURES OF CAPACITY FOR LIQUIDS. 
 
 ' ‘The Mnid for liquids (Paris measure) contains 8 French cubic feet, or 16744-7071 
 English cubic inches. 
 Six French cubic inches make a poincon, or by corruption poisson, = 7'2677 
 
 English cubic inches. 
 Eng. cub. inches. 
 
 2 poissons make eccscccccccccoreoereesses | demi-setier  14°5353 
 
 D demi-setiers sesersssecescecoscessscee - 1 chopine 29 O707 
 1 2 chopines eeesecsccccececcccrecccrsececs 1 pinte 58:1413 
 : 2 pintes Setelcintelccieis eau ae ces ee 6 e.snees ce a | quarte 116°2827 
 4 quartes AAO SOC CRU OCUDOOONCO COmcmOurr: | grand setier 465°1308 
 } 86 grand setiers ++cecsescesscsecescccccccs 1 muid 16744:7071 
 
 Litre escereeeeececcsecesseesess acubical decimeter = 14, pinte. 
 A muid therefore is equal to 724871 English wine gallons, or about 11 hogshead. 
 
 FRENCH DRY MEASURES. 
 The litron contains 36 French cubic inches, or 43°606 English cubic inches. 
 
 Eng. cub. inches. 
 
 16 litrons make seesceesecesesereseseere 1 boisseau 697°696 
 
 3 boisseaux eee eee eons eee ssseeeees eres 1 minot 2093:088 
 D MINOtS evcceecccsesnssoasreersersesosea: 1 mine 4186'176 
 2 mives weer rece eenesereseeereeseeaeesesene 1 setier 8372°352 
 
 12 setiers wpe oe dese eset sees ones eeseoeoereed 1 Paris muid 100468224 
 
(192 , MEASURES. 
 
 Hence the French muid for things dry is equal to 46-72 English bushels, or 5 quar- 
 
 ters 6 bushels 2:88 pecks. 
 The following tables of ancient measures, have been added from Arbuthnot. 
 
 ROMAN MEASURES OF LENGTH. 
 
 Digitus transversus 0°72525 Eng. in. Cubitus »eeesssees 14505 Eng. ft. 
 Uncia, the ounce -+ 0:967 — Gradius esesserees) 2:4175 SoM 
 Palmus minor ++: = 2°901 sie Passus e*+ecees »»  0°967 paces. 
 Pes, the foot«+++*: 11604 — Stadium++++eeecee+s 120°875 Ag 
 
 Milliare seoevaace de 967:0 a 
 
 Palmipes ++++-+++ 1°20875 Eng. ft: 
 
 SCRIPTURE MEASURES OF LENGTH. 
 
 Digit -+++eeeee- ~-  0°7425 inches. Arabian pole «-++«- 4-62 yards, 
 Palin -sahese ress «2507 ~ Schenus «++=> veeee 462 Dai 
 Span cereers coese §68O9] ae Stadium-+-+++see. 231-0 — 
 Lesser cubit +++-+- 1-485 Eng ft. | Sabbath day’s journey1155°0 = 
 Greater cubit :-+++> 17523 4 Eastern mile +--+. . 1886 miles. — 
 Fathom +--+ +++- «+ 231° yards, | Parasang +--+: feces? £158" am 
 Ezekiel’s reed ++++s 3465 — Day’sjourney +--+... 33264 — 
 GRECIAN MEASURES OF LENGTH. 
 Dactylos  ++:s+++- 0°75546 inches. Pygme* «+-seecere 1°13203 Eng. ft. 
 Doron Pygon -ereeee ecese 125911 — 
 De phine ' creas >» 302187 — Pechys-~++++ 05 wee. 151003408 
 Dichagess+sess ve -- 755468 — Orgya eerece eeee s = 1:00729 — 
 Orthodoron-++>+«-s 831015  — Stadios 
 Spithame-+-++++++ 9-06562 Eng in, | Dulos §°7777'7*, 10072016 paces, 
 Pougi ees <=> +> ceeee 12:0875 es Milion eseseesess 805°8333 — 
 Pous Misvelesé 6.665) 6a eisce 100729 Eng. ft. 
 
 ROMAN DRY MEASURES. 
 
 Hemina se ee ee rere? 0:5074 Eng. pints, | Modis ee cevenaee 1:0141 Eng. peck. 
 Sextarius «*++++++» 1:0148 —_ 
 
 ATTIC DRY MEASURES. 
 
 Mestes seeererees 0:9903 Eng. pints, | Medimnus ++-+++ 1:0906 Eng. bush. 
 Chenix +++e--++e: 1-486 — 
 
 JEWISH DRY MEASURES, ACCORDING TO JOSEPHUS. 
 Gachal -«++++e+-+ 0°1949 Eng. pints. Ephah +++++++e++ 1/0961 Eng. bush. 
 Cab csccceseeeee 3874 sia Latech -++e+see++ 54807 Bi 
 Gomer «erersseee T0152 ees Coron 
 Seah ««ce-eseeee+e 14615 Eng. peck. | Chomer { apes 13702 Eng. qr. 
 
 ROMAN MEASURES FOR LIQUIDS. 
 Hemina:++s++++ 0°59759 Eng. pints. Urna+s seecceee 3°5857 Eng. gals. 
 
 Sextarius -++es « 1:19518 — Amphora +++«-. = 7-1712 == 
 Congius:++s++++ F-17712 — Culeus ++++++++ 2-2766 Eng. hogs. 
 ATTIC MEASURES FOR LIQUIDS. 
 
 Cotyle -+++++- - 0°5742 Eng. pints. | Chous++++++++.-.. — 6:8900 Eng. pints. 
 Xestes sseees-s 11483 we | Meteotes ++-++++» 10-3350 Eng. gall. 
 
 * From this measure is derived the English word pigmy. 
 
MEASURES, | 193 
 
 | JEWISH MEASURES FOR LIQUIDS. 
 | 
 
 Caph +s+e+e+s+.  0°8612 Eng. pints. j Seah ........... 3°4450 Eng. gall. 
 Log seescesecs 11483 — Bath ike. ee 10°3350 -- 
 
 Od ee 4°5933 —_ Corny: aks: 16405 Eng hogs. 
 Hin .....0....  1°7225 Eng. gall. 
 
 FRENCH MEASURES. 
 The Paris foot is to the English foot, as 1 to 1:065977 
 The Paris square foot isto the English, as 1 to 1:136307 
 The Paris cube foot isto the English, as 1 to 1:211277 
 The French wine pint contains 58-1413 English cubical inches ; and the English wine 
 pint contains 28:875 cubical inches. 
 
 NEW FRENCH MEASURES. 
 
 ‘The new French measures were established by a decree of the national convention, 
 on the 7th of April, 1795. The elementary measure on which they are founded, is 
 a decimal part of the distance from the pole to the equator; that is to say, a decimal 
 part of a quarter of the terrestrial meridian: for the met7e, which is the element of 
 all the rest, is the ten millionth part of that distance, and is equal, in the old French 
 measures, to 36 inches and 11°296 lines. A metre in length, is the element of all 
 the lineal measnres; a square metre is the element of all the superficial measures : 
 and a cubie metre is the element of all the measures of capacity. 
 
 MEASURES OF LENGTH. 
 
 Eng. inches. 
 Millimetres:,vueteet- axel cess => ‘03937 
 Gentimetrerscaalssltevetn eerie ce 39378 
 DECIMGLE ER cin bac cover ese eee 3°93786 
 MSU Oniiodetrzstt. on torent 39°37860 
 Diecametreicny Go ceeh seca 393°78605 
 Hectometre eC hace Ue take eh 3937°86059 
 Chiliometre baocg)) oh ae) BOOTS O0H99 
 Myriometre Seek fe faatteed Voda ( GG,UD09 1 
 
 Miles. fur. yds. 
 
 ft. 
 PA MNEGLIE 1S oh. lucas casi cee 3 «3:37 
 A. Decametre: 005° sas) ses 10 a2 oeors 
 A, Hectometre =... 7° sca5. ses W921 S86 
 DRCMUIGMetrel fren cede 4 FSD #266 
 ] 
 
 AP Myriometre “,.. ..0) 4 6 17158 6:05 
 The distance from the pole to the equator, or fourth part of the terrestrial meridian, 
 according to the late French measurement, is 32815504 English feet. 
 Centesimal degree = 328155'04 English feet. 
 
 MEASURES OF CAPACITY. 
 Eng. cub. inches. 
 
 16 UR TAVES Cana a ne Tan eee 06106 
 MV ATITTIVELE as lacs Mee oe ek eee vle.s *61063 
 ICCTILEO Ee, ok hens ek Nl ge Tien 6°10634 
 GCM.) ak lull’ cocks se, cP saa Dh weakness 61:06345 
 reer ooh ie: eed dh seas 610 63450 
 PLeetouetre,, Sc... :.. cto Peery es 6106°34504 
 Chiliolitre (cubic metre) ~ eee oe  61063°45042 
 Myriolitre ... ... win Mees Unden G1 0634250427 
 ) 
 
Bit. - n: - ee : 
 litre is 2° 114, or nearly 2} English win 
 
 A Hectolitre is 2°6434 wine gallons, or 2 gallons 2 anerte V4 pint. Beet: 
 \ Chiliolitre is 4 hogsheads, 12 gallons, 1:36 quart; or 1 tun 12°34 gallons. ; 
 ahaa is 10 amiss: 1 Aen a 60 ‘4 he or el 103 tuns. | 
 
 ¥ oe ee ey 
 
 SQUARE OR SUPERFICIAL MEASURES. ety 
 Eng. square fects 
 Sanat millimetre Wi: ‘01076 © 
 
 Square centimetre ikter *10768 
 _ Square decimetre -1:07685 
 Centiare (square metre) ‘ 10°76856 
 Deeidren G00. . 107°68564 
 Are Pe ee - 1076-85645 
 Decare .. 10768°56454 
 TIGCLAre 1, . epee het 107685-64540 
 eOhiliard). 2.7 107685645407 
 /s"Myriare.. .. oe +e oe  10768564°54070 
 : A Hectare is 2° 47 2 English vir aa acres; or 2 acres, 1 rood, oe 5 oe 
 
 MEASURES FOR FIRE-WOOD. 
 
 pee | ae cubic. feet. 
 Decistere .. «2 4. ae + owe 85338764 
 Stere (cubic Het RP TS + 35°387045. 
 
MECHANICS, 195 
 
 PART THIRD, 
 
 CONTAINING VARIGUS PROBLEMS IN MECHANICS. 
 
 Arrrr Arithmetic and Geometry, Mechanics is the next of the physico-mathematical 
 sciences, having their certainty resting on the simplest foundations. It is a science 
 also, the principles of which, when combined with geometry, are the most fertile 
 and of the most general use in the other parts of the mixed mathematics. All those 
 mathematicians therefore who have traced out the development of mathematical 
 knowledge, place mechanics immediately after the pure mathematics, and this method 
 we shall here adopt also. 
 
 We suppose, as in every other part of the mathematics introduced into this work, 
 that the reader is acquainted with the first principles of the science of which we 
 treat. Thus, in regard to mechanics, we suppose him acquainted with the principles 
 of equilibrium and of hydrostatics; with the chief laws of motion, &e. For it is 
 not our intention to teach these principles; but only to present a few of the most 
 curious and remarkable problems which arise from them. 
 
 PROBLEM If. 
 
 To cause a ball to proceed in a retrograde direction, though it meets with no apparent 
 obstacle. 
 
 Place an ivory. ball on a billiard table, and give it a stroke on the side or back 
 part, with the edge of the open hand, in a direction perpendicular to the table, or 
 downward. It will then be seen to proceed a few inches forward, or towards the 
 side where the blow ought to carry it; after which it will roll in a retrograde direc- 
 tion, as it were of itself, and without having met with any obstacle. 
 
 Remark.—This effect is not contrary to the well-known principle in mechanics, 
 that a body once put in motion, in any direction, will continue to move in that di- 
 rection until some foreign cause oppose and prevent or turn it. For, in the present 
 case, the blow given to the ball, communicates to it two kinds of motion; one of 
 
 - rotation about its own centre, and the other direct, by which its centre moves pa- 
 tallel to the table, as impelled by the blow. The latter motion, on account of the 
 friction of the ball on the table, is soon annihilated; but the rotary motion about 
 the centre continues, and when the former has ceased, the latter makes the ball roll 
 in the retrograde direction. In this effect, therefore, there is nothing contrary to 
 the well known laws of mechanics, 
 
 PROBLEM II. 
 
 To make a false ball, for playing at nine pins. 
 
 Make a hole in a common ball used for playing at the above game; but in such a 
 manner as not to proceed entirely to the centre; then put some lead into it, and 
 close it with a piece of wood, so that the joining may not be easily perceived. When 
 this ball is rolled towards the pins, it will not fail to turn aside from the proper 
 direction, unless thrown by chance or dexterity in such a manner, that the lead shall 
 
 turn exactly at the top and bottom while the ball is rolling. 
 o 2 
 
196 MECHANICS. 
 
 Remark.—The fault of all balls used for billiards depends on this principle. For, 
 as they are all made of ivory, and as, in every mass of that substance, there are 
 always some parts more solid than others, there is not a single ball perhaps which 
 has the centre of gravity exactly in the centre of the figure. On this account_every 
 ball deviates more or less from the line in which it is impelled, when a slight motion 
 is communicated to it, in order to make it proceed towards the other side of the 
 billiard table, unless the heaviest part be placed at the top or bottom. We have 
 heard'an eminent maker of these balls declare, that he would give two guineas for a 
 ball that should be uniform throughout; but that he had never been able to find one 
 perfectly free from the above-mentioned fault. 
 
 Hence it happens, that when.a player strikes the ball gently, he often imagines 
 that he has struck it unskilfully, or played badly: while his want of success is en- 
 tirely the consequence of a fault in the ball. A good billiard player, before he 
 engages to play for a large sum, ought carefully to try the ball, in order to discover 
 the heaviest and lightest parts. This precaution was communicated to us by a first- 
 rate player. 
 
 PROBLEM ITf. 
 
 How to construct a balance, which shall appear just when not loaded, as well as when 
 loaded with unequal weights. 
 
 We certainly do not here intend to teach people how to commit a fraud, which 
 ought always to be condemned; but merely to shew that they should be on their 
 guard against false balances, which often appear to be exact ; and that in purchasing 
 valuable articles, if they are not well acquainted with the vendor, it is necessary to 
 examine the balance, and to subject it to trial. It is possible indeed to make one, 
 which when unloaded shall be in perfect equilibrium, but which shall nevertheless be 
 false. The method is as follows: 
 
 Let a ands be the two scales of a balance, and let a be heavier than ps: if the 
 arms of the balance be made of unequal lengths, in the same ratio as the weights 
 of the two scales, and if the heavier scale a be suspended from the shorter arm, and 
 the lighter scale B from the longer, these scales when empty will be in equilibrium. 
 They will be in equilibrium also when they contain weights which are to each other 
 in the same ratio as the scales. A person therefore unacquainted with this artifice 
 will imagine the weights to be equal; and by these means may be imposed on. 
 
 Thus, for example, if one of the scales weighs 15, and the other 16; and if the 
 arms of the balance from which they are suspended be, the one 16 and the other 15 
 inches in length; the scales when empty will be in equilibrium, and they will re-’ 
 main so when loaded with weights which are to each other in the ratio of 15 to 16, 
 the heaviest being put into the heaviest scale. It will even be difficult to observe 
 this inequality in the arms of the balance. Every time therefore that goods are 
 weighed with such a balance, by putting the weight into the heavier scale and the 
 merchandise into the other, the purchaser would be cheated of a sixteenth part, or 
 an ounce in every pound. 
 
 But, this deception may be easily detected by transposing the weights; for if they 
 are not then in equilibrium, itis a proof that the balance is not just. 
 
 And indeed in this way the true weight of any thing may be discovered, even by 
 such a false balance, namely, by first weighing the thing in the one scale, and then in 
 the other scale; for a mean proportional between the two weights will be the true 
 quantity ; that is, multiply the numbers of these two weights together, and take the 
 square root of the product. Thus, if the thing weigh 16 ounces in the one scale, 
 and only 14 in the other: then the product of 16 multiplied by 14 is 224, the square 
 root of which gives 1439 for the true weight, or nearly 15 ounces. Or indeed the 
 just weight is found nearly by barely adding the twonumbers together, and dividing 
 
CENTRE OF GRAVITY. - 197 
 
 the sum by. 2. Thus 16 and 14 make 30, the half of which, or 15, is the true weight 
 very nearly. 
 
 , 
 
 PROBLEM IV. 
 To find the centre of gravity of several weights. 
 
 As the solution of various problems in mechanics depends on a knowledge of the 
 nature and place of the centre of gravity, we shall here explain the principles of its 
 theory. 
 
 The centre of gravity of a body is that point around which all its parts are balanced, 
 in such a manner, that if it were suspended by that point, the body would remain at 
 rest in every position in which it might be placed around that point. 
 
 It may be readily seen that, in regular and homogeneous bodies, this point can be 
 no other than the centre of magnitude of the figure. Thus, the centre of gravity 
 in the globe and spheroid, is the centre of these bodies; in the cylinder it is in the 
 middle of the axis. 
 
 The centre of gravity between two weights, or bodies of different gravities, is 
 found by dividing the distance between their points of suspension into two parts, 
 which shall be inversely proportional to the weights; so that the shorter part shall 
 be next to the heavier body, and the longer part towards the lighter. This is the 
 principle of balances with unequal arms, by means of which any bodies of different 
 weights may be weighed with the same weight, as in the steel-yard. 
 
 When there are several bodies, the centre of gravity of two of them must be found 
 by the above rule: these two are then supposed to be united in that point, and the 
 
 common centre of gravity between them and the third is to be found in the same man- 
 
 ner, and so of the rest. 
 Let the weights a,B, and c, for example, be suspended from three points of the line 
 or balance pF (Fig. }.), which we shall suppose to have no weight. Let the body 
 Fig. 1. A weigh 108 pounds; zs 144, c 180; and let the dis- 
 tance D E be 1] inches, and & F, 9. 
 a= First find the common centre of gravity of the 
 c ae A AO bodies z and c, by dividing the distance & F, or 9 inches, 
 into two parts, which are to each other as 144 to 180, 
 oras 4to 5. These two parts will be 4. and 5 inches; the greater of which must 
 be placed towards the smaller weight: the body B being here the smaller we shall 
 have = c equal to 5 inches, and r G to 4; consequently p eG will be 16. 
 
 If we now suppose the two weights B and c, united into one in the point c, and 
 consequently equal in that point to 324 pounds ; the distance pc, or 16 inches, must 
 be divided in the ratio of 108 to 324, or of | to3. One of these parts will be 12 and the 
 other 4; and as a isthe less weight, p H must be made equal to 12 inches, and the 
 point u will be the common centre of gravity of all the three bodies, as required. — 
 
 The result would have been the same, had the bodies a and B been first, united. 
 
 In short, the rule is the same whatever be the number of the bodies, and whatever 
 be their position in the same straight line or in the same plane. 
 
 This may suffice here in regard to the centre of gravity. But for many curious - 
 truths deduced from this consideration, recourse may be had to books which treat 
 on mechanics. We shall however mention one beautiful principle in this science, 
 deduced from it, which is as follows : 
 
 If several bodies or weights be so disposed, that by communicating motion to each 
 other, their common centre of gravity remains at rest, or does not deviate from the hori- 
 zontal line, that is to say neither rises nor falls, there will then be an equilibrium. 
 
 The demonstration of this principle is almost evident from its enunciation ; and it 
 
 may be employed to demonstrate all the properties of machines. But we shall leave 
 the application of it to the reader. 
 
198 , MECHANICS, 
 
 Remark.— As this is the proper place, we shall here discharge a promise made at 
 * p. 183, viz. to resolve a geometrical problem, the solution of which, as we said, seems 
 to be only deducible from the property of the centre of gravity. 
 
 Let the proposed irregular polygon then be AaBcDE (Fig. 2.), 
 the sides of which are each divided into two equal parts, in 
 a, b, c, d, and e, from which results a new polygona bcdea; 
 let the sides of the latter be each divided also into two equal 
 parts by the points a’, 0’, c’, d’, e’, which when joined will give a 
 third polygon a’ b’ c’ d' e’ a’; and soon. Inwhat point will this 
 division terminate ? 
 
 To solve this problem, if we suppose equal weights placed at a, b, c, d, e, their 
 common centre of gravity will be the point required. © 
 
 But, to find this centre of gravity, we must proceed in the following manner, — 
 
 Fig..3. which is exceedingly simple. First draw ab (Fig. 3.) and let the 
 ; middle of it be the point f; then draw fe, and divide it in g, in 
 such a manner that fg shall be one third of it; draw also gd, and 
 let gh be the fourth of it; in the last place draw he, and let hi be 
 the fifth of it: the weight e being the last, the point 7, as may 
 be demonstrated from what has been before said, will be the 
 ‘centre of gravity of the five equal weights placed at a, b, c, d, and 
 e; and will solve the proposed problem. 
 
 PROBLEM V. 
 When two persons carry a burthen, by means of a lever or pole, which they support at 
 the extremities; to find how much of the weight is borne by each person. 
 It may be readily seen that, if the weight c were exactly in 
 
 aa yr the middle of the lever an (Fig.4.), the two persons would 
 
 a= each bear one half. But if the weight is not in the middle, it 
 
 ah 
 
 can be easily demonstrated, that the parts of the weight borne 
 by the two persons, are in the reciprocal ratio of their distance 
 from the weight. Nothing then is necessary but to divide the weight according to 
 this ratio; and the greater portion will be that supported by the person nearest the 
 weight, and the least that supported by the person farthest distant. The calculation 
 may be made by the following proportion : 
 
 As the whole length of the lever a8, is to the length a x, so is the whole weight 
 to the weight supported by the power or person at the other extremity B; or as aB 
 is to B £, so is the whole weight to the part supported by the power or person — 
 placed at a. 
 
 If az, for example, be 6 feet, the weight c 150 pounds, az 4 feet, and pr 2, we 
 shall have this proportion: as 6 is to 4, so is 150 to a fourth term, which will be 100. 
 The person placed at the extremity B, will therefore support 100 pounds, and conse- 
 quently the one placed at.a will have to support only 50. - 
 
 Remark.—The solution of this problem affords the means of dividing a burthen 
 or weight proportionally to the strength of the agents employed to raise it, Thus, 
 for example, if the one has only half the strength of the other, nothing is necessary 
 but to place him at a distance from the weight double to that of the other. 
 
 PROBLEM VI. 
 How 4, 8, 16, or 32 men may be distributed in such a manner as to carry a considerable 
 burthen with ease. 
 If the burthen can be carried by four men, after having made it fast to the 
 middle of a large lever az (Fig 5.), cause the extremities of this lever to rest on 
 two shorter ones cp and EF, and place a man at each of the points c, D, x, and F: 
 
 sl 
 
MECHANICAL PROBLEMS. . 199 
 
 it is evident that the weight will then 
 be equally distributed among these four 
 
 eoffof.f persons. 
 n bara If eight men are required, pursue the 
 ; Wi; same method with the levers cp and EF, 
 . WT as was employed in regard to the first; 
 A that is, let the extremities of c p be sup- 
 
 ported by the two shorter ones ab and 
 ed; and those of eF by the levers ef and gh: if aman be then stationed at each 
 of the points a, b, c,d, e, f, g, h, they will be all equally loaded. 
 
 The extremities of the levers or poles a b, ed, ef, and gh, might, in like manner, 
 be made to rest on others placed at right angles to them: by means of ‘this artifice 
 the weight would be equally distributed among sixteen men. And so of any other 
 number. 
 
 We have heard that this artifice is employed at Constantinople, to raise and carry 
 the heaviest burthens, such as cannons, mortars, enormous stones, &c, The velocity, 
 it is added, with which burthens are transported from one place to another, by this 
 method, is truly astonishing. 
 
 ; PROBLEM VII. 
 
 A rope acB (fg. 6.), of a determinate length, being made fast by both ends, but not 
 stretched to two points of unequal height, aand B; what position will be assumed by 
 the weight P, suspended from a pulley which rolls freely on that rope. 
 
 Fig. 6. From the points a and B, let fall the indefinite vertical lines 
 
 “® ap and sq; then from the point a, with an opening of the 
 
 “compasses equal to the length of the rope, describe an arc of a 
 
 circle, intersecting the vertical line BG inE: and from the point 
 
 B deseribe a similar arc of a circle, intersecting the vertical line 
 
 AD inp: if the lines az and BD be.then drawn, the point o, 
 
 where they cut each other, will give the position of the rope 
 
 AcB, when the weight has assumed that position in which it 
 
 - must rest; and the point c will be that in which the pulley 
 
 will settle. For it may be easily demonstrated, that in this 
 
 situation the weight Pp will be in the lowest position possible, 
 which is an invariable principle of- the centre of gravity. 
 
 PROBLEM VIII. 
 
 To cause a pail full of water to be supported by a stick, one half of which only, or 
 less, rests on the edge of a table. | 
 To make the reader comprehend properly the method of performing this trick, in 
 regard to equilibrium, we have given, in the annexed figure, a section of the table 
 and the bucket. 
 In this figure let aB be the top of the table, on 
 Fig. 7. which is placed the stick cp. Convey the handle of 
 ¢ R11”? the bucket over this stick, in such a manner that it 
 may rest on it in an inclined position; and let the 
 middle of the bucket be within the edge of the table. 
 That the whole apparatus may be fixed in this situ- 
 ation, place another stick GFE, with one of its ends 
 resting against the corner G of the bucket, while the 
 middle part rests against the edge F of the bucket, and its other extremity against 
 the first stick cp, in 5, where there ought to be a notch to retain it. By these 
 means the bucket will remain fixed in that situation, without being able to incline to 
 
 Ff = 
 
200 MECHANICS. 
 
 either side; and if not already full of water, it may be filled with safety; for its 4 
 centre of gravity being in the vertical line passing through the point 4,which itself meets 
 with the table, it is evident that the case is the same as if the pail were suspended — 
 from the point of the table where it is met by that vertical. It is also evident that | 
 the stick cannot slide along the table, nor move on its edge, without raising the | 
 centre of gravity of the bucketyand of the water it contains. The heayier therefore | 
 it is, the greater will be the stability. | 
 
 Remark.—According to this principle, various other tricks of the same kind, which 
 are generally proposed in books on mechanics, may be performed. For example, 
 provide a-bent hook pg F, as seen at the opposite end of the same figure, and insert 
 the part rv, in the pipe of a key at p, which must be placed on the edge of a table; 
 from the lower part of the hook suspend a weight c, and dispose the whole in such 
 a manner that the vertical line Gp may be a little within the edge of the table. 
 When this arrangement has been made, the weight will not fall, and the case will be 
 the same with the key, which had it been placed alone in that situation would per- 
 haps have fallen; and this resolves the following mechanical problem, proposed in_ 
 the form of a paradox: A body having a terdency to fall by its own weight, how to 
 prevent it from falling, by adding to it a weight on the same side on which it tends 
 to fall. ; 
 
 The weight indeed appears to be added on that side, but in reality it is on the op- 
 posite side. 
 
 PROBLEM IX. 3 
 To hold a stick upright on the tip of the finger, without its being able to fall. 
 
 Fig. 8. Affix two knives, or other bodies, to the extremity of the 
 stick, in such a manner that one ef them may incline to one 
 side, and the second to the other, as seen in the figure 
 (Fig. 8.): if this extremity be placed on the tip of the finger, 
 the stick will keep itself upright, without falling; and if it be 
 made to incline, it will raise itself again, and recover its former 
 situation. ™ 
 For this purpose, the centre of gravity of the two weights 
 added, and of the stick, must be below the point of suspension, 
 or the extremity of the stick, and not at the extremity, as as- 
 serted by Ozanam; for in that case there would be no stability. — 
 
 It is the same principle that keeps in an upright position 
 those small figures furnished with two weights, to counter- 
 balance them; and which are made to turn and balance, while 
 the point of the foot rests on a small ball, loosely placed on a 
 sort of stand. Of this kind is the small figure p (Fig. 9.), 
 supported on the stand 1, by a ball =, through which passes a 
 bent wire, having affixed to its extremities two balls of lead, c 
 and Fr. The centre of gravity of the whole, which is at a con- 
 siderable distance below the point of support, maintains the 
 figure upright, and makes it resume its perpendicular position, 
 after it has been inclined to either side; for this centre tends 
 to place itself as low as possible, which it cannot do without 
 making the figure stand upright. 
 
PROBLEMS IN BALANCING. 201 
 
 By the same mechanism, three knives may be disposed in 
 such a manner as to turn on the point of a needle; for being 
 disposed as seen_in the figure (Fig. 10.), and placed in equilibrio 
 on the point of a needle held in the hand, they cannot fall, be- 
 cause their common centre of gravity is far below the point of 
 the needle, which is above the point of support. 
 
 PROBLEM X. 
 
 To construct a figure, which, without any counterpoise, shall always raise itself upright 
 and keep in that position, or regain tt, however it may be disturbed, 
 
 Make a figure resembling a man, of any substance exceedingly light, such as the pith 
 of the elder tree, which is soft and can be easily cut into any form at 
 pleasure. Then provide for it an hemispherical base of some very 
 heavy substance, such as lead. The half of a leaden bullet, made 
 very smooth on the convex part, will be proper for this purpose. 
 If the figure be cemented to the plane part of this hemisphere; then, 
 in whatever position it may be placed, as soon as it is left to itself, it. 
 will rise upright (Fig. 11.); because the centre of gravity of its he- 
 mispherical base being in the axis, tends to approach the horizontal 
 plane as much as possible, and this can never be the case till the axis 
 becomes perpendicular to the horizon; for the small figure above 
 scarcely deranges it from its place, on account of the disproportion 
 between its weight and that of its base. 
 
 In this manner were constructed those small figures called Prussians, sold at Paris 
 some years ago. ‘Tbey were formed into battalions, and being made to fall down by 
 drawing a rod over them, they immediately started up again as soon as it was 
 removed. 
 
 Screens of the same form-have been-since invented, which always rise up of them- 
 selves, when they happen to be pressed down. 
 
 PROBLEM XI. 
 
 If a rope AcB, to the extremities of which are affixed the given weights P and Q, be 
 made to pass over two pulleys A and B; and if a weight R be suspended from the 
 point c, by the cord Rc; what position will be assumed by the three weights and the 
 ropeacse? (fig. 12.) a ee 
 
 Fig. 12. In the line ab, perpendicular to the horizon, as- 
 
 me sume any part ac, and on that part as a base, de- 
 
 scribe the triangle adc, in such a manner, that ae 
 shall be to ed, as the weight r to the weight Pp; 
 and that ac shall be toad, as R to Q; then through | 
 
 A, draw the indefinite line ac, parallel to ed; and 
 
 through B, draw BC, parallel toad: the point c, 
 
 where these two lines intersect each other, will be 
 the point required, and will give the position a cB of the rope. 
 
 For, if in R c continued we assume c D, equal toa c, and describe the parallelogram 
 EDF; it is evident that we shall have c¥F and cE equal to ed andad; and 
 therefore the three lines Ec, cD, andc F will be as the weights P, Rn,and Q ; consequently 
 the two forces acting from c to F, and from c to &, or in the direction of the 
 lines c A and cB, will be in equilibrio with the force which acts from c towards R. 
 
 Remarks.—Ist. If the ratio of the weights were such, that the point of intersection 
 c should fall on the line a 8, or above it, the problem in this case would be impossible. 
 
 4 
 
202 MECHANICS. 
 
 The weight @, or the weight Pp, would overcome the other two in such a manner, that 
 the point c would fall in B or 4; so that the rope would form no angle. | 
 These weights also might be such that it would be impossible to construct the trie 
 angle a c d, as if one of them were equal to or greater than the other two taken to- 
 gether ; for, to make a triangle of three lines, each of them must be less than the other 
 two. In that case the weight equal or superior to the other two would overcome 
 
 them both, so that no equilibrium could take place. ‘ 
 
 2d, If instead of a knot at c, we should suppose the weight r suspended from a 
 pulley capable of rolling on the rope a c B, the solution would be still thesame; for 
 it is evident that, things being in the same state as in the first case, if a pulley 
 were substituted for the knot c, the equilibrium would not be destroyed. But there 
 would be one limitation more thanin the preceding case. It would be necessary that 
 the point ofintersection, c, determined as above, should fall below the horizontal line, 
 
 drawn through the point B; otherwise the pulley would roll to the point B, as if on an 
 inclined plane. 
 
 PROBLEM XII. - 
 
 Calculation of the time which Archimedes would have required to move the earth, with 
 the machine of which he spoke to Hiero. 
 
 The expression which Archimedes made use of to Hiero, king of Sicily, is well 
 known, and particularly to mathematicians. ‘‘ Give mea fixed point,” said the philo- 
 sopher, “and I will move the earth from its place.” This affords matter for a very 
 curious calculation, viz. to determine how much time Archimedes would have required 
 to move the earth only one inch, supposing his machine constructed and mathemati- 
 cally perfect ; that is to say, without friction, without gravity, and in complete 
 equilibrium. 
 
 For this purpose, we shall suppose the matter of which the earth is composed 
 to weigh 300 pounds the cubic foot; being the mean weight nearly of stones mixed 
 with metallic substances, such in all probability as those contained in the bowels 
 of the earth. If the diameter of the earth be 7930 miles, the whole globe will 
 be found to contain 261107411765 cubic miles, which make 14234991] 20882544640000 
 cubic yards, or 38434476263828705280000 cubic feet ; and allowing 300 pounds to” 
 each cubic foot, we shall have 11530342879148611584000000 for the weight of the 
 earth in pounds. 
 
 Now, we know by the laws of mechanics that, whatever be the construction of a 
 machine, the space passed over by the weight, is to that passed over by the moving — 
 power, in the reciprocal ratio of the latter to the former, It is known also, that a 
 man can act with an effort equal only to about 80 pounds for eight or ten hours, 
 without intermission, and with a velocity of about 10000 feet per hour. If we sup- 
 pose the machine of Archimedes then to be put in motion by means of a crank, and 
 that the force continually applied to it is equal to 30 pounds, then with the velocity 
 of 10000 feet per hour to raise the earth one inch, the moving power must pass over 
 the space of 384344762638287052800000 inches ; and if this space be divided by 10000 
 feet, or 120000 inches, we shall have for quotient 3202873021985725440, which will 
 be the number of hours required for this motion. But asa year contains 8766 hours, 
 acentury will contain 876600; and if we divide the above number of hours by the 
 latter, the quotient, 3653745176803, will be the number of centuries during which it 
 would be necessary to make the crank of the machine continually turn, in order to 
 move the earth only one inch. We have omitted the fraction of a century, as being of 
 little consequence in a calculation of this kind.* 
 
 * The machine is here supposed to be constantly in action: but if it should be worked only 8 
 hours each day, the time required would be three times as long. 
 
HYDROSTATICAL PARADOX. 203 
 
 PROBLEM XIII. 
 
 With a very small quantity of water, such as a few pounds, produce the effect of 
 several thousands. (Fig. 13.) 
 
 Placea cask on one of its ends, and make a holein the other end, capable of admit- 
 ting a tube, an inch in diameter and from 12 to 15 feet in length; which must be fitted 
 Fig. 13. closely into the aperture by means of pitchor tow. Then load the upper 
 
 é end of the cask with several weights, so that it shall be sensibly bent 
 
 downwards ; and haying filled the cask with water, continue to pour some 
 
 | in through the tube. The effort of this small cylinder of water will be so 
 great, that not only the weights which pressed the upper end of the cask 
 
 | downwards will be raised up, but very often the end itself will be bent up- 
 wards, and form an arch in a contrary direction. 
 
 Care however must be taken that the lower end of the cask rest on the 
 ground; otherwise the first effort of the water would be’ directed down- 
 L wards, and the experiment might seem to fail. 
 
 By employing a longer tube, the upper-end of the cask might certainly 
 be made to burst. 
 
 The reason of this phenomenon may be easily deduced from a property pe- 
 culiar to fluids, of which it isan ocular demonstration, viz. that when they press upon 
 a base they exercise on it an effort proportioned to the breadth of that base multiplied 
 by the height. Thus, though the tube used in this experiment contains only about 
 
 _150 or 180 cylindric inches of water, the effort is the same as if the tube were equal 
 in breadthto the cask, and at the same time 12 or 15 feet in height. 
 
 Another Method. (Fig. 14.) 
 
 Suspend from a hook, well fixed in a wall, or any other firm support, a body weigh- 
 ing 100 pounds or more ; then provide a vessel of such dimensions, that between that 
 body and its sides, there shall be room for only one 
 pound of water; and let the vessel be suspended to 
 Baa one of the arms of a balance, the other arm of which 
 has suspended from it a scale, containing a weight of 
 100 pounds, Pour a pound of water into the vessel 
 suspended from the one arm of the balance, and it will 
 raise the scale containing the 100 pounds. 
 
 Those who have properly comprehended the pre- 
 ceding explanation, will find no difficulty in conceiv- 
 ing the cause and necessity of this effect ; for they 
 are both the same, with this difference only, that the 
 water, instead of being collected in a cylindric tube, 
 is in the narrow interval between the body i and the 
 vessel which surrounds it: but this water exercises on the bottom of the vessel the 
 same pressure that it would experience if entirely full of water. 
 
 Another Method. 
 
 Provide a cubic foot of very dry oak, weighing about 60.pounds, and a cubical ves- 
 sel about a line or two larger every way. If the cubic foot of wood be put into the 
 vessel, and water be poured into it, when the latter has risen to nearly two thirds of 
 its height, the cube will be detached from the bottom, and float. Thus we see a 
 weight of about 60 pounds overcome by half-a-pound of water and even less, 
 
 Remark.—Hence it appears that the vulgar are in an error, when they imagine a’ 
 body floats more readily in a large quantity of water than ina small one. It will 
 always float, provided there be a sufficiency to prevent it from touching the bottom- 
 If vessels are lost at the mouths of rivers, it isnot because the water is tooshallow. 
 
 A 
 rane / 
 < 
 
_ 204 MECHANICS. 
 
 but because the vessels are loaded so much, as to be almost ready to sink, even in salt 
 water. But asthe water ofthe sea is nearly a thirtieth part heavier than fresh water, 
 when a ship passes from the one into the other, it must sink more and go to the bottom. 
 
 Thus an egg, which sinks in fresh water, will float in water which holds in solution a 
 
 great deal of salt. 
 
 The principle of the foregoing experiments is what has been called the Hydrostatic 
 
 paradox, and it is on the same principle that Brahma’s Hydrostatic press is constructed. 
 With this press, it has been justly observed, that ‘‘a prodigious force is obtained 
 with the greatest ease and in a very small compass; so that with a machine of the 
 size of a teapot standing on the table before him, a man shall cut through a thick bar 
 of iron as easily as he could clip a piece of pasteboard with a pair of scissars.”” 
 The following isa description of Brahma’s press, as commonly constructed. A 
 (Fig. 15.) isa strong metal cylinder bored perfectly smooth and cylindrical, and into 
 it is fitted the piston B, made perfectly water-tight, by packing it in the usual way. 
 In the bottom of the cylinder is inserted the end of the 
 tube c, the aperture of which communicates with the 
 inside of the cylinder under the piston B, where it is shut 
 by a small valve, as in the suction pipe of a common 
 pump. The other end of the tube ccommunicates with 
 the small forcing pump ez, worked by the lever H. Now, 
 suppose the diameter of the cylinder a to be J2 inches, 
 and that of the smaller pump £ and the tube c only a 
 . quarter of an inch, the proportion between the covers of 
 ‘S the ends of the pistons will be as ] to 2304,—and sup- 
 posing the intermediate span between them to be filled. 
 with water, the force of one piston will act on the 
 other in that proportion. Suppose the small piston to be 
 forced down in the act of injecting water with a force of 
 20 cwt.—which may easily be done by the lever u—the 
 — piston 8 would be forced upwards with a force 2304 times 
 
 as great. Hence, with this machine, by the aid ofa simple lever, a weight of 2304 tons 
 may be raised through any given space better than by any other apparatus constructed 
 on the known principles of mechanics. It may be observed, too, that the effect of all 
 
 other mechanical combinations is counteracted by an accumulated complication of — 
 
 parts which limits the intent to which they can be usefully carried,—but in machines 
 
 depending on the principle under consideration, the power may be extended ad libitum . 
 
 either by increasing the proportion between the diameters of © and a, or applying 
 greater power to the lever H. 
 
 Fig. 16. is a section shewing how, by means of fluids, the power and motion of one 
 machine may be communicated to another, let their distance and relative situations be 
 
 whatthey may. aand Bare two small smooth cylindrical ~ 
 
 Fig. 16. tubes, inside of each being a portion made air-tight and 
 
 water-tight. cc is a tube conveyed underground or 
 otherwise, from the one cylinder to that of the other, — 
 and this tube is filled with water till it touch the bot- 
 tom of both pistons. Then by depressing the piston a, 
 the piston B will be raised, and vice versa. Thus bells 
 may be rung, wheels turned, or other machinery put 
 motion, by invisible agency. 
 
 Fig. 17. isa section shewing another instance of action and force communicated from 
 one machine to another,—and how water may be raised from wells of any depth, and 
 a’ any distance from the place where the power is applied. a is a cylinder of any re- 
 quired dimensions, in which is the working piston B of the Brahma press, described 
 above. Into the bottom of this cylinder is inserted thetube c, which may be of less 
 
 # 
 
TO WEIGH A FOOT OF WATER. 205 
 
 dimensions than the cylinder a. This tube is conveyed 
 in any direction down to the pump cylinder p in the 
 deep well £ E, and forms a junction with p above the pis- 
 ton F, which piston has a rod @ working through a stuff- 
 ing box. To this rod Gc a weight # is connected over a 
 pulley or otherwise, sufficient to overbalance the weight 
 of the water in the tube c, and to raise the piston F when 
 the piston B is lifted. Suppose, now, the piston B is 
 drawn up by its rod, there will bea vacuum made in D 
 below the piston F, which vacuum will be filled with 
 water, through the suction pipe, by the pressure of the 
 atmosphere. . 
 
 The piston B, being pressed downwards in the cylinder 
 a, will make a stroke in the pump cylinder p, which may 
 be repeated in the usual way by the motion of the pis- 
 ton B and the action of the water in the tube c. The 
 small tube and cock in the cistern 1 are for the purpose of charging the tube c. 
 
 By these meansa commodious machine of prodigious power and of the greatest 
 strength may readily be formed: all that is required is accurate execution, which in 
 the present state of the mechanical arts is quite attainable. 
 
 PROBLEM XIV. 
 To find the weight of a cubic foot of water. 
 
 To know the weight of a cubic foot of water is one of the most essential elements 
 of hydrostatics and hydraulics; and for that reason we shall here show how it may 
 be accurately determined. 
 
 Provide a vessel, capable of containing exactly a cubic foot, and having first 
 weighed it empty, weigh it again when filled with water. But as liquids always rise 
 considerably above the edges of the vessel that contains them, the result in this case 
 will not be very correct. There are means indeed to remedy this defect; but we 
 are furnished with a very accurate method of doing it by hydrostatics. 
 
 rovide a cube of some very homogeneous matter, such: as metal, each side of 
 which is exactly four inches; weigh it by a good balance, in order to ascertain its 
 weight within a few grains ; then suspend it by a hair, or strong silk thread, from one 
 of the scales of the same balance, and again find its weight when immersed in water. 
 We are taught by hydrostatics that it will lose exactly as much in weight as the 
 weight of an equal volume of water. The difference of these two weights therefore 
 will be the weight of a cube of water, each side of which is tour inches, or of the 
 twenty-seventh part of a cubic foot. 
 
 If very great precision is not required, provide a cube or rectangular parallelopipe- 
 don, of any homogeneous matter, lighter than water—such, for example, as wood; 
 and having weighed it as accurately as possible, immerse it gently in water, in such a 
 manner. that the water may not wet it above that point at which it ought to float 
 above the liquid. We shall here suppose that imp (Fig. 18.) 
 is the line, which exactly marks how much of it is immersed. 
 Find the content: of the solid azn cpm1, by multiplying its base 
 by the height; the product will be the volume of water dis- 
 placed by the body; and this volume, according to the principles 
 of hydrostatics, must weigh as much as the body itself. If this 
 volume of water be 720 cubic inches, for example, and if the weight of the body be 
 26-0416 pounds, we consequently know that 720 cubic inches of water weigh 26°0416 
 pounds. Hence it will be easy to determine the weight of a cubic foot, which con- 
 tains 1728 cubic inches. Nothing is necessary but to make this proportion: as 720 
 
 a 
 
206 MECHANICS. 
 
 cubic inches are to 1728, so are 26°0416 pounds to a fourth term, which will be 62°5 | 
 pounds, or 62 pounds and a half; which therefore is the weight of a cubic foot | } 
 water. | 
 
 PROBLEM XV. | 
 Two liquors being given ; to determine which of them is the lightest. | 
 
 This problem is generally solved by means of: a well known in : 
 ment called the areometer or hydrometer. This instrument is nothing 
 else than a small hollow ball, joined toa tube 4 or 5 inches in lengtill 
 (Fig. 19.); a few grains of shot, or a little mercury, being put into the — 
 ball, the whole is so combined, that in water of mean gravity, the small 
 ball and part of the tube are immersed. 
 
 It may now be easily conceived that when the instrument is put into 
 any fluid, for example river water, care must be taken to observe how 
 far it sinks into it; if it be then placed in another kind of water, such 
 as sea water, for instance, it will sink less; and if immersed in any 
 liquor lighter than the first, such as oil for example, it will sink farther, 
 Thus it can be easily determined, without a balance, which of two 
 liquors is the heavier or lighter. This instrument has commonly on the 
 tube a graduated scale, in order to shew how far it sinks in the fluid. 
 
 But this instrument is far inferior to that presented, in 1766, by M. 
 
 : de Parcieux, to the Academy of Sciences, and yet nothing is simpler. 
 
 This instrument consists of a small glass bottle, two inches, or two inches and a 
 half at most, in diameter, and from six to eight inches in length. The bottom must 
 not be bent inwards, lest: air should be lodged in the cavity when it is immersed in 
 any liquid. ‘The mouth is closed with a very tight cork stopper, into which is fixed, 
 without passing through it, a very straight iron wire, 25 or 30 inches in length, and 
 about a line in diameter. The bottle is then loaded in such a manner, by introducing | 
 into it grains of small shot, that the instrument, when immersed in the lightest of 
 the liquors to be compared, sinks so as to leave only the end of the iron wire above 
 its surface, and that in the heaviest the wire is immersed some inches. This may be 
 properly regulated by augmenting or diminishing either the weight with which the 
 bottle is loaded, or the diameter of the wire, or both these at the same time. The 
 instrument, when thus constructed, will exhibit, in a very sensible manner, the least 
 difference in the specific gravities of different liquors, or the changes which the same 
 liquor may experience, in this respect, under different circumstances; as by the effect 
 of heat, or by the mixture of various salts, &c. 
 
 It may be readily conceived, that to perform experiments of this kind, it will be 
 necessary to have a vessel of a sufficient depth, such as a cylinder of tin-plate, 3 or 
 4 inches in diameter, and 3 or 4 feet in length. 
 
 We have seen an instrument of this kind, the movement of which was so sensible, 
 that when immersed in water, cooled to the usual temperature, it sank several inches, 
 while the rays of the sun fell upon the water; and immediately rose on the rays of 
 that luminary being intercepted. A very small quantity of salt or sugar, thrown into 
 the water, made it also rise some inches. 
 
 By means of this instrument, M. de Parcieux examined the gravity of different 
 kinds of the most celebrated waters; among which was that drank at Paris; and he 
 found that the lightest of all was distilled water. The next in succession, according 
 to their lightness, were as in the following order; viz., the water of the Seine, that 
 of the Loire, that of the Yvette, that of Arcueil, that of Sainte-Reine, that of 
 Ville d’Avray, the Bristcl water, and well water. ; 
 
 We hence see the error of the vulgar, who imagine that the water of Ville d’ Arvay, 
 that of Sainte-Reine, and that of Bristol, particularly the last brought to France, at 
 
 a 
 
é 
 
 WEIGHT OF LIQUORS. 207 
 
 so great expense, are better than common river water; for they are, on the contrary, 
 worse, since they are heavier. 
 
 If different kinds of water differ in their gravity, the case is the same with wines 
 also. ‘The lightest of all the known wines, at least in France, is the Rhenish. The 
 next in succession are Burgundy, red Champagne, the wines of Bourdeaux, Langue- 
 doc, Spain, the Canaries, Cyprus, &c. 
 
 The lightest of all the known liquors is ether. The others, which follow in the 
 order of gravity, are: alcohol, oil of turpentine, distilled water, rain water, river 
 ‘water, spring water, well water, mineral waters. Among the tables, annexed to 
 this part of the work, the reader will find one containing the specific gravity of 
 various liquors, compared with that of rain water; which, being the easiest procured, 
 may serve as a common standard, and also the specific gravity of the different solid 
 bodies, whether belonging to the mineral, vegetable, or animal kingdom; which will 
 doubtless be found very useful, as it is often necessary to have recourse to tables of 
 this kind. 
 
 As the following rules for calculating the absolute gravity, in English troy weight, 
 of a cubic foot and inch, English measure, of any substance whose specific gravity 
 is known, may be of use to the reader, the translator has thought proper to subjoin 
 them to this article of the original. 
 
 In 1696, Mr. Everard, balance-maker’ to the Exchequer, weighed before the com- 
 missioners of the House of Commons, 2145°6 cubical inches, by the Exchequer 
 standard foot, of distilled water, at the temperature of 55° of Fahrenheit, and found 
 that it weighed 1131 oz. 14 drs. troy, of the Exchequer standard. The beam 
 turned with 6 grains, when loaded with 30 pounds in each scale. Hence, supposing 
 the pound averdupois to weigh 7000 grs. troy, a cubic foot of water weighs 623 
 pounds averdupois, or 1000 ounces averdupois, wanting 106 grs. troy. If the spe- 
 cific gravity of water therefore be called 1000, the proportion of specific 
 gravities of all other bodies will express nearly the number of averdupois ounces 
 in a cubic foot. Or, more accurately, supposing the specific gravity of water ex- 
 pressed by 1, and that of all other bodies in proportional numbers, as the cubic foot 
 of water weighs, at the above temperature, exactly 437489:4 grains troy, and the 
 cubic inch of water 253175 grains, the absolute weight of a cubical foot or inch of 
 any body, in troy grains, may be found by multiplying its specific gravity by either 
 of the above numbers respectively. 
 
 By Everard’s experiment, and the proportions of the English and French foot, as 
 established by the Royal Society and French Academy of Sciences, the following 
 numbers have been ascertained : 
 
 Paris grains, in a Paris cube foot of water .. .. .. «2 «oe «. 6455/1 
 
 English grains, in a Paris cube foot of water wietp sien Oo Behy qsetitan Baa aoe 
 
 Paris grains, in an English cube foot of water .. .. «2 ss e* 533247 
 
 English grains, in an English cube foot of water ..  .. .. «. «+ 4374894 
 
 English grains in an English cube inch of water .. .. + p 253307 
 
 By an experiment of Picard, with the measure and weight igs the Cha- 
 
 telet, the Paris cube foot of water contains of Paris grains .. .. 641326 
 
 By one of Du Hamel, made with great care .. .. «2 1. «. oe 641876 
 
 By Homberg all nilse BANA cael of) Wel cole oo ee BLIOBS 
 
 These results shew some ie PRUE RE in measure or in weights; but the above 
 computation from Everard’s experiment may be relied on; because the comparison 
 of the English foot with that of France, was made by the joint labour of the Royal 
 Society of London, and the French Academy of Sciences. It agrees likewise, very 
 
 nearly, with the weight assigned by Lavoisier, which is 70 Paris pounds to the cubical 
 foot of water. 
 
Cm te a > 
 Beas ee 
 
 208 -MECHANICS. 
 
 PROBLEM XVI. 
 To determine whether a mass of gold or silver, suspected to be mized, is pure or not. | 
 
 If the mass or piece, the fineness of which is doubtful, be silver for example, pro- 
 vide another mass of good silver equally heavy; so that the two pieces, when put into 
 the scales of a very accurate balance, may remain in equilibrio in the air. Then. 
 suspend these two masses of silver from the scales of the balance, by two threads 
 or two horse-hairs, to prevent the scales from being wetted when-the two masses are 
 immersed in. the water: if the masses are of equal fineness, they will remain in 
 equilibrio in the water, as they did when in the air; but if the proposed mass weighs 
 less in water, it is adulterated; that is to say, is mixed with some other metal, of 
 less specific gravity than that of silver, such as copper for example; and if it weighs 
 more, it is mixed with some metal of greater specific gravity, such as lead. 
 
 Remarks.—1. This problem is evidently the same as that whose solution gave so 
 much pleasure to Archimedes. Hiero, king of Syracuse, had delivered to a gold- 
 smith a certain quantity of gold, for the purpose of making a crown. When the 
 crown was finished, the king entertained some suspicion in regard to the fidelity of 
 the goldsmith, and Archimedes was consulted respecting the best means of detecting 
 the fraud, in case one had been committed. The philosopher, having employed the 
 above process, discovered that the gold, of which the crown consisted, was not pure. 
 
 If a large mass of metal were to be examined, as in the case of Archimedes, it 
 would be sufficient to immerse the mass of gold or silver, known to be pure, ina 
 vessel of water, and then the suspected mass. If the latter expelled more water 
 from the vessel it would be a proof of the metal being adulterated by another = 
 and of less value. 
 
 But notwithstanding what Ozanam says, the difference between the weights in air 
 and that in water will indicate the mixture with more certainty; for every body 
 knows that it is not so easy, as it may at first appear, to measure the quantity of 
 water expelled from any vessel. 
 
 2. According to mathematical rigour, the two masses ought first to be weighed 
 in vacuo; for since air is a fluid, it lessens the real gravity of bodies by a quantity 
 equal to the weight of a similar volume of itself. Since the two masses then, the 
 one pure and the other adulterated, are unequal in volume, they ought to lose un- 
 equal quantities of their weight in the air. But the great tenuity of air, in regard 
 to that of water, renders this small error insensible. 
 
 PROBLEM XVII. 
 The same supposition made ; to determine the quantity of mixture in the gold. 
 
 The ingenious artifice employed by Archimedes, is contained in the solution of this 
 problem, and is as follows. 
 
 Suspecting that the goldsmith had substituted silver or copper for an equal quan- 
 tity of gold, he weighed the crown in water, and found that it lost a weight, which 
 we shall call a: he then weighed in the same fluid a mass of pure gold, which in air 
 was in equilibrio with the crown, and found that it lost a weight, which we shall call 
 B; he next took a mass of silver, which in air was equal in weight to the crown, and 
 weighing it in water, found that it lost a quantity c. He then employed this propor- 
 tion: as the difference of the weights B and c, is to that of the weights a and B, so 
 is the whole weight of the crown to that of the silver mixed in it. The answer, in 
 this case, may be obtained by a very short algebraigal calculation, thongh the reason- 
 ing is rather too prolix ; we shall however explain,t after having illustrated this rule 
 by an example. 
 
 Let us suppose that Hiero’s crown weighed 20 pounds in the air, and that when 
 
 , 
 
 i 
 
 q 
 
fea 
 . 
 
 TO DISCOVER GOLD AND SILVER. . ; 209 
 
 weighed in water it lost a pound and a half. Archimedes, by weighing in air and in 
 water, a mass of gold containing 20 pounds, must have found a difference of 1}; 
 pound; and by weighing in like manner a mass of silver of 20 pounds, he must have 
 found a difference of 1 ~ pound. As 4, in this case, is equal to 3, B to 22, and c to 22; 
 hence the difference of a and B is 4, and that of B and c is 389: we must therefore use 
 the following proportion : as 4§$ are to4Z, so is 20 toa fourth term, which will be 187 = 
 1] lbs. 8 oz. 5 dwts. 
 
 The reasoning which conducted, or might have conducted, the Syracusan philoso- 
 pher to this solution, isas follows. If the whole mass were of pure gold, it would lose, 
 when weighed in water, 7, of its weight; and if it were of pure silver it would 
 lose, when weighed in water, }, of its weight: consequently, if it loses less than the 
 latter quantity, and more than the former, it must be a mixture of gold and silver; 
 and the quantity of silver substituted for gold will be greater as the quantity of 
 weight which the crown loses in water approaches nearer to 4, and vice versa. This 
 mass of 20 pounds then must be divided into two parts, in the ratio of the following 
 differences: viz. the difference between the loss which the crown experiences and 
 that experienced by the pure gold; and the difference between the loss experienced 
 by the crown and that experienced by the pure silver; these will be the proportions 
 of the gold and silver mixed together in the crown ; and from'this reasoning is deduced 
 the preceding rule. 
 
 We must here observe, that it is not necessary to take two masses, one of gold and 
 another of silver, each equal in weight to the crown. — It will be sufficient to ascer- 
 tain that gold loses a nineteenth of its weight, when weighed in water; and silver 
 one eleventh, and perhaps this was really the method employed by Archimedes, 
 
 PROBLEM XVIII. 
 
 _ Suppose there are two boxes exactly of the same size, similar and of equal weight, the 
 one containing gold and the other silver: is it possible, by any mathematical means, 
 to determine which contains the gold, and which the silver? Or, if we suppose 
 two balls the one made of gold and hollow, the other of solid silver gilt, is it pos- 
 sible to distinguish the gold from the silver ? 
 
 In the first case, if the masses of gold and silver are each placed exactly in the middle 
 of the box which contains it, so that their centres of gravity coincide, whatever may 
 be said in the old books on Mathematical Recreations, we will assert that there are no 
 means of distinguishing them, or at least that the methods proposed are defective. 
 
 The case is the same in regard to the two similar globes of equal size and weight, 
 
 If we were however under the necessity of making a choice, we would endeavour 
 to distinguish the one from the other in the following manner. 
 
 We would suspend both balls by as delicate a thread as possible to the arms of a 
 very accurate balance, such as those which, when loaded with a considerable weight, 
 are sensibly affected by the difference of a grain. We would then immerse the 
 two balls in a large vessel filled with water, heated to the degree of ebullition, and 
 that which should preponderate we would consider as gold For, according to the 
 experiments made on the dilatation of metals, the silver, passing from a mean tem- 
 perature, to that of boiling water, would probably increase more in volume than the 
 gold; in that case the two masses, which in air and temperate water were in equilibrio, 
 would not be soin boiling water. 
 
 Or, we might make a round hole in a plate of copper, of such a size, that both 
 balls should pass exactly through it with ease; we might then bring them to a strong 
 degree of heat, superior even to that of boiling water. Now, if we admit that silver 
 expands more than gold, as above supposed, we might apply each of them to the hole 
 in question, and the one which experienced the greater difficulty in passing, ought to 
 
 beaccounted silver. “ 
 
210 . MECHANICS. 
 
 PROBLEM XIX. 
 
 Two inclined planes AB and AD being given, and two unequal spheres p and p; to 
 bring them to an equitibrium in the angle, as seen in the figure. (Fig. 20.) 
 
 The globes p and p, wiil be in equilibrio, if the powers with which they repel each 
 
 other, in the direction of the line c c, which joins their centres, are equal. 
 But the force with which the globe P tends to descend 
 Fig. 20. along the inclined plane Ba, which is known, the inclina- 
 tion of the plane being given, is to the force with which 
 it acts in the direction c c, as radius is to the cosine of 
 the angle ¢ c F; and in like manner, the force with 
 which the weight p descends along D A, is to that with 
 which it tends to move in the direction ec c, as radius is 
 to the cosine of the anglece f: hence it follows, that 
 as these second forces must be equal, the cosine of 
 the angle c must have the same ratio to the cosine of the angle c, as the force with 
 which the globe P tends to roll along B a, has to that with which p tends to roll along 
 pa. Theratio of these cosines therefore isknown; and as in the triangle ce ¢ the 
 angle G is known, since it is equal to the angle D a B, it thence follows that its sup- 
 plement, or the sum of the two angles c and c, is alsoknown; and hence the problem 
 is reduced to this, viz. to dividing a known angle into two such parts, that their cosines 
 shall be in a given ratio; which isa problem purely geometrical. 
 
 But, that we may confine ourselves te the simplest case, we shall suppose the angle 
 A to be a right-angle. Nothing then will be necessary but to divide the quadrant 
 into two arcs, the cosines of which shall be in the given ratio, which may be done 
 with great ease. 
 
 Let the force then with which p tends to move along its inclined plane be equal to 
 m; and that of p to roll alongits plane equaltom. Draw a line parallel to the plane 
 A B, ata distance from it equal to the radius of the globe p, and another parallel to 
 the plane p 4, at a distance from it equal to the radius of p, which will intersect 
 each other in a; having then made cu to GJ, asm tom, employ the following pro- 
 portion: as LJ is to LG, so is the sum of the radii of the two globes to gc; and 
 from the point c, draw ce parallel tou 1: the points c and c will be the places of 
 the centres of the two globes, and in this situation alone they will be in equilibrio. 
 
 PROBLEM XX. 
 Two bodies, Pp and Q, depart at the same time from two points a and B, of two lines 
 given in position, and move towards a and b, with given velocities: required their’ 
 position when they are the nearest to each other possible? (Fig. 21.) 
 
 If their velocities were to each other in the ratio of the lines Bp and a D, it is 
 
 evident that the two bodies would meetin p. But sup posing their velocities different 
 
 from that, there will be a certain point where, with- 
 
 Fig. 21. out meeting, they will be at the least distance from 
 
 iT a each other possible; and after that they will conti- 
 
 nually recede from each other. Here, for example, 
 
 the lines B p and a D are nearly equal. If we sup- 
 
 pose then that the velocity of pis to that of @, in 
 
 the ratio of 2 to 1, required the point of the nearest 
 approach. 
 
 Through any point R, in a D, draw the line RS 
 parallel to B D, and in sucha manner, that a R shall 
 be to R s, as the velocity of Pp is to that of q; that is to say, in the present case, as 
 2 to 1; produce indefinitely the line a s T, and from the point B draw B c perpendi- 
 
GRAVITY OF BODIES. 211 
 
 cular to A T; through the point c draw c £ parallel to b D, till it meet aD in E; and 
 
 having drawn £ F parallel to c B, meeting 8 D in F, the points F and £ will be those 
 required. 
 
 PROBLEM XXI, 
 
 To cause a cylinder to support itself on a plane, inclined to the horizon, without roll- 
 ing down ; and even to ascend a little along that plane. (Fig. 22.) 
 
 If a cylinder be homogeneous, and placed on an inclined plane, its axis being in a 
 horizontal situation, it is evident that it will roll down; because its centre of gravity 
 being the same as that of the figure, the vertical line, drawn from this centre, will 
 always fall beyond the point of contact of the lowest side; consequently the body 
 must of necessity roll down towards that side. 
 
 But, if the cylinder be heterogeneous, so that its centre of gravity is not that of the 
 figure, it may support itself on an inclined plane, provided the angle which the plane 
 makes with the horizon does not exceed certain limits. 
 
 Let there be a cylinder, for example, of which u F p is a section perpendicular to 
 the axis. To remove its centre of gravity from the centre of the figure, make a groove 
 in it parallel to its axis, of a semi-circular form, and 
 fill it with some substance F much heavier, so that 
 the centre of gravity of the cylinder shall be removed 
 from cto £. Let the inclined plane be a zB, and let 
 BG be to Ga ina less ratio thancrtocEr. The 
 cylinder may then support itself on the inclined 
 plane, without rolling down; and if it be moved from that position, in a certain 
 direction, it will even resume it by rolling a little towards the summit of the plane. 
 
 For, let us suppose the cylinder placed on the inclined plane with its axis horizontal, 
 and its centre of gravity ina line parallel to the plane, and passing through the centre, 
 in such a manner that the centre of gravity shall be towards the upper part of the 
 
 Fig. 23. plane (Fig. 23.) Through the point.of contact, p, draw 
 CDH perpendicular to the inclined plane, and 1D e per- 
 pendicular to the horizon. We shall then have BG to 
 G A, or BItOT D, as DItO TH, oras D ctoce; and since 
 the ratio of B G to G A is less than that of cF orcp to 
 cE, it follows that c e is less than c E, consequenly the 
 vertical line drawn from the point = will fall without the point of contact towards A; 
 the body therefore will have a tendency to fall on that side, and will roll towards it, 
 ascending a very little till its centre of gravity E has assumed a position as seen Fig. 
 22, where it coincides with the vertical line passing through the point of contact. 
 When the cylinder arrives at this situation, it will maintain itself in it, provided 
 neither its surface nor that of the plane be so smooth as to admit of its sliding parallel 
 to itself. In this situation it will even have greater stability, according as the ratio of 
 B cto cis less than that of cr or cD to CE, or as the angle a BGor Cc De is less 
 than cDE. 
 
 This is also a truth which we must demonstrate. For this purpose, it is to be 
 remarked that gE, thecentre of gravity of the cylinder, in rolling along the inclined 
 
 plane, describes a curve, such as is seen in 
 
 Fig. 24. "Fig. 24.; this is what geometricians call an 
 
 elongated cycloid, which rises and descends 
 alternately below the line drawn parallel to 
 the inclined plane, through the centre of 
 the cylinder. But the cylinder being in 
 the position represented in Fig, 24, if the 
 
 line = p be drawn from the centre of gravity to the point, of contact, it may be 
 P 
 
 Fig. 22. 
 
912 MECHANICS. 
 
 demonstrated that the tangent to the point E of that curve is perpendicular to D &: 
 if the inclination of the plane therefore is less than the angle c p £, that tangent will 
 meet the horizontal line towards the ascending side of the plane; and the centre of 
 gravity of the cylinder will then be as on an inclined plane 1 K; consequently it 
 must descend to the point L of the hollow of the curve, which it describes, where 
 that curve is touched by the horizontal line. 
 
 When it reaches this point it cannot deviate from it, without ascending on the one 
 side or the other: if it be then removed a little from this point, it will return to 
 its former position. 
 
 PROBLEM XXII. ‘ 
 To construct a clock which shall point out the hours, by rolling down an inclined plane. 
 
 This small machine, invented by an Englishman named Wheeler, is exceedingly 
 ingenious, and is founded on the principle contained in the solution of the preceding 
 problem. 
 
 Fig. 25. It consists of a cylindrical box, made 
 of brass, four or five inches in diameter, 
 and having on one side a dial plate, di- 
 vided into 12 or 24 hours. In the inside, 
 represented by Fig. 25, is a central wheel, 
 which by means of a pinion moves a se~ 
 cond wheel, and the latter moves a third, 
 &c., while a scapement, furnished with a 
 balance or spiral spring, acts the part of 
 a moderator, as in common watches. To 
 { the central wheel is affixed a weight Pp, 
 
 which must be sufficient, with a moderate 
 inclination, as 20 or 30 degrees, to move that wheel, and those which receive 
 motion from it. But, as the machine ought to be perfectly in equilibrio around its 
 central axis, a counteracting weight, of such a nature that the machine shall be 
 absolutely indifferent to every position around this axis, must be placed diametrically 
 opposite to the small system of wheels 2, 3,4, &c. When this condition has been 
 obtained, the moving weight P must be applied; the effect of which will be to make 
 the central wheel, 1, revolve, and by its means the clock movement 2, 3,4, &c.; but, 
 at the same time that this motion takes place, the cylinder will roil down the plane 
 a little, which will bring the weight P to its primitive position, so that the effect of 
 
 this continual pressure will make the cylinder roll while the weight P changes its - 
 place relatively, in regard to the cylinder, but not in regard to the vertical line. The 
 weight P, or the inclination of the plane, must be regulated in such a manner, that 
 
 the machine shall perform a whole revolution in twenty-four or twelve hours. The 
 handle must be affixed to the common axis of the central wheel and weight P; so 
 that it shall always look towards the zenith or the nadir. If more ornaments are 
 required, the axis may support a small globe with a figure placed on it, to point out 
 the hours with its finger raised in a vertical position. It may be readily conceived, 
 that when the machine hag got to the lowest part of the inclined plane, to make it 
 continue going, nothing will be necessary but to cause it to ascend to the highest. If 
 it goes rather too slow, its movement may be accelerated by raising up the inclined 
 plane, and vice versa. 
 
 PROBLEM XXIII. 
 To construct a dress, by means of which it will be impossible to sink in the water, 
 
 and which shall leave the person, who wears it, at full freedom to make every hind 
 
 of movement. 
 
4 
 
 i 
 
 LIFE BOAT. 213 
 
 As a man weighs very nearly the same as an equal volume of water, it is evident 
 that a mass of some substance much lighter than that fluid may be added to his body, 
 by which means both together will be lighter than water, and of course must float. 
 It is in consequence of this principle that, in order to learn to swim, some people tie 
 to their breast and back two pieces of cork, or affix full blown bladders below their 
 
 arms. But these methods are attended with inconveniences, which may be remedied 
 
 in the following manner. 
 
 Between the cloth and lining of a jacket, without arms, place small pieces of cork, 
 an inch and a half square, and about half or three quarters of an inch in thickness. 
 They must be arranged very near to each other, that as little space as possible may 
 be lost; but yet not so close as to affect in any degree the flexibility of the jacket, 
 which must be quilted to prevent their moving from their places. The jacket must 
 be made to button round the body, by means of strong buttons, well-sewed on; and 
 to prevent its slipping off, it ought to be furnished behind with a kind of girdle, so 
 as to pass between the thighs and fasten before. 
 
 By means of such a jacket, which will occasion as little embarrassment as a com- 
 mon dress, people may throw themselves into the water with the greatest safety ; 
 for if it be properly made the water will not rise over their shoulders. They will 
 sink so little that even a dead body in that situation would infallibly float. The 
 wearers therefore need make no effort to support themselves; and while in the water 
 they may read or write, and even load a pistol and fire it. In the year 1767 an ex- 
 periment was made of all these things, by the Abbé de la Chapelle, fellow of the 
 Royal Society of London, by whom this jacket was invented. 
 
 It is almost needless to observe how useful this invention might be on land as 
 well as at sea. A sufficient number of soldiers, provided with these jackets, might 
 pass a deep and rapid river in the night time, armed with pistols and sabres, and 
 surprise a corps of the enemy. If repulsed, they could throw themselves into 
 the water, and escape without any fear of being pursued. 
 
 During sea voyages, the sailors, while employed in dangerous manceuvres, often fall 
 overboard and are lost; others perish in ports and harbours by boats oversetting in 
 consequence of a heavy swell, or some other accident; in short, some vessel or 
 other is daily wrecked on the coasts, and it is not without difficulty that only a part 
 of the crew are saved. If every man, who trusts himself to this perfidious element, 
 were furnished with such a cork jacket, to put on during the moments of danger, it 
 is evident that many of them might escape death. 
 
 PROBLEM XXIV. 
 To construct a boat which cannot be sunk, even if the water should enter it on 
 all sides. 
 Cause a boat to be made with a false bottom, placed at such a distance from the 
 real one, as may be proportioned to the length of the boat, and to its burthen and 
 the number of persons it is intended to carry. According to the most accurate 
 
 . calculation, this distance, in our opinion, ought to be one foot, for a boat eighteen feet 
 
 in length, and five or six in breadth. The vacuity between this false bottom and 
 the real one ought to be filled up with pieces of cork, placed as near to each other 
 as possible: and as the false bottom will lessen the sides of the boat, they may he 
 raised proportionally ; leaving large apertures, that the water thrown into tne vessel 
 may be able to run off. It may be proper also to make the stern higher, and to 
 furnish it with a deck, that the people may take shelter under it, in case the boat 
 should be thrown on its side by the violence of the waves. 
 
 Boats constructed in this manner might be of great utility for going on board a 
 
 vessel lying in a harbour, perhaps several miles from the shore; or for going on 
 
 shore from a ship anchored at a distance from the land. Unfortunate accidents too 
 
214 MECHANICS. 
 
 often happen on such occasions, when there is a heavy surf, or in consequence of 
 some sudden gust of wind; and it even appears that sometimes the greatest danger of 
 a voyage is to be apprehended under circumstances of this kind. But boats con- 
 structed on the above principle would prevent such accidents. 
 
 Much we confess is to be added to this idea, presented here in all its simplicity; 
 
 for some changes perhaps ought to be made in the form of the vessel; or heavy — 
 
 bodies ought to be added in certain places to increaseits stability. This is a subject 
 of research well worth attention, as the result of it might be the preservation of 
 thousands of lives every year. 
 
 For this invention we are indebted to M. de Bernieres, one of the four controllers 
 general of bridges and causeways: who in 1769 constructed a boat of this kind for 
 the king. Heafterwards constructed another with improvements for the Duke de 
 
 Chartres; and a third for the Marquis de Marigny. The latter was tried by filling "4 
 it with water, or endeavouring to make it overset; but it righted as soon as left to 
 
 itself; and though filled with water, was still able to carry six persons. 
 
 By this invention the number of accidents which befall those who lead a sea- 
 faring life, may in future be diminished; but the indifference with which the inven- 
 tion of M. de Bernieres was received, shews how regardless men are of the most 
 useful discoveries, when the general interests of humanity only are concerned, and 
 when trouble and expense are required to render them practically useful.* 
 
 PROBLEM XX¥. 
 How to raise from the bottom of the sea a vessel which has sunk. 
 
 This difficult enterprise has been several times accomplished by means of a very 
 simple hydrostatical principle, viz., that if a boat be loaded as much as possible, and 
 then unloaded, it tends to raise itself with a force equal to that of the weight of the 
 volume of water which it displaced when loaded. And hence we are furnished 
 with the means of employing very powerful forces to. raise a vessel that has been 
 sunk. 
 
 The number of boats employed for this purpose, must be estimated according to 
 the size of the vessel, and by considering that the vessel weighs in water no more 
 than the excess of its weight over an equal volume of that fluid; unless the vessel 
 is firmly bedded in the mud; for then she must be accounted of her full weight. 
 The boats being arranged in two rows, one on each side of the sunk vessel, the ends 
 of cables, by means of divers, must be made fast to different parts of the vessel, so 
 that there shall be four on each side, foreach boat. The ends of these cables, which 
 
 ’ remain above water, are to be fastened to the head and stern of the boat for which 
 
 they are intended. Thus, if there are four boats on each side, there must be thirty- 
 two cables, being four for each boat. 
 
 When every thing is thus arranged, the boats are to be loaded as much as they will 
 bear without sinking, and the cables must be stretched as much as possible. The 
 boats are then to be unloaded, two and two, and if they raise the vessel, it is a sign 
 
 that there is a sufficient number of them; but in raising the vessel, the cables affixed — 
 
 to the boats which remain loaded will become slack, and for this reason they must be 
 again stretched as muchas possible. The rest of the boats are then to be nnloaded, 
 by shifting their lading into the former. The vessel will thus be raised a little more, 
 and the cables of the loaded boats will become slack; these cables being again 
 stretched, the lading of the latter boats must be shifted back into the others, which 
 
 will raise the vessel still a little higher; and if this operation be repeated as long as 
 
 * Vessels constructed on this principle, known under the name of Life Boats, are in very general 
 use ; and have been the means of saving the lives of many who would otherwise have perished by 
 
 shipwreck. They were first constructed at Shields by Mr. Greathead ; but a humble mechanic of the 
 
 name of Wouldhave is said to have been the original inyentor. 
 
—— 
 
 TO RAISE SUNKEN VESSELS. 215 
 
 necessary, she may be brought to the surface of the water, and conveyed into port, or 
 into dock. 
 
 An account of the manceuvres employed to raise, in this manner, the Tojo, a 
 Spanish ship belonging to the Indian fleet, sunk in the harbour of Vigo, during the 
 battle on the 10th of October 1702, may be seen in the ‘‘ Mémoires des Academi- 
 ciens étrangers,” vol. Ii. But as this vessel had remained more than thirty-six years 
 in that state, it was imbedded in a bank of tenacious clay, so that it required incre- 
 dible labour to detach it; and when brought to the surface of the water, it con- 
 tained none of the valuable articles expected. It bad been one of those unloaded by 
 the Spaniards themselves, before they were sunk, to prevent them from falling 
 into the hands of the English. 
 
 Additions.—On the same principle is constructed the camel, a machine employed 
 by the Dutch for carrying vessels heavily laden over the sand banks in the Zuyder- 
 Zee. In that sea, opposite to the mouth of the river Y, about six miles from the 
 city of Amsterdam, there are two sand banks, between which is a passage, called 
 the Pampus, sufficiently deep for small vessels, but not for those which are large and 
 heavily laden. On this account ships which are outward bound, take in before the 
 city only a small part of their cargo, receiving the rest when they have got through 
 the Pampus. And those that are homeward bound must ina great measure unload 
 before they enter it. For this reason the goods are put into lighters, and in these 
 transported to the warehouses of the merchants in the city; and the large vessels 
 are then made fast to boats, by means of ropes, and in that manner towed through 
 the passage to their stations. . 
 
 Though measures were adopted, so early as the middle of the sixteenth century, 
 by forbidding ballast to be thrown into the Pampus, to prevent the farther accumu- 
 lation of sand in this passage, that inconvenience increased so much, from other 
 causes, as to occasion still greater obstruction to trade; and it at length became im- 
 possible for ships of war and others heavily laden to get through it. About the year 
 1672, no other remedy was known, than that of making fast to the bottoms of ships 
 large chests filled with water, which was afterwards pumped out, so that the ships 
 
 were buoyed up, and rendered sufficiently light to pass the shallow. By this method, 
 
 which was attended with the utmost difficulty, the. Dutch carried out their numerous 
 fleet to sea in the above-mentioned year. This plan however gave rise soon after 
 to the invention of the camel, by which the labour was rendered easier. The camel 
 consists of two half ships, constructed in such a manner that they can be applied, 
 below water, on each side of the hull ofa large vessel. On the deck of each part 
 of the camel are a great many horizontal windlasses; from which ropes proceed 
 
 through apertures in the one half, and, being carried under the keel of the vessel, 
 
 enter similar apertures in the other, from which they are conveyed to the windlasses 
 on its deck. When they are to be used, as much water as may be necessary is 
 suffered to run into them; all the ropes are cast loose, the vessel is conducted 
 between them, and large beams are placed horizontally through the port holes of 
 the vessel, with their ends resting on the camel, on each side. When the ropes are 
 made fast, so that the ship is secured between the two parts of the camel, the water 
 is pumped from them, by which means they rise, and raise the ship along with them. 
 Each half of the camel is generally 127 feet in length; the breadth at one end 1s 22, 
 and at the other 13. The hold is divided into several compartments, that the 
 machine may be kept in equilibrio, while the water is flowing into it. An East- 
 India ship that draws 15 feet of water, can by the help of the camel be made to draw 
 only 11; and the heaviest ships of war, of 90 or 100 guns, can be so lightened, as 
 to pass without obstruction all the sand hanks of the Zuyder-Zee. 
 
 Leupold, in his ‘‘ Theatrum Machinarum,” says that the camel was invented by 
 
a 
 
 My 
 
 216 MECHANICS. 
 
 ascribe this invention to a citizen of Amsterdam, called Meeuves Meindertszoon 
 Bakker. Some make the year of the invention to have been 1688, and others 1690. 
 However this may be, we are assured, on the testimony of Bakker himself, written 
 in 1692, and still preserved, that in the month of June, when the water was at its 
 usual height, he conveyed, in the course of twenty-four hours, by the help of the 
 camel, a ship of war called Maagt van Enkhuysen, which was 156 feet in length, from 
 Enkhuysen Hooft, to a place where there was sufficient depth; and that this could 
 have been done much sooner, had not a perfect calm prevailed at the time. In the 
 year 1693, he raised a ship called the Unie, six feet, by the help of this machine, and 
 conducted her to a place of safety. 
 
 As ships built in the Newa cannot be conveyed into harbour, on account of the 
 sand banks formed by the current of that river, camels are employed also by the 
 Russians, to carry ships over these shoals: and they have them of various sizes. 
 Bernoulli saw one, each half or which was 217 feet in length, and 36 in breadth. 
 Camels are used likewise at Venice.* : 
 
 Cornelius Meyer, a Dutch engineer. But the Dutch writers, almost unanimously, | 
 | 
 
 PROBLEM XXVI. 
 
 To make a body ascend as tf of itself along an inclined plane, in consequence of its 
 own gravity. 
 
 Provide a double cone (Fig. 26.), that is to say 
 two right cones united at their bases, so as to have 
 a common axis. Then make a supporter, consisting 
 of two branches, forming an angle at the point c, 
 which must be placed in such a manner that the 
 summit c shall be below the horizontal line, and 
 that the two branches or legs shall be equally in- 
 clined to the horizon. The line as must be equal 
 to the distance between the summits of the double 
 cone, and the height ap a little less than the radius of the base. These conditions 
 being supposed, if the double cone be placed between the legs of this angle, it will 
 be seen to roll towards the top; so that the body, instead of descending, will seem 
 to ascend, contrary to the affection of gravity: this however is not the case; for its 
 centre of gravity really descends, as we shall here shew. 
 
 Fig. 27 Let ac (Fig. 27.) be the inclined plane, 
 eats _ containing the angle acs; ce the hori- 
 zontal line, passing through the summit ¢,. 
 and consequently ea will be the elevation 
 of the plane above the horizontal line, 
 which is less than the radius of the circle 
 x C forming the base of the double cone. It 
 is evident that when this double cone is at 
 the summit of the angle, it will be as seen at cd; and when it reaches the highest 
 part of the plane, it will have the position seen at a f: its centre then will have 
 passed from d to a, and since dc is equal to a f, and ce is the horizontal line, ¢ f will 
 be a line declining below the horizon; and consequently da, which is parallel to it, 
 will be so also. The centre of gravity of the cone will therefore have descended, 
 while the cone appeared to ascend. But, as has been already seen, it is the descent 
 or ascent of the centre of gravity that determines the real descent or ascent of a 
 body. As long as the centre of gravity can descend, the body therefore really 
 moves in that direction, &c. 
 
 * An engraving of the camel may be seen in“ LL’ Art de batir les Vaisseaux ;” Amsterdam, 1719. 
 4to. vol. ii. p,93. See also the ‘* Encyclopedie,’”’ Paris edition, vol. iii. p. 67. 
 
ee 
 
 rie 
 ’ 
 
 WATER CLOCK. 217 
 
 It will be found, in the present case, that the course of the centre of gravity, in 
 its whole descent, is a straight line. But a parabola or hyperbola might be situated 
 in the same manner, with its summit downwards, and in that case the course of the 
 centre of gravity of the double cone would be a curve. This may furnish a subject 
 of exercise for young geometricians. 
 
 PROBLEM XXVII. 
 To construct a clock with water. (Fig. 28.) 
 
 If the water which issues from a cylindric vessel, through a 
 hole formed in its bottom, flowed in a uniform manner, nothing 
 would be easier than to construct a clock, to indicate the hours 
 by means of water. But it is well known that the greater the 
 height of the water above the orifice through which it issues, 
 the greater is the rapidity with which it flows; so that the 
 vertical divisions ought not to be equal: the solution of the 
 problem therefore consists in determining their ratio. 
 
 It is demonstrated in hydraulics, that the velocity with 
 which water flows from a vessel, through a very small orifice, 
 is proportional to the square root of the height of the water 
 above the aperture. And hence the following rule, for dividing 
 4 the height of the vessel, which we suppose to be cylindrical, 
 
 SS == has been deduced. 
 
 If we suppose that the whole water can flow out in twelve 
 hours; divide the whole eight into 144 parts; then 23 of these will be emptied in 
 the first hour; so that there will remain 121 for the other eleven. Of these 121 
 parts, 21 will be emptied during the second hour; then 19 will be emptied in the 
 third, 17 in the fourth, and so on. As the 144th division therefore corresponds to 
 twelve hours, the 121st will correspond to eleven; the 100th to ten; the 8lst to 
 nine, &c.; till the last hour, during which only one division will be emptied. These 
 divisions will comprehend in the retrograde order, beginning at the lowest, the first, 
 1 part; the second, 3; the third, 5; the fourth, 7, &c.; which is exactly the ratio 
 of the spaces passed over in equal times by a body falling freely in consequence 
 of its gravity. 
 
 But, if it were required that the divisions, in the vertical direction, should be 
 equal in equal times, what figure ought to be given to the vessel ? 
 
 The vase, in this case, ought to be a parabolvuid, formed by the circumvolution of 
 a parabola of the fourth degree ; or the biquadrates of the ordinates ought to be as 
 the abscissas. If an orifice of a proper size were made in the summit of this para- 
 
 - boloid; and if it were then inverted ; the water would flow from it in such a manner, 
 
 that equal spaces of the vertical height would be emptied in equal times. 
 The method of describing this parabola is as follows. Let ~ 
 a Bs (Fig. 29.) be a common parabola, the axis of which is ps, 
 and the summits. Draw,in any manner, the line rrr, parallel 
 to that axis, and then draw any ordinate of the parabola a P, inter- 
 secting R Tink; make PQ a mean proportional between PR and 
 pa; and let pq be a mean proportional also between pr and pa; 
 and soon. ‘The curve passing through all the points q, q, &c., 
 will be the one required ; and it may be employed to form a mould 
 for constructing a vessel of the required concavity. To what- 
 ever height it shall be filled with water, equal heights will always 
 be emptied in equal.times. 
 
 In another part of this work, we shall give a method of making equal quantities 
 
218 © MECHANICS. 
 
 of water flow from a vessel of any form in equal times. As this depends on the 
 property of the siphon, it belongs to a different head. 
 
 PROBLEM XXVIII. 
 
 A point being given, and a line not horizontal, to find the position of the inclined 
 plane along which, if a body descend, setting out from the given point, it shall reaen 
 that line in the least time. 
 
 The following solution to this curious problem is by W. Rutherford, Esq., of the 
 Royal Military Academy, Woolwich, and we give it here as much preferable to that 
 by Montucla. 
 
 Fig. 30 Let a be the given point, and 8 c the given line. 
 A yz Through a draw a B parallel to the- horizon meeting 
 
 \ c Bin B; make B F = BA, and join a F, andit is the line 
 
 of quickest descent. 
 
 Now draw F H perpendicular to B c meeting the verti- 
 cal through ainw. Then the angles Ba Hand BrH 
 being right-angles, are equal,—and because B a and BF 
 are equal, B AF and BF A are equal ; therefore the angles 
 H AF and HF a are equal,—and so therefore are HAandHF. Hence acircle described 
 from H on a centre, with radius H A or H F, will touch ABandBFataandr. And 
 drawing any line 4 K = from ato Bc, and cutting the circle at K, the time of descent 
 down A F is equal to that down a K, and it is therefore less than the time down a 5. 
 
 Mr. Rutherford solves also the kindred problem, ‘ ‘To determine the slope of aroot 
 of a given width down which the rain may descend in the least time.” 
 
 Let a B be the breadth of the building, and c its point of bisection ; and through a B 
 draw the vertical line cp. Makec p= ac, and draw A P, PB: Aneel ap Bis the 
 roof down which the rain or any heavy body will descend in the least time. 
 
 Fig. 31. For draw P8 parallelto a B meeting the vertical drawn through 
 6 aatu. Then p=Ha,and a circle described from nH, with 
 
 > radius HA or HP, wil] touch a Band cDat A and P: now the time 
 
 = of descent from P to A is the same as that from F toa, and is 
 
 cousequently less than that from ¢ to a, and a P B is therefore 
 the roof down which rain will descend the quickest. 
 
 PROBLEM XXIX. 
 
 Two points aand B being given in the same horizontal line ; required the position of 
 two planes a c and CB, of such an inclination, that two bodies descending with acce- 
 lerated velocity from a to c, and then ascending along c B with the acquired velocity, 
 
 » shall do so in the least time possible. (Fig. 32.) 
 
 It is evident that a body placed at a, on the horizontal line a B, would remain there 
 eternally without moving towards B. To makeit proceed therefore by the effect of 
 Fig. 32. its own gravity from a to B, it must fall along an inclined plane 
 i ora curve; so that, after having descended a certain space, it 
 A i B shall ascend along a second plane, or the remainder of the curve, 
 rua as far as B. But we shall suppose that this is done by means 
 of two inclined planes. It is here to be observed, that the time 
 employed to descend and ascend, must be longer or shorter ac- 
 cording to the inclination and the length of these planes. The question then is, to 
 determine what position of them is most advantageous, in order that the time may be 
 the least. Now it will be found that to obtain the required position, the two planes 
 must be equal and inclined to the horizon at an angle of 45° ; that is to say, the tri- 
 
 angle ac B ought to be isosceles and right-angled at c. 
 
J 
 
 * 
 
 WATER BUCKETS. 219 
 
 This solution is deduced from that of the preceding problem; for if we conceive 
 a vertical line drawn through the point c, it has been shewn that the plane a c, 
 inclined at an angle of 45 degrees, is the most favourably disposed to make the 
 body, sliding along it, arrive at the vertical line in the least time possible ; but the 
 time of the ascent along c B, is equal to that of the descent; whence it follows that 
 their sum, or the double of the former, is also the shortest possible. 
 
 PROBLEM XXx. 
 
 Ifa chain and two buckets be employed to draw up water from a weli of very great 
 depth ; it ts required to arrange the apparatus in such a manner, that in every posi- 
 tion of the buckets, the weight of the chain shall be destroyed ; so that the weight to 
 be raised shall be that only of the water contained in the ascending bucket. (Fig. 33.) 
 
 If two buckets be suspended from the two ends of a rope 
 or chain, so as to ascend and descend alternately, while the 
 rope rolls round the axis or wheel of the windlass which serves 
 to raise them, it is evident that when one of the buckets is 
 at the bottom, the person who begins t@ raise it has not only 
 the weight of the bucket to support, but that also of the 
 whole chain or rope fromthe top to the bottom of the well; 
 and there are some cases, as in mines of three or four hun- 
 dred feet in depth; where the weight of several quintals 
 must be overcome to raise only two or three hundred pounds 
 to the mouth of the mine. Such were the mines of Pontpean, 
 until M. Loriot suggested a remedy for this inconvenience, 
 
 This remedy is so simple, that it is astonishing no one 
 ever thought of it before. Nothing indeed is necessary but 
 to convert the rope or chain into a complete ring, one of 
 the ends of which descends to the depth where the water or 
 the ore is to be drawn up, and to affix the buckets to two points of the rope in such 
 a manner that when one of them is at the highest part, the other shall be at the lowest. 
 For it is evident that, as equal parts of the chain ascend and descend, these parts will 
 counterbalance each other; and the weight to be raised, were the pit several thou- 
 sand feet in depth, will be that only of the ore or other substances drawn up. 
 
 The case would evidently be the same if there were only one bucket: in every 
 position, the only weight to be raised would be that of the bucket, and the matter 
 it contained; but the machine would be attended with only one half of its advantage ; 
 for, by having no more than one bucket, the time which the bucket. when emptied 
 would employ in descending would be lost. 
 
 Remark.—In the Memoirs of the Academy of Sciences for 1731, M. Camus gave 
 another method of remedying the above inconvenience. It consists, when there 
 is only one bucket, in employing an axis nearly in the form of a truncated cone ; 
 so that when the bucket is at the lowest depth, the rope is rolled round the part 
 which has the least diameter; and when the bucket is at the top, it is rolled round that 
 which has the greatest. By these means, the same force is always required. Butitis 
 evident that, in every case, more must be applied than is necessary. 
 
 When there are two buckets, M. Camus proposes that one half of the rope 
 should be rolled round one half of the axis, which he divides into two equal parts ; 
 so that one half is covered by the rope belonging to the bucket raised up, while the 
 other is uncovered, the bucket which corresponds to it being at the bottom. By 
 these means the two efforts are combined in such a manner, that nearly the same 
 force is always required toovercome them. But these inventions, though ingenious, 
 are inferior to that of M. Loriot. 
 
220 MECHANICS. 
 
 PROBLEM XXXI. 
 Method of constructing ajack which moves by means of the smoke of the chimney. 
 (Fig. 34.) 
 
 The construction of this kind of jack, which is vem 
 ingenious, is as follows. An iron bar fixed in the back 
 of the chimney, and projecting from it about a foot,| 
 serves to support a perpendicular spindle, the extremity| 
 of which turns in a cavity formed in the bar; while the 
 other extremity is fitted-into a collar in another bar, | 
 placed at some distance above the former. This spindle 
 is surrounded with a helix of tin plate, which makes a 
 couple of revolutions, or turns round the spindle, and 
 which is about a foot in breadth. But instead of this 
 helix, it will be sufficient to cut several pieces of tin 
 plate, or sheet-iron, and to fix them to the spindle in 
 such a manner that their planes shall form with it an angle of about 60 degrees ; they 
 must be disposed in several stories, above each other; so that the upper ones may 
 stand over the vacuity left by the lower ones. The spindle, towards it summit, bears 
 a horizontal wheel, the teeth of which turn a pinion having a horizontal axis, and 
 the latter, at its extremity, is furnished* with a pulley, around which is rolled the 
 endless chain that turns the spit. Such is the construction of this machine, the 
 action of which may be explained in the following manner. When a fire is kindled in 
 the chimney, the air which by its rarefaction immediately tends to ascend,meeting with 
 the helicoid surface, or kind of inclined vanes, causes the spindle, to which they are 
 affixed, to turn round, and consequently communicates the same motion to the spit. 
 The brisker the fire becomes, the quicker the machine moves, because the air ascends 
 with greater rapidity. 
 
 When the machine is not used, it may be taken down, by raising the vertical 
 ‘spindle a little, and removing the point from its cavity; which will allow the summit 
 to be disengaged from the collar in which it is made to turn. When wanted for 
 use, it may be put up with the same ease. 
 
 MET 
 
 Remarks.—\st.. The following mechanical amusement is founded on the same prin= 
 ciple. Cut out from a card as large a circle as possible; then cut in this circle a 
 spiral, making three or four revolutions, and ending at a small circle, reserved 
 around the centre, and of about a line or two in diameter; extend this spiral by 
 raising the centre above the first revolution, as if it were cut into a conical surface 
 or paraboloid; then provide a small spit made of iron, terminating in a point, and rest 
 ing on asupporter. Apply the centre orsummit of the helix to this point: and if 
 the whole be placed on the top of a warm stove, the machine will soon put itself in 
 motion, and turn without the assistance of any apparent agent. The agent however 
 in this case is the air, which is rarefied by the contact of a warm body, and which 
 ascending forms a current. 
 
 2d. There is no doubt that a similar invention might be applied to works of ore 
 utility: it might be employed, for example, in the construction of wheels to be 
 always immersed in water, their axis being placed parallel to the current: to give 
 the water more activity this helicoid wheel might be inclosed in a hollow cylinder, 
 where the water, when it had once entered, being impelled by the current above it, 
 would in our opinion act with a great force. 
 
 If the cylinder were placed in an erect position, so as to receive a fall of water 
 through the aperture at the top, the water would turn the wheel and its axis, and 
 
BALANCING, 221 
 
 might thus drive the wheel of a mill, or of any other machine. Such is the principle 
 of motion employed in the wheels of Basacle, a famous mill at Toulouse. 
 
 3d. The smoke jacks here in England are made somewhat 
 different from that above described ; being mostly after the 
 manner of that exhibited in Fig. 34*, where a Bis a circle 
 containing the smoke vanes, of thin sheet iron, all fixed in 
 the centre, but set obliquely at a proper angle of inclination. 
 The other end of the spindle has a pinion c, which turns 
 the toothed wheel p, on the spindle of which is fixed the 
 vertical wheel £, over which passes the chain EF, which turns 
 the spit below. There are other forms of this useful machine 
 also made; but all or most of them having the same kind of 
 vanes in the circle a B, instead of the spiral form in the 
 
 original. 
 
 PROBLEM XXXII. 
 
 What is it es supports, in an upright position, a top or tetotum, while it ts 
 
 revolving ? 
 
 It is the iether force of the parts of the top, or tetotum, put in motion. For 
 a body cannot move circularly without making an effort to fly off from the centre ; 
 so that if it be affixed toa string, made fast to that centre, it will stretch it, and 
 in a greater degree according as the circular motion is more rapid. 
 
 The top then being in motion, all its parts tend to recede from the axis, and with 
 greater force the more rapidly it revolves; hence it follows that these parts are like 
 so many powers acting in a direction perpendicular to the axis. But as they are 
 all equal, and as they pass all round with rapidity by the rotation, the result must 
 be that the top is in equilibrio on its point of support, or the extremity of the 
 axis on which it turns. 
 
 PROBLEM XXXIII. 
 
 How comes it that a stick, loaded with a weight at the upper extrenuty, can be kept in 
 equilibrio, on the point of the finger, much easier than when the weight is near the 
 lower extremity ; or that a sword, for example, can be balanced on the finger much 
 better, when the hilt is uppermost 2 
 The reason of this phenomenon, so well known to all those who perform feats of 
 
 balancing, is as follows. When the weight is at a considerable distance from the 
 
 point of support, its centre of gravity, in deviating either on the one side or the 
 other from a perpendicular direction, describes a larger circle than when. the weight 
 is very near to the centre of rotation, or the point of support. But in a large circle 
 an arc of a determinate magnitude, such as an inch, describes a curve which deviates 
 much less from a horizontal direction than if the radius of the circle were less. The 
 centre of gravity of the weight then may, in the first case, deviate from the perpen- 
 dicular the quantity of an inch, for example, without having a tendency or force to 
 deviate more, than it would in the second case; for its tendency to deviate altogether 
 from the perpendicular is greater, according as the tangent to that point of the are 
 where it happens to ‘be approaches more to a vertical direction. The greater there- 
 fore the circle described by the centre of gravity of the weight, the less is its ten- 
 dency to fall, and consequently the greater the facility with which it can be kept in 
 equilibrio. 
 
 PROBLEM XXXIV. 
 
 What ts the most advantageous position of the feet for standing with firmness, in an 
 erect posture ? 
 
 It is customary among well-bred people to turn their toes outwards; that is to 
 
929 MECHANICS, 
 
 say, to place their feet in such a manner, that the line passing through the middle of 
 the sole is more or less oblique to the direction towards which the person is turned,, 
 Being induced by this circumstance to inquire whether this custom, to which an| 
 idea of gracefulness is attached, be founded on any physical or mechanical reason,| 
 we shall here examine it according to the principles of mechanics. 
 
 Every body whatever rests with more stability on its base, according as its centre) 
 of gravity, on account of its position and the extent of that base, is less exposed to 
 be carried beyond it by the effect of any external shock. The problem then, in con- 
 sequence of this very simple principle, is reduced to the following: To determine 
 whether the base, within which the line drawn perpendicular to the horizon from’ 
 the centre of gravity of the human body ought to fall, is susceptible of increase and 
 diminution, according to the position of the feet; fe what is the position of the 
 feet which gives to that base the greatest extent. But this becomes a problem of 
 
 : pure geometry, which might be thus expressed: Two lines 
 
 Fig. 35. AD and BC (Fig. 35.) of equal length, and moveable around the | 
 
 the points a and 8, as centres, being given ; to determine their 
 position when the trapezium or quadrilateral a Bc D is the! 
 
 . greatest possible. ‘This problem may be solved with the’ 
 \ greatest facility, by methods well known to geometricians ; | 
 and from the solution the following construction is deduced. | 
 
 On the line a d equal to aD, or BC, construct the isosceles triangle am d, right- | 
 angled at H; and make AKequal to Au. Having then assumed a 1 equal to one 
 half of AG, or one fourth of aB, draw the line K 1, and make I = equal to1K: on 
 G &, if an indefinite perpendicular, intersecting in D, the circle described from the 
 point a asa centre, with the radius A D, be then raised, the point pD, or the angle 
 D AB, will determine the position of aD, and consequently of pc. If the line as, 
 and consequently ac or a1, be nothing, or vanish, a& will be found equal to a m; 
 and the angle p a®& will be half a right one. Thus, when the heels absolutely 
 touch each other, the angle which the longitudinal lines of the soles of the feet | 
 ought to form, is half a right one, or nearly so, on account of the small distance 
 which is then between the two points of rotation, in the middle of the heels. 
 
 If the distance a Bis equal to aD, the angle DAE ought to be 60 degrees; if a B 
 is equal to twice a D, the angle pA E ought to be nearly 70 degrees; and in the 
 last place, if aB be equal to three times the line aD, it will be found that pax 
 ought to be nearly 74° 30. 
 
 It is thence seen, that in proportion as the feet are at a greater distance from 
 each other, their direction, in order to stand or walk with more stability, ought 
 to approach nearer to parallelism. But, in general, mechanical principles accord 
 with what is taught by custom and gracefulness, as it is called; that is to say, to 
 turn the toes outwards. 
 
 PROBLEM XXXv. 
 Of the game of Billiards. 
 
 It is needless to explain here the nature of billiards. It is well known that 
 this game is played on a table covered with green cloth, properly stretched, and 
 surrounded by a stuffed border, the elasticity of which forces back the ivory balls 
 that impinge against it. The winning strokes at this game, are those which, by 
 driving your ball against that of your adversary, force the latter into one of the 
 holes at the corners, and in the middle of the two longer sides, which are called 
 pockets. 
 
 The whole art of this game then consists in being able to know in what manner 
 you must strike your adversary’s ball with your own, so as to make it fall into 
 
one of the pockets, without driving your own into it also. This problem, and 
 ‘some others belonging to the game of billiards, may be solved by the following 
 principles. 
 
 Ist. The angle of the incidence of the ball against one of the edges of the table, 
 is equal to the angle of reflection. 
 
 2d. When a ball impinges against another, if a straight line be drawn between 
 their centres, which will consequently pass through the point of contact, that line 
 will be the direction of the line described after the stroke. 
 
 These things being premised, we shall now give a few of the problems which 
 arise out of this game. 
 
 GAME OF BILLIARDS. 223 
 
 I, The position of the pocket and that of the two balls m and n being given (Fig. 36.) 
 to strike your adversary’s ball m in such a manner, that it shall fall into the 
 pocket. 
 
 Fig. 36. Through the centre of the given pocket and that of the ball, 
 draw, or conceive to be drawn, a straight line; the point 
 where it intersects the surface of the ball, on the side oppo- 
 site to or farthest from the pocket, will be that where it 
 ought to be touched, in order to make it move in the required 
 direction. If we then suppose the above line continued from 
 one of the radii of the ball, the point 0, where it terminates, 
 will be that through which the impinging ball ought to pass. 
 It may be readily conceived, that it isin this that the whole dex- 
 terity of the game consists; nothing being. necessary, but to 
 strike the ball in the proper manner. It is easy to see what ought to be done, 
 but it is not so easy to perform it. 
 
 In the last place, it is evident from what has been said, that provided the angle 
 Nos exceeds aright angle ever so little, it is possible to drive the ball m into the 
 pocket. 
 
 II.—To strike the ball by reflection. 
 
 Fug. 37. The ball m (Fig. 37 ) being concealed, or almost concealed, 
 
 = behind the iron, in regard to the ball nN, so that it would be 
 impossible to touch it directly, without running the risk of 
 striking the iron and failing in the attempt; it is necessary, in. 
 that case; to try to touch it by reflection. For this purpose, cou- 
 ceive the line mo, drawn perpendicular from m to the edge 
 pc, to be continued to m: so that om shall be equal to om. 
 If you aim at the point m, the ball Nn, after touching the edge 
 Dc, will strike the ball m. 
 
 If it were required to strike the ball m (Fig. 38.) by 
 two reflections, the geometrical solution, in this case, is as 
 - follows. Conceive the line mo, drawn perpendicular from the - 
 point M, to the edge 3, to’ be continued till om become equal 
 toom. Conceive also the line mp, drawn perpendicular from 
 the point m to the edge continued, to be continued to q, until 
 pq be equal to pm; if the ball N be directed to the point q, 
 after impinging against the edges Dc and cB, it will strike the 
 ball m. 
 
 To those in the least acquainted with geometry, the de- 
 monstration of this problem will be easy. 
 
924 MECHANICS. | 
 
 IIL.—ZJf a ball strikes against another in any direction whatever, what is the direction 
 of the impinging ball after the shock ? | 
 
 It is of importance, at the game of billiards, to be able to know what will be the 
 direction of your own ball after it strikes that of your adversary obliquely ; for every 
 one knows that it is not sufficient to have touched the latter, or to have driven if 
 into the pocket ; you must also prevent your own from falling into it. 
 
 Let m and Nn (Fig. 39.) be the two balls, the latter of which! 
 Fig. 39. is to strike the former, touching it in the point 0. Through| 
 this point 0, let there be drawn the tangent op; and through: 
 the centre n, of the ball nN, when it arrives at the point of con-| 
 tact, draw or conceive to be drawn 7 p, parallel to oP: the di-| 
 rection of the impinging ball, after the shock, will benp. A 
 bad player would here be infallibly lost ; and indeed this is often 
 the case in this position of the balls. Expert players, when 
 they find that they have to do with novices, often give them 
 this deceitful chance, which makes them lose, by driving their 
 ball into one of the corner pockets. In this case you must not 
 take the ball of your adversary. by halfs, according to the technical term of the 
 game, to drive it to one of the corners at the other end of the table; for in doing 
 so, you will not fail to lose yourself in the other corner. 
 
 Remark.—In reasoning on this game, we set out from common principles; but we 
 must confess that we have some doubts on this subject, the reason of which we 
 shall here explain. 
 
 If the balls had only one progressive movement forwards, without rotation around 
 their centres, the above principles would be evidently and sufficiently demonstrated. 
 But every one knows that, independently of this progessive motion of the centre, a 
 billiard ball rolls upon the table in a plane which is perpendicular to it. Whena 
 ball then touches the edge, and is repelled with a force nearly equal to that with 
 which it impinged, it would appear that this motion ought to be compounded of the 
 rotary motion it had at the moment of the shock, and that which it has in a direction 
 parallel to the edge. But since the first of these motions compounded with the 
 latter, gives the angle of reflection equal to the angle of incidence, what then be- 
 comes of the second, which ought to alter the first result? In our opinion, this is a 
 dynamical problem, which has never yet been solved, though it deserves to be so. 
 
 However, this rotary motion, in certain circumstances, gives a result which seems 
 contrary to the laws of the impinging of elastic bodies; for, according to this law, 
 when an elastic body impinges directly and centrally against another which is equal 
 to it, the first ought to stop, in consequence of having communicated, as is supposed, 
 all its velocity to the second. But at the game of billiards, this does not take place; 
 for here the impinging ball continues to move, instead of stopping short. This 
 effect is partly a consequence of the motion of the impinging ball around its centre; 
 a motion which subsists in a great measure after the shock, and it is this motion 
 partly which makes the ball still move forwards. Another cause of the striking 
 ball’s moving forward, is the want of perfect elasticity in them both, on which ac- 
 count that ball still retains some portion of its direct forward motion, the other ball, 
 which is struck, receiving the rest of the motion. 
 
 PROBLEM XXXVI. 
 To construct a Water Clock. 
 This name is given to a clock shaped like a drum or barrel, as aBeD (Fig. 40.), 
 made of metal well soldered, and put in motion by a certain quantity of water 
 
 ., 
 
WATER CLOCKS. 225 
 
 %tg.40. contained in the inside of it. The hours are indicated on two 
 vertical pillars, between which it is suspended by small strings 
 | or cords, rolled round an axis, every where of the same thickness. 
 1 Aa The internal mechanism is exceedingly ingenious, and deserves 
 1 a better explanation than what has been given of it in the pre- 
 ' ceding editions of the Mathematical Recreations, where Oza- 
 nam does not tell us how the machine goes and is supported, as 
 we may say, in the air, without falling, as it seems it ought 
 to do. 
 
 Let the circle 1234 (Fig. 41.) represent a section of the 
 drum or cylinder, by a plane perpendicular to its axis. We 
 shall here suppose the diameter of it to be six inches; and let 
 A, B, C, B, E, F, G, represent seven cells, the partitions of which 
 are formed of the same metal, and are well soldered to the two 
 circular ends, and to the circular band which forms the circum- 
 ference. These partitions ought not to proceed from the centre 
 Fig.41 to the circumference, but to be placed in a somewhat transverse 
 i direction, so as to be tangents to an interior circle, of about an 
 inch and a half in diameter: the small square 8 is a section of 
 the axis, which in that part ought to be square, and to fit very 
 exactly into holes, of the same form, made in the centre of each 
 end of the cylinder. Each partition also ought to have in it a 
 small round hole, as near as possible to the circumference of the 
 cylinder, all pierced with the same piercer, that there may be 
 no difference among them. 
 
 Let us now suppose that a certain quantity of water, about 
 eight or nine ounces, has been put into the cylinder, and that it 
 has already distributed itself as shewn by the horizontal shading 
 lines Fig. 41. If the line 1K represent the two strings, Gu and EF (Fig. 40.), rolled 
 Mead. the axis of the cylinder, it may be easily seen that the centre of gravity, 
 which, if the machine were empty, would be in the centre of the figure, being 
 thrown out of the line of suspension, and towards the side where the machine has a 
 tendency to fall, it would indeed fall; but the effect of the water behind the parti- 
 tion D, is to throw back the centre of gravity, so that if it were on this side the 
 vertical line K 1 continued, the cylinder would revolve from p to &, in order to be in 
 that vertical; and in this position the machine would remain in equilibrio, if the 
 water could not proceed from the one cavity to the other; for the cylinder cannot 
 revolve in the direction a GF, without making the centre of gravity ascend towards 
 -p: in the like manner it cannot revolve in the direction Bc p, without the centre 
 rising on the opposite side. ‘The machine must then remain in equilibrio, until some- 
 thing is changed. 
 
 But, if the water flows gradually through the hole in the partition p, which is 
 between the cells p and £, it is evident that the centre of gravity will advance a 
 little beyond x r continued, and the machine will imperceptibly revolve in the 
 direction aGF: and since by descending in this manner, the centre of gravity is 
 thrown towards the vertical line KI produced, the equilibrium will at the same time 
 be restored, and this motion will continue until the whole of the cord be unrolled 
 from the axis. This movement indeed will not be altogether uniform; for it is 
 evident that when the water is almost entirely behind the partition p, the cylinder 
 will revolve faster than when it has nearly flowed off; and the periods of these 
 inequalities, during a whole revolution of the cylinder, will be equal in number to 
 the cells ; a circumstance which seems not to have been observed by those who have 
 
 written on clocks of this kind. 
 
 we @ 
 
276 MECHANICS. 
 
 To have an exact division of time by these means, it will therefore be necessary — 
 to make a mark on the circumference of the cylinder. If the machine be then 
 wound up as high as possible, and disposed in such a manner that the mark shall be | 
 at the top of the cylinder, you must have a good clock, with which to mark, during 
 a whole revolution, the points of the hours elapsed. But care must be taken that. 
 the number of hours shall be an integer number, as 2, 4, 6, &c. : and for that purpose 
 the movement of the machine must be retarded or accelerated till the proper pre- 
 cision has been obtained ; otherwise it might err some minutes, and perhaps a quar- 
 ter of an hour. How this movement may be accelerated or retarded, we shall shew 
 hereafter. 
 
 In the last place, in winding up the clock, great care must be taken that when the 
 axis is placed opposite to the arst division, the mark made in the cylinder shall be in 
 the same position; otherwise there may be an error, as already said, of some minutes. 
 We shall now add some useful observations in regard to this object. 
 
 I. It is absolutely necessary that the water employed be distilled water; otherwise 
 it will soon become corrupted, so as to stop up the holes through which it ought to 
 flow; and the machine will consequently stand still. 
 
 II. The substance most proper for constructing the cylinder of these machines, 
 is gold or silver; or what is cheaper, copper well tinned on the inside, or even tin 
 itself. 
 
 Ill. This machine is apt to go a little faster in summer than in winter, and there- 
 fore ought to be regulated from time to time, and retarded or accelerated. For this 
 
 Fig. re purpose it will be necessary to add to it a small weight as a counterpoise, 
 tending to make it revolve outward. This weight ought to have the form 
 of a bucket (Fig. 42.), and to be of some light substance, so that it ean be 
 charged more or less by means of small drops of lead. To accelerate the 
 machine, two or three drops of lead may be added; and-when it is neces- 
 sary to retard it, they may be removed; which will be much more conve- 
 
 : nient than adding or taking away water. 
 ra IV. The place where the axis passes through the cylinder must be well 
 @ cemented ; otherwise the water would gradually evaporate, by which 
 means the machine would be continually retarded, and at length would stop. 
 
 V. Notwithstanding all these precautions, it may be readily seen that a machine 
 of this kind is rather an object of curiosity, than calculated to measure time with 
 accuracy. It may be proper for the cell of a convent, or a cabinet of mechanical 
 curiosities ; but it will certainly never be used by the astronomer. 
 
 VI. The inventor of this kind of clock is not known. Ozanam, who wrote in 
 
 . 1693, says that the first seen at Paris about that period had been brought from Bur- 
 gundy; and he adds that Father Timothy, a Barnabite, who excelled in mechanics, 
 had given to this machine all the perfection of which it was susceptible. This monk 
 had constructed one about five feet in height, which required winding up only once a 
 month. Besides the hours, which were marked on a regular dial-plate, at the top of 
 the frame, it indicated the day of the month, the festivals throughout the year, the 
 sun’s place, and his rising and setting, as well as the length of the day and night. 
 This was performed by means of asmall figure of the sun, which gradually descended, 
 and which, when it reached the bottom of the frame, was raised to the top at the end 
 of every month. 
 
 Father Martinelli has treated, at great length, on these clocks, in an Italian work, 
 entitled ‘‘ Horologi Elementari,” in which he delivers methods of making clocks by 
 means of the four elements, water, earth, air, and fire. This work was printed at 
 Venice in 1663, and is very rare. The author shows in it how striking machinery 
 may be adapted to a water-clock; with other curiosities, which are sometimes addems 
 to common clocks. 
 
s MECHANICAL PARADOX. 227 
 
 PROBLEM XXXVI. 
 MecHANICAL PArapox.—How equal weights, placed at any distance from the point of 
 support of a balance, shall be in equilibrio. 
 
 Provide a frame in the form of a parallelogram, such as p E F G (Fig. 43.), con- 
 structed of four pieces of wood, joined together in such a manner as to move freely 
 at the angles, so that the. frame can change its rectangular form into that represented 
 _bytheletters efgd. The long sides ought to be about twice the length of the others. 
 Fig. 48. This frame inserted in a cleft formed in the perpendi- 
 
 Q _cular stand Bc, so as to be moveable on the two 
 points 1 and u, where it is fixed in the stand by two 
 small.axes: in the fast place, two pieces of wood, MN 
 and K L, pass through the shorter sides, in which they 
 are well fixed, and the whole apparatus rests on the 
 stand a 3B. 
 
 Now if the weight p be suspended from the point 
 mM, which is almost at the extremity of the arm MN, 
 the most distant from the centre or centres of motion ; 
 and if the weight 0, equal to the former, be suspended 
 from any point R, of the other arm K L, nearer the centre, and even within the frame, 
 these two weights will always be in equilibrio; though unequally distant from the 
 point of support or of motion in this kind of balance; and they will remain oe what- 
 ever situation may be given to the machine, ase fd g. 
 
 The reason of this effect which at first seems to contradict the principles of statics, 
 is however very simple. For two equal bodies will be in equilibrio, whatever move- 
 ment may be made by the machine trom which they are suspended, if the spaces 
 passed over by these two bodies or weights are equal and similar. But it may be 
 readily seen that this must necessarily be the case here, since the two weights, what- 
 ever be their position, are obliged to describe equal and parallel lines. 
 
 It may be readily seen also, that,in such a machine, whatever be the position of 
 the weights along the arms mN and K 1, the case will always be the same, as if they 
 were suspended from the middle of the short sides rE p and FG. But in the latter 
 case, the weights would be in equilibrio, therefore in former also. 
 
 PROBLEM XXXVIITI. 
 
 What velocity must be given to a machine, moved by water, in order that it may produce 
 the greatest effect ? 
 
 That this is not a matter of indifference, will readily appear from the following 
 observation. If the wheel moved with the same velocity as the fluid, it would 
 experience no pressure; consequently the weight it would be capable of raising 
 would be nothing, or infinitely small. On tbe other hand, if it were immoveable, 
 it would experience the whole pressure of the current; but in this case there would 
 be an equilibrium, and as no weight would be raised there would consequently be 
 no effect. There is therefore a certain mean velocity, between that of the cur- 
 rent and no velocity at all, which will produce the greatest effect—an effect propor- 
 tional, in a given time, to the product of the weight multiplied by the height to 
 which it is raised. 
 
 We shall not here give the analytical reasoning which conducts to the solution of 
 the problem. We shall only observe, that in a machine of the above nature, the 
 velocity of the wheel ought to be equal to one half of that of the current. Conse- 
 quently the resistance or the weight must be increased, until the velocity be in this 
 ratio, The machine will then produce ue greatest effect possible. 
 
 Q 
 
928 MECHANICS. 
 
 PROBLEM XXXIXx. 
 What is the greatest number of float-boards, that ought to be applied to a wheel moved 
 by a current of water, in order to make it produce the greatest effect ? 
 
 It was long believed that the float-boards of such a wheel ought to be so propor- 
 tioned, that when one of them was in a vertical position, or at the middle of its immer- 
 sion, the next one should be just entering the water. A great many reasons were 
 assigned for this mode of construction, which however are contradicted by calculation, 
 ‘as well as by experience. 
 
 It is now demonstrated, that the more float-boards such a wheel has, the greater 
 and more uniform will be its effect. This result is proved by the researches of the 
 Abbé de Valernod, of the academy of Lyons, and those of M. du Petit-Vandin, to be 
 found in the first volume of the ‘‘ Mémoires des Scavans Etrangers.” 
 
 The Abbé Bossut, who examined, by the help of experiments, the greater part of 
 the hydraulic theories, has demonstrated also the same thing. According to the expe- 
 riments which he made, 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 immersion in the water being equal. M. du Petit Vandin therefore observes, 
 that in Flanders, where running water is so exceedingly scarce, as to render it neces- 
 sary to turn it to the greatest possible advantage, the wheels of water-mills are fur- 
 nished with 32 float-boards at least, and even with 48, when the wheel is from 16 to 
 19 feet in diameter. 
 
 PROBLEM XL. 
 
 Lf there be two cylinders, containing exactly the same quantity of matter, the one solid 
 and the other hollow, and both of the same length ; which of them will sustain, with. 
 out breaking, the greatest weight suspended from one of its extremities, the other 
 being fixed ? 
 
 Some, and perhaps several of our readers, may be inclined to think that, the base 
 of rupture being the same, every thing else ought to be equal. On the first view, 
 one might even be induced to hand the solid cylinder as capable of presenting 
 greater resistance to being broken: this however would be a mistake. 
 
 Galileo, who first examined mathematically the resistance of solids to being broken 
 by a weight, bas shewn that the hollow cylinder will present the most resistance ; 
 und that this resistance will be greater in the transverse direction, according as the 
 hollow part is greater. He even shews, from a theory which approaches very near 
 the truth, that the resistance of the hollow cylinder will be to that of the solid one, 
 as the whole radius of the hollow is to that of the solid. Thus the resistance of a 
 hollow cylinder, having as much vacuity as solid, will be to the resistance of a solid 
 one, as »/ 2 to I, or as 1:141 to 1:000: for the radius of the former will be ./ 2, 
 while that of the latter is unity. The rasistance of a hollow cylinder, having twice 
 as much vacuity as solid, will be to a solid one, as ,/ 3 to 1, or as 1:73 to 1:00; for 
 their radii will be in the ratio of ,/ 3 tol. The resistance of a hollow cylinder, 
 the solidity of which forms only a twentieth part of the whole volume, will be to 
 that of a solid cylinder of the same mass, as ,/ 2] to 1, or as 4°31 to 1:00; See sO - 
 forth. 
 
 Remark.—-It may be readily observed, and Galileo does not fail to take notice of 
 it, that this mechanism is that which nature; or its Supreme Author, has employed, 
 on various occasions, to combine strength with lightness. Thus the bones of the 
 greater part of animals are hollow: by being solid with the same quantity of matter, 
 they would have lost much of their strength; or to give them the same power of 
 
resistance, it would have been necessary to render them more massy; which would 
 have lessened the facility of motion. 
 
 The stems of many plants are hollow also, for the very same reason In the last 
 place, the feathers of birds, in the formation of which it was necessary that great 
 strength should be united with great lightness, are also hollow: and the cavity even oc- 
 cupies the greater part of their whole diameter ; so that the sides are exceedingly thin. 
 
 SUBAQUEOUS LANTERN, ° 229 
 
 PROBLEM XUI. 
 To construct a lantern, which shall give light at the bottom of the water. 
 
 This lantern must be made of leather, which will resist the waves better than 
 any other substance; and must be furnished with two tubes, having a commu- 
 nication with the air above. One of these tubes is destined to admit fresh air for 
 maintaining the combustion of the candle or taper; and the other to serve as a 
 chimney, by affording a passage to the smoke: both must rise to a sufficient height 
 above the surface of the water, so as not to be covered by the waves when the sea is 
 tempestuous. It may be readily conceived, that the tube which serves to admit fresh 
 
 air, ought to communicate with the lantern at the bottom; and that the one which 
 serves as a chimney, must be connected with it at the top. Any number of holes at 
 pleasure, into which glasses are fitted, may be made in the leather of which the 
 lantern is constructed ; and by these means the light will be diffused on all sides. 
 In the last place, the lantern must be suspended from a piece of cork, that it may 
 rise and fall with the waves. 
 
 A lantern of this kind, says Ozanam, might be employed for catching fish by 
 means of light; but this method of fishing has, in some countries, been wisely for- 
 bidden under severe penalties. 5. 
 
 PROBLEM XLII. 
 To construct a lamp, which shall preserve its oil in every situation, however moved 
 or inclined, 
 
 To construct a lamp of this kind, the body of it, or the vase that contains the oil 
 and the wick, must have the form of a spherical segment, with two pivots at the 
 edge, diametrically opposite to each other, and made to turn in two holes at the ex- 
 tremities of the diameter of a brass or iron circle. This circle must, in like manner, 
 be furnished with two pivots exactly opposite to each other, and at the distance of 
 90° from the holes in which the former are inserted. These second pivots must be 
 made to turn in two holes diametrically opposite in a second circle; and this second: 
 circle must likewise be furnished with two pivots, inserted in some concave body, 
 proper to serve as a covering to the whole lamp. 
 
 It may be readily seen that, by this method of suspension, whatever motion be 
 given to the lamp, unless too abruptly, it will always maintain itself in a horizontal 
 position. 
 
 This method of suspension is that employed for the mariner’s compass, so useful 
 to navigators; and which must always be preserved in a horizonal situation. We 
 have read, in some author, that Charles V. caused a carriage to be suspended in this. 
 manner, to guard against the danger of being overturned. 
 
 PROBLEM XLIII. 
 
 Method of constructing an anemoscope and an anemometer, 
 
 These two machines, which in general are confounded, are not however the same. 
 The anemoscope serves for pointing out the direction of the wind, and therefore, 
 properly speaking, is a weather-cock ; but in common this term is used to denote a 
 more complex machine, which indicates the direction of the wind by means of a kind: 
 of dial plate, placed either on the outside of a house or in an apartment. In regard 
 
230 - MECHANICS. 
 
 to the anemometer, it is a machine which serves to indicate, not only the direction, 
 
 but the duration and force of the wind. 
 
 The mechanism of the anemoscope is very simple, (Fig. 44.) 
 It consists, in the first place, of a weather-cock, raised above 
 the building,.and supported by an axis, one end of which, pass- 
 ing through the roof, is made to turn in a socket fitted to re- 
 ceive it, and with such facility as to obey the least impulse of 
 the wind. On this axis is fixed a crown wheel, the teeth of 
 which being turned downwards, fit into those of a vertical 
 wheel, exactly of the same size, placed on a horizontal axis, 
 which at its extremity is furnished with an index. It is hence 
 evident, that when the vane makes one turn, the index will 
 make one exactly-also. If this index then be placed in such a 
 manner as to be vertical, when the wind is north; and if care 
 be taken to observe in what direction it turfns, when it changes 
 to the west; it will be easy to divide the dial plate into 32 parts. 
 
 An anemometer, if it be required only to measure the inten- 
 
 We would propose the following. Let a B (Fig. 45.) be an iron 
 bar, fixed in a horizontal direction to the vertical axis of a vane. 
 The extremities of this bar, which are bent at right angles, 
 serve to support a horizontal axis, around which turns a move- 
 able frame aBCD, of a foot square. To the middle of the 
 lower side of the frame is fastened a very fine but strong silk 
 thread, which passes over a pulley F, fitted into a cleft in the 
 
 an apartment below the roof. The distance ¢F must be equal 
 toaxE. To the end of the silk thread is suspended a small 
 weight, just sufficient to keep it stretched. When the frame 
 which, by the turning of the vane, will be always presented to 
 the wind, is raised up, as will be the case, more or less, according 
 to the force of the wind, the small weight will be raised up also, and will thus 
 indicate, by means of a scale adapted to the axis of the vane, the strength of the 
 
 wind. 
 or nothing, when the small weight is at its lowest point ; and that its maximum, or 
 
 greatest degree, will be when it is at its highest, which will indicate that the wind 
 keeps the frame in a horizontal position, or very nearly so. 
 
 The force of the wind, according to the different inclination of the frames may be 
 determined with still greater precision: for this force will always be equal to the 
 absolute weight of the frame, which is known, multiplied by the sine of the angle 
 which it makes with the vertical line, and divided by the square of the same angle. 
 Nothing then will be necessary, but to ascertain, by the motion of the small weight 
 affixed to the thread erp, the inclination of the frame. But this is easy ; for it 
 may be readily seen that the quantity which it rises above the lowest point, will 
 always be equal to the chord of the angle formed by the frame with the 
 vertical plane, or to double the sine of the half of that angle. The extent therefore 
 of this angle may be marked along the scale, and also the force of the wind, calcu- 
 lated according to the foregoing rule. 
 
 In the Memoirs of the Academy of Sciences, for the year 1734, may be found the 
 description of an anemometer invented by M. d’Ons-en-Bray, to indicate at the same 
 time the direction of the wind, its duration in that direction, and its strength, This 
 anemometer merits that we should here give some idea of it. 
 
 _It consists of three parts, viz., a common clock, and two other machines, one of 
 
 v 
 
 sity or force of the wind, may be constructed with equal ease. 
 
 vertical axis of the vane, whence it descends along the axis to 
 
 It may readily be perceived that the force of the wind will be equal to Zero - 
 
: 
 ¥ 
 ’ 
 
 ANEMOMETER, 231 
 
 which serves to mark the direction of the wind and its duration; the other to aah 
  eate its force. 
 
 The first of these machines consists, like the common anemoscope, of a vertical 
 axis bearing a vane, which by means of some wheels indicates, on a dial-plate, from 
 what quarter the wind blows; the lower part of this axis passes through a cylinder, 
 in which are implanted thirty-two pins, in a spiral line; and these pins, by the 
 manner in which they present themselves, press against a piece of paper, properly 
 prepared and stretched, between two vertical columns or axes, on one of which it is 
 rolled up, while it is unrolled from the other. This rolling up and unrolling are 
 performed by the simultaneous motion of two axes, which are made to move by the 
 clock above mentioned. It may now be readily conceived that, according to the 
 position of the vane, one of the pins will present itself to the prepared paper, 
 and by pressing gently against it will leave a mark, the length of which will indicate 
 the duration of the wind. If two neighbouring pins make a mark, at the same time, 
 this will indicate that the wind followed a middle direction. 
 
 The part of the anemometer which indicates the force of the wind, consists of a 
 mill, after the Polish manner, which revolves faster, according as the wind is stronger. 
 Its vertical axis is furnished with a wheel that drives a small machine, which, after a 
 certain number of turns, forces a pin against a frame of paper, having a motion 
 similar to that of the anemometer above described. The number of these strokes, 
 each of which is marked by a hole, on a determinate length of this moveable paper, 
 denotes the force of the wind, or rather the velocity of the circulation of the mill, 
 which is nearly in the same proportion. But, for a complete explanation of the 
 mechanism, we must refer to the Memoirs of the Academy of Sciences, above 
 
 quoted; as want of room will not allow us to give a more minute description of it 
 in this place. 
 
 Professor Leslie, in his Essay on Heat, has suggested an anemometer, founded on 
 different principles. He found, by experiment, that the cooling power of a stream 
 of air is proportional to its velocity ; and putting Tr for the time in which a body loses 
 an aliquot part of its heat in still air, ¢ the time in which it loses the same quantity 
 when exposed to the wind, and v the velocity of the wind in miles per hour, he gives 
 the following formula: 
 
 v=l1— a . 44, 
 : 2 
 T and ¢ are found by means of a thermometer, whose bulb is a little more than ial 
 an inch in diameter, and filled with tinged alcohol. 
 
 When the thermometer is held in still air its temperature is marked; it is then 
 heated by the application of the hand, till the alcohol rises a certain number of de- 
 grees, and the time which it takes to descend through half that number of degrees is ° 
 carefully marked. Mr. Leslie calls this time the fundamental measure of cooling. 
 The same observation is made when the ball is exposed to the wind, and the time 
 which the alcohol takes to descend through half the number of degrees that it rose 
 is called the occasional measure of cooling. The former of these is tT, and the latter 
 t,in the above formula, which may be thus expressed in words: 
 
 Divide the fundamental measure of cooling by the occasional measure of cooling, 
 and multiply the difference between unity and the quotient by 44, the product is the 
 velocity of the wind, in miles per hour. 
 
 As an example, let us suppose that in still air the temperature is 50°, and when 
 warmed by the hand it rises to 70°, and that in 100 seconds it falls to 60°; and 
 farther, that when exposed to the wind, and heated by the hand, it takes only 10 
 seconds to fall through the same number of degrees; then we have the velocity of 
 wind 19 — 1.41 = 401 miles per hour. 
 
Vie MECHANICS, 
 
 Remark.—Many other forms of anemometers have been invented, in various 
 countries. Of several of these the descriptions may be seen, with their figures and 
 the calculation of their effects, in Dr. Hutton’s Dictionary, under the several articles 
 ANEMOMETER, RESISTANCE, WIND, and WIND-GAGE. 
 
 PROBLEM XLIVv. 
 
 Construction of a Steel-yard, by means of which the weight of a body may be ascer- 
 tained, without weights. 
 
 We shall here describe two instruments of this kind ; the one portable and adapted 
 
 Fig. 46. for ascertaining moderate weights, such as from I to 25 or 30 pounds; the 
 
 other fixed, and employed for weights much more considerable, and even 
 
 custom house at Paris; and could be employed with great convenience, 
 for weights between 100 and 3000 pounds. 
 
 The first of these steel-yards is represented Fig.46. It consists of a 
 metal tube a B, about six inches in length, and eight lines in diameter, a 
 section of which is here given, to shew in the inside of it a spiral steel 
 spring. The upper end a, is pierced with a square hole, to afford a pas- 
 sage to a metal rod, which is also square; and which passes through the 
 spring, so that it is impossible to draw it upwards, without compressing 
 the spring against the upper end within the tube. To the lower part 
 of the tube is affixed a hook, from which the body to be weighed is sus- 
 pended. 7 
 
 It is here evident, that if bodies of different weights be applied to the hook, while 
 
 the steel-yard is suspended by its ring, they will draw down the tube more or less, by 
 
 forcing the upper end of it against the spring. The rod therefore must be divided, 
 by suspending successively fromthe hook different weights, suchas one pound, two 
 pounds, &c., to the greatest which it can weigh; and if the part of the rod drawn 
 out of the tube each time be marked by a line, accompanied witha figure denoting 
 the weight, the instrument will be complete. When you intend to use it, nothing is 
 necessary but to put your finger into the ring, to raise up the article you intend to 
 weigh, suspended from the hook, and to observe, on the divided face of the rod, the 
 division exactly opposite to the edge of the hole: the figure belonging to this division 
 will indicate the number of pounds which the proposed body weighs. 
 
 The second steel-yard consists of two bars, placed back to back, or of a single one 
 
 a BCDE bent in the form seen Fig. 47. The part a B is suspended by a ring froma 
 
 strong beam, aid the part D E terminates in a hook at £, from which the articles to 
 be weighed are suspended. To the part © p is fixed a rack, 
 fitted into a pinion, connected with a wheel, the teeth of which 
 are fitted into another pinion, having on its axis an index ; 
 
 3000 pounds is suspended from the hook ©. For it may be 
 readily seen, that when any weight is suspended from 8, the 
 spring Bc D must be more or less stretched; this will give 
 motion to the rack p F, and the latter will turn the pinion 
 into which itis fitted ; and consequently will give motion to 
 the wheel and second pinion, having on its axis the index. 
 It is also evident, that in constructing the machine, such a 
 force may be given to the spring, or its wheels may be com- 
 bined in suchamannner, that a determinate weight, as 3009 
 pounds, shall cause the index to perform a complete revolu- 
 tion. The centre of motion of this index isin the centre of a 
 
 of several thousand pounds. One of the latter kind was used in the 
 
 and this index makes just one revolution, when the weight of 
 
THE TUMBLER, : 933 
 
 circular plate, marked with the divisions, that serve to indicate the weight. These 
 divisions must be formed by suspending, in succession, weights less than the greatest, 
 in the arithmetical progression, as 29 hundred weight, 28, 27, &c. This wil! give the 
 principal divisions, which without any considerable error may be then subdivided into 
 equal parts. 
 
 When the instrument is thus constructed ; then to find the weight of any article that 
 weighs less than 3000 pounds, nothing is necessary but to suspend it to the hook 5; 
 and the index will point out, on the circular plate, its weight in quintals, or hun- 
 dreds, quarters, and pounds. 
 
 Remark.—It may be proper here to observe, that this method of weighing cannot. 
 be perfectly exact, unless we suppose that the temperature of the air always remains 
 the same; for during cold weather springs are stiffer, and during hot weather are less 
 so. On this account, we have no doubt that thure is a difference between the same 
 article weighed at the custom-house at Paris in winter and insummer. In winter it 
 must appear to weigh less than it does in summer. 
 
 PROBLEM XLV. 
 
 Method of constructing a small figure, which when left to itself descends along a small 
 stair on its hands and its feet. 
 
 This small machine, the mechanism of which is very ingenious, was a few years 
 ago brought from India. It is called the tumbler, because its motion has a great re- 
 semblance to that of those performers at some of the public places of amusement, 
 who throw themselves backwards resting on their hands; then raise their feet, and 
 complete the circle by resuming their former position ; but the figure can perform this 
 movement only descending, and along a sort of steps. ‘The artifice of this small ma- 
 chine is as follows: 
 
 a B (Fig. 48.) is a small piece of light wood, about two inches in length, 2 lines in 
 thickness, and 6 in breadth. At its two extremities are two holes c and p, which 
 
 serve to receive two smallaxes, around which the legs and arms 
 
 - Fig. 48. of the figure are made to play. At each extremity ofthe piece 
 of wood there is also a small receptacle, of the form seen iu the 
 figure, that is to say, nearly concentric with the holes c and p; 
 having an oblique prolongation towards the middle of the piece 
 of wood, and from the ends of these two prolongations proceed 
 
 two grooves cg and F f, formed in the thickness of the wood, and nearly a line in 
 diameter. 
 Quicksilver being put into one of these receptacles till 
 both c!osed up by means of very light pieces of pasteboard, applied on the sides. 
 To the axis, passing through one of the holes c, are affixed two supporters, cut into 
 the form of legs, with feet somewhat lengthened, to give them more stability. And 
 to the other axis, passing through », are affixed two supporters shaped like arms, 
 with their hands placed in such a manner as to become a base, when the machine is 
 turned backwards. In the last place, to the part c H is applied a sort of head and 
 visage, made of the pith of the elder tree, and dressed after the manner of tumblers. 
 A belly is constructed of the same substance, and the whole figure is clothed ina 
 silk dress, which descends to the middle of the thighs, Having thus given a general 
 account of the construction of this small machine, we shall now proceed to explain 
 
 it is nearly full, they are 
 
 its mode of action. “ 
 Let us first suppose the machine to be placed upright on its legs, as seen Fig. 49 
 
 or 50. No. 1. As all the weight ison one side of the axis of rotation c, because the 
 receptacle of the quicksilver on that side is filled, the machine must incline to that 
 side, and would be thrown entirely backwards, did not the arms or supporters, turn- 
 ing around the axis D, present themselves in a vertical direction; but as they are 
 
234 MECHANICS. 
 
 shorter than the legs, the machine assumes the position 
 represented Fig. 50, No. 2; and the quicksilver finding 
 the small groove Gc g, inclined to the horizon, flows with 
 impetuosity into the receptacle placed on the side pb. 
 
 Let us now suppose that at this moment the mdchine 
 rests on the supports or arms D L, which turn around the’ 
 axis D: it is evident that, if the empty part of the ma- 
 chine is very light, the quicksilver being entirely beyond 
 the point of rotation pb, will, by its considerable pre- 
 ponderance overcome it, and cause the machine to revolve 
 around the axis D, which will raise it, and make it turn on 
 the other side. But as the supporters c K must neces- 
 sarily be longer than the others p L that the line c D 
 may have the inclination which 
 is necessary to cause the quick- 
 silver to flow by the small 
 groove Gg, from the one recep- 
 tacle to the other, the base must 
 make a jump double in height 
 to the difference of these sup- 
 porters; otherwise the line G 
 g, instead of assuming a hori- 
 zontal position, would remain 
 inclined in a direction contrary 
 to that which it ought to have. 
 
 The machine having then at- 
 tained to the situation p c, Fig. 
 50, No. 3, and the quicksilver 
 having passed into the recep- 
 tacle on the side c; it is evident that the same mechanism which will raise it up, 
 by making it turn round the point ‘c, will overturn it on the other side, where the 
 two supporters, which revolve round the axis c, present it a base: this will make it 
 resume the position of Fig. 50, No.2: and soon. Hence this motion will be per- 
 petual, as long as the machine meets with steps like the first. 
 
 Remarks.—-Some particular conditions are required, in order that the supporters 
 of the small figure, that is to say its legs and arms, may present themselves in a proper 
 manner, to keep it in the position in which it ought to be. 
 
 Ist. It is necessary that the great supporters, or legs, when they have arrived at 
 that point at which the figure, after having thrown itself topsy-turvey, rests upon 
 them, should meet with some obstacle, to prevent them or the figure from turning 
 any more: this may be done by two small pegs, which meet a prolongation of the 
 thighs. : 
 
 2d. While the figure is raising itself on its legs, it is necessary that the arms should 
 perform, on their axis, a semi-revolution ; that they may present themselves perpen- 
 dicular to the horizon, and in a firm manner, when the figure throws itself back ward. 
 This may be accomplished by furnishing the arms of the figure with two small pulleys, 
 concentric to the axis of the motion of these arms, over which are conveyed two silk 
 threads, that unite under the belly of the figure, and are fixed to a small cross bar, 
 joining the thighs towards the middle: this will greatly contribute to their stability. 
 These threads must be lengthened, or shortened, till this semi-revolution of the arms 
 is exactly performed ; and until the figure, when placed on its four supporters, with 
 its face turned either up or down, does not waver; which it would do if these sup- 
 
MECHANICAL PHENOMENA. 235 
 
 porters were not bound together in this manner; and if the large ones, or legs, did 
 not meet with an obstacle to prevent them from inclining any farther. 
 
 PROBLEM XLVI. 
 
 To arrange three sticks, on a horizontal plane, in such a manner, that while the 
 lower extremities of each rest on that plane, the other three shall mutually support 
 each other in the air. 
 
 This depends merely on a little mechanical address, and may be performed in the 
 
 following manner. , 
 
 Take the first stick a B (Fig. 51), and rest the end a on the 
 
 Fig. 51. table, holding the other raised up, so that the stick shall be in- 
 
 clined at a very acute angle. Place above it the second stick, 
 
 | 82 with the end ¢ resting on the+able. And then dispose the third 
 
 x , Sticke F, in such a manner, that while the end E rests on the 
 
 table, it shall pass below the stick a B, towards the upper end 
 
 Oa g, and rest on the stick cp. These three sticks, by this ar- 
 
 | rangement, will be so connected with each other, that the ends 
 p, B and F will necessarily remain suspended, each supporting the other. 
 
 PROBLEM XLVII. 
 To make a soft body, such as the end of a candle, pierce a board. 
 
 Load a musket with powder, and instead of a ball put over it the end of a candle; 
 if you then fire it against a board, not very thick, the latter will be pierced by the 
 ~ eandle-end, as if by a ball. 
 
 The cause of this phenomenon, no doubt, is that the rapid motion with which the 
 candle-end is impelled, does not allow it time to be flattened, and therefore it acts as 
 ahard body. It is the effect of the inertia of the parts of matter, as may be easily 
 proved by experiment. Nothing is easier to be divided than water; yet if the palm 
 
 _ of the hand be struck with some velocity against the surface of water, a considerable 
 degree of resistance, and even of pain, is experienced from it, as if a hard body had 
 been struck. Nay, a musket ball, when fired against water, is repelled by it, and 
 
 even flattened. If the musket is fired with a certain obliquity, the ball will be re- 
 flected; and, after this reflection, is capable of killing any person who may be in its 
 
 way. ‘This arises from a certain time being necessary to communicate to any body a 
 sensible motion. "When a body then, moving with great velocity, meets with another 
 of asize much more considerable, it experiences almost as much resistance as if the 
 latter were fixed. 
 
 \ 
 
 PROBLEM XLVIII. 
 On the principles by which the possible effect of a machine can be determined. 
 It is customary for quacks, and those who have not a sufficient knowledge of me- 
 _ chanics, to ascribe to machines prodigious effects, far superior to such as are consistent 
 with the principles of sound philosophy. It may therefore be of utility to explain 
 ' here those principles by which we ought to be guided, in order to form a rational: 
 opinion respecting any proposed machine, 
 
 Whatever may be the construction of a machine, even supposing it to be mathema- 
 tically perfect, that is, immaterial and without friction, its effect, that is to say, the 
 weight put in motion, multiplied by the perpendicular height to which it may be 
 raised, in a determinate time, cannot exceed the product of the moving power, mul- 
 tiplied by the space it passes over in the same time. Consequently, since every 
 _ machine is material, and as it is impossible to get entirely rid of friction, which will 
 
 necessarily destroy a part of the power, it is evident that the first product will Biwaye 
 
 be less than the latter. Let us apply this to an example. 
 
236 MECHANICS, 
 
 Should a person propose a machine, which by the strength of one man applied to ¢ 
 crank, or the lever of a capstan, shall raise in an hour 3500 gallons of water, to the 
 height of 24 feet; we might tell him, that he was ignorant of the principles 0. 
 mechanics. | 
 
 For the strength of a man applied to a crank, or to draw or push any weight, is 
 only equal to about 26 or 28 pounds, with a velocity at gost of 11000 feet per hour; 
 and he could labour no more than 7 or 8 hours in succession. Now, as the product 
 of 11000 by 28 is 308000, if this product be divided by 24, the height to which the 
 water is to be raised, the quotient will be 12833 pounds of water, or 206 enbic feet = 
 1540 gallons raised to that height; which makes about 60 gallons, per minute, to the 
 height of 10 feet. This is all that could be produced by such a power in the most 
 favourable case. But the more complex the machine, the greater is the resistance to 
 be surmounted; so that the product would never be nearly equal to the above 
 effect. | 
 
 In amachine, where a man should act by his own weight, and in walking, the ad- 
 vantage would not be much greater: for all that a man could do by walking, with. 
 out any other weight than that of his body, ona plane inclined at an angle of 30 
 degrees, would be to pass over 6000 feet per hour, especially if he had to walk in this 
 manner for 7or Shours. But here it isthe perpendicular height alone, which in this 
 case is 3000 feet, that is to be considered: the product of 3000 by 150 pounds, which 
 is the average weight of a man, is 450000; the greatest effect therefore of sucha 
 machine, would be 450000 pounds, raised to the height of one foot, or 18750 to the 
 height of 24 feet, or about 90 gallons per minute, to the height of 10 feet. By taking 
 an arithmetical mean between this determination and the preceding, it will be found 
 that the mean product possible of the strength of a man, employed to put in motion 
 a hydraulic machine, is at most 75 gallons per minute; especially if continued for 7 or 
 8 hours in the day. 
 
 If the power were to act only for a very short time, as 3,4, or 5 minutes, the 
 product indeed might appear more considerable, and about double. Thisis one of the 
 artifices employed by mechanicians, to prove the superiority of their machines. They 
 put them in motion for some minutes, by vigorous people, who make a momentary 
 effort, and thus cause the product to appear much greater than it really is. | 
 
 The above determination agrees pretty well with that given by Desaguliers in his 
 Treatise on Natural Philosophy: for he assured himself, he says, by calculation, that 
 the effect of the simplest and most perfect machines, put in motion by men, never 
 gives, in the ratio of each man, above 72 gallons of water per minute raised to the 
 height of 10 feet. Si 
 
 A circumstance, very necessary to be known in regard to machines which are to b 
 moved by horses, is as follows: a horse is equal to about seven men*, or can make an 
 
 *C. Regnier, in his description of the Dynanometer, an instrument invented by him for the pur- 
 
 ose of determining the relative strength of men and horses, published in the “ Journal de |’Ecole 
 Dolptechaiquey? vol. ii. p. 160, says, that from the result of aJ] his experiments it appears, that the 
 mean term of the maximum of the strength of ordinary men, to raise a we'ght, is about 285 pounds 
 averdupois ; which agrees with the experiments of Delahire, but which Desaguliers consid-red as 
 too smal). In regard to horses, he says, that by taking the mean results given by four horses, of the 
 middle size, subjected to trial one after the other, the strength of ordinary horses may be estimated 
 at 794 pounds averdupois. - 
 
 In comparing the relative force of men with that of horses, when the former draw a cart or boat by 
 the help of a rope, after various trials, he found that the maximum of the strength of ordinary men, 
 in dragging a horizontal weight, by the help of a rope, is equal to 110 pounds averdupois, and that 
 of the strongest does not exceed 132 poundsaverdapvis. These different trials agree pretty well with 
 the general received opinion, thata horse is seven times as strong as aman. ‘his principle, however, 
 cannot be admitted in ail cases; for itis known by experiment that a horse would sink under a burden 
 seven times as heavy 2s that which aman can support when standing upright. It may readily be 
 conceived that what has been here said respectivg men and horses, is not applicable to daily and 
 incessant labour ; but we may deduce from it this very just consequence, that both can act for a 
 whole day, when employing a fifth of their absolute forces. According to the above results, there- 
 fore, the power which an ordinary man can exert for a continuance in dragging or pulling, is equal 
 to no more than about 22 pounds, and that of the stiongest to about 26 pounds. 
 
j 
 
 effort in a horizontal direction of 210 pounds, moving with the velocity of 10000 or 
 11000 feet per hour, supposing he is to work 8 or 10 hours perday. Desaguliers even 
 gives less, and thinks that the force of a man is to be only quintupled to find that 
 of the horse. 
 
 Those who are acquainted with these principles, will run no risk of being deceivea 
 by ignorant or pretended mechanicians; and it is no small advantage to be able to 
 avoid becoming the dupe of such men, whose aim is often to pick the ee of those 
 who are so simple as to lister'to them. 
 
 PERPETUAL MOTION. 237 
 
 PROBLEM XLIX. 
 Of the Perpetual Motion. 
 
 The perpetual motion has been the quicksanx’ of mechanicians, as the quadrature 
 of the circle, the trisection of an angle, &c., aveheei that of geometricians: and as 
 those who pretend to have discovered the solution of the latter problems are, in 
 general, persons scarcely acquainted with the principles of geometry, those who search 
 for, or imagine they have found, the perpetual motion, are always men to whom the 
 most certain and invariable truths in mechanics are unknown. 
 
 It may be demonstrated indeed, toall those capable of reasoning in a sound manner 
 on those sciences, that a perpetual motion is impossible: for, to be possible, it is 
 necessary that the effect should become alternately the cause, and the cause the 
 effect. It would be necessary, for example, that a weight, raised to a certain height 
 by another weight, should in its turn raise the second weight to the height from which 
 it descended. But, according to the laws of motion, all that a descending weight 
 could do, in the most perfect machine which the mind can conceive, is to raise another 
 in the same time to a height reciprocally proportional to its mass. But it is impos- 
 sible to construct a machine in which there shall be neither friction nor the resistance 
 of some medium to be overcome; consequently, at each alternation of ascent and 
 descent, some quantity of motion, however small, will always be lost: each time 
 therefore the weight to be raised will ascend toa less height; and the motion will 
 gradually slacken, and at length cease entirely. 
 
 A moving principle has been sought for, but without success, in the magnet, in the 
 gravity of the atmosphere, and in the elasticity of bodies. If a magnet be disposed 
 in such a manner as to facilitate the ascension of a weight, it will afterwards oppose 
 its descent. Springs, after being unbent, require to be bent by a new force equal to 
 that which they exercised; and the gravity of the atmosphere, after forcing one side 
 of the machine to the lowest point, must be itself raised again, like any other weight, 
 in order to continue its action. 
 
 We shall however give an account of various attempts to obtain a perpetual motion, 
 because they may serve to shew how much some persons have suffered themselves 
 to be deceived on this subject. 
 
 Fig. 52 represents a large wheel, the circumference of 
 which is furnished, at equal distances, with levers, each 
 bearing at its extremity a weight, and moveable ona hinge, 
 so that in one direction they can rest upon the circum- 
 ference, while on the opposite side, being carried away by 
 the weight at the extremity, they are obliged to arrange 
 themselves in the direction of the radius continued. 
 This being supposed, it is evident that when the wheel 
 turns in the direction a b c, the weights A, B and c will 
 recede from the centre; consequently, as they act with 
 more force, they will carry the wheel towards that side ; 
 and as a new lever will be thrown out, in proportion as 
 the wheel revolves, it thence follows, say they, that the wheel will continue to move 
 
athe Ne 
 
 238 MECHANICS. 
 
 in the same direction. But notwithstanding the specious appearance of this reasoning, 
 
 experience has proved that the machine will not go; and it may indeed be demon- 
 strated that there is a certain position, in which the centre of gravity of all these 
 weights is in the vertical plane passing through the point of suspension, and that 
 therefore it must stop. 
 
 The case is the same with the following machine, which it 
 would appear ought to move also incessantly. In a cylindrie 
 drum, in perfect equilibrium on its axis, are formed channels 
 as seen in Fig. 53, which contain balls of lead, or a certain 
 quantity of quicksilver. In consequence of this disposition, 
 the balls or quicksilver must, on the one side, ascend by ap- 
 proaching the centre ; and on the other must roll towards the 
 circumference. The machine then ought to turn incessantly 
 towards that side. 
 
 A third machine of this kind is represented Fig. 54. It con- 
 sists of a kind of wheel formed of six or eight arms, proceeding 
 from a centre, where the axis of motion is placed. Each of 
 these arms is furnished with a receptacle in the form of a pair 
 of bellows; but those on the opposite arms stand in contrary 
 directions, as seen in the figure. The moveable top of each 
 receptacle has affixed to it a weight, which shuts it in one si- 
 tuation, and opens itin the other In the last place, the bellows 
 of the opposite arms have a communication by means of a canal, 
 and one of them is filled with quicksilver. 
 
 These things being suppused, it is visible, that the bellows on the one side must 
 open, and those on the other must shut; consequently the mercury will pass from the 
 latter into the former, while the contrary will be the case on the opposite side, 
 
 It might be difficult to point out the deficiency of this reasoning; but those ac. 
 quainted with the true principles of mechanics will not hesitate to bet a hundred to 
 one that the machine, when constructed, will not answer the intended purpose. 
 
 The description of a pretended perpetual motion, in which bellows, to be alter. 
 nately filled with and emptied of quicksilver, were employed, may be seen in the 
 “* Journal des Scavans,” for 1685. It was refuted by Bernoulli, and some others, 
 and it gave rise to a long dispute. The best method which the inventor could have 
 employed to defend his invention, would have been to construct it, and shew it in 
 motion; but this was never done. 
 
 We shall here add another curious anecdote on this subject. One Orfyreus 
 announced at Leipsic, in the year 1717, a perpetual motion, consisting of a wheel, 
 which would continually revolve. This machine was constructed for the landgrave 
 of Hesse-Cassel, who caused it to be shut up in a place of safety, and the door to 
 be sealed with his own seal. At the end of forty days, the door was opened, and 
 the machine was found in motion. This however affords no proof in favour of a 
 perpetual motion: for as clocks can be made to go a year without being wound 
 up, Orfyreus’s wheel might easily go forty days, and even more. 
 
 The result of this pretended discovery is not known: we are informed, by one 
 ‘of the journals, that an Englishman offered 80000 crowns for this machine; but | 
 Orfyreus refused to sell it at that price ; in this he certainly acted wrong, as there is 
 reason to think that he obtained by bis invention neither mene nor even the ho- 
 nour of having discovered the perpetual motion. 
 
 The Academy of Painting, at Paris, possessed a clock, which tae no need of being 
 wound up, and which might be considered asa perpetual motion, though it was not so. 
 But this requires some explanation. The ingenious author of this clock employed © 
 the variations in the state of the atmosphere, for winding up his moving weight: 
 
HEIGHTS AND DEPTHS BY PENDULUMS. 239 
 
 various artifices might be devised for this purpose; but this is no more a perpetual» 
 _ motion, than if the flux and reflux of the sea were employed to keep the machine 
 
 continually going for this principle of motion is exterior to the machine, and forms 
 - no part of it. 
 
 But enough has been said en this chimera of mechanics. We sincerely hope that 
 
 none of our readers will ever lose themselves in the ridiculous and unfortunate laby- 
 _Finth of such a research. 
 
 To conclude, it is false that any reward has been promised by the European 
 
 powers to the person who shall discover the perpetual motion ; and the case is the 
 _ same in regard to the quadrature of the circle. It is this idea, no doubt, that excites 
 
 | so many to attempt the solution of these problems; and it is proper they should be 
 
 _ undeceived. ~ 
 
 PROBLEM L. M 
 To determine the height of the arched ceiling of a church, by the vibrations of the 
 lamps suspended from it. 
 For this invention we are indebted, it is said, to Galileo, who first ascertained the 
 
 ' ratio of the duration of the oscillations made by pendulums of different lengths.* 
 ' But in order that this method may have a certain degree of exactness, the weight 
 
 of the lamp ought to be several times greater than that of the cord by which it is 
 
 _ suspended. 
 
 This being supposed, put the lamp in motion by removing it a very little from its 
 perpendicular direction, or carefully observe that communicated to it by the air, 
 which is very common ; and with a stop-watch find how many seconds one vibration 
 
 continues, or, if a stop-watch is not at hand, count the number of vibrations per- 
 
 formed in a certain number of minutes: the greater the number of minutes, the 
 more exact will the duration of each vibration be determined; for nothing will then 
 be necessary, but to divide those minutes by the number of vibrations, and the quo- 
 
 — tient will be the duration of each in minutes or seconds. 
 
 We shall here suppose that it has been found, by either of these methods, that the 
 
 _ time ofeach vibration is 5} seconds; square 54, which is 301, and multiply by it 392 
 
 inches, the length of a pendulum that swings seconds in the latitude of London, the 
 
 - product will be 98 ft. 7 in. 6 lin., which will be nearly the height from the point of 
 
 suspension to the bottom or rather centre of the lamp. 
 
 If the distance from the bottom of the lamp to the pavement be then measured, 
 which may be done by means of a stick, and added to the former result, the sum 
 will give the height of the arch above the pavement. 
 
 ‘This solution is founded ona pioperty of pendulums, demonstrated in mechanics ; 
 which is, that the squares of the times of the vibrations are as the lengths; so that 
 a pendulum four times the length of another, performs vibrations which last twice 
 as long. 
 
 But on account of the irregular form of the lamp, and the weight of the rope 
 which sustains it, we must confess that this method is rather curious than exact. 
 We shall, however, present the reader with another problem of the same kind. 
 
 PROBLEM LI. 
 
 Lo measure the depth of a well, by the time elapsed between the commencement of the 
 fall of a heavy body, and that when the sound of its fall is conveyed to the ear. 
 
 Have in readiness a small pendulum that swings half seconds, that is to say, 
 
 * Indeed, it seems it was by that author accidentally observing the uniformity in the intervals of 
 the swing of the suspended lamps, that he first took the hint of empleying the oscillations of pen- 
 
 dulous hodies, or pendulums, for the purpose of measuring time. And hence the invention of pen- 
 dulum clocks. 
 
240 MECHANICS. ; =| 
 
 25 inches in length, between the centre of the ball, and the point of suspensiaa| 
 You must also employ a weight of some substance as heavy as possible, such for) 
 example as lead; as a common stone or pebble experiences a considerable retardation 
 in falling, and therefore would not answer the purpose so well. | 
 
 Let go the weight and the ball of the pendulum at the same moment of time, and. 
 count the number of vibrations the latter makes, till the moment when you hear the 
 sound. We shall here suppose that there were ten vibrations, which make five 
 seconds. . | 
 
 As a heavy body near the earth’s surface falls about.16j, feet in one second of 
 time, or for this purpose 16 feet will be exact enough; and as sound moves at the 
 rate of 1142 feet per second; multiply together 1142, 16, and 5, which will give 
 91360, and to 4 times this product, or 365440, add the square of 1142, which is | 
 1304164, and the sum will be 1669604 ; and if from the square root of the last number | 
 = 1292 the number 1142 be subtracted, the remainder 150, divided by 32, will give 
 4-69 for the number of geconds which elapsed during the fall of the body: if this | 
 remainder be subtracted from 5, the number of seconds during which the body was | 
 falling and the sound returning, we shall have 0°31 for the time which the sound | 
 alone employed before it reached the ear; and this number multiplied by 1142, will | 
 give for product 354 feet =the depth of the well. | 
 
 This rule, which is rather complex, is. founded on the property of falling bodia | 
 which are accelerated in the ratio of the times, so that the spaces passed over increase. 
 as the squares of the times*. But as the resistance of the air, which in considerable 
 heights, such as those of several hundred feet, does not fail to retard the fall in a 
 sensible manner, has been neglected, the case of this problem is nearly the same as 
 with the preceding; that is to say, the solution is rather curious than useful. 
 
 « For the sake of our algebraical readers we shall here shew how to find the formula from which 
 the above rule is deduced: Let a = 5, p16 3s € == 1142, and let a be the time which the body 
 employs in falling, consequently @a—.2 will be the time of the sound returning. Then as 12:3 
 
 °° 42: 6 x2= depth of the well; andl : ¢!! a—z2!ca—cx= depth of the well also; 
 
 ca 
 therefore 6 x2 == ca —c x, and by transposition and division, a “i+ b v==—, Completing the 
 
 b 
 ram c — tocar c c_ , 4bcea+c? 
 square, x + $e fh Sa ae work +5 her Pig od Behe Hence, oe Gr Ya / A bie and XU moe 
 4bca+c ¢ _/ e+4abc—e ac 
 — = YY = neal the time of descent. Consequentl 
 poe preci 2b %ab-e mia 
 acu. ceth 
 
 —_— —— — —— js nearly the time of the sound’s ascent. 
 abc ab+e M 
 
 Hence, from the expression — a much simpler rule is obtained for the time of the descent, 
 
 foe | 
 which is as follows: Multiply 1142 by 5, which gives for product 5710; then multiply also 16 by 5, 
 which gives 80, to which add 1142, this gives 1222, by which sum divide the first product 5710, and 
 the quotient 4-68 will be the time of descent, nearly the same as before. This taken from 5 leaves 
 0°32 for the time of the ascent; which multiplied by 1142, chia 365 for the depth, differing but 
 little from the former more exact number. 
 
ey 
 
 AUTOMATONS,.—CURIOUS CLOCKS. 24) 
 
 HISTORICAL ACCOUNT OF SOME EXTRAORDINARY AND CELEBRATED 
 MECHANICAL WORKS. 
 
 AN essential part might seem wanting to this work if we neglected to give some 
 account of the various machines most celebrated both among the ancients and moderns. 
 We shall therefore take a cursory view of the rarest and most singular inventions, 
 produced by mechanical genius, in different ages. 
 
 I.—Of the machines or automatons of Archytac. Archimedes, Hero, and Ctesibius. 
 
 Some machines of this kind are mentioned mi deay history, in terms of the ut- 
 most admiration. Such were the tripod automatons of Vulcan; and the dove of 
 Archytas, which,, as we are told, could fly like areal animal. We have no doubt, 
 however, that the wonderful properties of these machines, if they ever really existed, 
 have been greatly exaggerated by credulity; and by the accounts of them being 
 handed down through such a long series of ages. We are told also of the moving 
 sphere of Archimedes, in which, as appears, that celebrated philosopher had re- 
 presented all the celestial motions, as they were then known; and this, no doubt, 
 was a master-piece of mechanism for that remote period. Every one is acquainted 
 with the famous verses of Claudian on this machine. 
 
 Several wonderful machines were constructed also by Hero and Ctesibius of 
 Alexandria. An account of some of those invented by Hero may be seen in a 
 book called Spiritalia. Some of them are very ingenious, and do honour to the . 
 talents of that mechanician. 
 
 II.—Of the machines ascribed to Albert the Great, and to Regiomontanus. 
 
 That ignorance, in the darkness of which all Europe was involved, from the 
 sixth or seventh century to the fifteenth, did not entirely extinguish mechanical 
 genius. We are told that the ambassadors sent by the king of Persia to Char- 
 lemagne brought, as a present to the latter, a machine, which, according to the 
 description given of it, would have done honour to our modern mechanicians ; 
 for it appears to have been a striking clock, which had. figures that performed 
 various movements. It is indeed true that, while Europe was immersed in igno- 
 rance, the arts and sciences diffused a gleam of light among the nations of the 
 East. In regard to those of the West, if we can believe what is related of 
 Albert the Great, who lived in the thirteenth century, that mathematician con- 
 structed an automaton in the human form, which when any one knocked at the 
 door of its cell, came to open it and sent forth some sounds, as if addregsing 
 the person who entered. At a period later by some centuries, Regiomontanus, 
 or John Muller of Konigsberg, a celebrated astronomer, constructed an automaton 
 in the figure of a fly, which walked around a table. But these accounts are 
 probably very much disfigured by ignorance and credulity. The following however 
 are instances of mechanical skill, in which there is much more of reality. 
 
 Iil.—Of various celebrated Clocks. 
 
 In the fourteenth century, James Dondi constructed for the city of Padua a clock, 
 which was long considered as the wonder of that period. Besides indicating the hours, 
 it represented the motion of the sun, moon, and planets, as well as pointed out the 
 different festivals of the year. On this account, Dondi got the surname of Eorologio, 
 which became that of his posterity. A little time after, William Zelandin constructed, 
 for the same city, one still more complex; which was repaired in the sixteenth 
 century by Janellus Turrianus, the mechanician of Charles VY. 
 
 R 
 
242 ¥ MECHANICS. 
 
 But the most celebrated works of this kind aré the clocks of the cathedrals of Stras- 
 purghand Lyons. That of Strasburgh was the work of Conrad Dasypodius, a mathe- 
 matician of that city, who lived towards the end of the sixteenth century, and who 
 finished it about the year 1573. Itis considered as the first in Europe. Atany rate 
 there isnone but that of Lyons which can dispute pre-eminence with it, or be com- 
 pared toit iu regard to the variety of its effects. 
 
 The face of the basement of the clock of Strasburgh exhibits three dial-plates; one 
 of which is round, and consists of several concentric circles; the two interior ones 
 of which perform their revolutions in a year, and serve to mark the days of the year, 
 the festivals, and other circumstances of the calendar. The two lateral dial-plates are 
 square, and serve to indicate the eclipses, both of the sun and the moon. 
 
 Above the middle dial-plate, and in the attic space of the basement, the days of 
 the week are represented by different divinities, supposed to preside over the planets 
 from which their common appellations are derived. The divinity of the current 
 day appears in a car rolling over the clouds, and at midnight retires to give place to 
 the succeeding one. 
 
 Before the basement is seen a globe, borne on the wings of a pelican, around 
 which the sun and moon revolved; and which in that manner represented the motion 
 of these planets; but this part of the machine, as well as several others, has been 
 deranged for a long time. 
 
 The ornamented turret, above this basement, exhibits chiefly a large dial, in the 
 form of an astrolabe; which shews the annual motion of the sun and moon through 
 the ecliptic, the hours of the day, &c. The phases of the moon are seen also marked 
 out on a particular dial-plate above. 
 
 This work is remarkable also for a considerable assemblage of bells and figures, 
 which perform different motions. Above the dial-plate last mentioned; for example, 
 the four ages of man are represented by symbolical figures: one passes every quarter 
 of an hour, and marks the quarter by striking on small bells; these figures are fol- 
 lowed by death, whois expelled by Jesus Christ risen from the grave ; who however 
 permits it to sound the lrour, in order to warn man that time ison the wing. Two 
 small angels perform movements also; one striking a bell with a sceptre, while the 
 other turns an hour-glass, at the expiration of an hour. 
 
 In the last place, this work was decorated with various animals, which emitted 
 sounds similar to their natural voices; but none of them now remain except the cock, 
 which crows immediately before the hour strikes, first stretching out its neck and 
 clapping its wings. The voice of this figure however is become so hoarse as to be 
 much less harmonious than the voice of that at Lyons, though the latter is attended,. 
 in a considerable degree, with the same defect. It is to be regretted that a great 
 part of this machine is entirely deranged. It would be worthy of the illustrious me- 
 tropolitan chapter of Strasburgh to cause it to be repaired: we have heard indeed 
 that it has been attempted; but that no artist could be found capable of performing it. 
 
 | 
 
 The clock of the cathedral of Lyons is of less size than that of Strasburgh; but is” | 
 
 not inferior toit in the variety of its movements; and it has the advantage also of 
 being in a good condition, It is the work of Lippius de Basie, and was exceedingly . 
 
 well repaired in the last century by an ingenious clock-maker of Lyons, named Nou- 
 risson. Like that of Strasburgh, it exhibits, on different dial-plates, the annual and 
 
 diurnal progress of the sun and moon, the days of the year, their length, and the — 
 
 whole calendar, civilas well as ecclesiastical, The days of the week are indicated by 
 symbols more analogous to the place where the clock is erected; the hours are an- 
 nounced by the crowing of the cock, three times repeated, after it has clapped its 
 wings, and made various other movements. When the cock has done crowing, angels 
 appear, who, by striking various bells, perform the air of ahymn; the annunciation of 
 the Virgin is represented also by moving figures, and by the descent of a dove from 
 the clouds; and after this mechanical exhibition, the hour strikes. On one of the 
 
 | 
 
 | 
 
AUTOMATION MACHINES. 243 
 
 _ sides of the clock is seen an oval dial-plate, where the hours and minutes are indi- 
 cated by means of an index, which lengthens or contracts itself, according to the 
 _ length of the semi-diameter of the ellipsis over which it moves. 
 
 A very curious clock, the work of Martinot, a celebrated clock-maker of the 
 seventeenth century, was to be seen in the royal apartments at Versailles. Before 
 ‘it struck the hour, two cocks on the corners of a small edifice crowed alternately; clap- 
 ping their wings ; soon after two lateral doors of the edifice opened, at which appeared 
 two figures bearing cymbals, beat upon by a kind of guards with clubs. When these 
 figures had retired, the centre door was thrown open, and a pedestal, supporting an 
 
 equestrian statue of Louis XIV., issued from it, while a group of clouds separating 
 
 _ gave a passage to a figure of Fame, which came and hovered over the statue. An 
 
 air was then performed by bells; after which the two figures re-entered; the two 
 
 _ guards raised up their clubs, which they lowered as if out of respect for the presence 
 of the king, and the hour was then struck. ‘Though all these things are easy for 
 ingenious clock-makers of the present day, when we come to treat of Astronomy 
 we shall give an account of some machines of this kind, purely astronomical, which 
 do honour to the inventive genius of those by whom they were constructed. 
 
 IV.—Automaton machines of Father Truchet, M. Camus, and M. de Vaucanson. 
 
 Towards the end of the seventeenth century, Father Truchet, of the Royal Academy 
 of Sciences, coustructed, for the amusement of Louis XIV., moving pictures, which 
 were considered as very remarkable master-pieces of mechanics. One of these pic- 
 tures, which that monarch called his little opera, represented an opera of five acts, and 
 changed the decorations at the commencement of each. The actors performed their 
 partsin pantomime. The representation could be stopped at pleasure; this effect was 
 produced by letting go a catch, and by means of another the scene could be made to 
 re-commence at the place where it had been interrupted. This moving picture was 
 ‘sixteen inches and a half in breadth, thirteen inches four lines in height, and one 
 inch three lines in thickness, for the play of the machinery. An account of this piece 
 of mechanism may be found in the eulogy on Father Truchet, published in the 
 Memoirs of the Academy of Sciences, for the year 1729. 
 
 Another very ingenious machine, and in our opinion much more difficult to be 
 conceived, is that described by M. Camus, a gentleman of Lorrain, who says he 
 constructed it for the*amusement of Louis XIV., when a child It consisted of a 
 small coach, drawn by two horses, in which was the figure of a lady, with a footman 
 and page behind. 
 
 If we can give credit to what is stated in the work of M. Camus, this coach being 
 placed at the extremity of a table of a determinate size, the coachman smacked his 
 whip, and the horses immediately set out, moving their legs in the same manner as 
 teal horses do. When the carriage reached the edge of the table, it turned at a right 
 angle, and proceeded along that edge. When it arrived opposite to the place where 
 the king was seated, it stopped, and the page getting down opened the door, upon 
 which the lady alighted, having in her hand a petition which she presented with a 
 curtsey. After waiting some time, she again curtseyed, and re-entered the carriage ; 
 
 the page then resumed his place, the coachman whipped his horses, which began to 
 move, and the footman, running after the carriage, jumped up behind it. 
 
 It is much to be regretted that M. Camus, instead of confining himself to a general 
 account of the mechanism which he employed to produce these effects, did not enter 
 into a more minute description; for, if they are true, it must have required a very 
 singular artifice to produce them, and the same means might be applied to machines 
 of greater utility. 
 
 About thirty or thirty-five years ago, three very curious machines were exhibited 
 
 by M. de Vaucanson, viz., an automaton flute-player, a player on the flageolet and tam- 
 HiZ 
 
244 MECHANICS, 
 
 bourine, and an artificial duck, The first played several airs on the flute, witha 
 precision greater perhaps than was ever attained to by the best living player, and 
 even executed the tonguing, which serves to distinguish the notes. According to 
 M. de Vaucanson, this part of the machinery cost him the greatest trouble. In a 
 word, the tones were really produced in the flute by the proper motion of the 
 fingers. 
 The player on the flageolet and tambourine performed also some airs on the first 
 of these instruments, and at the same time kept continually beating on the latter. 
 But the motion of the artificial duck, in our opinion, was still more astonishing ; 
 for it extended its neck, raised up its wings, and dressed its feathers with ifs bill; it 
 _ picked up barley from a trough, and swallowed it; drank from another, and, after 
 various other movements, voided some matter resembling excrements. The first 
 time I saw these machines I immediately discovered some of the artifices employed 
 in regard to the two former, but I confess that the latter baffled my penetration. 
 We have also of late been amused, by M. Droz and M. Maillardet, &c., with the 
 surprising performances of the chess-players, the small but sweet singing-bird, the 
 writing figure, the musical lady, the conjurer, the tumbler, &c. &e. 
 
 V.—Of the Machine at Marly. 
 
 It will doubtless be allowed, that the machines above mentioned are, in general, 
 more curious than useful; but there are other two, the celebrity and utility of which 
 require that we should here give them a place. These are the machine of Marly, and | 
 that known under the name of the steam engine. We shall begin with the former, 
 of the construction and effects of which the following brief description will give 
 some idea, 
 
 The machine of Marly consists of 14 wheels, each about 36 feet in diameter, 
 moved by a stream of water, confined by an estacade, and received into so many 
 separate channels. Each wheel has at the extremities of its axis two cranks, and. 
 this forms 28 powers, distributed in the following manner. 
 
 It must however be first observed, that the water is raised, to the place to which | 
 it is to be conveyed, by three different stages; first from the river toa reservoir, at , 
 the elevation of 160 English feet above the level of the Seine; then toa second | 
 reservoir 346 feet higher; and from the latter to the summit of a tower, somewhat 
 more than 533 feet above the river. 
 
 Of the 28 cranks above mentioned, eight are employed to give motion to 64 
 pumps; which is done by means of working beams, having four pistons at each ex- 
 tremity of their arms: this makes eight to each working beam, which are drawn up 
 and pushed down aiternately. These 64 pumps force up the water to the first reser- 
 voir; and this reservoir furnishes water to the first well, on which is established the _ 
 second set of pumps. af 
 
 Eleven more cranks are employed to force the water from the first well to the | 
 second reservoir. This is done by means of long arms adapted to these cranks, 
 which move large frames, to one of the arms of which are attached strong iron 
 chains, that extend from the bottom of the mountain to the first well. These , 
 chains, called chevalets, are formed of parallel bars of iron, the extremities of which | 
 are bound together by iron bolts, and are supported at certain intervals by transversal | 
 pieces of wood, moveable on an axis, that passes through the middle of each; so | 
 that when the upper bar of iron, for example, is drawn down by the lower end, all | 
 these pieces of wood incline in one direction, and the lower bar moves backwards | 
 and pushes in a direction contrary to the upper one, These bars or chains serve to | 
 put in motion the working beams, or squares, and the latter move the pistons of 80 - 
 sucking and forcing pumps, which raise the water from the first well to the second - 
 reservoir, 
 
 ane = eee a ne. 
 
CELEBRATED MACHINES. 945 
 
 ‘In the last place, nine other cranks, by a similar mechanism, put in motion those 
 chains, called the grands chevalets, which move the pumps of the second well, 
 and raise the water from it to the summit of the tower. These pumps are in num- 
 ber seventy-two. 
 
 Such, in a few words, is the mechanism of the machine of Marly. Its mean product, 
 as said, is from 30000 to 40000 gallons of water per hour. We make use of the 
 term mean product, because at certain times it raises 60000 gallons, but only under 
 very favourable circumstances. During inundations, when the Seine is frozen, when 
 the water is very low, or when any repairs are making, the machinery stops, either 
 entirely or in part. We have read that in the year 1685 it raised 70000 gallons per 
 hour; but this we can scarcely believe, if by that quantity is understood its mean 
 product ; as it would be above 1090 gallons per minute. 
 
 However this may be, the following calculation is founded on details collected on 
 purpose, The annual expense of the machine, including the salaries of those who 
 superintend it, and the wages of the workmen employed, together with repairs, ne- 
 cessary articles, &c., may amount to about £3300 sterling, or £9 per day; which 
 makes about 1 farthing per 90 gallons. But if we take into this account the in- 
 terest of the £333000 which, it is said, were expended in the construction of it, 90 
 gallons will cost 3 halfpence, which is at the rate of a farthing for 15 gallons. This 
 is very far from the price which the king of Denmark thought he might set on this 
 water; for that prince, when he paid a visit to Marly, in the year 1769, being as- 
 tonished, no doubt, at the immensity of the machine, the multitude of its move- 
 ments, and the number of the workmen it employed, observed that the water 
 perhaps cost as much as wine. By the above calculation, the reader may see how 
 far his majesty was mistaken. 
 
 It is an important question to know, whether the machine of Marly could be sim- 
 plified. On this subject we shall give a few observations, which from some experi- 
 ments made, and a minute examination of the different parts of the machine, appear 
 to be founded on probability. 
 
 People in general are surprised that the inventor of this machine should cause the 
 water, in some measure, to make two rests before it is conveyed to the summit of 
 the tower. It has been humorously said, that he no doubt thought the water would 
 pe too much fatigued to ascend to the perpendicular height of more than 533 feet, all 
 at one breath. It is more probable that he thought his moving force would not be 
 sufficient to raise the water to that height; but this is not agreeable to theory; for 
 it is found by calculation, that the force of one crank is more than sufficient to raise 
 a cylinder of water of that altitude, and above 8 inches in diameter. Able mechani- 
 cians -however are of opinion, that though this be not impossible, to carry it into 
 execution would be attended with great inconveniences, which it would be too tedious 
 to explain. a 
 
 But it appears certain at present, that the water might be raised in one jet to the 
 second well. This results from two experiments, one made in 1738, and the other 
 in 1775. - In the first, M. Camus, of the Royal Academy of Sciences, endeavoured to 
 make the water rise in one jet to the tower: his attempt was not attended with 
 success, but he made it rise to the foot of the tower, which is considerably higher 
 than the second reservoir; hence it follows, that if he had confined himself to making 
 the water rise in one jet to the second reservoir, he would have succeeded. It is 
 said that, during this experiment, the machine was prodigiously strained; that it 
 was even found necessary to secure some parts of it with chains; that it required 
 twenty-four hours to force it to that height, which is about 480 feet, and that it was 
 not possible to make it go farther. The object of the second trial, made in 1775, 
 was to raise the water only to the second well. It indeed ascended thither at differ- 
 ent times, and in abundance; but the pipes were exceedingly strained at the bottom, 
 
945 MECHANICS. 
 
 so that several of them burst; and it was necessary to suspend and recommence the — 
 experiment several times. It is however evident that this arose from the age of the 
 tubes and their want of strength, as they had not the proper thickness; a fault which 
 might have been easily remedied. Here then we have one step towards the improye- 
 ment of the machine; and it results from this trial, that the chains which proceed from 
 the river to the first well, might be suppressed, and even the first well itself. 
 
 It still remains to be determined, whether the water could be made to ascend, in 
 one jet, to the summit of the tower. This would be a very curious experiment ; but 
 no doubt difficult and expensive, because it would be necessary to make considerable 
 changes in different parts of the machine; and even in the case of its succeeding, the 
 water raised might perhaps be in such small quantity, that it would be better to retain 
 the present mechanism. 
 
 It is probable that various improvements might be made in different parts of the 
 machine. Inseveral positions, the moving forces act only obliquely, which occasions : 
 a great loss of power, and must tend to render the machine less effectual. The 
 form of the pistons, valves, and aspiration tubes, might perhaps admit also of 
 some change. But as this is not the place for entering into these details, we 
 shall proceed to the Steam Engine, of which we promised to give a short 
 description. 
 
 ! 
 
 VI.—Of the Steam Engine. 
 
 The Steam Engine is that perhaps in which the genius of mechanism has been 
 manifested in the highest degree; for no idea could be more happy than that of 
 employing alternately, as moving powers, the expansive force of the steam of 
 water, and the weight of the atmosphere. Such indeed is the principle of this inge- 
 nious machine, which is at present employed with so much success in pumping water 
 from mines, and for a variety of other purposes in the arts and manufactures. 
 
 The first part of this machine is a large boiler, to the cover of which is adapted a 
 hollow cylinder, two, three, cr four feet in diameter. A communication is formed 
 between the boiler and the cylinder by an aperture, capable of being opened or shut. 
 Into this cylinder is fitted a piston, the rod of which is made fast to the extremity of 
 one of the arms of a working beam, having at the extremity of its other.arm the weight 
 to be raised, which is generally the piston of a second pump, adapted to raise water 
 froma great depth. The whole must be combined in such a manner that when the air 
 or steam has free access into the cylinder, which communicates with the boiler, 
 the weight alone of the apparatus affixed to the opposite arm shall be capable of 
 raising that piston. . ae 
 
 Let us now suppose the boiler filled with water to a certain height, and that it is 
 brought to a state of complete ebullition by a large fire kindled below the boiler, 
 As a part of this water will continually rise in steam, when the communication 
 between the boiler and the cylinder is opened, this vapour, which is elastic, will 
 introduce itself into it, and raise the piston; as its force is equivalent to that of air. 
 Let us suppose also that the piston, when it attains to a certain height, by means 
 of some mechanism, which may be easily conceived, moves a certain part of the 
 machine, which intercepts the communication between the boiler and the cylinder ; 
 and, in the last place, that by the same cause a jet of cold water is thrown beneath 
 the bottom of the piston in the cylinder, so as to fall down through the vapour in 
 the form of rain. At that moment the steam will be condensed into water ; a vacuum 
 will be formed in the cylinder ; and consequently the piston will be then charged with 
 the weight of the atmosphere above it, or a weight equivalent to a column of water 
 of the same base and 32 feet in height. If the piston, for example, be 52 inches in 
 ‘ diameter, as is the case in the steam-engines of Montrelais, near Ingrande, this 
 weight will be equal to 29450 pounds: the piston will consequently be obliged to 
 descend with a force equal to nearly 30000 pounds, and the other arm of the working 
 
 we 
 
hs 
 3 
 ' 
 
 THE STEAM ENGINE. 247 
 
 beam, if it be of the same length, will act with an equal force to overcome the 
 resistance opposed to it. When the piston has made this first stroke, the communi- 
 eation between the boiler and the cylinder is restored; the steam of the boiling 
 
 water again enters it, and the equilibrium between the air of the atmosphere and the 
 
 _jnside of the cylinder being re-established, the weight of the apparatus affixed to the 
 other end of the working beam descends, and raises the piston; the same play as 
 
 before is renewed; the piston again falls, and the machine continues to produce its 
 effect. 
 
 It may be readily conceived, that we must here confine ourselves to this short 
 sketch; for a long description and a variety of figures would be necessary to give a 
 correct idea of the many different parts requisite to produce this effect; such as that 
 which opens and shuts the communication between the boiler and the cylinder ; that 
 which injects cold water into the cylinder; those which serve to evacuate the air 
 and water formed in the inside of the cylinder; the regulator necessary to prevent 
 the steam, when it becomes too strong, from bursting the machine, &c. For farther 
 details therefore we must refer the reader to those authors who have purposely 
 treated of this machine; such as Belidor in his ‘* Architecture Hydraulique,” vol. 
 ii.; Desaguliers, in his “* Cours de Physique Experimentale,” vol. ii-; M. Prony, in 
 his ‘‘ Nouvelle Architecture Hydraulique ;” and several others. 
 
 The machine here described is very different from that mentioned by Muschen- 
 brock, in his ‘* Cours de Physique Experimentale.” In the latter, the steam acts by 
 its compression on a cylinder of water, which it causes to ascend. This requires 
 steam highly elastic, and very much heated ; but in this case there is great danger of 
 the machine bursting In the new machine, that above described, it is sufficient if 
 the steam has the elasticity of the air: this it will acquire if the water boils only 
 briskly ; and therefore the danger of the machine bursting is not nearly so great: it 
 is not even said that this accident ever happened to any of the large steam-engines, 
 which have been long established. 
 
 The largest steam-engine with which I am acquainted, is that of Montrelais, near 
 Ingrande, which is employed in freeing the coal mines from water. The cylinder is 
 524 inches in diameter.* It raises per hour, to the height of 652 feet, by eight 
 different stages, 1145 cubie feet of water, or 10800 gallons; and as it is estimated, 
 after deducting the time lost by putting it in motion, during accidental repairs, which, 
 are necessary from time to time, &c., that it works 22 hours’in the 24, its daily effect 
 is to raise, to the above height, and evacuate, 237600 gallons of water. In the same 
 time it consumes about 266 cubic feet of coals. The other expenses attending it 
 must also be considerable. 
 
 In the same place is another machine which, in some respects, appears to be con- 
 structed on a better principle. ‘Though the cylinder is only 34 inches in diameter, 
 it raises, in 22 hours, to the same height, and at one jet, 22000 cubic feet, or about 
 165000 gallons, which is above two thirds of the quantity raised by the former, 
 while the moving power, which is in the ratio of the squares of the diameters of the 
 pistons, is only about 2 of that of the other. 
 
 An attempt was eons some years ago, to employ the steam-engine to move car- 
 riages, and an experiment on this subject was tried at the arsenal of Paris. The 
 carriage indeed moved, but in our opinion this idea must be considered rather as 
 ingenious, than susceptible of being put in practice It would not be very 
 
 agreeable to travellers to hear, behind them, the noise of a machine capable, if it 
 
 should burst, of blowing them to atoms; and we much doubt whether this invention 
 would meet with encouragement. A boat also which, it is said, could be made to 
 
 * In some steam-engines in England the cylinder is 62, and even 72 inches in diameter, and. 
 their power is equal to that of 250 horses. 
 
248 MECHANICS. 
 
 move against the current by means of a steam-engine, was seen for a long time in 
 the middle of the Seine, opposite to Passy. Nothing less was hoped from this 
 invention, than to be able to convey a boat, laden with merchandise, in two or three 
 days, from Rouen to Paris; but scarcely was the machine in motion, when the 
 wheels, the float-boards of which were to serve as oars, were broken in pieces by the 
 effect of the too violent and sudden impression they received. Such was the result 
 of this attempt, the failure of which had been predicted by the greater part of those 
 mechanicians who had seen the preparations. 
 
 [ We have retained in this edition the above paragraph on the application of steam 
 to navigation and locomotive engines on land, as a curious record of the opinions on 
 those subjects entertained by men the most eminent in science, at a very recent 
 period. What would these men say, could they be permitted to view the achieve- 
 ments of modern science, on the application of steam to travelling by sea and by 
 land? Hundreds of people conveyed by the power of a single steam-engine from 
 London to Liverpool at the rate of from 20 to 30 miles an hour; and steam vessels 
 sailing from various ports in England to New York, with the regularity of mail 
 coaches, and completing their voyages in about a fortaight; would doubtless strike 
 them with amazement. ] 
 
 Remark,—As Montucla has given but a short and imperfect account of that truly 
 noble English invention, we have subjoined the following brief history of it. The 
 Steam Engine was invented by the Marquis of Worcester, in the year 1655; and an 
 account of it was printed in a little book, entitled “A Century of the Names 
 and Scantlings of such Inventions as at present I can call to mind,” &c.; in the 
 year 1663. 
 
 In the 68th article of that work, the Marquis describes the invention in the follow- 
 ing words :—‘‘ An admirable and most forcible way to drive up water by fire. Not 
 by drawing or sucking it upwards, for that must be as the philosopher calleth it, 
 intra speram activitatis, which is but at such a distance; but this way hath no 
 bounds, if the vessel be strong enough; for I have taken a piece of a whole cannon, 
 whereof the end was burst, and filled it three quarters full of water, stopping and — 
 securing up the broken end, as also the touch-hole, and making a constant fire under 
 it, within 24 hours it burst and made a great crack ; so that having a way to make my 
 vessels, so that they are strengthened by the force within them, and the one to fill 
 after the other. I have seen the water run like a constant fountain stream forty feet 
 high; one vessel of water rarefied by fire driveth up forty of cold water. Anda - 
 man that tends the work is but to turn two cocks, that one vessel of water being con- 
 sumed, another begins to force and refill with cold water, and so successively, the 
 fire being tended and kept constant, which the self-same person may likewise abun- 
 dantly perform in the interim between the necessity of turning the said cocks.” 
 
 But although the above description is a distinct and intelligible one, of the manner 
 of applying steam for raising of water, yet no person, that I have heard of, attempted 
 to erect a machine on these principles until the year 1699; when Captain Savary 
 produced, the 14th of June in that year, a model which was worked before the Royal 
 Society, at their weekly meeting at Gresham College. He afterwards published 
 an account of this machine in the year 1702, in a work entitled ‘* The Miner’s 
 Friend.” 
 
 In Savary’s machine, the steam is used for making a vacuum in a vessel placed near 
 to the water to be raised, and communicating with it by a pipe, which has a-cock or 
 valve adapted toit. This valve or cock being opened when there is a vacuum in the 
 vessel, the atmosphere presses the water into the vessel ; and when this is filled, the 
 yalve or cock is shut; and steam being let into it, this presses on the surface 
 
a 
 . THE STEAM ENGINE. 249 
 
 of the water, and forces it upwards through a pipe adapted to the vessel for this 
 purpose. 
 
 _ The disadvantages attending this method of construction were so great, that Capt. 
 Savary never succeeded further thanin making some engines for the supply of gentlemen’s 
 seats; but he did not succeed for mines, or the supplying of towns with water. This 
 discouragement stopped the progress and improvement of the Steam Engine, till Mr. 
 Newcomen, an ironmonger, and John Ceudley, aglazier at Dartmouth, about the year 
 1712, invented what is called the Lever or Newcomen engine. Inthis machine, the 
 steam is made to act in a cylinder distinct from the pumps, and is used merely for the 
 purpose of making and unmaking a vacuum, in this manner—namely, there is a piston 
 in the cylinder, fitted so nicely to it, that it can slide easily up and down without the - 
 admission of any air, or other fluid, to pass between its edge and the cylinder. The 
 steam isadmitted below the piston, which, being ofa strength equal to the atmosphere, 
 brings it into a state of equilibrium, when the weight of the pump rods and volumes . 
 of water, at the other end of the lever or balance, raises it up; when the piston has 
 got to the top of the cylinder, a jet of cold water is thrown amongst the steam, 
 which condenses it, and forms a partial vacuum. The atmosphere then acting on the 
 upper side of the piston, forces it down, and raises the column of water at the other 
 end of the beam. 
 
 No improvement on this principle took place for above half a century, except in the 
 construction of a variety of contrivances for the purpose of opening and shutting the | 
 different cocks and valves, necessary to admit the steam into the cylinder, the water 
 to condense it, to carry off the condensed steam, to make the piston more air-tight, 
 and in general to improve the various working parts of the engine. 
 
 Machines of this kind have been constructed in a variety of places; particularly 
 in Great Britain, for the purpose of raising water from mines or for supplying towns, 
 and for raising water to turn wheels. One of the largest of this kind is that which 
 was constructed by the late ingenious Mr. Smeaton, for raising water to turn the 
 wheels of the Blast Furnaces at Carron—the cylinder of this engine is 72 inches in 
 diameter, and I believe it is reckoned the most perfect engine that has been con- 
 structed on Newcomen’s principle. But although Mr. Smeaton spent much time 
 in the improvements of these engines, and succeede@ to a very considerable extent, 
 yet the manner of employing the steam in a cylinder where cold water is to be ad- 
 mitted, for the purpose of condensing it at each stroke, and the piston and cylinder 
 
 being exposed to the atmosphere, render it so imperfect that above one half of the 
 
 i 
 t 
 
 power of the steam is lost by this construction. And therefore, even with Mr. Smea- 
 ton’s ingenious improvements, the Steam Engine at that time was but a very imper- 
 fect machine, and by no means applicable to such a variety of purposes as it is now in 
 its improved state. 
 
 The ingenious Mr. James Watt of Glasgow, perceiving the great loss of steam © 
 which was sustained in its use,in Newcomen’s engine, about 1768 made a variety of 
 experiments on this subject, and in 1770 obtained a patent for a new mode of apply- 
 
 ing it; in which the cylinder was made close both at bottom and top, and the rod 
 ' which connected the piston with the lever, was made to work through a collar of hemp 
 
 and tallow, so as to be perfectly air-tight. The atmosphere being thus excluded from 
 the cylinder, both the vacuum is made by the steam, and the piston is moved by it. 
 Also the steam is not condensed by throwing cold water into the cylinder, but it is 
 taken out by au air-pump, and condensed ina separate vessel; and in order to keep 
 the cylinder as hot as possible, it is surrounded with steam, and covered with non-, 
 
 conducting substances. By this construction, the engine has been made to perform at 
 least double the effect with the same quanity of fuel, as the best engines on New- 
 
 comen’s construction. Mr. Watt obtained an extension of his patent right in the 
 year 1775, by an act of parliament, fo: 25 years; and was joined by the ingenious Mr. 
 
250 _MECHANICS. 
 
 Boulton of Soho, near Birmingham; since which, the same principle has still been 
 followed; but the working parts have undergone various modifications, by the joint 
 abilities of these able mechanicians. The principle which was applied to the work- 
 ing of the piston, only one way, that is, by pushing it downwards, as the atmosphere 
 did in Newcomen’s engine, has also been applied to the forcing it up; by which 
 means, engines, where cylinders are of a given diameter, are now made to perform 
 double the effect. This has not only saved great expence in the original construction 
 of the engines, but has enabled them to be applied in cases where immense power 
 has been wanted, and which could not have been performed at all by them on New- 
 comen’s construction. By the same mode of applying the steam, it can now not 
 only be used of the strength of the atmosphere, but as much stronger as necessity or 
 convenience may require ; which is a still further consolidation of the power. The 
 eelerity also with which the condensation of the steam, and the discharging of the 
 condensed steam and water, are performed, enables them to work quicker, and so 
 to be applied to all kinds of mill work, which are used in the numerous manufac- 
 tories of this country. Corn is ground by them, cotton spun, silk twisted, the 
 immense machinery used in the new manufactories are worked, and including every 
 kind of mill work to which water can be applied. They are also used in the various 
 branches of the civil engineer. Thus the water is taken from the foundations of 
 Locks, Bridges, Docks, &c. The piles are driven fur the foundations, as the mortar 
 manufactured for the building of the walls; earth taken from their canals; and docks 
 and works have been of late performed by their means, which could not have been 
 executed without them. 
 
 They are also made so portable for some purposes, that they are even constructed 
 on boats and carriages, to be moved from one place to another; while in others they 
 are made on a large and magnificent scale. Messrs. Boulton and Watt have made 
 them from the power of one, to that of 250horses; and by their late contrivances 
 in the execution of their different parts, they are so manageable, that evena lad may 
 attend and direct their operations; and so regular in their motions, that water itself 
 cannot be more so. 
 
 The quantity of fuel which they consume is comparatively small, to the effect 
 they produce. One bushel of the best Newcastle coal applied to the working of an 
 engine for pumping, will raise about thirty millions of pounds one foot high.—But 
 in these engines, when the steam acts on the piston, both in its ascent and descent, 
 the same quantity of fuel will not produce quite so great an effect, as there is not so 
 much time for performing the condensation, on which account the vacuum is not 
 so complete. 4 
 
 In the application of the steam ‘engine to driving machinery, it is important thata 
 uniform motion should exist. To equalize the variable force communicated by the 
 engine, a large and heavy metal wheel, called a fly wheel, is fixed on the axis 
 turned by the crank which converts the reciprocating motion into a rotatory one, 
 —and this wheel revolves with the axis on which it is fixed. The tendency of 
 this heavy rotating wheel, to retain the velocity which it receives, renders the 
 motion sufficiently uniform for all practical purposes, when the supply of steam 
 from the boiler is nearly uniform, and the resistance to be overcome is also nearly 
 uniform. 
 
 To ensure a uniform velocity, however the load or resistance may be varied, it is 
 necessary soto proportion the supply of steam tothe resistance, that upon the least 
 _change in the velocity the supply of steam may be correspondently raised, so as to 
 
 keep the engine always goingat the same rate. 
 
 _ One of the most remarkable appendages of the steam engine is an apparatus 
 called the governor, invented by Mr. Watt, for effecting this object,—viz., tor regu- 
 lating the supply of steam to the engine. In the pipe which conducts steam from 
 
 ; 
 
- 
 i 
 Ti : 
 ae 
 
 in , 
 
 STEAM ENGINE. 251 
 
 the boiler to the cylinder is placed a thin circular plate, which, when its face is 
 presented towards the length of the pipe, nearly stops it,—and when it is inclined 
 ‘more or less, a greater or less quantity of steam is permitted to pass. This plate 
 or valve is called the throttle valve; and the following account of the mechanism, 
 
 which Mr. Watt contrived for making the engine itself turn the plate exactly and 
 always into the precise position in which it is required to be, will, we have no doubt, 
 be interesting to our readers. 
 
 A (Fig. 55.) is a 
 perpendicular axle, 
 to which a grooved 
 wheel B is attached, 
 and which turns with 
 the shaft in the pi- 
 vots at the top and 
 bottom of it. A strap 
 rolled on the axis of 
 the fly-wheel passes 
 round the groove in 
 the wheel B, as the 
 strap acts ina turning 
 lathe. Thus the ro- 
 tation of the fly wheel 
 and that of the shaft 
 a will always vary inthe same proportion. c and pare two heavy balls at the 
 ends of the rods ¢ and d, which play on an axis fixed on the revolving shaft 
 at £, and extend beyond the axis torr. Connected with these rods by joints 
 at F, F, are two other rods F, R, attached to a broad ring of metal which moves freely 
 up and down the revolving shaft; and to this ring a lever is attached, whose 
 centre is at G; and it is connected by a series of levers with the throttle valve T. 
 When the speed of the fly-wheel becomes considerably increased, the spindle a is 
 wheeled more rapidly round ; and the balls acquiring greater centrifugal force recede 
 from the axis, depress the metal ring which slides on the spindle, and with it the 
 adjoining end of the lever, raising at the same time the opposite end, and thus par- 
 tially closing the throttle valve by means of the connecting apparatus, the supply of 
 steam from the boiler to the cylinder is diminished, and a corresponding retardation 
 of motion takes place in consequence. And the contrary effect is produced when 
 the rotation of the fly wheel is diminished. 
 
 It will thus be perceived, that when, from any alteration of the load or resistance 
 to be overcome, the velocity of the fly wheel becomes increased or diminished, a 
 corrective is supplied immediately and in accurate perfection, by the action of the 
 governor, which has justly been characterised as one of the most elegant and inge- 
 nious of mechanical inventions. 
 
 Another most important mechanical device, for converting the straight in and out 
 motion of the piston rod into a circular motion at the end of the working beam of 
 the engine, merits notice in this place. 
 
 Fig. ti. Mr. Watt conceived two straight rods, as, cp (Fig. 56.) 
 moving on pivots at a and c, their extremities B and p being 
 
 connected by a third rod Bp, by pivots at B and p, on which 
 BD can turn freely. Now the pivots Band p will move in 
 circular arcs, whose centres are at A andc; but the middle. 
 
 point of the connecting rod Bp will move upwards and down- 
 wards in a straight line. 
 
 Fig. be 
 
952° MECHANICS. 
 
 The apparatus which, on this principle, Mr. Watt devised, for accomplishing, in — 
 practice, the object in question, may be thus described. 
 
 The beam moving on its centre c (Fig. 57.), every 
 point in its arm describes the arc of a circle round ¢ 
 asacentre. Let B be the middle of ac, and let 
 DE be a straight rod equal in length to a Bor Be, 
 and playing on the pivot p. The end & of the rod is 
 connected by the pivots at B and £, with the straight 
 barr B. Hence, according to what has just been 
 stated, while the beam and pe revolve round @ 
 and p as centres, F, the middle of Bx, will move 
 up and down in a straight line. 
 
 Again, if a rod ac, equal in length to B EF, be 
 attached to a, the end of the beam, by a pivot on which it moves freely: and if G 
 be connected with © by a rod equal in length to AB, the point eG will move in a 
 straight line parallel to that in which F moves. 
 
 The piston-rod of the steam cylinder is attached to the point G, and that of the 
 air pump to the point r: so that they move in parallel straight lines, the piston-rod 
 having double the stroke of that of the air-pump, being twice the distance from c, 
 the centre of motion. 
 
 BALLOONS, TELEGRAPHS, &c. 
 
 Tue latter part of the last century, among many ingenious mechanical inventions, 
 has produced the two remarkable ones relating to air balloons, and to telegraphs, 
 with other means of distant, quick, or secret intelligence; concerning which a brief 
 account may here be added; and first of Aerostation and Air Balloons. 
 
 The fundamental principles of aerostation have been long and generally known, as 
 well as speculations on the theory of it; but the successful application of them to 
 practice seems to be altogether a modern discovery. These principles chiefly respect 
 the pressure and elasticity of the air, with its specific gravity, and that of the other 
 bodies to be floated in it. Now any body that is specifically, or bulk for bulk, lighter 
 than the atmosphere, is buoyed up by it, and ascends to such height where the air, 
 by always diminishing in its density upward, becomes of the same specific gravity as 
 the rising body; here this body will float, and move along with the wind or current 
 of air, like clouds at that height. This body then is an aerostatic machine, whatever © 
 its form or nature may be; such as an air-balloon, the whole mass of which, including 
 its covering and contents, with the weights annexed to it, is of less weight than the 
 ~ same bulk of air in which it rises. 
 
 We know of no solid bodies however that are Went enough thus to ascend and float 
 in the atmosphere; and therefore recourse must be had to some fluid or aeriform sub- 
 stance. Among these, that which is called inflammable air is the most proper for 
 that purpose: it is very elastic, and is six, eight, or ten times lighter than common 
 air. So that, if a sufficient quantity of that kind of air be inclosed in any thin bag 
 or covering, the weight of the two together will be less than the weight of the same 
 bulk of common air: consequently this compound mass will rise in the atmosphere, 
 till it attain the height at which the atmosphere is of the same specific gravity as it« 
 self; where it will remain or float with the current of air, as long as the inflammable 
 gas does not too much escape through the pores of its covering. And this is an 
 inflammable-air balloon. 
 
 Another way is, to make use of common air rendered lighter, by heating it, instead 
 of the inflammable air. Heat rarefies and expands common air, and consequently 
 lessens its specific gravity. So that, if the air, inclosed in any kind of a bag or cover- 
 
 7 
 
BALLOONS. 253 
 
 ing, be heated, and this dilated, to such a degree that the excess of the weight of 
 an equal volume of common air, above the weight of the heated air, be greater than 
 the weight of the covering and its appendages, the whole compound mass will ascend 
 inthe atmosphere, till it arrive at a height where the atmosphere has the same spe- 
 cific gravity with it; where it will remain till, by the cooling and condensation of 
 the included air, the balloon shall gradually contract, and descend again, unless the 
 heat be:renewed or kept up. And this is a heated-air balloon, which is also called a 
 Montgolfier, after the name of its inventor. 
 Various schemes for rising up in the air, and passing through it, have been devised 
 and attempted, both by the ancients and the moderns, on different principles, and with 
 various success. Of these attempts, some have been on mechanical principles, or by 
 the powers of mechanism; and such, it is conceived, were the instances related of 
 the flying pigeon made by Archytas, also the flying eagle, and the fly by Regiomon- 
 tanus, with many others, both among the ancients and moderns. 
 
 Other projects have been vainly formed, by attaching wings to some part of the 
 human body, to be moved either by the hands or the feet, by means of mechanical 
 powers; so that striking the air with them, after the manner of the wings of a bird, 
 the person might raise himself in the air, and transport himself through it, in imitation 
 of that animal. But these attempts belong rather to that species or principle of 
 motion called artificial flying, than to the subject of aerostation, which is properly 
 the sailing or floating in the air by means of a machine rendered specifically lighter 
 than that element, in imitation of aqueous navigation, or the sailing on the water in 
 a ship or vessel, which is specifically lighter than this element. 
 
 The first rational account to be found on record, for this sort of sailing, is perhaps 
 that of our countryman Roger Bacon, who died in the year 1292. He not only 
 affirms that the art is feasible, but assures us that he himself knew how to make a 
 machine, in which a man sitting might be able to convey himself through the air like 
 a bird: and he farther affirms that there was another person who had tried it with 
 success. ‘The secret it seems consisted in a couple of large thin shells, or hollow 
 globes, of copper, exhausted of air; so that the whole being thus rendered lighter 
 than air, they would support a chair, in which a person might sit. 
 
 Bishop Wilkins too, who died in 1672, in several of his works, makes mentign of 
 similar ideas being entertained by divers persons. ‘‘ It is a pretty notion to this 
 purpose,” says he, (in his Discovery of a New World), mentioned by Albertus de 
 Saxonia, and out of him by Francis Mendoza, “ that the air is in some part of it navi- 
 gable. And that upon this static principle, any brass or iron vessel, suppose a kettle, 
 whose substance is much heavier than that of the water; yet being filled with the 
 lighter air, it will swim upon it, and not sink.” And again, in his Dedalies, he says, 
 “ Scaliger conceives the framing of such volant automata to be very easy. ‘Those 
 ancient motions we thought to be contrived by the force of some included air As 
 if there had been some Jamp or other fire within it, which might preduce such a 
 forcible rarefaction as should give a motion to the whole frame.” From whence it 
 would seem that Bishop Wilkins had some confused notion of such a thing as a 
 heated-air balloon. x 
 
 Again, father Francisco Lana, in his Prodroma, printed in 1670, proposes the same 
 method with that of Roger Bacon, as his own thought. He considered that a hollow 
 vessel, exhausted of air, would weigh less than when filled with that fluid. He 
 also reasoned that, as the capacity of spherical vessels increases much faster than 
 their surface, the former increasing as the cube of the diameter, but the latter only 
 as the square of the same, it is therefore possible to make a spherical vessel of any 
 given matter and thickness, and of such a size as, when emptied of air, it will be 
 lighter than an equal bulk of that air, and consequently that it will ascend in the 
 atmosphere. After stating these principles, father Lana computes that a round 
 
254 MECHANICS. 
 
 vessel of plate brass, 14 feet in diameter, weighing 3 ounces the square foot, will 
 only weigh 1848 ounces; whereas a quantity of air of the same bulk will weigh’ 
 2156 ounces, allowing only one ounce to the cubic foot; so that the globe will not 
 only ascend in the air, but will also carry up a weight of 308 ounces: and by 
 increasing the bulk of the globe, without increasing the thickness of the metal, he 
 adds, a vessel might be made to carry up a much greater weight. 
 
 Such then were the speculations of ingenious men, and the gradual approaches to- 
 wards this art. But one thing more was yet wanting: although in some degree ac- 
 quainted with the weight of any quantity of air, considered as a detached substance, 
 it seems they were not aware of its great elasticity, and the universal pressure of the 
 atmosphere; a pressure by which a globe, of the dimensions above described, and 
 exhausted of its air, would immediately be crushed inwards, for want of the 
 equivalent internal counter pressure, to be sought for in some element, much 
 lighter than common air, and yet nearly of equal pressure or elasticity with it; 
 a property and circumstance attending inflammable gas, and also common air when 
 considerably heated. , 
 
 It is evident then that the-schemes of ingenious men hitherto must have gone no 
 farther than mere speculation; otherwise they could never have recorded fancies 
 which, on the first attempt to be put in practice, must have manifested their own 
 insufficiency, by an immediate failure of success, For, instead of exhausting the 
 vessel of air, it must be filled either with common air heated, or with some other 
 equally elastic but lighter air. So that on the whole it appears, that the art of 
 traversing the atmosphere, is an invention of our own time ; and the whole history 
 of it is comprehended within a very short period. . 
 
 The rarefaction and expansion of air by heat, is a property of it that has been long 
 known, not only to philosophers, but even to the vulgar. By this means it is, that 
 the smoke is continually carried up our chimneys: and the effect of heat upon air 
 is made very sensible by bringing a bladder, only partially filled with air, near a fire; 
 when the air presently expands with the heat, and disteuds the bladder so as almost 
 to burst it. Indeed, so well are the common people acquainted with this effect, 
 that it is the constant practice of those who kick about blown bladders, for foot 
 ballg, to bring them from time to time to the fire, to restore the spring of the air, 
 and the distension of the ball, lost by the continual cooling and waste of that fluid. 
 
 But the great levity, or rather small. weight, of inflammable gas, is a modern. 
 discovery, namely, within the last 70 or 80 years; a discovery chiefly owing to our 
 own countrymen, Mr. Cavendish and Dr, Black, the latter of whom frequently 
 mentioned also the feasibility of inclosing it in a very thin bag, so as that it might — 
 ascend into the atmosphere; an idea which was first put in practice, on a very small 
 scale, by Mr. Cavallo, another ingenious philosopher. 
 
 It was however two brothers, of the name of Montgolfier, near the city of Lyons 
 in France, who, in the year 1782, first exhibited to the world what may properly be 
 called air-balloons, of large dimensions, being silken bags of many feet in diameter. 
 These were on the principle of common air heated, by passing through a fire, made 
 near the orifice or bottom of the balloon. This heated air and the smoke thus 
 ascended straight up into the bag, and gradually distended it, till it became quite 
 full, and so much lighter than the atmosphere, that the balloon rapidly ascended, and 
 carried up other weights with it to very great heights. After attaining its utmost 
 height however, partly by the cooling of the included air, and partly by its escape 
 through the pores of the covering, the balloon gradually descends very slowly, and. 
 comes at length to the ground, after being sometimes carried to great distances by 
 the wind, or currents of air in the atmosphere. 
 
 Other balloons were also soon made by the philosophers in France, and after them 
 in other countries ; namely, by filling the balloon case with inflammable gas; a more 
 
TELEGRAPHS, 255 - 
 
 troublesome and expensive process, but of much better effect ; because, having only 
 to guard against the waste of the fluid through tne pores, but not its cooilng, these 
 balloons continue much longer in the air, sometimes for the space of many hours, 
 enabling the passengers to pass over large tracts of country. On one of these occa- 
 sions, Mr. Blanchard, a noted operator, with a favourable wind, passed ‘over from 
 Dover to Calais, accompanied by another gentleman. 
 
 Many other persons exhibited balloons, of large 
 dimensions, particularly in France and other parts of 
 the continent, with various success. The people of 
 that country have also successfully applied balloons 
 to the examination of the state of the higher regions 
 of the atmosphere; and also in their armies, to dis- 
 cover the dispositions and operations of an enemy’s 
 position and camp. In England they have been less 
 attended to, perhaps owing at first to an unfortunate 
 prejudice, and an idea thrown out, that they could 
 not be turned to any useful purpose in life. 
 
 Balloons have become so common within the last 
 few years, that their appearance must be familiar to 
 all our readers, and we therefore give only the an- 
 nexed cut of Blanchard’s balloon with the parachute. 
 
 TELEGRAPHS. 
 
 A Telegraph is a machine lately brought into use by the French nation, namely in 
 the year 1793; being contrived to communicate words or signals, from one person 
 to another, at a great distance, and in a very short time, a 
 
 The object proposed is, to obtain an intelligible figurative ianguage, to be distin~. 
 guished at a distance, to avoid the obvious delay in the dispatch of orders or informa- 
 tion by messengers. es hye 
 
 On first reflection, we find the practical. modes of such distant communication 
 must be confined to sound and vision, but chiefly the latter. Each of these is in a 
 great degree affected by the state of the atmosphere : as, independent of the wind’s 
 direction, the air is sometimes so far deprived of its elasticity, or whatever other 
 
 "quality the conveyance of sound depends on, that the heaviest ordnance is scarcely 
 heard farther than the shot flies : and, on the other hand, in thick hazy weather, the 
 largest objects become quite obscured at a short distance. No instrument therefore, 
 designed for the purpose, can be perfect. We can only endeavour to diminish these 
 defects as much as may be. ae 
 
 Some kind of distant signals must have been employed from the earliest antiquity. 
 It seems the Romans had a method in their walled cities, either by a hollow formed 
 
 in the masonry, or by tubes affixed to it, so to confine and augment sound, as to 
 
 | convey information to any part they wished ; and in lofty houses it is now sometimes 
 the custom to have a pipe, by way of speaking trumpet, to give orders from the 
 | upper apartments to the lower: by this mode of confining sound, its effect may be 
 carried ‘to a very great distance; but beyond a certain extent, the sound, losing 
 articulation, would only convey alarm, and not give directions. - 
 
 Every city among the ancients had its watch-towers ; and the castra stativa of the 
 | Romans had always some spot, elevated either by art or nature, from whence signals 
 _ were given to the troops cantoned or foraging in the neighbourhood. But they had 
 
 probably not arrived at greater refinement than that, on seeing a certain signal, 
 they were immediately to repair to their appointed stations. 
 
 A beacon, or bonfire made of the first inflammable materials that offered, as the 
 
256. - MECHANICS. 
 
 Sa 
 
 most obvious, is perhaps the most ancient mode of general alarm, and by being pre- 
 
 viously concerted, the number or point where the fires appeared might have its par- 
 
 ticular intelligence affixed. ‘The same observations may be referred to the throw- 
 ing up of rockets, whose number or the point from whence thrown, may have its affixed 
 signification. 
 
 Flags or ensigns, with their various devices, are of earliest invention, especially at 
 sea; where, from the first idea, which was probably that of a vane to shew the direc- 
 tion of the wind, they have been long adopted as the distinguishing mark of nations, 
 and are now so neatly combined by the ingenuity of a great naval commander, that 
 by his system every requisite order and question is received and answered by the most 
 distant ships of a fleet. 
 
 To the adopting this, or a similar mode, in land service, the following are objec- 
 tions: that in the latter case, the variety of matter necessary to be conveyed is 
 so exceedingly great, that the combinations would become too complicated. And if 
 the person for whom the information is intended should be in the direction of the 
 wind, the flag would then present a straight line only, and at a little distance be 
 invisible. ‘The Romans were so well aware of this inconvenience of flags, that many 
 of their standards were solid; and the name manipulus denotes the rudest of their 
 modes, which was a truss of hay fixed on a pole. . 
 
 The principle of water always keeping its own level has been suggested, as a pos- 
 sible mode of conveying intelligence, by an ingenious gentleman, and put in practice 
 on a small scale, with a very pleasing effect. As for example, suppose a leaden pipe 
 to reach between too distant places, and to have a perpendicular tube connected to 
 each extremity. Then, if the pipe be constantly filled with water to a certain height, 
 it will always rise to its level on the opposite end; and if but one inch of water be 
 added at one extremity, it will almost instantly produce a similar elevation in the 
 tube at the other end; so that by corresponding letters being adapted to the vertical 
 tubes, at different heights, intelligence may be quickly conveyed. But this method 
 is liable to such objections, that it is not likely it can ever be adopted to facilitate 
 the object of very distant communications. 
 
 Fullas many, if not greater objections, will perhaps operate against every mode of 
 electricity being used as the vehicle of information. And the requisite magnitude of 
 painted or illuminated letters, offers an insurmountable obstacle; besides in them one 
 object would be lost, that of the language being figurative. 
 
 Another idea is perfectly numerical, which is to raise and depress a flag or curtain 
 a certain number of times for each letter, according toa previously concerted system; 
 as, suppose one elevation to mean a, two to meanB, and so on through the alphabet. 
 But in this case, the least inaccuracy in giving or noting the number, changes the 
 letters; and besides the last letters of the alphabet would be a tedious operation. 
 
 Another method that has been proposed, is an ingenious combination of the mag- 
 netical experiment of Comus, and the telescopic micrometer. Butas this is only an 
 imperfect idea of Mr. Garnet’s very ingenious machine, described below,.no farther 
 notice need be taken of it here. 
 
 Mr. Garnet’s contrivance is merely a bar or plank, turning on a centre like the 
 arm of a windmill; which being moved into any position, an observer or correspondent 
 at a distance turns the tube of a telescope round its axis, into the same position, by 
 bringing a fixed wire within it to coincide with, or become parallel to, the bar, which 
 isa thing extremely easy todo. The centre of motion of the bar has a small circle 
 fixed on it, with letters and figures around the circumference, and a moveable index 
 turning together with the bar, pointing to any letter or mark the operator may wish 
 to set the bar to, or to communicate to the observer. The eye end of the telescope 
 has a like index and circle fixed on the outside of it with the corresponding letters or 
 other marks, The consequence is obvious; the telescope being turned round its axis, 
 
j 
 
 1 
 | 
 J 
 
 TELEGRAPHS. 957 
 
 till its wire cover, or become parallel to the bar, the index of the former necessarily 
 points out the same letter or mark on its circle as that of the latter, and the communi- 
 ‘eation of sentiment is immediate and perfect. The use of this machine :s so easy, 
 
 - that we have seen it put into the hands of two common labouring men, who had never 
 
 seen it before, when they have immediately held a quick and distant conversation 
 together. 
 
 Fig. 59. represents the principal parts of this telescope: a Bp & is the telegraph or 
 bar, having on the centre of gravity c, about which it turus, a fixed pin, going through 
 a hole or socket in the firm upright post G, and on the op- 
 posite side is fixedan index cr. Concentric toc, on the same 
 post, is fixeda brass circle, of 6 or 8 inches diameter, divided 
 into 48 equal parts, 24 of which represent the letters of 
 the alphabet, and in the other 24, between the letters 
 are numbers. So that the index, by means of the arm 
 AB, may be set or moved to any letter or number. The 
 length of the arm or bar should be 24 or 3 feet for every mile 
 of distance. ‘Two revolving lamps of different colours, sus- 
 pended occasionally at a and Bb, the ends of the arm, would 
 serve equally at night. 
 
 Let ss (Fig. 60.) represent a transverse section of the outward tube of a telescope, 
 and x x the like section of the sliding or adjusting tube, on which is fixed an index 
 11. On the part of the outward tube next to the observer, is fixed a circle of letters 
 and numbers, similarly divided and situated as the former circle in Fig. 59; so that the 
 index 1 1, by means of the sliding or adjusting tube, may be 
 turned to any other letter or number. Now there being a hair, 
 or fine silver wire, fg, fixed in the focus of the eye-glass ; when 
 the arm a B of the telegraph is viewed at a distance through the 
 _telescope, the hair may be turned, by means of the sliding tube, 
 to the same position as the arm a B; then the index11 (Fig. 60.) 
 will point to the same letter or number on its own circle, as the 
 index 1 (Fig 59.) points to on the telegraphic circle. 
 
 If, instead of using the letters and numbers to form words at length, they be 
 used as signals, three motions of the arm will give a hundred thousand different 
 signals. 
 
 But a telegraph, combined witha telescope, it seems was originally the invention of 
 M. Amontons, an ingenious French philosopher, about the middle of the 17th cen- 
 tury; when he pointed out a method to acquaint people at a great distance, and 
 in a very little time, with whatever we please. This method was as follows: Let 
 persons be placed in several stations, at such distances from each other, that, by 
 the help of a telescope, a man in oue station may see a signal made by the next 
 before him; this person immediately repeats the same signal to the third man; and 
 this again to a fourth, and so on through all the stations, to the last. 
 
 This, wiih considerable improvements, it seems has lately been brought into use 
 
 by the French, and called a Telegraph. It is said they have availed themselves of 
 this contrivance to good purpose, in the late war; which has induced the English — 
 also to employ a like instrument, in a different form. 
 
 The new invented telegraphic language of signals, says a French author, is an art= 
 ful contrivance to transmit thoughts, in a peculiar way, from one distance to another, 
 
 _ by means of machines, which are placed at different distances, of from 12 to 15 miles 
 each, so that the expression reaches a very distant place in the space of a few 
 minutes. The only thing which can interrupt their effects is, if the weather be so 
 
 bad and turbid, that the objects and signals cannot be distinguished. By this inven- 
 
 tion, remoteness and distance almost disappear; .and all the communications of 
 Ss 
 
258 MECHANICS, 
 
 correspondence are effected with the rapidity of the twinkling of an eye. The 
 
 greatest advantage which can be derived from this correspondence, is that, if we - 
 
 choose, its object shall be known to certain individuals only, or to one individual 
 alone, or to the extremities of any distance. 
 Fig. 61. represents the form of the French Telegraph. 
 Fig. 61. A Ais a beam or mast of wood, placed upright on arising 
 ground, and is 15 or 16 feet high. BB is a beam or 
 balance moving on the centre a A. This balance beam 
 may be placed vertically, or horizontally, or any how in- 
 clined, by means of strong cords, which are fixed to the 
 wheel D, on the edge of which is a double groove, to 
 receive the twocords. This balance is 11 or 12 feet long, 
 and 9 inches broad, having at the end two bars c o, 
 which likewise turn on the angles by means of four other 
 cords passing through the axis of the main balance. The 
 pieces c are each about three feet long, and may be 
 turned and placed either to the right or left, straight or square with the balance beam, 
 By means of these three, the combination of movements is said to be very extensive, 
 remarkably simple, and easy to perform. Below is a small wooden hut, in which a 
 person is employed to attend the movements of the machine. In the mountain near- 
 est to this, another person is to repeat these movements, and athird to write them 
 down. The signs are sometimes made in words, and sometimes in letters; when in 
 words a small flag is hoisted; and as the alphabet may be changed at pleasure, it is 
 only the corresponding person who knows the meaning of the signs. The alphabet, 
 as well as the numbers to 10, are exhibited in the middle of Fig. 61, annexed to the 
 different forms and positions into which the bars of the machine may be put. 
 Many improvements and additional coutrivances have been since made in England, 
 The following one is by the Rev. J.Gamble. The principle of it is simply that of a 
 Venetian window-blind, or rather what are called the lever 
 
 Fig. 62. boards of a brewhouse, which when horizontal, present so small 
 
 A, a a surface to the distant observer, as to be lost to his view, but 
 
 e ih are capable of being in an instant changed into a screen of a 
 Bue : : : = 
 
 et magnitude adapted to the required distance of vision. arBDFC 
 
 cod (Fig. 62), isa firm upright frame, supporting nine lever boards 
 
 fb ia: working on centres in BEand D F, and opening in three divisions 
 
 byironrods, Andabcd,efgh, are two lesser frames, fixed to 
 
 by iron rods, in the same manner as the others. If all these 
 rods be brought so near the ground, as to be in the management 
 of the operator, he will then have five keys to play on. Now 
 as each of the handles ik1 mn commands three lever boards, by raising any one of 
 them, and fixing it in its place by a catch or hook, it will give a different ap- 
 pearance in the machine ; and by the proper variation of these five movements, there 
 will be more than 25 of what may be called mutations, in each of which the machine 
 exhibits a different appearance, and to which any letter or figure may be annexed at 
 pleasure. 
 
 Should it be required to give intelligence in more than one direction, the whole 
 machine may easily be made to turn to different points, on a strong centre, after the 
 
 manner of a single post windmill. To use this machine by night, another frame must. 
 
 be connected With the back part of the Telegraph, for raising five lamps, of different 
 colours, behind the openings of the lever boards; these lamps by night answering for 
 the openings by day. 
 
 Fig. 63, represents a front view of the latest form of the Telegraph, now em- 
 
 x 
 fin 
 cag ae 
 
 the great one, having also three lever boards in each, and moving- 
 
TELEGRAPHS., 259 
 
 ployed by the English government, by which a signal is conveyed between 
 
 London and Deal, being 72 miles, by repetition, in 
 Fig. 63. three minutes. The corresponding boards forming a 
 
 ee a scale for the alphabet, and for numbering, is annexed 
 ean lj mcd in the engraving. 
 
 fe ta We shall limit ourselves to what has been here said 
 iy 
 
 respecting those machines, which have acquired the 
 greatest celebrity; but we shall point out a few books 
 which those who are fond of machines, and who wish 
 to instruct themselves by example, may consult for 
 that purpose. The first of these, which we shall 
 mention, is the Theatrum Mechanicum of Leupold, in 
 several volumes folio, the last of which appeared in 
 1725. This isa curious work, but the author’s theory 
 is not always well founded; for he seems not be en- 
 
 alll 
 
 iia eh 
 
 ye 
 
 l 
 
 B Sab 
 eat cabin ee eae tirely convinced of the impossibility of the perpetual 
 ae foeaaa motion. The next is the Thédtre des Machines of 
 
 James Besson, in Italian and French. And to these 
 
 _we shall add, Bockler’s work, in Latin; that of Ramelli in Italian and French, which 
 
 is rare, and in great request. The Cabinet des Machines of de Servieres, 4to, 
 Paris 1733, is one of the most curious works of this kind, on account of the great 
 number of machines described in it, and which were invented by the author. Some 
 of them are very ingenious, and the principles on which they are constructed de- 
 served to have been better explained; but, in general, they are more curious than 
 useful. 
 
 The description of the method in which the Chevalier Carlo Fontana raised the 
 famous obelisk, now before St. Peter’s at Rome, is likewise a work worthy of a place 
 in the library of every person fond of mechanics, M. Loriot, who has a collection of 
 machines, the invention of which displays great ingenuity, has promised to publish 
 some day a description of them. This, in our opinion, would be a curious and useful 
 work ; for the most of his machines bear the stamp of genius. We have seen one, 
 invented by him for driving piles, which acts by a motion always in the same di- 
 rection, without being obliged to stop or to retrograde, iu order to raise up again 
 the weight. Nothing, in our opinion, can be more ingenious than the method in which, 
 
 after the fall of the weight or rammer, the hook that serves to raise it again, 
 lays hold of it, and by which the cable lengthens itself in order to reach lower 
 and lower, in proportion as the pile sinks deeper. If this mode of construction 
 
 be compared with those hitherto employed, no one can refuse to give it the pre- 
 ference, 
 
 There is also the Collection, in 6 vols. 4to, of Machines and Inventions approved 
 by the Royal Academy of Sciences, containing the engravings and descriptions of 
 
 _a great multitude of machines. In English too we have Desaguliers’s Course of 
 
 } 
 . 
 
 > 
 y 
 
 i 
 
 Experimental Philosophy, in 2 vols, 4to.; also Emerson’s Mechanics, both con- 
 taining the figures and descriptions of many curious and useful machines. Besides 
 
 _ some others of less note. 
 
 28 
 
260 
 
 TABLE OF SPECIFIC GRAVITIES. 
 
 A TABLE 
 
 OF THE SPECIFIC GRAVITIES OF DIFFERENT BODIES, THAT OF RAIN 
 OR DISTILLED WATER BEING SUPPOSED 1000. 
 
 METALS. 
 Gold. 
 Spec, Grav. 
 Pure gold of 24 carats, melted The same wire-drawn <+ee+s 
 but not hammered °-++«+--- 19258 Brass, not hammered --++-+- 
 The same hammered :+++++ 19362 The same wire-drawn -++-«« 
 Gold, of the Parisian standard, Common cast brass :+-++++e: 
 22 carats fine, not hammered* 17486 
 The same hammered 17589 Iron and Steel, 
 Gold of the standard of French Cast iron -++ecesseeere nes 
 coin, 2122 carats fine, not Bar iron, either hardened or 
 hammered: 2e cee. Stee a os 17402 not -+-+-- beeceecesceces 
 The same coined +-++--+> 17647 Steel, neither tempered nor 
 Gold of the French trinket hardened «-+++-++eee- soee 
 standard, 20 carats fine, not | Steel hardened under the ham- 
 hammered «+++ --«cecsees 15709 mer, but not tempered -+-- 
 The same hammered:see-+++ 15775 Steel tempered and hardened 
 Steel tempered and not har- 
 Silver. Gened ~.+esss- 0s erecces 
 Pureor virgin silver, 12 deniers 
 fine, not hammered ---+-+- 10474 Other Metals. 
 The same hammered ------ 10511 Pure tin from Cornwall, melted 
 Silver of the Paris standard, and not hardened «---++«- A 
 11 deniers 10 grains fine, not The same hardened----+e«- 
 hammered ¢ ---+~+++ee+s 0175 Malacca tin, not hardened .. 
 The same hammered -+-+++++ 10377 The same hardened ----«see 
 Silver, standard of the French Molten lead -.--+-sesecees 
 coin 10 deniers 21 grains Molten zinc -+s+e-see- woes 
 fine, not hammered --+-+ 10048 Molten bismuth »+seecereeees 
 The same coined --++sceece 10408 Molten cobalt--+e2+ee..eee e 
 . Molten arsenic +-++e+es-veee 
 Platina. Molten nikel +2 cecessscvues 
 Crude platina, in grains ---+» 15602 Molten antimony +e+++s-e- 
 Purified platina, not hammered 19500 Crude antimony -++---+-+ee. 
 The same hammered ------ 20337 Glass of antimony --+++--++-+ 
 The same drawn into wire-- 213042 Molybdena -++--+++> AEE ADE 
 The same rolled ---+---+e6. 22069 Tungsten vate eae atelevere 
 Mereury -««.sscceccccs cerefats 
 Copper and Brass, nih ae 
 Copper, not hammered ----+» 7788 
 
 White oriental diamond :--- 
 Rose coloured ditto 
 
 3521 
 3531 
 
 * This is the same as sterling gold. 
 
 PRECIOUS STONES. 
 
 Oriental ruby a suererert staraieletetate 
 Spinell ditto ce eeseevnecece ee 
 
 Spec. Gray. 
 8879 
 
 3760 
 
 t This is 10 gre. finer than sterling. 
 
TABLE OF SPECIFIC GRAVITIES. 
 
 Spec. Grav. 
 
 Ballas ditto -+++++e+eerees 3646 
 Brasilian ditto se+ee-sseeeeee 8531 
 Oriental topaz’ ......cceccce 4011 
 Saxon ditto -+-++++.- eoese § §=3564 
 Oriental sapphire ----.--++ - 3994 
 Brasilian ditto +++ seee-+++ 3131 
 Girasol se«scecscevscecece 4000 
 Jargon of Ceylon «+--+: sooe §=64416 
 Hyacinth-+++-e--«- SC . 38687 
 
 Vermilion o+.-s.e+% gulatderne 
 Bohemian garnet ++++++-+ $ 
 Syrian ditto ---++e-+-++e-+-- 
 Volcanic ditto with 24 sides 
 Peruvian emerald ---.+e+ ese 
 Chrysolite of the jewellers -- 
 Brasilian ditto +++. + oe eeeee 
 Beryl or oriental aqua-marine 
 Occidental ditto:+++++++ +e+ 
 
 SILICEOUS STONES. 
 
 Pure rock crystal of Madagascar 2653 
 
 Brown Jasper .cccessescocce 
 
 mrito-Of Enrope ......ceece. 2600 Yellow ditto wscccccsseceee 
 Crystallized quartz -.-.. Age Gute Violet ditto’ ........ AP sin: 
 Oriental agate ....6e...e cee 2590 Grey ditto ...:.. eoneee 
 BEHGG. OLX. oe cn cc's ne ayes 2638 Black prismatic hexaedral schorl 
 Transparent calcedony ...... 2664 } Black amorphus.schorl, called 
 BEMCLIAN Tac cls setae oe eden > 8 2614 antique basaltes .....+s0. 
 BALCONY X - sc csesccccscsencne, 2000 Paving stone ...cccreccooes 
 BUSTED ase inra.sain era als, wie 8 "er ok Grind-stone -...ccccceecsseee 
 SUVS CUI e.6 cence ais.» «68's 2664 Cutler’s/stone 2 ..s.ceen: seas 
 YS ATTLESS 6G OS Oe ire ears 2950 Mill-stone...secssccccecsece 
 "Green dittd «oes. eovens ee 2966 White flint ...csccccccccecs 
 Red Jasper ....ssccccceceee 2661 Blackish ditto ....e0-cecvece 
 VARIOUS STONES, &c. 
 Opake green Italian serpentine 2430 Violet uot. asasseddien aces 
 Coarse Briancon chalk ...... 2727 Red porphyry ..... sae Relaene x 
 PSpanish chalk .......... biweN Zio Red Egyptian granite ........ 
 MMREEOVY AIC | no > 5 sin, oy cGeis).s 2792 Pamice stoned ss 0.00 0ce0 3% ola 
 Common schist or slate ...ee- 2672 Obsidian stone ............6- 
 OE ae 2854 Basaltes from the Giant’s Cause- 
 White razor hone .......++ 2876 Wel Vita ca elapse ye baleen ee 
 Black and white ditto ...... 3131 smouch=stOne@. oss ccc cee cee ae 
 ‘Icelandic crystal ...-.+.00e0+ 2715 Bottle glass ........ Zislaldtetetete 
 Pyramidal calcareous spar .... 2730 GACT C1ARS Las, «5 dane cinslne alot 
 _ Oriental or white antique ala- VTEC PAGS 51a oa; ora esti ece seatnents 
 ASCOT (he o.0.¢.0.- 00,0,0,6,0,3,0:9 #1 oN, SEE Tieith crystals jel nacsea enn 'ees 
 Green Campanian marble .... 2742 eMiin Colas Ons om s,01 ofose,nsereye steele 
 BEEUPOILED <y0's'c oe esa e'a's Pe eee Sevres porcelain es 6.55 . oma 
 White Casara marble .... ... 2717 China ditto ..... #600 stately ete 
 White Parian marble ....... - 2838 Nati ve sulphur ec os (3s, datee as 
 Ponderous spar .«...- secese 4430 belted UUEO claus; nivssarns, sbeteqeieds 
 White fluor ..,..-...sesee 2. 3156 PROSpROLUS ns. can cvee 30 aes 
 med ditto ...... osaccecreses. S19) Hard peat issn 5 wejeisssins saneee 
 BECO vein a 5b id odinis «nino, OOS Mmberpris.. ince annlcete Saas 
 Biue ditto -..... Datwieds stants 3169 Yellow transparent amber .... 
 LIQUORS. 
 Distilled water ........ «-.. 1000 Rain water 
 
 4230 
 
4 
 
 es 
 
 ae 
 
 262 TABLE OF SPECIFIC GRAVITIES. 
 Spec. Grav. Spec. Grav. 
 
 Saarwater *. scvenshocncse eeecthuce Distilled acetous acid ........ 1010 
 Burgundy wine ..se...e. eee 992 Acetic ditto ...ccccvccessese 1063 
 Malmsey Madeira..... coccewe 1088 Formic ditto ...cce.sseee a @),) Dom 
 Cyder  .-oeese Seclosscvcese a LOIS Solution of caustic ammonia, or 
 Red Deer vices wacs.e s a Moan og! 1034 volatile alkali fluor ........ 897 
 White ditto vcthic atc... we enews Essential oil of turpentine.... 870 
 Highly rectified alcohol ...... 829 Liquid turpentine ........-. 991 
 Common spirit of wine ...... 837 Volatile oil of lavender ...... 894 
 Sulphuric ether ‘..eee..-.-. 739 Volatile oil of cloves ........ 1036 
 Nitric ditto .....+. ea terete 909 Volatile oil of cinnamon 1044 
 Miuriatic ditto fear eans ees cece ua Fou Oil-of} olives 205 sh ose sca cece en 
 A Colic CittOwawes cess cb ene cas OOO Oil of sweet almonds ........ 917 
 Highly concentrated sulphuric Linseed 011.42. c.0+ ss eae Nos 940 
 
 BCL Ws wee aise sist once ctashe es 2125 Whale ollv.ses% vsscuecleus 923 
 Common sulphuric acid ...... 1841 Woman’s milk .........- soem 1m 
 Highly concentrated nitric acid 1580 Cow's milk \2s. an ase eeu - 1082 
 Common nitric acid.....-.... 1272 Mare’s milk ...... cc ccce 1035 
 IVLOTIAGIC CIC, “x's eis o.csianas aoa 1194 ASS NUK a tates op eae tar ee - 1036 
 TEMIOUIG UICLO.s o.0'¢ one ste eee ie one - 1500 Goat’s milk ....... cies Ciera 1034 
 Red-acetous ditf{o ajcless. 1025 Exwemilk/ ss Jia veeceee Loe 
 White acetous ditto .....-.. 1014 
 
 RESINS AND GUMS. 
 Common yellow resin .....+.. 1073 Gumi arabic “fu... shy eee - 1452 
 Mastic’... sets ces wats gehen y 1074 UV ragacauth Gat... Pe Pi oe I13IG5 
 SUOTAX |. eals\ seat ss Seca 1110 Terra Japonica... s.:..eeeese 1398 
 Opake.copal wes e¢s see ueerlos 1140 Socatrine aloes.......2-0--.. 1380 
 Madagascar ditto «..........- 1060 Opium ss cecsecerccces eoeee 1337 
 Chinese ditto *..6.6.20ccedse) 1063 Indigo » «deus deuetiee eae 769 
 Blemit woes des oc Cette ee 1018 Yellow wax, sssssssessteten de Gam 
 Labdanumil).. 954 ..ctns 2 eee. 1186 White ditto, o0ss.00s sess ae ore 
 Dragon's blooda. isa eee! 1205 Spermaceti «cecscescoovess O40 
 Gumilaces.) see eee «1189 Beef fat. sicvessicsvssber oe (925 
 Guin ‘elastic, ....5546 48 bee tea UN Oo4 Veal fat* -...0% os eeetwaans 934 
 Caniphora, 3 caacak ees uui 989 Mutton fat ....... o'yece Be eclde 2am 
 Gum ammoniac .......e0- 2, 1207 Tallow: v's <'0'cstoawte cet e blot . 9428 
 Gand bogey ts lereelortis's-ole's sind 1292 Hop's fat 020s Seeiee tae eee 937 
 NE YUL E iiesoscrcswlees torenctate erate HOt -. 1860 Tard :Fe2i.‘ektigte Bateanttn. secs 948 
 Galbanum ...... erasers SORES 1212 Butter ...... wriviclac'eslsios 942 
 Assafoetida ..... een k Sheu ee 1828 
 woops. 
 
 Heart of oak, 60 years old 1170 Willow ss sicoo. seat sey oeeees | em 
 Cork >.Syaureaienie eh cic eee 240 Male fir 250%.'.shc0re te ee ee 550 
 Elm plank ...... oro ata etal bale eave 671 Female gitto > <3... he. coos 498 
 Ash ditto. Warren eee wie ne eae 845 Pontarteasts siete 'si>!s 70 ‘aie 07s 'aseleiee a 
 Beech j-i5) see oc lsusre sieht s Ant = 5)" White Spanish ditto ........ 529 
 Alder 00 7ecvsins si pig's Oe SOO Apple tree .....+ ied 793 
 WU AIIUL = oer tse encenee, eri L Pear tree 2. asics see 85 cs 661 
 
 * Sea water differs in weight, according to the climate. It is heavier in the torrid zone, and at 
 a distance from the coasts, than in the northern seas and near land. 
 
 tie 
 
TABLE OF WEIGHTS. 
 
 Spec. Grav 
 Quince tree ......... iMveven vf00 
 I Sha cue ete ote! oferta toon semen 944 
 Plum tree ..)...'s.. ag vodictotl ols 785 
 Cherry thee v. Sow cecccse eens 715 
 RE TT OG or. chile et nlite 3 > 600 
 PMTCT UOXCm ie wp bles 8 be dee oe 912 
 mutch ditto. ..+...s WAP AL ED 1328 
 Mptch yew ..-.....06 Serie. IL TOO 
 
 263 
 
 Spec. Gray. 
 Spanish ditto’ J0i..0000% cree’ 807 
 Spanish cyprus ...... coseee 644 
 American ‘cedar’ J22i....0- 6. 561 
 Spanish Mulberry tree ...... 897 
 Pomegranate tree .......... 1354 
 ASN V Ita ies oe pace eta es al 1333 
 Orange tre@ ose. scencsesees 705 
 
 Note.—We may here observe, that the numbers in the above table express nearly 
 the absolute-weight of an English cubic foot, of each substance, in averdupois 
 
 ounces. 
 
 TABLE OF WEIGHTS, 
 
 BOTH ANCIENT AND MODERN, AS COMPARED WITH THE ENGLISH TROY POUND, 
 WHICH CONTaINs 12 ouNCczES, oR 5760 GRAINS. 
 
 As we gave, at the end of that part which relates to Geometry, a comparative 
 table of the principal longitudinal measures, we think it our duty to give here a 
 similar table of the ancient Hebrew, Greek, and Roman weights; and also of the 
 modern weights of different countries, particularly in Europe, as compared with the 
 
 English troy pound. 
 
 ANCIENT WEIGHTS. 
 Hebrew Weighis. 
 
 Grs. Troy. lib. oz. dwt. gr. 
 The obolus called gerah .........+.00. 1066 .. 0 O O 10°66 
 Half shekel or beka ......... rite eee 103-377 YO Os 4a ieae 
 SUS 6 See be mt tate ee ate 206-74 .. O 0 8 1474 
 MUREIAN POINI BITCH I “c sSa'c w's'So'deie skp de ce wee 1245367... | QIN IS O16? 
 Mealent OF CicaY’,.4 4's. os Hees on aces 2 OeeboacG 108 1 5 36 
 
 Attic Greek Weights.* 
 
 Grs. Troy. lib. oz. dwt. gr. 
 MmAlCUS “ects ss. RT Rac ue ake ate ene ot oe "82 0 0 0 O82 
 URIS MEE PON onc Nx hie. e: sre xia ain ieee cieretats 8°20 0 0 0 820 
 DoreCHMGA sets theses oteis Maes Boe tes SEBO ee nO OSS ag 
 PHULACHIGA e's a's v's a ce els HAs An 103°76) Oe 0 0 SENT 78 
 Petradrachma °:....23 22% etiam teaae tets 207706! 4 OF O28 115 OG 
 feesserimina Of (5 drachms)...-cccassse. SOOl77 .. O 8°:2> S77 
 Greater mina of 100 drachms ..e-...2+- 5189703 .. 01016 5:03 
 Lesser Talent of 60 lesser mine .eccoce 23300620 .. 40 6 9 10:20 
 Greater talent of 60 greater mine ...... 3113418 .. 54 O 12 13°8 
 
 Roman Weights. 
 
 EETiG COvIOUAITG th ctsls ¢ o's" bb. ues eo 
 Ounce, equal to 8 denarii Peeece 
 
 Grs. Troy. lib. oz. dwt. pr. 
 51°89 0 0 2 389 
 415°12 O20 17.712 
 
 * It may be proper hire to observe, that these weights were at the same time money. 
 
264 TABLE OF WEIGHTS, 
 Grs. Troy. lib. oz. dwt. grs. 
 As or pound, equal to 12 ounces ...... 4981-44 .. 010 7 13°44 - 
 Another pound of 10 ounces .......... 415)°2 oe... 0, 8 12 288 
 The lesser talent srcccceseesscscsceee 233506'20 .. 40 6 9 10°20 
 The greater talent .... vececsssceieses 3113418... 54 0 12 138 
 
 The above tables are taken from a work by M. Christiani, entitled, ‘‘ Delle Misure ~ 
 d’ogni genere, antiche é moderne,” &c.; printed in quarto, at Venice, in the year 
 1760. As this is an obscure subject, and as some difference prevails among the 
 learned in regard to the value of the ancient weights, the translator has added the 
 following tables from Arbuthnot, in order to render this article more complete. 
 
 Jewish weights reduced to English Troy weight. 
 lib. oz. dwt. gr. 
 
 Thevehekclee tr ee, Caen nce cae socesees 0 0 9 2 
 Maneh-ee+sececece soe ececs vane eeeeesee eccees 9 3.6 102 
 Talenteccccccccsscces er eccescccces eeoees 113 10 1 102 
 
 The most ancient Grecian weights, reduced to English Troy weight. 
 Drachma cesses cevepenscosscscsencs ane oe QO OO: 6 32 
 IVET Tad «latin e's ai0ie we! blelsieié s\ee\celjs\s1 01 e'/s sfels/sls ee clesel Lem) 444 
 Talent ccvceccccccccerecccsnccssesvere e*e G6 O 12 543 
 
 Less ancient Grecian and Roman weights reduced to English Troy weight. 
 Lentes eecsccevrecevessccasaressecne ereeseoe OF DO O 083 
 Siliquze sereee. Wiata stele a kia 5 We be eea taal or aia 0°00, .345 
 Obolus + +++ scree vesercesesereneesesers 0 0.0 92 
 Scriptulum-+ +--+. Hes: RE TR RA Ce Ri 00 018% 
 CAG Ninaea he «ee He eae eee - 00 2 62 
 
 SOx tls eo aCoe ies cee eee seseeseee 0 0 8 08 
 Sicilicus eves ccecccceccee re ceersreccce eo» QO QO 4 13 2 
 Duella «eccccsccceresnceccerercereacres «> 0 0 6 1 3 
 Uncia eereccere eoceccove ecoces cooesse () OI8 5 4 
 Tisiraaecieriesis Gaitute pees seseereereressees 0-10 18 13 § 
 
 The Roman ounce is the English averdupois ounce, which they divided into 7 
 denarii, as well as 8 drachms: and since they reckoned their denarius equal to the 
 Attic drachm, this will make the Attic weights } heavier than the correspondent 
 Roman weights. ; 
 
 We shall here observe, that the Greeks divided their obolus into chalci and lepta: 
 thus, Diodorus and Suidas divide the obolus into 6 chalci, and every chalcus into 7 
 
 lepta: others divided the obolus into 8 chalci, and every chalcus into 8 lepta, or 
 minuta. ; ; 
 
 The greater Attic weights, reduced to English Troy weight. 
 . Ib. oz. dwt. grs. 
 Libra or pound so cccorecee Gale elsiriniate etavers «-« 01018 138 
 Common Attic mina°:<::«-« Cie aushureia Phare ess @° 3 6 os (Ob i | 7 162 
 Another mina used in medicine +++++++++++e+s 1 211 102 
 ‘he common Attic talent ¢++es++++se+ee-s++ 56 11 0 17h 
 
 Itis here to be remarked, that there was another Attic talent, said by some to con- 
 sist of 80, and by others of 100 mine. Every mina contains ]00 drachme, and every 
 
 \ 
 
bal 
 hs 
 
 } 
 
 ; 
 
 TABLE OF WEIGHTS. 
 
 265 
 
 “talent 60 mine ; but the talents differ in weight, according to the different standard 
 of the Pichras and minz of which they are composed. The value of different mins 
 and talents, in English Troy weight, is exhibited in the following tables: 
 
 Table of different Mine. 
 
 lib. oz. dwt. grs. 
 
 Egyptian mina......... ecco eeene en esccene Poe 6 22% 
 Antiochic..... saa vevate ees be Seta twa see LS 622% 
 Ptolemaic of Cleopatra .....ssecececsvecses 1 6 14 1632 
 Alexandrian of Dioscorides ....... cssosceee 1 816 7H 
 Table of different Talents. 
 Beyptian qos ese Be Le. CE eoceeee 86 816 8 
 PL TIBICCIC) Ware se «RNa as uate «as sseoeeseee 86 816 8 
 BP LOleMalG. OL CICODALIA ican cad sscnececdarcue ..90.11 Lit 0 
 Alexandrian ..... Ves ee slows 50 ais concccccsoe 104 019 14 
 Of the Islands ..... Wares sc duets « caeneees lol. Pe eaele 
 Antiochian, <1... ES OAM RIO RICE > 390 3 13 11 
 
 Modern Weights of the principal countries in the world, and particularly in Europe. 
 
 Aleppo, the pound, called rotolo .. 
 Alexandria in ee AHO Toa aie slesiae ala 
 
 aeeece 
 
 PV IACAL UE wale ue cates Aaa ree 6 loess one c's 
 Amsterdam ....... Shaile « niehazey ace 
 Antwerp, and the Netherlanda: een ete tas 
 AViIgnon 2.60 cseeae Setee si a's ce eccceces 
 PealGuehint och cons oe seers hess Ro sseeee 
 PAVOUHCE -igu's'cerics ons sie tuale. ¥en 44s voce 
 WSGFEAIION: 2166... oes Mio 4ae ele vote tepece 
 EOP CHEN Wiss 40.2 & cs. S'e'tie'eio'g'sl ve p sina’ se 
 rerhGe o.oo sods ceeewee ae acim inielas ase 
 BRTOG outa s tie vs s14.0 0 Gctteinless 50% cleo eee 
 Bois-le-Duc © ~.......0ee008 Brn 
 Bourdeaux, see Bayonne. 
 RI Bi teh on og ¢ «okey eee os 9 vs owee 
 Btescia ....... eh tak bdenites's eerataeines ° 
 BRONZE L ole xis ssn pp 2 alesse mabe sieve sl dsiaee 6 
 "China (the kin) ++-+> dyeisiatets.« Ry ettoies ks 
 Cologne ........ 2 Aaa aeiewecns = 
 Constantinople ......-.ceeeee coves 3 ats 
 Copenhagen ...... wie noes Sable Sie ele aie 6 
 Damascus ..... Weve tict te pasar sens: 
 Dantzic  ......0:- ess Slew ea sale ea An 
 CUTS ARES A OO hap O OC aerinacn Pease s. 
 PPIDTCHCE. A Sole oo lsc Wide e v's onolncleuenve 
 Genoa secovcceccvcensccesccvccoes eee 
 Geneva stk. Reh. Sas hore de sie Pacis 
 Hamburgh) ..s0e--eees AChE Rete ee 
 Konigsberg ..-...e++see-ee COS : 
 Leghorn ...cserecncccrsrertecssssccecs 
 
 Grs. Troy. lib. oz. dwt. gr. 
 30984°86 5 411 0°86 
 6158°74 .. 1 016 14°74 
 6908°58 . USF) 20°58 
 7460°71 1 3 10 20°71 
 704815 .. Lec De la 4ct5 
 6216°99 1 019 0°99." 
 TELS OL oat RL eeoL ao al 
 7460°71 1 3 10 20°71 
 AGGSOT tee, Ore 94 77-97 
 11659°52 2 O45 19°52 
 Teale. «s- LE645.G,) Op 
 ay.) iss Saree a ess: 
 746071 +... be 310 20°F] 
 710548 .. J 216 1:48 
 (OTS Shs. od ae lahore 
 449661 .- 0 9 7 86l 
 7038°21 - 1 213 6:21 
 9222'°93 *- 1 7 4 6-93 
 722034... 1 8 0 20:34 
 7578:°03 1 3 15 18°03 
 6940°58 1 2:9 4°58 
 2561288 .« -4 5 7 488 
 6573°86 .. 1 1 13 21°86 
 7774-11 1 3 19 181 
 5286°65 -» O11] O 665 
 442605 .. O 9 4 10°05 
 6637°85 1 116 3°85 
 8407°45 1 510 7:45 
 731468 +» 1 3 4 18°68 
 5968°41 1 O 8 16°41 
 5145°54 01014 9°54 
 
266 
 
 TABLE OF WEIGHTS. 
 
 LAC CA eee t HOSS Sieh coc cic ee wie aes oes bela 
 Lyons Silk weight ...ccccosecceeeees 
 Town weight sa eececence eocrccee 
 
 Malo, St., see Bayonne. 
 
 Marseilles Men ard tees teas Sorta cone a tot 
 Mechlin, see Antwerp. 
 
 Wee lun he. Ss Sate Watt ee oie he etaverete’ s sate ote . 
 Messina.......+ Pls MaMa ed ANS Aedes 
 Montpellier oof. c..scedivedssessevrces 
 Namur?..¢ en. see's BOA nr SA Rn 
 Nancy cccerscesssescedecoees da otew 
 Nantes, see Bayonne. 
 
 Naples ceesevceccecccersececres eeeee 
 Nuremberg eccceccccccccccrecsevececs 
 Paris... os ete neee eee ceerccccerececees 
 Pisa, see Florence. 
 
 TieVel cece seve neces atic cs ot ales alate laitelale 
 Riga «+++: ERE UI NRCS S ets cues erent 
 TRONS) | CISD OOOO ORO O UDO OOD Cone clea: 
 Rotien oo cs-ccsene pee ee Aire ene 
 SaragoSSa ccscccccrsesecresccsvece oe 
 Seville oe cccvcssccvcerscerssce ott ee 
 Smyrna ...... BAO RC OLE ig Oe 
 Settle we cases +e sce esas Prior A abet 
 Stocknoln’ e2.. e-se ee Gees <e BA wis te ica 
 Strasburg «cn eee eereerceee aonesoree 
 
 Toulouse, and Upper Languedoc --+-+. 
 
 To reduce any of the weights in the preceding table to English averdupois pounds 
 nothing will be necessary but to divide the grains Troy, in the first column, by 
 
 Turin and Piedmont, in general .eccoc 
 Tunis and Tripoli, in Barbary ....ces. 
 
 Venice—lesser pound ....esereecseee ° 
 Some prenver GittO. ss sess > 25% ¥'s 447 
 VETONA, - ca ccscconscass ns pee Senne ace 
 Vicenza—lesser pound ..cecccessoe eos 
 — greater dittO .2.-.seeeessave ° 
 
 7000, 
 
 The Paris pound, poids de mark of Charlemagne, contains 9216 Paris grains: it 
 is divided into 16 ounces, each ounce into eight gros, and each gros into 72 grains*. 
 
 Grs. Troy. 
 
 7038°21 
 7089:07 
 6544°33 
 7005°39 
 5272-71 
 6946°32 
 6431-93 
 6544°33 
 
 6041°42 
 
 4440°82 
 4844-46 
 6217°8] 
 7174:39 
 7038 °21 
 
 4951°93 
 7870°91 
 7560°80 
 
 6573'86 
 6148-89 
 5257°12 
 7771°64 
 4707°45 
 703821 
 654433 
 678224 
 9211 °45 
 7276:94 
 6322'82 
 4939-62 
 7139:94 
 4215-21 
 6826°54 
 537444 
 4676:28 
 6879°05 
 
 FRENCH WEIGHTS, 
 
 It is equal to 7561 English Troy grains. 
 
 The English Troy pound, of 12 ounces, contains 5760 English grains, and is equal 
 
 to 7021 Paris grains. 
 
 - 
 
 lib. oz. dwts. grs. 
 
 1 213 6:21 
 
 1 
 uf 
 0 
 1 
 0 
 1 
 
 ] 
 it 
 
 1 
 1 
 1 
 0 
 1 
 0 
 1 
 0 
 0 
 1 
 
 — 
 
 7 23:93 
 12 16°33 
 
 11 17°42 
 
 5 0°82 
 1 20:46 
 
 0 19.181 
 
 2 
 2 
 
 _ 
 or © 
 
 ia 
 No ROO 
 
 — 
 
 woorewnNn Ne 
 
 — 
 doo —p ®@ 
 
 18 22°39 
 13 6:21 
 
 6 7:93 
 7 22°91 
 15 08 
 
 13 21°86 
 16 4°89 
 19 1°12 
 3 19°64 
 16 3°45 
 13 6:21 
 12 16°33 
 2 14°24 
 3 19°45 
 3 4:94 
 3 10°82 
 5 19°62 
 17 11:94 
 15 15:21 
 4 10°54 
 3 22°44 
 
 14 20:28 ~ 
 
 6 15°05 
 
 * Sometimes the gros is divided into 3 deniers, and each denier into 24 grains. 
 
TABLE OF WEIGHTS. 
 
 267 
 
 The English averdupois pound, of 16 ountes, contains 7000 English Troy grains ; 
 
 ‘and is equal to 8538 Paris grains. 
 
 NEW FRENCH WEIGHTS, 
 
 PAA Pr Annie cere cadesascesesssieccess 
 Centigramme...... ESE moweeee igs 
 DlCcigramiMee cscs ve snsaned «ne odo 20s so 
 GraMMG pat cedat vadseresceveccusg nsees 
 Decagramme..... Be ifidnssenke gee de se 0% 
 ACC ORTAIN EIGN ctyccnave season tose 
 CCHILNOPTAMING cc sasses-cesssosesanesed 
 MY PIAS TAMING 1.0) soc as-peceseces ase sas 
 
 Eng. Troy grains. 
 
 A decagramme is 6 dwts. 10°45 grs. Troy, or 2drs. 1 scr. 14°45 grs. apoth. weight, 
 
 or 5°648 drams averdupois. 
 
 A hectogramme is 3 oz. 8°48 drams averdupois. 
 
 A chiliogramme is 2 lbs. 3 oz. 4°87 drams averdupois. 
 A myriagramme is 22 lbs. 1 0z. 0°73 drams averdupois. 
 
268 OPTICS. 
 
 PART FOURTH, 
 
 CONTAINING MANY CURIOUS PROBLEMS IN OPTICS. 
 
 THE properties of light, and the phenomena of vision, form the object of that part 
 of the mixed mathematics, called Optics ;- which is commonly divided into four 
 branches, viz. Direct Optics, or vision, Catoptrics, Dioptrics, and Perspective. 
 
 Light indeed may reach the eye three ways: either directly, or after having been 
 reflected, or after having been refracted. Considered under the first point of view, 
 it gives rise to the first branch of optics, called Direct Optics, or vision: in which is 
 explained every thing that relates to the direct propagation of light, or by a straight 
 line from the object to the eye, with the manner in which objects are perceived, &c. 
 
 Catoptrics treat of the effects of reflected light, and the phenomena produced by 
 the reflection of light from surfaces of different forms,—plane, concave, convex, 
 &c. 
 
 When light, by passing through transparent bodies, is turned aside from its direct 
 course, which is called refraction, it becomes the object of Dioptrics. Itis this branch 
 of optics that explains the effects of refracting telescopes, and of microscopes. 
 
 Perspective ought to form a part of direct optics, as it is merely a solution of the 
 different cases of the following problem: On a given surface to trace out the image 
 of an object in such a manner, that it shall make on the eye, when placed in a proper 
 station, the same impressiou as the object itself—a problem purely geometrical, and 
 in which nothing is required but to determine, on a plane given in position, the points 
 where it is intersected by straight lines drawn to the eye from every point of the 
 object. Consequently, the only thing here borrowed from optics, is the principle of 
 the rectitude of the rays of light, as long as they pass PBtaee re the same medium; 
 the rest is pure geometry. 
 
 Without confining ourselves to any other order than that of method, we shall now 
 take a view of the most curious problems and phenomena in this interesting part of 
 the mathematics. 
 
 ON THE NATURE OF LIGHT. 
 
 Before we enter into any details respecting optics, we cannot help saying a few 
 words on the nature and properties of light in general. 
 
 Philosophers are still divided, and in all probability will be so for a long time to 
 come, in regard to the nature of light. Some are of opinion, that it is produced by 
 an extremely fine and elastic fluid, in consequence of an undulatory motion communi- 
 cated to it by the vibrations of luminous bodies, and which is propagated circularly 
 to immense distances, and with an inconceivable rapidity. Light, according to this 
 hypothesis, is entirely analogous to sound, which, as is well known, consists in a 
 similar undulation of the air, the vehicle of it. Several very specious reasons give 
 to this opinion a eehsidecnele degree of probability, nothwithstanding some physical 
 difficulties which it is not easy to obviate. 
 
 According to Newton, light is produced from luminous bodies by the emission of 
 particles highly rarefied, and projected with prodigious velocity. ‘The physical difii- 
 
NATURE OF LIGHT. 269 
 
 _ culties which militate against the former opinion, seem to serve as proofs of the 
 | present one; for the nature and propagation of light can be conceived only in these 
 two ways. 
 
 But whatever may be the nature of light, it is proved that it moves with astonish- 
 ing velocity, since it is well known that it employs only seven or eight minutes in 
 passing from the sun to the earth; and as the distance of the sun from the earth, 
 according to the best observations, is 24000 semi-diameters of the latter, or about 
 95 millions of miles, light moves at the rate of about two hundred iioteand miles 
 per second: at which rate it goes from the earth to the moon, and returns from the 
 moon to the earth, in less than three seconds. 
 
 The principal properties of light, or those which form the foundation of Optics, are 
 the following : 
 
 I —Light moves ina straight line, as long as it passes through the same transparent 
 medium, 
 
 This property is a necessary consequence of the nature of light ; for whatever it 
 may be, it is a body inmotion. Buta body moves in a straight line if nothing ob- 
 structs or tends to turn it aside from its course; and as every thing in the same 
 medium is equal in all directions, the light which passes through it must move in a 
 straight lined course. 
 
 This principle of optics, as well as the following, may be proved by experiment. 
 
 Il.—Light, when it meets with a polished plane, is reflected, making the angle of 
 reflection equal to the angle of incidence ; and the reflection always takes place in a 
 plane perpendicular to the reflecting surface, at the point of reflection. 
 
 That is to say, if a B (Fig. 1), be a ray of light, falling ona 
 plane surface p £; and if B be the point of reflection, to find 
 
 x ; C, the direction of the reflected ray Bc, we must conceive to 
 1 be drawn through the line a B, a plane perpendicular to the 
 es surface D E, and intersecting it in the point B; if the angle 
 a 
 
 CBE in this plane be then made equal to a BD, the linecs 
 | will be the reflected ray. € 
 
 d el If the reflecting surface be a curve, asd Be, a plane touching 
 
 that surface must be conceived passing through B, the point of 
 
 reflection: the reflection will take place the same as if it were produced by the 
 
 point 8; for itis evident that the curved surface and the plane, a tangent to it in the 
 
 point B, coincide in that infinitely small part, which may be considered as a plane 
 
 common to the curved surface and to the tangent plane: the ray of light therefore 
 
 ‘must be reflected from the curved surface, in ae same manner as from the point B 
 of the plane which touches it. 
 
 Tele e 
 
 IL—Light, in passing obliquely from one medium into another of a different density, 
 is turned aside from its rectilineal direction, so as to incline towards the perpen- 
 dicular when it passes from a rare medium into one that is denser, as from the air 
 into glass or water, and vice versa. 
 
 This proposition may be proved by two experiments, which are a kind of optical 
 illusions, 
 Experiment 1. 
 
 Fig. 2. 
 
 Expose to the sun, or to any other light, a vessel aBcD 
 (Fig. 2.), the sides of which are opake, aud examine at what 
 point of the bottom the shadow terminates. We shall here 
 suppose that itisate. Then fill it to the brim with water or oil, 
 and it will be found that the shadow, instead of terminating 
 at the point £, will reach no farther than to Fr. This difference 
 
270 OPTICS. 
 
 can arise only from the inflection of the ray of light s a, which touches the edge of 
 the vessel. When the vessel is empty, this ray, proceeding in the straight line 
 s A E, makes the shadow terminate at the point £; but when the vessel is filled 
 with a fluid denser than air, it falls back to ar. This inflection of a ray of light, in 
 passing obliquely from one medium into another, is called refraction. 
 
 Experiment 2. 
 
 Fig. 3. Place at the bottom of a vessel, the sides of which are 
 
 ° opake, at c for example (Fig.3.), a piece of money, or any 
 other object, and move backwards from the vessel till the 
 object disappears; if wate: be then poured into the vessel, 
 the object will immediately become visible, as well as that 
 part of the bottom which was concealed from your sight. 
 The reason of this is as follows : 
 
 When the vessel is empty, the eye at o can see the point ¢ 
 only by the direct ray ca 0, which is intercepted by the edge a of the vessel; but 
 when the vessel is full of water, the ray cD, instead of continuing its course directly 
 to £, is refracted into po, by diverging further from the perpendicular pp. This 
 ray conveys to the eye the appearance of the point c, which is seen as at c, in the 
 straight line op continued: the bottom therefore, in this case, appears to be raised. 
 For the same reason, a straight stick or rod, when immersed in water, appears 
 to be bent at the point where it meets with the surface, unless it be immersed ina 
 perpendicular direction. 
 
 Philosophers have carefully examined the law according to’ 
 which this inflection takes place, and have found that when a 
 ray, as EF (Fig.4.) passes from air into glass, it is refracted 
 into FI, in such a manner, that the sine of the angle crx, 
 and that of DF1I, are in aconstant ratio. Thus, if the ray 
 EF be refracted into F1, and the ray e¥F into Fz, the sine 
 of the angle crx will have the same ratio to the sine 
 of pF, as the sine of the angle cre has to that of pri. 
 This ratio, when the ray passes from air into common glass, 
 is always as 3 to 2; that is to say, the sine of the angle 
 which the refracted ray forms with the perpendicular to the refracting substance, is 
 always two thirds of the sine of the angle formed by the incident ray with the same 
 perpendicular. 
 
 It is to be observed, that when the latter angle, that is the distance of the inci- ~ 
 dent ray from the perpendicular, which is called the angle of inclination, is very 
 small, the angle of refraction may be considered as two thirds of it, because small 
 angles have nearly the same ratio as their sines. We here suppose that the ray 
 passes from air into glass; for it is well known, and may be easily proved by the 
 table of sines, that when two angles are very small, that is if they do not exceed 5 
 or 6 degrees, they are sensibly in the same ratio as their sines. Thus, in the case 
 above, the angle of refraction 1rD, will be two thirds of the angle of inclination 
 GFE; and-consequently the angle formed by the refracted ray and the incident, con- 
 tinued in a straight line, will be one third of it. 
 
 When the passage takes place from air into water, the ratio of the sine of the 
 angle of inclination, and that of the angle of refraction, is that of 4 to 3; that is to 
 say, the sind of the angle p F1 is constantly 3 of the sine of arn, the angle of in- 
 clination of .the ray incident-in air. Consequently, when these angles are very 
 small, they may be considered as being in the same ratio; and the angle of refrac- 
 tion will be 3 of the angle of inclination. 
 
 This proportion is the basis of all the calculations of dioptries; and on that 
 
' NATURE OF LIGHT.—CAMERA OBSCURA. 271 
 account ought to be well imprinted in the memory. For the discovery of it we are 
 ‘indebted to the celebrated Descartes; though it appears certain, by the testimony 
 ‘of Huygens, that a law of refraction equally constant, and which in fact is the same, 
 was discovered before by Willebrod Snell, a Dutch mathematician. But Vossius is 
 wrong when he asserts, as he does in his book ‘‘ De Natura Lucis,” that the expres- 
 sion of Snellius was more convenient. This learned man did not know what he 
 said, when he attempted to speak of natural philosophy. 
 
 PROBLEM I. 
 
 To exhibit, in a darkened room, external objects, in their natural colours and pro- 
 portions. 
 
 Shut the door and windows of the apartment, in such a manner, that no light can 
 enter it, but through a small hole very neatly cut in one of the window shutters, 
 opposite to some well frequented place or landscape; then hold a white cloth or 
 piece of white paper opposite to the hole, and if the external objects are strongly 
 illuminated, and the room very dark, they will appear as if painted on the cloth or 
 _ paper, in their natural colours, but inverted. 
 
 The experiment, performed in this simple manner, will succeed well enough to 
 surprise those who see it for the first time; but it may be rendered much more 
 striking by means of a lens. 
 
 Adapt to the hole of the shutter, which in this case must be some inches in 
 diameter, a tube having at its internal extremity a convex lens, of 4, 5, or 6 feet 
 focus; if a piece of white cloth, or a sheet of paper, be then held at that distance 
 
 from the glass, and in a direction perpendicular to the axis of the tube, the external 
 objects will be painted on the cloth or paper, with much more distinctness and 
 vivacity of colouring, than in the preceding experiment; and in so accurate a man- 
 ner, that the features of the person seen may be distinguished. This spectacle is 
 highly amusing, especially when a public place, a promenade filled with people, &c. 
 are exhibited. | 
 
 This painting. indeed is inverted, which destroys a little of the effect; but 
 different methods may be employed to make it appear in its natural position: it is 
 however to be regretted that this cannot be done without injuring the distinctness, 
 or lessening the field of the picture. ‘Those who may be desirous of seeing the objects 
 erect, must proceed in the following manner : 
 
 At about half the focal distance of the lens place a plane mirror, inclined at an. 
 
 rar angle of 45°, so that it may reflect downwards the rays 
 ig. 9. : ; é 
 
 proceeding from the lens; if you then place horizon- 
 
 tally below it asheet of paper, the image of the external 
 objects will appear painted on the paper, and in their 
 natural situation to those who have their backs turned 
 towards the window. Fig. 5. represents the mechanism 
 of this inversion, of which a clear idea cannot be 
 formed without some knowledge of catoptrics. ‘ 
 
 The sheet of paper may be extended on a table, 
 and nothing will be necessary but to dispose the glass and mirror at such a height 
 from the paper, that the objects may be distinctly painted on it. By these means a 
 landscape, or edifice, &c., may be exactly delineated with great ease. 
 
 PROBLEM II. 
 ‘o construct a portable Camera Obscura. 
 
 Construct a wooden box aB cp (Fig. 6.) about a foot in height, ee much in 
 breadth, and two or three feet in length, according to the focus of the lenses em- 
 
972 OPTICS. 
 
 ployed. To one of the sides adapt a tube = F, consisting of two, one thrust within — 
 the other, that it may be lengthened or shortened at pleasure ; and in the anterior 
 aperture of the first tube fix two lenses, convex on both sides, 
 and about seven inches in diameter, so as almost to touch each 
 other; place another of about 5 inches focal distance in the 
 | al Hi interior aperture; and at about the middle of the box, taken 
 
 le = lengthwise, dispose in a perpendicular direction a piece of oiled 
 paper G H, stretched on a frame: in the last place, make a round hole J, in the side 
 opposite to the tube, and sufficiently large to receive both eyes. 
 
 When you are desirous of viewing any objects, turn the tube, furnished with the 
 lenses, towards them; and adjust it either by drawing it out or pushing it in, till the 
 image of the objects is painted distinctly on the oiled paper. 
 
 The following is the description of avother camera obscura, invented by Gravesande, 
 and of which he gave an account at the end of his Essay on Perspective. 
 
 This machine is shaped alnmiost like a hackney chair; the top of tt is rounded off 
 towards the back part, and before it swells out into an arch at about the middle of 
 its height. (See Fig. 7. where this machine is represented with the side opposite 
 to the door taken off, in order that the interior part of it may be exhibited to 
 
 view.) 
 
 fig. 6. 
 
 Fig. 7. Ist. The board a, in the inside, serves asa table: 
 
 it turns on two iron hinges fastened to the fore part 
 of the machine, and is supported by two small 
 chains, that it may be raised to facilitate entrance 
 into the machine. 
 
 2d. To the back of the machine, on the outside, 
 are affixed four small staples c, c, c, c, in which 
 slide two pieces of wood D £, D £, three inches in 
 breadth; and through these pass two other pieces, 
 serving to keep fast a small board ¥, which by their 
 means can be moved forwards or backwards. 
 
 3d. At the top of the machine is a slit o Q, 
 nine or ten inches in length, and four in breadth, 
 to the edges of which are affixed two rules in the 
 form of a dove tail: between these slides a board 
 of the same length, having a round hole, of about 
 = three inches diameter, in the middle, furnished with 
 a nut, that serves to raise or lower a tube about four inches in height, which has a 
 screw corresponding to the nut. This tube is intended for reeciving a convex glass. 
 
 4th. The moveable board, above described, supports a square box x, about seven 
 and a half inches in breadth, and ten in height, the fore part of which can be opened 
 by a small door, and in the back part of the box towards the bottom isa square aperture 
 w, of about four inches in breadth, which may be shut at pleasure by a moveable board. 
 
 5th. Above this square aperture is aslit parallel to the horizon, and which occupies 
 the whole breadth of the box. It serves for introducing into the boxa plane mirror, 
 which slides between two rules so that the angleit makes with the horizon towards 
 the door B is 1]24°, or § of a right angle. : 
 
 6th. The same mirror, when necessary, may be placed in a direction perpendicular 
 to the horizon, as seen at H, by means of a small iron plate adapted to one of its sides, ~ 
 and furnished witha screw which enters a slit formed in the top of the machine, and 
 which may be screwed fast by a nut. 
 _%th. Within the box is another small mirror, LL, which turns on two pivots, fixed 
 a little above the slit of No. 5, and which being drawn up or pushed down by, the 
 small rod s, may be inclined to the horizon at any angle whatever. 
 
CAMERA OBSCURA. 273 
 
 8th. That the machine may be supplied with air, a tube of tin-plate, 
 bent at both ends, as seen Fig. 8, may be fitted into one of the sides, 
 this will give access to the air without admitting light. But if this 
 should not be sufficient, a small pair of bellows, to be moved by the foot, 
 may be placed below the seat, and in this manner the air may be conti- 
 nually renewed. 
 
 The different uses of this machine are as follow. 
 
 I.—To represent objects in their natural situation. 
 
 When objectsare to be representedin this machine, extend asheet of paper on the 
 
 table, or rather stretched on a frame, or you may employ a piece of strong card, and 
 ‘fix it in such a manner as to remain immoveable. 
 
 In the tube c, (Fig. 7.) place a convex glass, the focus of which is nearly equal to 
 
 the height of the machine above the table; open the back part of the box x, and 
 having removed the mirror H, as well as the board Fr, and the rules p &, incline the 
 -moveable mirror & 1, till it make with the horizon anangle of nearly 45°, if you intend 
 to represent, objects at a considerable distance, aud which forma perpendicular land- 
 -scape. When this is done, all those objects which transmit rays to the mirror L L, so 
 
 as to be reflected on the convex glass, will appear painted on the paper frame: the 
 
 _ point where the images are most distinct may be found, if the tube which contains 
 
 the lens be lowered or raised, by screwing it up or down. 
 By these means any landscape, view of a city, &c., may be exhibited with the 
 
 greatest precision. 
 
 Il —To represent objects in such a manner, as to make that which is on the right 
 appear on the left, and vice versa. 
 
 The box x being in the situation represented in the figure, open the door B, and 
 having placed the mirror 8 in the slit, and in the situation already mentioned No. 5, 
 
 raise the mirror Lt till it make with the horizon an angle of 224 degrees; if the fore 
 
 part of the machine be then turned towards the objects to be represented, which we 
 here suppose to be at a considerable distance, they will be seen painted on the paper, 
 
 but transposed from right to left. 
 
 It may sometimes be useful to make a drawing where the objects are transposed 
 in this manner; for example, in the case when it is intended to be engraved; for as 
 the impression of the plate will transpose the figures from right to left, they will 
 
 _ then appear in their natural position. 
 
 I1I.—7To represent in succession all the objects in the neighbourhood, and quite 
 around the machine. 
 
 Place the mirror H in a vertical position, as seen in the figure, and incline the 
 mirror L at an angle of 45 degrees; if the former be then turned round vertically, 
 
 ' the lateral objects will be seen painted in succession on the paper, in a very pleasant 
 , Manner. 
 
 ee 
 
 It must here be observed, that it will be necessary to cover the mirror H with a 
 kind of box made of pasteboard, open towards the objects, and also towards the 
 aperture n of the box x; for if the mirror H were left entirely exposed, it would 
 
 _ reflect on the mirror L a great many lateral rays, which would considerably weaken 
 _ the effect. 
 
 IV.—To represent the image of Paintings or Prints. 
 
 Affix the painting or print to the side of the board Fr, which is next to the mirror 
 L, and in such a manner that it may be illuminated by the sun. But as the object 
 
hs 
 
 274 OPTICS, 
 
 is 
 in this case must be at a very small distance, the tube must be furnished with a 
 glass, having its focal distance nearly equal to half the height of the machine above 
 the paper: if the distance of the painting from the glass be then equal to that of the 
 glass from the paper, the figures of the painting will be represented on the paper 
 exactly of the same size. 
 
 The point at which the figures have the greatest distinctness, may be found, by 
 moving backwards or forwards the board F, till the representation be very distinct. 
 
 Some attention is necessary in regard to the aperture of the convex glass. 
 
 In the first place, the same aperture may in general be given to the glass as toa 
 telescope of the same length. 
 
 Secondly, this aperture must be diminished when the objects are very much 
 iluminated; and vice versa. 
 
 Thirdly, as the traits appear more distinct when the aperture is small, than when 
 it is large, if you intend to delineate the objects, it will be necessary to give to the 
 glass as small an aperture as possible; but taking care not to extenuate the light; 
 it will therefore be proper to have different circles of copper or of blackened paste. 
 board to be employed for altering the size of the aperture, according to circum- 
 stances. 
 
 A small portable camera obscura is shewn in sec. 
 Fig. 9. tion (Fig. 9.), where paxnc is a rectangular box, in 
 which slides another EF GM, open at the end EM, 
 and in the centre of the end Fa is a double convex 
 lens H. AX isa mirror inclined at an angle of 45° 
 and 1A is a piece of ground glass, on which the 
 image formed in the focus of the lens is depicted, 
 the rays which would have formed an image on 
 AB being reflected to ait by the mirror ax. The 
 box F Mis drawn out till the image on the ground 
 glass is quite distinct. Objects seen in this instrument are erect, but inverted; 
 but this defect may be remedied by placing the camera before a mirror, and forming 
 . the picture on a1 from the object as seen in the mirror, where already right has been 
 exchanged for Jeft; and a like change being made in the camera, the picture appears 
 as In nature. 
 
 Remark.--On the top of the Royal Observatory, at Greenwich, was an excellent 
 camera ‘obscura, capable of containing five or six persons, all viewing the exhibition 
 together. All the motions of the glasses were easily performed by one of the per- 
 sons within, by means of attached rods; and the images were thrown on a large and 
 smooth concave table, cast of plaster of Paris, and moveable up and down, so as to 
 suit the distances of the objects. But this fine instrument has been removed since 
 Mr. Airy became Astronomer Royal. 
 
 THE CAMERA LUCIDA. 
 
 The Camera Lucida was invented by Dr. Wollaston in 1807, and has for its purpose’ 
 the facilitation of the operations of the draughtsman. It consists of a quadrangular 
 glass prism, by which the rays from an object are twice reflected; the second re- 
 
 flection correcting ‘the transposition of right for left, caused by 
 
 Fig 10. the first reflection. The form of the prism is shewn in the 
 
 annexed figure. The faces A B and a D are at right angles to 
 
 each other, and a c and cp make with a B and Aap respectively, 
 
 angles of 674 degrees, and therefore with each other an angle 
 of 135 degrees. 
 
a 
 
 CAMERA LUCIDA. 275 
 
 Fig. 11. The object F (Fig. 11), being opposite the face ac of the 
 E prism; the rays proceeding from Fr pass through it, and are 
 reflected from the surface c p to p 8B, and thence they are re- 
 | flected to the eye at x. The rays proceeding upwards to the 
 F eye by the last reflection, the observer is led to imagine the 
 d image below the instrument at, and placing his paper there 
 the image may be traced with a pencil. 
 
 3 The prism is mounted ina brass frame supported by brass tubes 
 sliding stiffly within one another, the tubes being slit at d and e 
 (Fig. 12), so as to form a spring for that purpose; and the 
 instrument is furnished with a clamp at 4, to fix it to the 
 drawing board or sketching book. The prism is furnished 
 at = witha piece of brass blackened, and perforated with 
 
 a small hole for directing the position of the eye. 
 
 The instrument should be leaned forward until the 
 prism is directly over the middle of the space intended 
 to be occupied by the drawing; the upper surface A B 
 being parallel to the paper. If the eye be altogether 
 over this surface, the pencil cannot be seen, the rays 
 from it do not pass directly through the prism; but 
 by placing the hole of the eye-piece so as to be divided 
 by the edge of the prism, and then applying the eye toit ; 
 with one part of the pupil the image of the object is 
 seen, and witb the other part the pencil. 
 
 It must be observed, that there is no image actually 
 formed upon the paper, as in the camera obscura, but the 
 image appears as far below the instrument as the object 
 is before it; and therefore the eye cannot, in the same state, see both the pencil and 
 the image distinctly. To remedy this inconvenience; to the brass frame of the 
 prism are attached two lenses a, b, (Fig. 12.), the one concave and the other convex, 
 ‘the former to be turned up in front of the instrument, for short-sighted persons, and 
 the latter to be turned below for long sights. The size of the picture always bears 
 the same relation to the size of the object, that the distance from the eye to the paper 
 does to the distance of the object from the eye. Then by increasing the distance of 
 the prism from the paper, by lengthening the tubes d,.e, the drawing is increased in 
 size, and vice versa. 
 
 Fig. 18. The above is the form of the instrument generally employ- 
 
 ed; but it is not the only one, nor perhaps the most ingenious. 
 
 There are two forms of it invented by S. Amici, which we 
 shall briefly describe. 
 
 The first consists of a parallel piece of plate glass, c D, joined 
 to a reflecting speculum a. The rays froman object at B are 
 thrown on the speculum a, which is inclined to the surface of 
 
 the glass c p, at an angle of 135°, and are reflected by it to the glass 
 at x, and thence to the eye above. 
 
 Fig. 14. In the other, which Amici esteems the best, the rays are made 
 a to pass through the plate glass before striking on the speculum. 
 Thus the ray from B (Fig. 12), passing through the glass is 
 reflected by the speculum to c, and thence by the surface of the 
 
 glass, to the eye. 
 
276 OPTICS. 
 
 THE KALEIDESCOPE. 
 
 The Kaleidescope was invented by Sir D. Brewster, in 1814, while experimenting 
 on the polarization of light. In a patent which he obtained for it, it was described 
 as a new optical instrument ‘‘ for creating and exhibiting beautiful forms.” The 
 extreme simplicity of the instrument however destroyed the effect of the patent, as 
 any one possessing the least mechanical skill can make one for himself. 
 
 The name of the Kaleidescope is derived from three Greek words, xadog beau- 
 tiful, «30 a form, and cxorew to see. Two reflecting plates, tapered from one end 
 to the other, to about half the breadth, are inclined to each other at an angle, which 
 must be an even aliquot part of 360°, and placed in a tube so that the point of 
 meeting E at the narrower ends of the plates is near the centre of the circular end 
 
 F G of the tube, in which there is a hole for the eye; and 
 
 Fig. 15. the edges c A and 4B of the mirrors are in the plane of 
 = the other end of the tube, to which is fitted a pox made of 
 Cars y—7 two circular pieces of glass, the outer. one being greyed 
 
 G6 to render the light uniform. This box contains the 
 
 objects, such as beads, small pieces of coloured glass, &c, 
 
 On looking in at ©, a beautiful coloured star is seen, consisting ofa number of iden- 
 
 tical sectors ; a greater number the smaller the angle between the reflecting plates. 
 
 There are several modifications of the instrument depending on the number and 
 position of the mirrors. 
 
 Fig. 16. The Polyangular Kaleidescope, for instance, is so con- 
 structed that by means of screws the angle of the mirrors can 
 be varied, thus increasing or diminishing the number of the 
 sectors. 
 
 In the Annular Kaleidescope, by which we may obtain 
 patterns for circular borders, the mirrors are not placed edge 
 to edge, but as ac and Bp (Fig. 16.), producing a pattern 
 surrounding a black circle. 
 
 Fig. 17. 
 
 When the plates ac and BD are parallel, 
 the pattern becomes rectilineal. (Fig. 17.) 
 
 Kaleidescopes may be constructed also, with three or four reflecting plates, but 
 those of three are limited to these three cases: | 
 The mirrors at angles of 90°, 45° and 45°.° 
 The mirrors at angles of 90°, 60° and 30°. 
 The mirrors at angles of 60°, 60° and 60°. | 
 Those of four reflecting plates must be of the form of a hollow square or rectangle. 
 And lastly, by taking off the object box, and placing a lens so that the focus falls, 
 in the plane of ca and a8, (Fig. 15.) a star is formed of the distant objects. 
 
 PROBLEM III. 
 To explain the nature of V sion, and its principal phenomena. 
 
 Before we explain in what manner objects are perceived, it will be necessary to 
 -begin with a description of the wonderful organ destined for that purpose. 
 
 The eye isa hollow globe, formed of three membranes, which contain humours 
 of different densities, and which produces, in regard to external objects, the same effect 
 as the Camera Obscura. The first or outermost of these membranes, called the 
 sclerotica, is only a prolongation of that which lines the inside of the eye-lids. 
 The second, called the choroides, is a prolongation of the membrane which covers 
 
ae 
 
 NATURE OF VISION. 277. 
 
 the optic nerve, as well as all the other nerves. And the third, which lines the 
 inside of the eye, is an expansion of the optic nerve itself: it is this membrane, 
 entirely nervous, which is the organ of vision; for, notwithstanding the experiments 
 in consequence of which this function has been ascribed to the choroides, we cannot 
 look for sensation any where else than in the nerves and nervous parts. 
 
 In the front of the eye the sclerotica changes its nature, and assumes a more 
 convex form than the ball of the eye, forming here what is called the transparent 
 cornea. ‘The choroides, by being prolonged below the cornea, must necessarily leave 
 asmall vacuity, which forms the anterior receptacle of the aqueous humour. This 
 prolongation of the choroides terminates at a circular aperture well known under 
 the name of the pupil. The coloured part which surrounds this aperture is called 
 the iris or uvea; it is susceptible of dilatation and contraction, so that when exposed 
 to a strong light, the aperture of the pupil contracts, and in a dark place it dilates. 
 
 This aperture of the pupil is similar to that of the camera obscura. Behind it is 
 suspended, by a circular ligament, a transparent body of a certain consistence, and 
 having the form of a lens: it is ealled the erystalline humour, and in this natural 
 camera obscura performs the same office as the glass in the artificial one. 
 
 By this description it may be seen, that between the cornea and the crystalline 
 humour there is a sort of chamber, divided into nearly two equal parts, and another 
 between the crystalline humour and theretina. The first is filled with a transparent 
 humour similar to water, on which account it has been called the aqueous humour. The 
 second chamber is filled with a humour of the same consistence almost as the white ~ 
 of an egg: it is known by the name of the vitreous humour. All these parts may be 
 seen represented Fig. 18; where a is the sclerotica, 6 the cornea, ¢ the choroides, d 
 the retina, e the aperture of the pupil, ff the uvea, h the crystalline humour, 7 z the 
 aqueous, & & the vitreous, and / the optic nerve. 
 
 The form of the cornea isa prolate spheroid, whose 
 axis is that of the eye ; the surfaces of the crystalline 
 are also spheroidal, the curvature of the inner sur- 
 face being much greater than that of the outer,— 
 
 b -¥ and their axes neither coincide with each other, nor 
 
 TE espa iN | with that of the cornea. The density of the crystal- 
 
 line lens increases from the surface to the centre. 
 
 This variable density aids in correcting aberration: 
 
 and the elliptic form of the surface is supposed to have the effect of correcting the 
 aberration of rays falling obliquely on the eye. 
 
 As it is evident, from the above description, that the eye is a camera obscura, but 
 more complex than the artificial one before described, it may readily be conceived that 
 the images of the external objects will be painted in an inverted situation, on the 
 retina, at the bottom of it; and these images, by affecting the nervous membrane, 
 excite in the mind the perception of light, colours, and figures. If the image 
 ‘be distinct and lively, the impression received by the mind is the same; but 
 if it be confused and obscure, the perception is confused and obscure also: this 
 is sufficiently proved by experiment. That such images really exist, may be easily 
 shewn by employing the eye of any animal, such as that of a sheep or bullock; for if 
 the back part of it be cut off, so as to leave only, the retina; and if the cornea of it 
 be placed before the hole of a camera obscura, the image of the external objects will 
 be seen painted on the retina at the bottom of it. 
 | But it may here be asked, since the images of the objects are inverted, how comes 
 it that they are seen in their proper position? This question can have no difficulty 
 but to those who are ignorant of metaphysics. ‘The ideas indeed which we have of 
 the upright or inverted situation of objects, in regard to ourselves, as well as of their 
 distance, are merely the result of the two senses, seeing and touching combined. 
 
278 ~ OPTICS. 
 
 The moment we begin to make use of our sight, we experience, by means of touch- 
 ing, that the objects which affect the upper part of the retina, are towards our feet in 
 regard to those that affect the lower part, which touching tells us are at a greater 
 distance. Heuice is established the invariable connection that.subsists between the 
 sensation of an object which affects the upper part of the eye, and the idea of the 
 lowness of that object. 
 
 But what is meant by lowness? It is being nearer the lower part of our body, 
 In the representation of any object, the image of the lower part is painted nearer 
 that of our feet than the image of the upper part: in whatever place the image of 
 our feet may be painted on the retina, this image is necessarily connected with the 
 idea of inferiority ; consequently, whatever is nearest to it necessarily produces in the 
 mind the sameidea. The two sticks of the blind man of Descartes prove nothing here 
 and Descartes would certainly have expressed himself in the same manner, had he not 
 adopted the doctrine of innate ideas, proscribed by modern metaphysics. 
 
 PROBLEM IV. 
 To construct an artificial eye, for exhibiting and explaining all the phenomena of vision. 
 
 This machine .may be easily constructed from the following description. a BDE 
 is a hollow ball of wood (Fig. 19.) five or six inches in diameter, formed of two hemi- 
 spheres joined together at L M, and in such a manner, that they can be brought nearer 
 
 Fi to or separated from each other about half aninch. The seg- 
 
 ig. 19. 3 : : : : 
 
 ment A B of the anterior hemisphere is a glass of uniform thick- 
 ness, like that of a watch; below which is a diaphragm, with 
 a round hole about six lines in diameter in the middle of it; F is 
 a lens, convex on both sides, supported by a diaphragm, and — 
 having its focus. equal to F c when the two hemispheres are at 
 their mean distance. In the last place, the part pc = is formed 
 of a glass of uniform thickness, and concentric to the sphere, 
 the interior surface of which, instead of being polished, is only rendered smooth, so 
 as to be semi-transparent. Such is the-artificial eye to which scarcely any thing is 
 wanting but the aqueous and vitreous humours; and these might be represented also, 
 by putting into the first cavity common water, and into the other water charged with 
 a strong solution of salt. But for the experiments we have in view, this is entirely 
 useless. 
 
 This small machine however may be greatly simplified, by reducing it to two tubes: 
 of an inch and a half or two inches in diameter, one thrustinto the other. The first, 
 or anterior one, ought to be furnished with a lens of about three inches focus; but 
 care must be taken to cover the whole of it except the part nearest the axis, which 
 may be done by means of a circular piece of card, having in the middle of it a hole 
 about half an inch in diameter. The extremity of the other tube may be covered with 
 oiled paper, which will perform the part of the retina. The whole must then be 
 arranged in such a manner that the distance of the glass from the oiled paper, 
 may be varied from about two to four inches, by pushing in or drawing out one of the 
 tubes. A machine of this kind may easily be procured by any one, and at a very small 
 expence. 
 
 Experiment 1. 
 
 The glass or the oiled paper being exactly in the focus of the lens, if the machine 
 be turned towards very distant objects, they will be seen painted with great distinct- 
 ness on the retina. If the machine be lengthened or shortened, till the bottom part 
 
 be no longer in the focus of the lens, the objects will be seen painted, not ina distintg 
 but in a confused manner. 
 
ARTIFICIAL EYRE. 279 
 
 Experiment 2. 
 
 Present a taper, or any other enlightened object, to the machine ata moderate dis- 
 tance, such as three or four feet, and cause it to be painted ina distinct manner on the 
 retina, by moving the bottom of the machine nearer to or farther from the glass. If 
 you then bring the object nearer, it will cease to be painted distinctly ; but the image 
 will become distinct if the machine be lengthened. On the other hand, if the object 
 beremoved to a considerable distance, it will cease to be painted distinctly, and the 
 image will not become distinct till the machine is shortened. 
 | Experiment 3. 
 A distinct image, however, may be obtained in another manner, without touching 
 the machine. In the first case, if a concave glass be presented to the eye, at a 
 distance which must be found by trial, the painting of the object will be seen’ to 
 become distinct. In the second case, if a convex glass be presented to it, the same 
 effect will be produced. 
 _ These experiments serve to explain, in the most sensible manner, all the phenomena 
 of vision, as well as the origin of those defects to which the sight is subject, and the 
 ‘means by which they ‘may be remedied. 
 Objects are never seen distinctly unless when they are painted in a distinct manner 
 on the retina; but when the conformation of the eye is such, that objects at a 
 ‘moderate distance are painted in a distinct manner, those which are much nearer, 
 or at a much greater distance, cannot be painted with distinctness. In tlie first case, 
 the point of distinct vision is beyond the retina; and if it were possible to change 
 the form of the eye, so as to move the retina farther from that ‘point, or the erystal- 
 line humour farther from the retina, the objects would be painted in a distinct manner. 
 In the second case, the effect is contrary: the point of distinct vision is on this side 
 of the retina; and, to produce distinct vision, the retina ought to be brought nearer 
 to the crystalline humour, or the latter nearer to the retina. We are taught by 
 experience that in either case a change is produced, which is not made without some 
 effort. But in what does this change consist? Is it. in a prolongation or flattening 
 of the eye; or is it in a displacement of the crystalline humour ? This has never yet 
 been properly ascertained. 
 
 In regard to sight, there are two defects, of a contrary nature, one of which con- 
 sists in not seeing distinctly any objects but such as are at a distance; and as this is 
 
 | generally a failing in old persons, those subject to it are called presbyte: the other 
 consists in not seeing distinctly but very near objects ; and those who have this failing 
 are called myopes. ; 
 
 The cause of the first of these defects, is a certain conformation of the eye, in con- 
 
 sequence of which the image of near objects is not painted in a distinct manner but 
 beyond the retina. But the image of distant objects is nearer than that of neigh- 
 bouring objects, or objects at a moderate distance; the image of the former may 
 | therefore fall on the retina, and distant objects will then be distinctly seen, while 
 neighbouring objects will be seen only in a confused manner, 
 _ But to render the view of neighbouring objects distinct, nothing else is necessary 
 than to employ a convex glass, as has been seen in the third experiment: for a con- 
 vex glass, by making the rays converge sooner, brings a distinct image of the objects 
 ‘Rearer ; consequently it will produce on the retina a distinct picture, which otherwise 
 would have fallen beyond it. 
 
 In regard to the myopes, the case will be exactly the reverse. As the defect of 
 their sight is occasioned by a conformation of the eye which unites the rays too soon, 
 and causes the point, where the image of objects moderately distant are painted with 
 distinctness, to fall on this side the retina, they will receive relief from concave 
 
 | 
 
280 OPTICS. 
 
 glasses interposed between the eye and the object; for these glasses, by causing the 
 rays to diverge, remove to a greater distance the distinct image, according tothe third — 
 experiment; the distinct image of objects which was before painted on this side the 
 retina, will be painted distinctly on that membrane when a concave glass is employed. 
 Besides, myopes will discern small objects within the reach of their sight much 
 
 better than the presbyte, or persons endowed with common sight; for an object 
 
 placed at a smaller distance from the eye, forms in the bottom of it a larger image, — 
 nearly in the reciprocal ratio of the distance. Thus a myope, who sees distinctly an 
 object placed at the distance of six inches, receives in the bottom of the eye an image 
 three times as large as that painted in the eye of the person who does not see dis- 
 tinctly but at the distance of eighteen inches: consequently all the small parts of 
 this object will be magnified in the same proportion, and will become sensible to the 
 myope, while they will escape the observation of the presbytz. If a myope were in 
 such a state as not to see distinctly but at the distance of half an inch, objects would 
 appear to him sixteen times as large as to persons of ordinary sight, whose boundary 
 of distinct vision is about eight inches: his eye would be an excellent microscope, 
 and he would observe in objects what persons of ordinary sight cannot discover 
 without the assistance of that instrument. 
 
 PROBLEM V. 
 To cause an object, whether seen near hand or at a great distance, to appear anit ys of 
 
 Fig. 20. 
 d 
 
 the same size. 
 
 _ The apparent magnitude of objects, every thing else being 
 alike, is greater according as the image of the object painted on 
 the retina occupies a greater space. But the space occupied by 
 an image on the retina, is nearly proportioned to the angle formed 
 by the extremities of the object, as may be readily seen by in- 
 specting Fig. 20; consequently it is on the size of the angle 
 formed by the extreme rays, proceeding from the object, which cross each other in 
 the eye, that the apparent magnitude of the object depends. 
 
 This being premised, let a B be the object, which is to be viewed at different dis- 
 tances, and always under the same angle. On a B, as a chord, describe any arc of a 
 circle, as ACDB: from every point of this arc, as a, c, D, B, the object a B will be 
 seen under the same angle, and consequently of the same size; for every one knows” 
 that all the angles having 4 8 for their base, and their summits in the segment 
 ACDB, are equal. 
 
 The case will be the same with any other arc, as a cd B. 
 
 PROBLEM VI. 
 
 Two unequal parts of the same straight line being given, whether adjacent or not ; 
 to find the point where they will appear equal. 
 
 On aB and Be (Fig. 21.), and on the 
 same side, construct the two similar isos- 
 celes triangles a FB and Bac; then from 
 the centre Fr, with the radius FB, describe 
 a circle, and from the point c, with the 
 radius GB, describe another circle; inter- 
 secting the former in p: the point p will 
 be the place required, where the two lines 
 appear equal. 
 
 For the circular ares AE BD and Be CD are, by construction, similar ; and hence 
 
 it follows, that the angle a pB is equal to BDC, as the point p is common to both 
 the ares. 
 
ae 
 
 ! . - APPARENT MAGNITUDE. 281 
 
 Remarks.—I\st. There are a great many points, such as p, which will answer the 
 problem ; and it may be demonstrated, that all these points are in the circumference 
 
 of a semicircle, described from the point 1 asa centre. This centre may be found 
 
 by drawing, through the snmmits F andc of the similar triangles a FB and BGC, the 
 
 line ra, till it meet ac produced, in 1. 
 
 2d. If the lines az and Bc form an angle, the solution of the problem will be 
 still the same; the two similar arcs described on a 8 and Bc will necessarily intersect 
 each other in some point b, unless they touch in B; and this point p will, in like 
 
 _ manner, give the solution of the problem. 
 
 3d. The solution of the problem will be still the same, 
 even if the unequal lines proposed, a B and bc (Fig. 22. ), 
 are not contiguous ; only care must be taken that the 
 radii FB and Gb, of the two circles, be such, that the 
 circles shall at least touch each other. If aB=a, 
 Bb=c, bc=b, and ac=d=a-+b-e, that the two 
 circles touch each other, F B must be at least = da 
 
 pysbcina tele bar dere ai) Shey ae ancien 
 a 
 
 | A or 3b nee If these lines be less, the two circles 
 
 Fig. 22. 
 
 will neither touch nor cut each other. If they be greater, the circles will intersect 
 each other in two points, which will each give a solution of the problem. Let a, 
 for example, be = 3, b= 2, andc = 1: in this case Gd will be found =,/ 2 = 
 14142, and pF = 3,/ 2=2°1213, when the circles just touch each other. 
 
 4th. In the last place, if we suppose 
 
 Fig. 23. three unequal and contiguous lines, as 
 
 aA /é AB, BC, CD (Fig. 23.), and if the point 
 from which they shall all appear under 
 
 ie the same angle, be required, find, by the 
 
 ; first article, the circumference BEF, &c., 
 Sean from every point of which the lines aB 
 : 4h Peet, and Bc appear under the same angle ; find 
 A also the are cEG from which Bc and cp 
 appear under the same angle; then the 
 point where these two arcs intersect each other will be the point required. But to 
 make these two circles touch each other, the least of the given lines must be between 
 the other two, or they must follow each other in this order, the greatest, the mean, 
 and the least. 
 If the lines aB, Bc, and cp be not contiguous, or in one straight line, the problem 
 becomes too difficult to be admitted into this work. We shall therefore leave it to 
 
 the ingenuity of such of our readers as have made a more considerable progress in 
 the mathematics. 
 
 PROBLEM VII. 
 
 If aw be the length of a parterre, situated before an edifice, the front of which is 
 CD, required the point in that front from which the apparent magnitude of the 
 parterre 1B will be the greatest. (Fig. 24.) 
 
 Fig. 24. 
 
 Take the height cz a mean proportional between cB and 
 ca: this height will give the point required. For if a circle 
 be described through the points a, 8, £, it will touch the line 
 CE, in consequence of a well-known property of tangents and 
 secants. But it may be readily seen that thé angle ans is 
 
282 OPTICS. 
 
 greater than any other ae B, the summit of which is in the line cp; for the angle — 
 
 aes is less than ag B, which is equal to AEB. 
 
 PROBLEM VIII. 
 A circle on a horizontal plane being given; it is required to find the position of 
 the eye where its image on the perspective plane will be still a circle. 
 We here suppose that the reader is acquainted with the fundamental principle of 
 perspective representation, which consists in supporting a vertical transparent plane 
 between the eye and the object, called the perspective plane. As rays are supposed 
 
 to proceed from every point of the object to the eye, if these rays leave traces on ~ 
 
 the vertical plane, it is evident that they will there produce the same effect on the 
 eye as the object itself, since they will paint the same image on the retina. The 
 traces made by these rays are called the perspective image. 
 
 Let ac (Fig. 25.) then be the diameter of the circle on 
 Fig. 25. the horizontal plane acp, perpendicular to the perspective 
 R plane, QR, a section of the perspective plane, and P o a plane 
 
 is required to find the point o, where, if the eye be placed, 
 the representation ac, of the circle ac, shall be also a circle. 
 If po be made a mean proportional between a P and cP, 
 the point o will be the one required. 
 For if pa be to Po, as Po to Pc, the triangles PAo and 
 PCO will be similar, and the angles Pao and cop will be equal: the angles Pao 
 and cc @, or Pao and Reco, will also be equal; hence it follows that in the small 
 
 triangle aco, the angle at ¢ will be equal to the angle o 4 c, and the angle at o being . 
 
 common to the triangles aoc and aoc, the other two, aco and cao, will be also 
 equal: ao then will be to co, as co toao; hence the oblique cone Aco will be cut 
 in a sub-contrary manner, or sub-contrary position, by the vertical plane qr, and 
 consequently the new section will be a circle, as is demonstrated in conic sections. 
 
 PROBLEM IX. 
 
 Why is the image of the sun, which passes into a darkened apartment through a square 
 or triangular hole, always circular ? 
 
 This problem was formerly proposed by Aristotle, who gave a very bad solution 
 of it; for he said it arose from the rays of the sun affecting a certain roundness, 
 which they resumed when they had surmounted the restraint imposed on them by the 
 hole being of a different figure. This reason is entirely void of foundation. 
 
 i et ee eee 
 
 perpendicular to the horizon and to the line a Pp, in which it - 
 
 To account for this phenomenon, it must be observed that the rays proceeding 
 
 _ from any object, whether luminous or not, which pass through a very small hole into 
 a darkened chamber, form there an image exactly similar to the object itself; for 
 these rays, passing through the same point, form beyond it a kind of pyramid similar 
 to the first, and having its summit joined to that of the first, and which, being cut 
 by a plane parallel to that of the object, must give the same figure, but inverted. 
 This being understood, it may be readily conceived that each point of the triangu- 
 lar hole, for example, paints on paper, or on the floor, its solar image round ; for 
 every one of these points is the summit of a cone of which the solar disk is the base. 
 Describe then on paper a figure simittr and equal to that of the hole, whether 
 square or triangular, and from every point of its periphery, as a centre, describe equal 
 circles; while these circles are small, you will have at first a triangular figure with 
 rounded angles; but if the magnitude of the circles be increased more and more, till 
 
 the radius be much greater than any of the dimensions of the figure, it will be ob- — 
 
 served to become rounder and rounder, and at length to be sensibly converted into a 
 
 circle. i 
 
OPTICAL PROPERTIES. 283 
 
 But this is exactly what takes place in the darkened apartment; for when the 
 paper is held very near to the triangular hole, you have a mixed image of the triangle 
 and the circle; but if it be removed to aconsiderable distance, as each circular image 
 of the sun becomes then very large, in regard to the diameter of the hole, the image 
 is sensibly round. If the disk of the sun were square, and the hole round, the image 
 at a certain distance would, for the same reason, be a square, or in general of the same 
 figure as the disk. The image of the moon therefore, when increasing, is always, at 
 asufficient distance, a similar crescent, as is proved by experience. 
 
 : PROBLEM xX. 
 To make an object which is too near the eye to be distinctly perceived, to be seen in a 
 
 : . distinct manner, without the interposition of any glass. 
 
 Make a hole in a card with a needle, and without changing the place of the eye 
 or of the object, look at the latter through the hole; the object will then be seen 
 distinctly, and even considerably magnified. 
 
 The reason of this phenomenon may be deduced from the following observations : 
 When an object is not distinctly seen, on account of its nearness to the eye, it is 
 because the rays proceeding from each of its points, and falling on the aperture of the 
 pupil, do not converge to a point, as when the object is at a proper distance: the 
 image of each point is a small circle, and as all the small circles, produced by 
 the different points of the object, encroach on each other, all distinction is destroyed. 
 But when the object is viewed through a very small hole, each pencil of rays, pro- 
 ceeding from each point of the object, has no other diameter than that of the hole; 
 and consequently the image of that point is considerably confined, inan extent which 
 scarcely surpasses the size it would have, if the object were at the necessary distance ; 
 it must therefore be seen distinctly. 
 
 PROBLEM XI. 
 
 When the eyes are directed in such a manner as to see a very distant object ; why do 
 near objects appear double, and vice versa? 
 
 The reason of this appearance is as follows. When we look at an object, we are 
 accustomed, from habit, to direct the optical axis of our eyes towards that point which 
 we principally consider. As the images of objects are, in other respects, entirely 
 similar, it thence results that, being painted around that principal point of the 
 retina at which the optical axis terminates, the lateral parts of an object, those 
 on the right for example, are painted in each eye to the left of that axis; and 
 the parts on the left are painted on the right of it. Hence there has been es- 
 tablished between these parts of the eye such a correspondence, that when an 
 object is painted at the same time in the left part of each eye, and at the same dis- 
 tance from the optical axis, we think there is only one, and on the right; but if 
 by a forced movement of the eyes we cause the image of an object to be painted 
 in one eye, on the right of the optical axis, and in the other on the left, we see 
 double. But this is what takes place when, in directing our sight to a distant - 
 object, we pay attention to a neighbouring object situated between the optical — 
 axes; it may be easily seen that the two images which are formed in the two 
 eyes are placed, one to the right and the other to the left, of the optical “axis ; 
 that is to say, on the right of it in the right eye, and on the left of. it in the 
 left. If the optical axis be directed to a near object, and if attention be at the 
 same time paid toa distant object, in a direct line, the contrary will be the case. 
 By the effect then of the habit above-mentioned, we must by one eye judge 
 the object to be on the right and by the other to be on the left; the two eyes 
 are thus in contradiction to each other, and the object appears double. 
 
 This explanation, founded on the manner in which we acquire ideas by sight, 
 
284 ee OPTICS. 
 
 is confirmed by the following fact. Cheselden relates that a man having sus- 
 tained a hurt in one of his eyes by a blow, so that he could not direct the op- 
 tical axes of both eyes to the same point, saw all objects’ double: but this 
 inconvenience was not lasting; the most familiar objects gradually began to appear 
 single, and his sight was at length restored to its natural state. 
 
 What takes place here in regard to the sight, takes place also in regard to the 
 touch; for when two parts of the body which do not habitually correspond, in 
 feeling one and the same object, are employed to touch the same body, we 
 imagine it to be double. This is a common experiment. If one of the fingers 
 be placed over the other, and if any small body, such as a pea for example, be 
 put between them, so as to touch the one on the right side, and the other on 
 the left, you would almost swear that you felt two peas. The explanation of 
 this illusion depends on the same principles. 
 
 PROBLEM XII. 
 To cause an object, seen distinctly, and without the interposition of any opake or dia- 
 phonous body, to appear to the naked eye inverted. 
 Construct a small machine, such as that represented Fig. 26. It consists of two parallel 
 ends, A B and C D, joined together bya third piece a c, halfaninch in breadth, andan 
 : inch and a half in length. This may be easily done by means of a 
 Fig. 26. slip of card. In the middle of the end a B makea round hole &, 
 
 [ about a line and a half in diameter; and in the centre of it fix 
 
 inal 
 
 the head of a pin, or the point of a needle, as seen in the figure : 
 opposite to it inthe other end make a hole F with a large pin; 
 
 _if you then apply your eye to E, turning the hole F towards _ 
 the light or the flame of a candle, you willsee the head of the 
 pin greatly magnified, and in an inverted position, as represented © 
 at G. 
 
 The reason of thisinversion is, that the head of the pin being exceedingly near the 
 pupil of the eye, the rays which proceed from the point F are greatly divergent, on 
 account of the hole F; and instead of a distinct and inverted image, there is painted 
 at the bottom of the eye a kind of shadow in an upright position. But inverted images 
 on the retina convey to the mind the idea of upright objects: consequently, as this — 
 kind of image is upright, it must convey to the mind the idea of an inverted object. 
 
 PROBLEM XIII. 
 
 To cause an object, without the interposition of any body, to disappear from the naked 
 eye, when turned towards it. 
 
 For this experiment we are indebted to Mariotte: and though the consequences he 
 deduced from it have not been adopted, it is no less singular, and seems to prove a 
 particular fact in the animal economy. 
 
 Fix, at the height of the eye, on a dark ground, a small round piece of white paper, 
 and a little lower, at the distance of two feet to the right, fix up another of about 
 three inches in diameter ; then place yourself opposite to the first piece of paper, and 
 having shut the left eye, retire backwards, keeping your eye still fixed on the first 
 object: when you have got to the distance of nine or ten feet, the second will 
 entirely disappear from your sight. 
 
 This phenomenon is accounted for by observing, that when the eye has got to the 
 above distance, the image of the second paper falls on the place where the optic nerve 
 is inserted into the eye, and that according to every appearance this place of the retina 
 does not possess the property of transmitting the impression of objects ; for while the — 
 nervous fibres in the rest of the retina are struck directly on the side by the rays pro- 
 ceeding from the objects, they are struck here altogether obliquely, which destroys 
 the shock of the particle of light. 
 
OPTICAL PROPERTIES. 985 
 
 The above experiment may be performed more neatly in the following manner. 
 _ Having made three dots, thus— 
 | © @ @ 
 close the left eye, and bring the right directly over the left-hand dot, and about three 
 or four inches from the paper. Though the eye be directed steadily to that dot, the 
 _ other two will be distinctly recognised as existing. But drawing the head gradually 
 _ back, when the eye is about five or six inches from the paper the middle dot will 
 disappear. Withdrawing a little further, it will re-appear; and all three will be again 
 visible: but moving a little further, the dot in the right will disappear: and on 
 withdrawing a little further, still it will again appear, and all three will be visible. 
 _If the right eye be closed, and the left one brought over the dot on the right, 
 similar phenomena will be observed. 
 
 PROBLEM XIV. 
 
 To cause an object to disappear to both eyes at once, though it may be seen by each of 
 them separately. 
 
 Affix to a dark wall a round piece of paper, an inch or two in diameter, and a little 
 lower, at the distance of two feet on each side, make two marks: then place your- 
 self directly opposite to the paper, and hold the end of your finger before your face, 
 in such a manner that when the right eye is open it shall ‘conceal the mark on the 
 left, and when the left eye is open the mark on the right; if you then look with both _ 
 eyes tothe end of your finger, the paper, which is not at all concealed by it from either — 
 of your eyes, will nevertheless disappear. : 
 
 This experiment is explained in the same manner as the former; for by the means 
 here employed, the image of the paper is made to fall on the insertion of the optic 
 nerve of each eye, and hence the disappearance of the object from both. 
 
 PROBLEM XV. 
 An optical game, which proves that with one eye a person cannot judge well of the 
 distance of an object. 
 
 Present to any one aring, or place it at some distance, and in such a manner that 
 the plane of it shall be turned towards the person’s face: then bid him shut one of his 
 eyes, and try to push through it a crooked stick, of sufficient length to reach it: he 
 will very seldom succeed. 
 
 The reason of this difficulty may be easily given: it depends on the habit we have 
 acquired of judging of the distances of objects by means of both our eyes; but when 
 we use only one, we judge of them very imperfectly. 
 
 A person with one eye would not experience the same difficulty : being accustomed 
 
 to make use of only one eye, he acquires the habit of judging of distances with great 
 . correctness. 
 
 PROBLEM XVI. 
 A person born blind, having recovered the use of his sight; tf a globe and a cube 
 which he has learnt to distinguish by the touch are presented to him, will he be able, on 
 the first view, without the aid of touching, to tell which is the cube, and which is the 
 globe ? : 
 This is the famous problem of Mr. Molyneux, which he proposed to Locke, and 
 which has much exercised the ingenuity of metaphysicians. 
 
 Both these celebrated men thought—not without reason, and it is the general opinion— 
 that the blind man, on acquiring the use of his sight, would not be able to distinguish 
 the cube from the globe, or at least without the aid of reasoning; and indeed, as Mr. 
 Molyneux said, though this blind man has learned by experience in what manner the 
 
286 OPTICS. 
 
 cube and the globe affect his sense of touching, he does not yet know how those 
 objects which affect the touch will affect the sight; nor that the salient angle, which 
 
 presses on his hand unequally when he feels the cube, ought to make the same impres- 
 sion on his eyes that it does on his sense of touching. He has no means therefore of 
 discerning the globe from the cube. : 
 
 The most he could do, would be to reason in the following manner, after carefully 
 examining the two bodies on all sides: ‘¢ On whatever side I feel the globe,” he would 
 say, ‘‘I find it absolutely uniform; all its faces in regard to my touch are the same; 
 
 one of these bodies, on whatever side I examine it, presents the same figure and the — 
 
 same face; consequently it must be the globe.” But is not this reasoning, which 
 supposes a sort of analogy between the sense of touching and that of seeing, rather 
 
 too learned for a man born blind? It could only be expected from a Saunderson. — 
 
 But it would be improper here to enter into farther details respecting this question, 
 which has been discussed by Molyneux, Locke, and the greater part of the modern 
 metaphysicians. ~ 
 
 What was observed in regard to the blind man restored to sight by the celebrated 
 Cheselden, has since confirmed the justness of the solution given by Locke and 
 Molyneux. 
 
 When this man, who had been born blind, recovered his sight, the impressions be — 
 
 experienced, immediately after the operation, were carefully observed; and the fol- 
 lowing is a short account of them. 
 
 When he began to see, he at first imagined that all objects touched his eyes, as 
 those with which he was acquainted by feeling touched his skin. He knew no figure, 
 and was incapable of distinguishing one body from another. He had an idea that soft 
 and polished bodies, which affeeted his sense of touching in an agreeable manner, 
 
 ought to affect his eyes in the same way; and he was much surprised to find that 
 
 these two things had no sort of connection. In a word, some months elapsed before 
 
 he was able to distinguish any form in a painting; for a long time it appeared to him 
 
 a surface daubed over with colours; and he was greatly astonished when he at length 
 saw his father ina miniature picture: he could not comprehend how so large a visage 
 
 could be put into so small a space; it appeared to him as impossible, says the author 
 
 from whom this account is extracted, as to put a cask of liquor into a pint bottle. 
 
 PROBLEM XVII. 
 
 To construct a machine by means of which any objects whatever may be delineated 
 in perspective, by any person, though unacquainted with the rules of that sci 
 ence. ° sa ale 
 
 we 
 
 The principle of this machine consists in making the point of a pencil, which © 
 
 continually presses against a piece of paper, to describe a line parallel to that de- 
 scribed by a point made to pass over the outlines of the objects, the eye being ina 
 fixed position, and looking through an immoveable sight. 
 
 Fig. 27. 
 
 perpendicular direction by the two pieces of wood 
 
 it, is adapted for receiving a sheet of paper, on 
 whieh the objects are to be traced out in perspec- 
 tive. The paper is extended on it, and kept in 
 that position by being cemented at the four corners. 
 EF is across bar perpendicular to the two pieces 
 8G, sc, and having at its extremity another piece 
 KD, moveable on an axis at kK. The latter serves to support the perpendicular rod 
 Dc, bearing a moveable sight B a, to which the eye is applied. 
 
 The frame T, T, T, T (Fig. 27.), supported in a 
 
 SG, SG, passing through the two lower corners of 
 
@ PERSPECTIVE MACHINES. 287 
 
 The piece of wood NP is moveable, and its extremity Pp is furnished with a slender 
 point, terminating ina small button. Near its two extremities are fixed two pulleys, 
 under which pass two small cords mm: these two cords are conveyed over the 
 pulleys L, 1, fixed at the corners T, T, of the frame, and then around two horizontal 
 ones R, R: by these means they fall on the other side of the frame, where they are 
 
 fastened to the weight @, which moves in a groove, so that when the weight @ rises 
 
 or falls, the moveable piece of wood NP remains always in a situation parallel to 
 itself. This piece of wood ought to be nearly in equilibrium with the weight, that 
 
 _it may be easily moved, when it is necessary to raise or to lower it a little: in the 
 ‘middle of it is fixed the pencil or crayon 1. 
 
 It may now be readily conceived that, if the eye be applied to the hole a, and 
 
 if the moveable piece of wood nP be moved with the hand, in such a manner as to 
 make the end P pass over the outlines of a distant. object, the point of the pencil 1 
 
 will necessarily describe a line parallel and equal to that described by the point P; 
 and consequently will trace out on the paper 0, 0, against which it presses, the image 
 
 of the object in exact perspective. 
 
 This machine was invented by Sir Cristopher Wren, a celebrated mathematician, 
 
 and the architect who built St. Paul’s.. But if it be required to trace out any object 
 _ whatever, according to the rules of perspective, the very simple means described in 
 
 the following problem may be employed. 
 
 PROBLEM XVIII. 
 
 _ Another method, by which a person may represent an object in perspective, without 
 
 any knowledge of the principles of the art. 
 
 This method of representing an object in perspective requires, in the same manner 
 as the preceding, no acquaintance with the rules of the art; and the kind of machine 
 employed is much simpler; but it supposes a considerable degree of expertness in 
 the art of drawing, or at least enough to be able to delineate in one small space what 
 is seen in another. 
 
 To put this method in practice, construct a frame of such a size, that when 
 
 ‘looking at the object from a determinate point, it may be contained within that 
 
 frame. Then fix the place of the eye before the frame, and, in regard to its plane, 
 in whatever manner you think proper. The best position for the eye, unless you 
 intend to make a drawing somewhat fantastical by the position of the objects, will 
 be in a line perpendicular to the plane of the frame, at a distance nearly equal to the 
 
 breadth of the frame, and at the height of about two-thirds of that of the frame. 
 
 ers 
 
 This place must be marked by means of a sight or hole, about two lines in diameter, 
 made in the middle of a square or circular vertical plane, of about an inch or two 
 in breadth. Then divide the field of the frame into squares of an inch or two in 
 size, by means of threads extended from the sides, and crossing each other at 
 right angles. 
 Then provide a piece of paper, and divide it, by lines drawn with a black lead 
 pencil, into the same number of squares as the frame. When these preparations 
 
 | have been made, nothing is necessary but to apply your eye to the sight above- 
 
 mentioned, and to draw in each square of the paper that part of the object observed 
 in the corresponding square of the frame. By these means you will obtain an exact 
 representation of the object in perspective; for it is evident that it will be deli- 
 neated such as it appears to the eye, and perfectly similar to the figure which would 
 remain on any transparent substance extended on the frame, if the rays, proceeding 
 from each point of the object to the eye, or the place of sight, should leave traces 
 on that substance. The object, or assemblage of objects, will therefore be repre- 
 sented in perspective with great accuracy. 
 
288 _ OPTICS. _.* 
 
 * 
 
 Remark.—The same means may be employed to demonstrate, in a sensible manner, — 
 without the least knowledge of geometry, the truth of the greater part of the rules 
 of perspective; for if a straight line be placed behind the frame, in a direction 
 perpendicular to its plane, you will see its image pass through the point 
 of sight, or through that point of the plane of the frame which corresponds to the 
 perpendicular let fall from the eye on that plane. If the line be placed horizontally, 
 and if you cause it to make an angle of 45 degrees with the plane of the picture, 
 you will see the image of it pass through one of those points called the points 
 of distance. If this line be placed in any direction whatever, you will see its image 
 concur with one of the accidental points. It is in these three rules that the whole 
 of perspective almost consists. 
 
 PROBLEM XIX. 
 
 Of the apparent magnitude of the heavenly bodies on the horizon. 
 
 It is a well known phenomenon that the moon and_ sun, when near the horizon, 
 appear much larger than when they are at a mean altitude, or near the zenith. This 
 phenomenon has been the subject of much research to philosophers; and some of 
 them have given very bad explanations of it. 
 
 Those indeed who reason superficially, ascribe it to a very simple cause, viz. 
 
 refraction ; for if we look obliquely, say they, at a crown piece immersed in water, 
 it appears to be sensibly magnified. But every body knows that the rays which 
 proceed from the celestial bodies, experience a refraction when they enter the 
 atmosphere of the earth. The sun and moon are then like the crown immersed in 
 water. ‘ 
 But those who reason in this manner do not reflect that, if a crown piece im- 
 mersed in a denser medium appears magnified to the eye situated in a rarer medium, 
 the contrary ought to be the case when the eye is situated in a dense medium, while 
 the crown piece is immersed in ararer. A fish would see the crown piece out of the 
 water much smaller than if it were in the water. But we are placed in the dense 
 medium of the atmosphere, while the moon and sun are inararer. Instead there- 
 fore of appearing larger, they ought to appear smaller; and this indeed is the case, 
 as is proved by the instruments employed to measure the apparent magnitude of the 
 celestial bodies: these instruments shew that the perpendicular diameter of the sun 
 and moon, when on the horizon, is shortened about two minutes, which gives them 
 that oval form, pretty apparent, under which they often appear. 
 
 The cause of this phenomenon must therefore be sought for in a mere optical 
 illusion; and in our opinion the following explanation is the most probable. 
 
 When an object paints on the retina an image of a determinate size, the object 
 appears to us larger, according as we judge it to be at a greater distance: and this 
 is theconsequence of a tacit reasoning pretty just; for an object which, at the 
 distance of six hundred feet, is painted in the eye under the diameter of the line, 
 must be much larger than that which is painted under the same diameter, though only _ 
 at the distance of sixty feet. But when the sun and moon are on the horizon, a 
 multitude of intervening objects give us an idea of great distance; whereas when 
 they are near the zenith, as no object intervenes, they appear, to be nearer us. In 
 the former situation then they must excite an idea of magnitude, quite different 
 from what they do in the latter. 
 
 We must however confess that this explanation is attended with some diffi- 
 culties. 
 
 Ist. When we look at the moon on the horizon through a tube, or through the 
 fingers bent into the form of one, the size of it appears to be much diminished, 
 though the fingers conceal the intervening objects in a very imperfect manner. 2d. 
 
CONVERGENCE OF PARALLELS. | 289 
 
 We often see the moon rising behind a hill at a small distance, and on such occasions 
 she appears to be exceedingly large. 
 
 These facts, which seem to overturn the explanation before given, have induced 
 yther philosophers to endeavour to find out a different one. The following is that 
 of Dr. Smith, a celebrated writer on optics. ; 
 
 The celestial arch does not exhibit to us the appearance of a hemisphere, but that 
 of a very oblate surface, the elevation of which towards the zenith is much less 
 chan its extension towards the horizon. The sun and moon also appear under the 
 same angle, whether at the horizon or near the zenith. But the intersection of a 
 leterminate angle, at a mean distance from the summit, is less than at a greater. 
 The projection therefore of the sun and moon, or their perspective image on the 
 selestial arch, is less at a great distance from the horizon than in the neighbourhood 
 of it. Consequently, when at a distance from the horizon they must appear less than 
 when they are near it. 
 
 This explanation of the phenomenon is very specious. But may it not here be 
 asked, why these two images, though seen under the same angle, appear ove greater 
 than the other? Are we not still obliged to have recourse to the former explana- 
 tion? But for the sake of brevity we shall leave the discussion of these two ques- 
 tions to the reader. 
 
 It is sufficient that it is fully demonstrated that this apparent magnification is not 
 produced by a larger image painted on the retina. In regard to the moon, it is even 
 somewhat less ; since that luminary, when on the horizon, is farther from us, by about 
 a semi-diameter of the earth, or a sixtieth part, than when she is very much elevated 
 above the horizon. Ina word, this phenomenon is merely an optical illusion, what- 
 ever may be the cause, which is still very obscure, but in our opinion it seems to 
 
 bigpend chiefly on the idea of great distance exited by the intervening objects. 
 
 PROBLEM XX. 
 On the converging appearance of parallel rows of trees. 
 
 The phenomenon which is the subject of this problem, is well known. Every per- 
 son must have observed, that when at the extremity of a very long walk, planted on 
 each side with trees, the sides instead of appearing parallel, as they really are, seem to 
 ‘converge towards the other end. The case is the same with the ceiling of a long gal- 
 lery; and indeed when it is necessary to represent these objects in perspective, the 
 sides of the walk or ceiling must be represented by converging lines; for they are 
 really soin the small image or picture painted at the bottom of the eye. 
 
 Other considerations however are necessary, in order to give a complete explana- 
 tion of the phenomenon; foras the apparent magnitude of objects is not measured by . 
 the real magnitude of the images painted in the eye, but is always the result of the 
 judgment formed of their distance by the mind, combined with the magnitude of the. 
 image present in the eye, the sides of a walk are far from appearing to converge with 
 so much rapidity, as the lines which form the image of them in the perspective plane, 
 ‘or inthe eye. M. Bouguer first gave a complete explanation of what takes place 
 ‘on this occasion; itis as follows: 
 
 As the ceiling of a long gallery appears to an eye, placed atone extremity of it, to 
 become lower, the case is the same with a long level walk, the sides of which are 
 parallel; the plane of that walk, instead of appearing horizontal, seems still to rise. 
 ‘For the same reason, as when on the sea shore, the water appears like an inclined 
 plane which threatens the earth with an inundation. Some superstitious persons,. 
 little acquainted with the principles of philosophy, have considered this inclination as 
 real, and the apparent suspension of the waters as a visible and continued miracle. In 
 like manner, in the middle of an immense plain we see it rise around us, as if we were 
 at the bottom of avery broad and shallow funnel. M. Boguer has taught us a very 
 
 : U 
 
290 OPTICS. 
 
 ingenious method of determining this apparent inclination; but it will be sufficient 
 here to say that, to most men, it is about 2 or 3 degrees. 
 
 Let us then suppose two horizontal and parallel lines, and an inclined plane of — 
 2 or 3 degrees passing below our feet: it is evident that these two horizantal lines 
 will appear to our eye as if projected on that inclined plane. But their projection on 
 that plane will be two lines concurring in one point, viz., that where the horizon- 
 tal drawn from the eye would meet it. We must therefore see these lines as con- 
 vergent. 
 
 It thence follows, that if, by any illusion peculiar to the sight, the plane where the 
 parallel lines are situated, instead of appearing inclined upwards, should appear de- 
 clined downwards, the sides of the walk would appear divergent. This, Dr. Smith, 
 in his Treatise of Optics, says is the case with the avenue at the seat of Mr. North, 
 in the county of Norfolk. But it is to be wished that Dr. Smith had described, ina 
 more minute manner, the position of the places. However we shall solve, according 
 to these principles, another curious problem, which has been much celebrated among 
 opticians. 
 
 PROBLEM XXI. 
 
 In what manner must we proceed to trace out an avenue, the sides of which, when seen 
 from one of its extremities, shall appear parallel ? 
 
 Suppose an inclined plane of two degrees and a half, and that two parallel lines are 
 traced out onit. From the eye suppose two planes passing through these lines, and 
 which being continued, cut the horizontal plane in two other lines; these two lines 
 will be convergent, and if continued backwards will meet behind the spectator. 
 
 Nothing then is necessary but to find this point of concurrence, which is very easy; 
 for any one in the least acquainted with geometry, must perceive that it is the point 
 where a line drawn through the eye, parallel to the above inclined plane, and in the 
 direction of the middle of the avenue, meets with the horizontal plane. Let a 
 line then inclined. to the horizon two or three degrees, be drawn through the eye of 
 the spectator, and in the vertical plane passing through the middle of the avenue; 
 the point where it meets the horizontal plane will be that where the two sides of the 
 avenue must unite. If from this point, therefore, two straight lines be drawn through 
 thetwo extremities of the initial breadth of the avenue, they will trace out where all 
 the trees ought to be planted, so as to appear to form parallel sides. " 
 
 If the height of the eye be supposed equal to 5 feet, and the breadth of the com- 
 mencement of the avenue to be 36, the point of concurrence will be found by calcula- 
 tion to be 102 backwards, and the angle formed by the sides of the avenue ought tobe 
 about 18 degrees. It is difficult however to believe, that lines which form so sen- 
 sible an angle will ever appear parallel to an eye within them, in babes 8 point it 
 may be placed. 
 
 PROBLEM XXII. 
 
 To form a picture which, according to the side on which tt is viewed, shall exhibit two 
 different subjects. 
 
 Provide a sufficient number of small equilateral prisms, a few lines only in breadth, 
 and in length equal to the height of the painting which you intend to make, and 
 place them all close to each other on the ground to be occupied by the painting. 
 
 Then cut the painting into bands equal to each of the faces of the prisms, and 
 cement them, inorder, to the faces of the same side. 
 
 When this is done, take a painting quite different from the former, and haying 
 divided it into bands in the same manner, cement them to the faces of the opposite side. 
 
 It is hence evident, that when cn one side you can see only the faces of the prism 
 turned towards that side, one of the paintings will be seen; and if the picture be 
 looked at on the opposite side, the first will disappear, and the second only wili be seen. 
 
 H 
 
DISTORTIONS OF FIGURES. 291 
 
 A painting may even be made, which when seen in front, and on the two sides, 
 shall exhibit three different subjects. For this purpose, the picture of the ground 
 must be cut into bands, and be cemented to that ground in such a manner, that a 
 space shall be left between them, equal to the thickness of a very fine card. On 
 these intervals raise, in a direction perpendicular to the ground, bands of the same 
 card, nearly equal in height to the interval between them; and on the right faces of 
 these pieces of card cement the parts of a second painting, cut also into bands. In 
 the last place, cement the parts of a third picture, cut in the same manner, on the 
 left faces of the pieces of card. It is evident that when this picture is viewed 
 in front, at a certain distance, the bottom painting only will be seen; but if you 
 stand on one side, in such a manner that the height of the slips of card conceals from 
 you the bottom, you will see only the picture cemented in detached portions to 
 the faces turned towards that side: if you move to the other side, a third paint- 
 ing will be seen. 
 
 PROBLEM XXIII. 
 
 To describe on a plane a distorted figure, which when seen from a determinate point 
 shall appear in its just proportions. 
 
 A figure, such for example as a head, may be disguised, that is to say distorted, in 
 such a manner, as to exhibit no proportion, when the plane on which it has been 
 drawn is viewed in front; but when viewed from a certain point it shall appear 
 beautiful, that is to say, in its just proportions. This may be done in the following 
 manner : 
 
 Having drawn on a piece of paper, in its just pro- 
 portions, the figure you intend to disguise, describe 
 a square around it, as a BCD (Fig. 28), and divide 
 it into several other small squares, which may be 
 done by dividing the sides into equal. parts, for ex- 
 ample seven, and then drawing straight lines through 
 the corresponding points of division, as the engravers 
 do when they intend to make a reduced drawing 
 from a picture. 
 
 Then describe, at pleasure, on the proposed plane, 
 | a parallelogram EMFG, and divide one of the two 
 ishorter sides, as E c, into as many equal parts as D c, one of the sides of the 
 square aBCD, which in this case are seven. Divide the other side mF, into two 
 ‘equal parts, in the point u, and draw from it to the points of division of the 
 opposite side E G, as many straight lines, the two last of which will be H B and Ha. 
 
 Having then assumed at pleasure, in the side mr, the point 1, above the point 
 H, as the height of the eye above the plane of the picture, draw from 1 to the 
 point xz, the straight line = 1, which will cut those lines proceeding from the 
 point u, in the points 1, 2, 3, 4, 5, 6, 7. Through these points of intersection 
 (draw straight lines parallel to each other, and to the base E G of the triangle 
 ¥ GH, which will thus be divided into as many trapeziums as there are little 
 ‘squares in the square ABCD. Hence, if the figure in the square a BCD be 
 transferred to the triangle r GB, by making those parts of the outline contained 
 in the different. natural squares of aBcD, to pass through the corresponding 
 trapeziums or perspective squares, the figure will be found to be distorted. But 
 it may be seen exactly like its prototype, that is to say as in the square aBcD, 
 if it be viewed through a hole k, which ought to be small towards the eye and 
 wide towards the object, made in a small board 1, placed perpendicularly in u, 
 ‘so that, the height 1. x shall be equal to H1, which must never be very great, 
 in order that the figure may be more distorted in the picture. 
 
 u 2 
 
 PM! 
 
 | 
 
292 OPTICS, f 
 
 In the convent of the Minimes de la Place Royale there is a Magdalen at prayers 
 distorted in the same manner, which has some celebrity. It is the work of 
 Father Niceron of that order, who frequently employed himself on this kind of 
 optical amusement. 
 
 Several other anamorphoses may be made in the same manner, by painting, for 
 example, on a curved surface, either cylindric, or conical, or spherical, a certain 
 figure, which when seen from a determinate point shall appear regular; but as 
 this does not succeed so well in practice as in theory, we think it needless to 
 say any thing further on the subject, while there are so many others much more 
 curious. Those persons who are fond of such optical curiosities may consult La 
 Perspective Curieuse of Father Niceron, where they will find the subject treated 
 of at full length. 
 
 PROBLEM XXIV. 
 
 Any quadrilateral figure being given; to find the different parallelograms or rectangles 
 of which it may be the perspective representation. Or any parallelogram, whether 
 right-angled or not, being given, to find its position, and that of the eye, which shall 
 cause its perspective representation to be a given quadrilateral. 
 
 Fig.29. Let the given quadrilateral be the trapezium 
 
 ABCD (Fig. 29), which we shall suppose the most 
 
 irregular possible, having none of its sides parallel, 
 
 Continue the sides AB and cD, till they meet in F, 
 
 and the sides ap and go, till they meet in E; then 
 
 draw © F, and through the point a, draw & 4, pa- 
 rallel to it. Whatever be the position of the eye, 
 provided what is called the point of sight be in the 
 line £ F, or not only in E F, but in the continuation 
 of it on both sides; the object, of which the quad- 
 
 rilateral ABCD is the perspective representation, will be a parallelogram. 4 
 For all persons acquainted with the rules of perspective, know that lines parallel 
 
 to each other on a horizontal plane, when represented in perspective meet in one 
 
 point of the line parallel to the horizon, drawn through the point of sight. Thus 
 all the lines perpendicular to the ground line, meet in the point of sight itself: 
 all those which form with that line an angle of 45°, concur in what is called the 
 points of distance; and those which form a greater or less angle, concur in other 
 points which are always determined by drawing from the eye to the picture a 
 line parallel to those of which the perspective representation is required. AJl the 
 lines then, which in the picture concur in points situated. in the line of the point of 
 sight, are images of horizontal and parallel lines. Thus, the lines on the horizontal 
 plane, which have as representatives in the picture the lines B c and a D, are 
 parallel ; and the case is the same with those which give the lineal images a B and 
 Dc. But two pairs of parallel lines necessarily form, by their intersection, a parallel- 
 ogram. The object then of which the quadrilateral a B c D is the image, to an eye 
 situated in the line y E, wherever the point of sight may be, is a parallelogram. 
 This being demonstrated, we shall first suppose that the required object is a rect- 
 angle.. To find in this case the place of the eye, divide the distance Fr & into two 
 equal parts in 1, and suppose the eye situated in such a manner that the perpendi- — 
 cular, drawn from its place to the painting, shall fall on the point 1; and that the — 
 distance is equal to 1 © or r¥: the points F and 1 will then be what, in the language 
 of perspective, is called the points of distance. Continue the lines c 8 and cp to 
 
 ‘Gand # in the ground line: the lines no ¥ and a BF will be the images of the lines 
 
 which form with the ground line angles of 45 degrees. The case will be the same 
 
 with those of which Gc £ and ab © are the images. If the indefinite lines ude and 
 
 y 
 x 
 
 a 
 
! 
 . 
 ! PERSPECTIVE DEVICES. 293 
 
 vA b, inclined to the ground line at an angle of 45 degrees, be then drawn on the one 
 side and on the other, and ina contrary direction, the lines eG 6 ¢ and a d, inclined 
 ‘also at half a right angle, these lines will necessarily meet at right angles, and 
 form the rectangle a be d. , 
 If the point of sight be supposed in another point, for example £, that is to say, 
 \iffwe suppose the eye to be directly opposite to the point =, and at a distance equal 
 ‘to EK, after drawing EL and FM perpendicular to the ground line in the plane of the 
 picture, we must draw to the same ground line, in the horizontal plane, the perpen- 
 dicular LN, equal to EK, and then the line Nm, making with the ground line the 
 ‘angletmN. If we then draw to the points cand a the indefinite perpendiculars 
 'a Aand GK, and through the points a and u the indefinite lines H K and a &, meking 
 with the ground line angles equal to LN, and in a contrary direction; these two 
 pairs of lines will meet in 6, x, A, and evidently form an oblique parallelogram, which 
 will be the object of which pcp a is the representation, to an eye situated opposite 
 ‘to BE, and at a distance from the picture equal to EK. 
 
 | If the sides ab and cd, in the rectangle a bed, were divided into equal parts by 
 ‘lines parallel to the other sides, it is evident that these parallels, being continued, 
 would cut the line 4G into as many equal parts. The case would be the same with 
 lines parallel to a4 and ed dividing into equal portions the sides ad and bc: the’ 
 line AH would likewise be divided by them into equal parts. Thus we have the 
 means of dividing the trapezium a BC D, if necessary, into lozenges, which would be 
 ‘the representation of the squares into which a bcd might be divided. 
 
 We shall give hereafter the solution of a very curious problem, in regard to orna- 
 
 mental gardening, which is a consequence of the one here solved. 
 
 OF PLANE MIRRORS. 
 
 Plane mirrors are those the reflecting surface of which is plane; as is the case 
 with the common glass mirrors used for decorating apartments. Plane mirrors may 
 be made also of metal. Of this kind were those of the ancients; but since the 
 invention of glass, metallic mirrors are never used, except small ones for certain 
 ‘optical instruments, where it is necessary to avoid the double refiection produced by 
 glass, one from the anterior and the other from the posterior surface. It is the 
 latter which gives the liveliest image; for if the silvering be scraped from the back 
 of a mirror, you will see the bright image immediately disappear almost entirely, 
 jand that which remains in its place will scarcely be equal to that produced by the 
 nearer surface. , ‘ 
 
 | But in catoptrics, in general, the two surfaces of a mirror are supposed to be at 
 ‘such a small distance from each other, as to produce only one image; otherwise the 
 determinations given by this science would require to be greatly modified. 
 
 | PROBLEM XXV. 
 A point of the object B, and the place of the eye A, being given; to find the point 
 of reflection on the surface of a plane mirror. (Fig. 30.) 
 Fig. 30 Through z, the given point of the object, and a, 
 pee) the place of the ey l ‘pendi 
 place of the eye, conceive a plane perpendicular 
 to the mirror, and cutting it in the linecp: from the 
 point B, draw BD perpendicular to cp, and continue 
 it tor, so that D F and vB shall be equal: if through 
 the points F and a, the line a ¥ be drawn, intersecting 
 cDin £, the point © will be the point of reflection ; 
 Be will be the incident ray; ua the reflected ray; 
 and » nD, the angle of incidence, and a EB c, the 
 angle of reflection, will be equal. 
 
 ne 
 
an 
 
 “wr 
 . 
 “ 
 ‘* 
 
 5 a 
 bs 
 
 294 OPTICS. 
 
 For it is evident, by the construction, that the angles BED and DEF are equal; 
 but the angles p  F and a EC are also equal, being vertical angles ; therefore, &c. 
 
 PROBLEM XXVI. 
 
 The same supposition being made as before; to find the place of the image of the 
 point B. 4 
 
 The place of the image of the point 'B is exactly in the pointr. But we shall not 
 assign as the reason what is commonly given in books on catoptrics, viz., that in 
 mirrors of every kind the place of the image is in the continuation of the reflected 
 ray, where it is intersected by the perpendicular drawn from the point of the object 
 to the reflecting surface: for what effect can this perpendicular, which is merely 
 imaginary, have to fix the image in this manner, in the point where it meets with 
 the reflected ray continued, rather than in any other point? This principle then is 
 ridiculous, and void of foundation. 
 
 It is however true that, in plane mirrors, the place where the object is perceived, 
 is in the point where the above perpendicular meets with the reflected ray produced ; 
 but this is accidental, and the reason is as follows. 
 
 All the rays which emanate from the object B, and are reflected by the mirror, 
 meet, if produced, in the point F: their arrangement then, in regard to the eye, is 
 the same asif they proceeded from the point F. Consequently they must make 
 the same impression on the eye, as if the object were in F; for the eye would 
 not be otherwise affected, if they really proceeded from that point. 
 
 Hence it may be concluded that, in a plane mirror, the object appears to be as 
 far behind, as it is distant from the mirror. 
 
 It therefore follows, that a F, the distance of the image F from the eye, is equal 
 to the sum of BB, the ray of incidence, and a £ the ray of reflection, since B £ and 
 E F are equal. 
 
 It thence follows also, that when the plane mirror is parallel to the horizon, as 
 c D, a perpendicular object, such as B D, must appear inverted. 
 
 In the last place, when we look at ourselves in a mirror, the left seems to be on 
 the right, and the right on the left. 
 
 PROBLEM XXVilI. 
 
 Several plane mirrors being given, and the place of the eye, and of the object ; to find 
 the course of the ray proceeding from the object to the eye, when reflected two, three, 
 or four times, 
 
 Let there be two mirrors, a B and cp, (Fig. 31.), and let o F be the perpendicu- 
 lar, drawn from the object o to the mirror a B, and continued beyond it, so that F E be 
 equal to o F; and let s urbe the perpendicular drawn from the eye to the mirror 
 c p, and continued till #1 be equal to Hs; join the points 1 and & by the line £1, 
 
 which will intersect the mirrors in G and «x; and 
 
 fig. 31. if the lines 0 G, Gk, and K 8 be then drawn, they 
 
 will represent the course of the ray, proceeding 
 from the pointo to the eye by two reflections. 
 
 Or, from the point £, the first part of the con- 
 struction remaining the same, let fall, onthe mirror 
 c D, the perpendicular = L mM, and continue it be-— 
 yond it, till 1 m be equal to L ©; draw the line 
 S M, intersecting c D ink; and from the point kK, 
 the line K 8, intersecting a B ing; if G o be also drawn, the lines 0 G, & K, and K$ 
 will represent the course of the ray proceeding from the point 0, and conveyed to 
 the eye by two reflections. 
 
 In this case, the point m will be the image of the point o, and the distance s M 
 will be equal tothe sum of the rays sK, KG, andGo. | : 
 
PLANE MIRRORS. 295 
 
 _ If we suppose three mirrors, and three reflections, the course which the incident ray 
 must pursue, in order toreach the eye, may be found in the same manner. For this 
 purpose, let o 1 (Fig. 32.) be the perpendicular drawn 
 from the object to the mirror a B, and let m 1 be 
 equal to Ho. From the point 1 draw 1 K perpendi- 
 cular to c B, produced if necessary, and make kK M 
 equal to mM I: from the point k let fall on pc pro- 
 duced the perpendicular K N, and continue it to L, so 
 that uN shall be equal to KN: draw s u, which 
 will intersect cD in G; and from the point e the line 
 G kK, which will intersect c gin F; if the line F 1, 
 intersecting a B in E, be then drawn from the point F, and also the line x o, then 
 the line © o will be that according to which the incident ray must fall on the 
 first mirror, to reach to eye s, after three reflections at BE, F, and a. 
 
 In this case the point L will be the apparent place of the image of the object, 
 to an eye situated in s; and the distance s u will be equal to 3 Gc, GF, FE, 
 and E£ o, taken all together. 
 
 The application of this problem is generally shewn at the game of billiards; but as 
 we have already treated that subject, under the head mechanics, the reader is referred 
 to that article. 
 
 Fig. 32. 
 
 PROBLEM XXVIII. 
 Various properties of plane mirrors. 
 
 I. In plane mirrors the image is always equal and similar to the object. For it may 
 be easily demonstrated, that as each point of the image seems to be as far within the 
 mirror as the object is distant from it, each point of the image is similarly situated, and 
 at an equal distance in regard to all the rest, asin the object: the result must there- 
 fore necessarily be the equality and similarity of the image and object. 
 
 II. In a plane mirror, what is on the right appears in the object to be on the 
 left, and vice versa. This may be easily proved in the following manner. If a 
 piece of common writing be held before a mirror, it cannot be read: as the word 
 
 ‘GENERAL, for example, will appear under the annexed form; but on the other hand, 
 
 if the latter word be presented to the mirror, GENERAL will 
 
 Fig. 33. appear. This affords the means of forming a sort of secret 
 
 IAAI O writing; for if we write from right to left, it cannot be read 
 
 by those ignorant of the artifice; but those acquainted with it, 
 
 by holding the writing before a mirror, will see it appear like common writing. 
 
 This method however must not be employed for concealing secrets of great import- 
 “ance, as there are few persons to whom it is not known. 
 
 III. In a plane mirror, when you can see yourself at full length, at whatever. dis- 
 tance you remove from it, you will always see your whole body; and the height of 
 the mirror occupied by your image will always be equal to the half of your height. 
 
 IV. If one of the sun’s rays be made to fall on a plane mirror, and if an angular 
 motion be given to the mirror, the ray will be seen to move with a double angular 
 motion; so that when the mirror has passed over 90°, the ray will have passed over 
 180°. 
 
 V. Ifa plane mirror be inclined to a horizontal surface, at an angle of 45°, its image 
 ‘ will be vertical. 
 
 VI. Iftwo plane mirrors be disposed parallel to each other; and if any object, such 
 asa lighted taper, be placed between them ; you will see in each along series of tapers 
 which would be extended zn infinitum, did not each image become fainter in propor- 
 tion as the reflections by which they are produced become more numerous. 
 
 WII. When two mirrors are disposed in such a manner as to form an angle of at 
 
296 OPTICS. 
 
 least 120°, several images will be seen, according to the position of the eye. If the 
 angle of the mirrors be diminished, without changing the place of the eye, these 
 images will be seen to increase in number, as if they emerged from behind an opake 
 body. 
 
 it must be observed, that all these images are in the circumference of a circle, 
 described from the point where the mirrors meet, and passing through the place 
 of the object. ; 
 
 Father Zacharias Traber, a Jesuit, in his Mervus Opticus, and Father Tacquet, 
 in his Optics, have carefully examined all the cases resulting from the different 
 
 angles of these mirrors, as well as from the different positions of the eye and the 
 
 object. To these we refer the reader. 
 
 VIII. When a luminous object, such as the flame of a taper, is viewed in a 
 plane glass mirror of some thickness, several images of that object are perceived ; 
 the first of which, or that nearest the surface of the glass, is less brilliant than 
 the second; the latter is the most brilliant of the whole; and after it, a series of 
 images less and less brilliant are observed, to the number sometimes of five or six. 
 
 The first of these images is produced by the anterior surface of the glass, and the 
 second by the posterior, which being covered with tin-foil, must produce a more lively 
 reflection: it is therefore the most brilliant of the whole. The rest are produced by 
 the rays of the object, which reach the eye after being several times reflected from 
 the anterior, as well as posterior, side of the mirror. This phenomenon may be 
 explained as follows. 
 
 Let v x (Fig. 34.) be the thickness of the glass, a 
 the object, and o the place of the eye, which we shall 
 suppose to be both equally distant from thg mirror, 
 Of all the small bundles of incident rays, there is one 
 A B, which being reflected by the anterior surface in B, 
 is conveyed to the.eye by the line Bo, and forms at a’ 
 the first image of the object. Another, as ac, pene- 
 trates the glass, and being refracted into the line cp, 
 is wholly reflected into pF, on account of the opacity 
 of the posterior side of the mirror, and being again — 
 refracted at E proceeds to o, and forms at a” the live- 
 liest image of the point a. 
 
 Another small bundle“ F penetrates also the glass, is refracted along the line F«, 
 and reflected in the direction of «B, from which a part of it issues, but cannot reach — 
 the eye; the other part is reflected in the direction BH, and then into HJ, from — 
 which a small part is still reflected, but the remainder issues from the glass and is 
 refracted in the direction of the line 10, by which it reaches the eye: consequently 
 it produces the third image, at a”, weaker than the other two, 
 
 The fourth image is formed by a bundle of incident rays, which experience twe 
 refractions like the rest, and five reflections, viz., three from the posterior surface 
 of the glass, and two from the anterior. In regard to the fifth, it requires two 
 refractions and seven reflections, viz., three from the anterior surface, and four from — 
 the posterior ; and so of the rest. It may hence be easily conceived how much the 
 brightness of the images must be dimished, and therefore it is very uncommon to see 
 _more than four or five. 
 
 Fig. 34. 
 
 PROBLEM XXIX. 
 Yo dispose several mirrors in such a manner, that you can see ERE in each 
 of them, at the same time. 
 To produce this effect, nothing is necessary but to dispose the mirrors on the cir- 
 cumference of a circle, in such a manner, that they shall correspond with the chords 
 
 i 
 
PLANE MIRRORS. 297 
 
 of that circle; if you then place yourself in the centre, you will see your image in 
 all the mirrors at the same time. ° 
 
 Remark.—lIf these mirrors are disposed according to the sides of a regular polygon, 
 of an equal number of sides, such as a hexagon or octagon, which seem to be 
 fittest for the purpose, and if all the mirrors are perfectly vertical and plane, they 
 will form a sort of cabinet, which will appear of an immense extent, and in whatever 
 part of it you place yourself, you will see your image, and immensely multiplied. 
 
 If this cabinet be illuminated in the inside, by a lustre placed in its centre, it will 
 exhibit a very agreeable spectacle, as you will see long rows of lights towards what- 
 ever side your sight is directed. 
 
 PROBLEM XXX. 
 
 To measure, by means of reflection, a vertical height, the bottom of which is 
 tnaccessible. 
 
 We shall here suppose that a 8 (Fig. 35.), the vertical 
 height to be measured, is that of a tower, steeple, or 
 such like. Place a mirror atc, ina direction perfectly 
 horizontal; or, because this is very difficult, and as the 
 least aberration might produce a great error in the mea- 
 surement, place in ca vessel containing water, which will 
 reflect the light in the same manner as a mirror. The 
 eye which receives the reflected ray being at 0, measure with care the height op 
 above the horizontal plane of the mirror at c; measure also pc as well as c 8, if the 
 latter is accessible, and then say: As cD, is to Do, so is cB to a fourth proportional 
 BA, which will be the height required. 
 
 But if the bottom of the tower be not accessible, to measure the height a B, we 
 
 must proceed as follows: 
 _ Having performed every part of the preceding operation, except measuring cB, 
 which by the supposition is impossible, take another station, as c, and place there a 
 mirror or vessel of water: then taking your station in d, from which you can sce the 
 point a, by means of the reflected ray co, measure cd and do. When this is done, 
 you must employ the following proportion: As the difference between cp and ed is 
 to cD, so is ce, the distance between the two points of reflection, to a fourth pro- 
 portional, which will be the distance Bc, before unknown. 
 
 When Bc is known, nothing is necessary but to make use of the proportion indi- 
 cated in the first case, which will give the height a B. 
 
 We do not consider this operation as susceptible of much accuracy in practice. 
 Methods purely geometrical, if good instruments are employed, ought always to be 
 preferred ; but we should perhaps have been considered as guilty of an omission, had 
 -we taken no notice of this geometrico-catoptric speculation, though it has never 
 perhaps been put in practice. 
 
 PROBLEM XXXI. 
 To measure an inaccessible height by means of its shadow. 
 
 Fix a stick in a perpendicular direction, on a plane perfectly horizontal, and mea- 
 sure the height of it above that plane, which we shall suppose to be exactly 6 feet. 
 When the sun begins to sink towards the horizon in the afternoon, mark on the 
 ground which is accessible the point c (Fig. 36.) where the shadow of the summit 
 of the tower falls, and also the point c the extremity of the shadow of the stick 
 erected perpendicularly on the same plane: at the end of two hours, more or less, 
 mark, as speedily as possible, the two points p and d, which will be the 
 summits of the shadows at that period; then join the two points of the shadow 
 
. oa 
 Ver 
 a ad 
 , 
 ¥ 
 
 298 OPTICS. 
 
 of the summit of the tower by means of a straight line, and 
 measure their distance; measure also, in like manner, the line 
 which joins the two points ec and d of theshadow of the stick ; 
 after which you will have nothing to do but to employ the fol- 
 lowing proportion: As the length of the line ¢ d which joins the 
 two points of the shadow of the stick, is to the height of the 
 stick a b, so is the length of the line c p which joins the two 
 points of the shadow of the tower, to the height of the tower 
 AB. 
 
 It requires only an acquaintance with the first principles of 
 geometry to be able to perceive, merely by inspecting Fig. 36, 
 that the pyramids B a pc and 6 a dc are similar; consequently, that cd is to a b 
 as c pv to A B, which is the height required. 
 
 PROBLEM XXXII. 
 Of some tricks or kinds of illusion, which may be performed by means of plane mirrors, 
 
 Many curious tricks, capable of astonishing those who have no idea of catoptries, 
 may be performed by the combination of several plane mirrors. Some of these we shall 
 
 here describe. 
 
 Ist. To fire a pistol over your shoulder and hit a mark, with as much certainty asif 
 you took aim at it in the usual manner. (Fig. 37.) 
 
 Fig. 37. To perform this trick, place before you a plane mirror, so dis- 
 posed, that you can see init the object you propose to hit; then 
 es er hoy er <7 rest the barrel of the pistol on your shoulder and take aim, looking 
 Ny gh at the image of the pistol in the glass as if it were the pistol 
 j itself; that is, in such a manner, that the image of the object 
 may be concealed by the barrel of the pistol: it is evident that 
 
 if the pistol be then fired, you will hit the mark. 
 
 2d. To construct a box in which heavy bodies, such as a ball of lead, will appear to 
 ascend contrary to their natural inclination. 
 
 Construct a square box,as AB cD (Fig. 38.), where one of the sides is supposed 
 to be taken off, in order to shew the inside; and fix init a boardu 6 D c,so as to 
 form a plane, somewhat inclined, with a serpentine groove in it 
 of such a size, thata ball of lead can freely rollin it and descend. - 
 Then place the mirror H G FTI in an inclined position, as seen in 
 
 “Sof the box, but so disposed that the eye, when applied to it, can 
 || [see only the mirror, and not the inclined plane u vp, It may be 
 i easily perceived that the image of this plane, viz. HL K G, will 
 wy af seem to be a plane almost vertical, and that a body which rolls 
 from cto c, along the serpentine groove, will appear to ascend in a similar direction 
 from G tou, Hence, if the mirror is very clean, so as not to be observed, orif only 
 a faint light be admitted into the box, which will tend to conceal the artifice, the 
 illusion will be greater, and those not acquainted with the deception wili have a good 
 deal of difficulty to discover it. 
 
 3d, To construct a box in which objects shall be seen through one hole, different from 
 what were seen through another, though in both cases they seem to occupy the whole 
 box. 
 
 Provide a square box, which, on account of its right angles, is the fittest for this 
 purpose, and divide it into four parts, by partitions perpendicular to the bottom, 
 
7S 
 il 
 
 PLANE MIRRORS. 299 
 
 crossing each other in the centre. To these partitions apply plane mirrors, and make 
 a hole in each face of the box, to look through; but disposed in such a manner, that 
 the eye can see only the mirrors applied to the partitions, and not the bottom of the 
 box. Ineachright angle of division formed by the partitions, place some object, which, 
 being repeated in the lateral mirrors, may form a regular representation, such as a par- 
 terre, a fortification or citadel, a pavement divided into compartments, &c. That 
 the inside of the box may be sufficiently lighted, it ought to be covered with a piece 
 of transparent parchment. 
 
 Itis evident that, if the eye be applied to each of the small apertures formed in 
 the sides of the box, it will perceive as many different objects, which however will 
 seem to occupy the whole inside of it. The first will be aregular parterre, the second 
 a fortification, the third a pavement incompartments, and thefourth some other object. 
 
 If several persons look at the same time through these holes, and then ask each 
 other what they have seen, ascene highly comic to those aequainted with the secret 
 may ensue, as each will assert that he sawa different object. 
 
 Remark.—To render the parchment employed for covering optical machines, such as 
 the above,-more transparent, it ought to be repeatedly washed in a clear ley, which must 
 be changed each time: it isthen to be carefully extended, and exposed to the air to dry. 
 
 If you are desirous of giving it some colour, you may employ, for green, verdigrise 
 diluted in vinegar, with the addition of a little dark green; for red, an infusion of 
 Brazil wood; for yellow, an infusion of yellow berries; the parchment afterwards 
 ought to be now and then varnished. 
 
 4th. In a room on the first floor, to see those who approach the door of the house, with- 
 out looking out at the window, and without being observed. 
 
 Under the middle of the architrave’ of the window place a mirror, with its face 
 downwards, and a little inclined towards the side of the apartment, so that it shall 
 reflect to the distance of some feet from the bottom of the window, or on the bot- 
 tom itself, objects placed before and near the door of the house. But as the objects 
 by these means will be seen inverted, in which case it will be difficult to distinguish 
 them, and as it is fatiguing and inconvenient to look upwards, fix another plane mirror 
 in a horizontal position, in the place to which the image of objects is reflected by the 
 first mirror. As this second mirror will exhibit the objects in their proper position, 
 they can be better distinguished. They will appear however at a much greater dis- 
 tance, andas if placed perpendicularly on a plane, somewhat inclined, and almost in 
 such a situation as they would be seen in if you looked downwards from the win- 
 dow; which will be sufficient in general to enable you to distinguish those with whom 
 you are acquainted, 
 
 Two mirrors, arranged in this manner, are represented figure 39. 
 
 Ozanam, and others before him, who published Ma- 
 thematical Recreations, propose by way of problem, to 
 shew a jealous husband what his wife is doing in ano- 
 ther apartment. To bore a hole near the ceiling in the 
 partition wall which separates two apartments, and fix 
 a horizontal mirror, half in the one room and half in the 
 other, to reflect, by means of another mirror placed 
 opposite to it, the image of what might take place in 
 one of these rooms, is certainly an ingenious idea; but 
 there is reason to think that neither Ozanam_ nor his 
 predecessor were jealous husbands, or that they had a 
 singular dependance on the folly and stupidity of the 
 two lovers. 
 
309 OPTICS. 
 
 . PROBLEM XXXIII. 
 To inflame objects, at a considerable distance, by means of plane mirrors. 
 
 Arrange a great number of plane mirrors, each about six or eight inches square, 
 in such a manner that the solar rays reflected from them may be united in one focus. 
 
 It is evident, and has been proved by experience, that if there are a sufficient number — 
 
 of these mirrors, as 100 or 150 for example, they will produce in their common 
 focus a heat capable of inflaming combustible bodies, and even at a very great 
 distance. 7 
 
 This was, no doubt, the invention employed by Archimedes, if he really burnt 
 the fleet of Marcellus by means of burning mirrors, as we are told in history; for 
 Kircher, when at Syracuse, observed that the Roman ships could not have been at a 
 less distance from the walls of the city than twenty-three paces. But it is well 
 
 known that the focus of a concave spherical mirror is at the distance of half its 
 
 radius; consequently the mirror employed by Archimedes must haye been a portion 
 
 of a sphere of at least 46 paces radius, the construction of which would be attended 
 
 with insurmountable difficulties. Besides, can it be believed that the Romans, at so 
 short a distance, would have suffered him to make use of his machine without 
 interruption? On the contrary, would they not have destroyed it by a shower of 
 missile weapons ? 
 
 Anthemius of Tralles, the architect and engineer who lived under Justinian, is 
 the first who, according to the account of Vitellio, conceived the idea of employing 
 plane mirrors for burning ;* but we are not told whether he ever carried this method 
 into execution. It is to Buffon that we are indebted for a proof of its being prac- 
 
 ticable. In the year 1747, this eminent naturalist caused to be constructed a ma- | 
 
 chine consisting of 360 plane mirrors, each 8 inches square, and all moveable on 
 hinges, in such a manner that they could be made to assume any position at pleasure. 
 By means of this machine he was able to burn wood at the distance of 200 feet. 
 Buffon’s curious paper on this subject may be seen in the Memoirs of the Academy 
 of Sciences for the year 1748. 
 
 That the ancients made use of burning glasses is evident from a passage in a play 
 of Aristophanes, called The Clouds, where Strepsiades tells Socrates, that he had 
 found out an excellent method to defeat his creditors, if they should bring an action 
 against him. His contrivance was, that he would get from the jewellers a certain 
 transparent stone, which was used for kindling fire, and then, standing at a distance, 
 he would hold it to the sun, and melt down the wax on which the action was written. 
 
 The astonishing philosophico-military exploit of Archimedes may deserve some 
 farther notice. That exploit has been recorded by Diodorus Siculus, Lucian, Dion, 
 Zouaras, Galen, Anthemius, Tzetzes, and other ancient writers. The account of 
 Tzetzes is so particular, that it suggested to father Kircher the specific method by 
 
 which Archimedes probably effected his purpose. ‘* Arthimedes,” says that author, | 
 
 ‘set fire to the fleet of Marcellus by a burning glass, composed of small square 
 mirrors, moving every way upon hinges; and which, when placed in the sun’s rays, 
 reflected them on the Roman fleet, so as to reduce it to ashes at the distance of a 
 bow-shot.” This account gained additional probability by the effeet which Zonaras 
 
 ascribes to the burning mirror of Proclus, by which he affirms, that the fleet of’ 
 
 Vitellius, when besieging Byzantium, now Constantinople, was utterly consumed. 
 But perhaps no historical testimony could have gained belief to such extraordinary 
 facts, if similar ones had not been seen in modern times. In the Memoirs of the 
 French Academy of Sciences for 1726, p. 172, we read of a plane mirror, of twelve 
 inches square, reflecting the sun’s rays to a concave mirror sixteen inches in diameter, 
 
 * Histoire des Mathematiques, par Montucla, vol. i. p- 328. 
 
SPHERICAL MIRRORS. ess 0) | 
 
 in the focus of which bodies were burnt at the distance of 600 paces. Speaking 
 of this mirror, father Regnault asks, (in his Physies, vol. 3. disc. 10.), ‘* What would 
 be the effect of a number of plane mirrors, placed in a hollow truncated pyramid, 
 and directing the sun’s rays to the same point? Throw the focus, said he, a little 
 farther, and you re-discover or verify the secret of Archimedes.’ This was actually. 
 effected by M. Buffon: in the year 1747 he read to the Academy an account of a 
 mirror, which he had composed of an assemblage of plane mirrors, which made the 
 sun’s rays converge to a point at a great distance. 
 
 OF SPHERICAL MIRRORS, BOTH CONCAVE AND CONVEX. 
 
 A spherical mirror is nothing else than a portion of a sphere, the surface of which. 
 is polished so as to reflect the light in a regular manner. If it be the convex surface 
 that is polished, it will form a convex spherical mirror ; if it be the concave surface, 
 it will be a concave mirror. 
 
 We must here first observe, that when a ray of light falls on any curved surface 
 whatever, it will be reflected in the same manner as from a plane touching the point 
 of that surface where it falls. Thus, if a tangent be drawn at the point of reflection 
 to the surface of a spherical mirror, in the plane of the incident ray and of the centre, 
 the ray_will be reflected, making with that tangent an angle of retlection equal to the 
 angle of incidence, 
 
 PROBLEM XXXIV. 
 The place of an object, and that of the eye, being given; to determine, ina spherical 
 mirror, the point of reflection, and the place of the image. 
 
 The solution of these two problems is not so easy in regard to spherical as to plane 
 mirrors; for when the eye and the object are at unequal distances from the mirror, 
 the determination of the point of reflection necessarily depends on principles which 
 require the assistance of the higher geometry; and this point cannot be assigned in 
 the circumference of the circle without employing one of the conic sections. For this 
 reason, we shall omit the construction, and only observe that there is one extremely 
 simple, in which two hyperbolas between their asymptotes are employed: one of 
 these determines the point of reflection on the convex surface, and the second the 
 point of reflection on the concave surface. 
 
 It will be sufficient for us here to take notice of one property belonging to this 
 point. Let 8 be the object (Fig. 40.), a the place of the eye, E the point of reflec- 
 
 tion from the convex surface of the spherical mirror D E L, 
 
 Fig. 40. the centre of which isc ; also let F & be a tangent to the point 
 
 B: A  &, in the plane of the lines Bc and ac, which it mects int 
 BN. a3 andi; and let the reflected ray AE, when produced, inter- 
 FT SE sect the line B cin H: the points H and I will be so situated, 
 
 tlat we shall have the following proportion: as B c is to cH, 
 so is BI to1H. 
 
 In like manner, if BE be produced till it meet ac inh, we 
 shall have ac: ch: ai: ih; proportions which will be 
 equally true in the case of reflection from a concave surface. 
 _- In regard to the place of the image, opticians have long admitted it as a priaciple 
 that it is inthe point u, where the reflected ray meets the perpendicular dfawn from 
 the object to the mirror. But this supposition (though it serves pretty well to shew 
 how the images of objects are less in convex, and larger in concave, than they are in 
 plane mirrors, ) has no foundation. 
 
 A more philosophical principle advanced by Dr. Barrow is, that the eye perceives 
 the image of the object in that point where the rays forming the small divergent 
 
302 OPTICS. 
 
 bundle, which enters the pupil of the eye, meet together. It is indeed natural to | 
 
 think that this divergency, as itis greater when the object is near and less when | 
 
 it is distant, ought to enable the eye to judge of distance. 
 
 By this principle also we are enabled to assign a pretty plausible reason for the di- — 
 
 minution of objects in convex, and their enlargement in concave mirrors; for the 
 convexity of the former renders the rays, which compose each bundle that enters the 
 eye, more divergent than if they fell on a plane mirror; consequently the point where 
 they meet in the central ray produced, is much nearer. It may even be demonstrated, 
 that in convex mirrors it is much nearer, and in concave much farther distant, than 
 the point H, considered by the ancients, and the greater part of the moderns, as the 
 place of the image. Ina word, it is concluded that in our convex mirrors this image 
 ~ will be still more contracted, and in concave ones more extended, than the ancients 
 supposed; which will account for the apparent enlargement of objects in the latter, 
 and their diminution in the former. 
 
 This principle however is attended with difficulties, which Dr. Barrow, the author 
 of it, does not conceal, and to which he confesses he never saw a satisfactory answer. 
 This induced Dr. Smith, in his Treatise on Optics, to propose another ; but we shall 
 not here enter into a discussion on this subject, as it would be too dry and abstruse 
 for the generality of readers. 
 
 PROBLEM XXXV. 
 The principal properties of Spherical Mirrors, both convex and concave. 
 
 1. The first and principal property of convex mirrors is, that they represent 
 objects less than they would be if seenin a plane mirror at the same distance. This 
 may be demonstrated independently of the place of the image; for it can be shewn 
 that the extreme rays of an object, however placed, which enter the eye after being 
 reflected by a convex mirror, form a less angle, and consequently paint a less image 
 on the retina than if they had been reflected by a plane mirror, which never changes 
 that angle. But, the judgment which the eye in genera) forms respecting the 
 magnitude of objects, depends on the magnitude of that angle, and that image, unless 
 modified by some particular cause. 
 
 On the other hand, in concave mirrors it may be easily demonstrated, that the 
 extreme rays of an object, in whatever manner situated, make a greater angle on ar- 
 riving at the eye, than they would do if reflected from a plane mirror ; consequently 
 the appearance of the object, for the above reason, must be much greater. 
 
 2. In a convex mirror, however great be the distance of the object, its image is 
 never farther from the surface than half the radius; sothat a straight line perpen-- 
 dicular to the mirror, were it even infinite, would not appear to extend farther 
 within the mirror, than the fourth part of the diameter of the circle of which it isa 
 segment. 
 
 But in a concave mirror, the image of a line perpendicular to the mirror is always 
 longer than the line itself; and if this line be equal to half the radius, its image 
 will appear to be infinitely produced. 
 
 3. In convex mirrors, the appearance of a curved line concentric to the mirror, is — 
 
 acircular line also concentric to the mirror; but the appearance of a straight line, or 
 
 plane surface, presented to the mirror, is always convex on the outside, or towards the ~ 
 
 eye. 
 Ina eoncave mirror, the contrary is the case: the image of a rectilineal or plane 
 object appears concave towards the eye. 
 4. A convex mirror disperses the rays; that is to say, if they fall on its surface 
 parallel, it reflects them divergent; if they fall divergent, it reflects them still more 
 divergent, according to circumstances. 
 
} 
 , 
 
 BURNING MIRRORS. 303 
 
 On this property of concave spherical mirrors, is founded the use made of them 
 for collecting the sun’s rays into a small space, where their heat, multiplied in the 
 ratio of their condensation, produces astonishing effects. But this subject deserves 
 to be treated of separately. 
 
 OF BURNING MIRRORS. 
 The properties of burning mirrors may be deduced from the following propo- 
 sition : 
 PROBLEM XXXVI. 
 If a ray of light fall very near the axis of a concave spherical surface, and pa- 
 rallel to that axis, it will be reflected in such a manner, as to meet it at a dis- 
 tance from the mirror nearly equal to half the radius. i 
 
 Fig. 41. For let apc (Fig. 41.) be the concave surface of a 
 BL well polished spherical mirror, of which p is the centre, 
 =r and pB the semi-diameter in the direction of the axis; 
 if EF be aray of light parallel to BD, it will be reflected 
 in the direction of F G, which will intersect the diameter 
 BD in a certain point gc. But the point c will always 
 be nearer to the surface of the mirror than to the centre. 
 For if the radius p F be drawn, we shall have the angles 
 DF Eand DFG equal ; consequently the angles DF E and 
 GD F will also be equal; since the latter, on account of the parallel linesr E and BD, 
 is equal to DFE: the triangle paF then is isosceles, and GD is equal toGF; but 
 GF is always greater than GB; whence it follows that DG also is greater than GB; 
 the point c therefore is nearer the surface of the mirror than the centre. 
 
 But when the are BF is exceedingly small, it is well known that the difference 
 between GF and GB will be insensible ; consequently, in this case, the point @ will» 
 be nearly in the middle of the radius. 4 
 
 This is confirmed by trigonometry ; for if the arc BF be only 5 degrees, and if we © 
 suppose the semi-diameter pB to be J00000 parts, the line BG will be 49809, which 
 differs from half the radius but 7/84, part only, or less then ;j;.* It is even found, 
 that as long as the are BF does not exceed 15 degrees, the distance of the point ¢ 
 from half the semi-diameter is scarcely a 56th part. Hence it appears, that all the 
 rays which fall on a concave mirror, in a direction parallel to its axis, and at a dis- 
 tance from its summit not exceeding 15 degrees, unite at a distance from the mirror 
 nearly equal to half the semi-diameter. Thus, the solar rays, which are sensibly 
 
 parallel when they fall on this concave surface, will be there condensed, if not into 
 
 one point, at least into a very small space, where they will produce a powerful heat, 
 
 _ so much the stronger as the breadth of the mirror is greater. For this reason, the 
 - place where the rays meet is called the focus, or burning point. 
 
 The focus of a concave mirror then is not a point: it has even a pretty sensible 
 magnitude. Thus, for example, if a mirror be the portion of a sphere of six feet 
 radius, and form an are of 30 degrees, which gives a breadth of somewhat more than 
 three feet, its focus ought to be about the 56th part of that size, or between seven 
 and eight lines. The rays, therefore, which fall ona cirele of three feet diameter, 
 will for the most part be collected in a circle of a diameter 56 times less, and which 
 consequently is only the 3136th part of the space or surface. It may hence be easily 
 
 * The calculation in this case is easy. For the arc BF being given, we have given also the 
 angle B D F, as well as G F D, which is equal to it, and consequently the angle p & F, which is the 
 supplement of their sum to two right angles. In the triangle p GF then, we have given the three 
 angles and a side, viz. D F, Which is the radius; and therefore, by a very simple trigonometrical 
 analogy, we can find the side DG or G F, which is equal to it, 
 
304 OPTICS. 
 
 conceived what degree of heat such mirrors must produce, since the heat of boiling 
 water is only triple that of the direct rays of the sun on a fine summer’s day. 
 
 Attempts however have been made to construct mirrors to collect all the rays _ 
 
 of the sun into one point. For this purpose it would be necessary to give to the 
 Fig. 42. _ polished surface a parabolic curve. For let cBD be a para- 
 bola (Fig. 42.), the axis of which is AB: we here suppose 
 
 that the reader has some knowledge of conic sections. It is — 
 well known that in this axis there is a certain point F, so” 
 situated, that every ray, parallel to the axis of this parabola, 
 will be reflected exactly to that point, which on this account 
 has been called the focus. If the concave surface therefore 
 of a parabolic spheroid be well polished, all the solar rays, 
 parallel to each other, and to the axis, will be united in one point, and will produce 
 
 there a heat much stronger than if the surface had been spherical. 
 
 Remarks.—I. As the focus of a spherical mirror is at the distance of a fourth part 
 of the diameter, the impossibility of Archimedes being able, with such a mirror, to 
 burn the Roman ships, supposing their distance to have been only 30 paces, as 
 Kircher says he remarked when at Syracuse, may be easily conceived ; for it would 
 have been necessary that the sphere, of which his mirror was a portion, should have 
 had a radius of 60 paces; and to construct such a sphere would be impossible. A 
 parabolic mirror would be attended with the same inconvenience. Besides, the 
 Romans must have been wonderfully condescending, to suffer themselves to be burnt 
 so near, without deranging the machine. If the mathematician of Syracuse therefore 
 burnt the Roman ships by means of the solar rays, and if Proclus, as we are told, 
 treated in the same manner the ships of Vitellius, which were besieging Byzantium, 
 they must have emploved mirrors of another kind, and could succeed only by an 
 invention similar to that revived by Buffon, and of which he shewed the possibility. 
 (See Prob. 33.) 
 
 The ancients made use of concave mirrors to rekindle the vestal fires. Plutarch, 
 in his life of Numa, says that the instruments used for this purpose, were dishes 
 which were placed opposite to the sun, and the combustible matter placed in the 
 centre ; by which it is probable he meant the focus, conceiving that to be at the 
 centre of the mirror’s concavity. 
 
 II. We cannot here omit to mention some mirrors celebrated on account of their 
 size, and the effects they produced ; one of them was the work of Settala, a canon 
 of Milan: it was parabolic, and, according to the account of father Schott, inflamed 
 wood at the distance of 15 or 16 paces.. 
 
 Villette, an artist and optician of Lyons, constructed three, about the year 1670, 
 one of which was purchased by Tavernier, and presented to the king of Persia; the 
 second was purchased by the king of Denmark, and the third by the king of France. 
 The one last mentioned was 30 inches in diameter, and of about 3 feet focus. The 
 rays of the sun were collected by it into the space of about half-a-guinea. I¢ im- 
 mediately set fire to the greenest wood; it fused silver and copper in a few seconds; 
 and in one minute, more or less, vitrified brick, flint, and other vitrifiable substances. 
 
 These mirrors however were inferior to that constructed by Baron yon Tchirn- 
 ‘hausen, about 1687, and of which a description may be found in the Transactions 
 of Leipsie for that year. This mirror consisted of a metal plate, twice as thick as 
 the blade of a common knife; it was about 3 Leipsic ells, or 5 feet 3 inches, in 
 breadth, and its focal distance was two of these ells, or 3 feet 6 inches: it produced 
 the following effects: 
 
 Wood, exposed to its focus, immediately took fire; and the most violent wind was 
 not able to extinguish it. 
 
 _ 
 
i... 
 
 BURNING MIRRORS. 305 
 
 Water, contained in an earthen vessel, was instantly thrown into a state of ebul- 
 lition; so that eggs were boiled in it in a moment, and soon after the whole water 
 was evaporated. 
 
 Copper and silver passed into fusion in a few minutes, and slate was transformed 
 into a kind of black glass, which, when laid hold of with a pair of pincers, could be 
 drawn out into filaments. 
 
 Brick was fused into a kind of yellow glass; pumice stone and fragments of cru- 
 cibles, which had withstood the most violent furnaces, were also vitrified, &c. 
 
 Such were the effects of the celebrated mirror of Baron von Tchirnhausen; which 
 afterwards came into the possession of the king of France, and which was kept in 
 the Jardin du Roi, exposed to the injuries of the air, which in a great measure 
 destroyed its polish. 
 
 But metal is not the only substance of which burning mirrors have been made. 
 We are told by Wolf, that an artist of Dresden, named Gertner, constructed one in 
 imitation of Tchirnhausen’s mirror, composed only of wood, and which produced 
 effects equally astonishing. But this author does not inform us in what manner 
 Gertner was able to give to the wood the necessary polish. 
 
 Father Zacharias Truber however seems to have supplied this deficiency, by in- 
 forming us in what manner a burning mirror may be constructed with wood and 
 leaf-gold ;- for nothing is necessary but to give to a piece of exceedingly dry and very 
 hard wood the form of the segment of a concave sphere, by means of a turning 
 machine ; to cover it ina uniform manner with a mixture of pitch and wax, and then 
 to apply bits of gold-leaf, about three or four inches in breadth. Instead of gold- 
 leaf, small plane mirrors, he says, might be adapted to the concavity ; and it will be 
 seen with astonishment, that the effect of such a mirror is little inferior to that of a 
 mirror made entirely of metal. 
 
 Father Zahn mentions something more singular than what is related by Wolf cf 
 the artist of Dresden, for he says that an engineer of Vienna, in the year 1699, made 
 a mirror of paste-board, covered on the inside with-straw cemented to it, which was 
 so powerful as to fuse all metals. 
 
 Concave mirrors of a considerable diameter, and which produce the same effect as 
 the preceding, may be procured at present at much less expense. For this advantage 
 we are indebted to M. de Bernieres, one of the controllers general of bridges and 
 causeways, who discovered a method of giving the figure of any curve to glass 
 mirrors; an invention which, besides its utility in Optics, may be applied to various 
 purposes inthe arts. The concave mirrors which he constructed, were round pieces 
 of glass bent into a spherical form; concave on one side and convex on the other, 
 and silvered on the convex side. M. de Bernieres constructed one for the king 
 of France, of 3 feet 6 inches in diameter, which was presented to his majesty in 
 1757. Forged iron exposed to its focus was fused in two seconds: silver ran in such 
 a manner that when dropped into water it extended itself in the form of a spider's 
 web; flint became vitrified, &e. 
 
 These mirrors have considerable advantage over those of metal. Their reflection 
 - from the posterior surface, notwithstanding the loss of rays, occasioned by their 
 ' passage through the first surface, is still more lively than that from the best polished 
 metallic surface; besides, they are not subject, like metallic mirrors, to lose their 
 _ polish by the contact of the air, always charged with vapours which corrode metal, 
 but which make no impression on glass: in a word, nothing is necessary but to pree 
 serve them from moisture, which destroys the silvering. 
 
 PROBLEM XXXVII. 
 Some properties of concave mirrors, in regard to vision, or the formation of images. 
 
 I, If an object be placed between a concave mirror and its focus, its image ia 
 x 
 
wy PO A oie rae 0 na a 
 E = . Sage fs 
 
 =? 
 
 306 — ; OPTICS. 
 
 seen within the mirror, and more magnified the nearer the object is to the focus; so 
 that when the object is in the focus itself, it seems to occupy the whole capacity 
 of the mirror, and nothing is seen distinct. 
 
 If the object, placed in the focus, be a luminous body, the rays which proceed | 
 from it, after being reflected by the mirror, proceed parallel to each other, so that they 
 form a cylinder of light, extended to a very great distance, and almost without 
 diminution. This column of light, if the observer stands on one side, will be easily 
 perceived when it is dark; and at the distance of more than a hundred paces from | 
 the mirror, if a book be held before this light, it may be read. | 
 
 II. If the object be placed between the focus and the centre, and if the eye be 
 either beyond the centre, or between the centre and the focus, it cannot be distinctly 
 perceived, as the rays reflected by the mirror are convergent. But if the object be 
 strongly illuminated, or if it be a luminous body itself, such as a candle, by the 
 union of its rays there will be formed, beyond the centre, an image in an inverted 
 situation, which will be painted on a piece of paper or cloth at the proper distance, 
 or which, to an eye placed beyond it, will appear suspended in the air. 
 
 IiIl. The case will be nearly the same when the object is beyond the centre, 
 in regard to the mirror; an inverted image of the object will be painted then be- 
 tween the focus and the centre; and this image will approach the centre in propor- 
 tion as the object itself approaches it; or will approach the focus as the object 
 removes from it. 
 
 In regard to the place where the image will be painted in both these cases, it 
 may be found by the following rule. 
 
 Let acs (Fig. 48.) be the axis of the mirror, indefinitely pro- 
 duced; F the focus, c the centre, and o the place of the object, 
 between the centre and the focus. If F « be taken a third pro- 
 portional to F o and F ©, it will represent the distance at which 
 the image of the point placed in o will be painted. 
 
 If the object be in w, by employing the same proportion, 
 with the proper changes, that is by making F oa third propor- 
 tional to F w and F ©, asin o, the image of it will be found 
 in 0. 
 
 In the last place, if the object be between the focus and the 
 glass, the place where it will be observed within the mirror, may 
 be found by making Fw toF A, asF a toFo. 
 
 Remarks.—1. This property which concave mirrors have, of forming between the 
 centre and the focus, or beyond the centre, an image of the objects presented to 
 them, is one of those which excite the greatest surprise in persons not acquainted 
 with this theory. For if a man advance towards a large concave mirror, presenting a 
 sword to it, then he comes to the proper distance, he will see a sword blade, with 
 the point turned towards him, dart itself from the mirror; if he retires, the image 
 of the blade will retire; if he advances in such a manner that the point shall be 
 between the centre and the focus, the image of the sword will cross the real sword 
 as if two people were engaged in fighting. 
 
 2. If, instead of a sword blade, the hand be presented at a certain distance, you 
 will see a hand formed in the air in an inverted situation; which will approach the 
 real hand, when the latter approaches the centre, so,that they will seem to meet each 
 other. 
 3. If you place yourself a little beyond the centre of the mirror, and then look 
 directly into it, you will see beyond the centre the image of your face inverted. If- 
 you then continue to approach, this phantastic image will approach also, so that you 
 can kiss it. 
 
LENSES. 307 
 
 4. If anosegay be suspended in an inverted situation 
 (Fig. 44.), between the centre and the focus, a little 
 below the axis, and if it be concealed from the view 
 of the spectator, by means of a piece of black paste- 
 board, an upright image of the nosegay will be formed - 
 above the pasteboard, and will excite the greater asto- 
 nishment, as the object which produces it is not seen; 
 for this reason those not acquainted with the deception 
 will take it fora real object, and attempt to touch it.* 
 
 5. If aconcave mirror be placed at the end of ahall, at an inclination nearly equal 
 to 45°, and if a print or drawing be laid on atable ares the mirror, with the bottom 
 part turned towards it, the figures in the print or drawing will be seen greatly mag- 
 nified; andif a proper arrangement be made so as to favour the illusion, that is if 
 the mirror be concealed, and only a small hole left for looking through, you will 
 imagine that you see the objects themselves. 
 
 | On this principle are constructed what are called Optical boxes, which are now very 
 common: the method of constructing them will be found in the following problem. 
 
 PROBLEM XXXVIUII. 
 
 To construct an optical box or chamber, in which objects are seen much larger than the 
 box itself. 
 
 Provide a square box, of a size proper to contain the concave mirror you intend to 
 employ; that is to say, let each side be a little less than the focal distance of the 
 mirror; and cover the top of it with transparent parchment, or white silk, or glass 
 made smooth, but not polished. 
 
 Apply the mirror to one of the vertical sides of the box, and on the opposite side 
 place a coloured print or drawing, representing a landscape, or seaport, or buildings, 
 &e. The print ought to be introduced into the box by means of a slit, so that it can 
 be drawn out, and another substituted in its place at pleasure. 
 
 At the top of the side opposite to the mirror, a round hole or aperture must be 
 made, for the purpose of looking through; and if the eye be applied to this hole, 
 the objects represented in the print will be seen very much magnified: those whe 
 look at them will think they really behold buildings, trees, &c. 
 
 We have seen some of these machines, which by their construction, the size of the 
 
 ‘mirror, and the correctness of the colouring, exhibited a spectacle highly agreeable and 
 amusing. 
 
 OF LENTICULAR GLASSES, OR LENSES, 
 
 Alens isa bit of glass having a spherical form on both sides, or at least on one side. 
 Some of them are convex on the one side and plane on the other; and others are 
 conyex on bothsides: some are concave on one side, or on both; and others are con-— 
 _Vex on one side and concave on the other. Those convex on both sides, as they resem- 
 ble a lentil, are in general distinguished by the name of lenticular glasses, or lenses. 
 The uses to which these glasses are applied, are well known. Those which are 
 convex magnify the appearance of objects, and aid the sight of old people; on the: 
 other hand, the concave glasses diminish objects, and assist those who are short-sighted. 
 |The former collect the rays of the sun around one point, called the focus ; and when 
 of a considerable size, produce heat and combustion The concave glasses, on the 
 contrary, disperse the rays of the sun. Both kinds are employed in the con- 
 ‘struction of telescopes and microscopes. 
 
 * Curious spectres and appearances, formed in this manner, have of late years ‘been exhibited as 
 | Baars to spectators in London. 
 > a 
 
308 OPTICS. 
 
 ‘PROBLEM XXXIX. 
 To find the focus of a Glass Globe. 
 
 As glass globes supply, on many occasions, the place of lenses, itis proper that we 
 should here say a few words respecting their focal distance. The method of determin- 
 ing itisas follows. 
 
 Let Bc pD (Fig. 45.) bea glass sphere, the centre of whichis Fr, and cpa dia- 
 meter to which the incident ray a B is parallel. This ray, when it meets with the 
 
 surface of the sphere in B, will not continue its course in a 
 Fig. 45. straight line, as would be the case if it did not enter a new 
 medium, but will approach the perpendicular drawn from the 
 centre F to B the point of incidence. Consequently, when it 
 issues from the sphere at the point 1, it would meet the diameter 
 ina point £, if it did not deviate from the perpendicalar F 1, 
 which makes it take the direction 1 0, and proceed to the 
 point o, the focus required. 
 
 To determine the focus 0, first find the point of meeting 5, 
 which may be easily done by observing that in the triangle FB x, 
 the side F B is to F Eas the sine of the angle FE B is to the 
 sine of the angle F B E; or, on account of the smallness of these 
 angles, as the angle F E B, or its equal G B E£, is to the angle 
 F BE; for we here suppose the incident ray to be very near the diameter c p; 
 consequently the angle a B H is very small, as well as its equal Fr BG; and angles 
 extremely small have the same ratio as their sines. But, by the laws of refraction, 
 when a ray passes from air into glass, the ratio of the angle of incidence a 8 4H, or 
 GBF, to the angle of refraction F B 1, if the angles be very small, is as 3 to 2, and 
 therefore the angle F B E is nearly the double of = B G: it thence follows that the 
 side Fr E, of the triangle F BE, is nearly the double of FB, or equal to twice the 
 radius ; Consequently p x is equal to the radius. 
 
 To find the point o, where the ray, when it issues from the sphere, and deviates 
 from the perpendicular, ought to meet the line pg, the like reasoning may be em- 
 ployed. In the triangle ro £, the side 1o is to of nearly as the angle 1£ 0, or its 
 equal IF E, is to the angle orE. Now these two angles are equal; for the angle 
 IF D is the one third of the angle of incidence FBG or ABH; but, by the law of re- 
 fraction, the angle o1E is nearly the half of the angle of incidence u1k, or of its 
 equal F1 8B, which is 2 of the angle rBG: like the preceding it is therefore the third 
 of FBG or HB4, and consequently the angles olf and OE1 are equal; whence it 
 follows that oF is equal to of, which is itself equal to D 0, on account of their very 
 great proximity. Therefore po, or the distance of the focus of a glass globe from 
 the surface, is equal to half the radius, or the fourth part of the diameter. Q@. E.D. 
 
 PROBLEM XL. 
 To find the focus of any lens. 
 
 _ The same reasoning, as that employed to determine the course of a ray passing 
 through a glass sphere, might be employed in the present case. But for the sake 
 ‘of brevity, we shall only give a genera] rule, demonstrated by opticians, which in- 
 cludes all the cases possible in regard to lenses, whatever combinations may be 
 formed of convexities and concavities. We shall then shew the application of it 
 to a few of the principal cases. It is as follows: 
 
 As the sum of the semi-diameters of the two convexities, is to one of them, so is 
 the diameter of the other, to the focal distance, 
 
 In the use of this rule, one thing in particular is to be observed. When one of the 
 
 i 
 
: 
 : 
 
 LENSES. 309 
 
 faces of the glass is plane, the radius of its sphericity must be considered as infinite ; 
 and when concave, the radius of the sphere, of which this concavity forms a part, 
 
 - must be considered as negative. This will be easily understood by those who are in 
 
 the least familiar with algebra. 
 
 Case 1.—When the lens is equally convex on both sides. 
 Let the radius of the convexity of each of the faces be, for example, equal to 12 
 
 inches. By the general rule we shall have this proportion: As the sum of the radii, 
 
 or 24 inches, is to one of them, or 12 inches, so is the diameter of the other, or 24 
 inches, to a fourth term, which will be 12 inches, the focal distance. Hence it ap- 
 
 _ pears that a lens equally convex on both sides unites the solar rays, or in general 
 
 rays parallel to its axis, at the distance of the radius of one of the two sphericities. 
 
 Case u1.—When the lens is unequally convex on both sides. 
 
 If the radii of the convexities be 12 and 24, for instance, the following proportion 
 must be employed: As 12-1 24, or 36, is to 12, the radius of one of the convexities, 
 
 so is 48, the diameter of the other, to 16; or as 12-4 24, or 36, is to 24, the radius 
 
 —s 
 
 of one of the convexities, so is 24, the diameter of the other, to 16: the distance 
 of the focus therefore will be 16 inches. 
 
 Case 111.—When the lens has one side plane. 
 
 If the sphericity on the one side be as in the preceding case, we must. say, by 
 applying the general rule: As the sum of the radii of the two sphericities, viz. 12, — 
 an infinite quantity, is to one of them, or the infinite quantity, so is 24, the diameter 
 of the other convexity, to a fourth term, which will be 24; for the two first terms 
 are equal, for an infinite quantity increased or diminished by a finite quantity, is 
 always the same: the two last terms therefore are equal; and it hence follows,’ 
 that a plano-convex glass has its focus at the distance of the diameter from its 
 convexity. — 
 
 Case 1v.—When the lens is convex on the one side, and concave on the other. 
 
 Let the radius of the convexity be still 12, and that of the concavity 27. As a 
 concavity is a negative convexity, this number 27 must be taken with the sign — 
 prefixed. We shail therefore have this proportion: 
 
 As 12 inches — 27, or — 15 inches, is to the radius of the coneavity — 27 (or as 
 15 is to 27), so is 24 inches, the diameter of the convexity, to 43!. This is the focal 
 distance of the lens, and is positive or real; that is to say, the rays falling parallel 
 to the axis, will really be united beyond the glass. The concavity indeed having a 
 greater diameter than the convexity, this must cause the rays to diverge less than 
 the convexity causes them to converge. But if the concavity be of a less diameter 
 than the convexity, the rays, instead of converging when they issue from the glass, 
 will be divergent, and the focus will be before the glass: in this case it is called 
 virtual. Thus, if the radius of the concavity be J2, and that of the convexity 27, 
 we shall have, by the general rule: As 27 — 12, or 15, is to 27, so is — 24 to — 
 
 431. The last term being negative, it indicates that the focus is before the glass, and 
 
 that the rays will issue from it divergent, as if they came from that point. 
 
 CasE v.— When the lens is concave on both sides. 
 
 If the radii of the two concavities be 12 and 27 inches, we shall have this propor- 
 
 ‘tion: As — 12 — 27 is to — 27, or as 39 is to 27, so is — 24 to 168. The last 
 
 term being negative, it shews that the focus is only virtual, and that the rays, when 
 they issue from the glass, will proceed diverging, as if they came from a point 
 situated at the distance of 168 inches before the glass. 
 
310 OPTICS. 
 
 Casr vi.—-When the lens is concave on one side, and plane on the other. 
 
 If the radius of the concavity be still 12, the above rule will give the follow- 
 ing proportion: As — 12 an infinite quantity, is to an infinite quantity, so is — 24 
 to — 24; for an infinite quantity, when it is diminished by a finite quantity, remains 
 still the same. Thus it is seen that in this case the virtual focus of a plano-con- 
 caye glass, or the point where the rays after their refraction seem to diverge, is at 
 a distance equal to the diameter of the concavity, as the point to which they con- 
 yerge is in the case of the plano-convex glass. 
 
 These are all the cases that can occur in regard to lenses: for that where the 
 two concavities might be supposed equal is comprehended in the fifth. 
 
 Remark.—In all these calculations, we have supposed the thickness of the glass to 
 be of no consequence in regard to the diameter of the sphericity, which is the most 
 common case; but if the thickness of glass were taken into consideration, the deter- 
 minations would be different. 
 
 OF BURNING GLASSES. 
 
 Lenticular glasses furnish a third method of solving the problem, already solved by 
 means of mirrors, viz. to unite the rays of the sun in such a manner as to produce fire 
 and inflammation: for a glass of a few inches diameter will produce a heat suffi-. 
 ciently strong to set fire to tinder, linen, black or grey paper, &e. 
 
 The ancients were acquainted with this property in glass globes, and they even 
 sometimes employed them for the above purpose. It was probably by means of a glass 
 globe that the vestal fire was kindled. Some indeed have endeavoured to prove 
 that they produced this effect by lenses: but De la Hire has shewn that this idea 
 is entirely void of foundation, and that the burning glasses of the ancients were 
 only glass globes, and consequently incapable of producing a very remarkable 
 effect. 
 
 Baron von Tchirnhausen, who constructed the celebrated mirror already men- 
 tioned, made also a burning glass, the largest that had ever been seen. ‘This 
 mathematician, being near the Saxon glass manufactories, was enabled, about the 
 year 1696, to procure plates of glass sufficiently thick and broad, to be converted into 
 lenses several feet in diameter. One of them, of this size, inflamed combustible 
 substances at the distance of 12 feet. Its focus at this distance was about 1} inch 
 in diameter. But when it was required to make it produce its greatest effects, the 
 focus was diminished by means of a second lens, placed parallel to the former, and at 
 the distance of four feet. In this manner the diameter of the focus was reduced 
 to 8 lines, and it then fused metals, vitrified flint, tiles and slate, earthen ware, 
 &c.; in a word, it produced the same effects as the burning mirrors of which we have 
 already spoken. ' 
 
 Some years ago a lens, which one might have taken for that of Tchirnhausen, 
 was exhibited at Paris. The glass of which it consisted was radiated and yellowish; 
 and the person to whom it belonged asked no less for it than £500 sterling. 
 
 For the means of obtaining, at a less expence, glasses capable of producing the 
 same effects, we are indebted to M. de Bernieres, of whom we have already spoken. 
 By his invention for bending glass, two round plates are bent into aspherical form, and 
 being then applied to each other the interval between them is filled with distilled water, 
 or spirit of wine. These glasses, or rather water lenses, have their focus a little farther 
 distant, and ceteris paribus ought to produce a somewhat less effect ; but the thin- 
 ness of the glass and the transparency of the water occasion less loss in the rays than 
 in lens of several inches in thickness. Ina word, it is far easier to procure a lens 
 of this construction than solid ones like that of Tchirnhausen. M. de Trudaine, 
 
TELESCOPES. Sih 
 
 some years ago, caused to be constructed, by M. de Bernieres, one of these water 
 lenses 4 feet in diameter, with which some philosophical experiments have been 
 already made in regard to the calcination of metals and other substances. The heat 
 produced by this instrument is much superior to that of all the burning glasses and 
 mirrors hitherto known, and even to that of all furnaces. We have reason to expect 
 from it new discoveries in chemistry. Weshall here add that with water lenses, of a 
 much smaller size, M. de Bernieres has fused metals, vitrifiable stones, &c. 
 
 PROBLEM XLI. 
 Of some other properties of Lenticular Glasses. 
 
 1. If an object be exceedingly remote, so that there is no proportion between its 
 distance and the focal distance of the glass, there is painted in the focus of the 
 lens an image of the object in an inverted situation, This experiment serves as the 
 basis of the construction of the camera obscura. In this manner the rays of the sun, 
 or of the moon, unite in the focus of a glass lens, and forma small circle, which is 
 nothing else than the image of the sun or moon, as may be easily perceived. 
 
 2. In proportion as the object approaches the glass, the image formed by the rays 
 proceeding from the object, recedes from the glass; so that when the distance of 
 the object is double that of the focus, the image is painted exactly at the double of 
 that distance ; if the object continues to approach, the image recedes more and more ; 
 and when the object is in the focus, no image is formed; for it is at an infinite 
 distance that it is supposed to form itself. In this case therefore the rays which fall 
 on the glass, diverging from each point of the object, are refracted in such a manner 
 as to proceed parallel to each other. 
 
 The method of determining, in general, 
 the distance from the lens at which the 
 image of the object is formed, is as follows. 
 Let oc be the object (Fig. 46.), pF its 
 distance from the glass, and E F the focal 
 distance of the glass ; if we make use of 
 this proportion : as F D is tor HE, so is EF 
 to E G, takingr Gc on the other side of the 
 glass when & Dis greater than E F, the 
 point c will be that of the axis to which the point p of the object, situated in the axis, 
 will correspond. 
 
 Hence it may be easily seen, that when the distance of the object from the 
 focus is equal to nothing, the distance © G must be infinite, that is to say there can 
 
 _. be no image. 
 
 It must also be observed, that when EF is greater than ED, or when the object is 
 between the glass and the focus, the distance x G must be taken in a contrary direc- 
 tion, or on this side of the glass, as EY; which indicates that the rays proceeding 
 from the object, instead of forming an image beyond the glass, diverge as if uley 
 proceeded from an object placed at g. 
 
 OF TELESCOPES, BOTH REFRACTING AND REFLECTING, 
 
 Of all Optical inventions, none is equal to that of the telescope: for without - 
 mentioning the numerous advantages derived from the common use of this wonderful 
 instrument, it is to it we are indebted for the most interesting discoveries in astro- 
 nomy. Itis by its means that the human mind has been able to soar to those regions 
 otherwise inaccessible to man, and to examine the principal facts which serve as the 
 foundation of our knowledge respecting the heavenly bodies. 
 
 The first telescope was constructed in Holland, about the year 1609; but there 
 
312 yrs 
 
 is much uncertainty in regard to the name of the inventor, and the means he em- 
 ployed in the formation of his instrument. A dissertation on this subject may be 
 seen in Montucla’s History of the Mathematics. We shall confine ourselves at 
 present to a description of the different kinds of telescopes, both refracting and 
 reflecting, and of the manner in which they produce their effect. 
 
 Of Refracting Telescopes. 
 
 Ist. The first kind of telescope, and that most commonly used, is composed of a 
 convex glass, called the object glass, because it is that nearest the objects, and a 
 concave one called the eye-glass, because it is nearest the eye. These glasses must 
 be disposed in such a manner, that the posterior focus of the object glass shall 
 coincide with the concave glass. By means of this disposition, the object appears 
 magnified in the ratio of the focal distance of the object glass, to that of the eye- 
 glass. Thus, if the focal distance of the object glass be ten inches, and that of the 
 eye-glass one inch, the instrument will be nine inches in length, and will magnify 
 objects ten times. 
 
 This kind of telescope is called the Batavian, on account of the place where it 
 was invented. It is known also by the name of the Galilean, because Galileo, 
 having heard of it, constructed one of the same kind, and by its means was enabled 
 to make those discoveries in the heavens which have immortalized his name. At 
 present, very short telescopes only are made according to this principle; because 
 they are attended with one defect, which is, that when of a considerable length they 
 have a very confined field. 
 
 2d. The second kind of telescope is called the astronomical, because employed 
 chiefly by astronomers. It is composed of two convex glasses, disposed in sucha 
 manner, that the posterior focus of the object glass and the anterior focus of the 
 eye-glass coincide together, or very nearly so. The eye must be applied to a small 
 aperture, at a distance from the eye-glass, equal to that of its focus. It will then 
 have a field of large extent, and it will shew the objects inverted, and magnified in 
 the ratio of the focal distances of the object glass and eye-glass. If we take, by 
 way of example, the proportions already employed, the astronomical telescope will be 
 12 inches in length, and will magnify ten times. 
 
 Telescopes of very great length may be constructed according to this combina- 
 tion, It is common for astronomers to have them of 12, 15, 20, and 30 feet. Huy- 
 gens constructed one for himself of 123 feet, and Hevelius employed one of 140. 
 But the inconvenience which attends the use of such long telescopes, in consequence 
 of their weight, and the bending of the tubes, has made them be laid aside, and 
 another instrument more commodious has been substitued in their stead. Hartso- 
 ecker made an object glass of 600 feet focus, which would have produced an extra- 
 ordinary effect had it been possible to use it. 
 
 3d. The inconvenience of the Batavian telescopes, which suffer only a small 
 quantity of objects to be seen at once, and that of the astronomical telescope, 
 which represents them inverted, have induced opticians to devise a third arrangement 
 of glasses, all convex, which represents the objects upright, gives the same field as 
 the astronomical telescope, and which is therefore proper for terrestrial objects: on 
 this account it is called the terrestrial telescope. It consists of a convex object 
 glass, and three equal eye-glasses. The posterior-focus of the object glass generally 
 coincides with the anterior one of the first eye-glass; the posterior focus of the 
 latter coincides also with the anterior focus of the second, and in like manner the 
 posterior focus of the second with the anterior one of the third, at the posterior 
 focus of which the eye ought to be placed. This instrument always magnifies in 
 the ratio of the focal distances of the object glass and one of the eye-glasses. But 
 
TELESCOPES. 313 
 
 it may be readily seen that the length is increased four times the focal distance of the 
 eye-glass. 
 
 4th. The image of objects might be made to appear upright by employing only 
 two eye-glasses: for this purpose it would be necessary that the first should be ata 
 distance from the focus of the object glass equal to twice its own focal distance; 
 and that the anterior focus of the second should be at twice that distance. Such 
 is the terrestrial telescope with three glasses; but experience has shewn that, by 
 this arrangement, the objects are somewhat deformed, for which reason it is no 
 longer used. 
 
 5th, Telescopes with five glasses have also been proposed, in order to bend the 
 rays gradually, as we may say, and to obviate the inconveniences of the too strong 
 refraction, which suddenly takes place at the first eye-glass; and also to increase © 
 the field of vision. We have even heard of some telescopes of this kind which were 
 attended with great success ; but we do not find that this combination of glasses has 
 been adopted. 
 
 6th. Some years ago a new kind of telescope was invented, under the name of the 
 achromatic, because it is free from those faults occasioned by the different refrangi- 
 bility of light, which in other telescopes produces colours and indistinctness. The only 
 difference between this and other telescopes is, that the object glass, instead of being 
 formed of one lens, is composed of two or three made of different kinds of glass, 
 which have been found by experience to disperse unequally the different coloured 
 rays of which light is composed. One of these glasses is of crown-glass, and the 
 other of flint-glass. An object glass of this kind, constructed according to certain 
 dimensions determined by geometricians, produces in its focus an image far more 
 distinct than the common ones; on which account much smaller eye-glasses may be 
 employed without affecting the distinctness, as is confirmed by experience. These 
 telescopes are called also Dollond’s telescopes, after the name of the English artist 
 who invented them. By the above means, the English opticians construct tele- 
 scopes of a moderate length, which are equal to others of a far greater size; and 
 small ones, not much longer than opera-glasses, with which the satellites of Jupiter 
 may be seen, are sold under Dollond’s name at Paris. M. Antheaume, according to 
 the dimensions given by M. Clairault, made, in that capital, an achromatic telescope 
 of 7 feet focal distance, which, when compared with a common one of 30 or 36 feet, 
 was found to produce the same effect. 
 
 This invention gives us reason to hope that discoveries will be made in the hea- 
 vens, which a few years ago would have appeared altogether impossible. It is not 
 improbable even that astronomers will be able to discover in the moon habitations 
 and animals, spots in Saturn and Mercury, and the satellite of Venus, so often seen 
 and so often lost. 
 
 [Since the publication of the former editions of this work, achromatic telescopes, 
 greatly exceeding in size any that had before been contemplated, have been con- 
 structed in Paris, and at Munich. M. Guinand, of Neufchatel, succeeded in 
 overcoming, to a certain extent, the difficulties which had been experienced in the 
 manufacture of large and homogeneous discs of flint glass; and some of the Parisian 
 glass-makers are understood to be in possession of processes by which the same 
 object may be effected. This has given a decided impulse to the efforts of opticians, _ 
 aud the result has been, that at the observatories of Cambridge, Munich, Dorpat, and 
 other places, astronomical telescopes, of more than ]2inches aperture and 20 feet focal 
 length, have been mounted, and are in action ; and equatcrial motion being commu- 
 nicated by clock work, they are as manageable as instruments of a very moderate size. 
 With the Dorpat telescope, Professor Struve has already rendered important services 
 
. 
 
 314° OPTICS. 
 
 to astronomical science. A full description of this instrument may be seen in the — 
 Memoirs of the Royal Astronomical Society, vol. ii. part 1.] 
 
 To give an accurate idea of the manner in which telescopes magnify the appearance 
 of objects, we shall take, by way of example, that called the astronomical telescope, 
 as being the simplest. If it be recollected that a convex lens produces in its focus 
 an inverted image of objects which are at a very great distance, it will not be diffi- 
 cult to conceive, that the object glass of this telescope will form behind it, at its 
 focal distance, an inverted image of any object towards which it is directed. But, 
 by the construction of the instrument, this image is in the anterior focus of the 
 eye-glass, to which the eye is applied; consequently the eye will perceive it dis- 
 tinctly ; for it is well known, that when an object is placed in the focus of a lens, 
 or a little on this side of it, it will be seen distinctly through the glass, and in the 
 same direction. 'The image of the object, which here supplies its place, being then 
 inverted, the eye-glass, through which it is viewed, will not make it appear upright, 
 and consequently the object will be seen inverted. 
 
 In regard to the size, it is demonstrated, that the angle under which the image is 
 seen is to that under which the object is seen from the same place, as the focal 
 distance of the object glass is to that of the eye-glass: hence the magnified appear- 
 ance of the object. 
 
 In terrestrial telescopes, the two first eye-glasses only invert the image; and this 
 telescope therefore must represent objects upright. But having said enough respect= — 
 
 ing refracting telescopes, we shall now proceed to reflecting ones. - 
 
 Of Reflecting Telescopes. 
 
 Those who are well acquainted with the manner in which objects are represented 
 by common telescopes, will readily conceive that the same effect may be nroduced 
 by reflection ; for a concave mirror, like a lens, paints i in its focus an image of distant 
 objects. If means then are found to reflect the image on one side, or backwards, in 
 such a manner as to be made to fall in the focus of a convex glass, and to view it 
 through this glass, we shall have a reflecting telescope. It need therefore excite no 
 surprise that before Newton, and in the time of Descartes and Mersenne, telescopes 
 ‘on this principle were proposed. 
 
 Newton was led to this invention while endeavouring to discover some method 
 of remedying the want of distinctness in the images formed by glasses; a fault 
 which arises from the different refrangibility of the rays of light that are decom- - 
 posed. Every ray, of whatever colour, being reflected under an angle equal to the 
 angle of incidence, the image is much more distinct, and better terminated in all its 
 parts, as may be easily proved by means of a concave mirror. On this account he 
 was able to apply an eye-glass much smaller, which would produce a greater mag- 
 nifying power; and this reasoning was confirmed by experience. 
 
 Newton never constructed telescopes of more than fifteen inches in length. Ac- 
 
 cording to his method, the mirror was placed in the bottom of the tube, and reflected 
 the image of the object towards its aperture: near this aperture was placed a plane 
 mirror, that is to say, the base of a small isosceles rectangular prism, silvered at the 
 back, and inclined at an angle of 45 degrees. This small mirror reflected the image 
 towards the side of the tube, where there was a hole, into which was fitted a lens 
 of a very short focal distance, to serve as the eye-glass. The object then was 
 viewed from the side, a method, in many cases, exceedingly convenient. Mr, Had- 
 ley, a fellow of the Royal Society, constructed, in the year 1723, a telescope of this 
 kind, 5 feet in length, which was found to produce the same effect as the telescope 
 of 123 feet, presented to the Royal Society by Huygens. 
 
TELESCOPES. 315. 
 
 The reflecting telescopes used at present are constructed in a manner somewhat 
 different. The concave mirror, at the bottom of the tube, has a round hole in the 
 middle, and towards the other end isa mirror, sometimes plane, turned directly 
 towards the other one, which, receiving the image near the middle of the focal dis- 
 tance, reflects it towards the hole in the other mirror. Against this hole is applied 
 a lens of a short focal distance, which serves as an eye-glass, or for viewing ter- 
 restrial objects, in order that they may appear-_upright ; and three eye-glasses are 
 used, arranged in the same manner as in terrestrial telescopes. 
 
 A telescope however may be made to magnify much more by the following con- 
 struction. The large mirror, as in all the others, is placed at the bottom, and has a 
 hole in the centre, before which the eye-glass is applied. At the other end of the 
 tube is another concave mirror, of a less focal distance than the former, and so dis- 
 posed that the image reflected by the former is painted very near its focus, but at a 
 little farther distance than the focus from its surface. This produces another image 
 beyond the centre, which is greater as the first one is nearer the focus: this image is 
 formed very near the hole in the centre of the large mirror, opposite to which the 
 eye-glass is in general placed. 
 
 This kind of reflecting telescope is called the Gregorian, because proposed by Mr. 
 James Gregory, even before Newton conceived the idea of his; and it this kind 
 which is at present most in use. 
 
 There is also the telescope of Cassegrain, who employs a convex mirror to magnify 
 the image formed by the first concave one. Dr. Smith thought it attended with so 
 many advantages, that he was induced to analyse it in his Treatise on Optics. Cas- 
 segrain was a French artist, who proposed this method of construction about the 
 year 1665, and nearly at the same time that Gregory proposed his. It is certain that 
 the length of the telescope is by these means considerably diminished. 
 
 The English, for a long time, have enjoyed a superiority in works of this kind. 
 The art of casting and polishing the metallic mirrors, necessary for these instruments, 
 is indeed exceediugly difficult. M. Passement, a celebrated French artist, and the 
 brothers Paris and Gonichon, opticians at Paris, are the first who attempted to vie 
 with them in this branch of manufacture; and both have constructed a great number 
 of reflecting telescopes, some of which are 5 or 6 feet inlength. Among the English, 
 no artist distinguished himself more in this respect than Short, though his telescopes 
 were not of great length: besides some of 4, 5, and 6 feet, he made one of 12, which 
 belonged some years ago to the physician of Lord Macclesfield. By applying a lens 
 of the shortest focal distance which it could bear, it magnified about 1200 times. The 
 satellites of Jupiter therefore, seen through this telescope, are said to have had a sen- 
 sible apparent diameter. But this telescope, as we have heard, is no longer in exist- 
 ence, the large mirror being lost. 
 
 The longest of all the reflecting telescopes ever yet constructed, if we except that 
 lately made by Herschel, is one in the king’s collection of philosophical and optical 
 
 instruments at La Meute; it is the work of Dom Noel, a Benedictine, the keeper 
 
 of the collection, and was begun several years before he was placed atthe head of 
 that establishment, where he finished it, and where the curious were allowed to see it, 
 and to contemplate with it the heavens. It is mounted on a kind of moveable 
 pedestal, and, notwithstanding its enormous weight, can be moved in every direc- 
 tion, along with the observer, by a very simple mechanism. But what would be 
 most interesting, is to ascertain the degree of its power, and whether it produces 
 an effect proportioned toits length, or at least considerably greater than the largest 
 and best reflecting telescopes constructed before; for we know that the effects of 
 these instruments, supposing the same excellence in the workmanship, do not increase 
 in proportion to the length. 
 
 Huygens’ telescope of 123 feet, which he presented to the Royal Society, did not 
 
 “a 
 
‘ 
 
 ¢ 
 
 316 OPTICS. 
 
 produce an effect quadruple that of a good telescope of 30 feet; and the case must 
 be the same in regard to reflecting telescopes, where the difficulties of the labour 
 ‘are still greater; so that if a telescope of 24 feet produced one half more effect 
 than another of 12, or only the double of one of 6 feet, it ought, in our opinion, to 
 be considered as a good instrument. 
 
 We have heard that Dom Noel was desirous of making this comparison, and the 
 method he proposed was rational. We have long considered it as the only one proper 
 for comparing suchinstruments. It is to place at the distance of several hundred feet 
 printed characters of every size, composingbarbarous words without any meaning, in 
 order that those wao make the experiment may not be assisted by one or two words 
 to guess the rest. The telescope, by means of which the smallest characters are 
 read, will undoubtedly be the best. We have seen stuck up, on the dome of the 
 Hospital of Invalids, pieces of paper of this kind, which Dom Noel had placed 
 there for the purpose of making this comparison; but unfortunately such instru- 
 ments cannot be brought to one place. Printed characters, such as above de- 
 scribed, might therefore be fixed up at a convenient distance from each without remove 
 
 ing the instruments, and persons appointed for the purpose ought to go to the different 
 
 observatories, at times when the weather is exactly similar, and examine what charac- 
 
 ters can be read by each telescope. By this method a positive answer to the above 
 
 question would be obtained. 
 
 But the largest and the most powerful of all the reflecting telescopes has been 
 lately made by Dr. Herschel, under the auspices of the British monarch; a conse- 
 quence of which was the discovery of his new primary planet, and of many additional 
 satellites. After a long perseverance in a series of improvements of reflecting teles- 
 
 copes, of the Newtonian form, making them successively larger and more accurate, 
 
 this gentleman came at length to make one of the amazing size of forty feet in length. 
 This telescope was begun in the year 1785, and completed in 1789. ‘The length of 
 the sheet-iron tube is 40 feet,and diameter 4 feet 10 inches. The great mirror is 
 491 inches in diameter, 33 inches thick, and weighs 21181b. The whole is managed 
 by a large apparatus of machinery, of wheels and pulleys, by means of which it is 
 easily moved in any direction, vertically and sideways. The observer looks in at the 
 outer or object end; from whence proceeds a pipe to a small house near the instru- 
 ment for conveying information by sound, backward and forward to an assistant, who 
 thus under cover sets down the time and observations made by the principal observer. 
 The consequences of this, and the other powerful machines of this gentleman, 
 have been new discoveries in the heavens of the most important nature. * 
 
 PROBLEM XLII. 
 
 Method of constructing a telescope, by means of which an object may be seen, evenwhen 
 then instrument appears to be directed towards another. 
 
 As it is not polite to gaze at any one, a sort of glass has been invented in 
 England, by means of which, when the person who uses it seems to be viewing one ob- 
 ject, he is really looking at another. The construction of this instrument is very simple. 
 
 Adapt to the end of an opera glass (Fig. 47.), the object glass of which in this 
 case becomes useless, a tube with a lateral aperture as large as the diameter of the 
 
 * This celebrated telescope, by which its maker, Sir William Herschel, LL.D., made discoveries 
 of the greatest importance in the heavens, has long been in a state unfit for use : the surface of the 
 great mnror having become covered with a crystallised crust. Sir J. Herschel, the distinguished son 
 of Sir William, uses a twenty-feet reflector, of his own construction ; and he has found its powers 
 quite sufficieut for the most delicate observations of astronomical phenomena. 
 
 The late Mr. Ramage, an Aberdeen tradesman, made a considerable number of excellent mirrors of 
 large dimensions,some of which were fitted up as telescopes, and mounted in a manner similar to 
 the great one of Sir William Herschel, with some simplifications. Une was for several years on 
 view at the Greenwich Observatory. Mr. Ramage had one fitted up for himself; and we believe Sir 
 John Ross, the northern navigator, had one set up at his house at Stranraer, with which he occa. 
 sionally made observations on the eclipses of Jupiter’s satellites. 
 
 “ 
 
MICROSCOPES. 317 
 
 tube will admit, and opposite to this aperture place a small 
 mirror inclined to the axis of the tube at an angle of 45 
 
 Fig. 47. 
 
 degrees, and having its reflecting surface turned towards the 
 object glass. It is evident that when this telescope is directed 
 
 straight forwards, you will see only some of the lateral objects, 
 viz. those situated near the line drawn from the eye in the direction of the axis of 
 the telescope, and reflected by the mirror. These objects will appear upright, but 
 transposed from right to left. To conceal the artifice better, the fore part of the 
 
 - telescope may be furnished with a plane glass, which will have the appearance of an 
 _ object glass placed in the usual manner. 
 
 This instrument, which is not very common in France, is exceedingly convenient 
 for gratifying one’s curiosity in the playhouse, and other places of public amusement, 
 especially if the mirror be so fixed as to be susceptible of being more or less 
 inclined ; for those who use it, while they seem to look at the stage and the per- 
 
 _ formers, may without affectation, and without, violating the rules of politeness, 
 examine an interesting figure in the boxes. 
 
 We must however observe that the first idea of this instrument is not very new ; 
 for the celebrated Hevelius, who it seems was afraid of being shot, proposed many 
 years ago his polemoscope, or telescope for viewing under cover, and without danger, 
 warlike operations, and those in particular which take place during the time of 
 a siege. It consisted of a tube bent in such a manner as to form two elbows, in 
 each of which was a plane mirror inclined at an angle of 45 degrees. The first 
 part of the tube was made to rest on the parapet towards the enemy; the image 
 reflected by the first inclined mirror passed through the tube in a perpendicular 
 direction, and meeting with the second mirror was reflected horizontally towards 
 the eye glass, where the eye was applied: by these means a person behind a strong 
 parapet could see what the enemy were doing without the walls. The chief thing 
 to be apprehended in regard to this instrument was, that the object glass might be 
 broken by a ball; but this was certainly a trifling misfortune, and not very likely 
 to happen. 
 
 OF MICROSCOPES, 
 
 What the telescope has performed in the philosophy of the heavenly bodies, the 
 microscope has done in regard to that of the terrestrial: for by the assistance of 
 the latter we have been able to discover an order of beings which would otherwise 
 have escaped our notice; to examine the texture of the smallest of the productions 
 of nature, and to observe phenomena which take place only among the most minute 
 parts of matter. Nothing can be more curious than the facts which have been ascer- 
 
 tained by the assistance of the microscope: but in this part of science much still 
 
 remains to be done. 
 
 There are two sorts of microscopes—simple and compound : we shall speak of 
 both, and begin with the former. 
 
 PROBLEM XLIII. 
 
 Method of constructing a Simple Microscope. 
 
 I. Every convex lens of a short focal distance is a microscope ; for it is shewn 
 that alens magnifies in the ratio of the focal distance to the least distance at which 
 the object can be placed to be distinctly seen ; which, in regard to most men who 
 are not short-sighted, is about 8 inches. Thus a lens, the focal distance of which 
 is 6 lines, will magnify the dimensions of the object 16 times ; if its focal distance be 
 only one line, it will magnify 96 times. 
 
 IJ. It is difficult to construct a lens of so short a focus; as it is necessary that 
 the radius of each of its convexities should be only a line; for this reason small 
 
$18 OPTICS. 
 
 glass globes, fused at an enameller’s lamp, or the flame of a taper, are employed in 
 their stead. The method by which this is done, is as follows. | 
 
 Break off a piece of very pure transparent glass, either by means of an instrument | 
 made for that purpose, or the wards of a key; thentake up one of these fragments | 
 by applying to it the point of a needle a little moistened with saliva, which will make it | 
 adhere, and present it to the blue flame of a taper, which must be kept somewhat. 
 inclined, that the fragment of glass may not fall upon the wax. As soonalmost as it. 
 is held to the flame, it will be fused into a round globule, and drop down: a. 
 piece of paper therefore, with a turned up border, must be placed below, in 
 order to receive it. 
 
 It is here to be observed that: there are some kinds of glass which it is difficult to 
 fuse: in this case it will be necessary to employ another kind. 
 
 Of these globules select the brightest and roundest; then take a plate of copper, 
 5 or 6 inches in length, and about 6 lines in breadth, and having folded it double, 
 make a hole in it somewhat less in diameter than the globule, and raise up the edges, 
 If you then fix one of these globules in this hole, between the two plates, and bind 
 them firmly together, you will have a wey microscope. 
 
 As it is easy to obtain globules of 4, 1, and 1 of a line in diameter, and as the focus 
 of a globule is at the distance of a quarter of its diameter without it, we are enabled 
 by this process to magnify objects in a very high degree; for if the dia- 
 meter of the globule be only 3 line, by employing this proportion : as § of half a 
 line, or &, are to 96 lines, so is 1 toa fourth term, we shall have as fourth term the 
 number 153, which will express the increase of the diameter of the object. The 
 object therefore, in regard to surface, will be magnified 23409 times, and in regard to 
 solidity 3581677 times. 
 
 The celebrated Lewenhoeck, so well known on account of his microscopical ob- 
 servations, never employed microscopes of any other kind. It is however certain 
 that they are attended with many inconveniences, and can be used only for objects 
 which are transparent, or at least semi-transparent, as it may be readily conceived 
 that it is not possible to illuminate a surface which is viewed in any other way than 
 from behind. By means of these microscopes Lewenhoeck made a great number of 
 curious observations, an account of which will be found hereafter, under the head 
 Microscopical Observations. 
 
 Ill. The water microscope of Gray, which is much simpler, may be constructed in 
 the following manner. 
 
 Provide a plate of lead, } of a line in thickness at most, and make a round hole in 
 it with a needle ora large pin; pare the edges of this hole, and put into it, with 
 the point of a feather, a small drop of water: the anterior and posterior surfaces 
 of the water will assume a convex spherical form, and thus you will have a mi- 
 croscope. 
 
 The focus of such a globule is at a distance somewhat prenter than that of a glass 
 globule of equal size; for the focus of a globule of water is at the distance of the ra- 
 dius from its surface. A globule of water, therefore, 4 a line in diameter, will magnify 
 only 128 times; but this deficiency is fully compensated by the ease with which a 
 globule of any diameter, however small, may be obtained. 
 
 If water be employed in which leaves, wood, pepper, or flour has been infused, in 
 the open air, the microscope will be both object and instrument; for by this means 
 the small microscopic animals which the water contains will be seen. Mr. Gray was 
 very much astonished, the first time he observed this phenomenon; but it afterwards 
 occurred to him that the posterior surface of the drop produced, in regard to those 
 animals placed between it and its focus, the same effects as a concave mirror, and 
 magnified their image, which was still farther enlarged by the kind of convex lens of 
 the anterior surface. 
 
 IY. Another kind of microscope may be also procured at a very small expense, by 
 
 a 
 
 ul 
 ne 
 A 
 
COMPOUND MICROSCOPES. 319: 
 
 making ahole of about the fourth or fifth part of a line in diameter, in a card or very 
 thin plate of metal. If very small objects be viewed through this hole, they will 
 appear magnified in the ratio of their, distance from the eye, to that at which an 
 
 object can be distinctly seen by the naked eye. 
 
 This kind of microscope is much extolled in the “‘ Journal de Trevoux;” but we 
 
 _ must confess that we never could see small objects distinctly through such holes, 
 
 unless at the distance of aninch, or half an inch; and even then they did not appear 
 tobe much magnified. 
 PROBLEM XLIV. 
 Of Compound Microscopes. 
 
 The compound microscope consists of an object glass, which is alens of a very short 
 focus, such for example as 4 or 6 lines, and an eye-glass of 2 inches focus, at the 
 distance from it of about 6 or 8 inches. The object must be placed a little beyond 
 the focus of the object glass, and the distance of the eye from the eye-glass ought to 
 be equal to the focal distance of the latter. Having formed such a combination of 
 
 glasses, if the object be made to approach gently to the object glass, there will be a 
 
 certain point at which it will appear to be considerably magnified. 
 
 If the focal distance of the object glass be 4 lines, for example, and if the object 
 be 43 lines from it, the image will be formed at the distance of 64 lines, or five inches 
 4 lines: it will therefore be 14 times as large as the object, for 64 is to44 nearly as 14 
 tol. Ifthe focal distance of the eye-glass, in the focus of which this image is formed, 
 be two inches, it will magnify about 4 times more: but 14 x 4 = 56, which ex- 
 presses the number of times that the diameter of the object will appear to be magnified. 
 
 If you are desirous that it should not be magnified so much, remove gradually the 
 object from the object glass, and bring the eye-glass nearer; the image will then be 
 seen not so large, but more distinct. 
 
 On the other hand, if you wishit to be magnified more, move the object gradually — 
 
 towards the object glass, or move the latter towards the object, and remove the eye- 
 glass: the object will then appear much larger; but there are certain limits beyond 
 which every thing seems confused. 
 
 Instead of one eye-glass, two are sometimes used to increase the field of vision; 
 the first of which has a focal distance of 4 or 5 inches, while that of the second is 
 much less; but this is still the same thing. The image of the small object must be 
 placed, in regard to this compound eye-glass, in the same point where an object 
 ought to be, to be seen distinctly when viewed through it. 
 
 A concave object glass might be employed by making its posterior foeus coincide 
 with the image: this would form a kind of microscope similar to the Batavian teles- 
 cope; but it would be attended with the same inconvenience, that of having tuo 
 
 contracted a field. 
 
 There are also reflecting microscopes as well as telescopes: the principle of both | 
 
 is the same, a minute object placed very near the focus of a concave mirror, and on 
 this side of it, in regard to the centre, reflects an image of it beyond the centre; 
 and this-image will be larger the nearer it is to the focus. The image is viewed 
 through a convex lens, and in this kind of microscope an object glass of a much 
 shorter focus may be employed, which will contribute to the amplification of the object. 
 
 Every thing relating to this subject may be found in a very curious work by 
 Baker, entitled the “‘ Microscope made Easy.” 'Thereader may consult also Smith’s 
 
 Optics, part 4. These works, and particularly the first, contain a great variety of . 
 
 enrious details respecting the method of employing microscopes, and the observations 
 made by means of them. See also ‘‘ Essais de Physique de Muschenbroeck.” 
 - We intend to give an account of the most curious observations which have 
 
 been made by the assistance of the microscope ; but to avoid confusion we shall — 
 
 reserve that article for the end of this part of our work. 
 
 wee 
 
5202) OPTICS. 
 
 PROBLEM LXV. 
 
 A very simple method of ascertaining the real magnitude of objects, seen through a 
 microscope. | 
 
 It is often useful, and may sometimes gratify curiosity, to be able to determine 
 the real magnitude of certain objects examined by means of the microscope: the 
 following very simple and ingenious method for this purpose was invented by Dr. | 
 Jurin, a celebrated philosopher, and a fellow of the Royal Society of London. | 
 
 Take a piece of the finest silver wire possible to be obtained, and roll it as close 
 as you can aroundan iron cylinder, a few inches in length. It will be necessary to 
 examine it with a microscope, in order to discover whether there be any vacuity or 
 opening between the folds: by these means you will ascertain, with great precision, 
 the diameter of the silver wire. For if we suppose that there are 520 turns in the 
 space of an inch, it is evident that the diameter of the wire will be the 520th part of 
 an inch ; a measure which cannot be obtained in any other manner, 
 
 Then cut this silver wire into very small bits, and scatter a certain quantity of 
 them over the small plate on which the objects, submitted to examination, are 
 placed: if you look at these bits of wire along with the objects, you will be enabled, 
 by comparing them together, to judge of the size of the latter. 
 
 It was by a similar process that Dr. Jurin determined the size of the globules 
 which give to blood its red colour. He first found that the diameter of his silver 
 wire was the 485th part or an inch, and then judged by comparison that the diameter 
 of a red globule of blood was the fourth part of that of the wire; from which he 
 concluded that the diameter of the globule was the 1940th part of an inch. 
 
 PROBLEM XLVI. 
 
 To construct a Magic Picture, which being seen in a certain point through a glass, shall 
 exhibit an object, different from that seen with the naked eye. 
 
 As this opticial problem is solved by means of a glass cut into facets, or what is 
 called a multiplying glass, we shall first explain the nature of such glasses. 
 Multiplying glasses are generally lenses, plane on 
 one side, and on the other cut into several facets in 
 the form of a polyedron, of this kind is the glass 
 represented Fig. 48. and 49. where it is seen in front, 
 and also edgewise. It consists of a plane hexagonal 
 facet in the centre, and six trapeziums arranged. 
 round the circumference, 
 These glasses have the property of representing the object as many times as there 
 are facets ; for if we suppose the object to be o, the rays which proceed from it fall 
 upon all the facets of the glass A D,pc,c B. Those which tra- 
 verse the facet p c, pass through it as through a plane glass in- 
 terposed between the eye and the object ; but the rays that pro- 
 ceed from o, tothe inclined facet aD, experience a double re- 
 fraction, which makes them converge towards the axis oz, nearly 
 as they would do if they fell upon the spherical surface, in which 
 the glass polyedron might be inscribed. The eye, being placed 
 in the common point of concurrence, sees the point o, at a, 
 in the continuation of the radius E F ; consequently an image of the point o, dif- 
 ferent from the former, will be observed. As the same thing takes placein regard 
 to each facet, the object will be seen as many times as there are facets on the glass, 
 and in different places, ; i 
 Now, 1f we suppose a luminous point in the axis of the glass, and at a proper 
 distance, all the rays which fall on one facet will, after a double refraction, proceed 
 
MAGIC PICTURES, oe 
 
 toa piece of white paper placed perpendicular to the axis continued, and paint on it 
 an image of that facet of a greater or less size, and which at a certain distance will be 
 inverted. Consequently, if we suppose the eye to be substituted instead of the lu- 
 minous point, and that the image itself is luminous or coloured, the rays which pro- 
 ceed from that image, or part of the paper, will terminate at the eye ; and they will 
 be the only ones that reach it after experiencing a double refraction on the same facet. 
 If the like reasoning be employed in regard to the rest, it may be easily seen that, 
 when the eye is placed ina fixed point, it will observe through each facet only a cer- 
 tain portion of the paper, and that the whole together will fill the field of vision, 
 though detached on the paper ; so that if a certain part of a regular and continued 
 picture be painted on each, they will all together represent that picture. 
 
 The artifice then of the proposed magic picture, after having fixed the place of the 
 eye, that of the glass and the field of the picture, is to determine those portions of 
 the picture which shall alone be seen through the glass ; to paint upon each the deter- 
 minate portion, according to a given subject, such as a portrait, so that when united 
 
 together they may produce the painting itself; and in the last place, to fill up the 
 remainder of the field of the picture with any thing at pleasure; but arranging the 
 whole in such a manner as to form a regular subject. 
 
 Having thus explained the principle of this optical amusement, we shall now shew 
 how it is to be put in practice. 
 
 Let a scp (Fig. 50.) represent a board, at the ex- 
 Fig. 50. tremity of which is fixed another in a perpendicular 
 direction, having at its edges two pieces of wood 
 with grooves, to receive a piece of pasteboard, co- 
 vered with white paper or canvass. This pasteboard, 
 which may be pushed in or drawn out at pleasure, 
 is the field of the intended picture: uD H is a verti- 
 cal board, the bottom part of which must be contrived 
 in such a inanner that it can be brought nearer to or farther from the painting; and 
 towards the upper part it is furnished with a tube, having at its anterior extremity a 
 glass cut into facets, and at the other a piece of card, in which is a small hole made 
 by means of aneedle, and to which the eye is applied. We shall here suppose the 
 glass to be plane on one side, and on the other to consist of six rhomboidal facets, 
 placed around the centre, and of six triangular ones which occupy the remainder 
 of the hexagon. 
 
 When every thing is thus prepared, fix the support z pu at a certain distance from 
 
 the field of the picture, according as you are desirous that the parts to be delineated 
 should be nearer to or farther from each other. But this distance ought, at least, to 
 be four times the diameter of the sphere in which the polyedron of the glass could 
 be inscribed ; and the distance from the eye to the glass may be equal to twice that 
 diameter. Then place the eye at the hole hole x, the distance of which has been 
 thus determined, and with a stick having a pencil at the end of it, if the hand cannot 
 ‘reach the pasteboard, trace out, in as light a manner as possible, the outline of the 
 ‘space observed through one facet, and do the same thing in regard to the rest. This 
 operation will require a great deal of accuracy and patience ; for, to render the work 
 perfect, no perceptible interval must be left between the two spaces seen through two 
 contiguous facets: it will be better on the whole if they rather encroach a little on 
 each other. Care must also be taken to mark each space with the same number as 
 that assigned to each facet, in order that they may be again known. This however 
 will be easy, by observing that the space corresponding to each facet is always 
 transferred parallel to itself from top to bottom, or from right to left, on the other 
 side of the centre. 
 
 The next thing is to delineate the regular picture intended to be seen, and to 
 
 a6 
 
322 OPTICS. 
 
 q 
 
 transpose it into the spaces where it appears distorted. According to mathematia| 
 accuracy, it would be necessary for this purpose to form a projection of the glass cut! 
 into facets, supposing the eye at the distance at which it is really placed ; but as we sup- 
 pose it a little more remote, we may without any sensible error’ 
 assume, as the field of the regular picture, the vertical projec. 
 tion, as seen Fig. 51, where it is represented such as it would 
 appear to the eye bined perpendicularly above its centre, and at 
 a very considerable distance. 
 
 Delineate in the field, which in this case will be hexagonal, 
 and composed of six rhomboids and six triangles, any figure 
 whatever, as a portrait for example, and then, considering that 
 the space abcd (Fig. 52.) is that where the portion of the pic. 
 . ture marked 1 ought to appear, it must be transferred thither with 
 
 oA as much care as possible ; do the same thing in regard to the 
 
 Cl gl 2] yest; and by these means the principal part of the picture will 
 
 <o> ee be completed. But as it is intended to shew something else 
 
 we Py oo beside what ought to be seen, it must be disguised hy means. 
 
 Ag ys of some other objects painted in the remaining part of the 
 
 N74 field, making them to harmonize with what is already painted, in 
 
 such a manner, that the whole shall appear to form one regular 
 
 and connected subject. All this however must depend on the taste and genius of the 
 artist. 
 
 In the “ Perspective Curieuse” of father Niceron, a much more minute explana- 
 tion of the whole process may be found. ‘Those to whom what is here said does 
 not seem sufficient, must consult that work. Niceron tells us that he executed, 
 at Paris, and deposited in the library of the Minimes, of the Place Royale, a picture 
 of this kind, which, when seen with the naked eye, represented fifteen portraits 
 of Turkish Siltane ; but, when viewed through the glass, was the portrait | 
 Louis XIII. | 
 
 A picture by Amadeus Vanloo, much more ingenious, was shewn in the year 1759, 
 in the exhibition room of the Royal Academy of Painting. To the naked eye, it 
 was an allegorical picture, which represented the Virtues, with their attributes, 
 properly grouped; but, when seen through the glass, it exhibited the portrait of 
 Louis XV. 2 
 
 Remarks.—\st. It is necessary to observe, that the place of the glass, when once 
 fixed, must be invariable; for as glasses perfectly regular cannot be obtained, if they 
 are moved, it will be almost impossible to replace them in the proper point ; hence it” 
 will be necessary to be assured that the glass is of a good quality; for, if it be too 
 alkaline, and happen to lose its polish by the contact of the air, another capable 
 of producing the same effect cannot bessubstituted in its stead. This is an accident 
 which, according to what we have heard, happened to the glass of Vanloo’s picture. _ 
 
 2d. Instead of a glass, like that employed in the above example, or of one more. 
 compounded, a plain pyramidal glass might be employed, by which the problem would 
 be greatly simplified. 
 
 3d. A glass, the portion of a prism, cut into a great number of planes parallel to 
 its axis, might also be employed; iv this case the painting to be viewed through the 
 glass ought to be delineated on parallel bands. 
 
 4th. A glass might be formed of several concentric conical surfaces, or of ceva ) 
 spherical surfaces of different diameters, likewise concentric: in this case the pic- ) 
 ture to be viewed through the glass ought to be distributed in different concentric 
 rings. | 
 Oth. A magic picture might be formed by reflection. For this purpose, provide a 
 
} MAGIC LANTERN. 323 
 
 metal mirror with facets well polished, and having very sharp edges; place before it, 
 
 ina direction parallel to its axis, a piece of white paper or card, and by means of the 
 
 principles above explained delineate a picture, which when viewed in front by the 
 naked eye, shall represent a certain subject; if you then make a hole in the middle 
 of the picture, and look through this hole ac the image of it formed by the mirror, it 
 will appear to be entirely different. 
 
 PROBLEM XLVII. 
 
 To construct a lantern, by means of which a book can be read at a great distance, 
 at night. 
 
 Construct a lantern of a cylindric form, or shaped like a small cask placed length- 
 | wise, so that its axis shall be horizontal; and in one end of it fix a parabolic mirror, 
 _or merely a spherical one, the focus of which falls about the middle of the length 
 _of the cylinder: if a taper or lamp be then placed in ‘this focus, the light will be 
 reflected through the open end, and will be so strong that very small print may be 
 read by it at a great distance, if looked at through a telescope. Those who see this 
 light at a distance, if standing in the axis of the lantern continued, will imagine that 
 
 _they see a large fire. 
 
 PROBLEM XLVIII, 
 To construct a Magic Lantern. 
 
 The name of magic lantern, as is well known, is given to an optical instrument, 
 by means of which figures greatly magnified may be represented on a white wall or 
 cloth. This instrument, invented, we believe, by Father Kircher, a jesuit, has be- 
 come a useful resource toa great number of people, who gain their livelihood by 
 exhibiting this spectacle to the populace. But though it has fallen into vulgar 
 hands, it is nevertheless ingenious, and deserves a place in this work. We shall 
 ‘therefore describe the method of constructing it, and add a few observations, which 
 
 may tend to improve it, and to render it more interesting. 
 
 : First, provide a box about a foot square (Fig. 53.) 
 ) Fig. 53. of tin-plate, or copper, or wood, and make a hole 
 towards the middle of the fore-part of it, about 
 three inches in diameter: into this hole let there be 
 soldered a tube, the interior aperture of which must 
 be furnished with a very transparent lens, having its 
 focus within the box, and at the distance of two- 
 thirds or three-fourths of the breadth of the box. 
 In this focus place a lamp with a large wick, in order 
 that it may produce a strong light; and that the 
 ‘machine may be more perfect, the lamp ought to be moveable, so that it can be 
 ‘placed exactly in the focus of the lens. To avoid the aberration of sphericity, the 
 lens in question may be formed of two lenses, each of a double focus. This, in our 
 | opinion, would greatly contribute to the distinctness of the picture. 
 
 ' Atasmall distance from the aperture of the box, let there be a slit in the tube, 
 for which purpose this part of it must be square, capable of receiving a slip of glass 
 surrounded by a frame, four inches in breadth, and of any length at pleasure. Va- 
 ‘rious objects, according to fancy, are painted on this slip of glass, with transparent 
 colours ;- but in general the subjects chosen are of the comic and grotesque kind. 
 
 Another tube, furnished with a lens of about three inches focal distance, must be 
 ‘fitted into the former one, and in such a manner, that it can be drawn out or pushed 
 in as may be found necessary. 
 
 Having thus given a description of the machine, we shall now explain its effect. 
 
 The lamp being lighted, and the machine placed on the table opposite to a white 
 ¥ 2 
 
324 OPTICS. 
 
 wall, if it be exhibited in the day time, shut the windows of the apartment, and 
 introduce into the slit above mentioned one of the painted slips of glass, but in such 
 a manner that the figures may be inverted: if the moveable tube be then pushed in 
 or drawn out, till the proper focus is obtained, the figures on the glass will be seen 
 painted on the wall, in their proper colours, and greatly magnified. | 
 If the other end of the moveable tube be furnished with a lens of a much greater 
 focal distance, the luminous field will be increased, and the figures will be magnified 
 in proportion. It will be of advantage to place a diaphragm in this moveable tube, 
 at nearly the focal distance of the first lens, as it will exclude the rays of the lateral 
 objects, and thereby contribute to render the painting much more distinct. | 
 
 We have already said that the small figures on the glass must be painted with 
 transparent colours. The colours for this purpose may be made in the following. 
 manner: red, by a strong infusion of Brasil wood, or cochineal, or carmine, according: 
 to the tint required; green, by a solution of verdigris; or for dark greens, of martial 
 vitriol (sulphate of iron); yellow, by an infusion of yellow berries; blue, by a’ 
 solution of vitriol of copper (sulphate of copper); these three or four colours, as is 
 well known, will be sufficient to form all the rest: they may be mixed up and 
 rendered tenacious by means of very pure and transparent gum-water, after which 
 they will be fit for painting on glass. In most machines of this kind, the paintings 
 are so coarsely executed, that they cannot fail to excite disgust; but if they are 
 neatly designed, and well finished, this small optical exhibition must afford a con- 
 siderable degree of pleasure. 
 
 PROBLEM XLIx. 
 Method of constructing a Solar Microscope. ; 
 
 The Solar Microscope, for the invention of which we are indebted to Mr. Lig 
 berkun, is nothing else, properly speaking, than a kind of magic lantern, where the 
 sun performs the part of the lamp, and the small objects exposed on a glass or the 
 point of a pin, that of the figures painted on the glass slips of the latter. But the 
 following is a more minute description of it. 
 
 Make a round hole in the window shutter, about three inches in diameter, and 
 place in it a glass lens of about twelve inches focal distance. To the inside of the 
 hole adapt a tube, having, at a small distance from the lens, a slit or aperture, capa- 
 ble of receiving one or two very thin plates of glass, to which the objects to be: 
 viewed must be affixed by means of a little gum-water exceedingly transparent. 
 Into this tube fit another, furnished at its anterior extremity with a lens of a short 
 focal distance, such for example as half an inch. If a mirror be then placed before 
 the hole in the window-shutter on the outside, in such a manner as to throw the 
 light of the sun into the tube, you will have a solar microscope. The method of em- 
 ploying it is as follows. 
 
 Having darkened the room, and by means of the mirror reflected the sun’s rays on 
 the glasses in a direction parallel to their axes, place some small object between the 
 two moveable plates of glass, or affix it to one of them with very transparent gum- 
 water, and bring it exactly into the axis of the tube: if the moveable tube be then 
 pushed in or drawn out, till the object be a little beyond the focus, it will be seen 
 painted very dictinctly on a card or piece of white paper, held at a proper distance; 
 and will appear to be greatly magnified. A small insect, such asa flea for example, 
 may be made to appear as large as a sheep, or a hair as large as a walking-stick: by 
 means of this instrument the eels in vinegar, or flour paste, will have the appearance 
 of small serpents. 4 
 
 Remark.— As the sunis not stationary, this instrument is attended with one incon- 
 venience, which is, that as this luminary moves with great rapidity, the mirror on 
 
; 
 
 t 
 
 ; COLOURS AND REFRACTIONS. 325 
 
 the outside requires to be continually adjusted. This defect however S’Gravesande 
 remedied by means of a very ingenious machine, which moves the mirror in sucha 
 manner that it always throws the sun’s rays into the tube. This machine, therefore, 
 has been distinguished by the name of the sol-sta. 
 
 Some curious details respecting the solar microscope may be seen in the French 
 ‘Translation of Smith’s Optics, where several useful inventions for improving it, and 
 for which we are indebted to Euler, are explained. A method, invented by Apinus, 
 of rendering it proper for representing opake objects, will be found there also. It 
 consists in reflecting, by means of a large lens and a mirror, the condensed light of 
 the sun on the surface of the object, presented to the object glass of the microscope. 
 _M. Mumenthaler, a Swiss optician, proposed a different expedient. But solar 
 microscopes are still attended with another inconvenience : as the objects are very 
 ‘near the focus of the first lens, they are subjected to a heat which soon destroys or 
 disfigures them. Dr. Hill, who made great use of this microscope, proposed tbere- 
 fore to employ several lamps, the light of which united into one focus is 
 ‘exceedingly bright and free from the above inconvenience; but we do not know 
 whether he ever carried this idea into practice, and with what success. 
 
 PROBLEM L. 
 Of Colours, and the different Refrangibility of Light. 
 One of the noblest discoveries of the 17th century, is that made by the celebrated 
 
 Newton, in 1666, respecting the compositicn of light, and the cause of colours. 
 Who could have believed that white, which appears to be a colour so pure, is the 
 
 result of the seven primitive unalterable colours mixed together ina certain propor- 
 tion! This however has been proved by his experiments. 
 
 The instrument which he employed for decomposing light in this manner, was the 
 
 prism, now well known, butat that timea mere object of curiosity on account of the 
 -eolonrs, with which every thing viewed through it seems to be bordered. But on this 
 subject we shall confine ourselves to two of Newton’s experiments, and a deduction 
 
 of the consequences which result from them, 
 
 If a ray of a solar light, an inch or half an 
 inch in diameter (Fig. 54.), be admitted into a 
 darkened room, so as to fall on a prism placed ho- 
 rizontally, with a piece of white paper behind it, 
 and if the prism be turned in.such a manner, that 
 the image seems to stop; instead of an image of 
 the sun nearly round, you will observe a long per- 
 pendicular band, consisting of seven colours, in this 
 invariable order, red, orange, yellow, green, blue, 
 indigo, violet. When the angle of the prism is turned downwards, the red will be 
 
 at the bottom, and vice versa; but the order will be always the same. 
 
 From this, and various other experiments ofa similar kind, Newton concludes : 
 
 1. That the light of the sun contains these seven primitive colours. 
 
 2. That these colours are formed by the rays experiencing different refractions ; and 
 the red, in particular, is that which is the least broken or refracted; the next is the 
 orange, &c.: in the last place, that the violet is that which, under the same inclina- 
 tion, suffers the greatest refraction. The truth of these consequences cannot be 
 denied by those who are in the least acquainted with geometry. 
 
 But the nicest experiment is that by which Newton proved that these differently 
 coloured rays are afterwards unalterable. To make this experiment in a proper man- 
 ner, it will be necessary to proceed as follows: 
 
 In the first place, the hole in the window shutter of the darkened room must be 
 
 reduced to the diameter of a lineat most ; and the light every where else must be 
 
326 OPTICS. 
 
 earefully excluded. When this is done, receive the solar rayson a large lens, of 7 or 
 8 feet focus, placed at the distance of 15 feet from the hole, and a little beyond the | 
 lens place a prism in such amanner that the stream of light may fall upon it. Then 
 holda piece of white card at such a distance that the image of the sun would be 
 painted upon. it without the interposition of the prism, and you. will see painted on 
 the card, instead of a roundimage, a very narrow coloured band, containing the seven 
 primitive colours, 
 
 Then pierce a hole in the card, about a line in diameter, and suffer any one of 
 the colours to pass through it, taking care that it shall do so in the middle of the — 
 space which it occupies, and receive it on a second card placed behind the former. If 
 intercepted by another prism, it will be found that it no longer produces a lengthened, 
 but a round image, and all of the same colour. Besides, if you hold in that colour 
 any object whatever, it will be tinged by it; and if you look at the object with a 
 third prism, it will be seen of no other colour but that in which it is immersed, and 
 without any elongation, as when it is immersed in light susceptible of decompo- 
 sition. 
 
 This experiment, which is now easy to those tolerably well versed in philosophy, 
 proves the third of the principal facts advanced by Newton. 
 
 3. That whena colour is freed from the mixture of others, it is unalterable ; that a 
 red ray, whatever refraction it may be made to experience, will always remain red, 
 and so of the rest. 
 
 It does no great honour to the French philosophers of the 17th century to have 
 disputed, and even declared false, this assertion of the English philosopher, especially 
 on no better foundation than an experiment so badly performed. and so incomplete 
 as that of Mariotte. We even cannot help accusing that philosopher, who in other 
 respects deserves great praise, of too much precipitation; for his experiment was 
 not the same as that described by Newton in the ‘‘ Philosophical Transactions” for 
 1666; and it may be readily seen that, if performed according to Mariotte’s manner, 
 it is impossible it should succeed, 
 
 However, it is at present certain, notwithstanding the remonstrances of Father 
 Castel and the Sieur Gautier*, that there are in nature seven primitive, homogeneous 
 colours, unequally refrangible, unalterable, and which are the eause of the different 
 colours of bodies; that white contains them all, and that all of them together com- 
 pose white; that what makes a body be of one colourrather than another, is the 
 configuration of its minute parts, which causes it to reflect in greater number the rays 
 of that particular colour; and in the last place, that black is the privation of all re- 
 flection; but this is understood of perfect black, for the material and common black is 
 only an exceedingly dark blue. 
 
 Some people, such as Father Castel, have admitted only three primitive colours, 
 viz., red, yellow, and blue; because red and yellow form orange, yellow and blue 
 green, and blue and red violet or indigo, according as the former or the latter predo- 
 minates. But this is another error. It is very true that with two rays, one yellow and 
 the other blue, green can be formed; and this holds good also in regard to material 
 colours ; but the green of the coloured image of the prism is totally different: it is 
 primitive, and stands the same proof as red, yellow, or blue, without being decom- 
 posed. The case is the same with orange, indigo, and violet. 
 
 We extract the following remarks on this subject from Dr. Brewster’s treatise on 
 Optics, in Dr. Lardner’s Encyclopedia. 
 
 * The Sieur Gautier, who pretended to be the inventor of the method of engraving in colours, 
 opposed with great violence, in the year 1750, the theory of Newton, both in regard to colours and 
 to the system of the universe. His reasoning and experiments, however, are as conclusive as eX- 
 periments made with a faulty air-pump would be against the gravity of the atmosphere. For this 
 
 _ reason, he never had any partizans but a few of his own countrymen, one of whom was a poct, 
 who had found out that objects are not painted on the retina in an inverted position, 
 
DECOMPOSITION OF LIGHT. Jey 
 
 Decomposition of Light by Absorption. 
 
 *‘ If we measure the quantity of light, which is reflected from the surfaces, and 
 transmitted through the substance of transparent bodies, we shall find that the sum 
 of these quantities is always less than the quantity of light that falls upon the 
 body. Hence we may conclude that a certain portion of light. is lost in passing 
 through the most transparent bodies. This loss arises from two causes.. A part 
 of the light is scattered in all directions by irregular reflection from the imperfectly 
 polished surface of particular media, or from imperfect union of its parts; while 
 another, and generally a greater portion, is absorbed, or stopped by the particles 
 of the body. Coloured fluids, such as black and red ink, though equally homo- 
 geneous, stop or absorb different kinds of rays, and when exposed to the sun they 
 become heated in different degrees, while pure water seems to transmit all the rays 
 equally, and scarcely receives any heat from the passing light of the sun. 
 
 ‘* When we examine more minutely the action of coloured fluids in absorbing 
 light, many remarkable phenomena present themselves, which throw much light 
 upon this curious subject. 
 
 ‘* If we take a piece of blue glass, like that generally used for finger glasses, and 
 transmit through it a beam of white light, the light will be a fine deep blue. This 
 blue is not a simple homogeneous colour, like the blue or indigo of the spectrum, 
 but is a mixture of all the colours of white light which the glass has not absorbed ; 
 and the colours which the glass has absorbed are those which the blue wants of white 
 light, of which, when mixed with this blue, would form white light. In order to 
 determine what these colours are, let us transmit through the blue glass the pris- 
 matic spectrum kK L (Fig. 54.) ; or what is the same thing, let the observer place his 
 eye behind the prism pac, and look through it at the sun, or rather at a circular 
 aperture made in the window-shutter of a dark room. He will then see through the 
 prism the spectrum KL, as far below the aperture as it was above the spot rp, when 
 shewn on the screen. Let the blue glass be now interposed between the eye and 
 the prism, and a remarkable spectrum will be seen, deficient in a certain number 
 of its differently coloured rays. <A particular thickness absorbs the middle of the red 
 space, the whole of the orange, a great part of the green, a considerable part of the 
 blue, a little of the indigo, and very little of the violet.’ The yellow space, which 
 has not been much absorbed, has increased in breadth. It occupies part of the space 
 formerly occupied by the orange on the one side, and part of the space formerly 
 covered by the green on the other. Hence it follows, that the blue glass has 
 absorbed the red light, which, when mixed with the yellow light, constituted orange ; 
 and has absorbed also the blue light, which, when mixed with the yellow, constituted 
 the part of the green space next to the yellow. We have, therefore, by absorption, 
 decomposed green light into yellow and blue, and orange light into yellow and red ; 
 and it consequently follows, that the orange aud green rays of the spectrum, though 
 they cannot be decomposed by prismatic refraction, can be decomposed by absorp- 
 tion, and actually consist of two different colours possessing the same degree of re- 
 frangibility. Difference of colour is therefore not a test of difference of refrangi- 
 bility, and the conclusion deduced by Newton is no longer admissible as a general 
 truth: ‘ That to the same degree of refrangibility ever belongs the same colour, and 
 to the same colour ever belongs the same degree of refrangibility.’ 
 
 “With the view of obtaining a complete analysis of the spectrum, I have ex- 
 _ amined the spectra produced by various bodies, and the changes which they undergo 
 by absorption when viewed through various coloured media; and I find that the 
 colour of every part of the spectrum may be changed, not only in intensity, but in 
 colour, by the action of particular media; and from these observations, which it 
 would be out of place here to detail, I conclude that the solar spectrum consists 
 
Mie) OPTICS. 
 
 of three spectra of equal lengths, viz., a red spectrum, a yellow spectrum, and a blue 
 spectrum.” 
 
 PROBLEM LI. 
 Of the Rainbow ; how formed ; method of making an artificial one. 
 Of all the phenomena of nature, none has excited more the admiration of man- 
 , kind, in all ages, than the rainbow; but there is none perhaps 
 Fig. 55. at present which philosophy can explain in a more satisfactory 
 AZ manner, 
 Ta S The rainbow is formed by the solar rays being decomposed 
 into their principal colours, in the small drops of rain, by 
 SS) means of two refractions, which they experience in entering 
 them and issuing from them. In the interior rainbow, which 
 often appears alone, the solar ray enters at the upper part 
 Ki 6 of the drop, is reflected against the bottom, and issues at the 
 ae lower side. This decomposition may be seen Fig. 55. 
 Fig. 56. In the exterior rainbow, the rays enter at the bottom 
 o° the drop, experience two reflections, and issue at 
 the upper part. Their progress and decomposition, 
 which produces colours in an order contrary to the 
 former, are represented Fig. 56. Hence the colours 
 of the exterior rainbow appear to be inverted, in 
 regard to those of the first. 
 
 The manner in which the eye perceives this double 
 series of colours is seen Fig. 57. 
 
 But the explanation would be incom- 
 plete if we did not shew that there is a 
 certain determinate inclination, under 
 which the red rays issue parallel, and as 
 close to each other as possible, while all 
 the rest are divergent; that there is 
 another under which the green rays 
 issue in this manner; and so of the rest. 
 It is by this alone that they can produce 
 an effect on a distant eye. 
 
 This explanation of the rainbow is. 
 confirmed by asimple experiment. When 
 the sun is very near the horizon, suspend in an apartment a glass globe filled with 
 water, in such a manner as to be illuminated by the sun; and place yourself 
 with your back to that luminary, so that the globe shall be elevated in regard 
 to your eye, about 42 degrees above the horizon. By advancing or retiring a 
 little, you will not fail to meet with the coloured rays, and it will be easily seen 
 that they issue from the bottom of the globe; it will be seen also that the red 
 ray issues frora it under the greatest angle with the horizon, and the violet, which 
 is the lowest one, under the least, so that the red must be without the axis, and 
 the violet within it. 
 
 Then raise the globe, in regard to your eye, to 54 degrees, or continue to approach 
 it till it be elevated at that angle, and you will meet with the coloured rays issuing . 
 from the top of it; first the violet, and then the blue, green, and red, in an order 
 altogether contrary to the preceding. If you cover, in the first case. the upper part 
 of the globe, and in the second the lower part, no colours will be produced ; which 
 is a proof of the manner in which they enter it, and issue from it. 
 
 The spectacle of an artificial rainbow may be easily obtained; it is seen in the 
 
RAINBOW. 329 
 
 vapour of a jet of water, when the wind disperses it in minute drops. For this pur- 
 pose, place yourself in a line between the jet of water and the sun, with your back 
 turned towards the latter. Ifthe sun be at a moderate elevation above the horizon, 
 by advancing towards the jet of water, or receding from it, you will soon find a point 
 from which a rainbow will be seen in the drops that fall down in fine light rain. 
 
 If there be not a jet of water in the neighbourhood, you may make one at a very 
 small expense. Nothing will be necessary but to fill your mouth with water, and 
 having turned your back to the sun when at a moderate elevation, to spurt the water 
 into the air as high as possible, and in a direction somewhat oblique to the horizon. 
 The imitation of this phenomenon may be greatly facilitated by employing a syringe, 
 which will scatter the water in very small drops. 
 
 If you are desirous of performing the experiment ina manner still easier, fill a very 
 transparent cylindric glass bottle with water, and place it on a table in an upright 
 position ; place a lighted candle at the same height, and at the distance from it of 10 
 or 12 feet, and then walk in a transversal direction between the light and the bottle, 
 keeping your eye at the same elevation. When you have reached a certain point, 
 you will see bundles of coloured rays issuing from one of the sides of the bottle, in 
 the following order: violet, blue, yellow, red; and if you continue to walk trans- 
 versely, you will meet with asecond series, in a contrary order, viz. red, yellow, blue 
 and violet, proceeding from the other side of the bottle. This is exactly what takes 
 place in regard to the drops of rain; and to imitate the phenomenon completely, 
 seven similar bottles might be ahora in such a manner, that the eye being placed 
 in the proper point, one of the seven colours should be seen in each; and seven 
 others might be arranged at some distance, so as to exhibit the same colours in an 
 inverted order. 
 
 Two rainbows would still be produced, even if the solar rays were not differently 
 refrangible ; but they would be destitute of colour, and would consist only of two 
 circular bands of white or yellowish light. 
 
 The rainbow always forms a portion of a circle around the line drawn from the sun, 
 and passing through the eye of the spectator; for this reason, when the sun is ele- 
 vated above the horizon, the rainbow is less thana semicircle; but when the sun is in 
 the horizon, it is equal to a semicircle. 
 
 A rainbow, however, has been seen larger than a semicircle, and which intersected 
 the common rainbow; but this phenomenon was produced by the image of the sun 
 reflected from the calm, smooth surface of a river. The image of the sun, in this 
 case, produced the same effect as if that luminary had been below the horizon. 
 
 Dr. Halley has calculated, from the ratio of the different refrangibilities of the sun’s 
 rays, that the semi-diameter of the interior rainbow, taken in the middle of its extent, 
 ought tobe 41° 10’; and that its breadth, which would be only 1° 45’, if the sun were 
 a point, ought to be 2° 15’ on account of the apparent diameter of that luminary. 
 This apparent diameter is the cause why the colours are not separated from each 
 other with the same distinctness as they would be, if the sun were a luminous 
 point: the radius of the exterior rainbow, taken in the same manner, that is to 
 say in the middle of its extent, is 52° 30’. ; 
 
 This geometrician and astronomer not only calculated the dimensions of that rain- . 
 bow which actually appears to us in the heavens, but of those also which would be 
 produced if the light of the sun did not issue from the drop of water till after 3, 4, 
 5, &c., reflections; whereas, in the principal and interior rainbow, it issues after one, 
 and in the second or exterior one, after two. By these calculations it is found that 
 the semi-diameter of the third rainbow, counted from the place of the sun, would 
 be 41°; that of the fourth, 48° 50’; &c. But geometry here goes much farther than 
 nature: for besides the continued weakness of the rays, which would render these 
 
330 OPTICS. 
 
 rainbows scarcely perceptible, being towards the sun, they would be lost amidst the 
 splendour of that luminary. If the drops which form the rainbow, instead of being 
 water, were glass, the mean semi-diameter of the interior rainbow would be 22° 52’, 
 and that of the exterior 9° 30‘, towards the side opposite to the sun, 
 
 PROBLEM LII. 
 
 Analogy between Colours and the Tones of Music. Of the Ocular Harpsichord of 
 Father Castel. 
 
 As soon as it had been observed that there were seven primitive colours in nature 
 there was some reason to conceive that there might be an analogy between these 
 colours and the tones of music; for the latter form a series of seven in the whole 
 extent of the octave. This observation did not escape Newton, who remarked also 
 that, in the coloured spectrum, the spaces occupied by the violet, indigo, blue, &c., 
 correspond to the divisions of the monochord, which gives the sounds ve, mi, fa, sol, 
 la, si, ut, re. 
 
 Newton on this subject proceeded no farther. But Father Castel, whose vi- 
 sionary scheme is well known, enlarged this idea; and on the above analogy of 
 sounds founded a system, in consequence of which he promised to the eyes, but 
 unfortunately without success, a new pleasure similar to that which the ears expe- 
 rience from a concert. 
 
 Father Castel, for reasons of analogy, first changes the order of the colours into 
 the following, viz. blue, green, yellow, orange, red, violet, indigo, and in the last 
 place blue, which forms as it were the octave of the first. These, according to his 
 system, are the colours which correspond to the diatonic octave of our modern music, 
 ut, re, mi, fa, sol, la, si, ut. ‘The flats and the sharps gave him no embarrassment ; 
 and the chromatic octave divided into its twelve colours, was blue, sea-green, olive- 
 green, yellow, apricot, orange, red, crimson, violet, agate, indigo, blue, which cor- 
 responded to wt, ut,* re, re,* mi, fa, fa,* sol, sol,* la, la,* si, ut. 
 
 Nowifa harpsichord be constructed in such a manner, says Father Castel, that on 
 striking the key u¢, instead of hearing a sound, a blue band shall appear; that on 
 striking re, a green one shall be seen, and so on, you will have the required instru- 
 ment; provided that for the first octave of ut a different blue be employed. But 
 what are we to understand by a blue an octave to another? We do not find that 
 Father Castel ever explained himself on this subject in a manner sufficiently clear. 
 He only says that, as there are reckoned to be twelve octaves appreciable by the 
 ear, from the lowest sound to the most acute, there are in like manner twelve octaves. 
 of colours, from the darkest blue to the lightest; which gives us reason to believe 
 that since the darkest blue is that which ought to represent the lowest key, the blue 
 corresponding to the octave must be formed of eleven parts of pure blue, and one of 
 white; that the lightest must be formed of one part of blue and eleven parts of 
 white, and so of the rest. 
 
 However, Father Castel did not despair of producing by these means an ocular 
 music, as interesting to the eyes as the common music is to well organised ears; and 
 he even thought that a piece of music might be translated into colours for the use 
 of the deaf and dumb. ‘‘ You may conceive (says he,) what a spectacle will be ex- 
 hibited by a room covered with rigadoons and minuets, sarabands and _passeailles, 
 sonatas and cantatas, and if you choose with the complete representation of an opera? 
 Have your colours well diapasoned, and arrange them on a piece of canvas according — 
 to the exact series, combination, and mixture of the tones, the parts and concords of 
 the piece of music which you are desirous to paint, observing all the different values 
 of the notes, minims, crotchets, quavers, syncopes, rests, &c.; and disposing all the 
 parts according to the order of counter-point. It may be readily seen that this is 
 
TABLE OF COLOURS. 331 
 
 uot impossible, nor even difficult, to any person who has studied the elements of 
 painting, and at any rate that a piece of tapestry of this kind would be equal to those 
 where the colours are applied as it were at hazard in the same manner as they are 
 in marble. 
 
 ‘*Such a harpsichord (continues he,) would be an excellent school for painters, 
 who might find in it all the secrets and combinations of the colours, and of that 
 which is called claro-obscuro. But even our harmonical tapestry would be attended 
 with its advantages ; for one might contemplate there at leisure what hitherto could 
 be heard only in passing with rapidity, so as to leave little time for reflection. And 
 what pleasure to behold the colours in a disposition truly harmonical, and in that infi- 
 nite variety of combinations which harmony furnishes! ‘The design alone of a paint- 
 ing excites pleasure. There is certainly a design in a piece of music; but it is not 
 so sensible when the piece is played with rapidity. Here the eye will contemplate it 
 at leisure ; it will see the concert, the contrast of all the parts, the effect of the 
 one in opposition to the other, the fugues, imitations, expression, concatenation of the 
 cadences, and progress of the modulation. And can it be believed that those pathetic 
 passages, those grand traits of harmony, those unexpected changes of tone, that 
 always cause suspension, languor, emotions, and a thousand unexpected changes in 
 the soul which abandons itself to them, will lose any of their energy in passing from 
 the ears to the eyes, &c. ? It will be curious to see the deaf applauding the same 
 passages as the blind, &c. Green, which corresponds to re, will no doubt shew that 
 the tone re is rural, agreeable, and pastoral; red, which corresponds to sol, will 
 excite the idea of a warlike and terrific tone; blue, which corresponds to ut, of a 
 noble, majestic, and celestial tone, &c. Itis singular that the colours should have the 
 proper characters ascribed by the ancients to the exact tones which correspond to them, 
 but a great deal might be said, &c. 
 
 ‘* A spectacle might be exhibited of all forms human and angelical, animais, birds, 
 reptiles, fishes, quadrupedes, and even geometric figures. By asimple game the whole 
 series of Euclid’s Elements might be demonstrated.’ Father Castel’s imagination 
 seems here to conduct him in the straight road to Bedlam. 
 
 These passages of Father Castel are so-singular, that we could not help 
 quoting them; but unfortuately all his fine promises came to nothing. He had con- 
 structed a model of his harpsichord, as he tells us himself, so early as the end of the 
 year 1734, and he spent almost the remainder of his life, till the time of his death, 
 which took place in 1757, in completing his instrument, but without success. This 
 harpsichord, constructed at a great expense, as we are told by the author of his life, 
 neither answered the author’s intention, nor the expectation of the public. And 
 indeed if there be any analogy between colours and sounds, they differ in 
 so many other points, that it needs excite no wonder that this project should 
 miscarry. 
 
 PROBLEM LIII. 
 
 To compose a table representing all the Colours ; and to determine their number. 
 
 Though Newton has proved the homogeneity of the colours into which the solar 
 rays are decomposed, and the orange, green, and purple produced by this decomposi- 
 tion are no less unalterable, by farther refraction, than the red, yellow, and blue; it 
 is however well known that with the three latter, the three former, and all the other 
 colours of nature, can be imitated: for red combined with yellow, in different pro- 
 portions, gives all the shades of orange ; yellow and blue produce pure greens; red and 
 blue violets, purples and indigoes ; in a word the different combinations of these com- 
 pound colours, give birth toall the rest. On these principles is founded the invention 
 of the chromatic triangle, which serves to represent them. 
 
332 OPTICS. 
 
 Construct an equilateral 
 triangle, as seen Fig. 58, and 
 divide the two sides adjacent 
 to the vertical angle into 13 
 equal parts: if parallel lines 
 be then drawn through the 
 points of division, in each 
 side, they will form 91 equal 
 rhombuses. 
 
 In the three angular 
 rhombs place the three pri- 
 mitive colours, red, yellow, 
 and blue, having an equal 
 degree of strength, and as 
 we may say of concentra- 
 tion ; consequently, between 
 the yellow and blue, there 
 will be left 1] rhomboidal 
 cells, which must be filled 
 up in the following manner : 
 in that nearest the yellow 
 put 11 parts of yellow and one of red; in the next, 10 parts of yellow and 2 of red; 
 &c.; so that in the cell nearest the red, there will be 1] part of yellow and 11 of red: 
 by these means we shall have all the shades of orange, from the one nearest red to 
 that nearest yellow. By filling up, in like manner, the intermediate cells between 
 red and blue, and between blue and yellow, the result will be all the shades of pur- 
 ple, and all those of green, in a similar gradation. 
 
 To fill up the other ceils, let us take for example those of the third row below 
 red, where there are three cells. The two extreme cells being filled up on the one 
 side with a combination of 10 parts of red and 2 of yellow, and on the other with a 
 combination of 10 parts of red and 2 of blue, the middle cell will be composed of 10 
 parts of red, 1 of blue, and i of yellow. 
 
 In the band immediately below, we shall have, for the same reason, in the first 
 cell towards the yellow, 9 parts of red and 3 of yellow; in the next, 9 parts of red, 
 2 of yellow, and 1 of blue; in the third, 9 parts of red, 1 of yellow, and 2 of blue; 
 in the fourth, 9 parts of red, and 3 of blue: and the case will be similar in regard 
 to the lower bands; but we shall here content ourselves with detailing the colours 
 in the last except one, or that above the band containing the greens, the cells of which 
 must be filled up as follows: In 
 
 The Ist on the left, 11 parts yellow and 1 red. 
 2 The 2d, 10 parts yellow, Il red, 1 blue. 
 The 3d, 9 parts yellow, lred, 2 blue. 
 The 4th, 8 parts yellow, lred, 3 blue. 
 The 5th, 7 parts yellow, lred, 4 blue. 
 The 6th, 6 parts yellow, l red, 5 blue. 
 The 7th, 5 parts yellow, lred, 6 blue. 
 The 8th, 4 parts yellow, J] red, 7 blue. 
 The 9th, 3 parts yellow, J] red, 8 blue. 
 The 10th, 2 parts yellow, I red, 9 blue. 
 The llth, 1 part yellow, 1 red, 10 blue. 
 The 12th, Opart yellow, J red, 11 blue. 
 This band, as may be seen, contains all the greens of the lowest band into which 
 
COLOUR OF THE SKY. 3338 
 
 one part of red has been thrown. In like manner, there will be found in the band 
 parallel to the purples, all the purples with which one part of yellow has been mixed ; 
 and in the band parallel and contiguous to the oranges, all the orange colours with 
 one part of blue. 
 
 In the central cell of the triangle there are 4 parts of red, 4 of- blue, and 4 of 
 yellow. 
 
 All these mixtures might be easily made with colours ground exceedingly fine; and 
 if the proper quantities were employed we have no doubt that they would produce 
 all the shades of the different colours. But if all the colours of nature, from the 
 lightest to the darkest, that is from black to white, be required, we shall find for 
 each cell 12 degrees of gradation to white, and 12 others to black. If 91 therefore 
 be multiplied by 24, we shall have 2184 perceptible colours; to which if we add 24 
 grays, formed by the combination of pure black and white, and white and black, 
 the number of compound colours which we believe to be distinguishable by the 
 senses, will amount to 2218. But we ought not perhaps to consider as real colours, 
 those formed by the pure colours with black ; for black only obscures, but does not 
 colour. In this case, the real colours, with their shades, from the darkest to the 
 lightest, ought to be reduced to 1092, which, with a black, and 12 grays, will form 
 1106 colours.- 
 
 PROBLEM Liv. 
 On the Cause of the Blue Colour of the Sky. 
 
 This is a very remarkable phenomenon, though little attention is paid to it, as our 
 eyes are so much accustomed to it from our infancy; and it would be difficult to 
 explain it had not Newton’s theory respecting light, by teaching us that it is decom- 
 posed into seven colours, of different degrees of refrangibility and reflexibility, 
 afforded us the means of discovering the cause. 
 
 To explain this phenomenon, we shall observe then, that according to Newton’s 
 theory, so well proved by experience, of the seven colours which the solar light pro- 
 duces when decomposed by the prism, the blue, indigo, and violet are those easiest 
 reflected, when they meet with a medium of different density, But whatever may be 
 the transparency of the air, that which surrounds our earth, and which constitutes 
 our atmosphere, contains always a mixture of vapours more or less combined with 
 it: bence it happens that the light of the sun and stars, sent back in a hundred 
 different ways into the atmosphere, must experience in it numberless inflections and 
 reflections. But as the blue, indigo, and violet rays are those chiefly sent back to 
 us, at each of these reflections, from the minute particles of the vapours which they 
 are obliged to pass through, it is necessary that the medium which sends them back 
 should appear to assume a blue tint. ‘This must even be the case if we suppose a 
 homogeneity in the atmosphere: for however homogeneons a transparent medium 
 may be, it necessarily reflects a part of the rays of light which pass through it. 
 But of all these rays, the blue are reflected with the greatest facility ; consequently 
 the air, even supposing it homogeneous, would assume a blue, or perhaps a violet 
 colour. 
 
 It is for the same reason that the water of the sea appears of a blue colour when 
 very pure, as is the case at a distance from the coasts. When illuminated by the 
 sun, a part of the rays enters the water, and another part is reflected; but the latter 
 is composed chiefly of blue rays, and consequently it must appear blue. 
 
 This explanation is confirmed by a curious observation of Dr. Halley. This 
 celebrated philosopher having de seended in a diving bell to a considerable depth in 
 the sea, while it was illuminated with the sun, was much surprised to see the back 
 of his hand, which received the direct rays, of a beautiful rose colour, while the 
 lower part, which received the reflected rays, was blue. This indeed is what ought 
 
334 OPTICS. 
 
 to take place, if we suppose that the rays reflected by the surface of the sea, as well 
 as by the minute parts of the middle of it, are blue rays. In proportion as the light 
 penetrates to,a greater depth, it must be more and more deprived of the blne rays, 
 and consequently the remainder must incline to red. 
 
 PROBLEM LV. 
 
 Why the Shadows of bodies are sometimes Blue, or Azure coloured, instead of being 
 
 Black. 
 
 Tt is often observed at sun-rise, during very serene mornings, that the shadows of 
 bodies projected ona white ground, at a small distance, are blue or azure coloured. 
 This phenomenon appears to us to be sufficiently curious to deserve here a place as 
 well as an explanation. 
 
 If the shadow of a body exposed to the sun were absolute, it would be perfectly 
 
 black, since it would be a complete privation of light; but this does not really take | 
 
 place: for to be so, the field of the heavens ought to be absolutely black ; whereas 
 it is blue, or azure coloured, and it is so only because it sends back to us chiefly blue 
 rays, as already observed. 
 
 The shadow therefore projected by bodies,exposed to the sun, is not a pure shadow, 
 but is itself illuminated by that whole part of the sky not occupied by the luminous 
 body. This part of the heaven being blue, the shadow is softened by the blue or 
 azure coloured rays, and consequently must appear of that colour. It is exactly in 
 the same manner that in painting, reflections are tinted with the colour of the sur- 
 rounding bodies. The shadow which we here examine, is nothing else than a shadow 
 
 mixed with the reflection of a blue body, and therefore it must participate in that 
 
 colour. 
 
 It is well known that this phenomenon was first observed and explained by Buffon. 
 
 But it may here be asked, why are not all shadows blue? In reply to this question, 
 we shall observe, that to produce this effect, the concurrence of several circumstances 
 are necessary. Ist. A very puresky, and ofa very dark blue colour; for ifthe heavens 
 be interspersed with light clouds, the rays reflected from them, falling on the bluish 
 shadow, will destroy its effect; if the blue be weak, as is often the case, the quan- 
 tity of the blue rays will not be sufficient to enlighten the shadow. 2d. The light of 
 the sun must be livelier than it usually is when that luminary is near the horizon, in 
 order that the shadows may be full and strong. But these circumstances are rarely 
 united. Besides, the:sun must be only at a small elevation above the horizon; for 
 even when at a moderate altitude there is too much splendour in the atmosphere, 
 to allow the blue rays to be sensible. This light renders the shadows less strong, 
 but does not tinge it blue. 
 
 PROBLEM LVI. 
 Experiment on Colours. 
 
 Hold before your eyes two glasses of different colours, suppose the one blue and 
 the otherred: and having placed yourself-at a proper distance from a candle, if you 
 shut one of your eyes, and look at the light with the other, that for example 
 before which the blue glass is held, the light will appear blue. If you next shut 
 this eye and open the other, the flame will appear red; and if you then open them 
 both, you will see it of a bright violet colour. 
 
 Every person almost, in our opinion, must have foreseen the success of this expe- 
 riment ; which we have mentioned merely because an oculist of Lyons, M. Janin, 
 thought he could deduce from it a particular consequence; which is, that the retina 
 may perform the part of a concave mirror, and reflect the rays of light, so that each 
 eye forms at a certain distance an aérial image of the object. Both eyes forming 
 each an image afterwards in the same place, the result is a double image, one blue 
 
 an 
 
. 
 
 PHOTOPHORUS.—SITUATION OF A LIGHT. 335 
 
 and the other red, which by their union produce a violet image, in the same manner 
 as when red and blue rays are mixed together. But this explanation will certainly 
 not bear to be examined according to the true principles of optics. How is it possible 
 to conceive that such an image can be formed by the retina? Is it not more probable, 
 and more agreeable to the well known phenomena of vision, that from the two im- 
 pressions received by the two eyes, there is produced, in the common sensortum, or in 
 the place where the optic nerves are united in the brain, one compound impression ? 
 In this experiment therefore the same thing must take place, as when a person looks . 
 at a candle with one eye, through two glasses, the one red and the other blue. In 
 this case the flame will be seen of a violet colour, and consequently it must have the 
 same appearance in the former. 
 
 PROBLEM LVII. 
 
 Method of constructing a Photophorus, very convenient to illuminate a table where a 
 person is reading or writing. : 
 
 Construct a cone of tin-plate, 43 inches in diameter at the base, and 72 inches in 
 height, measured on the slant side ; which may be easily done by cutting froma circle, 
 of 73 inches radius, a sector of 1093 degrees, and bending itinto the formofacone. Then 
 through a point in the axis, 24 inches distant from the summit, cut off the upper part 
 of the cone by a plane inclined to one of its sides at an angle of 45°, The result will 
 be an elongated elliptical section, which must be placed before a candle or other light, 
 as near to it as possible, the plane of the section being vertical, and the greatest 
 diameter in a perpendicular direction. When disposed in this manner, if the flame 
 of the candle or lamp be raised 12 or 13 inches above the plane of the table, you will 
 be astonished to see the vivacity and uniformity of the light which it will project over 
 an extent of 4 or 5 feet in length. 
 
 M. Lambert, the inventor of this apparatus, observes that it may be used with 
 great advantage to those who read in bed; for by placing a lamp or taper furnished 
 with this photophorus upon a pretty high stand, at the distance of 5, 6, or 8 feet, 
 from the bed, it will afford a sufficiency of light without any danger. He says he tried 
 this apparatus also in the street, by placing a lamp furnished with it in a window 
 raised 15 feet above the pavement, and that its effect was so great, that at the dis- 
 tance of 60 feet, a bit of straw could be distinguished much better than by moon 
 light, and that writing could be read at the distance of 35 or 40 feet. A few of these 
 machines, placed on each side of a street, and arranged in a diagonal form, would 
 consequently light it much better than any of the means hitherto employed. See 
 Mémoires de 1 Academie de Berlin, ann. 1770. ' 
 
 PROBLEM LVIII, 
 
 The place of an object, such for erample as of a piece of paper on a table, being given ; 
 and that of a candle destined to throw light upon it; to determine the height at 
 which the candle must be placed, in order that the object may be illuminated the most 
 possible. 
 
 That we may exclude from this problem several considerations, which would 
 render the solution of it very difficult, we shall suppose the object destined to be 
 illuminated to be very small, or that it is only required that the middle of it shall 
 be illuminated as muchas possible. We shall suppose also that the light is entirely 
 concentrated into one point, where the splendor of all its different parts is united. 
 
 But it is well known that the light diffused by a luminous point, over any sur- 
 face which it illuminates, decreases, the angle being the same, in the*inverse ratio 
 of the square of the distance; and that when the angle of inclination varies, it is 
 as the sine of that angle. Hence it follows that it decreases in the compound 
 ratio of the square of the distance taken inversely, and the sine of the angle of incli- 
 
336 OPTICS. 
 
 nation taken directly. To solve this problem, then, we must find that height of the 
 luminous point in the given perpendicular, which will render this ratio the greatest 
 possible. 
 
 ‘But it will be found that this ratio is greatest when the perpendicular height, and 
 the distance of the object to be illuminated from the bottom of the candle, are to 
 each other as the side of a square is to the diagonal. On this given and invariable 
 distance as hypothenuse, if a right-angled isosceles triangle be therefore described, 
 the side of this triangle will be the height’at which, if the flame of the candle be 
 placed, the given point or centre of the paper will be illuminated in the highest de- 
 gree possible. 
 
 On this subject, the following problem, which i is of a similar nature, might also be 
 proposed : 
 
 Two candles of unequal height, placed at the extremities of a horizontal line, being 
 given; to find in that line a point so situated, that the object placed in it shall be 
 illuminated the most possible 2 
 
 But we shall not give the solution, that our readers may exercise their own inge- 
 nuity in discovering it. 
 
 PROBLEM LIX. 
 On the Proportion which the Light of the Moon bears to that of the Sun. 
 
 This is a very curious problem: but it is only within these few years that philoso- 
 phers began to turn their attention to the principles, and means, which ean lead to 
 the solution of it. We are indebted for them to M. Bougner, who has explained 
 them in his ‘“ Treatise on the Gradation of Light ;” a work that contains many 
 curious particulars, a few of which we shall here extract. ~ 
 
 To obtain this measure of the intensity of light, M. Bouguer sets out with a 
 fact, founded on experience, which is, that the eye judges pretty exactly by habit, 
 whether two similar and equal surfaces are equally illuminated. Nothing then is 
 necessary, but to place at unequal distances two unequal lights, or by means of concave 
 glasses, the focal distances of which are unequal, to make them be unequally dilated, 
 so that the surfaces which are illuminated by them shall appear to be so in an equal 
 degree. The rest depends merely on calculation: for if two lights, one of which is 
 
 four times nearer than the other, illuminate equally two similar surfaces, it is evident — 
 
 that, as the degrees of illumination of the same light decrease in the inverse ratio 
 of the squares of the distances, we ought to conclude that the splendour of the first 
 
 light is sixteen times as great as that of the second. In like manner, if a light dilated — 
 
 into a circular space, double in diameter, illuminates as much as another direct light, 
 there is reason to conclude that the former is quadruple the second. 
 
 By employing these means, M. Bouguer found that the light of the sun diminished 
 
 11664 times was equal to that of a flambeau, which illuminates a surface at the 
 distance of 16 inches ; and that the same flambeau illuminating a similar surface, at the 
 distance of 50 feet, gave it the same light as that of the moon, when diminished 64 
 times. By compounding these two ratios he concludes, that the light of the sun is 
 to that of the moon, at their mean distances and at the same altitude, as 256289 to 
 1; that is to say 250,000 times greater. From some other experiments, he is even 
 inclined to think, that the light of the moon is only equal to the 300,000th part 
 of that of the sun. 
 
 The result of a celebrated experiment, made by Couplet and La Hire, two acade- 
 micians of Paris, needs therefore excite no surprise. These two philosophers col- 
 lected the lunar rays by means of the burning mirror at the Observatory, which is 
 35 inches in diameter, and made the focus fall on the bulb of a thermometer, but no 
 motion was produced in the liquor.. And indeed this ought to be the case, for if we 
 
CERTAIN OPTICAL ILLUSIONS. 337 
 
 suppose a mirror like the above, which collects the rays that fall on its surface into a 
 space 1200 or 1400 times less, the heat thence resulting will be 1200 or 1400 times 
 greater ; but, on account of the dispersion of the rays, it will be sufficient to suppose 
 this light to be a thousand times denser than the direct light, and the heat in propor- 
 tion. A mirror of this kind then, by collecting the lunar rays, would produce in its 
 focus a heat 1000 times greater than that of the moon. If 300,000 therefore 
 be divided by 1000, we shall have for. quotient 300, which expresses the ratio 
 of the direct solar heat to that of the moon thus condensed. But a beat 300 times 
 less than the direct heat of the sun is not capable of producing any effect on the 
 liquor in the thermometer. This fact then is far from being inexplicable, as we are 
 told by the author of the ‘‘ History of the Progress of the Human Mind in Philo- 
 sophy,”* for it is a necessary consequence of Bouguer’s . calculations, which this 
 writer no doubt overlooked. 
 
 We shall, in the last place, observe, that Bouguer found, by a meaa calculation, 
 that the splendour of the sun, when on the horizon, supposing the sky to be free from 
 clouds or fog, is about 2000 times less than when elevated 66°. The case ought to be 
 the same also with the light of the moon. 
 
 PROBLEM LX. 
 Of certain Optical Illusions. 
 
 I. If you take a seal with a cipher engraved on it, and view it through a convex 
 glass of an inch or more focal distance, the cipher or engraving will be seen sunk in 
 the stone as it really is; but if you continue to look at it, without changing your 
 situation, you will soon see it in relief; and by still continuing to look at it, you 
 
 will see it once more sunk, and then again in relief. Sometimes, after having dis- 
 continued to look at it, instead of seeing it sunk, it will appearin relief ; it will then 
 appear sunk,and soon. When the side of the light is changed, this genera!ly causes 
 achange in the appearance. 
 
 Some have taken a good deal of pains to discover the cause of this illusion ; which 
 in our opinion may be explained without much difficulty. When an object is viewed 
 
 _ with a lens of a short focal distance, and consequently with one eye, we judge very 
 imperfectly of the distance, and the imagiuation has a great share in that assigned 
 ‘to the image which we perceive. On the other hand, the position of the shadow can 
 never serve to rectify the judgment formed of it: forif the engraving is hollow, and 
 if the light comes from the right, the shadow ison the right ; it is also on the right 
 if the engraviug is in relief, and if the light comes from the left. But when an 
 engraved stone is attentively viewed with a magnifying glass, we do not pay atten- 
 tion to the side from which the light proceeds. Here then, every thing, as we may 
 say, is ambiguous and uncertain; consequently it is not surprising that the organ of 
 sight should form an undecisive and continually variable judgment; but we are 
 fully persuaded that an experienced eye will not fall into these variations. 
 
 The same phenomenon is not observed when the experiment is performed with 
 a piece of money. ‘The reason of this probably is, that we are accustomed to handle 
 such pieces, and to see the figures on them in relief, which does not permit the 
 mind to form, in consequence of the image painted in the eye, any other idea than 
 that which it always has had on seeing a piece of money, viz. that of figures in 
 relief. 
 
 II. If a glass decanter, half filled with water, be presented to a concave mirror, 
 and at a proper distance, that is to say between the centre and the focus, it will be 
 seen inverted before the mirror. But it is very singular, in regard to many persons, 
 though it is not general, that they imagine they see the water in the half of the bottle 
 
 * Saverien Histoire du Progres de |’ Esprit humain dans les Sciences physiques, 
 
 Z 
 
338 OPTICS. 
 
 next the neck, which is turned downwards. For our part, we are of opinion that 
 the cause why some think they see the water in this situation, arises from their knows | 
 ing by experience that if a bottle half filled with water be inverted, the fluid will 
 descend into the lower part, or that next the neck. 
 
 Another cause concurs to make us judge in this manner. When a decanter is half | 
 full of very pure water, each half is as transparent as the other, and the pre-— 
 sence of the water is perceived only by the reflection of the light which takes place 
 at itssurface; but in the inverted image this surface reflects the light below, and 
 even with the same force; by which means we are led to conclude that the fluid 
 is at the bottom. 
 
 This subject may be farther exemplified as follows : 
 
 Take a glass bottle; fill it partly with water, and cork it in the usual way: 
 piace this bottle opposite a concave mirror, and beyond its focus, that it may 
 appear reversed : then place yourself still farther distant than the bottle, and this 
 will be seen inverted in the air, and the water, which | 
 is really in the lower part of the bottle, will appear 
 to bein the upper. (See Fig. 59. 60.) If the bottle 
 be inverted while it is before the mirror, the image 
 will appear in its natural erect position, and the water 
 will appear in the lower part of the bottle. While it 
 is in this inverted state, uncork the bottle, then while 
 the water is running out the image is filling. But as 
 soon as the bottle is empty, the illusion ceases. The 
 illusion also ceases when the bottle is quite full. 
 
 The remarkable circumstances in this experiment are, Ist. Not only to see an 
 object where it is not, but also where its image is not. 2d. That of two objects 
 which are really in the same place, as the surface of the bottle and the water it con- 
 tains, the oneis seen inone place, and the other in another, &c. Itis conceived that 
 this illusion arises, partly from our not being accustomed to see water suspended in 
 a bottle with the neck downward, and partly from the resemblance there is between 
 the colour of water and the air. 
 
 Additional Amusements and Experiments with Concave Mirrors, &c., as described by 
 Mr. Adams, Mr. Jones, &c. 
 
 1. Placing yourself before a concave mirror, but farther from it than the centre, 
 you will see an inverted image of yourself, but smaller, in the air between you and 
 the mirror: holding out your hand towards the mirror, the hand of the image will 
 come out towards your hand, and when at the centre of concavity, be of an equal 
 size with it; and you may as it were shake hands with this aérial image. On ad- 
 vancing your hand farther, the handof the image passes by your hand, and comes 
 between it and your body: on moving your hand towards one side, the hand of the 
 image moves towards the other, the image moving always contrariwise to the object. 
 Allthis while, the bye-standers see nothing of the image, because none of the reflected 
 rays which form it enter their eyes. To render this effect more surprising, and more 
 vivid, the mirror is often concealed in a box, after the manner as we shall shew pre- 
 sently. In fact, the appearance of the image in the air, between the object and the 
 mirror, has been productive of many agreeable deceptions, which, when exhibited 
 with art and an air of mystery, have been very successful, and the source of emolu- 
 ment to many of our public showmen. In this manner they have exhibited the 
 images of animated and other objects, in such a way, as to surprise the ignorant, and 
 please the scientific, or better informed. ‘ 
 
 2. Mr. Ferguson mentions two pleasing experiments to be made with a concave 
 mirror, which may be easily tried. If a fire be made in a large room, and a smooth, 
 
well-polished mahogany table be placed at a good distance, near the wall, before a 
 large concave mirror, so that the light of the fire may be reflected from the mirror 
 to its focus on the table; if you stand by the table, you will see nothing but a long 
 beam of light; but if you stand at some distance, as towards the fire, you will see, 
 
 on the table, an image of the fire, large and erect: and if another person, who knows 
 nothing of the matter before hand, should chance to enter the room, he will be 
 startled at the appearance, for the table will seem to be on fire, and being near the 
 wainscot, to endanger the whole house. For the better deception, there ought to 
 be no light in the room but what proceeds from the fire. 
 
 3. If the fire be darkened by a screen, and a large candle be placed at the back 
 of the screen ; then a person standing by the candle will see the appearance of a fine 
 large star, or rather planet, on the table, as large as Jupiter or Venus; and if a small 
 
 wax taper be placed near the candle, it will appear as a satellite to the planet; 
 
 if the taper be moved round the candle, the satellite will be seen to go round the 
 planet. 
 
 * DIOPTRICAL PARADOX.—OPTICAL PARADOX. 339 
 
 4. The Dioptrical Paradox, or Optical Deception. 
 
 Fig. 61. No. 1. Fig. 61. No. 1. represents the dioptrical paradox. It consists 
 of a mahogany base azcD, about eight inches square, with a 
 groove, in which slides various coloured prints, or ornamental 
 drawings: and connected with the base are, a pillar £, a hori- 
 zontal bar F, with a perspective G, which is placed exactly over 
 the centre of the base, and containing a glass of a particular 
 form. The curious and surprising effect of this instrument is, 
 that an ace of diamonds, in the centre of one of the drawings, 
 when placed on the base, shall through the perspective @ be 
 actually represented as the ace of clubs; a figure of a cat in another, seen as an owl; 
 a letter a, as an 0; and a variety of others equally astonishing. 
 
 The principle of this machine is very simple, and is as follows. The glass in the 
 tube G, which produces this change, is somewhat on the principle of the common 
 multiplying glass, and is represented at Fig.61. No.2. The only difference is, that 
 its sides are flat, and diverging from its hexagonal base upwards, to a point in the 
 axis of the glass, like a pyramid, each side forming an isosceles triangle. Its dis- 
 tance from the eye is to be so adjusted, that each angular side, by its refractive 
 power on the rays of light coming from the border of the print, and such a portion 
 designedly there placed, will refract to the eye the various parts as one entire figure 
 to be represented; the shape of the glass preventing any appearance of the original 
 figure 1 in the centre, such as the ace of diamonds being seen: so that the ace of clubs 
 being previously and mechanically drawn on the circle of refraction, 
 at six different parts of the border, 1, 2, 3,4,5,6, (Fig. 62.), and 
 artfully disguised there by blending them with it; then the glass in 
 the tube c will change, in appearance, the ace of diamonds into the 
 ace of clubs. And in like manner for the other prints. 
 
 5. The Optical Paradox. 
 
 Fig. 63. is a representation of the double perspective, or 
 ‘optical paradox. One of the perspectives of the instrument 
 being placed before the eye, an object will be seen directly 
 through both. A board a, or any opake object, being 
 interposed, will not make the least obstruction to the rays ; 
 and the observer will be surprised that he sees through a 
 perspective having the property of penetrating as it were 
 either solid metal or wood. 
 
 z2 
 
340 OPTICS. 
 
 The artifice in this instrument consists chiefly in four small plane mirrors, a, 5, c, d, 
 
 of which a and d are placed at an angle of 45 degrees in the two perspectives, and 6 
 and c parallel to them in the trunk below; this being so formed as to appear only as 
 a solid handle to the two perspectives. It is hence obvious that, on the principle 
 of catoptrics, the object T, falling on the first mirror d, will be reflected down to e, 
 
 thence to b, then up to a, and so out to the eye, giving the appearance of the straight — 
 
 lineal direction ad. 
 
 6. The Endless Gallery. 
 
 Fig. 64, represents a box, about 18 inches in length, 
 
 12 in width, and 9 deep, or any others that are — 
 
 nearly in the same proportions. Against each of its 
 
 opposite ends a and B within place a plane true-ground 
 
 glass mirror, as free from veins as possible, of the di- 
 ‘A mensions nearly equal to the faces, only allowing a 
 
 small space for a transparent paper, or other cover, at 
 top. From the middle of the mirror c, placed at B, take off neatly a round surface 
 of the silvering, about an inch and half in diameter, against which, in the end of 
 the box, must be cut a hole of the same size, or less. The top of the box should be 
 of glass, covered with gauze, or of oiled transparent paper, to admit as much light 
 as possible into the box. On the two longer sides within must be cut or placed two 
 grooves, at E and Ff, to receive various drawings or paintings. Indeed many grooves 
 may be cut in the sides, for the reception of a variety of objects. Two paintings or 
 good drawings, of any perspective subject, must be made on the two opposite 
 faces of a pasteboard, such as a forest, gardens, colonnades, &c.; after having 
 cut the blank parts neatly out, place them in the two grooves, 5, F, of the box. 
 Take also two other boards, of the same dimensions, painted on one side only with 
 similar subjects, to be placed at the opposite ends c and p; observing that the one 
 which is to be placed against c should have nothing drawn there to prevent the 
 ‘sight, and that the other, for the opposite end p, should also not be very full of 
 figures, that after being neatly cutout, and placed against the glass, it may cover 
 but a small part of it. The top being then closed over with its transparent cover, 
 the instrument is ready for use. 
 
 The effect is very striking and entertaining. The eye, being applied to the hole 
 c, will see the various objects drawn on the scenes reflected in a successive and end- 
 less manner, by being reflected alternately from each mirror to that which is opposite. _ 
 As for instance, if they be trees, they will appear an entire grove, very long, seem- 
 ingly without end; each mirror-repeating the objects more faintly, as the reflections 
 are more numerous, and so contributing still more to the illusion. 
 
 Ingenuity will suggest a variety of amusing figures, of men, women, &e., to 
 increase the effect. And two mirrors may also be placed on the longer sides, to con- 
 vey an idea of great breadth, as well as length. 
 
 7. The Real Apparition. 
 
 Behind a partition a B, (Fig. 65), place somewhat 
 inclined a concave mirror E F, which must be at 
 least 10 inches in diameter, and its distance equal to 
 three-fourths from its centre. In the partition is cut 
 
 =>) a square or circular opening, of 7 or 8 inches in dia- 
 meter, directly opposite to the mirror. Behind this 
 a strong light is so disposed as to illuminate strongly 
 an object placed at c, without shining on the mirror, 
 and without being seen at the opening. Beneath the 
 
APPARITION.—REFLEXION, 341 
 
 aperture, and behind the screen is placed any object at c, which is intended to be 
 represented, but in an inverted position, which may be either a flower, or figure, or 
 picture, &c. Befcre the partition, and below the aperture, place a flowerpot D, or 
 _ other pedestal suitable to the object c, so as the top may be even with the bottom 
 of the aperture, and that the eye placed at c may see the flower in the same position as if 
 its stalk came out of the pot. The space between the mirror and the back part of 
 the partition being painted black, to prevent any extraneous light being reflected on. 
 _ the mirror; and indeed the whole disposed so as to be as little enlightened as 
 _ possible-—Then a person placed at c will perceive the flower, or other object, placed 
 behind the partition, as if standing in the flowerpot or pedestal: but on putting forth 
 his hand to pluck it, he will find that he grasps only at a phantom. 
 
 Fig. 66. represents a different position of the mirror and par- 
 tition, and better adapted for exhibiting effect by various ob- 
 jects. ABcisa thin partition of a room, down to the floor, 
 with an aperture for a good convex lens turned outwards into 
 the room, nearly in a horizontal direction, proper for viewing 
 by the eye of a person standing upright from the floor or foot- 
 stool. pb isa large concave mirror, supported at a proper angle, 
 to reflect upwards through the glass in the partition B, images 
 of objects at £, presented towards the mirror below. A strong 
 light from alamp, &c. being directed on the object E, and no where else; then to the 
 eye of a spectator at F, in a darkened room, it is truly surprising and admirable to 
 what effect the images are reflected up into the air at c. 
 
 It is from this arrangement that a showman, both in London and the country, 
 excited the people to the surprise of wonderful apparitions of various kinds of ob- 
 jects, such as a relative’s features for his own, paintings of portraits, plaster figures, 
 flowers, fruit, a sword, dagger, death’s head, &c. 
 
 The phenomena to be produced by concave mirrors are endless ; what have been 
 just described, will be a sufficient specimen of what might be exhibited to elucidate 
 the principles of that curious machine. 
 
 PROBLEM LXI. 
 
 Is it true that Light is reflected with more Vivacity from Air than from Water ? 
 
 This assertion is certainly true, provided it be understood in a proper sense, that 
 is as follows: when light tends to pass from air into water, under a certain obliquity, 
 such as 80° for example, the latter reflects fewer rays, than when the light tends, 
 under the same inclination, to pass from the water into air. But what is very sin- 
 gular, if the air were entirely removed, so as to leave a perfect vacuum in its stead, 
 the light, so far from passing with more facility through this vacuum, which could 
 oppose no resistance, would experience more difficulty, and more rays would be 
 reflected in the passage. 
 
 We do not know why this has been given in the Philosophical Transactions as a 
 paradoxical novelty; for this kind of phenomenon is a necessary consequence of the 
 law of refraction. When light indeed passes from a rare medium into a denser, as 
 from air into water, the passage is always possible; because the sine of the angle of 
 refraction is less than that of the angle of incidence; that is to say, these sines, in the. 
 present case, are in the ratio of 3 to4. But, on the other hand, when light tends 
 to pass obliquely from water into air, the passage under a certain degree of obliquity 
 is impossible, because the sine of the angle of refraction is always much greater than 
 that of the angle of incidence, their sines, in this case, being in the ratio of 4 to 3. 
 There is therefore a certain obliquity of such a nature, that the sine of the angle 
 of refraction would be much greater than the radius ; and this will always happen when 
 the sine of the angle of incidence is greater, however small the excess, than 3 of the 
 
342 | OPTICS. 
 
 radius, which corresponds to an angle of 48° 36’. But a sine can never exceed 
 radius; consequently it is impossible, in this case, that the ray of light should pene- 
 trate the new medium. Thus while light passes from a rare medium into a denser, 
 from air into water for example, under every degree of inclination, there are some 
 rays, viz. all those which form with the refracting substance an angle less than 
 41° 24’, that will not admit of the passage-of light from water into air: it is then 
 under the necessity of being reflected, and refraction is changed into reflection. But 
 *thongh light may pass from water into air, under greater angles of inclination, this 
 tendency to be reflected, or this difficulty of proceeding from one medium into another, 
 is continued at all these angles,in such a manner, that fewer rays are reflected when they 
 tend to pass from air into water under an angle of 60°, than when they tend to pass 
 from water into air under the same angle. In the last place, when light tends in a 
 perpendicular direction from water into air, it is more reflected than when it tends to 
 pass in the same direction from air into water. . 
 
 This truth may be proved by a very simple experiment. Fill a bottle nearly two- 
 thirds with quicksilver, and fill up the other third with water; by which means you 
 will have two parallel surfaces, one of water, and the other of quicksilver. If you 
 then place a luminous object at a mean height between these two surfaces, and the" 
 eye on the opposite side at the same height as the object, you will see the object 
 through the bottle, and reflected almost with equal vivacity from the surface of the 
 quicksilver, and from that which separates the water and the air. The air then, in 
 this case, reflects the light with almost as much vivacity as the quicksilver. 
 
 Remarks.—\1st. We have reason therefore to conclude, that the surface of the ~ 
 water, to beings immersed in that fluid, is a much stronger reflecting mirror, than it 
 is to those beings which are in the air. Fishes see themselves much more dis- 
 tinctly, and clearly, when they swim near the surface of the water, than we see 
 ourselves in the same surface. 
 
 2d. Nothing is better calculated than this phenomenon to prove the truth of the 
 reasons assigned by Newton for reflection and refraction. Light passing froma — 
 dense fluid into a rarer, is, according to Newton, exactly in the same case as a stone — 
 thrown obliquely into the air; if we suppose. that the power of gravitation does not 
 act beyond a determinate distance, such for example as 24 feet; for it may be de- 
 monstrated, that in this case the deviation of the stone would be exactly the same, 
 and subject to the same law, as that followed by light in refraction. There would 
 also be certain inclinations under which the stone could not pass from this atmo- ~ 
 sphere of gravity, though there were nothing beyond it capable of resisting it, and 
 even though there were a perfect vacuum. 
 
 In this case however we must not say, as a certain celebrated man when ex- 
 plaining the Newtonian philosophy, that a vacuum reflects light: this is only a mode 
 of speaking. To express our ideas correctly, we ought to say that light is sent back 
 with greater force to the dense medium, as the medium beyond it is rarer. 
 
 We are far from being satisfied with what is said on this subject in the “ Diction- ~ 
 naire d’Industrie,” into which one may be surprised to see optical phenomena intro- 
 duced ; for it is there asserted, that this phenomenon depends on the impenetrability 
 of pee and the high polish of the reflecting surface. But when light is strongly 
 reflected, during its passage from water into a vacuum, or a space almost free from 
 air, where is the impenetrability of the reflecting substance, since such a space has 
 less impenetrability than air or water? In regard to the polish of the reflecting 
 surface, it is the same, both for the ray which passes from air into water, and that 
 which peases from water into air. 
 
CURIOUS PHENOMENA. 343 
 
 PROBLEM LXII. 
 
 _ Account of a Phenomenon, either not observed, or hitherto neglected by Philo- 
 
 sophers. 
 
 If you hold your ‘finger i in a perpendicular direction rey near your eye, that is to 
 say, at the distance of a few inches at most, and look at a candle in such a manner 
 that the edge of your finger shall appear to be very near the flame, you will see the 
 border of the flame coloured red. If you then move the edge of your finger before 
 the flame, so as to suffer only the other border of it to be seen, this border will ap- 
 pear tinged with blue, while the edge of your finger will be coloured red. 
 
 If the same experiment be tried with an opake body surrounded by a luminous 
 medium, such for example as the upright bar of a sash window, the colours will 
 appear in a contrary order. When a thread of light only remains between your 
 * finger and the bar, the edge of the finger will be tinged with red, and the edgé next 
 the bar will be bordered with blue; but when you bring the edge of your finger near 
 the second edge of the bar, so that it shall be entirely concealed, this second edge 
 will be tinged red, and the edge of the finger would doubtless appear to be coloured 
 blue, were it possible that this dark colour could be seen on an obscure and brown 
 ground. 
 
 This phenomenon depends no doubt on the different refrangibility of light; but a 
 proper explanation of it has never yet been given. . 
 
 PROBLEM LXIII. 
 Of some curious Phenomena in regard to Colours and Vision. 
 
 I. When the window is strongly illuminated with the light of the day, look at it 
 steadily and with attention for some minutes, or until your eyes become a little 
 fatigued; if you then shut your eyes, you will see in your eye a representation 
 of the squares which you looked at; but the place of the bars will be luminous and 
 white, while that of the panes will be black and obscure. If you then place your 
 hand before your eyes, in such a manner as absolutely to intercept the remainder 
 of the light which the eye-lids suffer to pass, the phenomenon will be changed ; for 
 the squares will then appear luminous, and the bars black: if you remove your hand, 
 
 ___ the panes will be black again, and the place of the bars luminous. 
 
 If. If you look steadily and with attention for some time at a luminous body, 
 such as the sun, when you direct your sight to other objects in a place very much 
 illuminated, you will observe there a black spot; a little less light will make the 
 spot appear blue, anda degree still less will make it become purple: in a place abso- 
 | Jutely ‘dark, this spot, which you have at the bottom of your eye, will become lu- 
 minous, 
 
 Ill. If you look for a long time, and till you are somewhat fatigued, at a printed 
 book through green glasses; on removing the glasses, the paper of the book will 
 appear reddish: but if you look at a book in the same manner, through red glasses ; 
 when you lay aside the glasses, the paper of the book will appear greenish. 
 
 IV. If you look with attention at a bright red spot on a white ground, as a red 
 wafer on a piece of white paper, you will see, after some time, a blue border around 
 the wafer; if you then turn your eye from the wafer to the white paper, you will see 
 around spot of delicate green, inclining to blue, which will continue longer accord- 
 ing to the time you have looked at the red object, and according as its splendour 
 and brightness have been greater. On directing your eyes to other objects, this 
 impression will gradually become weaker, and at length disappear. 
 
 If, instead of a red wafer, you look at a yellow one; on turning your eye to the’ 
 white ground you will observe a blue spot. 
 
 A green wafer on a white ground, viewed in the same manner, will produce in 
 
344 OPTICS. 
 
 the eye a spot of a pale purple colour: a blue wafer will produce a spot of a pale 
 red, 
 
 In the last place, if a black wafer on a white ground be viewed in the same man- 
 ner, after looking at it for some time with attention, you will observe a white bor- 
 der form itself around the wafer ; and-ifyou then turn your eye to the white ground, 
 you will observea spot of abrighter white than the ground, and well defined. When 
 you look at a white spot on a black ground, the case will be reversed. 
 
 In these experiments, red is opposed to green, and produces it, as green produces 
 red; blue and yellow are also opposed, and produce each other; and the case is the 
 same with black and white, which evidently indicates a constant effect depending 
 on the organization of the eye. 
 
 This is what is called the Accidental colours, an object first considered by Dr. 
 Jurin, which Buffon afterwards extended, and respecting which he transmitted a 
 memoir in 1743 to the Royal Academy of Sciences. This celebrated man gave no 
 explanation of these phenomena, and only observed that, though certain in regard to 
 the correctness of his experiments, the consequences did not appear to be so well 
 established as to admit of his forming an opinion on the production of these colours. 
 There is reason however to believe that he would have explained the cause, had he 
 not been prevented by other occupations. But this deficiency has been supplied by 
 Dr. Godard of Montpellier; for the explanation which he has given of these pheno- 
 
 mena, and several others of the same kind, in the ‘‘ Journal de Physique” for May 
 
 and July 1776, seems to be perfectly satisfactory. 
 
 PROBLEM LXIV. 
 To determine how long the Sensation of Light remains in the Eye. 
 The following phenomenon, which depends on this duration, is well known. If 
 a fiery stick be moved round in a circular manner, with a motion sufficiently rapid, 
 you will perceive a circle of fire. It is evident that this appearance arises merely 
 from the vibration impressed on the fibres of the retina not being obliterated, when 
 the image of the fiery end of the stick again passes over the same fibres; and there- 
 
 fore, though it is probable that there is only one point of light on the retina, you 
 
 every moment receive the same sensation as if the luminous point left a continued 
 trace. 
 
 But it has been found, by calculating the velocity of a luminous body put in motion, 
 that when it makes its revolution in more than 8 thirds, the string of fire is inter- 
 
 rupted; and hence there is reason to conclude, that the impression made on the fibre 
 continues during that interval of time. Butit may be asked whether this time is the - 
 
 same for every kind of light, whatever be its intensity ? We do not think it is; 
 for a brighter light must excite a livelier and more durable impression. 
 
 This affection of the eye has been used by Dr. Paris 
 in constructing a toy called the Thaumatrope. It is 
 shewn in Fig. 67., where 4 B is a circle cut out of card, 
 and having two strings c D, fixed to it, by twisting 
 which with the finger and thumb of each hand it may be 
 twirled round rather rapidly. On one side of the cord 
 
 a charioteer in the attitude of driving,—so that when the 
 cord is twisted round, the charioteer is seen driving the 
 chariot as in the figure,—in consequence of the dura- 
 tion of the impressions of sight on the retina, we see 
 at once what is drawn on both sides of the card. 
 
 there is drawn any object, as a chariot, and on the other | 
 
MICROSCOPICAL OBSERVATIONS. 345 
 
 SUPPLEMENT, 
 
 CONTAINING A SHORT ACCOUNT OF THE MOST CURIOUS 
 MICROSCOPICAL OBSERVATIONS. 
 
 PwHILOSOPHERS were no sooner in possession of the microscope, than they began to 
 employ this wonderful instrument in examining the structure of bodies, which, in con- 
 sequence of their minuteness, had before eluded their observation. There is 
 scarcely an object in nature to which the microscope has not been applied ; and seve- 
 ral have exhibited such a spectacle as no one could have ever imagined. What 
 indeed could be more unexpected than the animals or molecule (for philosophers 
 are not yet agreed in regard to their animality) which are seen swimming in vinegar, 
 in the infusions of plants, and in the semen of animals? What can be more curious 
 than the mechanism in the organs of the greater part of insects, and particularly those 
 which in general escape our notice; such as the eyes, trunks, feelers, terebre or 
 augres, &c.? What more worthy of admiration than the composition of the blood, 
 the elements of which we are enabled to perceive by means of the microscope; the 
 texture of the epidermis, the structure of the lichen, that of mouldiness, &c.? We 
 shall here take a view of the principal of these phenomena, and give a short account 
 of the most curious observations of this kind. 
 
 I.—Of the Animals, or Pretended Animals, in Vinegar and the infusions of Plants. 
 
 lst. Leave vinegar exposed for some days to the air, and then place a drop of it on 
 the transparent object-plate of the microscope, whether single or compound: if the 
 object-plate be illuminated from below, you will observe, in this drop of liquor, ani- 
 mals resembling small eéls, which are in continual motion. On account of the 
 circumvolutions which they make with their long, slender bodies, they may be 
 justly compared to small serpents. 
 But it would be wrong, as many simple people have done, to ascribe the acidity of 
 vinegar to the action of these animalcules, whether real or supposed, on the tongue 
 and the organs of taste; for vinegar deprived of them is equally acid, if not more so. 
 These eels indeed, or serpents, are never seen but in vinegar which, having been for 
 some time exposed to the air, is beginning to pass from acidity to putrefaction. 
 
 2d. If you infuse pepper, slightly bruised, in pure water for some days, and then 
 expose a drop of it tothe microscope, you will behold small animals of another kind, 
 almost without number. They are of a moderately oblong, elliptical form, and are 
 
 seen in continual motion, going backwards and forwards in all directions ; turning 
 aside when they meet each other, or when their passage is stopped by any immove- 
 able mass. Some of them are observed sometimes to lengthen themselves, in order 
 to pass through a narrow space. Certain authors of a lively imagination, it would 
 appear, even pretend to have seen them copulate, and bring forth ; but this assertion 
 we are not bound to believe. 
 
 If other vegetable bodies be infused in water, you will see animalcules of a diffe- 
 rent shape. In certain infusions they are of an oval form, with a small bill, and a 
 long tail: in others they have a lengthened shape like lizards: in some they exhibit 
 the appearance of certain caterpillars, or worms, armed with long bristles ; and some 
 devour, or seem to devour, their companions. 
 
 When the drop in which they swim about, and which to them is like a capacious 
 basin, becomes diminished by the effect of evaporation, they gradually retire towards 
 the middle, where they accumulate themselves, and at length perish, when entirely 
 deprived of moisture. They then appear to be in great distress ; writhe their bodies, 
 and endeavour to escape from death, or that state of uneasiness which they experience. 
 
346 OPTICS. 
 
 In general, they have a strong aversion to saline or acid liquors. If asmall quantity 
 of vitriolic acid be put into a drop of infusion which swarms with these insects, they 
 immediately throw themselves on their backs and expire; sometimes losing their 
 skin, which bursts, and suffers to escape a quantity of small globules that may be 
 often seen through their transparent skin. The case is the same if a little urine be 
 thrown into the infusion. ; 
 
 A question here naturally arises: ought these moveable molecule to be considered 
 as animals? On this subject opinions are divided. Buffon thinks they are not ani- 
 mals; and consigns them, as well as spermatic animals, to the class of certain bodies . 
 which he ealls organic molecule. But what is meant by the expression organic mole- 
 cule? As this question would require too long discussion, we must refer the reader 
 to the Natural History of that learned and celebrated writer. 
 
 Needham also contests the animality of these small bodies, that is to say perfect 
 animality, which consists in feeding, increasing in size, multiplying, and being — 
 endowed with spontaneous motion; but he allows them a sort of obscure vitality, 
 and from all his observations he deduces consequences on which he has founded a 
 very singular system. He is of opinion that vegetable matter tends to animalise — 
 itself. As the eels produced in flour paste act a conspicuous part in the system of 
 this naturalist, a celebrated writer has omitted no opportunity of ridiculing his ideas, 
 by calling these animals the eels of the jesuit Needham, and representing him asa 
 partisan of spontaneous generation, which has been justly exploded by.all the modern 
 philosophers. But ridicule is not reasoning: we are so little acquainted with the 
 boundaries between the vegetable and animal kingdoms, that it would be presuming ~ 
 too much to fixthem, But if must be allowed that Needham’s ideas on this subject 
 are so obscure, that in our opinion few have been able to comprehend them. 
 
 Other naturalists and observers assert the animality of these small beings: for they 
 ask, by what can an animal be better. characterised. than spontaneity of motion ? 
 But these molecule, when they meet each other in the course of their movements, 
 retire backwards, not by the effect of a shock, as two elastic bodies would do; but 
 the part which is generally foremost turns aside on the approach of the body that 
 meets it: and sometimes both move a little from their direction, in order to avoid 
 running against each other. They have never yet indeed been seen for certain to 
 copulate, to produce eggs, or even to feed; but the last-mentioned function they — 
 may perform, without any apparent act, like the greater part of the other animalcules. — 
 The smallness and strange form of these molecule can afford no argument against 
 their animality. That of the water_polypes is at present no longer doubted, though - 
 their form is very extraordinary, and perhaps more so than that of the moving mole- 
 cule of infusions. Why then should animality be refused to the latter ? 
 
 It might however be replied, in opposition to this supposed similarity, that the 
 polype is seen to increase in size, to regenerate itself, in a way indeed very different 
 from that of the generality of animals, and in particular to feed. The pretended 
 microscopic animals do nothing of the kind, and consequently ought not to be ranked 
 in the same class. But it must be allowed that this subject is still involved in very 
 great obscurity; and therefore prudence requires that we should suspend our opinion 
 respecting it. 
 
 I1.—Of Spermatic Animals. 
 
 Of the microscopic discoveries of the last century, none has made a greater noise 
 than that of the moving molecule observed in the semen of animals, and which are 
 called Spermatic animalcules. This singular discovery was first made and announced 
 by the celebrated Lewenhoek, who observed in the human semen a multitude of — 
 small bodies, most of them with very long slender tails, and in continual motion. 
 In size they were much less than the smallest grain of sand, and even so minute in — 
 
 . 
 + 
 
 i 
 
7 
 
 some seminal liquors, that a hundred thousand, and even a million of them, were not 
 equal to a poppy seed. By another calculation, Lewenhoek has shewn that in the 
 milt of a cod-fish there are more animals of this species than human beings on the 
 whole surface of the earth. 
 
 Lewenhoek examined also the prolific liquor of a great many animals, both 
 quadrupeds and birds, and that even of some insects. In all these he observed 
 nearly the same phenomenon ; ; and these researches, since repeated by many other 
 observers, have given rise to a system in regard to generation, which it is pee 
 here to explain. 
 
 ‘No one however has made more careful, or more correct, observations on this 
 subject than Buffon ; and for this reason we shall give.a short view of them. | 
 
 This celebrated naturalist, having procured a considerable quantity of semen, 
 extracted from the seminal vessels of a man who had perished by a violent death, 
 observed in it, when viewed through an excellent microscope, longish filaments, which 
 
 _had akind of vibratory motion, and which appeared to contain in the inside small 
 bodies. ‘The semen having assumed a little more fluidity, he saw these filaments 
 swell up in some points, and oblong elliptical bodies issue from them; a part of which 
 remained at first attached to the filaments by a very slender long tail. Some time 
 after, when the semen had acquired a still greater degree of fluidity, the filaments 
 disappeared, and nothing remained in the liquor but these oval bodies with tails, by 
 the extremity of which they seemed attached to the fluid, and on which they 
 balanced themselves like a pendulum, having however a progressive motion, though 
 slow, and as it were embarrassed by the adhesion of their tails to the fluid; they 
 exhibited also a sort of heaving motion, which seems to prove that they had nota 
 flat base, but that their transverse section was nearly round. In about twelve or 
 fifteen hours after, the liquor having acquired a still greater degree of fluidity, the 
 small moving molecule had lost their tails, and appeared as elliptic bodies, moving 
 with great vivacity. In short, as the matter became attenuated in a greater degree, 
 they divided themselves more and more, so as at length to disappear; or they were 
 precipitated to the bottom of the liquor, and seemed to lose their vitality. 
 
 Buffon, while viewing these moving molecule, once happened to see them file 
 off like a regiment, seven by seven, or eight by eight, proceeding always in very 
 close bodies towards the same side. Having endeavoured to discover the cause 
 of this appearance, he found that they all proceeded from a mass of filaments 
 accumulated in one corner of the spermatic drop, and which resolved itself suc- 
 
 — eessively in this manner into small elongated globules, all without tails. This cir- 
 cumstance reminds us of the singular idea of a naturalist, who observing a similar 
 phenomenon in the semen of a ram, thought he could there see the reason of the 
 peculiar propensity which sheep have to follow each other, when they march together 
 
 in a flock. 
 
 Buffon examined, in like manner, the spermatic liquor of various other animals, 
 
 _ such as the bull, the ram, &c., and always discovered the same molecule, which at 
 
 - first had tails, and then gradually lost them as the liquor assumed more fluidity. 
 
 Sometimes they seemed to have no tails, even on their first appearance and forma- 
 tion. In this respect, Buffon’s observations differ from those of Lewenhoek, who 
 
 _ always describes these animalcules as having tails, with which he says they seem to 
 
 assist themselves in their movements; and he adds, that they are seen to twist 
 themselves in different directions. Buffon’s observations differ also from those of the 
 | Dutch naturalist in another respect, as the latter says that he never could discover 
 any trace of these animalcules in the semen or liquor extracted from the ovaria 
 of females; whereas Buffon saw the same moving molecule in that liquor, but not 
 so often, and only under certain circumstances. 
 
 It appears, from what has been said, that many. researches still remain to be made 
 
 SPERMATIC ANIMALS. _ 347 
 
348 OPTICS. 
 
 in regard to the nature of these moving molecule; since two observers so celebrated” 
 do not agree in all the circumstances of the same fact. 
 
 Nothing of this kind is observed in the other animal fluids, such as the blood, 
 lymph, milk, saliva, urine, gall, and chyle; which seems to indicate that these ani- 
 malcules, or living molecule, act a part in generation. 
 
 TII.—Of the Animals or Moving Molecule in spoiled corn. 
 
 This is another microscopic observation, which may justly be considered as one 
 of the most singular; for if we deduce from it all those consequences which some 
 authors do, it exhibits an instance of a resurrection, repeated, as we may say, at 
 pleasure. 
 
 The disease of corn, which produces this phenomenon, is neither smut nor blight, 
 as some authors, for want of a sufficient knowledge in regard to the specific differ- 
 ences of the maladies of grain, have asserted, but what ought properly to be called 
 abortion, or rachitis. If a grain of corn, in this state, be opened with caution, it will 
 be found filled with a white substance, which readily divides itself into a multitude 
 of small, white elongated bodies, like small eels, swelled up in the middle. While 
 these molecule, for we must be allowed as yet to remain neuter in regard to their 
 pretended animality, are in this state of dryness, they exhibit no signs of life; but 
 if moistened with very pure water, they immediately put themselves in motion, and 
 shew every mark of animality. If the fluid drop, in which they are placed, be 
 suffered to dry, they lose their motion; but it may be restored to them at pleasure, 
 even some months after their apparent death, by immersing them in water. Fon- 
 tana, an Italian naturalist, does not hesitate to consider this phenomenon as a real 
 resurrection. If this circumstance should be verified by repeated observations, and 
 that also of the Peruvian serpent, which may be restored to life by plunging it 
 in the mud, its natural element, several months after it has been suffered to dry at 
 the end of a rope, our ideas respecting animality may be strangely changed. But 
 we must confess that we give very little credit to the latter fact ; though Bouguer, 
 who relates it on the authority of Father Gumilla, a Jesuit and a French surgeon, 
 does not entirely disbelieve it. Some other observers, such as Roffredi, pretend to 
 have distinguished, in these eel-formed molecule, the aperture of the mouth; that 
 of the female parts of sex, &c. They assert also that they have perceived the 
 motion of the young ones contained in the belly of the mother eel, and that having 
 opened the body, the young were seen to disperse themselves all over the object-plate 
 of the microscope. ‘These observations deserve to be further examined, as a con- 
 firmation of them would throw great light on animality. 
 
 Note.—It appears from the careful observation of Francis Bauer, Esq., that the 
 molecules in question, the nature of whose existence so puzzled Montucla, are minute 
 parasitical fungi of the genus wredo. See an interesting article on the subject in 
 the Penny Magazine for March 31st, 1833, and continued in the No. for May 11th, 
 1823. 
 
 IV.—On the Movements of the Tremella. 
 
 The tremella is that gelatinous green plant, which forms itself in stagnant water, 
 and which is known to naturalists by the name of conferva gelatinosa, omnium tener- 
 rima et minima, aquarum limo innascens. It consists of a number of filaments inter- 
 woven through each other, which when considered singly are composed of small parts, 
 about aline in length, united by articulations. 
 
 This natural production, when viewed with the naked eye, exhibits nothing re- 
 markable or uncommon; but by means of microscopic observations, two very extra- 
 ordinary properties have been discovered in it. One is the spontaneous motion with 
 
 ae 
 
‘ CIRCULATION OF THE BLOOD, 349 
 
 which these filaments are endowed. Ifa single one, sufficiently moistened, be placed 
 _on the object-plate of the microscope, its extremities are seen to rise and fall alter- 
 
 nately, and to move sometimes to the right and sometimes to the left: at the 
 same time, it twists itself in various directions, and without receiving any external 
 impression. Sometimes, instead of appearing extended like a straight line, it forms 
 itself into an oval or irregular curve. If two of them are placed side by side, they 
 become twisted and twined together, and bya sort of imperceptible motion, the one 
 
 from one side, and the other from the other. This motion has been estimated by 
 Adansou to be about the 400th part of a line per minute. 
 
 The other property of this plant is, that it dies and revives, as we may say, seve- 
 yal times; for if several filaments, or a mass of tremella, be dried, it entirely loses the 
 faculty above mentioned. It will remain several months in that state of death or 
 sleep; but when immersed in the necessary moisture, it revives, recovers its power 
 _of motion, and multiplies as usual. 
 
 _ The Abbé Fontana, a celebrated observer of Parma, does not hesitate, in conse- 
 
 quence of these facts, to class the tremella among the number of the Zoophytes; 
 and to consider it as the link which connects the vegetable with the animal king- 
 
 dom, or the animal with the vegetable; in a word, as an animal or a vegetable 
 endowed with the singular property of being able to die and to revive alter- 
 nately. But is this a real death, or only a kind of sleep, a suspension of all the 
 -faculties in which the life of the plant consists? To auswer this question, it would 
 be necessary to know exactly what is the nature cf death; a great deal might be 
 said on this subject, were not such disquisitions foreign to the present work. 
 
 V.—Of the Circulation of the Blood. 
 
 Those who are desirous to observe the circulation of the blood by means of the 
 microscope, may easily obtain that satisfaction. The objects employed chiefly for this 
 purpose, are the delicate, transparent member which unites the toes of-the frog, and 
 the tail of the tadpole. If this membrane be extended, and fixed on a piece of glass 
 illuminated below, yeu will observe with great satisfaction the motion of the blood 
 in the vessels with which it is interspersed: you will imagine that you see an archi- 
 pelago of islands witha rapid current flowing between them. 
 
 Take a tadpole, and having wrapped up its bodyin a piece of thin, moist cloth, 
 place its tail on the object-plate of the microscope, and enlighten it below: you will 
 
 then see very distinctly the circulation of the blood; which in certain vessels pro- 
 ceeds by a kind of undulations, and in others with an uniform motion. The former 
 
 are the arteries, in which the blood moves in consequence of the alternate pulsation 
 of the heart ; the latter are the veins. 
 
 The circulation of the blood may be seen also in the legs and tails of shrimps, by 
 putting these fish into water with a little salt; but their blood isnot red. The 
 wings of the locust are also proper for this purpose: in these the observer will see, 
 not without satisfaction, the green globules of their blood carried away by the serosity 
 in which they float. The transparent legs of small spiders, and those of small bugs, 
 will also afford the means of observing the circulation of their blood. The latter 
 
 exhibit an extraordinary vibration of the vessels, which Mr. Baker says he never saw 
 
 any where else. 
 But the most curious of all the spectacles ot this kind, is that exhibited by the 
 
 | mesentery of a living frog, applied in particular to the solar microscope, which Mr. 
 _ Baker tells us he did incompany with Dr. Alexander Stuard, physician to the queen. . 
 _ It is impossible to express, says he, the wonderful scene which presented itself to our 
 
 eyes. We saw at the same moment the blood, which flowed in a prodigious number 
 
 _ of vessels, moving insome to one side, and in others to the opposite side. Several of 
 _ these vessels were magnified to the size of an inch diameter; and the globules of 
 
350 | OPTICS. 
 
 blood seemed almost as large as grains of pepper, while in some of the vessels, which 
 were much smaller, they could pass only one by one, and were obliged to change their 
 figure into that of an oblong spheroid. 
 
 VI.— Composition of the Blood. 
 
 With the end of a quill, ora very soft brush, take upa small drop of blood just drawn 
 from a vein, and spread it as thin as possible over a bit of tale. If you then apply 
 to your microscope one of the strongest magnifiers, you will distinctly see its 
 globules. 
 
 By these means it has been found, that the red globules of the human blood are 
 each composed of six smaller globules, united together; and that when disunited by 
 any cause whatever, they are no longer of ared colour. Thesered globules are so 
 exceedingly minute, that their diameter is only the 160th part cf a line, so that a 
 sphere of a line in diameter would contain 4096000 of them. 
 
 VII.—Of the Skin ; its Pores, and Scales. 
 
 If you cut off a small bit of the epidermis by means of a very sharp razor, and 
 place it on the object-plate of the microscope; you will see it covered with a mul- 
 titude of small scales, so exceedingly minute, that, according to Lewenhoek, a graia 
 of sand would cover two hundred of them; that is to say, in the diameter of a grain 
 
 -of sand there are 14 or 15. These scales are arranged like those on the back 
 of fishes, or like the tiles of a house; that is, each covering the other. 
 
 If you are desirous of viewing their form with more convenience, scrape the epi- 
 dermis with a pen-knife, and put the dust obtained by these means into a drop 
 of water: you will then cbserve that these scales, in general, have five planes, and 
 that each consists of several strata. 
 
 Below these scales are the pofes of the epidermis which when the former are re- 
 moved may be distinctly perceived, like smali holes pierced with an exceedingly fine 
 needle. Lewenhoek counted 120 in the length of a line; so that a line square, 10 
 of which form an inch, would contain 14400; consequently a square foot would 
 contain 144000000; and as the surface of the human body may be estimated at 14 
 square feet, it must contain 2016 millions. 
 
 Each of these pores corresponds in the skin to an excretory tones the edge 
 of which is lined with the epidermis. When the epidermis has been detached from 
 ‘the skin, these internal prolongations of the epidermis may be observed in the same 
 manner as we see, in the reverse of a piece of paper, pierced with a blunt needle, 
 the rough edge formed by the surface, which has been torn and turned inwards. 
 
 The pores of the skin are more particularly remarkable in the hands and the feet. 
 If you wash your hands well with soap, and look at the palm with a common mag- 
 nifier, you will see a multitude of furrows, between which the pores are situated. 
 If the body be in a state of perspiration at the time, you will see issuing from 
 these pores a small drop of liquor, which gives to each the appearance of a 
 fountain. 
 
 VIII.—Of the Hair of Animals. 
 
 The hairs of animals, seen through the microscope, appear to be organized bodies, 
 like the other parts; and, by the variety of their texture and conformation, they — 
 afford much subject of agreeable observation. In general, they appear to be eom-— 
 posed of long, slender,hollow tubes, or of several small hairs covered with a common 
 bark ; others, such as those of the Indian deer, are hollow quite through. . The 
 bristles of a cat’s whiskers, when cut transversely, exhibit the appearance of a me- 
 dullary part, which occupies the middle, like the pith ina twig of the elder-tree. 
 Those of the hedge-hog contain a real marrow, which is whitish, and formed of radii. 
 
 re 
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| 
 
 | 
 
 Bg 
 7 
 
 EYES OF INSECTS.—MITES IN CHEESE. 351 
 
 As yet however we are not perfectly certain in regard to the organization of the 
 human hair. Some observers, seeing a white line in the middle, have concluded that 
 it is a vessel which conveys the nutritive juice to the extremity. Others contest 
 this observation, and maintain that it is merely an optical illusion, produced by the 
 covexity of the hair. It appears however that some vessel must be extended length- 
 wise in the hair, if it be true that blood has been seen to issue at the extremity 
 of the hair cut from persons attacked with that disease called the Plica Polonica. 
 But quere, is this observation certain ? 
 
 IX.—Singularities in regard to the Eyes of most Insects. 
 
 The greater part of insects have not moveable eyes, which they can cover with 
 eye-lids at pleasure, like other animals. These organs, in the former, are absolutely 
 immoveable; and as they are deprived of that useful covering assigned to others for 
 defending them, nature has supplied this deficiency by forming them of a kind 
 of corneous substance, proper for resisting the shocks to which they might be 
 exposed. 
 
 But it is not in this that the great singularity of the eyes of insects consists. We 
 discover by the microscope that these eyes are themselves divided into a prodigious 
 multitude of others much smaller. If we take a common fly, for example, and 
 examine its eyes by the microscope, we shall find that it has on each side of its head 
 a large excrescence, like a flattened hemisphere. This may be perceived without a 
 microscope ; but by means of this instrument these hemispheric excrescences will be 
 
 seen divided into a great number of rhomboids, having in the middle a lenticular con- 
 
 vexity, which performs the part of the crystalline humour. Hodierna counted more 
 than 3000 of these rhomboids on one of the eyes of a common fly; M. Puget reckoned 
 8000 on each eye of another kind of fly, so that there are some of these insects 
 which have 16000 eyes; and there are some which even have a much greater num- 
 ber ; for Lewenhoek counted 14000 on each eye of another insect. 
 
 These eyes however are not all disposed in the same manner: the dragon fly, for 
 example, besides the two hemispherical excrescences on the sides, has between these 
 two other eminences, the upper and convex surface of which is furnished with a 
 multitude of eyes, directed towards the heavens. The same insect has three also in 
 front, in the form of an obtuse and rounded cone. The case is the same with the 
 fly, but its eyes are less elevated. 
 
 It is an agreeable spectacle, says Lewenhoek, to consider this multitude of eyes in 
 insects ; for if the observer is placed in a certain manner, the neighbouring objects 
 appear painted on these spherical eminences of a diameter exceedingly small, and by © 
 
 means of the microscope they are seen multiplied, almost as many times as there are 
 
 eyes, and in such a distinct manner as never can be attained to by art. 
 
 A great many more observations might be made in regard to the organs of insects, 
 and their wonderful variety and conformation, but these we shall reserve for another 
 place. 
 
 X.—Of the Mites in Cheese, and other Insects of the same kind. 
 
 If you place on the object-plate of the microscope some of the dust which is formed ~ 
 on the rind and other neighbouring parts of old cheese, it will be seen to swarm with 
 a multitude of small transparent animals, of an oval figure, terminating in a point, 
 and in the form of a snout. These insects are furnished with eight scaly, articulated 
 legs, by means of which they move themselves heavily along, rolling from one side 
 to the other; their head is terminated by an obtuse body in the form of a truncated 
 cone, where the organ through which they feed is apparently situated. Their bodies, 
 particularly the lateral parts, are covered with several long sharp-pointed hairs, and 
 the anus, bordered with hair, is seen in the lower part of the belly. 
 
352 OPTICS. 
 
 There are mites of another kind which have only six legs, and which consequently 
 are of a different species. 
 
 Others are of a vagabond nature, as the observer calls them, and are found in all 
 places where there are matters proper for their nourishment. 
 
 This animal is extremely vivacious ; for Lewenhoek says that some of them, which 
 he had attached to a pin before his microscope, lived in that manner eleven weeks. 
 
 XI.—Of the Louse and Flea. 
 
 Both these animals are exceedingly disagreeable, particularly the latter, and do not 
 seem proper for being the subject of microscopic observation; but to the philosopher 
 no object in nature is disagreeable, because deformity is merely relative, and the 
 most hideous animal often exhibits singularities, which serve to make us better 
 acquainted with the infinite variety of the works of the Creator. ; 
 
 If you make a louse fast for a couple of days, and then place it on your hand, you 
 will see it soon attach itself to it, and plunge its trunk into the skin. If viewed in 
 this state by means of a microscope, you will sée, through its skin, your blood 
 flowing under the form of a small stream, into its ventricle, or the vessel that supplies 
 its place, and thence distributing itself to the other parts, which will become dis- 
 tended by it. 
 
 This animal is one of the most hideous in nature: its head is triangular, and ter- 
 minates in a sharp point, to which is united its proboscis or sucker. On each side 
 of the head, and ata small distance from its anterior point, are placed two large an- 
 tenne, covered with hair; and behind these, towards the two other obtuse angles of 
 the triangle, are the animal’s two eyes. The head is united by a short neck to the 
 corslet, which has six legs furnished with hair at the articulations, and with two 
 hooks each at the extremity. -The lower part of the belly is almost transparent, and 
 on the sides has a kind of tubercles, the last of which are furnished with two hooks, 
 Dr. Hook, in his ‘‘ Micrographia,” has given the figure of one of these animals, about 
 half a foot in length. ‘Those who see the representation of this insect will not be 
 surprised at the itching on the skin which it occasions to persons, who in consequence 
 of dirtiness are infested with it. 
 
 The flea has a great resemblance to the shrimp, as its back is arched in the same 
 manner as the back of that animal.’ It is covered as it were with a coat of mail, 
 consisting of large scales laid over each other; the hind part is round, and very large 
 in regard to the rest of the body; its head is covered by a single scale, and at the 
 extremity has a kind of three terebra, by means of which the insect sucks the blood 
 of animals. Six legs, with thighs exceedingly thick, and of which the first pair 
 are remarkably long, enable it to perform all its movements. The great size of the 
 thighs is destined, no doubt, to contain the powerful muscles which are necessary to 
 carry the insect to a height or distance equal to several hundred times its length. 
 Being destined to make such large leaps, it was also necessary that it should be 
 strongly secured against falls to which it might be exposed, and nature has made 
 ample provision against accidents of this kind, by supplying it with scaly armour. 
 Figures: of the flea and louse, highly magnified, will be found in the works of Hook 
 and Joblot. 
 
 XII.—Mouldiness. 
 
 Nothing can be more curious than the appearance exhibited by mouldiness, when 
 viewed through the microscope. When seen by the naked eye, one is almost induced 
 to consider it as an irregular tissue of filaments ; but the microscope shews that 
 it is nothing else than a small forest of plants, which derive their nourishment from 
 the moist substance, tending towards decomposition, which serves them as a base. 
 The stems of these plants may be plainly distinguished; and sometimes their buds, 
 
LYCOPERDON.—FARINA OF FLOWERS.—LEAVES OF PLANTS, 353 
 
 some shut and others open. Baron de Munchausen has even done more: when 
 carefully examining these small plants, he observed that they had a great similarity 
 to mushrooms. They are nothing, therefore, but microscopic mushrooms, the tops 
 of which, when they come to maturity, emit an exceedingly fine kind of dust, which 
 is their seed. It is well known that mushrooms spring up in the course of one night ; 
 but those of which we here speak, being more rapid, almost in the inverse ratio of 
 _their size, grow up in a few hours. Hence the extraordinary progress which moul- 
 _ diness makes in a very short time. 
 
 Another very curious observation of the same kind, made by M. Ahlefeld of 
 Giessen, is as follows: Having seen some stones covered with a sort of dust, he had 
 the curiosity to examine it with a microscope, and found, to his great astonishment, 
 that it consisted of small microscopic mushrooms, raised on very short pedicles, the 
 heads of which, round in the middle, were turned up at the edges. They were 
 striated also from the centre to the circumference, as certain kinds of mushrooms 
 -are. He remarked likewise that they contained, above their upper covering, a mul- 
 _ titude of small grains, shaped like cherries, somewhat flattened; which in all proba- 
 bility were the seeds. In the last place, he observed, in this forest of mushrooms, 
 several small red insects, which no doubt fed upon them, (See Act. Leips. for the 
 year 1739.) 
 
 XIII.—Dust of the Lycoperdon. 
 
 The lycoperdon, or puff-ball, is a plant of the fungus kind, which grows in the 
 form of a tubercle, covered with small grains like shagreen. If pressed with the 
 foot, it bursts, and emits an exceedingly fine kind of dust, which flies off under the 
 appearance of smoke; but commonly a pretty large quantity remains in the half 
 opened cavity of the. plant. If some of this dust be placed on the object-plate of 
 the microscope, it appears to consist of perfectly round globules, of an orange colour, 
 the diameter of which is only about the 50th part of a hair; so that each grain of 
 this dust is but the 125000th part of a globule equal in diameter to the breadth of 
 a hair. Some lycoperdons contain browner spherules, attached to a small pedicle. 
 This dust no doubtis the seed of this anomalous plant. 
 
 XIV.—Of the Farina of Flowers. 
 
 It is not long since the utility of this farina in the vegetable economy was known. 
 Betore this discovery, it was thought to be nothing else than the excrement of the 
 juices of the flower; but it is shewn by the microscope that this dust is regularly and 
 uniformly organised in each kind of plant. In the mallow, for example, each grain is 
 an opake ball, entirely covered with points. The farina of the tulip, and of most of 
 the lily kind of flowers, has a resemblance to the seeds of cucumbers and melons. 
 That of the poppy resembles a grain of barley, with a longitudinal groove in it. 
 
 But weare taught by observation still more; for itis found that this dust or farina 
 
 is only a capsule, which contains another far more minute; and it is the latter which 
 is the real fecundating dust of plants. 
 
 XV.—Of the apparent holes in the Leaves of some Plants. 
 
 There are certain plants, the leaves of which appear to be pierced with a multitude | 
 of small holes. Of this kind, in particular, is that called by the botanists hypericum, 
 and by the vulgar St. John’s wort. But if a fragment of one of these leaves be 
 viewed through a microscope, the supposed holes are found to be vesicles, contained 
 in the thickness of the leaf, and covered with an exceedingly thin membrane: in a 
 word, they are the receptacles which contain the essential and aromatic oil peculiar 
 to that plant. 
 
 rt 
 
 2A 
 
354 OPTICS. 
 
 XVIL—Of the Down of Plants. : i ; 
 
 The spectacle exhibited by those plants which have down, such as borage, nettles, | 
 &c., is exceedingly curious.. When viewed through the microscope, they appear to 
 be so covered with spikes as to excite horror. Those of borage are for the most part | 
 bent so as to form an elbow ; and, though really very close, they appear by the mi-, 
 eroscope to be at a considerable distance from each other. Persons who are not: 
 previously told what substance they are looking at, will almost be induced to be. | 
 lieve that they see the skin of a porcupine. 
 
 XVII.---Of the Sparks struck from a piece of Steel by means of a Flint. 
 
 If sparks struck from a piece of steel by a flint be made to fall on a leaf of paper, 
 they will be found, for the most part, to be globules, formed of small particles of 
 stee], detached by the shock, and fused by the friction. Dr. Hook observed some | 
 which were perfectly smooth, and reflected with vivacity the image of a neighbour- 
 ing window. When in this state, they are susceptible of being attracted by the 
 magnet; but very often they are reduced by the fusion to a kind of scoria, and in 
 that case the magnet has no power over them. The cause of this we shall explain 
 hereafter. This fusion will excite no surprise, when it is known that the bodies 
 most difficult to be liquefied need only, for that purpose, to be copneas to very minute 
 particles. 
 
 XVIII.— Of the Asperities of certain bodies, which appear to be exceedingly sharp 
 and highly polished. 
 
 If a needle, apparently very sharp, be viewed through the microscope, it will seem to 
 have a very blunt, irregular point, much resembling that of a peg broken at the end, 
 
 The case is the same with the edge of the best set razor. When viewed through 
 the microscope, it will appear like the back of a penknife, and at certain distances 
 exhibit indentations like the teeth of a saw, but irregular. 
 
 If a piece of the highest polished glass be exposed to the microscope, you will be 
 much astonished at its appearance: it will be seen furrowed, and filled with asperi- 
 ties, which reflect the light in anirregular manner, making it assume different colours. 
 The case is the same with the best polished steel. 
 
 Art, in this respect, is far inferior to nature; for if works which have been made 
 and polished, as we may say, by the latter, are exposed to the microscope, instead of 
 losing their polish, they appear with greater lustre. When the eyes of a fly, if illu- 
 minated by means of a lamp or taper, are viewed through this instrument, each of 
 them exhibits an image of the taper with a precision and vivacity which. nothing 
 can equal. 
 
 XIX.—O/f Sand seen through the Microscope. 
 
 It is well known that there are some kinds of sand calcareous, and others vitrifiable. 
 The former, seen through the microscope, resembles in a great measure large irregular 
 fragments of rock. The most curious spectacle however is exhibited by the vitreous — 
 kind: when it consists of rolled sand, it appears like so many rough diamonds, and 
 sometimes like polished ones. One kind of sand, when seen through the micro- | 
 scope, appears to be an assémblage of diamonds, rubies, and emeralds: another pre- 
 sents the embryos of shells, exceedingly small. 
 
 XX.—Of the Pores of Charcoal. 
 
 Dr. Hook had the curiosity to examine with a microscope the texture of charcoal, 
 which he found to be filled with pores regularly arranged, and passing through its 
 whole length: hence it appears that there is no charcoal into which the air does not 
 
are 
 
 | LINES IN THE PRISMATIC SPECTRUM. . 355 
 introduce itself. This observer, in the 18th part of an inch, counted 150 of these 
 pores; from which it follows, that in a piece of charcoal, an inch in diameter, there 
 are about 5720000. 
 
 On this subject we have been obliged, agreeably to our plan, to be exceedingly 
 brief; but, to supply this deficiency, we shall here point out the principal works 
 which contain micrographic observations, and the authors who have particularly applied 
 themselves to this kind of study. The first we shall mention is Father Bonnani, 
 a Jesuit, author of a book entitled ‘‘ Ricreazione dell’ochio é della mente,” part 
 of which is entirely devoted to this subject. The celebrated Lewenhoek spent 
 almost the whole of his life in the same occupation, and published the observations 
 he made in his ‘‘ Arcana Nature.” A great many observations of this kind may be 
 found scattered here and there throughout all the Journals and Memoirs of learned 
 Societies. But few have made so many researches on this subject as M. Joblot, 
 author of a quarto volume, entitled “ Description et usages de plusieurs nouveaux 
 Microscopes, &c., avec de nouveaux observations sur un multitude innombrable 
 dinsectes, &¢c., qui naissent dans les liqueurs, &c. Paris, 1716.’’ He infused in 
 water a great number of different substances, and caused the small animals produced 
 by these infusions to be engraved: to the greater part of them he has even given 
 names, derived from their resemblance to known bodies, or from other circumstances. 
 But we must refer the reader to the work itself, which was republished in 1754, 
 considerably enlarged, under this title: ‘‘ Observations d’Histoire Naturelle, faites 
 avec le Microscope sur un grand nombre d’Insectes, et sur les Animalcules qui se 
 trouvent dans les liqueurs preparées et non preparées, &c.,”’ 4to, with a great number 
 of plates. Needham, in the year 1750, published his work, called ‘“‘ New Micro- 
 scopical Observations.” Buffon’s observations on spermatic molecule may be seen 
 in his work on Natural History. We have also Baker’s works, entitled ‘‘ The 
 
 . Microscope made Easy, and Employment for the Microscope.” The first part con- 
 
 tains a description of the apparatus and the method of using different kinds of mi- 
 croscopes, and the second a very long detail of microscopical observations made on va- 
 
 rious natural objects. This work was attended with great success, and is exceedingly 
 
 instructive. The Abbé Spallanzani caused his microscopical observations, in which 
 
 _ he several times contradicts Needham, to be printed in Italian ; a French translation, 
 entitled ‘‘ Nouvelles Observations Microscopiques,” was published, in octavo, at 
 
 Paris, in 1769, with notes by the above philosopher. If to these be added various 
 
 Memoirs by Fontana, Roffredi, Spallanzani, &c., published in the ‘* Journal de 
 
 Physique,” we shall have enumerated all the writings, or at least the principal ones, 
 which have hitherto appeared on this subject. 
 
 We shall add to the preceding problems on Optics in Montucla’s work, a brief 
 
 _ summary of the modern discoveries in this branch of science. 
 
 Fired Lines in the Prismatic Spectrum. 
 
 By viewing through a telescope the spectrum formed from a narrow beam of solar 
 light by a fine prism of flint glass, Fraunhofer of Munich discovered that the spec- 
 
 _ trum was covered throughout by dark lines of different widths, perpendicular to the 
 direction of the length of the spectrum, none of the lines coinciding with the 
 _ boundary of any of the coloured spaces. There are not less than 600 of these lines. 
 
 Several of them are sensibly broader than the others, and may be discovered with 
 
 | comparative ease. One is near the outer end of the red space; a broad and dark 
 
 DA IN He 
 
356 OPTICS. 
 
 one is near the middle of the red, a strong double line is in the middle of the orange, 
 one in the green consists of several lines, a very strong one is in the blwe, one in the 
 indigo, and one in the violet. 
 
 Similar bands are seen in the light of the planets, fixed stars, the electric spars 
 and coloured flowers, but they are not found in the spectrum formed by the light ofa 
 lamp; but in the orange portion of the lamp- light spectrum, there is a line brighter 
 than the rest of the spectrum. ) 
 
 On the Heating Power of the Spectrum. 
 
 Dr. Herschel found by experiments that the heating power of the spectrum gra- 
 dually increased from the violet to the red extremity, and that the thermometer 
 continued to rise beyond the red end, where no light whatever could be perceived, 
 Hence he drew the conclusion that there are invisible rays in the light of the sun, 
 which produce heat, and which have a less refrangibility than red light. 
 
 Mr. Seebeck however, who has recently experimented on the subject, shews that 
 the place of maximum heat in the spectrum varies with the substance of which 
 the prism is made. Thus with water the maximum of heat was in the yellow portion 
 of the spectrum; with sulphuric acid, it was in the orange; with crown glass, in the 
 middle of the red ; and with flint glass, (the material which Herschel used) beyond 
 the red. 
 
 On the Chemical Influence of the Spectrum. 
 
 It has long been known that lunar caustic becomes black when exposed to the 
 light, and very speedily so when exposed to the light of the sun. When exposed to 
 the light of the spectruin, it is found to become very soon black beyond the violet 
 extremity, less readily so in the violet, and so on towards the red end; and 
 when a little blackened by exposure near the violet end, its colour is partially 
 restored by exposure in the red rays. 
 
 On the Magnetizing power of the Solar Rays. 
 
 Some years ago Dr. Morichini announced that by collecting the violet rays in the 
 focus of a convex lens, and carrying the focus from the middle of one half of a needle 
 to the extremities of that half, and continuing the operation for an hour, the needle 
 acquired perfect polarity. ‘The experiment was repeated sometimes with, and some- 
 times without, success, by various scientific persons; but the truth of the fact an- 
 nounced has recently been put beyond dispute by some experiments made by our 
 distinguished countrywoman, Mrs. Somerville. 
 
 Having covered with paper one half of a sewing needle, quite devoid of magnetism, 
 Mrs. 8. exposed the uncovered half to the violet rays, and in about two hours the 
 needle had become decidedly magnetized, the exposed end exhibiting north polarity. 
 The indigo rays magnetized a needle with nearly the same facility as the blue ones; 
 and the blue and green produced also a small analogous effect, but the other rays pro- 
 duced no sensible effect whatever. 
 
 When the rays were concentrated by means of a lens the effect was produced 
 more speedily ; and what is very remarkable, the magnetic effect was produced by 
 exposing the needle half covered with paper to the sun’s rays transmitted through 
 green glass, or through glass coloured b/ue with cobalt. The light transmitted 
 through a blue or green riband produced the same effect, and when the needles thus 
 covered had hung a day or two in the sun’s rays, behind a pane of glass, their 
 exposed ends were north poles, as when the effect was produced by the rays of the 
 spectrum, 
 
 These peculiar properties ot light have acquired increased importance from 
 the singular and most wonderful application that has been recently made of 
 
7 ies $ ’ 
 
 PHOTOGENIC DRAWING. 307 
 
 some of them to taking drawings and views of matchless accuracy, and of delicacy 
 
 beyond the reach of all human art. 
 
 _ About twelve months ago it was announced that M. Daguerre, well known for the 
 beautiful and interesting dioramie pictures executed by him, and exhibited in most of 
 the principal cities of Europe, had discovered a mode of fixing the images formed in 
 
 the Camera Obscura and of producing pictures which exceeded in delicacy any thing ~ 
 
 that had ever before been seen. 
 On this announcement reaching England, Mr. Fox Talbot immediately laid before 
 the Royal Society an account of a method which he had practised for some years, 
 and which it was thought might possibly be the same in substance as the process 
 discovered by the ingenious Frenchman. A comparison of the results shewed, how- 
 ‘ever, that the processes of M. Daguerre and Mr. Talbot were by no means identical. 
 Both methods, indeed, gave permanent pictures from images formed in the Camera; 
 ‘but Mr. Talbot’s pictures had dark shades where light colours were in the foe 
 ‘copied, and lights where in nature the colours were dark. On the contrary, i 
 ‘Daguerre’s pictures, light colours were represented. by light shades, and dark ae 
 ‘by dark shades ; and the gradations of the shades—the translation, soto speak, from 
 ‘colour in nature to shade in the picture, exhibited a degree of perfection utterly un- 
 ‘attainable byart. Still, however, the process of Mr. Talbot has its ewn useful and 
 independent application in cases where that of M. Daguerre, admirable as it is, does 
 not apply at all. We shall give such an account of both processes as, we trust, will 
 enable an attentive reader to practise either with success ; beginning with 
 
 | 
 ‘S 
 } 
 | 
 
 | PHOTOGENIC DRAWING. 
 
 Having pasted a bottle carefully over with paper, dissolve in it a quantity of 
 nitrate of silver, (lunar caustic) in distilled water, putting one drachm of the nitrate to 
 four table spoonfuls of water; and taking care that neither the nitrate in its solid 
 state, nor the solution, is exposed to the light of day. Put the bottle away in a dark 
 closet for use. Fill another bottle with a saturated solution of common salt and 
 water; it is not necessary that the water should be distilled. ‘Take a sheet of stout 
 writing paper, such as Bath post, dip it in a solution composed of from ten to twenty 
 parts of pure water to one of the saturated sclution of salt. Press the wetted paper 
 ' between sheets of blotting paper, and then dry it at the fire. 
 
 Ina room from which day-light is carefully excluded, wash over one side of the 
 paper twice with the solution of nitrate of silver, using for the purpose a large camel’s 
 hair pencil, and dry the paper after each washing. 
 
 The paper may now in general be considered as prepared, and it will be found 
 sufficiently sensitive for making photogenic pictures by the direct action of the sun’s 
 jrays. Butif it is intended for taking pictures by meansof reflected light, as in the 
 
 Camera Obscura, it will be necessary to go over the whcle process of preparation 
 again; dipping in the salt and water, drying,—washing with the solution of nitrate 
 of silver, and drying again. By proceeding in this way, the sensibility of the paper 
 ‘may be increased almost indefinitely: but it is very difficult fo fix images on paper 
 | of extreme sensibility, as the darkening processis apt to proceed, whatever process 
 may beadopted for checking it. 
 
 Having prepared paper of the requisite sensibility for the object in view, we shall 
 _ suppose that it is desired to have on the paper a representation of some subject, 
 _as shewn in the Camera Obscura. 
 | Having fixed upon the point of view, draw out the slider of the Camera till a 
 _ distinct image of the object in view is obtained in the focus, or a piece of common 
 
 paper. Then, inaroom from which day-light is shut out, replace the piece of com- 
 /mon paper in the Camera with a piece of prepared, highly sensitive photogenic 
 paper (the object glass being covered) ; and shut down the posterior flap of the 
 
 Aaa ahi ee Sa ile Pon ee he 
 
 > 
 
 as P 
 ek 
 
358 OPTICS. 
 
 Camera ; and removing the instrument to the selected point of view, take off the cover — 
 of the object-glass, and allow the image of the object which it is desired to have — 
 represented to be formed on the photogenic paper. If the day is bright, a distinct | 
 image of the object will be found imprinted on the paper in from 15 to 30 minutes, 
 Then, the cover being put again upon the object-glass, remove the Camera with the — 
 paper initto the dark chamber ; and, withdrawing the picture, immerse it immediately | 
 in cold water for 15 or 20 minutes. It may beadvisable to change the water 
 occasionally, to wash out as much as possible of the nitrate of silver remaining un- 
 changed on the paper. Dry the paper at the fire, and make a solution of one table — 
 spoonful of the saturated solution of salt and water, and three or four spoonfuls of 
 pure water,—adding the bulk of two pins’ heads of iodide of potassium. Soak the 
 paper in the solution, and dry it at the fire, and in all ordinary cases the image will 
 be found to be fixed, being altogether insensible to the further action of light. 
 
 To make an accurate Drawing of any object, by means of Solar Light. 
 
 Take a piece of prepared paper large enough for the object in view, and in a 
 darkened room place the object on the side of the paper which has been washed with 
 the nitrate of silver, and place over the object a piece of good window glass. 
 
 It may be well to place the paper on a book, and to keep the object in accurate 
 contact with the paper, by pressure on the edge of the glass ; for absolute contrast is 
 essential to the perfection of the picture. 
 
 ‘The arrangements may be more conveniently made by means of a small drawing 
 board, with a cushioned back-board to lay the paper and the object upon, and a 
 pane of glass adapted to the board to lay upon the object; and the whole can be 
 fastened into the frame by the cross-sticks at the back of the cushioned board, as 
 
 the paper is fastened into the frame for drawing. 
 
 Let the object under the glass be suddenly placed in the direct rays of the sun, and 
 in a few minutes a perfect image of it will be formed on the paper ; the parts of the 
 paper uncovered will be black, or nearly so; those parts of the object through 
 which the solar rays pass most freely will be darkest in the picture, and such parts - 
 as totally exclude the rays of light will be white; the other parts of different 
 degrees of shade, depending on the transparency of different parts of the object. 
 
 It may be stated, generally, withrespect to colour, that lights in the object give 
 darks in the picture, and vice versa. 
 
 If the picture be fired as directed above, zt may be used, as the object was, to pro- 
 duce another picture in which lights, shades, and direction will be again reversed; but 
 this second picture, is always less distinct than the first one, because all the im- 
 perfections of the paper through which the rays penetrate to the prepared paper be- 
 neath, are represented in the transferred picture. Copies of engravings may be taken 
 in the same way, by laying the face of the engraving on the prepared side of the 
 photogenic paper; but here again both lights, shadows, and ‘positions will be re- 
 versed, and the want of uniformity in the texture of the paper on which the 
 engraving is made will impair the distinctness of the photogenic copy; and if this 
 copy be fixed and used to obtain a reverse, the indistinctness in the reverse will be still 
 further increased from the same cause. But both the copies from the original, and 
 the transfer from the copy, are often very beautiful. Feathers, finely veined leaves, 
 and many grasses, form very interesting photogenic pictures ; and the use of the art in 
 preserving pictures, fac-similes of rare and delicate plants, &c., is very obvious. 
 
 A pleasing application of the art, to persons who draw with confidence and taste, 
 and are desirous of multiplying copies of their sketches, may be called 
 
 Photogenic Etching. 
 Smoke one side of a piece of window glass over the flame of a candle, and 
 
DAGUERREOTYPE. 359 
 
 with a pointed implement make a drawing of any object on the smoked side of the 
 lass. 
 
 4 Lay the clean side of this glass on a piece of prepared paper, and expose the 
 
 drawing for a short time to the sun, when a perfect copy of the drawing will be 
 
 imprinted on the paper, the part hid from the sun by the soot being the white ground 
 
 of the copy. 
 
 This smoked glass, with the drawing upon it, may be used to take any number 
 of copies in succession; but the pictures, as they are taken, must be kept from the 
 light till they are fixed. 
 
 It may again be noticed, that any imperfection in the contact of the glass and the 
 paper injures the distinctness of the impression. 
 
 Many other applications of the process will readily suggest themselves to persons 
 who may be inclined to practise it. 
 
 We now proceed to give directions for producing pictures with the aid of the Ca- 
 mera, by the more delicate process of M. Daguerre; availing ourselves chiefly of the 
 official account published by the discoverer, in compliance with an agreement made — 
 with the French government, which has settled an annuity of 6000 francs on M. 
 Daguerre, and one of 4000 frances on his associate, M. Neipce, as a recompence 
 for making the particulars of the discovery known for the benefit of the public at 
 large. 
 
 DAGUERREOTYPE. 
 
 The pictures are formed on thin plates of the purest silver plated on copper ; 
 the copper being of sufficient thickness to maintain perfect smoothness and flatness 
 in the silver plate; but the thickness of both ought not to exceed that of a stout 
 eard, and the size of the plate will depend on that of the Camera. 
 
 Powder the surface of the plate with the finest pumice stone; then laying the 
 plate on several folds of paper, take some cotton dipped in a little olive oil, rub the 
 plate gently, rounding the strokes. The pumice stone and cotton must be changed ~ 
 several times. When the plate is well polished, it must be cleaned by powdering it 
 over again with pumice stone, rounding and crossing the strokes to obtain a flat 
 
 ' surface. 
 
 Roll up a little pledget of cotton, and moisten it with a diluted solution-of nitric 
 acid (one part of acid to eight of distilled water), applying the cotton to the mouth 
 of the phial containing the solution, so that the centre of the cotton only may be _ 
 slightly wetted; and with the pledget so moistened rab the surface of the plate 
 equally. Change the cotton and rub on, rounding the strokes till the acid is per- 
 fectly spread, and forms a thin film on the surface. Again powder with pumice, 
 and clean with fresh cotton, rubbing as before, but very slightly. 
 
 Put the plate with the silver side upwards in a wire frame, and to the copper 
 side apply the flame of a spirit lamp, the flame playing upon and touching the copper 
 as the lamp is carried round. Continuing this for about five minutes, a strong white 
 
 _ coating will be formed on the surface of the silver. Withdraw the lamp, and cool 
 the plate suddenly by placing it on a cold substance, as a marble table. 
 
 When perfectly cooled, polish it again with dry pumice stone and cotton, repeated 
 several times. Repeat the operations with the acid; and polishing afterwards with 
 dry pumice stone and cotton thrice; taking care not to breathe upon the plate, or 
 to touch it with the fingers, or even with the cotton on which the fingers have 
 rested. 
 
 Put the plate into a frame, and invert it, so that the silver face may be downward 
 on the top of a box, at the bottom of which is a quantity of todine broken into small 
 pieces, and contained in a little dish. Let the plate remain in this position, till the 
 
360 OPTICS. | | 
 
 : 
 
 condensation of the vapour of the iodine has covered the surface with a fine coating 
 of a yellow gold colour. The time for effecting this may vary from five minutes to 
 balf an hour, according to the operation ; the operator examines from time to time 
 how the process is proceeding. This process is conducted in a darkened apartment, 
 and the examinations are made by a little light admitted sideways, not from the roof. 
 Lifting the plate with the frame with both hands, and turning it up quickly, the 
 operator sees at a glance the true colour of the coating; very little light sufficing for 
 the purpose. If too pale, the plate is instantly replaced; if the gold tint is 
 passed, the coating is useless, and the whole of the operations must be gone over 
 de novo. 
 
 Having previously adjusted the Camera to focus, place the plate in the Camera; 
 and placing the Camera in front of the landscape, uncover the lens, and allow the 
 light to form a picture of what is before the Camera on the coated surface of the 
 silver; all light except that from the object glass being rigidly excluded. The lens 
 of the Camera ought to be periscopie. 
 
 The time which the plate must. remain in the Camera depends entirely on the 
 intensity of the light of the objects whose image is to be depicted. At Paris it may 
 vary from three to thirty minutes. 
 
 When it is conceived that the plate has remained long enough, the Camera with 
 the plate in it must be removed toa darkened chamber ;.the only light admissible 
 being that of a taper. 
 
 The plate removed, nothing whatever will be perceived upon its surface ; its ap- 
 pearance is an absolute blank. 
 
 Put it into a box, the face inclining forwards 45 degrees from the perpendicular ; 
 and at the bottom of the box place a cup containing -mercury. Putting on the lid 
 of the box, place a spirit lamp below the mercury, till it is raised to a temperature 
 of 60° centrigrade. Withdraw the lamp immediately, and after continuing to rise 
 some time, the thermometer, by which the heat of the mercury is measured, will 
 begin to fall. When it falls to 45° centrigrade, withdraw the plate. 
 
 A plate of glass is placed in this mercurialising box in front of the plate; so that 
 by means of a feeble taper the operator can see the gradual development of the 
 picture under the influence of the mercurial vapour. 
 
 The impress of nature is on the plate when it is removed from the Camera, but it 
 is invisible; and it is not till after several minutes’ exposure to the mercurial vapour 
 that the faint traces of objects begin to appear. 
 
 After the mercurialising process has been completed, the next object is to fix the 
 image. 
 
 Plunge the plate into a plate of common water, and withdraw it immediately, the 
 surface merely requiring to be moistened. Then plunge it into a saturated solution 
 of common salt; or, which is better, into a saturated solution of hyposulphate of 
 soda, moving the plate about in the solution by means of a hook of copper wire. 
 When the yellow colour is quite gone, lift up the plate with both hands, taking care 
 not to touch the drawing; and plunge it again into a trough of pure water. 
 
 Lay the plate immediately upon an inclined plane; pour over it, in a stream, hot 
 but not boiling water, to carry off what may remain of the saline solution. 
 
 If any drops of water remain on the drawing, they must be blown off; for by drying 
 they would leave stains on the drawing. 
 
 It remains only now to place the plate in a square of strong pasteboard, covered 
 by a glass; and to frame the whole in wood; and if all the operations have been 
 successfully conducted, a production will be the result which, with respect to deli- 
 cacy and faithfulness, is unapproachable by any art previously known. 
 
 ; 
 * , ag 
 q _ ai 
 
COLOURS OF PLATES.—GROOVED SURFACES. 361 
 
 On the Inflexion and Diffraction of Light. 
 
 To observe the action of bodies on light passing near them, let a lens of short focus 
 be fixed in a window shutter, anda beam of sun light be transmitted through the lens. 
 This light will diverge from the focus of the lens and form a circular image of light 
 on the opposite wall. The shadows of all bodies held in this light will be found to 
 be surrounded with three coloured fringes. The first, reckoning from the shadow, 
 will be, violet, indigo, pale blue, green, yellow, red. The second, blue, yellow, red ; 
 and the third, pale blue, pale yellow, pale red. 
 
 These fringes may be conveniently examined by means of a lens, and they present 
 various interesting phenomena, according as the light is the direct solar ray, or the dif- 
 ferent primitive portions of the spectrum; and even according to the shape of the 
 aperture through which the light is admitted. 
 
 Of the Colours of Thin Plates. 
 
 The thinnest transparent film that can be generally met with will both reflect and 
 transmit light which is white or colourless; but if the thickness be diminished to a 
 very great degree, the reflected and transmitted light are both colvured. 
 
 A soap bubble is a familiar and beautiful illustration of the colours produced by 
 reflection and refraction from and through thin plates. The colours of the oxidated 
 film on glass which has been long exposed to the weather, is another example; and if 
 a piece of sealing wax be stuck toa plate of mica, and detached with a jerk, extremely 
 thin filaments will adhere to the wax, and they will exhibit the most brilliant colours 
 by reflected light. 
 
 If we blow a soap bubble, and cover it with a clear glass to protect it from cur- 
 rents of air; after it has grown thin by standing awhile, a great many concentric 
 coloured rings will be observed round the top of it. As the bubble grows thinner 
 the rings will dilate; the central spot will become in ‘succession white, bluish, and 
 then black, after which from the extreme thinness of the black part, the bubble 
 will burst. 
 
 Of the Colours of Thick Plates. 
 
 The colours of thick plates may be seen with a candle held before the eye, ten 
 or twelve feet from a pane of crown glass in a window on which has been a fine 
 deposition of moisture or of dust. 
 
 But these colours may be seen to great advantage by means of two equally 
 thick plates of glass placed near to and above each other, and nearly, but not quite, 
 parallel. 
 
 If a ray of condensed light, subtending about 2°, fall nearly perpendicularly on the 
 upper plate, and the eye be placed behind the plate, several reflected images will be 
 seen in a row, besides the direct one. The field will be seen crossed by numerous 
 
 beautiful bands of colour; the central bands consist of blackish and whitish stripes, 
 and the exterior of brilliant bands of red and green light; the direction of the bands 
 
 being parallel to the common section of the inclined planes. 
 
 On the Colours of Grooved Surfaces. 
 It has long been known that the beautiful play of colours exhibited by mother of 
 
 | pearl and some other substances is derived from their surfaces being covered with 
 _ minute grooves; and the late ingenious Mr. Barton, of the Royal Mint, made a beau- 
 
 tiful application of this property in the production of what he very appropriately 
 called Jris ornaments. By means of a delicate machine he was enabled to cut, with 
 a diamond or polished steel, parallel grooves at the distance of from the 2000th to the 
 
362 OPTICS. 
 
 10000th part of an inch apart; and the light reflected from the finely grooved sur- 
 face exhibited the most beautiful prismatic colours. ‘They were formed into buttons 
 for dress coats for gentlemen, and into articles of ornament for ladies, arranged in 
 patterns. 
 
 In forming the buttons, patterns were cut in steel, which was afterwards har. 
 dened and used as a die to stamp the impressions on buttons made of brass. In sun- 
 light, gas-light, or even brilliant lamp or candle-light, the brilliancy of the colours 
 of these ornaments was scarcely surpassed by the flashes of the diamond. 
 
 Perhaps it was because these very beautiful articles were soon supplied at a cheap 
 rate, that they have ceased to be much used in fashionable life; they are, however, 
 very elegant; and, asa branch of Optics, the phenomena which they exhibit form a 
 most interesting object of contemplation. 
 
 On the Absorption of Light. 
 
 All bodies absorb light. On the summits of high mountains, where light from 
 celestial objects has to pass through a thinner stratum of air, a much greater number of 
 stars are visible to the eye than onthe plains below; and through great depths of — 
 matter objects become almost invisible. The absorptive power of air is finely dis- 
 played in the colour of the morning and evening clouds; and that of water, in the red 
 colour of even the meridian sun, when seen from a diving bell at a great depth in 
 the sea. 
 
 These appearances are caused by the absorption of one class of rays in passing 
 _ through the air or the water, while the rest make their way, either directly or by 
 reflection, to the eye. 
 
 Charcoal, in its ordinary state, is the most absorbent of all bodies; but in some 
 particular states of combination—in gas or in flame—as forming the essential © 
 constituent of the diamond, it is very transparent. Metals are transparent in a state 
 of solution; and silver and gold, when beaten very thin, transmit light, the former 
 blue, the latter green.-: 
 
 Some clouds absorb blue rays, and transmit red ; others absorb all rays in equal pro- 
 portions, and exhibit the sun through them perfectly white. The image of the sun, 
 as seen through diluted ink, is also quite white. 
 
 The absorbing power of different bodies is variously modified by heat and other 
 circumstances. Pure phosphorus, which is of a slightly yellow colour, transmits 
 freely almost all the coloured rays. When melted, and gradually cooled, it absorbs 
 all the colours of the spectrum at thicknesses at which it formerly transmitted them. 
 
 On the Polarisation of Light. 
 
 When light emitted from the sun or any self-luminous body, is reflected from the 
 surface, or transmitted through the surface, of any homogeneous uncrystallized body, 
 the reflected or transmitted light continues the same when we turn round the body ; 
 so that the light falls upon it at the same angle, or the different sides of the rays from 
 the same paper lie with reference to the angle of incidence. Such light is called com- 
 mon light. 
 
 But a kind of light has been discovered which exhibits different properties with 
 respect to the angle of incidence, according as the reflecting or transmitting surface is 
 presented to one side or another of the incident ray. Such light is called polarised 
 light. 
 
 Whenever this light is obtained it must have previously existed in the form of 
 common light; from which it may be obtained hy reflection from the surface of tran- 
 sparent and opake bodies, by transmission through plates or planes of uncrystallized 
 bodies, or by transmission through bodies regularly crystallized, and possessing the 
 property of double refraction. 
 
THE POLARISATION OF LIGHT. 363 
 
 Fig. 68. To explain the difference between common and 
 EH; polarised light, let a (Fig. 68.), be a plate of glass 
 
 | so placed at the end of atube mn, that aray of 
 light incident at a, may be reflected along the axis 
 of the tube mn. At the end of another smaller 
 tube NP, which can turn round within mn, place 
 a similar piece of glass capable of reflecting a ray a c,to the eye at 5. 
 
 Let aray of light,.r a, fall upon the vertical plate of glass a, at an angle of 56°, 
 and incline the plate a to the axis ac, so that the ray may be reflected along a c, 
 and from c again at an angle of 56°tox. Then, when the first reflection is horizon- 
 tal, and the second vertical, or when R A C is a horizontal plane, and ac Fa vertical 
 one, the ray c £ will be so weak as to be scarcely visible. But if we turn the tube np 
 within m N without moving a orm N, or altering the inclination of c to ac, the raycE 
 will become stronger and stronger, till it has been turned round 90°, or that acE 
 is in a horizontal plane as well as R ac, when the light of the beam c £ attains its 
 maximum—continuing to turn the tube it will become fainter and fainter, till, after 
 being turned 90° more, when the plane a CE is in the opposite vertical ck, it will again 
 be invisible; at the next 90° the brightness of the ray will be at its maximum, and on 
 completing the revolution it will vanish again. 
 
 This experiment shews that when either the upper or the under side of the ray 
 is nearest c, the plate is incapable of reflecting it; but that when the right or the left 
 side is nearest the reflecting plate, the plate reflects the ray as if it were common 
 light, and in intermediate positions, intermediate portions of light are reflected. The 
 ray a Chas therefore properties different from common light, and we hence conclude 
 that a ray of common light, as R a, reflected from glass at an angle of 56°, becomes 
 polarized by the reflection. If the original beam R a has considerable intensity, the 
 reflected pencil c £ does not wholly vanish, and the part remaining visible is coloured. 
 
 This branch of Optics is fertile in striking phenomena ; but it is of such extent 
 that we must, in this place, content ourselves with referring to works in which room 
 has admitted its being treated in requisite detail. Dr. Brewster’s Volume in Lard- 
 ner’s Cyclopedia, and the article Optics, apparently by the same author, in the 
 Library of Useful Knowledge, may be consulted with advantage. 
 
 7 
 
S| rt” ‘/ .« ae tee A ae eek 
 a ee! Oe ee ee 4. | OS eee Are oe 
 
 364 ACOUSTICS, 
 
 PART FIFTH. 
 
 CONTAINING EVERY THING MOST CURIOUS IN REGARD TO 
 ACOUSTICS AND MUSIC. 
 
 Tur ancients seem to have considered sounds under no other point of view than 
 that of Music; that is to say, as affecting the ear in an agreeable manner. It is 
 even very doubtful whether they were acquainted with any thing more than melody, 
 and whether they had any art similar to what we call composition. The moderns, 
 however, by studying the philosophy of sounds, have made many discoveries in this 
 department, so much neglected by the ancients; and hence has arisen a new science, 
 distinguished by the name of Acoustics. Acoustics have for their object the nature 
 of sounds, considered in general, both in a mathematical and a philosophical view. 
 This science therefore comprebends music, which considers the ratios of sounds, 
 so far as they are agreeable to the ear, either by their succession, which constitutes 
 melody, or by their simultaneity, which forms harmony. We shall here give an _ 
 account of every thing most curious and interesting in regard to this science. 
 
 ARTICLE I. 
 
 Definition of Sound ; how diffused and transmitted to our organs of hearing ; ex- 
 periments on this subject ; different ways of producing Sound. 
 
 Sound is nothing else but the vibration of the particles of air, occasioned either 
 by some sudden agitation of a certain mass of the atmosphere violently compressed 
 or expanded; or by the communication of the vibration of the minute parts of hard 
 and elastic bodies. 
 
 These are the two best known ways of producing sound. The explosion of a 
 pistol, or any other kind of fire arms, produces a report or sound, because the air or. 
 elastic fluid contained in the gunpowder, being suddenly dilated, compresses the 
 external air with great violence: the latter, in consequence of its elasticity, re-acts 
 on the surrounding atmosphere, and produces in its molecule an oscillatory motion, 
 which occasions the sound, and which extends to a greater or less distance, according 
 to the intensity of the cause that gave rise to it. 
 
 To form a proper idea of this phenomenon, let us conceive a series of springs, all 
 maintaining each other in equilibrium, and that the first is suddenly compressed in a 
 violent manner by some shock, or other cause. By making an effort to recover its 
 former situation, it will compress the one next to it, the latter will compress the 
 third; and the same thing will take place to the last, or at least to a very great 
 distance; for the second will be somewhat less compressed than the first, the third 
 a little less than the second, and so on; so that, at a certain distance, the com- 
 pression will be almost insensible, and at length it will totally cease. But each 
 of these springs. in recovering itself, will pass a little beyond the point of equili- 
 
 -brium, and this will occasion, throughout the whole series put in motion, a vibration, 
 
 which will continue for a longer or shorter time, and at length cease. Hence it 
 
 happens that no sound is instantaneous, but always continues, more or less, according 
 to circumstances. 
 
 : 
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 NATURE OF SOUND. 365 
 
 The other method of producing sound, is to excite, in an elastic body, vibrations 
 sufficiently rapid to occasion, in the surrounding parts of the air, a similar motion. 
 Thus, an extended string, when struck, emits a sound, and its oscillations, that is to 
 say, its motion backward and forward, may be distinctly seen. The elastic parts 
 of the air, struck by the string during the time it is vibrating, are themselves put 
 into a state of vibration, and communicate this motion to the neighbouring ones. 
 Such is the mechanism by which a bell produces its sound: when struck, its vibra- 
 tions are sensible to the hand which touches it. 
 
 Should these facts be doubted, the following experiments will exhibit the truth 
 of them in the clearest point of view. 
 
 Experiment 1. 
 
 Half fill a vessel, such as a drinking glass, with water; and having made it fast, 
 moisten your finger a little, and move it round the edge. By these means a sound 
 will be produced, and at the same time you will see the water tremble, and form 
 undulations so as to throw up small drops. .What but the vibration of the particles 
 of the water can produce in it such a motion? 
 
 Experiment 2. 
 
 If a bell be suspended in the receiver of an air-pump, so as not to touch any part 
 of the machine; it will be found, ’on the bell being made to sound, that as the air is 
 evacuated and becomes rarer, the sound grows weaker and weaker, and that it ceases 
 entirely when as complete a vacuum as possible has been effected. If the air be 
 gradually re-admitted, the sound will be revived, as we-may say, and will increase in 
 proportion as the air contained in the machine approaches towards the same state as 
 that of the atmosphere. 
 
 From these two experiments it results, that sound considered in the sonorous 
 bodies, is-nothing else than rapid vibrations of their minute parts: that air is the 
 vehicle of it; and that it is transmitted so much the better when the air by its 
 density is itself susceptible of a similar motion. 
 
 In regard to the manner in which sound affects the mind, we must first observe 
 that at the interior entrance of the ear, which contains the different parts of the 
 organ of hearing, there is a membrane extended like that of a drum, and which on 
 that account is called the tympanum. It is very probable that the vibrations of the 
 air, produced by the sonorous body, excite vibrations in this membrane ; that these 
 produce similar ones in the air with which the internal cavity of the ear is filled; 
 
 and that the sound is increased by the peculiar construction of the parts, and the 
 circumvolutions both of the semicircular canals and of the helix: herice there is 
 occasioned in the nerves that cover the helix, a motion which is transmitted to the 
 brain, and by which the mind receives the perception of sound. Here however we 
 must stop; for it is not possible to ascertain how the motion of the nerves can affect 
 the mind; but it is sufficient for us to know, by experience, that the nerves are 
 as it were the mediators between our spiritual part, and the external and sensible 
 objects. 
 
 Sound always ceases when the vibrations of the sonorous body cease, or become too 
 weak. This is proved also by experiment, for when the vibrations of a sonorous 
 body are damped by any soft body, the sound seems suddenly to cease. In a piano- 
 forte, therefore, the quills are furnished with bits of cloth, that by touching the 
 strings when they fall down, they may damp their vibrations. On the other hand, 
 when the sonorous body, by its nature, is capable of continuing its vibrations for a 
 considerable time, as is the case with a large bell, the sound may be heard for a long 
 time after. 
 
366 ACOUSTICS. 
 
 ; 
 
 ARTICLE II. 
 
 On the Velocity of Sound ; experiments for determining it ; method of measuring 
 distances by it. 
 
 Light is transmitted from one place to another with incredible velocity; but this 
 is not the case with sound: the velocity of sound is very moderate, and may be mea- 
 sured in the following manner. 
 
 Let a cannon be placed at the distance of several thousand yards, and let an ob- 
 server, with a pendulum that vibrates seconds, or rather half seconds, put the pendu- 
 lum in motion as soon as he sees the flash, and then count the number of seconds or 
 half seconds which elapse between that period and the moment when he hears the 
 explosion. It is evident that, if the moment when the flash is seen be considered as 
 the signal of the explosion, nothing will be necessary, to obtain the number of yards 
 which the sound has passed over in a second, but to divide the number of the yards 
 between the place of observation and the cannon, by the number of seconds or half 
 seconds which have been counted. 
 
 Now the moment when the flash is perceived, whatever be the distance, may be 
 considered as the real moment of the explosion ; for so great is the velocity of light, 
 that it employs scarcely a second to traverse 60000 leagues.* 
 
 By similar experiments the members of the Royal Academy of Sciences found, that 
 sound moved at the rate of 1172 Parisian feet ina second. Gassendus makes its 
 velocity to be 1473 feet in a second; Mersenne 1474; Duhamel, in the History of 
 the Academy of Sciences, 1338; Newton 968; and Derham, in whose measure 
 Flamsteed and Halley concurred, 1142. Though it is difficult to determine among 
 so many authorities, the last estimate, viz. 1142 per second, has been generally adopted 
 in this country. 
 
 Recent experiments on the velocity of sound, made with all the advantages of 
 modern science, give results differing considerably from that of Derham; and from 
 the close agreement which they present, they seem entitled to great confidence. 
 
 The results being reduced to the temperature of freezing, Arago and others found 
 the velocity per second, in English feet, to be 1086-1; Professor Moll and assistants, 
 1089-42; Dr. O. Gregory 1088°05; Myrbach 1092°1; and Goldingham, at Madras, 
 mean of two results ]084°9. 
 
 It may therefore be stated, in round numbers, that, in dry air, at the freezing tem- 
 perature, sound travels at the rate of 1090 feet in a second. 
 
 ‘* That sounds, of all pitches and of every quality, travel with equal speed, we have 
 a convincing proof in the performance of a rapid piece of music by a band at a dis- 
 tance. Were there the slightest difference of velocity in the sounds of different notes 
 they could not reach our ears in the same precise order, and at the exact intervals of 
 time, in which they are played; nor would the component notes of a harmony, in 
 which several sounds of different pitches concur, arrive at once.”—(Sir J. HERsScHEL ; 
 Sound, Encyc. Metrop.) 
 
 It is to be observed that, according to Derham’s experiments, the temperature of 
 the air, whether dry or moist, cold or hot, causes no variation in the velocity of 
 sound. This philosopher had often an opportunity of seeing the flash and hearing 
 
 * The velocity of the particles and rays of light is truly astonishing, as it amounts to nearly two 
 hundred thousand miles in a second of time, which is nearly a million times greater than the velocity 
 of acannon-ball. It has been found, by repeated experiments, that when the earth is exactly between 
 Jupiter and the sun, his satellites are seen eclipsed” 8} minutes sooner than they could be accord- 
 ing to the tables; but when the earth is nearly in the opposite point of its orbit, these eclipses 
 happen about 84 minutes later than the tables predict them: hence it is certain that the motion 
 of light is not instantaneous, but that it takes up about 16} minutes of time in passiny over a space 
 equal to the diameter of the earth’s orbit, which is at least 190 millions of miles in length, or 
 
 moves at the rate of nearly 200000 miles per second. Hence therefore light takes about 84 minutes 
 in passing from the sun to the earth. 
 @ 
 
ON THE VELOCITY OF SOUND. 36] 
 
 the report of cannon fired at Blackheath, nine or ten miles distant, from Upminster, 
 the place of his residence ; but whatever might be the state of the weather, he always 
 counted the same number of half seconds, between the moment of seeing the flash 
 and that of hearing the report, unless any wind blew from either of these places; in 
 which case the number of the seconds varied from 111 to 112. It may be readily 
 conceived, that if the wind impelled the fluid put into a state or vibration, towards 
 the place of the observer, the vibrations would reach him sooner than if the fluid 
 
 had been at rest, or had been impelled in a contrary direction. 
 
 | 
 } 
 4 
 
 ea 
 
 But notwithstanding what Derham has said, we can hardly be persuaded that the 
 velocity of sound is not affected by the temperature of the air; for when the air is 
 heated, and consequently more rarefied or elastic, the vibrations must be more rapid: 
 observations on this subject ought to be carefully repeated. 
 
 A remarkable instance of the effect of a low temperature and a dry atmosphere in 
 facilitating the transmission of sound, is recorded in the account of Sir E. Parry’s 
 voyage, in which he wintered at Port Bowen. Ona particular occasion there was 
 found no difficulty i in making a man hear, at the distance aE amile, directions which 
 were given in an ordinary tone of voice. 
 
 An inaccessible distance then may be measured by means of sound. For this pur- 
 pose provide a pendulum that swings half seconds, which may be done by suspending 
 from a thread a ball of lead, half an inch in diameter, in such a manner, that there 
 shall be exactly 923 inches, or 93 between the centre of the ball and the point of sus- 
 pension: then fie moment you perceive the flash of a cannon, or musket, let go the 
 pendulum, and count how many vibrations it makes till the instant when you hear 
 the report: if you then multiply this number by 571 feet, you will have the dis- 
 tance of the place where the musket or cannon was fired. 
 
 We here suppose the weather to be calm, or that the wind blows only in a trans- 
 versal direction ; for if the wind blows towards the observer from the place where 
 the cannon or gun is fired, and if it be violent, as many times twelve feet as there 
 have been counted half seconds must be added to the distance found; and in the 
 contrary case, that isto say, if the wind blows from the observer, towards the quarter 
 where the explosion is made, they must be subtracted. It is well known that a> 
 violent wind makes the air move at the rate of about twenty-four feet per second, 
 which is nearly the 48th part of the velocity of sound. Ifthe wind be moderate, a 
 96th may be added or subtracted ; and if it be weak, but sensible, a 192nd: but this 
 correction, especially in the latter case, seems to be superfluous; for can we ever 
 flatter ourselves that we have not erred a 192d part in the measuring of time ? 
 
 This method may be employed to determine the distance of ships at sea, or ina 
 
 harbour, when they fire guns, provided the flash can be seen, and the explosion heard. 
 
 During a storm also, the distance of a thunder cloud may be determined in the same 
 manner. But as a pendulum is not always to be obtained, its place may be sup- 
 plied by observing the beats of the pulse, for when in its usual state, each interval 
 between the pulsations is almost equal to a second; but when quick and elevated, 
 each pulsation is equal to only two thirds of a second. 
 
 ARTICLE III. 
 How Sounds may be propagated in every direction without confusion. 
 
 This is. a very singular phenomenon in the propagation of sounds ; for if several 
 persons speak at the same time, or play on instruments, their different sounds are 
 heard simultaneously, or all together, either by one person or by several persons, 
 without being confounded in passing through the same place in different directions, 
 or without damping each other. Let us endeavour to account for this phenomenon. 
 
 The cause no doubt is to be found in the property of elastic bodies. For let us 
 conceive a series of globules equally elastic, and all contiguous, and let us suppose 
 
ey eee ALS eee | ee ee ee CL CS Fe en ae ‘es 
 a eg : : 4 “hE FP ae ek Ne a 
 4 ‘ oye S : + “ 
 
 368 ACOUSTICS. 
 
 that a globule is impelled with any velocity whatever against the first of the series ; 
 it is well known that ina very short time the motion will be transmitted to the 
 other extremity, so that the last globule will have the same mo- 
 Fig. 1. tion communicated to it asif it had been itself immediately im- 
 ea pelled. Now if two globules with unequal velocities impel at the 
 
 g Beas a same time the two extremities of the series, the globule a, for 
 o example, the extremity a, and the globule b the extremity 8 
 
 (Fig. 1.), it is certain, from the well known properties of elastie 
 
 bodies, that the globules a and 4, after being a momentat rest, — 
 will be repelled, making an exchange of velocity, as if they had been immediately 
 
 impelled against each other. 
 
 If we suppose a second series of globules, intersecting the former in a transversal 
 direction, the motion of this second series will be transmitted by means of the com- 
 mon globule, from one end to the other of this series, in the same manner as if it had 
 beenalone. The case will be the same if two, three, four, or more series cross the first 
 one, eitherin the same point or in different points. The particular motion commu- 
 nicated to the beginning of each series will be transmitted to the other end, as if 
 that series were alone. 
 
 This comparison may serve to shew how several sounds may be transmitted in 
 
 _all directions, by the help of the same medium; but it must be allowed that there are 
 some small differences. 
 
 For, in the first place, we must not conceive the air, which is the vehicle of sound, 
 to be composed of elastic globules, disposed in such regular series as those here sup- 
 posed ; each particle of air is no doubt in contact with several others at the same 
 time, and its motion is thereby communicated in every direction. Hence it hap- 
 pens that the sound, which would reach toa very great distance almost without dimi- 
 nution, if communicated as here supposed, experiences a considerable decrease, in 
 propertion as it recedes from the body which produced it. Though the movement 
 by which sound is transmitted be more complex, there is reason to believe that it is 
 reduced, in the last instance, to something similar. to what has been here described. 
 
 The second difference arises from the particles of air by which the organ of hear- 
 ing is immediately affected, not having a movement of translation, like the last 
 globule of the series, which proceeds with a greater or less velocity, in consequence 
 of the shock that impels the other extremity of the series. But the movement 
 in the air consists merely of an undulation or vibration, which, in consequence 
 of the elasticity of its aérian particles, is transmitted to the extremity of the series, 
 such as it was received at the other. It must be observed that the sonorous body 
 communicates to the air, which it touches, vibrations isochronous with those which 
 
 ’ it experiences itself; and that the same vibrations are transmitted from the one end 
 to the other of the series, and always with the same velocity: for we are taught by 
 experience that a grave sound, ceteris paribus, does not employ more time than an 
 acute one to pass through a determinate space. 
 
 ARTICLE IV. 
 
 Of Echoes ; how produced ; account of the most remarkable echoes, and of some phe- 
 nomena respecting them. 
 
 Echoes are well known; but however common this phenomenon may be, it must 
 be allowed that the manner in which it is produced is still involved in considerable 
 obscurity, and that the explanation given of it does not sufficiently account for all the 
 circumstances attending it. 
 
 All philosophers almost have ascribed the formation of echoes to a reflection of 
 sound, similar to that experienced by light, when it falls on a polished body. But, as 
 D’Alembert observes, this explanation is false ; for if it were not, a polished surface 
 
ECHOES, 369 
 
 would be necessary to the production of an echo; and it is well known that this is 
 not the case. Echoes indeed are frequently heard opposite to old walls, which are 
 far from being polished; near huge masses of rock, and in the neighbourhood of 
 forests, and even of clouds. This reflection of sound therefore is not of the same 
 nature as that of light. , 
 
 It is evident however, that the formation of an echo can be ascribed only to the 
 
 -repercussion of sound; for echoes are never heard but when sound is intercepted, 
 
 and made to rebound by one or more obstacles. The most probable manner in which 
 this takes place, is as follows. 
 
 For the sake of illustration, we shall resume our comparison of the aérian molecule to 
 a series of elastic globules. If a series of elastic globules then be infinite, it may readily 
 be conceived, that the vibrations communicated to oue end, will be always propagated 
 in the same direction, and continually recede; but if the end of the series rest against 
 any fixed point, the last globule will re-act on the whole series, and communicate to 
 
 - it, in the contrary direction, the same motion as it would have communicated to the 
 rest of the series, if it had not rested against a fixed point. This ought indeed to be 
 the case whether the obstacle be in a line with the series, or oblique to it, provided 
 the last globule be kept back by the neighbouring ones; only with this difference, that 
 the retrograde motion will be stronger in the latter case, according as the obliquity is 
 less. If the aérian and sonorous molecule then rest against any point at one end; and if 
 the obstacle be at such a distance from the origin of the motion, that the direct and re- 
 percussive motion shall not make themselves sensible at the same instant, the ear will 
 distinguish the one from the other, and there will be anv echo. 
 
 But we are taught by experience, that the ear does not distinguish the succession 
 of two sounds, unless there be between them the interval of at least one-twelfth 
 of a second: for during the most rapid movement of instrumental music, each 
 measure of which cannot be estimated at less than a second,* twelve notes are the 
 utmost that can be comprehended in a measure, to render the succession of sounds 
 distinguishable; consequently the obstacle which reflects the sound must be at such 
 a distance, that the reverberated sound shall not succeed the direct sound till after 
 one-twelfth of a second; and as sound moves at the rate of about 1142 feet in a 
 second, and consequently about 95 feet in the twelfth of a second, it thence follows 
 
 that, to render the reverberated sound distinguishable from the direct sound, the 
 obstacle must be at the distance at least of about 48 feet. 
 
 There are single and compound echoes. In the former only one repetition of the 
 
 ' sound is heard; in the latter there are 2,3, 4,5, &c., repetitions. We are even told 
 of echoes that can repeat the same word 40 or 50 times. 
 
 Single echoes are those where there is only one obstacle: for the sound being » 
 impelled backwards, will continue its course in the same direction without returning ; 
 but double, triple, or quadruple echoes may be produced different ways. If we 
 suppose, for example, several walls one behind the other, the remotest being the 
 
 _ highest ; and if each be so disposed as to produce an echo; as many repetitions of the 
 
 | same sound as there are obstacles will be heard. 
 
 Another way in which these numerous repetitions may be produced, is as follows: 
 
 F Let us suppose two obstacles, a and B (Fig. 2.) op- 
 Fig. 2. posite to each other, and the productive cause of the 
 
 A S : B sound to be placed between them, in the point s; the 
 
 sound propagated in the direction from s to a, after 
 
 | returning from a to s, will be driven back by the obstacle B, and again return to s; 
 
 having then traversed the space s A, it will experience a new repercussion, which 
 
 * If a piece of music, consisting of 60 measures, were executed in a minute, this, in our opinion, 
 would be a rapidity of which there are few instances in the art. 
 
 2B 
 
fe Pes Tat 
 
 SF ACOUSTICS. 
 
 will carry it to s, after it has struck the obstacle n; and this would be continued in 
 infinitum if the sound did not always become weaker. On the other hand, since the 
 
 sound is propagated as easily from 3 to B as from s to a, it will at first be sent back 
 
 also from 8 towards s; having then passed over the space s a, it will be repelled from” 
 a towards s; then again from zB towards s, after having traversed the distance s zB, 
 and so on in succession, till the sound dies entirely away. 
 
 The sound therefore produced in g will be heard after times, which may be ex- 
 pressed by 285A; 28B,28B-—-28A; 484-+258B; 48B-+ 28a; 48a-+ 4588; 
 6sa+4sB; 6sBp+484; 684+68B,&c.; which will form a repetition of the 
 sound after equal intervals, when s 4 is equal tos B, and even when sB is double 
 sa; but when sa isa third, for example, of s3,.this remarkable circumstance will 
 take place, that after the first repetition, there will be a kind of double silence; 
 three repetitions will then follow, at equal intervals; there will then be a silence 
 double one of these intervals; then three repetitions after intervals equal to the 
 former; and so on till the sound is quite extinguished. The different ratios of the 
 distances s A, $B, will also give rise to different irregularities in the succession of these 
 sounds, which we have thought it our duty to notice, as being possible, though we 
 do not know that they have been ever observed. , 
 
 There are some echoes that repeat several words in succession ; but this is rl 
 astonishing, and must always be the case when a person is at Bach a distance from 
 the echo, that there is sufficient time to pronounce several words before the repetition 
 of the first has reached the ear. 
 
 There are some echoes which have been much celebrated on account of their 
 singularity, or of the number of times that they repeat the same word. Misson, in 
 his Description of Italy, speaks of an echo at the Villa Simonetta, which repeaaaa | 
 the same word 40 times. 
 
 At Woodstock in Oxfordshire, there is an echo which cepa the same sound 50 
 times. * 
 
 The description of an echo still more singular near Roseneath, some miles distant 
 from Glasgow, may be found in the Philosophical Transactions for the year 1698. Ifa 
 person, placed at the proper distance, plays 8 or 10 notes of an air with a trumpet, 
 
 the echo faithfully repeats them, but a third lower; after a short silence another 
 repetition is heard in a tone still lower; and another short silence-is followed by a 
 third repetition, in a tone a third lower. 
 
 A similar phenomenon is perceived in certain halls; where, if a person stands in a - 
 certain position, and pronounces a few words with alow voice, they are heard only - 
 by another person standing in a determinate place. Muschenbroeck speaks of a hall 
 of this kind in the castle of Cleves; and most of those who have visited the Ob« 
 servatory at Paris have experienced a similar phenomenon in the hall on the first 
 story. 
 
 Philosophers unanimously agree in ascribing this phenomenon to the reflection 
 of the sonorous rays; which, after diverging from the mouth of the speaker, are 
 reflected in such a manner as to unite in another point. But it may be readily con 
 ceived, say they, that as the sound by this union is concentrated in that point, a per- 
 son whose ear is placed very near will hear it, though it cannot be heard by those 
 who are at a distance. 3 
 
 We do not know whether the hall in the castle of Cleves, of which Muschen- 
 broeck speaks, is elliptical, and whether the two points where the speaker and the 
 person who listens ought to be placed are the two foci; but in regard to the hall in 
 the Observatory at Paris, this explanation is entirely void of foundation. For, 
 
 Ist. The echoing hall. or as it is called the Hall of Secrets, is not at all elliptical ; 
 
 “ This seems to be a mistake: the echo at Woodstock, according to Dr. Plat, repeats in the day 
 time yery distinctly 17 eee i and in the night time 20.— Nat. Hist. of Oxf. chap. i. p. 7. 
 
— ; . VIBRATIONS OF MUSICAL STRINGS. 371 
 
 it is an octagon, the walls of which at a certain height are arched with what are 
 called in architecture clozster arches ; that is to say, by portions of a cylinder which, 
 ‘in meeting, form re-entering angles, that continue those formed by the sides of the 
 octagonal plan. 
 
 2d. The person who speaks does not stand at a moderate distance from the wall, 
 as ought to be the case in order to make the voice proceed from one of the foci of the 
 supposed ellipsis: he applies his mouth to one of the re-entering angles, very near 
 the wall, and the person whose ear is nearly at the same distance from the wall, on 
 the side diametrically opposite, hears the one who speaks on the other side, even when 
 he does so with a very low voice. 
 
 It is therefore evident that, in this case, there is no reflection of the voice according 
 to the laws of catoptrics; but the re-entering angle continued along the arch, from 
 one side of the hall to the other, forms a sort of canal, which contains the voice, and 
 transmits it to the other side. This phenomenon is entirely similar to that of a very 
 
 long tube, to the end of which if a person applies his mouth and speaks, even witha 
 
 low voice, he will be heard by a person at the other end. 
 
 The Memoirs of the Academy of Sciences, for the year 1692, speak of a very 
 remarkable echo in the court of a gentleman’s seat called Le Genetay, in the neigh- 
 bourhood of Rouen. It is attended with this singular phenomenon, that a person 
 who sings or speaks in a low tone, does not hear the repetition of the echo, but only 
 his own voice; while those who listen hear only the repetition of the echo, but with 
 surprising variations; for the echo seems sometimes to approach and sometimes to 
 recede, and at length ceases when the person who speaks removes to some distance 
 in a certain direction. Sometimes only one voice is heard, sometimes several, and 
 sometimes one is heard on the right, and another on the left. An explanation of all 
 
 these phenomena, deduced from the semicircular form of the court, may be seen in 
 
 the above collection. 
 
 ARTICLE V. 
 
 Experiments respecting the vibrations of Musical Strings, which form the basis o/ 
 the theory of Musie. 
 
 If a string of metal or catgut, such as is used for musical instruments, made fast at 
 one of its extremities, be extended in a horizontal direction over a fixed bridge; and 
 if a weight be suspended from the other extremity, so as to stretch it; this string, 
 when struck, will emit a sound produced by reciprocal vibrations, which are sensible 
 to the sight. 
 
 If the part of the string made to vibrate be shortened, and reduced to one half of 
 its length, any person who has a musical ear will perceive, that the new sound is the 
 octave of the former, that is to say twice as sharp. 
 
 If the vibrating part of the string be reduced to two thirds of its original length, 
 the sound it emits will be the fifth of the first. 
 
 If the length be reduced to three fourths, it will give the fourth of the first. 
 
 If it be reduced to 4, it wil! give the third major; if to 3, the third minor. If re- 
 duced to 8, it will give what is called the tone major; if to #, the tone minor; 
 and if to 48, the semi-tone, or that which in the gamut is between mi and fa or si 
 and sol. 
 
 _ The same results will be obtained if a string be fastened at both ends, and 4, 3, 
 and 3 of it, be successively intercepted by means of a moveable bridge. - 
 
 As this subject will be better understood if the reader has a clear idea of the rela- 
 tion of the sounds in the diatonic progression, we shall here insert the following 
 table. 
 
 2B 2 
 
‘ ve 
 ) 
 
 “qaqa agg 
 : *“dABIO er1qnop oq} jeren i “Yyunoy out a 3 
 IOUT q)XxIs ogy ——, ; “IOUTUU piryy oq} = 9 
 ¢ 
 
 . . ss ; hs r f “= i . 3 g 
 9Ab100 9[dts} a4} 0} | | “‘Wyanoj 9q3 
 
 942490 OY} SAaJoUDP Z 0} 
 
 sofeur q}xIs a4} sayouap § | csofeu psig oy sayouap at Lhe Na 
 
 sny L—: siaquinu Surmozfoy ayy Aq pautmiagep 
 pe) RN “q38u9] [enba jo sSursqys uasos yuasoidas saut] UaAVS asaq} JI eq} Ueas eq oq ALLE IT 
 
 -9q TIEM spr09u09 atuouey pedroursd ay} Jo 
 
 7 XQ y 
 
 Bal 
 | 
 PPL 
 vy0 
 
 " 
 : 
 
 "QAB}IO a[qnog 
 
 uy 
 
 TBI 4}XIS 
 
 ‘10fe 
 VINDS 
 
 J 
 a. 
 4e) 
 =— 
 | O 
 ° 
 ie) 
 ob 
 ph] 
 << 
 < 
 
 4 
 
 “UTIAPS 
 
 . IOUT TIXIS 
 qtp. | 
 
 ‘uoissasborg 91u0jmIg” ay} mu. spunog: ay? f0 u0IInI24 ay) sassaudaxe nnawDY yoryN ur sauunpy snowabuy 
 
 , 
 
MUSICAL STRINGS, . 373 
 
 Such is the result of a determinate degree of tension given to a string, when the 
 length of it has been made to vary. Let us now suppose that the length of the 
 string is constantly the same, but that its degree of tension is varied. The following 
 is what we are taught by experiment on this subject. 
 
 If a weight be suspended at one end of a string of a determinate length, made fast 
 
 by the other, and if the tone it emits be fixed; when another weight quadruple 
 
 of the first has been applied, the tone will be the octave of the former; if the weight 
 be nine times as heavy, the tone will be the octave of the fifth; and if it be only a 
 fourth part of the first the tone will be the octave below. Nothing more is necessary 
 to prove that the effect produced, by successively reducing a string to one half, 2, 3, 
 &e., will be produced also by suspending from it in succession weights in the ratio 
 of 4, 9, 18, &c. ; that is to say, the squares of the weights, or the degrees of tension, 
 must be reciprocally as the squares of the lengths proper for emitting. the same 
 tones. 
 
 Pythagoras, we are told, was led to this discovery by the following circumstance. 
 Harmonious sounds proceeding from the hammers striking on an anvil in a smith’s 
 shop happening one day to reach his ear, while walking in the street, he entered the 
 shop, and found, by weighing the hammers which had occasioned these sounds, that 
 the one which gave the octave was exactly the half of that which produced the lowest 
 tone; that the one which produced the fifth, was two thirds of it; and that the one 
 which produced the third major, was four fifths. When he returned home, medi- 
 tating on this phenomenon, he extended a string, and after successively shortening it 
 to one half, two thirds, and four fifths, perceived that it emitted sounds which were 
 the octave, the fifth, and the third major of the tone emitted by the whole string. 
 He then suspended weights from it, and found that those which gave the octave, the 
 fifth, and the third major, ought to be respectively as 4, 9, 23, to that which emitted 
 the principal tone; that is to say, in the inverse ratio of the squares of 4, 2, 4. 
 
 Whatever may be the degree of credit due to this anecdote, which is appreciated 
 as it deserves in the History of the Mathematics, such were the first facets that enabled 
 mathematicians to subject the musical intervals to calculation. The sum of what 
 the moderns have added to them, is as follows: 
 
 It can be demonstrated at present by the principles of mechanics: 
 
 Ist. That if a string of a uniform diameter, extended by the same weights, be 
 lengthened or shortened, the velocity of its vibrations, in these two states, will be in 
 the inverse ratio of the lengths. If this string then be reduced to one half of its 
 length, its vibrations will have a double velocity ; that is to say, it will make two 
 vibrations for one which it made before ; if it be reduced to two thirds, it will make 
 three vibrations for two which it made before. When a string therefore performs 
 two vibrations, while another performs one, the tones emitted by these strings will 
 be octaves to each other; when one vibrates three times while another vibrates 
 twice, the one will be the fifth to the other, and so on. 
 
 2d. The velocity of the vibrations performed by a string, of a determinate length, 
 
 _ and distended by different weights, is as the square roots of the stretching weights : 
 
 quadruple weights therefore will produce double velocity, and consequently double 
 the number of vibrations in the same time; a noncuple weight will produce vibra- 
 
 _ tions of triple velocity. or a triple number in the same time. 
 
 3d. If two strings, differing both in length and in weight, be stretched by different 
 
 | weights, the velocities of their vibrations will be as the square roots of the distending 
 | weights divided by the lengths and the weights of the strings: thus, if the string a, 
 
 stretched by a weight of 6 pounds, weigh six grains, and be a foot in length ; while 
 the string B, stretched by a weight of 10 pounds, weighs five grains, and is half a 
 
 _ foot iu length ; the velocity of the vibrations of the former will be to that of the 
 
374 ACOUSTICS. 
 
 vibrations of the latter, as the square root of 6 X 6 x 1 to that of 5 x 10 x 4, that . 
 is, as the square root of 36, which is 6, to that of 25, or 5: the first therefore will 
 perform 6 vibrations while the second performs 5. 
 
 From these discoveries it follows, that the acuteness or gravity of sounds, is 
 merely the effect of the greater or less frequency of the vibrations of the string 
 which produces them; for since we know by experience, on the one hand, that a 
 string when shortened, if subject to the same degree of tension, emits a more ele- 
 vated tone; and on the other, both by theory and experience, that its vibrations are 
 more frequent the shorter it is, it is evident that it is only the greater frequency 
 of the vibrations that can produce the effect of elevating the tone. 
 
 It thence results also, that a double number of vibrations produces the octave 
 of the tone produced by the single number ; that a triple number produces the octave 
 of the fifth; a quadruple number, the double octave; a quintuple, the third major 
 above the double octave, &c.; and if we descend to ratios less simple, three 
 vibrations for two will produce the concord of fifth; four for three, that of the 
 fourth, &e 
 
 The ratios of tones therefore may be expressed, either by the lengths of the 
 equally stretched strings which produce them, or by the ratio of the number of the 
 vibrations performed by these strings; thus, if the principal tone be denoted by I, the 
 octave above is expressed mathematically by 4, or by 2; the fifth by 2 or 3; the third 
 major by 4 or §, &c. In the first case, the respective lengths of the strings are de- 
 noted; in the second, the respective numbers of vibrations. In calculation, the 
 results will be the same, whichever method of denomination be adopted. 
 
 PROBLEM. 
 
 To determine the number of the vibrations made by a string, of a given length and 
 size, and stretched by a given weight; or, in other words, the number of the vibra- 
 tions which form any tone assigned. 
 
 Hitherto we have considered only the ratios of the number of the vibrations, 
 performed by strings which give the different concords; but a more curious, and far 
 more difficult problem, is, to find the real number of the vibrations performed by 
 a string which gives a certain determinate tone; for it may be readily conceived that 
 their velocity will not admit of their being counted. Geometry, however, with the 
 help of mechanics, has found means to resolve this question, and the rule is as 
 
 ‘a follows: 
 
 Divide the stretching weight by that of the string ; multiply the quotient by the 
 length of the pendulum that vibrates seconds, which at London is 394 inches, or 
 4693 lines, and divide the product by the length of the string from the fixed point 
 to the bridge; extract the square root of this new quotient, anid multiply it by the 
 ratio of the circumference of the circle to the diameter, viz. 3} nearly, or the frac- 
 tion 333, in decimals 3°1416 nearly; the product will be the number of the vibrations 
 performed by the string in the course of a second. 
 
 Let a string of a foot and a half in length, for example, and weighing 8 grains, be 
 stretched by a weight of 4 pounds Troy weight, or 23040 grains: the quotient 
 of 23040 divided by 8 is 2880; and as the length of the pendulum which swings 
 seconds is 4693 lines, the product of 2880 by this number will be 1352160; if this 
 product be divided by 216, the lines in a foot and a half, we shall have 6260, the 
 square root of which will be 79°1201: this number multiplied by 33 or 3°1416, gives 
 248°563, which is the number of the vibrations made by the abate string in the 
 course of a second. 
 
 A very ingenious method, invented by M. Sauveur, for finding the number of 
 these vibrations, may be seen in the Memoirs of the Academy of Sciences for 1700. 
 
MUSICAL STRINGS. 375 
 
 Having remarked, when two organ pipes, very low, and having tones very near to 
 each other, were sounded at the same time, that a series of pulsations or beats was - 
 heard in the sounds; by reflecting on the cause of this phenomenon, he found that 
 these beats arose from the periodical meeting of the coincident vibrations of the two 
 pipes. Hence he concluded, that if the number of these pulsations, which took 
 place in a second, could be ascertained by a stop watch, and if it were possible also 
 
 _ to determine, by the nature of the consonance of the two pipes, the ratio of the vi- 
 
 brations which they made in the same time, he should be able to acertain the real 
 number of the vibrations made by each. 
 We shall here suppose, for example that two organ pipes are exactly tuned, the 
 
 | one to mi flat, and the other to mz, as it is well known that the interval between 
 
 these two tones is asemi-tone minor, expressed by the ratio of 24 to 25, the higher pipe 
 
 _ will perform 25 vibrations while the lower performs only 24; so that at each 25th 
 
 vibration of the former or 24th of the latter, there will be a pulsation; if 6 pulsa- 
 
 - tions therefore are observed in the course of one second, we ought to conclude that 
 
 _ 24 vibrations of the one and 25 of the other are performed in the 10th part ofa 
 
 second: aud consequently that the one performs 240 vibrations, and the other 250 
 in the course of a second. 
 
 M. Saveur made experiments according to this idea, and found that an open organ- 
 pipe, 5 feet in length, makes 100 vibrations per second; consequently one of 4 
 feet, which gives the triple octave below, and the lowest sound perceptible to the 
 ear, would make only 124: on the other hand, a pipe of one inch less 7, being the 
 shortest the sound of which can be distinguished, will give in a second 6400 vibra- 
 tions. The limits therefore of the slowest and quickest vibrations, appreciable by 
 the ear, are according to M. Sauveur 12} and 6400. 
 
 We shall not enlarge farther on these details, but proceed to a very curious phe- 
 nomenon respecting strings in a state of vibration. 
 
 Make fast astring at both its extremities, and by means of a bridge divide it. into 
 aliquot parts, for example 3 on the one side, and | on the other; and put the 
 larger part, that is tosay the 3, in astate of vibration; if the bridge absolutely inter- 
 cepts all communication from the one part to the other, these 3 of the string, as is well 
 known, will give the tone of the fourth of the whole string; if 4 be intercepted, 
 the tone will be the third major. 
 
 But if the bridge only prevents the whole of the string from vibrating, without 
 intercepting the communication of motion from the one part to the other, the greater 
 part will then emit only the same sound as the less ; and the 3 of the string, which 
 in the former case gave the fourth of the whole string, will give only the double octave, 
 which is the tone proper to the fourth of the string. The case is the same if this 
 fourth be struck: its vibrations, by being communicated to the other three fourths, 
 will make them sound, but in such a manner as to give only this double octave. 
 
 The following reason, which may be rendered plain by an experiment, is assigned 
 for this phenomenon: when the bridge absolutely intercepts all communication 
 between the two parts of the string, the whole of the largest part vibrates together ; 
 and if it be 3 of the whole string, it makes, agreeably to the general law, 4 vibra- 
 tions in the Fete that the whole string would make 3: its sound therefore is the 
 fourth of the whole string. 
 
 But in the second case, the larger part of the string divides itself into as many 
 portions as the number of times it contains the less, which in the present example is 
 3, and each of these portions, as well asthe fourth, performs its particular vibrations : 
 at the points of division, as B,c,p (Fig. 3.), there are established fixed points, 
 between which the portions of the string a B, BC, CD, D £, each vibrate separately, 
 forming alternate bendings in a contrary direction, as if these parts were alone and 
 
 _ invariably fixed at their extremities. 
 
376 ACOUSTICS, 
 
 This explanation is founded on a fact which M. Saveur ren- 
 
 Fig. 3. dered sensible to the eyes in the presence of the Royal Academy 
 
 A B pRB of Sciences. (‘‘Hist. de l’Acad. année 1700.) On the 
 > < 
 
 Cc points c and p (Fig. 3.), he placed small bits of paper; and 
 
 having put the small part of the string A B in astate of vibra- 
 tion, the vibrations being communicated to the remaining part B £, the spectators 
 saw, with astonishment, the small bits of paper placed on the points c and D remain 
 motionless, while those placed on the other parts of the string were thrown down. 
 
 If the part a B of the string, instead of being exactly an aliquot part of the re- 
 mainder B £, be for example 2 of it, the whole string a E will divide itself into 7 por- 
 tions, of which « B will contain two, and each of these portions will vibrate separately, 
 and emit only that sound which belongs to the } of the string. 
 
 If the parts a B and BE be incommensurable, they will emit a sound absolutely 
 discordant, and which almost immediately ceases on account of the impossibility of 
 bendings and invariable points of rest being established. 
 
 ARTICLE VI. 
 Method of adding, subtracting, muitiplying, and dividing Concords. 
 It is necesary for those. who wish to understand the thecry of music, to know what 
 concords result from two or more concords, either when added or subtracted, &c., 
 by each other. For this reason we shall give the following rules. 
 
 PROBLEM I. 
 To add one concord to another. 
 
 Express the two concords by the fractions which represent them, and then mul- 
 tiply these two fractions together; that is to say, first the numerators, and then the 
 denominators; the number thence produced will express the concord resulting from 
 the sum of the two concords given. ; 
 
 Example 1.—Let it be rae to add the fourth and fifth together. 
 
 The paler for the fifth is 2; and that for the fourth 3; the product of these 
 two is §, = 4, being the peressicn for the octave. It is indeed well known that the 
 octave is composed of a fifth and a fourth. 
 
 Example 2.—What is the concord arising from the addition of the third major and the 
 third minor ? 
 
 The expression for the third major is 4, and that of the third minor is 8, the pro- 
 duct of which is 39 or 2, which expresses the fifth; and this concord indeed is com- 
 posed of a third major and a third minor. 
 
 Example 3.—What ts the concord produced by the addition of two tones major ? 
 
 A tone major is expressed by 3 cD consequently, to add two tones major, § must be 
 multiplied by §. The product ${ is a fraction less than 84 or 4, which expresses the 
 third major ; hence it follows, oat the concord expressed by § is greater than the 
 third major; and consequently two tones major are more than a “third major, or form 
 a third major false by excess. 
 
 On the other hand, by adding ve tones minor, eee are ee expressed by 
 fo, it will be found that their sum }, is greater than 8% or 4, which denotes the third 
 major: two tones minor erick added together, make more than a third major. 
 This third indeed is composed of a tone major and a tune minor, as may be proved by 
 adding together the concords $ and ~,, which make bo = to OF $- 
 
 It might be proved,-in fies manner, that two semi-tones major make more than a 
 
 - 
 
: ti ‘ ? ; ; a ie ey ‘ 4 qu x x A : 2 \ ‘ 
 P ae s ‘i =.’ > us : ; oh - ‘ oe 
 PRINCIPLES OF HARMONY. e: 
 
 ‘tone major, and two semi-tones minor less than even a fone minor; and in the last | 
 place, that a semi-tone major and a semi-tone minor make exactly a tone minor. 
 
 » 
 
 a . PROBLEM II. 
 To subtract one concord from another. 
 
 Instead of multiplying together the fractions which express the given concords, 
 they must here be divided; or invert that which expresses the concord to be _ 
 
 subtracted from the other, and then multiply them together as before: the BRAC i 
 will give a fraction expressing the quotient, or concord required. 
 
 aecnels 1.—What is the concord which results Srom the fifth subenaceed from x 
 the octave ? 
 
 _ The expression of the octave is 4, that of the fifth 3, which inverted gives 8; and 
 if $ be multiplied by 3, we shall have 3, which expresses the fourth. 
 
 Example 2.—What is the difference between the tone major and the tone minor? 
 
 tae tone major is expressed by §, and the tone minor by #, which when inverted 
 gives 5 ; the product of § by = is §f, which expresses the difference between the tone : 
 major ae the tone Rance: net is what is called the great comma. 
 
 PROBLEM III. i 
 To double a concord, or to multiply it any number of times at pleasure. 
 In this case, nothing is necessary but to raise the terms of the fraction, which 
 _ expresses the given concord, to the power denoted by the number of times it is to be 
 multiplied ; that is, to the square if it is to be doubled, to the cube if to be mie 
 and so on. 
 
 Thus, the concord arising from the tone major tripled is $}3: for as the expression 
 of the tone major is 8, we shall have 8 X 8 X 8= 512, and 9 xX 9 x 9=729. This 
 concord $12 corresponds to the interval between wé and a fa higher than fa sharp 
 of the gamut. 
 
 PROBLEM Iv. 
 
 _ To divide one concord by any number at pleasure, or to find a cone which shall 
 be the half, third, &c. of a given concord. 
 | 
 
 To answer this problem, take the fraction which expresses the given concord, and 
 
 extract that root of it which is denoted by the determinate divisor: that is to say, 
 the square root, if the concord is to be divided into two; the cube root, if it is to 
 be divided into three, &c. ; and this root will express the concord required. 
 
 Ezxample.—As the octave is expressed by }, if the square root of it be extracted 
 
 it will give ¥ nearly; but jis less than #, and greater than 3; consequently the | a 
 
 middle of the octave is between the fourth and the fifth, or very near fa sharp. 
 
 ARTICLE VII. 
 Of the resonance of sonorous bodies; the fundamental principle of harmony and 
 melody ; with some other harmonical phenomena. 
 Experiment 1. 
 
 If you listen to the sound of a bell, especially when very grave, however in- 
 
 } 
 
 different your ear may be, you will easily distinguish, besides the principal sound, 
 several others more acute; but if you have an ear accustomed to appreciate the 
 musical intervals, you will perceive that one of these sounds is the twelfth, or fifth 
 
378 ACOUSTICS. 
 
 above the octave, and another the seventeenth major, or third major above the double ~ 
 octave. If your ear be exceedingly delicate, you will distinguish also its octave, 
 its double, and even its triple octave: the latter indeed are somewhat more difficult to 
 be heard, because the octaves are almost confounded with the fundamental sound, in 
 consequence of that natural sensation which makes us confound the octave with unison. 
 
 The same effect will be perceived if the bow of a violoncello be strongly rubbed 
 against one of its large strings, or the string of a trumpet-marine. 
 
 In short, if you have an experienced ear, you will be able to distinguish these 
 different sounds, either in the resonance of a string, or in that of any other sonorous 
 body, and even in the voice. 
 
 Another method of making this experiment. 
 
 Suspend a pair of tongs by a woollen or cotton cord, or_any other kind of small 
 string, and twisting the extremities of it around the fore finger of each hand, put 
 these two fingers into your ears. If the lower part of the tongs be then struck, you 
 will first hear a loud and grave sound, like that of a large bell at a distance; and 
 this tone will be accompanied by several others more acute; among which, when 
 they begin to die away, you will distinguish the twelfth and the seventeenth of the 
 lowest tone. : 
 
 The truth of this phenomenon, in regard to the multiplicity of sounds, is con- . 
 firmed by another experiment, mentioned by Rameau, in his ‘‘ Harmonical Genera- 
 tion.” If you take, says he, those stops of the organ called bourdon, prestant or 
 flute, nazard and tierce, which form the octave, the twelfth and seventeenth major of 
 the bourdon, and if you draw out in succession each of the other stops, while the. 
 bourdon alone is sounding, you will hear their sounds successively mixed with each 
 ‘other ; you may even distinguish them while they are all sounding together ; but if 
 you prelude for a moment, by way of amusement, on the same set of keys, and then 
 return to the single key first touched, you will think you hear only one tone, that 
 of the bourdon, the gravest of all which corresponds to the sound of the whole 
 system. 
 
 Remark.—This experiment, respecting the resonance of bodies, is not new. It 
 was known to Dr. Wallis, and to Mersenne, who speak of it in their works ; but it 
 appeared to them a simple phenomenon, with the consequences of which they were 
 entirely unacquainted. Rameau first discovered its use in deducing from it all the | 
 rules of musical composition, which before had been founded on mere sentiment, and 
 on experience, incapable of serving as a guide in all cases, and of accounting for every 
 effect. It forms the basis of his theory of fundamental bass, a system which has been 
 opposed with much declamation, but which however most musicians seem at present 
 to have adopted. 
 
 All his harmony then is multiple, and composed of sounds which would be produced 
 by the aliquot parts of the sonorous body 4, 1,1, 1, 3, and we might add },1, &c. But 
 the weakness of these sounds, which go on always decreasing in strength, renders it 
 difficult to distinguish them. Rameau, however, says that he could distinguish very 
 plainly the sound expressed by 3, which is the double octave of a sound divided nearly 
 into two equal parts, being the interval between Ja and si.flat below the first octave: 
 he calls it a lost sound, and totally excludes it from harmony. It would indeed be 
 singularly discordant with all the sounds given by the fundamental tone. 
 
 We must however observe that the celebrated Tartini, in regard to this sound, 
 was not of the same opinion as Rameau. Instead of calling it a lost sound, he 
 maintains that it may be employed in melody as well as harmony; he distinguishes 
 it by the name of the seventh consonant. But we shall leave it to musicians to ap- * 
 preciate this idea of Tartini, whose celebrity in composition, as well as execution, 
 
be Wey ee 6 eee ee en Pe Ber A a, re Pets C0 UP Oe) ee Pare er oe. Pe 
 7 lA aa Ps 4 y, . é : = ~ ‘ mer 
 
 PRINCIPLES OF HARMONY. 379 
 
 required a refutation of a different kind, from that to be found at the end of a work 
 printed in 1767, entitled ‘‘ Histoire de la Musique.” 
 
 Experiment 2. 
 
 If you tune several strings to the octave, to the twelfth, and to the seventeenth 
 major, of the determinate sound emitted by another string, both ascending and de- 
 scending; as often as you make that which gives the determinate sound to resound 
 strongly, and with continuance, you will immediately see all the rest put themselves 
 in a state of vibration: you will even hear those sound which are tuned lower, 
 if care be taken to damp suddenly, by means of a soft body, the sound of the former. 
 
 Most persons have heard the glasses on a table sound, when a person near them 
 has been singing with a strong and a loud voice. The strings of an instrument 
 though not touched, are often heard to sound, in consequence of the same cause, 
 especially after swelling notes long continued. 
 
 This phenomenon arises, no doubt, from the vibrations of the air being commu- 
 nicated to the string, or to the sonorous body, elevated to the above tones: for it may 
 be easily conceived that the vibrations of strings, tuned to unison or to the octave, 
 or to the twelfth, &c., of that put in motion, are disposed to recommence regularly, 
 and at the same time as those of that string, one vibration corresponding to another, 
 in the case of unison; two to one, in case of the octave; or three to one in that of 
 the twelfth: the small impulsions therefore of the vibrating air, produced by the 
 string put in motion, will always concur to increase those movements, at first insen- 
 sible, which they have occasioned in the other strings; because they will take place 
 in the same direction, and will at length render them sensible. Thusa gentle breadth 
 of air, continued always in the same direction, is at length able to elevate the waters 
 of theocean. But when the strings in question are stretched in such amanner, that 
 their vibrations can have no correspondence with those of the string which is struck, 
 they will in this case be sometimes assisted and sometimes opposed, and the small 
 movement which can be communicated to them, will be annihilated as soon as pro- 
 duced, consequently they will remain at rest. ‘ 
 
 Question.—Do the sounds heard with the principal sound derive their source imme- 
 diately from the sonorous body, or do they reside only in the air or the organ? 
 
 It is very probable that the principal sound is the only one that derives its origin 
 immediately from the vibrations of the ‘sonorous body. Philosophers of eminence 
 have endeavoured to discover whether, independently of the total vibrations made 
 by the body, there are not also partial vibrations; but hitherto they have been able 
 to observe only simple vibrations. Besides, how can it be conceived that the whole 
 of a string should be in vibration, and that during its motion it should divide itself 
 into two or three parts that perform also their distinct vibrations ? 
 
 It must then be said that these harmonical sounds of octave, twelfth, seventeenth, 
 &c., are in the air or the organ—both suppositions are probable,—for since a deter- 
 minate sound has the property of putting intoa state of vibration bodies disposed to 
 give its octave, its twelfth, &c., we must allow that this sound may put in motion the 
 particles of the air susceptible of vibrations of double, triple, quadruple, and quin- 
 tuple velocity. What, however, appears most probable in this respect is, that these 
 vibrations exist only in the ear: it seems indeed to be proved, by the anatomy of 
 this organ, that sound is transmitted to the soul only by the vibrations of those ner- 
 yous fibres which cover the interior part of the ear; and as they are of different 
 lengths, there are always some of them which perform their vibrations isochronous 
 to those of a given sound., But, at the same time, and in consequence of the pro- 
 perty above mentioned, this sound must put in motion those fibres which are suscep- 
 tible of isochronous vibrations, and even those which can make vibrations of double, 
 
Op a YS Soe nee a eee 
 Fae Ta re RL ee 
 
 380 ACOUSTICS. 
 
 triple, quadruple, &c., velocity. Such, in our cpinion, is the most probable long 
 tion that can be given of this singular phenomenon. 
 
 Experiment 3. 
 
 For this experiment we are indebted to the celebrated Tartini of Padua. If you 
 draw from two instruments, at the same time, any two sounds whatever, you will 
 hear in the air a third, which will be the more perceptible the nearer your ear is 
 placed to the middle of the distance between the two instruments. Let us suppose 
 then, for example, two sounds which succeed each other in the order of consonances, 
 as the octave and the twelfth, the double octave and the seventeenth major, &c. ; 
 the sound resulting, says Tartini, will be the octave of the principal sound. 
 
 This experiment was repeated in France with the same success, as we are assured 
 by M. Serres, in his ‘* Principes de l’Harmonie,”’ printed in 1753; but with this ex- 
 ception, that M. Serres found the latter sound to be lower by an octave. As the 
 octaves are easily confounded, this difference needs excite no surprise. We must 
 however here observe, that the celebrated musician of Padua established on this 
 phenomenon a system of harmony and composition ; but it does not seem to have met 
 with so favourable a reception as that of Rameau. 
 
 ARTICLE VIII. 
 
 Of the different systems of Music; the Grecian and the Modern, together with their 
 peculiarities. 
 
 I1.—Of the Grecian Music. 
 
 During the infancy of music among the Greeks, their lyre had four strings, the 
 sounds of which would have corresponded to sz, ut, re, mi; but they afterwards added 
 other three fa, sol, la. The first diatonic scale therefore of the Greeks, translated 
 into our musical language, was st, ut, re, mi, fa, sol, la, and was composed of two 
 tetrachords, or systems of four sounds, si, wt, re, mi; mi, fa, sol, la ; in which the 
 last of the one and the first of the other were common, and on this account they 
 were called conjunct tetrachords. 
 
 We must here observe that, however singular this disposition of sounds may ap- 
 pear to those who are acquainted only with the modern diatonic order, it is no less 
 natural and agreeable to the rules of harmony; for Rameau has shewn that it is 
 nothing else than a chant, the fundamental bass of which would be, sol, ut, sol, ut, fa, 
 ut, fa. It possesses also the advantage of having only one altered interval, viz. the 
 third minor from re to fa, which, instead of being in the ratio of 5 to 6, is in that of 
 27 to 32; which is somewhat less, and consequently too low by a comma of from 
 80,to 81. ; 
 
 But this perfection in the Grecian gamut was counter-balanced by two great im- 
 perfections, viz. Ist. that it did not complete the octave ; 2d. that it did not termi- 
 nate by a rest, which leaves to the ear that kind of uneasiness resulting from a song 
 begun and not finished. It could neither ascend to sz, nor descend to la; and there- 
 fore the musicians who, to complete the octave, added the latter note below, con- 
 sidered it to be foreign, as we may say, and gave it the name of proslambanomenos. 
 
 For this reason they endeavoured to discover another remedy for this detect, and 
 Pythagoras, as is said, proposed the succession of sounds mi, fa, sol, la; si, ut, re, mi, 
 composed, as it appears, of two disjunct tetrachords. This diatonic scaleis almost the 
 same as ours, with this difference, that ours begins and ends with the tonic note, 
 while the former begins and ends with the mediante, or third major. This termina- 
 tion, alinost reprobated at present, was very common among the Greeks, and is still 
 so in the chants or vocal music of our churches. . 
 
 But here, in consequence of the harmonic generation, the values of the sounds and 
 
Ry tae ee Pa ee eee re CO ye Os ine a es BN A aa Hl) 1 Oe sar 
 OF THE DIFFERENT SYSTEMS OF MUSIC, 381 
 
 intervals are not the same as in the first scale. In the first, the interval from sol to 
 la was a tone minor, in the second it is a tone major. In the last place, according to 
 this second arrangement there are three intervals altered or false, viz. the tierce 
 major, from fa to la, too high; the tierce minor, from da to ut, too low ; and the fifth, 
 from Ja to mi, too high. These are the same faults as those of our diatonic scale ; 
 but the temperament corrects them. 
 
 To these sounds the Greeks afterwards added a conjunct tetrachord descending, si, 
 ut, re, mi, and another ascending, mz, fa, sol, la; by which they nearly supplied all 
 the wants of melody, so far as it was confined to one tone: Ptolemy speaks ofa 
 combination, by means of which they joined the second primitive tetrachord to the 
 first, lowering the si a semi-tone, which made si flat, ut, re,mi. This, no doubt, answered 
 the purpose when they passed from the tone of wé to that of its lower fifth fa ; a tran- 
 sition common inthe Grecian music, as well as in our church music: for in that casea 
 si flat is required. Plutarch also speaks of a combination where the two last tetra- 
 chords were disjoined, by raising the fa a semi-tone, and that no doubt of its lower 
 octave. Who does not here perceive our fa $, which is necessary when we pass from 
 the tone of ué to that of its upper fifth sol? ‘The strings, which corresponded to si flat 
 and fa sharp, wereno doubt merely added, and not substituted in the room of si and fa. 
 
 It is well known that in the Grecian music there were three genera, viz. the dia- 
 tonic, chromatic, and enharmonic. What has been hitherto said relates only to the 
 diatonic. 
 
 What characterises the enharmonic is, that it employs, either ascending or descend- 
 ing, several semi-tones in succession, The chromatic gamut of the Greeks was 
 si, ut, ut sharp, mi, fa, fa sharp, la. This disposition, by which they passed imme- 
 diately from ut sharp to mz, omitting the re, must no doubt appear very strange; but 
 it is certain that this was the gamut employed by the Greeks in the chromatic 
 genus. It is however not known whether the Greeks had considerable pieces of 
 music of this kind, or whether, like us, they employed it only in very short passages 
 of cantatas; for we also have a chromatic kind, though in a different acceptation. 
 This transition from semi-tones to semi-tones, is less natural than the diatonic suc- 
 cession; but it has more energy to express certain peculiar sensations: the Italians 
 therefore, who are great colorists in music, make frequent use of it in their airs. 
 
 In regard to the enharmonic music of the Greeks, though considered by the an- 
 cients as the most perfect kind, it is to us still an enigma. To give some idea of it, 
 let us assume the sign %, as that of the enharmonic diesis or sharp, which raises the 
 note a quarter of a tone: the enharmonic scale then was si, si 3%, ut, mi, mi ®, fa, la, 
 where it appears that, after two-fourths of atone, from sz to ut, or from mi to fa, they 
 proceeded to mi or la. It can hardly be conceived how there could be ears so well 
 exercised as to appreciate fourths of a tone, and even if we suppose that there were, 
 what modulation could they make with these sounds? It is however very certain 
 that this kind of music was long held in high estimation in Greece ; but on account 
 of its difficulty it was at length abandoned, so that not even a fragment of Grecian 
 music in the enharmonic kind has been handed down to us; nor any in chromatic, 
 though we have some in the diatonic. 
 
 We must however here observe, that this enharmonic music of the Greeks is not 
 perhaps so remote from nature as has been hitherto supposed; for does not Tartini, 
 in proposing the use of his consonant seventh, which is nearly a mean sound between 
 la and si flat, pretend that this intonation, la, sib, szb, re, re, sib, sib, la, is not 
 only supportable, but highly agreeable. Tartini does more; for he assigns to this 
 succession the sounds of its bass, fa, wé, sol, sel, ut, fa, marking ut with this sign 67, 
 which signifies consonant seventh. If this pretension of Tartini should find 
 partizans, may we not say that the enharmonic music of the Greeks has been 
 revived ? 
 
 < 
 
 Ben: 
 i 
 
382 ACOUSTICS, 
 
 It now remains that we should say a few words respecting the modes of the 
 Grecian music. However obscure this matter may be, if we can believe the author 
 of ‘“ Histoire des Mathematiques,” who founds his ideas on certain tables of Ptolemy, 
 these modes are nothing else than the tones of our music, and he gives the following 
 comparison. , 
 
 The Dorian being taken hypothetically for the mode of ut, these modes, some 
 lower than the Dorian and others higher, were: 
 
 The Hypodorian ..+++.corresponding to sol 
 Tie DY POPEryOlan: svaso suse e's seos5 0) 14 Hab. 
 The Hypophrygian acutior...... Dae tas la - 
 The Hypolydian or Hypo-elian .. ... si flat. 
 The=Hvnoudian acuti0res as ccess css aos. St 
 The’ Dorian | .28.%. sie 
 
 eocrccecersccese Ut 
 The Iastian or Toniag .... +00 seseces (22 8000p: 
 Aa i SL SRE ARR RAGA Pit SOR seonse TE 
 TREC EGIGN tiratet oe ek vote te eee seee- re Siiarp. 
 Them Lydiin Wess 4 pcan ok cee lives esto wal 
 ARCO VRerdOFiGn aatsie an a tistie dg os ens fa 
 
 The Hyperiastian or Mizxolydian ...... fa sharp. 
 The Hypermixolydian ..seee+eee+eeeee Sol the replicate of the first. 
 
 But this question might be asked: If the difference of the Grecian modes con- 
 sisted in the greater or less height of the tone of the modulation, how can we ex- 
 plain what is told us of the characters of these different modes, some of which 
 excited fury, others appeased it, &c.? There is reason therefore to think that they 
 depended on something more; and it is not improbable, that besides differences 
 of tone, there was a character of modulation peculiar to each. The Phrygian, for 
 example, which originated among a hardy and warlike people of that name, had a 
 masculine and warlike character, while the Lydian, which was derived from a soft 
 and effeminate people, had an analogous character, and consequently was proper for 
 calming the transports excited by the former. . 
 
 As we have here said enough respecting the Grecian music, we shall now proceed 
 to the modern. 
 
 Il.—Of the Modern Music. 
 
 Every person acquainted with music knows, that the gamut, or diatonic scale 
 of the moderns, is represented by these sounds ut, re, mi, fa, sol, la, si, ut, which 
 complete the whole extent of the octave ;* and, we shall add, that from its genera- 
 tion, as explained by Rameau, it follows that between ut and re there isa tone major; 
 from re to mi, a tone minor ; from mi to fa a semi-tone major; from fa to sol a tone 
 major, as well as from sol to /a; from la to si a tone minor, and from si to ut a semi- 
 tone major. 
 
 Hence it is concluded, that in this scale there are three intervals which are not 
 entirely just, viz., the third minor from re to fa, and indeed, being composed of a 
 tone minor and a semi-tone major, it is only in the ratio oi 27 to 32, which is some- 
 what less, viz. an 80th, than that of 5 to 6, the just ratio of the sounds which com- 
 pose the third minor. 
 
 In like manner, the third major, from fa to Ja, is too high, being composed of two 
 tones major, whereas it ought to be composed of a tone major and a tone minor, to 
 
 * Of the seven notes in the French scale wtf, re, mt, fa, sol, la, si, four oniy are generally 
 used among us, as mi, fa, sol, la, which are applied to the scale in this order, fa, sol, la, fa, sol, 
 la, mi, fa, and express the natural series from c. It is of little consequence however which method 
 be used ; the principles still remain the same. : 
 
 J 
 
MODERN MUSIC. 383 
 
 be exactly in the ratio of 4to05. In the last place, the third minor, from /e to ut, 
 is also altered, and for the same reason as that from re to fa. 
 
 If this disposition of tones major and minor were arbitrary, they might no doubt 
 be arranged in such a manner that fewer intervals should be altered; it would be 
 sufficient for this purpcse, to make the tone from ut to ve minor, and that from re to 
 mi major; the tone from sol to la might also be made minor, and that from la to st 
 major. For it will be found, that by this method there would be no more than a 
 single third altered ; whereas, according to the other disposition, there are three. This 
 circumstance has given rise to disputes among the musicians respecting the distribu- 
 tion of the tones minor and major; some being desirous, for example, that there 
 should be a tone major between ué and re, and others atone minor. The harmonic 
 generation of the diatonic scale, as explained by Rameau, will not however allow 
 this disposition, but only the former, which is that indicated by nature; and not- 
 withstanding its imperfections, which the temperament corrects in the execution, it 
 is preferable to the first of the Grecian scales, a scale very deficient, as it did not 
 comprehend the whole extent of the octave; it is superior also to the second, mi, fa, 
 sol, &c., ascribed to Pythagoras, because its desinence is more: perfect, and conveys 
 to the ear a rest, which is not in that of Pythagoras, on account of its fall on the 
 tonic note, announced and preceded by the note si, the third of the fifth sol, the 
 effect of which is so striking to our musical ears, that it has been distinguished by 
 the name of the senstble note. 
 
 Two modes, properly so called, are known in music, the characters of which are 
 exceedingly striking to ears possessed of any musical sensibility : these are the major 
 mode and the minor mode. 'The major mode is, when in the diatonic scale the third 
 of the tonic note is major; such is the third from ué to mi. The above gamut, or 
 diatonic scale, therefore is in the major mode. 
 
 But if the third of the tonic note be minor, it indicates the minor mode. This 
 mode has its scale as well as the major. Thus, for example, if we assume da as the 
 tonic note, the scale of the minor mode ascending will be la, si, ut, re, mi, fa, sol f, 
 la. We here make use of the term ascending, because it isa singularity of the minor 
 mode, that its scale descending, is different from what it is ascending ; and indeed in 
 descending we ought to say la, sol, fa, mi, re, ut, si, la. Ifthe tone were in wt, the 
 ascending scale would be ut, re, mi, fa, sol, la», si, ut, and descending wt si», la b, 
 sol, fa, mio, re, ut. Hence the reason why, in airs in the minor mode, we so often 
 find, without the tone being changed, accidental flats cr sharps, or naturals, which 
 soon destroy their effect, or that of those which are in the clef. This is one of those 
 singularities, of the necessity-of which the ear made musicians sensible: the cause of 
 
 it however, which depends on the progress of the fundamental bass, was first ex- 
 plained by Rameau. 
 
 To these two modes shall we add a third, proposed by M. de Blainville, under the 
 name of the mixed mode, the generation and properties of which he explains in his 
 History of Music? His scale is mi, fa, sol, la, si, ut, re, mi. We shall here only 
 observe, that musicians do not seem to have given a very favourable reception to this 
 new mode, and we confess that we are not sufficiently versed in these matters to be 
 able to decide whether they are right or wrong. 
 
 But however this may be, the character of the major mode is sprightliness and 
 gaiety ; while in the minor mode there is something gloomy and sad, which renders 
 it peculiarly fitted for expressions of that kind. 
 
 The modern music has its genera as well as the ancient. The diatonic is the most 
 
 - common; and is that most agreeable to what is pointed out by nature; but the 
 
 moderns have their chromatic also, and even in certain respects their enharmonic, 
 » though in a sense somewhat different from that assigned to these words by the 
 ancients. 
 
 ‘ 
 
384 ACOUSTICS. 
 
 The modulation is chromatic when several semi-tones are passed over in succession, 
 as if we should say fa, mi, mi, re, or sol, fag, fa mi. It is very rare to have more 
 than three or four semi-tones following each other in this manner; yet in an air of 
 the second act of la Zingara, or the Gypsey, an Italian intermede, there is a whole 
 lower octave almost from ut to re in consecutive semi-tones. It is the longest chro- 
 matic passage with which we are acquainted. 
 
 Rameau finds the origin of this progression in the nature of the fundamental bass, 
 which, instead of proceeding from fifth to fifth, which is its natural movement, pro- 
 ceeds from third to third. But it must here be remarked, that in the first passage 
 from mi to mi, there ought strictly to be only a semi-tone minor, and from mi» to 
 re asemi-tone major; but the temperament and constitution of most instruments, 
 by confounding the ref with mi, divide into equal parts the interval from re to mi, 
 aud the ear is affected by them exactly in the same manner, especially by means of 
 the accompaniment. 
 
 There are two enharmonic genera, the one called the diatonic enharmonic, and the 
 other the chromatic enharmonic, but they are very rarely employed by musicians. 
 These genera are not so called because quarters of a tone are employed in them, as 
 in the ancient enharmonic; but because, from the progress of the fundamental bass, 
 _ there result sounds, which, though taken one for the other, really differ a quarter of 
 a tone, called by the ancients enharmonic, or are in the ratio of 125 to 128. In the 
 diatonic enharmonic, the fundamental bass goes on alternately by fifths and thirds, 
 and in the chromatic enharmonic it goes on alternately by third major and minor. 
 This progression introduces, both into the melody and the harmony, sounds which, — 
 belonging neither to the principal tone nor its relatives, convey astonishment to the 
 ear, and affect it in a harsh and extraordinary manner, but which are proper for cer- 
 tain terrible and violent expressions. It was for this reason that Rameau employed 
 the diatonic enharmonic in the trio of the Fates, in his opera of Hippolitus and Aricia ; 
 and. though he was not able to get it executed, he was firmly persuaded that it would 
 have produced a powerful effect, had he found performers disposed to fall into his 
 ideas, so that he suffered it to remain in the partition which was printed. He mentions, 
 as a piece of the enharmonic kind, a scene of the Italian opera of Coriolano, beginning 
 with these words, O iniqut Marmi! which he says is admirable. Specimens of this 
 genus are to be found also in two of his own pieces for the harpsichord, the Triwm- 
 phante and the Enharmonique, and he did not despair of being able to employ the chro- 
 matic enharmonic at least in symphonies. And why indeed might he not have done 
 so, since Locatelli, in his first concertos, employed this genus, leaving the flats and 
 sharps to exist, and distinguishing for example the re~{ from mi». This, says a 
 modern historian of music, M. de Blainville, is a piece truly infernal, which throws 
 the soul into a violent state of apprehension and terror. 
 
 We cannot terminate this article better than by giving a few specimens of the 
 music of different nations. For this purpose we have given, on the opposite page, 
 Grecian, Persian, Chinese, Armenian and Tartar airs, which will serve to form an 
 idea of the modulation that characterises the music of these people. 
 
a 
 
 385 
 GREEK AIR OF A HYMN TO NEMESIS. 
 
386 ACOUSTICS. 
 
 ARTICLE IX. 
 
 Musical Paradozes. 
 
 I.—It is impossible to intonate justly the following intervals, sol, ut, Ja, re, sol; that is 
 to say, the interval between sol and ut ascending, that from ut to la redescending 
 from third minor, then ascending from fourth to re, and that between re and sol de- © 
 scending from fifth, and to make the second sol in unison with the first. 
 
 It will be found indeed by calculation, that if the first sol be represented by 1, the 
 ut, ascending from fourth, will be 3; consequently the la, descending from third 
 minor, will be #; the Ke above them will be 3%; and inthe last place the sol, descend- 
 ing from fifth, will be 8. But the sound represented by 4, is lower than that repre- 
 sented by 1, therefore Ke last sol is lower than the first. 
 
 But how comes it that experience is contrary to this calculation? In answer to 
 this question we shall observe, that the difference arises merely from the remem- 
 brance of the first tone sol. If the ear however were not affected by this tone, and 
 if the performer’s whole attention were directed to the just intonation of the above 
 intervals, it is evident that he would end with a lower sol. It therefore often hap- 
 pens that a voice, without an accompaniment, after having chanted a long air, in 
 which several tones are passed through, remains, in ending, higher or t lower than the 
 . tone by which it began. 
 
 This arises from the necessary alteration of some intervals in the diatonic scale. 
 In the preceding example, from la to ut, there is only a third minor in the ratio of 
 _ 27 to 32, and not of 5 to 6; but it is the latter which is intonated if the voice be. 
 true and well exercised; consequently the person who chants, lowers by a comma 
 more than is necessary, and therefore it is not astonishing that the last sol should 
 always be lower, by a comma, than the first. 
 
 II.—Jn instruments constructed with keys, such as the harpsichord, it is impossible that 
 the thirds and the fifths should be both just. 
 
 This may be easily demonstrated in the following manner.—Let there be a series 
 of tones, fifths to each other ascending, as ut, sol, re, la, mi; if ut be denoted by 1, 
 sol will be 2, re 4, la 8, mi 1$: this mi ought to form the third major with the double 
 octave of wt or 3, that is to say they ought to be in the ratio of 1 to 4, or of 5 to 4, 
 or of 80 to 64; fine this is not the case, for } and }§ are to each other as 81 to 64: _ 
 this mi therefore does not form the third ADs. ‘with the double octave of ut; or if 
 both are lowered from the double octave, ut and mi are not thirds to each other, if 
 mi is a just fifth to la. 
 
 In instruments with keys then, such as the harpsicord, however well tuned, all the 
 intervals, the octaves excepted, are either false or altered. This necessarily follows 
 from the manner in which that instrument is tuned; for when all the wé’s are made 
 octaves to each other, as they ought to be, the sol is made the fifth to ut, re the 
 fifth to sol, and the octave is lowered, because it is too high; da is then made the 
 fifth to re, thus lowered, and mi the fifth to da, and this mi is lowered from octave. 
 By continuing in this manner to ascend twice from fifth, and then to descend 
 from octave, the series of sounds si, fa, ut, solf, ref, lat, mi, si, are obtained. 
 But the latter si, which ought at most to be in unison with the wt, the octave of 
 _ the first, is found to be higher; for calculation shews that it is expressed by 
 28744, which is less than } the value of the octave of ut: this renders necessary what 
 is called temperament, which consists in lowering gently and equally all the fifths, so 
 that the latter si $, is found to be exactly the octave of the first ué. Such at least is” 
 the method taught by Rameau, and it is no doubt the most rational. But whatever 
 may be the method employed, it always consists in rejecting, in a more or less equal 
 
_. ae Vets ee ae 5 
 it 4 re 
 ey : ¥ a 
 
 EFFECTS OF MUSIC. 387 
 
 manner from the notes of the octave, this excess of si { above ut, which cannot be 
 done without altering, in some measure, the fifths, thirds, &c. 
 pit have just seen that the sz a, given by the progression of fifths, i is higher than 
 ; but if the following progression’ of thirds be employed, ut, mz, sol ff, si #, this 
 mt will be very different from the former; for it will be found that it is expressed 
 by #%, while the octave of ut is 3. But 4 is less than $f, consequently this si# is 
 below ut expressed by 4, and the interval of these two sounds is expressed by the 
 ratio of 128 to 125, which is the fourth of the enharmonic tone. 
 
 Ill.—A lower note, for example re, affected by a sharp, is not the same thing as the 
 higher note mi, affected by a fiat ; and the case is the same with other notes which are 
 a whole tone distant from each other. 
 
 The sharps are generally given by the major mode, and even by the minor, pro- 
 vided the sub-tonic note is not distant from the tonic more than a semi-tone major, as 
 the sz is from ut, in the tone of ut; then, as from re to m2 there is a tone minor, which 
 is composed of a semi-tone. major and a semi-tone minor, if we take away a semi- 
 tone major, by which re $ ought to be lower than mi, the remainder will be a semi- 
 tone minor, by which the same ref ought to be higher than re. If the distance 
 between the notes were a tone major, the sharp would raise the lower note by an 
 interval equal to a semi-tone minor, plus a comma of 80 to 81, which is a mean semi- 
 tone betweem the major and the minor. 
 
 The note therefore is raised by the sharp only a mean semi-tone, or a semi-tone 
 minor. 
 
 Flats are generally introduced in modulation by the minor mode, when it is neces- 
 sary to lower the note a third, so that it shall form with the tonica third minor: me 
 flat therefore ought to form with ut a third minor; consequently if from the third 
 major ut mt, which is 4, we take the third minor, which is 3, the remainder 3$ is the 
 quantity which expresses how much the flat lowers the mi below the natural tone : mz 
 flat then is higher than re sharp. 
 
 In practice however the one is taken for the other, especially in instruments con- 
 structed with keys: the flat in these is lowered, and the sharps gradually raised, till 
 they coincide with each other; and we do not know whether practice would gain 
 much by making a distinction between them. 
 
 ARTICLE X. 
 
 On the cause of the pleasure arising from music—The effects of it on man and on 
 
 animals. 
 
 It has often been asked, why two sounds, which form to each other the fifth and 
 the third, excite pleasure, while the ear experiences a disagreeable sensation by hear- 
 ing sounds which are no more than a tone or a semi-tone distant from each other. 
 Though it is difficult to answer this question, the following observations may tend to 
 throw some light on it. 
 
 Pleasure, we are told, arises from the perception of relations, as may be proved by 
 various examples taken from the arts. The pleasure therefore derived from music, 
 consists in the perception of the relations of sounds. But are these relations suffi- 
 ciently simple for the soul to perceive and distinguish their order? Sounds will 
 please when heard together in acertain order; but on the other hand, they will dis- 
 please if their relations are too complex, or if they are absolutely destitute of order. 
 
 This reasoning will be sufficiently proved by an enumeration of the known concords. 
 In unison, the vibrations of two sounds continually coincide throughout the whole 
 time of their duration; this is the simplest kind of relation. Unison also is the first 
 concord. In the octave, the two sounds, of which it is composed, perform their vi- 
 brations in such a manner, that two of the one are completed in the same time as 
 
 aes 974 
 
388 ACOUSTICS. 
 
 one of the other. Thus unison is succeeded by the octave. It is so natural to man, 
 that he who, through some defect in his voice, cannot reach a sound too grave or too 
 acute, falls into the higher or lower octave. 
 
 When the vibrations of two sounds are performed in such a manner, that three of 
 the one correspond to one of the other, these give the simplest relation, next to 
 those above mentioned. Who does not know, that the concord most agreeable to 
 the ear is the twelfth, or the octave of the fifth? In this respect it even surpasses 
 the fifth, the ratio of which, a little more compounded, is that of 2 to 3. 
 
 Next tothe fifth is the double octave of the third, or the seventeenth major, which 
 is expressed by the ratio of 1 to3. This concord therefore, next to the twelfth, is 
 the most agreeable; and if it be lowered from the double octave, to obtain the third, 
 it will still be in consonance; the ratio of 4 to 5, by which it is then expressed, - 
 being very simple. 
 
 In the last place the fourth, expressed by 3, the third minor, expressed by 3, and 
 the sixths, both major and minor, expressed by 3 and 3, are concords, and for the 
 same reason. 
 
 But it appears that all the other sounds, after these relations, are too complex for 
 the soul to perceive their order: of this kind are the intervals called the tone major 
 and the tone minor, expressed by § and $,, and much more so the semi-tones major 
 and minor, expressed by 43 and 34. Such also are the concords of third and fifth, 
 however little they may be altered; for the third major, raised a comma, is eX- 
 pressed by 33; and the fifth diminished by the same quantity, has for its expression 
 73: in the last place, the tritone, as from ut to fa, is one of the most disagreeable 
 discords, and is expressed by 18. 
 
 The following very strong objection however may be made to this reasoning. 
 How can the pleasure arising from concords consist in the perception of relations, 
 since the soul often does not know whether such relations exist between the sounds ? 
 The most ignorant person is no less pleased with a harmonious concert than he who 
 has calculated the relation of all its parts: what has hitherto been said may there- 
 fore be more ingenious than solid. 
 
 We cannot help acknowledging that we are rather inclined to think so; and it 
 appears to us that the celebrated experiment on the resonance of sonorous bodies, 
 may serve to account, in a still more plausible manner, for the pleasure arising from 
 concords ; because, as every sound degenerates into mere noise, when not accom- 
 panied by its twelfth and its seventeenth major, besides its octaves, is it not evident - 
 that, when we combine any sound with its twelfth or its seventeenth major, or with 
 both at the same time, we only imitate the process of nature, by giving to that 
 sound, in a fuller and more sensible manner, the accompaniment which nature itself 
 gives it, and which cannot fail to please the ear on account of the habit it has ac- 
 quired of hearing them together? This is so agreeable to truth, that there are only 
 two primitive concords, the twelfth and the seventeenth major; and that the rest, 
 as the fifth, the third major, the fourth, and the sixth, are derived from them. We 
 know also that these two primitive concords are the most perfect of all, and that 
 they form the most agreeable accompaniment that can be given to any sound; though 
 on the harpsichord, for example, to facilitate execution, the third major and the fifth 
 itself, which with the octave form what is called perfect harmony, are substituted in 
 their stead. But this harmony is perfect only by representation, and the most per- 
 fect of all would be that in which the twelfth and the seventeenth were combined 
 with the fundamental sound and its octaves. Rameau therefore adopted it as often 
 as he could in his chorusses, and particularly in his Pygmalion. We might enlarge 
 farther on this idea, but what has been already said will be sufficient for every intel- 
 ligent reader. 
 
 Some very extraordinary things are related in regard to the effects produced by 
 
EFFECTS OF MUSIC. 389 
 
 the music of the ancients, which, on account of their singularity, we shall here men- 
 tion. We shall then examine them more minutely, and shew that, in this respect, 
 the modern music is not inferior to the ancient. . 
 
 Agamemnon, it is said, when he set out on the expedition against Troy, being 
 desirous to secure the fidelity of his wife, left her under the care of a Dorian mu- 
 sician, who, by the effect of his airs, rendered fruitless, tor a long time, the attempts 
 of Agisthus to obtain her affection; but that Prince having discovered the cause 
 of her resistance, got the musician put to death, after which he triumphed without 
 difficulty over the virtue of Clytemnestra. 
 
 We are told also that, at a later period, Pythagoras composed songs or airs capable 
 of curing the most violent passions, and of recalling men to the paths of virtue and 
 moderation: while the physician prescribes draughts for curing bodily diseases, an 
 able musician might therefore prescribe an air for rooting out a vicious passion. 
 
 The story of Timotheus, the director of the music of Alexander the Great, is 
 well known,—One day, while the prince was at table, Timotheus performed an air 
 in the Phrygian mode, which made such an impression on him that, being already heated 
 with wine, he flew to his arms, and was going to attack his guests, had not Timotheus 
 immediately changed the style of his performance to the Sub-Phrygian. This mode 
 calmed the impetuous fury of the monarch, who resumed his place at table. This 
 was the same Timotheus who, at Sparta, experienced the humiliation of seeing 
 publicly suppressed four strings which he had added to his lyre. The severe Spar- 
 tans thought that this innovation would tend to effeminate their manners, by intro- 
 ducing a more extensive and more variegated kind of music. This at any rate proves 
 that the Greeks were convinced that music had a peculiar influence on manners; and 
 that it was the duty of government to keep a watchful eye over that art. 
 
 Who indeed can doubt that music is capable of producing such an effect? Let us 
 only interrogate ourselves, and examine what have been our sensations on hearing a 
 majestic or warlike piece of music, or a tender and pathetic air sung or played with 
 expression. Who does not feel that the latter tends as much to melt-the soul, and 
 dispose it to pleasure, as the former to rouse and exalt it? Several facts in regard 
 to the modern music place it on a level in this respect with the ancient. 
 
 The modern music indeed has also had its Timotheus, who could excite or calm, 
 at his pleasure, the most impetuous emotions. Henry III. king of France, says ‘‘ Le 
 Journal de Sancy,” having a concert on occasion of the marriage of the Duke de 
 Joyeuse, Claudin Je Jeune, a celebrated musician of that period, executed certain 
 airs, which had such an effect on a young nobleman, that he drew his sword, and 
 challenged every one near him to combat; but Claudin, equally prudent as Timo- 
 theus, instantly changed to an air, apparently Sub-Phrygian, which appeased the 
 furious youth. 
 
 But, what shall we say of Stradella, the celebrated composer, whose music made 
 the daggers drop from the hands of his assassins? Stradella having carried off the 
 mistress of a Venetian musician, and retired with her to Rome, the Venetian hired 
 three desperadoes to assassinate him; but fortunately for Stradella, they had an ear 
 sensible to harmony. These assassins, while waiting for a favourable opportunity 
 to execute their purpose, entered the church of St. John de Lateran, during the per- 
 formance of an Oratorio composed by the person whom they intended to destroy, and 
 were so affected by the music, that they abandoned their design, and even waited on 
 the musician to forewarn him of his danger. Stradella, however, was not always so 
 fortunate ; other assassins, who apparently had no ear for music, stabbed hitn some 
 time after at Genoa: this event took place about the year 1670. 
 
 Every person almost has heard that music is a cure for the bite of the tarantula. 
 This cure, which was formerly considered as certain, has by some been contested ; 
 but, however this may be, Father Schott, in his ‘‘ Musurgia Curiosa,” gives the- 
 
390 ACOUSTICS. 
 
 tarantula air, which appears to be very dull, as well as that employed by the Sicilian 
 fishermen to entice the thunny fish into their nets. 
 
 Various anecdotes are related respecting persons whose lives have been preserved, 
 by music affecting a sort of revolution in their constitutions. A woman being at- 
 tacked for several months with the vapours, and confined to her apartment, had 
 resolved to starve herself to death: she was however prevailed on, but not without 
 difficulty, to see a representation of the Servo Padrona, at the conclusion of which 
 she found herself almost cured, and, renouncing her melancholy resolution, was en- 
 tirely restored to health by a few more representations of the like kind. 
 
 There is a celebrated air in Switzerland, called Ranz des Vaches, which had such 
 an extraordinary effect on the Swiss troops in the French service, that they always 
 fell into a deep melancholy when they heard it: Louis XIV. therefore forbade it ever 
 to be played in France, under the pain of a severe penalty. We are told also 
 of a Scotch air (Lochaber no more) which has a similar effect on the natives of 
 Scotland. 
 
 Most animals, and even insects, are not insensible to the pleasure of music. There 
 are few musicians perhaps who have not seen spiders suspend themselves by their 
 threads in order to be near the instruments. We have several times had that satis- 
 faction. We have seen a dog who, at an adagio of a sonata by Sennaliez, never 
 failed to shew signs of attention, and some peculiar sensation by howling. 
 
 The most singular fact however is that mentioned by Bonnet, in his History of 
 
 Music. This author relates that an officer, being shut up inthe Bastile, had permis- 
 _ sion to carry with him a lute, on which he was an excellent performer: but he had 
 scarcely made use of it for three or four days, when the mice issuing from their holes, 
 and the spiders suspending themselves from the ceiling by their threads, assembled 
 around him to participate in his melody. His aversion to these animals made their visit 
 at first disagreeable, and induced him to lay aside his recreation; but he was soon so 
 accustomed tothem, that they became a source of amusement. We are informed by 
 the same author, that he saw, in 1688, at the country seat of Lord Portland, the 
 English ambassador in Holland, a gallery in a stable, employed, as he was told, for 
 giving a concert once a week to the horses, which seemed to be much affected by the 
 music. This, it must be allowed, was carrying attention to horses to a very great 
 length. But it isnot improbable that this anecdote was told to Bonnet by some per- 
 son, in order to make game of him. 
 
 ARTICLE XI. 
 Of the properties of certain Instruments, and particularly Wind Instruments. 
 
 I. We are perfectly well acquainted with the manner in which stringed instru- 
 ments emit their sounds; but erroneous ideas were long entertained in regard to 
 wind instruments, such as the flute; for the sound was ascribed to the interior sur- 
 face of the tube. The celebrated Euler first rectified this error, and it results from 
 his researches : 
 
 Ist. That the sound produced by a flute, is nothing else than that of the cylinder of 
 air contained in it. / : 
 
 2d. That the weight of the atmosphere which compresses it, acts the part of a 
 stretching weight. | 
 
 3d. That the sound of this cylinder of air, is exactly the same as that which would 
 be produced by a spring of the same mass and length, extended by a weight equal to 
 that which compresses the base of the cylinder. 
 
 This fact is confirmed by experiment and calculation: for Euler found that a cylin- 
 der of air, of 73 Rhinlandish feet, at a time when the barometer is at a mean height, 
 must give C—sol__ut ; and such is nearly the length of the open pipe of an organ 
 which emits that sound. The reason of its being made generally 8 feet is, be- 
 
cause that length is required at those times when the weight of the atmosphere is 
 greater. 
 
 Since the weight of the atmosphere produces, in regard to the sounding cylinder 
 of air, the same effect as that produced by the weight which stretches a string, the 
 more that weight is increased, the more will the sound be elevated ; it is therefore 
 observed that during serene warm weather, the tone of wind instruments is raised ; 
 and that during cold and stormy weather it is lowered. These instruments also emit 
 a higher sound, in proportion as they are heated; because the mass of the cylinder 
 of heated air becoming less, while the weight of the atmosphere remains unchanged, 
 the case is exactly the same as if a string should become less, and be still stretched 
 by the same weight: every body knows that such a string would emit a higher 
 tone. 
 
 But as stringed instruments must become lower, because the elasticity of the strings 
 insensibly decreases, it thence follows that wind and stringed instruments, however 
 well tuned they may be to each other, soon become discordant: for this reason the 
 Italians never admit the former into their orchestras. 
 
 II. A very singular phenomenon is observed in regard to wind instruments, such as 
 the flute and huntsman’s horn: with a flute, for example, when all the holes are 
 stopped, if you blow faintly into the mouth aperture, a certain tone will be pro- 
 duced; if you blow a little stronger, the tone instantly rises to the octave; and by — 
 blowing successively with more force, you will produce the twelfth, or fifth above the 
 octave; then the double octave or seventeenth major. 
 
 The cause of this effect is the division of the cylinder of air contained in the 
 instrument : when you breathe into the flute gently, the whole column resounds, and 
 it emits the lowest tone; but if you endeavour, by a stronger inspiration, to make it 
 perform quicker vibrations, it divides itself into two parts, which perform their vibra- 
 tions separately, and which consequently must give the octave; a still stronger 
 inspiration makes the column divide itself into three portions, which give the 
 twelfth, &c. 
 
 III. It remains for us to speak of the trumpet marine. This instrument is only a 
 monochord of a singular construction, being composed of three boards that form a 
 triangular body. It hasa very long neck, and one thick string, mounted on a bridge, 
 which is firm on the one side, and tremulous on the other. It is struck by a bow 
 with one hand, and with tbe other the string is stopped or pressed on the neck by 
 means of the thumb, applied to the divisions indicated for the different tones. The 
 trembling of the bridge, when the string is struck, makes it imitate the sound of the 
 trumpet; and this it does to such perfection that it is scarcely possible to distinguish 
 the one from the other. Henceit haditsname; but whereas, in the common stringed 
 instruments the tone becomes lower as the part of the string struck is longer, the 
 case here is the contrary ; for if the half of the string, for example, gives ut, the two 
 thirds give the sol above, and the three fourths give the octave. 
 
 M.: Saveur first assigned the reason of this singularity, and proved it in a sensible 
 manner, by shewing that when the string, by the gentle application of the finger, is 
 divided into two parts which are to each other as 1 to 2, whatever part be touched, 
 the greater immediately divides itself into two equal portions, which consequently 
 perform their vibrations in the same time, and give the same sound as the less. But 
 the less being the third of the whole, and the two thirds of the half, it must give 
 the fifth or sol when the half gives ut. In like manner, the three fourths of the 
 string divide themselves into three portions, each equal to the remaining fourth, and 
 as they perform their vibrations separately they must emit the same sound, which 
 can be only the octave of the half. The caseis the same with the other sounds of the 
 trumpet marine, which may be easily explained on the same principle. 
 
 MUSICAL INSTRUMENTS. 391. 
 
Ben 
 “m 
 
 392 ACOUSTICS. 
 
 ARTICLE XII. 
 Of a fixed Sound ; method of preserving and transmitting it. 
 
 Before the effects of the temperature of the air on sound, and on the instruments 
 by which it is produced, were known, this would not have formed the subject of a 
 question, but to the few possessed of an ear exceedingly fine and delicate, and in 
 which the remembrance of a tone is perfect: to others no doubt would remain that 
 a flute, not altered, would always give the same tone. Such an opinion however 
 would be erroneous, and if the means of transmitting to St. Domingo, for example, 
 or to Quito, or only to posterity, the exact pitch of our opera were required, to 
 solve this problem would be attended with more difficulty than might at first be 
 imagined. 
 
 Notwithstanding what may be generally said in this respect, we shall here begin 
 by a sort of paradox. It is every where said that the degree of the tone varies 
 according to the weight of the atmosphere, or the height of the barometer. This 
 we can by no means admit: and we flatter ourselves that we can prove the 
 contrary. 
 
 It has been demonstrated by the formule of Euler, and no one entertains any 
 doubt in regard to their truth, that if @ represents the weight which compresses the 
 column of air in a flute, L the length of that column, and w its weight, the number of 
 
 the vibrations it makes will be expressed by Vo that is to say, will be in~ the 
 CL 
 
 compound ratio of the square root of c, or the compressing weight taken directly, 
 and the product of the length by the weight taken inversely. Let us suppose then 
 that the length of the column of air put in vibration is invariable, and that the gravity 
 of the atmosphere only, or G, is variable, as well as the weight of the vibrating 
 column. In this case we shall have the number of the vibrations proportional to 
 
 the expression e/a, But the density of any stratum of air being proportional to 
 W 
 the whole weight of that part of the atmosphere immediately above it, it thence 
 
 follows that w, which in equal lengths is as the density, is as c. The fraction © there- 
 Ww 
 
 fore is constantly the same, when difference of heat does not alter. the density. The 
 square root of < then is always the same; consequently there will be no variation 
 
 in the number of the vibrations, or in the tone, at whatever height in the atmosphere 
 the instrument may be situated, or whatever be the gravity of the air, provided its 
 temperature has not changed. 
 
 This reasoning, in our opinion, 1s unanswerable ; and if the gravity of the air has 
 hitherto been reckoned among those causes which alter the tone of wind instruments, 
 it is because it has been implicitly believed that the weight of the column of air put in 
 vibration is invariable. It is however evident that under the same temperature it 
 must be more or less dense, according to the greater or less density of the atmo- 
 sphere; since it has a communication with the surrounding stratum of air, the density 
 of which is proportional to that gravity. But the gravity in equal volumes is pro- 
 portional to the density ; therefore, &c. 
 
 Nothing then remains to be considered but the temperature of the air, which is 
 the only cause that can produce variations in the tone of a wind instrument. But 
 whatever may be the degree of heat or of cold, the tone might be fixed in the fol- 
 lowing manner. For this purpose provide an instrument, such as a German flute, 
 the cylinder of air in which can be lengthened or shortened by moving the joints closer 
 to or farther from each other; and have another so constructed, as to remain inva- 
 riable, and which ought to be preserved in the same temperature, such as 54 degrees 
 
wd 
 
 MUSIC AND MECHANICS. 393 - 
 
 of Fahrenheit’s thermometer. The first flute being at the same degree of tempera- 
 ture, bring them both into perfect unison, and then heat the first to 74° degrees of 
 Fahrenheit, which will necessarily communicate to the cylinder of air contained in it 
 the same degree of heat, and lengthen it by the quantity necessary to restore perfect 
 unison: it is evident that if this elongation were divided into twenty parts, each 
 of them would represent the quantity by which the flute ought to be lengthened for 
 each degree of Fahrenheit’s thermometer. 
 
 But it may be readily conceived that the quantity of this elongation, which at 
 most would be but a few lines, could not be divided into so many parts; and there- 
 fore it ought to be executed by the motion of a screw, that is to say one of the joints 
 of the instrument should be screwed into the other ; for it would then be easy to make 
 this elongation correspond to a whole revolution, and hence it might be divided into 
 a great number of equal parts. 
 
 By these means the opera at Lima, if required, where the heat frequently rises to 
 110° of Fahrenheit’s thermometer, might be made to have exactly the same pitch or 
 tone asat Paris. But this is sufficient on’a subject the utility of which would not be 
 worth the trouble necessary for attaining to such a degree of precision. 
 
 ARTICLE XIII. 
 Singular application of Music to a question in Mechanics. 
 
 This question was formerly proposed by Borelli; and though we do not think that 
 it can at present be a subject of controversy, it has occasioned some difference of 
 opinion among a certain class of mechanicians. 
 
 Fasten a string at one end to a fixed point; and having stretched it over a kind 
 of bridge, suspend from it a weight, such as 10 pounds for example. 
 
 Now, if instead of ‘the fixed point, which maintains the string in its place in 
 opposition to the action of the weight, a weight equal to the former be substituted, 
 will the string in both cases be equally stretched ? 
 
 We have no doubt that every well informed mechanician will readily~ believe that 
 in both cases the tension will be the same ; and this necessarily follows from the prin- 
 ciple of equality between action and re-action. According to this principle, the im- 
 moveable point, which in the first case counteracts the weight suspended from the 
 other end of the string, opposes to it a resistance exactly equal to the action which 
 it exercises: if a weight equal to the former be therefore substituted instead of the 
 fixed point, every thing remains equal in regard to the tension experienced by the 
 parts of the string, and which tends to separate them. 
 
 But music furnishes us with a method of proving this truth to the reason, by means 
 of the sense of hearing; for as the tone is not altered while the tension remains the 
 same, nothing is necessary but to make the following experiment. Take two strings 
 of the same metal, and the same size, and having fastened one of them by one end to 
 a fixed point, stretch it over a bridge, so as to intercept between it and the fixed 
 
 * point a determinate length, such as a foot for example; and suspend from the other 
 
 end of it a given weight, such as ten pounds. Then extend the second string over 
 two bridges, a foot distant from each other, and suspend from each extremity of it a 
 weight of ten pounds; if the tone of these two strings be the same, there will be 
 reason to conclude that the tension also is the same. We do not know whether this 
 experiment was ever made; but we will venture to assert that it will decide in 
 favour of equality of tension. 
 
 This ingenious application of music to mechanics, is the invention of Diderot, who 
 proposed it in his ‘*‘ Mémoires sur differentes sujets de Mathematique et de Physique,” 
 printed at Paris, in octavo, in the year 1748. 
 
 aa 
 
394 ' ACOUSTICS. 
 
 ARTICLE XIV. 
 
 Some singular considerations in regard to the Flats and Sharps, and to their progression 
 on their different tones. 
 
 Those in the least acquainted with music know that, according to the different keys 
 employed in modulation, a certain number of sharps or flats are required ; because in 
 the major mode, the diatonic scale, with whatever tone we begin, must be similar 
 to that of ut, which is the simplest of all, as it has neither sharp nor flat. These 
 flats or sharps have a singular progress, which deserves to be observed; it is even 
 susceptible of a sort of analysis, and as we may say algebraic calculation. 
 
 To give some idea of it, we shall first remark, that a flat may and ought to be con- 
 sidered as a negative sharp, since its effect is to lower the note a semi-tone ; whereas 
 the sharp raises it the same quantity. This consideration alone may serve to deter- 
 mine all the sharps and flats of the different tones. 
 
 It may be readily seen that when a melody in ut major is raised a fifth, or 
 brought to the tone of sol, asharp is required on the fa. It may therefore be thence 
 concluded, that this modulation, lowered a fifth or brought to fa, will require a 
 flat ; and indeed one is required on the si. 
 
 It hence follows also, that if the air be raised another fifth, that is to say to re, one 
 sharp more will be required ; and this is the reason why two are necessary. But to 
 ‘raise two fifths, and then descend an octave to approach the primitive tone, is to 
 rise only one tone; consequently to raise the air one tone, two sharps must be added. 
 The tone of re indeed requires two sharps, and for the same reason the tone of mi 
 requires four. . 
 
 The tone of fa requires one flat, and that of mi requires four sharps; therefore 
 when an air is raised a semi-tone, five flats must be added; for, a flat being a negative 
 sharp, it is evident that such a number of flats must be added to the four sharps of 
 mi, as shall efface these four sharps, and leave one flat remaining; which cannot be 
 done but by five flats; for, according to the language of analysis, — 52 must be added 
 to 4x, to leave asremainder — x. For the same reason, if the modulation be lowered 
 a semi-tone, five sharps must be added: thus, as the tone of ut has neither sharps. 
 nor flats, five sharps will be found necessary for sz, which is indeed the case. If the 
 modulation be still lowered a tone, to be in la, we must add two flats, in the same 
 manner as two sharpsare added when we rise a tone. But five sharps plus two flats, 
 is the same thing as five sharps minus two sharps, or three sharps. We still find 
 therefore, by this method, that the tone of Ja requires three sharps. 
 
 But, before we proceed farther, it will be necessary to observe, that all the chro- 
 matic tones, that is to say all those inserted between the tones of the natural diatonic 
 scale, may be considered as sharps or flats; for it is evident that ut or re» are the 
 same thing. It is very singular however, that according as this note is considered an 
 inferior one affected by a sharp, or a superior one affected by a flat, the number of 
 sharps required by the tone of the first, ut #, for example, and that of the flats re- 
 quired by the tone of the second, re», always make 12; which evidently arises from 
 the division of the octave into 12 semi-tones: therefore, since re b, as above shewn, 
 requires five flats, if, instead of this tone, we consider it as ut, seven sharps will be 
 required ; but for the facility of execution it is much better, in the present case, to 
 consider this tone as re b, than ut §. 
 
 This change therefore ought always to be made when the number of the sharps 
 exceeds six; so that, since ten sharps, for example, would be found in the tone of 
 la =, we must call it sib, and we shall have for that tone two flats, because two 
 flats are the complement of ten sharps. On the other hand, in following the pro- 
 gression of the semi-tones descending, if we should find a greater number of sharps 
 than 12, we ought to reject 12, and the remainder will be that of the tone proposed: 
 
SHARPS AND FLATS. 395 
 
 for example, as wt has neither sharp nor flat, we have five sharps for the lower tone 
 si; ten sharps for the semi-tone below la#; fifteen sharps for the still lower semi- 
 tone /a:; if twelve sharps therefore be rejected, there will remain three, which are 
 indeed the number of sharps necessary in the tone of A—mi—la. 
 
 The tone of sol § ought to have 8 or 4 flats, if we call it Jab, ’ 
 
 The tone of sol will have 13 sharps, from which if 12 be deducted, one sharp will 
 remain, as is well known. 
 
 The tone of fa will have 6 sharps, or 6 flats, if we call it solb. 
 
 The tone fa ought to have 6 flats plus 5 sharps; that is to say, ] flat, as the 5 
 sharps destroy the same number of flats. 
 
 That of mi will have | flat plus 5 sharps; that is, 4 sharps, as the flat destroys one of 
 them. 
 
 That of re will have 9 sharps or 3 flats, if it be considered as mib. 
 
 That of re will have 14 sharps; that is to say 2, by rejecting 12, or $ flats plus 
 5 sharps = 2 sharps. 
 
 That of ut will have 7 sharps, or 5 flats, if we call it reb, 
 
 In the last place, the tone ut natural will have 12 sharps; that is to say none, or - 
 5 flats plus 5 sharps, which destroy each other. 
 
 The very same results would be obtained in ascending by semi-tone after semi- 
 tone from ut, and adding 5 flats for each; taking care to reject 12 when they 
 exceed that number. Our readers, by way of amusement, may make the calcula- 
 tion. By calculating the number of the semi-tones, either ascending or descending, 
 we might in like manner find that of the sharps or flats of any tone given. 
 
 Let us take, for example, that of fa: from ué ascending there are six semi- 
 tones, and six times 5 flats makes 30 flats; from which if we deduct 24, a multiple 
 of 12, the remainder will be 6: solb therefore will have 6 flats. 
 
 - The same fa is 6 tones lower than ut; consequently there must be six times 5 or 
 30 sharps; from which if 24 be deducted, 6 sharps will remain, as we have found 
 by another method. 
 
 The tone of sol is 5 semi-tones lower than ut; consequently there must be five 
 times 5, or 25 sharps; from which if 24 be deducted, there will remain only one 
 sharp. 
 
 As the same tone is 7 semi-tones higher than wt, there must be seven times 5 or 
 35 flats; from which if 24 be deducted, the remainder will be 11 flats, that is to say 
 one sharp. 
 
 This progression appeared to us so curious as to be worthy of this notice; but 
 in order that it may be exhibited under a clearer and more favourable point of view, 
 we shall form it into a table, which will at any rate be useful to those who are 
 beginning to play on the harpsichord. For this purpose we shall present each 
 chromatic note as flattened or sharpened, and on the left of the former we shall 
 mark the sharps it requires, and the flats on the right of the latter. 
 
 0 sharp ut* 0 flats. 1 sharp sol * 
 7 sharps ut # © or reb* 5 flats. 8 sharps solf or la * 4 flats. 
 2 sharps re* 3 sharps la* 
 9 sharps re# or mib 3 flats. 10 sharps laf or — szb* Q flats. 
 4 sharps mi * 5 sharps 924 
 11 sharps fas 1 flat. 0 sharp ut* 0 flat. 
 
 6 sharps fa or solb* 6 flats. 
 
 Of these tones, we have marked those usually employed with a*; for it may be 
 easily conceived, that by employing re { under this form, we should have 9 sharps, 
 which would give two notes with double sharps, viz. fa $#, ut $H ; so that the gamut 
 
re 
 oe any 
 
 396 ACOUSTICS. 
 
 woud be ref, mit or fa, fade or sol, solf, lah, si% or ut, utZE or re, regs 
 which it would be exceedingly difficult to execute: but by taking mib, instead ¢ 
 re #, we have only 3 flats, which renders the gamut much simpler, as. it then become. 
 mib, fa, sal, lab, sif, ut, re, mib. 
 
 We are almost inclined to ask pardon of our readers for having amused them wit) 
 this frivolous speculation; but we hope the title of our work will plead our excuse 
 
 ee 
 
 ARTICLE XV. 
 
 | 
 
 Method of improving Barrel-instruments, and of making them fit to execute airs ofa 
 every kind. 
 
 The mechanism of that instrument called the barrel organ, is well known. It con! 
 sists of a great number of pipes, graduated according to the tones and semi-tone/ 
 of the octave, or at least those semi-tones which the progress of modulation it 
 general requires. But these pipes never sound except when the wind of a bellows! 
 kept in continual action, is made to penetrate to them by means of a valve. Thi 
 valve is shut by a spring, and opened when necessary by a small lever, raised by 
 spikes implanted in a wooden cylinder, which is put in motion bya crank. The cran] 
 serves also to move the bellows, which must continually furnish the air, destine: 
 to produce the different sounds by its introduction into the pipes. 
 
 But in order that the subject of this article may be properly comprehended, it wil 
 be necessary that the reader should have a perfect idea of the manner in which thy 
 notes are arranged on the cylinder. 
 
 The different small levers, which must be raised to produce the different tones 
 being placed at a certain distance from each other, that of half an inch for example. 
 circular lines are traced out at that distance on the cylinder. One of these lines i 
 intended for receiving the spikes that produce the sound wf, the next for those tha’ 
 sound wt #, the next for those that give ve, and soon. ‘There are as many lines 0 
 this kind as there are pipes; but it may be easily conceived that the duration of ar 
 air or tune can not exceed one revolution of the cylinder. : 
 
 Let us suppose then that the air consists of twelve measures. Each of these cir. 
 cumferences is divided into twelve equal parts at least, by twelve lines drawr 
 parallel to the axis of the cylinder ; and if we suppose that the shortest note of the ai 
 is a quaver, and that the air isin triple time, denoted by 3, each interval must be di: 
 vided into six equal portions; because, in this case, a measure will contain six quavers, 
 Let us now suppose that the first notes of the air are la, ut, si, re, ut, mi, re, &c., al 
 equal notes, and all simple crotchets. At the beginning of the line for receiving the 
 la, and of the first measure, a spike must be placed of such a construction as t¢ 
 keep raised up during the third of a measure the small lever that makes the Ja sound; 
 then, in the line destined for the ut, at the end of the second division or beginning of the 
 third,a spike similar to the first must be fixed in the cylinder; and in the line destined 
 for the sz, another of the same kind must be placed: it is evident that, when the 
 cylinder begins toturn, the first spike will make Ja sound during the third of a mea- 
 sure. The second, as the first third of the measure is elapsed, will catch the lever 
 and make ut sound; and the third will in like manner make st sound during the ss 
 third. The instrument therefore will say da, ut, si, &c. 
 
 If, instead of three crotchets, there were six quavers, which in this measure are the 
 first long, the second short, the third long, and so on alternately, which are called 
 dotted quavers, it may be easily perceived, that, after the spikes of the first, third, and 
 fifth notes have been fixed in the respective places of the division where they ought 
 to be, nothing will be necessary but to take care that the spike of the first quaver, 
 which in this time ought to be equal to a quaver and a half, shall have its head con- 
 structed in such manner as to raise the lever during one part and a half of the six 
 
> 
 
 BARREL INSTRUMENTS. 397 
 
 divisions into which the measure is divided; which may be done by giving it a tail 
 
 behind of the necessary length. In regard to the short quavers, the spikes repre- 
 senting them ought to be removed back half a division, and to be formed in sucha 
 manner as to keep the lever corresponding to them raised up only during the revolu- 
 tion of a semi-division of the cylinder. By these examples it may be easily seen what 
 must be done in the other eases, that is, when the notes have other values. 
 
 Were the cylinder immoveable in the direction of its axis, only one air could be 
 performed; but as the spikes move the small levers merely by touching them beneath 
 in a very narrow space, such as the breadth of a line at most, which is a mechanism 
 that may be easily conceived, it will be readily seen that, by giving to the cylinder 
 the small lateral motion of a line, none of the spikes can communicate motion to the 
 levers. Another line therefore, to receive spikes arranged so as to produce a diffe- 
 rent air, may be drawn close to each of the first set of lines, and the number of the 
 different sets of lines may be six or seven, according to the interval between the first 
 lines, which is the same as that between the middle of one pipe and the middle of the 
 neighbouring one: by these means, if the cylinder be moved a little in the direction 
 of its axis, the air may be changed. 
 
 Such is the mechanism of the hand or barrel organ, and other instruments con- 
 structed on the same principle; but it may be easily seen that they are attended with 
 this inconvenience, that they can perform only a very small number of airs. But as 
 aseries of five, six, eight, or a dozen of tunes, is soon exhausted, it might be a 
 matter of some importance to discover a method by which they might be cnanged at 
 pleasure. 
 
 We agree in opinion with Diderot, who has given some observations on this subject, 
 in the work above quoted, that this purpose might be answered by constructing the 
 cylinder in the following manner. Let it be composed of a piece of solid wood, 
 covered with avery hard cushion, and let the whole be pushed into a bollow cylinder, 
 of about a line in thickness, On this inner cylinder draw the lines destined to re- 
 ceive the spikes, placed at the proper intervals for producing the different tones ; 
 and let holes be pierced in these lines at certain distances, six for example in each 
 division of the measure if it be triple time, or eight in the measure if it be common 
 time, denoted by c: we here suppose that no air is to be set that has shorter notes 
 than plain quavers. Twelve holes per measure will be required in the first case, 
 and sixteen in the second, if the air contains semi-quavers. 
 
 It may now be readily conceived that ona cylinder of this kind any air whatever might 
 be set; nothing will be necessary for this purpose, but to thrust into the holes of the 
 
 exterior cylinder spikes of the proper length, taking care to arrange them as above 
 explained ; they will be sufficiently firm in their places in consequence of the elasticity 
 of the cushion*, strongly compressed between the inner cylinder and the hollow outer 
 one. When the air is to be changed, the spikes may be drawn out, and put intoa 
 box divided into small cells, in the same manner as printing types when distributed 
 in the cases. The interior cylinder may then be made to revolve a little, in order to 
 separate the holes in the cushion from those in the exterior cylinder, and a new air 
 may then be set with as much facility as the former. 
 
 We shall not examine, with Diderot, all the advantages of such an instrument, be- 
 cause it must be allowed that it never can be of much utility, and will have no value | 
 in the eyes of the musician. It is however certain that it would be agreeable, for 
 those who possess such instruments, to be able to give more variety to the airs they 
 are capable of performing; and this end would be answered by the construction here 
 described. 
 
 * Might not cork be employed instead of the cushion here proposed ? 
 
398 ACOUSTICS. 
 
 ARTICLE XVI. 
 
 Of some Musical Instruments, or Machines, remarkable for their singularity of | 
 construction. 
 
 At the head of all these musical’ instruments, or machines, we ought doubtless to| 
 place the organ; the extent and variety of the tones of which would excite much) 
 more admiration, were it not so common in our churches; for, besides the artifice | 
 necessary to produce the tones by means of keys, what ingenuity must have been. 
 required to contrive mechanism for giving that variety of character to the tones, which | 
 is obtained by means of the different stops, such as those called the voice stop, flute 
 stop, &c. ? A complete description therefore of an organ, and of its construction, | 
 would be sufficient to occupy a large volume. : 
 
 The ancients had hydraulic organs, that is, organs the sound of which was occa-_ 
 sioned by air produced by the motion of water. These machines were invented by 
 Ctesibius of Alexandria, and his scholar Hero. From the description of these hydrauli¢ 
 organs given by Vitruvius, in the tenth book of his Architecture, Perrault con- 
 structed one which he deposited in the King’s Library, where the Royal Academy of 
 Sciences held their sittings. This instrument indeed is not to be compared to the 
 modern organs; but it is evident that the mechanism of it has served as a basis for 
 that of ours, St. Jerome speaks with enthusiasm of an organ which had twelve pair 
 of bellows, and which could be heard at the distance of a mile. It thence appears 
 that the method employed by Ctesibius, to produce air to fill the wind box, was soon 
 laid aside, for one more simple; that is, for a pair of bellows. 
 
 The performer on the tambour de basque, and the automaton flute-player of Vau- 
 canson, which were exhibited and seen with admiration in most parts of Europe, in 
 the year 1749, may be classed among the most curious musical machines ever in- 
 vented. We shall not however say any thing of the former of these machines, be- 
 cause the latter appears to have been far more complex. 
 
 The automaton flute-player performed several airs on the flute, with the precision 
 and correctness of the most expert musician. It held the flute in the usual manner, 
 and produced the tone by means of its mouth; while its fingers, applied on the 
 holes, produced the different notes. It is well known how the fingers might be 
 raised by spikes fixed in a cylinder, so as to produce these sounds; but it is difficult 
 to conceive how that part could be executed which is performed by the tongue, and 
 without which the music would be very defective. Vaucanson indeed confesses that 
 this motion in his machine was that which cost him the greatest labour. Those 
 desirous of farther information on this subject may consult a small work, in quarto, 
 which Vaucanson published respecting these machines. 
 
 A very convenient instrument for composers was invented some years ago in 
 Germany: it consists of a harpsichord which, by certain machinery added to it, 
 notes down any air or piece of music, while a person is playing it. This is a great 
 advantage to composers, as it enables them, when hurried away by the fervour 
 of their imagination, to preserve what has successively received from their fingers a 
 fleeting existence, and what otherwise it would often be impossible for them to re- 
 member. A description of this machine may be found in the Memoirs of the Aca- 
 demy of Berlin, for the year 1773. 
 
 ARTICLE XVII. 
 Of a new instrument called the Harmonica. 
 
 This new instrument was invented in America, by Dr. Franklin, who gave a 
 description of it to Father Beccaria, which the latter published in his works, printed 
 in 1773. 
 
 It is well known that when the finger, a little moistened, is rubbed against the 
 
THE HARMONICA. 399 
 
 edge of a drinking glass, a sweet sound is produced; and that the tone varies 
 
 according to the form, size, and thickness of the glass. The tone may be raised or 
 lowered also by putting into the glass a greater or less quantity of water. Dr. 
 Franklin says that an Irishman, named Puckeridge, first conceived the idea, about 
 twenty years before that time, of constructing an instrument with several glasses 
 of this kind, adjusted to the various tones, and fixed to a stand in such a manner, that 
 different airs could be played upon them. Mr. Puckeridge having been afterwards 
 burnt in his house along, with this instrument, Mr. Delaval constructed another 
 of the same kind, with glasses better chosen, which he applied to the like purpose. 
 Dr. Franklin, hearing this instrument, was so delighted with the sweetness of its 
 tones, that he endeavoured to improve it; and the result of his researches was the 
 instrument which we are now going to describe. 
 
 Cause to be blown on purpose glasses of different sizes, and of a form nearly 
 hemispherical, having each in the middle an open neck. The thickness of the glass, 
 
 ~ near the edge, should be at most one tenth of an inch, and ought to increase gradually 
 
 to the neck, which in the largest glasses should be an inch in length, and an inch and 
 a half in breadth in the inside. In regard to the dimensions of the glasses them- 
 selves, the largest may be about nine inches in diameter at the mouth, and the least 
 three inches, ‘each glass decreasing in size a quarter of an inch. It will be proper to 
 have five or six of the same diameter, in order that they may be more easily tuned to 
 the proper tones; for a very slight difference will be sufficient to make them vary 
 a tone, and even a third. 
 
 When these arrangements are made, try the different glasses, in order to form 
 
 of them a series of three or four chromatic octaves. To elevate the tone, the edge 
 
 towards the neck ought to be ground, trying them every moment, for if they be raised 
 
 _too high, it will afterwards be impossible to lower them. 
 
 When the glasses have been thus graduated, they must be arranged on a common 
 axis. For this purpose, put a cork stopper very closely into the neck of each, so as 
 to project from it about half an inch; then make a hole of a proper size in all these 
 corks, and thrust into them an iron axis, but not with too much force, otherwise the 
 necks might burst. Care must also be taken to place the glasses in such a manner, 
 that their edges may be about an inch distant from each other, which is nearly the 
 distance between the middle of the keys of a harpsichord. 
 
 To one of the extremities of the axis affix a wheel of about eighteen inches in 
 diameter, loaded with a weight of from twenty to twenty-five pounds, that it may 
 retain for some time the motion communicated to it. This wheel, which must be 
 
 turned by the same mechanism as that employed to turn a spinning wheel, commu- 
 
 nicates, as it revolves, its motion to the axis, which rests in two collars, one at the 
 extremity, and the other at some distance from the wheel. The whole may be fitted 
 into a box of the proper form, placed on a trame supported by four feet. The glasses 
 corresponding to the seven tones of the diatonic octave, may be painted of the seven 
 prismatic colours in their natural order, that the different tones to which they cor- 
 respond may be more readily distinguished. 
 
 The person who plays on this instrument, is seated before the tow of glasses, as 
 if before the keys of a harpsichord; the glasses are slightly moistened, and the 
 wheel being made to revolve, communicates the same motion to the glasses: the 
 fingers are then applied to the edges of the glasses, and the different sounds are by 
 these means produced. It may be easily seen that different parts can be executed 
 with this instrument, as with the harpsichord. 
 
 About fourteen or fifteen years ago, an English lady at Paris performed, it is said, 
 exceedingly well on this instrument. The sounds it emits are remarkably sweet, 
 and would be very proper as an accompaniment to certain tender and pathetic airs. 
 It is attended with one advantage, which is, that the sounds can be maintained of 
 
 *e 
 
400 ACOUSTICS. P| 
 prolonged, and made to swell at pleasure ; and the instrument, when once tuned, 
 never requires to be altered. It afforded great satisfaction to many amateurs ; but 
 we have heard that the sound, on account of its great sweetness, became at last 
 somewhat insipid, and for this reason perhaps it is now laid aside, and confined to 
 cabinets, among other musical curiosities. ; 
 
 A few years ago, Dr. Chladni, who has made various researches respecting the 
 theory of sound, and the vibrations of sonorous bodies,* invented a new kind of in- 
 strument of this kind, to which he gave the name of euphon. This instrument has 
 some resemblance to a small writing desk, and contains in the inside 40 glass tubes 
 of different colours, of the thickness of the barrel of a quill, and about sixteen 
 inches in length. They are wetted with water by means of a sponge, and stroked 
 with the fingers in the direction of their length; so that the increase of the tone 
 depends merely on the stronger or weaker pressure, and the slower or quicker move- 
 ment of the fingers. In the back part there is a perpendicular sounding board, 
 through which the tubes pass. In sweetness of sound, this instrument approaches: 
 near to harmonica; but seems to be attended with advantages which the other does 
 not possess. . 
 
 Ist. Itis simpler, both in regard to its construction, and the movement necessary 
 to produce the sound; as neither turning nor stopping is required, but merely the 
 motion of the finger. 
 
 2d. It produces its sound more speedily; so that as soon as touched the tone may 
 be made as full as the instrument is capable of giving it: whereas in the harmonica 
 the tones, and particularly the lower ones, must be made to increase gradually. 
 
 3d. It has more distinctness in quick passages, because the tones do not resound 
 so long as in the harmonica, where the sound of one low tone is often heard when 
 you wish only to hear the following one. 
 
 4th. The unison is purer than is generally the case in the harmonica; where it Is 
 difficult to have perfect glasses, which in every part give like tones with mathemati- 
 cal exactness. Itis however as difficult to be tuned as the harmonica. 
 
 Sth. It does not affect the nerves of the performer; for a person scarcely fecls a 
 weak agitation in the fingers; whereas in the harmonica, particularly in concords of 
 the lower notes, the agitation extends to the arms, and even through the whole body 
 of the performer. 
 
 6th, The expense of this instrument will be much less than that of the harmonica, 
 
 7th. When one of the tubes breaks, or any other part is deranged, it can be easily 
 repaired: whereas when one of the glasses of the harmonica breaks, it requires 
 much time, and is difficult to procure another capable of giving the same tone as 
 the former, and which will correspond sufficiently with the rest. 
 
 For farther particulars respecting this instrument, and the history of its invention, 
 see ‘‘ The Philosophical Magazine,’’ No. 8, or vol. ii. p. 391. 
 
 ARTICLE XVIII. 
 Of some singular ideas in regard to Music. 
 
 Ist. One perhaps would scarcely believe it possible for a person to compose 
 an air, though entirely ignorant of music, or at least of composition. This secret, 
 however, was published a few years ago in a small work ,entitled “Le Jeu Dez 
 Harmonique,” or ‘ Ludus Melothedicus,”’ containing various calculations, by means 
 of which any person, even ignorant of music, may compose minuets, with the accoiw- 
 paniment of a bass. 8vo. Paris 1757. In this work, the author shows howa minuet 
 
 * He published a work on this subject entitled, ‘ Entdeckungen uber die Theorie des Klanges.” 
 Leipsic, 1787. 4to 
 
FIGURES IN SAND ON VIBRATING PLATES. 401 
 
 and its bass may be Seiniiowed according to the points thrown with two dice, by 
 means of certain tables. 
 
 This author gives a method also of performing the same thing by means of a pack 
 of cards. We do not remember the title of this work; and we confess that we 
 ought to attach no more importance to it than the author does himself. 
 
 We shall therefore content ourselves with having mentioned works to which the 
 reader may have recourse for information respecting this kind of amusement, the 
 combination of which must have cost more labour than the subject deserved. We 
 shall however observe, that this author published: another work, entitled ‘‘ Invention 
 d’une Manufacture et Fabrique de Vers au petit metier,” &c. Svo. 1759, in which 
 he taught a method of answering, in Latin verse, by means of two dice and cer- 
 tain tables, any question proposed. This, it must be confessed, was expending 
 much labour to little purpose. 
 
 2d. A physician of Lorrain, some years ago, published a small treatise, in which he 
 employed music in Hetoriunioy the state of the pulse. He represented the beats of 
 -a regular pulse by minuet time, and those of the other kinds of -pulse by different mea- 
 sures, more or less accelerated. If this method of medical practice should be intro- 
 duced, it will be a curious spectacle to see a disciple of Hippocrates feeling the pulse 
 of his patient by the sound of an instrument, and trying airs analogous by their time 
 to the motion of his pulse, in order to discover its quality. If all other diseases 
 should baffle the physician’s skill, there is reason to believe that low spirits will not 
 be able to withstand such a practice. 
 
 ARTICLE XIX. 
 On the nies Sormed by Sand and other light substances on vibrating surfaces. 
 
 Dr. Chladni of Wittenberg, by his experiments on vibrating surfaces, published 
 in 1787, opened a new field in this department of science, viz., the consideration of the 
 curves formed by sand and other light bodies, on surfaces put into astate of motion. 
 As this subject is curious, and seems worthy of farther research, we shall present the 
 reader with a few observations on the method of repeating these experiments, 
 taken from Gren’s Journal of Natural Philosophy, vol. iii.* 
 
 Vibration figures, as they are called, are produced on vibrating surfaces, because 
 some parts of these surfaces are at rest, and others in motion. The surfaces fittest for 
 being made to vibrate, are panes of glass; though the experiments will succeed equally ~ 
 well with plates of metal, or pieces of board, a line or two in thickness. If the surface 
 of any of these bodies be strewed over with substances easily put in motion; such for 
 example as fine sand; these, during the vibration of the body, will remain on the 
 parts at rest, and be ‘thrown from the parts in motion, so as to form mathematical 
 figures. To produce such figures, nothing is necessary but to know the method of 
 bringing that part of the surface, which you wish not to vibrate,into a state of 
 rest ; and of putting in motion that which you wish to vibrate; on this depends the 
 whole expertness of producing vibration figures. 
 
 ~ “Fig. 4. Those who have never tried these experiments might 
 imagine, that to produce Fig. 4. it would be necessary 
 to damp, in particular, every point of the part to be kept at 
 rest, viz., the two concentric circles and the diameter, and to 
 put in motion every part intended to vibrate. This however 
 is not the case; for you need damp only the points a and 4, 
 and cause to vibrate one part c, at the edge of the plate; 
 for the motion is soon communicated to the other parts, 
 
 * See also Phil. Mag. No. 12. 
 2D 
 
402 
 
 ACOUSTICS. 
 
 which you wish to vibrate, and the required figure will in this manner be 
 
 produced. 
 
 The damping Ps be best effected by laying hold of the place to be danipel be- 
 
 Fig. 5. 
 
 end, brought into contact with the glass in such a manner, as to supply the place ~ © 
 
 tween two fingers, or by supporting it only by one finger. 
 This will be more clearly comprehended by turning to 
 Fig. 5., where the hand is represented in that, position 
 necessary to hold the plate. In order to produce Fig. 6. 
 you must hold the plate horizontally, placing the thumb 
 above at a, with the second finger directly below it; and 
 besides this, you must support the point 6 on the under 
 side of the plate. If the bow of a violin be then rubbed 
 against the plate at c, you will produce on the glass the 
 figure which is delineated Fig. 6. When the point to be 
 supported or damped lies too near the centre of the 
 plate, you may rest it on acork, not too broad at the 
 
 of the finger. It is convenient also, when you wish to damp several points at the 
 
 Fig. 7. 
 di 
 q 
 Fig: 8. 
 
 Fig. 11. 
 
 circumference of the glass, to place your thumb on the cork, 
 and to use the rest of your fingers for touching the parts which 
 you wish to keep at rest. For example, if-you wish to produce 
 Fig. 7. on an elliptic plate, the larger axis of which is to the 
 less as 4 to 8, you must place the cork under ¢, the centre of the 
 plate; put your thumb upon this point, and then damp the two 
 points of the edge p and q, as may be seen Fig. 8., and make 
 the plate to vibrate by rubbing the violin bow against it at r. 
 There is still another convenient method of damping several 
 points at the edge, when large plates are employed. Fig. 9. re- 
 presents a strong square bit of metal a 8, a line in circumference, 
 which is screwed to the edge of the table, or made fast in any 
 other manner; and a notch, about as broad as the edge of the 
 plate, is cut into one side of it witha file. You then hold the 
 plate resting against this bit of metal, by two or more fingers 
 when requisite, as at c and d; by which means the edge of the 
 plate will be damped in three points d, c, e ; and in this manner, 
 by putting the vibration at f, you can produce Fig. 10. In 
 cases of necessity you may use the edge of a table, instead 
 of the bit of metal; but it will not answer the purpose so 
 well. 
 
 To produce the vibration at any required place, a common 
 violin bow, rubbed with rosin, is the most proper instrument to 
 be employed. The hair must not be too slack, because it.is — 
 sometimes necessary to press pretty hard on the plate, in order 
 to produce the tone sooner. 
 
 When you wish to produce any particular figure, you must 
 first form it in idea on the plate, in order that you may be able 
 to determine where a line at rest, and where a vibrating part, 
 will occur. The greatest rest will always be where two or 
 more lines intersect each other, and such places must in par- 
 ticular be damped. For example, in Fig. 11. you must damp 
 the part x,-and stroke with the bow in p. Fig. 12. may be 
 produced with no less ease, if you hold the plate at r, and 
 stroke with the bow at f. The strongest vibration seems 
 
FIGURES IN SAND ON VIBRATING PLATES. 403 
 
 Fig. 12. - always to be in that part of the edge which is bounded by a 
 curve: for example, in Fig, 8. and Fig. 2, at n. To produce 
 these figures, therefore, you must rub with the bow at 7, and 
 
 s not at r. 
 You must however damp, not only those points where two 
 lines intersect each other; but endeavour to support at least one 
 ee which is suited to that figure, and to no other. For example, 
 f when you support a and A, Fig. 4., and rub with the bow at e, 
 Fig. 13. 
 
 Fig. 11. also may be produced ; because both these figures have 
 these two points at rest. To produce Fig.4., you must support 
 Z with one finger the part e, and rub with the bow inc; but Fig. 
 
 11. cannot be produced in this manner, because it has not the 
 
 Fig. 14, point e at rest. 
 A One of the greatest difficulties in producing the figures, is to 
 determine before-hand the vibrating and resting points which 
 (O\ belong to a certain figure, and to noother. Hence, when one is 
 = , “ not able to damp those points which distinguish one figure from 
 
 another, if the violin bow be rubbed against the plate, several 
 
 hollow tones are heard, without the sand forming itself as ex- 
 pected. You must therefore acquire by experience a readiness, in being able to search 
 out among these tones that which belongs to the required figure, and to produce it 
 on the plate by rubbing the bow against it. When you have acquired sufficient 
 expertness in this respect, you can determine before-hand, with a considerable degree 
 of certainty, the figures to be produced, and even the most difficult. It may be 
 easily conceived, that you must not forget what part of the plate, and in what man- 
 
 ner, you damped ; and you may mark these points by making a scratch on the plate 
 
 with a bit of flint. cio : 
 When the plate has acquired the proper vibration, you must endeavour to keep it 
 
 in that state for some seconds; which can be best done by rubbing the bow against 
 it several times in succession. By these means the sand will be formed much more 
 accurately. 3 : : 
 
 Any sort of glass may be employed for these experiments, provided its surface be 
 smooth; otherwise the sand will fall into the hollow parts, or be thrown about 
 in an irregular manner. Common glass plates, when cut with a stone, are very sharp 
 on the edge, and would soon destroy the hair of the violin bow: on this account 
 the edge must be rendered somewhat smooth, by means of a file, or a piece of coarse 
 hard free-stone. , 2 ; 
 
 You must endeavour to procure such plates as are pretty uniform in thickness ; and 
 you ought to have them of different sizes; such as circular ones of from four to 
 twelve inches in diameter. You must not employ sand too fine, but rather that 
 which is somewhat coarse. The plate must be equally strewed with it, and not too 
 thick ; as the lines will then be exceedingly fine, and the figures will acquire a better 
 
 defined appearance. 
 
 2n2 
 
404 ASTRONOMY AND GEOGRAPHY, 
 
 PART SIXTH. 
 
 CONTAINING THE EASIEST AND MOST CURIOUS PROBLEMS, AS WELL AS 
 THE MOST INTERESTING TRUTHS, IN ASTRONOMY AND GEOGRAPHY, 
 BOTH MATHEMATICAL AND PHYSICAL. 
 
 Or all the parts of the mathematics, none are better calculated to excite curiosity 
 than astronomy and its different branches. Nothing indeed can be a stronger proof 
 of the power and dignity of the human mind, than its having been able to raise 
 itself to such abstract knowledge as to discover the causes of the phenomena 
 exhibited by the revolution of the heavenly bodies; the real construction of the 
 universe ; the respective distances of the bodies which compose it, &c, At all times 
 therefore this study has been considered as one of the sublimest efforts of genius ; and 
 Ovid himself, though a poet, never expresses his thoughts on this subject but with 
 a sort of enthusiasm. Thus, when speaking of the erect posture of man, he says: 
 
 Cunctaque cum spectent animalia cetera terram, 
 Os homini sublime dedit, coelumque tueri 
 
 Jussit, et erectos in sidera tollere vultus. 
 Metamorph. Lib. I. 
 
 In another place, speaking of astronomers, he says: 
 
 Felices animze! quibus hzc cognoscere primis, 
 Inque domos superas scandere cura fuit. 
 
 Credibile est illos pariter vitiisque, jocisque, 
 Altius humanis exeruisse caput. 
 
 Non venus aut vinum sublimia pectora fregit, 
 Officiamve fori, militizeve labor ; 
 
 Nec levis ambitio perfusaque gloria fuco, 
 Magnarumve fames sollicitavit opum. 
 
 Admovere oculis distantia sidera nostris, 
 /Etheraque ingenio supposuere suo. 
 
 If astronomy at that period excited admiration, what ought it not do at present, 
 when the knowledge of this science is far more extensive and certain than that of the 
 ancients, who, as we may say, were acquainted only with the rudiments of it! How 
 great would have been the enthusiasm of the poet, how sublime his expressions, had he 
 foreseen only a part of the discoveries which the sagacity of the moderns has enabled 
 them to make with the assistance of the telescope! The moons which surround Jupi- 
 terand Saturn; the singular ring that accompanies the latter; the rotation of the sun and 
 planets around their axes ; the various motions of the earth; its immense distance from 
 the sun; the still more incredible distance of the fixed stars ; the regular course of the 
 comets ; the discovery of new planets and comets ; and in the last place, the arrange- 
 ment of all the celestial bodies, and their laws of motion, now as fully demonstrated as 
 the truths of geometry. With much more reason would he have called those who 
 have ascended to these astronomical truths, and who have placed them beyond all 
 doubt, privileged beings, and of an order superior to human nature. 
 
t 
 
 MERIDIAN LINE. 405 
 
 CHAPTER. I. 
 
 ELEMENTARY PROBLEMS IN ASTRONOMY AND GEOGRAPHY. 
 
 PROBLEM I, 
 To find the Meridian Line of any place. 
 
 The determination of the meridian line, is certainly the basis of every operation, 
 both in astronomy and geography; for which reason we shall make it the first 
 problem relating to this subject. 
 
 There are several methods of determining this line, which we shall here describe. 
 
 I.—On any horizontal plane, fix obliquely, and in a firm manner, a spike or sharp 
 pointed piece of iron, with the point uppermost, as a B, Fig 1. Then provide a 
 double square, that is to say, two squares joined together so as to form an angle, 
 and by its means find, on the horizontal plane, the point c, corresponding in a perpen- 
 dicular direction with the summit of the style. From this point 
 describe several concentric circles, and mark, in the forenoon, 
 where the summit of the shadow touches them. Do the same 
 thing in the afternoon; and the two points D and £ being thus 
 determined in the same circle, divide into two equal parts the ~ 
 arc intercepted between them. If a straight line be then 
 drawn through the centre, and this point of bisection, it will 
 be the meridian line required. 
 
 By taking two points in one of the other circles, and re- 
 peating the same operation ; if the two lines coincide, it will 
 be a proof, or at least afford a strong presumption, that the 
 operation has been accurately performed; if they do not coincide, some error must 
 have arisen; and therefore it will be necessary to recommence the operation with 
 more care. 
 
 Two observations, the least distant from noon, ought in general to be preferred ; 
 both because the sun is then more brilliant and the shadow better defined, and be- 
 - cause the change in the sun’s declination is less; for this operation supposes that 
 the sun neither recedes from nor approaches to the equator, at least in a sensible 
 manner, during the interval between the two observations. : 
 
 In short, provided these two observations have been made between 9 o’clock in 
 the morning and 8 in the afternoon, even if the sun be near the equator, the meridian 
 found by this method will be sufficiently exact, in the latitude of from 45 to 60 degrees; 
 for we have found that in the latitude of Paris, and making the most unfavourable 
 suppositions, the quantity which such a meridian may err will not be above 20”. If 
 it be required with perfect exactness, nothing is necessary but to make choice of a 
 time when the sun is either in one of the tropics, particularly that of Cancer, or very 
 near it, so that in the interval between the two operations his declination may not 
 have sensibly changed. 
 
 We are well aware that, for the nice purposes of astronomy, something more 
 precise will be necessary ; but the object of this work is merely to give the simplest 
 and most curious operations in this science. The following however is a second 
 method of finding the meridian by means of the pole star. 
 
 Il.—To determine the meridian line in this manner, it will be necessary to wait 
 till the pole star, which we here suppose to be known, has reached the meridian. 
 But this will be the case when that star and the first in the tail of the Great Bear, 
 or the one nearest the square of the constellation, are together in the same line 
 perpendicular to the horizon; for about the year 1700 these two stars passed over 
 
406 ASTRONOMY AND GEOGRAPHY. 
 
 the meridian exactly at the same time; so that when the star in the Great Bear was 
 below the pole, the polar star was above it; but though this is not precisely the case 
 at present, these stars, as we shall here shew, may be still employed in obtaining an 
 approximate meridian. 
 
 Having suspended a plumb line in a motionless state, wait till the pole star, and 
 that in the Great Bear above described, are together concealed by the thread; and 
 at that moment suspend a second plumb line, in such a manner that it shall hide the 
 former and the two stars. These two threads will then comprehend between them 
 a plane which will be that of the meridian ; and if the two points on the ground, cor- 
 responding to the extremities of the two plumb lines, be joined by a straight line, 
 you will have the direction of the meridian. 
 
 The time at which the pole star, or any other star, passes the meridian on a given 
 day, may be found in the following manner : 
 
 Subtract the right ascension of the sun from the right ascension of the star (in- 
 creased by 24 hours if necessary), and the remainder is the apparent time of the 
 star’s passing the meridian. The sun’s right ascension, as well as that of any star 
 likely to be used in the ordinary procedure of astronomy, may be feund in the 
 Nautical Almanac, or White’s Ephemeris, one of which no English astronomer will 
 be without. : 
 
 To trace out a meridian line by means of the pole star, find as above the time 
 when the star is on the meridian; and about six hours before that time, the star will 
 be at its greatest elongation east of the meridian: and about six hours after at its 
 greatest elongation west of the meridian. 
 
 This greatest elongation may be found by adding together the log. cosine of the 
 declination, and the log. secant of the latitude, and the sum (rejecting 10) is the 
 log. sine of the greatest elongation. 
 
 Thus on Feb. 10th, 1889, in lat. 51° 29 N., to find the greatest elongation of the 
 pole star one hour:— 
 
 Declination 88° 27’ 24” cosine 8:430279 
 Latitude 51° 29 0” secant 10:205692 
 
 Required elongation 2° 28’ 43” sine 18°635971 
 
 Now for some time, about the greatest elongation, the azimuth does not sensibly 
 
 change ; there is therefore sufficient time to note from a given place the direction — 
 
 of the star at the time, and to draw on the ground a line in that direction. 
 
 From this line let an angle be laid off equal to the greatest elongation computed 
 as above; towards the west of the star is east of the meridian, but towards the east 
 of the star is west of the meridian; and the line forming this angle with the line 
 drawn in the direction of the star, will be the meridian line. 
 
 But the object may be effected with much more neatness by a theodolite ; for, 
 having adjusted the instrument, and directed the telescope to the star at the time 
 of its greatest elongation, it is only requisite to turn round the instrument in 
 
 azimuth 2° 28’ 43”, or such other angle as may result from the computation, and then 
 
 any object on the horizon bisected by the cross wires of the telescope will be ina 
 meridian line from the place of the instrument. 
 
 PROBLEM It.. 
 To find the Latitude of any place. 
 
 The latitude of any place on. the earth is its distance from the equator; and is 
 measured by an arc of the celestial meridian, intercepted between the zenith of the 
 place and the equator; for this arc is similar to that comprehended on the earth 
 between the place and the terrestrial equator. This is equal to the elevation of the 
 
LATITUDE AND LONGITUDE. 407 
 
 pole, which is the arc of the meridian intercepted between the pole and the horizon. 
 To those therefore who live under the equator, the poles are in the horizon; and 
 if there were inhabitants at either pole, the equator would be in their horizon. 
 
 The latitude of any place on the earth may be easily found by various methods. 
 
 Ist. By the meridian altitude of the sun on any given day. For if the sun’s decli- 
 nation for that day, when the sun is in any of the northern signs, and the given 
 place in the northern hemisphere, be subtracted from the altitude, the remainder 
 will be the elevation of the equator, the complement of which is the elevation of the 
 pole, or the latitude. If the sun be in any of the southern signs, it may be readily seen 
 that, to find the elevation of the equator, the declination must be added. 
 
 2d. If the meridian altitude of one of the circum-polar stars, which do not set, 
 be taken twice in the course of the same night, namely,once when directly above the 
 pole, and again when exactly below it; and if from each of these altitudes the refrac- 
 tion be subtracted; the mean between these two altitudes will be that of the pole, 
 or the latitude. Or, take any two altitudes of such a star at the interval of 11? 58™ 
 of time, correcting them by subtracting the refractions as before; then the mean be- 
 tween them will be the height of the pole, or the latitude of the place. 
 
 3d. Look, in some catalogue of the fixed stars, for the distance of any star from 
 the equator, that is to say its declination; then take its meridian altitude, and by 
 adding or subtracting the declination, you will have the elevation of the equator, the 
 complement of which, as before said, is the latitude. 
 
 PROBLEM III. 
 To find the Longitude of any place on the earth. 
 
 The longitude of any place, or the second element of its geographical position, is 
 the distance of its meridian froma certain meridian, which by;common consent is con- 
 sidered as the first. This first meridian was formerly supposed to be that passing 
 through the island of Ferro, the most eastern of the Canaries. But the meridian of 
 the observatory of Paris is now used by the French, and that of the Royal Observa- 
 tory of Greenwich by the English. 
 
 Formerly the longitude was reckoned, from west to east, throughout the whole 
 circumference of the equator; but at present-it is almost the general practice to 
 reckon both ways from the first meridian, or the meridian accounted as such; that is 
 to say east and west, so that the longitude according to this method can never 
 exceed 180 degrees: and in the tables it is marked whether it be east or west. 
 We shall now proceed to shew in what manner the longitude is determined. 
 
 If two terrestrial meridians, distant from each other 15°, for example, be supposed 
 to be continued to the heavens; it is evident that they will intercept, in the equator 
 and all its parallels, arcs of the same number of degrees. It may be readily seen 
 also that the sun will arrive first at the more eastern meridian, and that he will 
 then have to pass over !5° in the equator, or the parallel which he describes that day 
 during his diurnal rotation, before he arrives at the more western meridian. But to 
 pass over 15° the sun requires one hour, since he employs 24 hours to pass over 360° ; 
 
 x 
 
 hence it follows, that when it is noon at the more eastern places it will be only 11 
 
 o’clock in the morning at the more western. If the distance of the meridians of the 
 two places be greater or less, the difference of the hours will be greater or less, in 
 the proportion of one hour for 15°; and consequently of 4 minutes for a degree, t 
 seconds for a minute, and so on. 
 
 Thus it is seen, that to determine the longitude of a place, nothing is necessary 
 but to know what hour itis there, when it is a certain hour in another place situated 
 under the first meridian, or the distance of which from the first meridian is known; 
 for if this difference of time be changed into degrees and parts of a degree, allowing 
 
A408 _ ASTRONOMY AND GEOGRAPHY. 
 
 15° for one hour of time, one degree for 4 minutes, and so on, then the longitude of 
 the proposed place will be obtained. 
 
 To find this difference of hours, the usual method is to employ the observation 
 of some celestial phenomenon that happens exactly at the same moment to every 
 place on the earth, such for example as eclipses of the moon. Two observers sta- 
 tioned at two places, the difference of the longitude of which is required, observe, 
 by means of a well-regulated clock, the moments when the shadow successively 
 reaches several remarkable spots on the moon’s disc; they then compare their ob- 
 servations, and by the difference of the time which they reckoned when the shadow 
 reached the same spot, they determine, as above explained, the difference of the lon- 
 gitude of the two places. 
 
 Let us suppose, by way of example, that an observer at London pater - obser- 
 vation, that the shadow reached the spot called Tycho at lh. 45m, 50s. in the 
 morning; and that another stationed at a place a madea similar observation at 24m. 
 30s. after midnight: the difference of this time is lh. 21m. 20s., which, reduced to 
 degrees and minutes of the equator, gives 20° 20’. This is the difference of longi- 
 tude: and as it was later at London when the phenomenon was observed, than at 
 the place a, it thence follows that the place a is situated 20° 20’ farther west than 
 London. 
 
 As eclipses of the moon are very rare, and as it is difficult to observe with preci- 
 sion when the shadow comes into contact with the moon’s disc, so as to determine 
 the commencement of the eclipse, and also the exact period when the shadow reaches 
 any particular spot, the modern astronomers make use of the immersions, that is to 
 say the eclipses, of Jupiter’s satellites, and particularly those of the first, which, as 
 it moves very fast, experiences frequent eclipses that end in a few seconds. The - 
 case is the same with the emersion or return of light to the satellite, which takes 
 place very quickly. For the sake of illustration we shall suppose that an observer, 
 stationed at the place a, observes an immersion of the first satellite to have hap- 
 pened on a certain day at 4h. 55m. in the morning ; and another stationed at a place 
 B at 3h. 25m. The difference being lh. 30m. it gives 22° 30’ for the difference of 
 longitude. We may therefore conclude that the place a is farther to the east than 
 B, since the inhabitants at the former reckoned an hour more at the time of the phe- 
 nomenon. 
 
 Remark.—These observations of the satellites, which, since the discovery of those 
 of Jupiter, have been often repeated in every part of the globe, have in some 
 measure made an entire reformation in geography; for the position in longitude of 
 almost all places was determined merely by itinerary distances very incorrectly mea- 
 sured; so that in general the longitudes were counted much greater than they 
 really were. Towards the end of the seventeenth century there were more than 
 25° to be cut off from the longitude assigned to the old continent from the. western 
 ocean to the eastern coast of Asia. 
 
 This method, so evident and demonstrative, was however criticised by the celebrated 
 Isaac Vossius, who preferred the itinerary results of travellers, or the estimated dis- 
 tances of navigators; but by this he only proved that, though he possessed a great 
 deal of erudition badly digested, he had a weak judgment, and was totally unac- 
 quainted with the elements of astronomy. 
 
 A knowledge of the latitude and longitude of the different places of the earth, is 
 of so much importance to astronomers, geographers, &c., that we think it our date 
 to give a table of those of the principal places of the earth. This table, which is 
 very extensive, contains the position of the most considerable towns both in England 
 and in France, as well as of the greater part of the capitals and remarkable places in 
 
’ LATITUDES AND LONGITUDES. 409 
 
 every quarter of the globe; the whole founded on the latest astronomical observa- 
 tions, and the best combinations of distances and positions. 
 
 The reader must observe, that the longitude is reckoned from the meridian of 
 Greenwich, both east and west. When east it is denoted by the letter ©, and when 
 west by the letter w. In regard to the latitude, it is distinguished, in the same 
 manner, by the letters N and s, which denote north and south. 
 
 A TABLE, . 
 
 CONTAINING THE LATITUDES AND LONGITUDES OF THE CHIEF TOWNS AND MOST 
 REMARKABLE PLACES OF THE EARTH. 
 
 “Names of Places. Lat. Long. Names of Places. Lat. .| Long. 
 Abbeville, France 50° 7’N| 1°50’ £ || Awatcha;Kamtschatka |52° 52’ N/158° 47’ EB 
 Aberdeen, Scotland 57 9N| 2 6w || Azoph, Crimea 47 ON| 39 14 5 
 Abo, Finland 60 27 Nj 22 17 E || Bagdad, Mesopotamia |33 20 N| 44 25 & 
 Acapulco, America 16 50 N| 99 49 w || Bahama l., America |26 43 n| 78 56w 
 Acheen, Sumatra 5 36 N| 95 19 E || Baldivia, Chili 39 50 s| 73 34Ww 
 Adrianople, Turkey [41 3 Nj 27 8 E|| Bale, Switzerland 47 34N| 7 35 E 
 Agra, India 27 13 N| 78 17 £ || Bangalore, India L2E5S Nie 
 Aleppo, Syria 36 11 N| 37 10 E || Bantry Bay, Ireland [51 34 nj 10 10 w 
 Alexandretta, Syria |36 45 nN} 36 15 E || Barcelona, Spain 41 22n| 2108 
 Alexandria, Egypt 31 13 N} 29.55 §E || Bassora, Arabia 30 32 N| 44 46 £ 
 Algiers, Algiers 36 49 N| 3 5 E || Batavia, Java I. 6 9 s/l106 52 E 
 Alicant, Spain 38 12 N| 0 29w || Bayeux, France 49 17 Nj 042W 
 Altona, Germany 53 33 N| 9 57 E || Bayonne, France 43 29 n| 1 28 Ww 
 Altorf, Germany 47 45. N| 9 34 E || Beechy Head, England |50 44 nj 015 & 
 
 - Amiens, France 49 54N| 2 18 £ || Belfast, Ireland 54 35 ni 5 57 WwW 
 Amboyna I. India 3 40 s|128 15 £ || Bencoolen, Sumatra I.| 3 48 s|102 O8 
 Amsterdam, Holland [52 22 N; 4 53 E | Belgrade, Turkey 44 43 nN} 20 10 E 
 Anabona I. Ethiopia 1 25 s| 5 45 & || Bender, Turkey 46 51 N| 29 46 E 
 Ancona, Italy 43 38 N| 13 29 E || Bergen Castle, Norway |60 24N| 5 20 E 
 Andrew’s St., Scotland|56 20 Nj 2 50w || Berlin, Germany 52 32N| 13 22 F 
 Angers, France 47 28 Ni 0 28w || Bermuda, Bahama I. 32 22Nn) 64 30 Ww 
 Angouleme, France 45 39N| O 9 £ || Berne, Switzerland 46 57N| 7 26 & 
 _Anapolis Royal, Nova/44 45 nN} 65 46 w || Berwick Tweed, Eng. |55 46n| 2 OW 
 
 Scotia Besancon, France 4714N| 6 3E 
 Antigua, Caribbee 17 4N| 61 55 w | Bezieres, France 43 21N) 3138 
 Antibes, France 43 35 N| 7 8 E || Bilboa, Spain 43 26n| 3 18W 
 Antiochetta, Syria 36 6N/ 32 20 & || Blois, France 47 35N] 1208 
 Antwerp, Flanders 5113N| 4 24 £]! Bologna, Italy 44 30N| 11 215 
 Archangel, Russia 64 34 nN} 40 43 E || Bolkereskoy, Kamts- |52 54 N/156 50 E 
 Arcot, India 12 54N| 79 22 E chatka 
 Arles, France 43 41 N} 4 38 E || Bombay, India 18 54N] 72 50 E 
 Arras, France 50 18 N| 2 46 E || Borneo, Borneo I. 4 55 N\114 55 & 
 Ascension I., Brazil 7 57 s| 13 59 w || Boston, England 53 10N| 0 258 
 Astracan, Siberia 46 21 nN} 48 8 E |! Boston, America 42 22x] 70 59 w 
 Athens, Greece 37 58N/ 23 46 x |} Botany Bay,N. Holland/34 0 s/151 14 £ 
 Auch, France 43 39 N| 0O 40 E || Boulogne, France 50 44 N] 137 & 
 Augustine St., Florida |29 48 N} 81 18 w || Bourdeaux, France 44 50N|} 0 34W 
 Augsburg, Germany |48 22 N| 10 55 £ || Bourges, France 47 5N| 2248 
 Avignon, France 43 57 N| 4 48 E || Bremen, Germany 53 5N| 848 EB 
 Avranches, France 48 41 N} 1 21 Ww || Breslaw, Silesia 51 ON] 17) 2k 
 Aurillac, France 44 55n| 2 32 E |} Brest, France 48 23 N| 4 29w 
 Auxerre, France 47 48 NI 3 34 E || Bridge Town, Barbad. 113 5 N! 59 41 w 
 
410 
 
 Names of Places. 
 
 Bristol Cathedral, Eng.|51° 27’ n| 2° 35’w 
 
 51 
 50 
 
 Bruges, Flanders 
 Brussels, Flanders 
 
 Buchan-ness, Scotland |57 
 Bucharest, Wallachia |44 
 Buda, Turkey 47 
 Buenos Ayres, Brazil |34 
 
 Cadiz Observ., Spain |36 
 
 Caen, France 49 
 Caffa, Crimea 45 
 Cagliari, Sardinia 39 
 Cairo, Egypt 30 
 Calais, France 50 
 Calcutta, India 22 
 Calicut, India 11 
 Callao, Peru 12 
 Camboida, India 10 
 Cambray, France 50 
 Cambridge, England {52 
 Canary I., Canaries 28 
 Candy, Ceylon 7 
 Canterbury, England 51 
 Cape Comorin, India | 8 
 
 Cape Finisterre, Spain |42 
 
 Cape Francois,-St. Do-|19 
 mingo 
 
 Cape Town, Caffraria |33 
 
 Cape Ortegal, B. of Bis.|43 
 
 C. St. Lucas, California/22 
 
 Cape Verd, Negroland |14 
 
 Caracas, S. America |10 
 Carcassone, France 43 
 Carlescrona, Sweden {56 
 Carlisle, England 54 
 Carthagena, Spain 37 
 
 Carthagena, S. America]10 
 
 Casan, Russia 55 
 Cassel, Germany 51 
 Castres, France 43 
 Cayenne I.,S.America | 4 
 
 Cephalonia 1., Turkey |38 
 Cette Light H., France|43 
 Ceuta, Barbary 35 
 Chalons-sur-Marne, Fr.|48 
 ChAalons-sur-Saone, Fr.|46 
 Chandernagor, Bengal |22 
 
 Charlestown Light, {32 
 Carolina 
 
 Chartres, France 48 
 
 Cherbourg, France 49 
 
 Chester, England 53 
 
 Christiana, Norway [59 
 
 Christianstadt, Sweden|56 
 Civita Vecchia, Italy |42 
 Clagenfurth, Carinhia |46 
 Clermont-Ferrand,Fr. |45 
 
 Cochin, India 9 
 Colchester, England {51 
 Collioure, France 42 
 Cologne, Germany 50 
 Compiegne, France 49 
 
 Lat. 
 
 13 
 
 36 
 
 eA AOL AAA A A Ae Ze Se A Beg eee Oh 
 
 43 
 N 
 N 
 
 N 
 55 N 
 1N 
 5 N 
 37 N 
 47 N 
 57 .N 
 53 -N 
 32 N 
 55 N 
 25 N 
 
 y day Aa Ay A Wey Ae say Ae Ae Oe a Se ee ee ee <= 
 
 Long. 
 
 14 £ 
 22 E 
 
 54 
 5 
 55 
 2 54 E}! 
 
 ASTRONOMY AND GEOGRAPHY. 
 
 — 
 
 Names of Places. Lat. Long. 
 Conception la, Chili |36°49’s| 73 5w 
 Congo R. Ent., Congo| 6 10 sj 11 15 £ 
 Constance, Switzerland|47 36 N| 9 8 E 
 Constantinople, Turkey|41 1N| 28 55 E 
 Copenhagen, Denmark |55 41 nN| 12 40 & 
 Cordova, Spain 37 52N| 4 46Ww 
 Corfu, Vido L, Turkey |39 38 N| 19 56 E 
 Corinth, Greece 37 58N| 23 28 & 
 Cork, Ireland 51 52 Nil 8 16 We 
 Coutances, France 49 3N! 1 26W 
 Cowes, Isle of Wight [50 45 nN} 1 19W 
 Cracow, Poland 50 4N|\ 19 57 & 
 Cremsmunster, Germ. [48 3N| 14 8 E 
 Cuddalore, India 11 43. N| 79 48 E 
 Curasgoa, West Indies [12 8 N| 69 OW 
 Dabul, India 17 45 N| 72 6308 
 Dantzic, Poland 54 21 N\ 18 388 
 Dartmouth, England (50 17 Nx) 3 35 Ww 
 Deseada, Caribbees 16 20 N} 61° 2Ww 
 Dieppe, France 49 56Ni 1 58 
 Dijon, France 47°19 .N| 95-2 
 Dillengen, Germany [48 34 N| 10 30 & 
 Dol, France © 44 33 N| 1 45 W 
 Dole, France 47 7N| 5 308 
 Domingo St., Antilles |18 30 nj 69 49 w 
 Dordrecht, Netherlands|51 49 nN} 4 40 B 
 Dover, England 51 8N! 119E 
 Dresden, Saxony 51 3 Nn 13 43 B 
 Drontheim, Norway /|63 26 N/ 10 23 & 
 Dublin Obs., Ireland (53 23x] 6 20 Ww 
 Dunbar, Scotland 55 58 nil 2 36W 
 Dundee, Scotland 56 25Ni 3° 2% 
 Dungeness, England [50 55 N| O 58 E 
 Dunkirk, France 51 2n| 2238 
 Durazzo, Turkey 41 19 N} 19 27 
 Edinburgh Obs., Scot. |55 57 N|} 3 11 Ww 
 Elba, P. Torrey, Italy|42 49 Nj 10 20 E- 
 Elbing, Poland 54 8N| 19 228 
 Elsinore, Denmark 56 2Nn| 12 388 
 Embden, Germany 53°22 nl 7:12 
 Enchuysen, Holland (52 42 N 5 18 E 
 Ephesus, Natolia 37 50 N| 27 37 EB 
 Erfurth, Germany 50 59 N| 11 2E 
 Erivan, Armenia 40 20 N| 44 35 ED 
 Erzerum, Armenia 39 57 Ni 48 36 EB 
 Eustatia, Caribbees 17 20 N| 63 5 Ww 
 Faenza, Italy 44 17 Nj 1] 218 
 Falmouth, Pend. Cas.,|50 8N; 5 2w 
 Fernambouc, Brazil 8 3s} 34 54wWw- 
 Ferrara, Italy 44 50 N| 11 368 
 Ferro I., Canaries 27-50 N| 17 58 Ww 
 Finisterre C., France |42 54 nN] 9 16W 
 Fladstrandt, Denmark |57 27 n| 10 33 BE 
 Florence, Italy 43 47 nN) 11 16 E 
 Flushing, Holland 51 27 -Ni 33am 
 Forbisher’s Straits, (62 5 N| 47 18 W 
 
 Greenland 
 Formoso I. N. p. China/25 11 n/121 56 & 
 —S.p. E. do./21 54N121 58 
 Frankfort on the Mayn,/50 7N| 8 36 & 
 
 Germany 
 
LATITUDES AND LONGITUDES. 
 
 _ —. 
 
 Names of Places. 
 
 Frejus, France 
 Gallipoli, Turkey 
 Gambia R. mouth, 
 Negroland 
 Geneva, Switzerland 
 Genoa, Italy 
 Ghent, Netherlands 
 Gibraltar, Spain 
 Glasgow, Scotland 
 Gloucester Cath. Eng. 
 Gluckstadt, Holstein ~ 
 ~ Goa, India 
 Gombroon, Persia 
 Good Hope C., Africa 
 Gottenburg, Sweden 
 Gottingen, Germany 
 Granville, France 
 Gratz, Styria 
 Greenwich, England 
 Grenoble, France 
 Guadaloupe, Caribbee 
 Guernsey, St. P. Eng. 
 Hague, Holland 
 Halifax, Nova Scotia 
 Halle, Saxony 
 Hamburgh, Germany 
 Harlem, Holland 
 Harwich, England 
 Hastings, England © 
 Havannah, Cuba I., 
 
 Helena St. I., Africa 
 Holy Head, Wales 
 
 Hull, England 
 Hydrabad, India 
 Jafnapatan,C.CeylonI 
 Jago, St. Cape Verd I. | 
 Jamaica, Kingston, W.L. 
 Jassey, Moldavia 
 Java Head, Java I. 
 Jeddo, Japan 
 Jena, Germany 
 
 _ Jersey I. St. Aubin, Eng. 
 
 | Jerusalem, Palestine 
 
 _ Jeniseik, Russ. Tartary 
 Ingolstadt, Germany 
 Inspruc, Tyrol 
 Inverness, Scotland 
 
 _ Joppa, Syria 
 
 | Ipswich, England 
 
 _ Ismail, Turkey 
 
 _ Ispahan, Persia 
 Juan Fernandez J.,Chili/33 
 Judda, Arabia 21 
 Ivica I., Spain 38 
 Kilda St. I., Scotland [57 
 Kinsale, Ireland 51 
 
 14° 
 
 Long. 
 
 33’ 5 
 
 1lE 
 44 E 
 
 Names of Places. 
 
 Konigsberg, Prussia 
 Lancaster Steeple, Eng.|54 
 Landau, France 49 
 Land’s End, England |50 
 Landscrona, Sweden |55 
 
 Langres, France 47 
 Lausanne, Switzerland |46 
 Leeds, England 53 
 Leghorn, Italy 43 
 Leipsic, Germany 51 
 Leostoff, England 52 
 Lepanto, Turkey 38 
 Leyden, Holland 52 
 
 Liverpool, St. Paul’s, [53 
 England 
 
 Liege, Germany 50 
 Lima, Peru 12 
 Limerick, Ireland 52 
 
 Lisbon Obs., Portugal |38 
 Lizard Light, England |49 
 London, St. Paul’s Ch. [51 
 Londonderry, Ireland 154 
 Loretto, Italy 43 
 Louisburg, Cape Breton |45 
 Louvain, Netherlands [50 
 
 Lubeck, Germany 53 
 Lucia, St. I. Caribbee |13 
 Lucca, Italy 43 
 
 Lunden Tower, Sweden|55 
 Luxembourg, Netherlds./49 
 
 Lynn, Old Tower, Eng.|52 
 Macao, China 22 
 Macassar, Celebes I. 5 
 Madras, India 13 
 Madrid, Spain 40 
 Madura, India 9 
 Mahon Port, Minorca |39 
 Malacca Fort, India 2 
 
 Malta I., Val. Obs.,Italy|35 
 Manchester, England |53 
 Manilla, Luconia I. 14 
 Mantua, Italy 45 
 Marseilles Obs., France|43 
 Martinico I., F., Royal,|14 
 West Indies 
 Masulipatam, India {16 
 Mauritius I.,Pt. Lewis,/20 
 Africa 
 
 Meaco, Japan 35 
 Meaux, France 43 
 Mecca, Arabia a1 
 Mechlin, Netherlands [51 
 Memel, Courland 55 
 
 Messina Light, Sicily {38 
 
 Metz, France 49 
 Mexico, Mexico 19 
 Milan Obs., Italy 45 
 Mocha, Arabia 13 
 Modena, Italy 44 
 Montpelier Obs., France/43 
 Montreal, Canada 45 
 
 Lat. 
 
 54° 42’n| 21° 
 
 3 N| 2 
 TZN SS 
 4N 
 52 N 
 52 N 
 31N 
 48N 
 
 —_ 
 on 
 
 wo 
 
 io) 
 
 Z 
 
 — 
 NHPnrw NOK DAY 
 
 a 
 or) 
 2 
 
 no 
 
 24 N|153 
 58 N 2 
 18 n| 40 
 2n| 4 
 42 N] 21 
 11 N| 15 
 7N| 6 
 26 N| 99 
 28 Ni 9 
 20 N| 43 
 34 nN} 11 
 36 N| 3 
 31 N) 73 
 
 411 
 
 Long. 
 
 29’ EB 
 48 W 
 
412 ASTRONOMY AND GEOGRAPHY. 
 
 : 2 
 —————— ee 
 
 Names of Places. lat. Long, Names of Places. Lat. Long. 
 
 Mosambique Harbour, |15° 1’ s| 40°47’ & || Quebec, Canada 46°47’ nN} 71°10’w. 
 Zangue Quiloa, Zanguebar 8 41 s| 39 47 E 
 Moscow, Russia 55 46 N| 37 33 £ || Quimper, France 47 58 Ni 4 6wW 
 Munich, Germany 48 34 E || Quito, Peru 013 Ss} 78 45 Wi 
 Munster, Germany 51 36 £ || Ragusa, Dalmatia. |42 39 N] 18 6 E, 
 Namur, Netherlands {50 51 § || Ramsgate, England 51 20 Nn} 1 25 | 
 Nangasaki, Japan oy) 52 E || Ratisbon, Germany 49 1N\ 12 4 at 
 Nankin, China 32 7 £ || Ravenna, Italy 44 25 N| 12 11 E| 
 Nantes, France 47 33 w || Rennes, France 48 7N| 1 41 WwW) 
 Naples, Italy 40 16 £ || Rheims, France 49 16N| 4 6 Ei 
 Narbonne, France ~ 43 0 £ || Revel, Livonia 59 27 N| 24 35 E 
 Narva, Livonia 59 14 £ || Riga, Livonia 56 57 N| 24 8 EB) 
 Naze, Norway 57 3 E || Rimini, Italy 44 4N| 12 33 E| 
 Negapatnam Port, Indiaj10 55 © || Rio Janeiro, Rat.1., }22 53 s| 43 12 w) 
 Nevis, 1.,S. Pt. Caribb.|17 33 Ww Brazil : 
 Newcastle, England [55 36 w || Rochelle, France 46 9N 
 
 Nice, Italy 
 Nieuport, Flanders 
 Nombre de Dios, S. A. 
 
 17 © || Rochester, England {51 23 N 
 45 E || Rome College, Italy 41 54N 
 35 w || Rotterdam, Holland |51 56N 
 37 w || Rouen, France 49 26N 
 1 £ || Rye, England 51 3N 
 4 £ || Saffia, Barbary 32 20N 
 14 £ || Saint-Flour, France |45 25 
 54 w || Saint-Malo, France 48 39N 
 9 E || Saint-Omer, France [50 45 N 
 
 — 
 
 OQRNNWHOOFPKRNOF 
 on 
 
 Ochotsk, Tartary 
 Olinda, Brazil 
 Olmutz, Moravia 
 
 Oneglia, Italy 43 4 Ej) Salerno, Italy 40 40 Ni} 14 35 E 
 Oporto Bar, Portugal {41 8 37 w || Sallee, Barbary 34 5N 43 W, 
 Oran, Barbary 3 39 w || Salonica, Turkey 40 38 Nj 22 56 BE 
 Orenburg, Astracan [51 5 E || Saragossa, Spain 41 38 N| 1 42w 
 Orleans, N. Louisiana |29 11 w || Scanderoon, Syria 36 35 N| 36 15 B 
 Orleans, France (47 55 E || Schamaki, Persia 40 27 n| 36 45 B 
 
 37 E || Scilly Isles, St. Mary’s,|49 54 N| 6 17 W, 
 55 EB England | 
 16 w || Selinginsk, Russ. Tart./51 6 N/106 39 E 
 52 © || Senegal R. ent. Negro./15 53 Nj 16 31 Ww 
 22 £ || Senlis, France 49 12 N| 2 35 EB 
 50 £ || Sens, France 48 12N| 317 E 
 41 w || Seringapatam, India |12 25 | 76 42 EB 
 27 w || Seville, Spain 37 24.N}| 5 38w 
 40 w || Sheerness Staff, Eng. |5111N 0 448 
 20 £ || Siam, India 14 21 N/100 50 E 
 
 Ostend, Flanders 
 Oxford, England 
 Padua Obs., Italy 
 Palermo Obs. Sicily 
 Palicaud, India 
 Pampeluna, Spain 
 Panama, Mexico 
 Para, South America 
 Paris Obs. France 
 
 Parma, Italy 44 27 £ | Sienna, Italy 43 22 N| 11 10 E 
 Passau, Germany 48 25 © || Sierra Leone, Guinea 8 31 N} 13 18 W 
 Patmos I., Natolia a7 40 £ || Shields, England 55 2N| 1 20 Wi 
 Pavia, Italy 45 10 E || Skalpolt, Iceland 64 ON| 16 OW 
 Pegu, India un 12 E£ || Smyrna, Natolia 38 25 N| 27 68 
 Pekin Obs., China 39 28 E || Socotra I., Africa 12 30N| 54 10 E 
 Perpignan, France 42 54 E || Soissons, France 49 23N| 3 208 
 Petersburgh, Russia {59 19 E || Southampton Spire, En./50 54 N, 1 24W 
 Philadelphia, America |39 11 w || Spoletto, Italy 42 45 N] 12 36 E 
 
 Pico I. Peak, Azores 
 Pisa Obs., Italy 43 
 Plymouth, N. ch. Eng./50 
 Pondicherry, India j11 
 Port Mahon, Minorca I./39 
 Porto Bello, New Spain| 9 
 Port Royal, Jamaica |17 
 Port Royal, Martinico |14 
 Portsmouth Obs., Eng.|!50 
 Prague, Bohemia 50 
 Presburg, Hungary 48 
 
 33 w || Start Point, England [50 13 N} 3 38 W 
 24 £ || Stettin, Pomerania 53 26N| 14 46 E 
 
 7 w || Stockholm, Sweden 59 21 N| 18 35 
 54 £ || Stockton, England 54 34 N] 1 16W 
 18 E || Stralsund, Germany [54 19 N| 13 328 
 43 w || Strasburgh, France 48 34N| 7 51E 
 52 w || Stromness, Orkneys (58 56N} 3 31 W 
 6 w || Stuttgard, Germany [48 46 Nn} 9 11 E 
 6 w |} Sunderland, England [54 55 Nn} 1 15 W 
 25 £ || Surat, India 21 4N/ 72 518 
 11 £ |) Surinam,ent.S.America’ 6 25 N] 55 40 W 
 
DIFFERENCE OF HOURS. 413 
 Names of Places. | Lat. Long Names of Places. Lat. Long. 
 le ansea, Wales bi 37’n| 3° 56’w || Valparaiso, Chili 33° O' NI 71°38’w 
 Syracuse Light, Sicily [37 3 N, 15 16 B Vannes, France 47 39 Ni 2 45 w 
 Tangier, Barbary 35 42 Ni} 5 50 w/|! Venice St Mark’s,Italy/45 26 Ni 12 21 & 
 Tarento, Italy 40 35 N| 17 29 & || Vera Cruz, New Spain [19 12 nN] 96 9w 
 Tauris, Persia 38 10 N| 46 37 £ |} Verona Obs. Italy 45 26n|/ 11 18 
 Tefflis, Georgia 41 43 nN; 62 40 £ || Versailles, France 48 48nl 2 7 E 
 Tellicherry, India 11 44N) 75 36 E Vienna, Germany 48 13 N| 16 23 EB 
 ‘Temeswar, Hungary (44 47 nj 29 0 £ || Vigo, Spain 42 13 NN) 8 33W 
 “Teneriff Peak, Canaries|28 17 n| 16 40 w || Vilna, Poland 54 41 N| 25 17 E 
 -Tetuan, Barbary 35 50 Ni 5 20w || Upsal, Sweden 59 52 N' 17 39 B 
 “Tinmouth. England. [55 3N/ 1 18w |} Uraniburg, Denmark |55 55 N| 12 43 & 
 ‘Thessalonica, Greece |48 38 N| 22 56 & || Urbino, Italy 43 44 n| 12 37 EB 
 ‘Tobago I., N.E. point,|11 10 n| 60 27 w || Wardhus, Lapland 70 23 N/ 31 7 & 
 _ Caribbees Warsaw, Poland 52 14 Ni 21 3 8 
 -Tobolsk, Siberia 58 12 N| 68 68] Waterford, Ireland 52 13N} 7 10wW 
 Toledo, Spain 39 52 N| 4 11w || Wells, England 51 11 N, 10 12Ww 
 Tonsberg, Norway 59 23 Ni 10 12 £ |) Wexford Harb., Ireland}/52 22 N| 6 19 W 
 Torbay, England 50.26 Nn} 3 31 w || Weymouth, England |50 37 Nn] 2 22w 
 Tornea, Sweden 65 51 N) 24 12 E Whitby, England 54 28 Nj 0 36W 
 ‘Toulon, France ya peeks, N| 5 56 £ || Whitehaven, W. mill, E.|54 33 n| 3 35 Ww 
 Toulouse, France 43 36 N) 1 26 £ || WicklowLight, Ireland |52 38 n) 6 OW 
 ‘Tours, France 47 24n| 0 42 £ || Wittenberg, Saxony {51 53.N; 12 46 E 
 Trente, Italy 46 6N\ 11 4 £ || Wurtzburg, Franconia|49 46 N| 9 55 & 
 Trieste, Carniola 45 38 n| 13 47 BE || Wybourg, Finland 60 43 Nj 28 46 E 
 Trincomalee, Ceylon I.| 8 33 Nj 81 22 w Yarmouth, England 52 46 N| 1 41 EB 
 ‘Tripoli, Syria 34 26 N| 35 51 E || Yellow River, China (34 3N/120 0 & 
 Tripoli, Barbary 32 54 Nj 13 12 E || Ylo, Peru 17 36 s| 71 10w 
 ‘Truxilla, Peru 8 6s! 79 3w| York New, Bat. Amer.|40 42 nl 73 59 w 
 Tunis, Barbary 36 48 nN] 10 11 £ || Youghal, Ireland 51 55.N| 7 48W 
 ‘Turin, Piaz. Cast., Italy|45 4N| 7 40 & || Zagrab, Croatia 45 49Nn|\ 16 5E 
 Tyrmau, Hungary 48 23 N| 17 35 E|| Zante I. Town,Italy [37 47N| 20 55 5 
 Valencia, Spain 36 29 Nn} O 23 w || Zara, Dalmatia 44 2N,) 15 10 E 
 Valladolid, Spain 41 42 n\| 5 34w/|) Zurich, Switzerland ‘47 23 | SUSicR 
 PROBLEM Iv. 
 To find what o'clock it is at any place of the earth, when it is a certain hour at 
 
 | a another. 
 
 As the earth makes one revolution on its axis in the course of a com- 
 Mon day, or 24 hours, every point of the equator will describe the whole 
 eircle of 360 degrees in that period: and therefore if 360 be divided by 24, the 
 quotient 15 will be the number of degrees that correspond to one hour of time. 
 | Hence it is evident that two places which are 15 degrees of longitude distant from 
 ‘each other, will differ one hour in their computation of time, one of them making it 
 earlier or later according asit is situated to the east or west of theother. To deter- 
 mine this problem therefore, find by the preceding table the difference of longitude 
 of the two places, which may be done by subtracting the longitude of the one from 
 that of the other if they are both east or both west of Greenwich, or by adding them if 
 the one is east and the other west, and then change the sum or difference into time: 
 ‘this time added to or subtracted from the hour at one of the given places, will 
 give for result the hour at the other. If Greenwich be one of the places proposed, the 
 difference of longitude will be found in the last column to the right in the preceding 
 table. 
 
q 
 
 =. 
 
 414 ASTRONOMY AND GEOGRAPHY. 
 
 Multiply the degrees by 4 for minutes of time, and the miles by 4 for seconds of 
 time, or find the hours and minutes corresponding to the given degrees and minutes 
 in the subjoined table, which will greatly facilitate operations of this kind. 
 
 Now let it be proposed to find what o’clock it is at Cayenne, when it is noon at 
 London, The difference of longitude, or of meridians, between London and Cayenne, 
 is 52° 7’; which converted into time, gives 3 hours 28 minutes 28 seconds; and as 
 Cayenne lies to the west of London, if 3h. 28m. 28s. be subtracted from 12 hours, 
 the remainder will be 8 hours 31 minutes 32 seconds: hence it appears that when itis 
 noon at London, it is only 8h. 31m. 32s. in the morning at Cayenne; consequently 
 when it is noon at Cayenne, it is 3h, 28m. 28s. in the afternoon at London. 
 
 When it is noon at London, required the hour at Pekin? The difference of me- 
 ridians between London and Pekin is 116° 36’, which is equal in time to 7 hours 
 46 minutes 24 seconds. But as Pekin lies to the east of London, these 7h. 46m. 24s. 
 must be added to 12 hours: and hence it is evident that when it is noon at London, 
 it is 7h. 46m. 24s. in the evening at Pekin. On the other hand, to find what 
 o’clock it is at London when it is noon at Pekin, these 7h. 46m. 24s. must be sub- 
 tracted from 12 hours, and the result will be 4h. 13m. 36s. in the morning, 
 
 When the two given places are both to the west of London, to find their difference 
 of meridians, the longitude of the one must be subtracted from that of the other. If 
 Madrid and Mexico, for instance, be proposed; as the longitude of the first is 3° 42’, 
 and that of the second 99° 5’, if the former be subtracted from the latter, the re- 
 mainder 95° 23’ will be their difference of longitude, which, changed into time, gives 
 6 hours 2] minutes 32 seconds. Hence, when itis noon at Madrid, it is 5h. 38m. 28s. 
 in the morning at Mexico. 
 
 If one of the proposed places lies to the east and the other to the west of London, 
 the longitude of the one must be added to that of the other, in order to have their 
 difference of longitude ; and the sum must then be converted into time, and added or 
 subtracted as before. 
 
 By way of example we shall take Constantinople and Mexico, the former of which 
 lies to the east of London. The longitude of Constantinople is 28°55’, and that of 
 Mexico 99° 5’, which added give for difference.of longitude 171°= in time to 8h. 32m.) 
 When it is noon therefore at Constantinople, it is only 3h. 28m. in the morning at 
 Mexico; and when itis noon at the latter, it is 8h. 32m. in the evening at Con- 
 stantinople. | 
 
 | 
 
LENGTH OF DAYS. e415 
 
 A table for changing degrees and minutes into hours, minutes, and seconds; 
 or the contrary. 
 
 HM D | HM D | HM D HM 
 
 M M § M M §s M M S M MS M MSs 
 1 ora 37 2 28 73 4 52 109 7 16 145 9 40 
 2 0 8 38 2 32 74 4 56 110 7 20 146 9 44 
 3 0 12 39 2 36 75 5 0 ie 7594 147 9 48 
 4 0 16 40 2 40 76 5 4 112 7,28 148 9 52 
 5 0 20 41 2 44 77 Pee: 113 39 149 9 56 
 6 | O 24 42 2 48 78 5 12 114 7 36 150 | 10 O 
 7 0 28 43 2 52 79 5 16 115 7 40 To?) 077s 
 8 0 32 44 > 56 80 5 20 116 7 44 152 | 10 8 
 9 0 36 45 3440 81 5 24 117 7 48 15S elon) 2 
 10 0 40 46 3 4 82 5 28 118 7 52 154 | 10 16 
 11 0 44 47 Ss 83 5 32 119 7 56 155 | 10 20 
 12 0 48 48 3.12 84 5 36 120 8 0 156 | 10 24 
 13 0 52 49 3 16 85 5 40 121 8 4 157 {2 10V2s8 
 14 0 56 50 3 20 86 5 44 122 8 8 158 | 10 32 
 15 10 51 3 24 87 5 48 123 8 12 159 | 10 36 
 16 wees 52 3 28 88 5 52 124 8 16 160 | 10 40 
 17 i 8 53 3 32 89 5 56 125 8 20 161 | 10 44 
 18 1212 54 3 36 90 6 0 126 8 24 162 | 10 48 
 19 1 16 55 3 40 91 6 4 ey 8 28 163 | 10 52 
 20 1 20 56 3 44 92 6 8 128 8 32 164 | 10 56 
 21 1 57 3.48 93 6 12 129 8 36 165 01196 
 22 1 58 3,52 94 6 16 130 8 40 1661 11° 4 
 Boi 59 3 56 95 6 20 131 8 44 167-1198 
 24 1 60 4 0 96 6 24 132 8 48 168 | 11 12 
 25 1 61 44 97 6 28 133 8 52 169 | 11 16 
 26 1 62 4 8 98 6 32 134 8 56 170 | 11 20 
 7 1 63 4 12 99 6 36 135 9 0 a | a oe 
 28 1 64 4 16 100 6 40 136 9 4 172) 1 298 
 29 1 65 4 20 101 6 44 137 97.8 L7SC tok by 32 
 | 30 2 66 4 24 | 102 6 48 138 9 12 174 | 11 36 
 31 2 67 4 28 103 6 52 139 9 16 175 } 71) 40 
 32 2 68 4 32 104 6 56 140 9 20 176°h 11 44@ 
 33 2 69 4 36 105 7 0 141 9 24 Lut Litas 
 34 2 70 4 40 106 ood. 142 9 28 178 | 11 52 
 35 2 71 444 # 107 Te. 8 143 9 32 179 | 11 56 
 36 2 72 448 # 108 ie ANG: 144. 9 36 180 | 12 0 
 
 ~ In the above table the narrow columns contain degrees or minutes, and the broad 
 ones hours and minutes, or minutes and seconds. Thus, if 4 in the first narrow column 
 represent degrees, the 16 opposite to it in the broad column will be minutes; and if 4 
 represent minutes, the 16 will be seconds. If it be required to change 4* 20’ into 
 time ; opposite to 4 will be found 16, which, in this case is minutes, and opposite to 
 
 20’ stands 1 minute 20 seconds, which added to 16 minutes, gives 17 minutes 20 
 Seconds, the time answering to 4° 20’. 
 
 PROBLEM V. 
 
 How two men may be born on the same day, die at the same moment, and yet the one may 
 | have lived a day, or even two days more than the other. 
 
 It is well known to all navigators, that if a ship sails round the world, going from 
 east to west, those on board when they return will count a day less than the inha- 
 bitants of the country. The cause of this is, that the vessel, following the course of 
 the sun, has the days longer, and in the whole number of the days reckoned, during 
 
 _ the voyage, there is necessarily oné revolution of the sun less. 
 
416 Sia ASTRONOMY AND GEOGRAPHY. 
 
 On the other hand, if the ship proceeds round the world from west to east, as it 
 goes to meet the sun, the days are shorter, and during the whole Sib damtea 3 the 
 people on board necessarily count one revolution of the sun more. 
 
 Let us now suppose that there are two twins, one of whom embarks on board a 
 vessel which sails round the world from east to west, and that the other has remained 
 athome. Whenthe ship returns, the inhabitants will reckon Thursday, while those 
 on board the vessel will reckon only Wednesday; and the twin who embarked will — 
 have a day less in his life. Consequently if they should die the same day, one of 
 them would count a day older than the other, though they were born at the same 
 hour. 
 
 But let us next suppose that, while the one circumnavigates the globe from east 
 to west, the other goes round it from west to east, and that on the same day they 
 return to port, where the inhabitants reckon Thursday, for example: in this case, 
 the former will count Wednesday, and the latter Friday, so that there will be two 
 days’ difference in their ages. 
 
 In short, it is evident that the one is as old as the other; the only difference is, that 
 in the course of their voyage the one has had the days longer and the other shorter. 
 
 If the latter returned on a Wednesday and the former ona Friday, the former 
 would count the day of his arrival Thursday: next day would be Thursday to the 
 inhabitants, and the day after would -be a Thursday to those who arrived in the 
 second vessel; which, notwithstanding the popular proverb, would give three 
 Thursdays in one week. 
 
 PROBLEM VI. 
 . To find the length of any given day in any proposed. latitude. 
 
 Let the circle ancx (Fig. 2.), represent a meridian, 
 and ac the horizon. Assume the arc cx, equal to the 
 elevation of the pole of the proposed place, for example 
 London, which is 51° 31’; and having drawn px, draw 
 DF perpendicular to it, or make the arc A F equal to the. 
 complement of cE, and draw Fp: it is here evident that — 
 Ep will represent. the circle of 6 hours and pF the | 
 equator. | 
 
 After this is done, find by the Ephemeris the sun’s 
 declination, when in the proposed degree of the ecliptic, | 
 or determine it by an operation which we shall shew how to perform hereafter, We 
 shall suppose that the declination is north: assume the arc rm toward the arctic | 
 pole, equal to the declination, and through the point m draw MN parallel to Fp, | 
 meeting the line DE in o, and the horizon acin Nx. Then from the point 0, as a 
 centre, with the radius om, describe an are of a circle MT, comprehended between 
 the point m and nT parallel top x. Having measured the number of the degrees 
 comprehended in this arc, which may be easily done by means of a protractor, and | 
 having changed them into time, at the rate of 1 hour to 15 degrees, &c., the double 
 of the result will be the length of the day. 
 
 Thus, if the length of the day at London, at the time when the sun has attained — 
 to his greatest northern declination, be required ; as the greatest declination is 23° | 
 28’, make FB equal to 23° 28’, and the arc Br will be found to be 123° 6’, which 
 corresponds to 8h. 12m., and this doubled gives 16h. 24m., as the length of the day. 
 
 If you have no fans. of the sun’s declination for each depree of the ecliptic, this 
 deficiency may be supplied in the following manner. Find the number of degrees 
 which the sun is distant from the nearest solstice, whether he has not yet reached © 
 it, or has passed it. We-shall suppose tuat he is in the 23d degree of Taurus. The 
 nearest solstice is that of Cancer, from which the sun, according to this supposition, © 
 
 - 
 
LATITUDE.—CLIMATES. 417 
 
 is distant 37°. Draw the line Bp representing a quarter of the ecliptic; and having 
 assumed, from the point B, the arcs BK and B&, each equal to 37°, draw kK &, inter- 
 secting BD in L: if MN be then drawn through the point L, it will give the position 
 of the parallel required. 
 
 All these things may be found much more correctly by trigonometrical calcula- 
 tion. Thus, in the case in hand, add together the log. tangents of the latitude and 
 declination, and the sum, rejecting 10 from the index, is the log. sine of an arc; 
 which, added to or taken from 90°, according as the latitude and declination are 
 of the same or contrary denominations, and the sum or remainder reduced to time, 
 the result is half the length of the day. 
 
 st PROBLEM VII. 
 The longest day in any place being given, to find the Latitude. 
 
 This problem is the converse of the preceding, and may be solved without much 
 difficulty ; for the longest day, in all places of the northern hemisphere, always 
 happeus when the sun has just entered the sign Cancer. Let 
 Fig. 3. FD (Fig. 3.) then represent the celestial equator, or rather its 
 diameter, and Bx that of the tropic of Cancer. On the latter 
 Caan describe a circle BK L; and having assumed the arc B K equal to 
 Lie ‘ the number of degrees corresponding to half the length of the 
 as given day, at the rate of 15° for one hour, draw K™ perpen- 
 we dicular to BL; if the diameter N Mo be then drawn through the 
 eS point m, the angle pco will be the elevation of the pole, or 
 
 at latitude of the place. 
 
 The same Trigonometrically. 
 
 From the sum of the log. cotangent of the declination, and the log. sine of half the 
 length of the day diminished by twelve hours, deduct 10, and the remainder is the 
 log. tangent of the latitude. 
 
 PROBLEM VIII. 
 The Latitude of a place being given, to find the Climate in which it is situated. 
 In astronomy, the name climate is given to an interval, on the surface of the 
 earth, comprehended between two parallels under which the difference of the longest 
 days is half an hour: thus the days in summer, under the parallel, whether north or 
 south, distant from the equator 8° 25’, being 12h. 30m, this interval, or the zone com- 
 _ prehended between the equator and that parallel, is called the first climate. 
 The limits of the different climates may therefore be easily determined, by finding 
 in what latitudes the days are 123 hours, 13, 13, 14, &c. The following is a table 
 of all these climates. 
 Climates. Latitude. Latitude. Climates. Latitude. Latitude. 
 meee trom. O°. 0 to Siz 2h All... from, 58° 29 -. to, 59°58 
 | RM as isichl) | Ae 2 ny oie vet pe hOr.s BO MEV ed bale OO., OS's. 4 ae ORaREeS 
 Seer AO.) DOs teats ot Zo OO XV seat: Ghu lSy eau ak Gomme 
 PM nina opt od" DO: iho ajo. OO 420 BOVE. 93. 6 asin 62... 25... ae GS eae 
 
 RE eet OO aU er oleie OU OO ANID oa are GS 22g Foss G4 
 Be lencsi dO, 28: cesien 41, 22 AVIIE ...- 64 G ... 64 49 
 Vil 4] 22 45 29 XIX 64 49 65 2] 
 Vill 45 29 Je 49 21 XX 65 21 65 47 
 Dea ge AO Zhe, eos OL, 128 DORE. oy ts.0ti OS) 4 haere 66.) 6 
 meets OL 2B ieee ne t04 2 AL icra 660 BGG e266 9220 
 XI 54 27 ot apes red. XXIII 66 20 66 28 
 XI 56 37 58 29 XXIV 66 28 66 31 
 
 2E 
 
4a 
 
 418 ASTRONOMY AND GEOGRAPHY. 
 
 As the longest day at the polar circle is 24 hours, and at the pole 6 oe there 
 
 are supposed to be six climates between that circle and the pole. 
 
 Climates. Latitude. Latitude. Climates. Latitude. Latitude. 
 XxXV. from’ 667731’: to’ ~ 672 30’ XXVIII from 73° 20 to 78° 20 
 XXVI OP HESO! geek: SOO M30 XXIX v2, 78/720) ee Be 
 KV LEE OG BO oo te pad: 20 XXX oé) ‘84° 20) Vig 00 ae 
 
 Now if it be asked in what climate London is, it may be easily replied that it is in 
 the tenth; its latitude being 51° 31’, and its longest day 16h 34m. 
 
 Remark.—The idea of climates belongs to the ancient astronomy ; but the modern 
 pays no attention to this division, which in a great measure is destitute of correctness, in 
 consequence of the refraction ; for if the refraction be taken into account, as it ought 
 to be, whatever Ozanam may say, it will be found that, under the polar circle, the 
 longest day, instead of 24, will be several times 24 hours; for as the horizontal re- 
 fraction elevates the centre of the sun 32’ at least, the centre of that luminary ought 
 consequently never to set between the 9th of June and the 3d or 4th of July; and 
 the upper limb, from the 6th of June to the 6th of July; this makes a complete 
 month, during which the sun would never be out of sight. 
 
 PROBLEM IX. 
 To measure a degree of a great circle of the earth, and even the earth itself. 
 
 The rotundity of the earth, that is to say its being a globe, or of a form approaching 
 
 very near to one, is proved by a number of astronomical phenomena; but we think | 
 
 it needless to enumerate these proofs, which must be known by those who are in the 
 least acquainted with the principles of philosophy and the mathematics. 
 
 We shall here then suppose that the earth is perfectly spherical, as it apparently is ; 
 and shall begin our reasoning on. that hypothesis. 
 
 What is called a degree of the meridian on the earth, is nothing else than the dis- 
 tance between two observers, the distance between whose zeniths is: equal to 
 a degree, or the geometrical distance between two-places lying under the same 
 meridian, the latitudes of which, or their elevation of the pole, differ a degree. 
 Hence, if a person proceeds along a meridian of the earth, measuring the way 
 he travels, he will have passed over a degree when he finds a degree of dif- 
 
 ference between the latitude of the place which he left, and that at which he | 
 
 has arrived; or when any star near the zenith of his first station has approached or 
 receded a degree. 
 
 Nothing then is necessary but to make choice of two places, situated under the 
 same meridian, the distance and latitudes of which are exactly known; for if the less 
 latitude be taken from the greater, the remainder will be the arc of the meridian 
 comprehended between the two places; and thus it will be known that a certain 
 number of degrees and minutes correspond to a certain number of toises, or yards, or 
 feet, &c. All then that remains to be done, is to make use of the following prepor- 
 
 tion. As the given number of degrees and minutes, is to the given number of toises, - 
 
 yards, or feet, so is one degree to a fourth term, which will be the toises, yards, or . 
 
 feet corresponding to a degree. 
 
 But as the statious chosen may not lie exactly under the same meridian, but nearly 
 so, as Paris and Amiens, the meridional distance between their two parallels must be 
 measured geometrically ; ; and when this distance, as well as the difference of latitude 
 of the two places, is known, the number of toises, yards, or feet corresponding toa 
 degree may be found by a proportion similar to the preceding. 
 
 This was the method employed by Picard to determine the length of a terrestrial 
 degree, or the meridian in the neighbourhood of Paris. By a series of trigonometrical 
 
3 , 
 . > 
 v 
 5 
 Pr “3 
 ’ 
 
 GREAT CIRCLE OF THE EARTH.—FORM OF THE EARTH. 419 
 
 operations, he measured the distance between the pavilion of Malvoisine, to the 
 south of Paris, as far as the steeple of Amiens, reducing it tothe meridian, and found 
 _ it to be 78907 toises. He found also, by astronomical obervations, that the cathe- 
 dral of Amiens was 1° 22’ 58” farther north than the pavilion of Malvoisine. By 
 _ making this proportion then: As 1° 22’ 58” are to one degree, so are 78907 toises to 
 57057; he concluded that a degree was equal to 57057 toises. 
 Picard’s measurement having been since rectified in some points, it has been found 
 that this degree is equal to 57070 toises. 
 
 Corollaries.—I. Thus, if we suppose the earth spherical, its circumference will 
 be 20545200 French toises = 24881°8 English miles, 
 
 II. Its diameter will easily be found by making use of the following proportion : 
 as the circumference of the circle is to its diameter, or as 314159 is to 100000, so is 
 the above number to a fourth term, which is 6530196 toises==the diameter of the 
 earth = 7920-12 English miles. 
 
 III. If we suppose its surface to be as smooth as that of the sea during a calm, its 
 superficial content will be found to be 134164182859200 square toises = 197063856 
 English square miles. The rule for obtaining this result is: Multiply the cireumfe- 
 rence by half the radius, and then quadruple the product; or still shorter, multiply 
 the circumference by the diameter. | 
 
 IV. To find the solidity: Multiply the superficial content, above found, by a third 
 of the radius, which will give 146019735041736067200 cubic toises = 260124289920 
 English cubic miles. 
 
 Remark.—The operation performed by Picard between Paris and Amiens, was 
 afterwards continued throughout the whole extent of the kingdom, both north and 
 south; that is to say, from Dunkirk, where the elevation of the pole is 51° 2’ 27”, to 
 Collioure, the latitude of which is 42° 31’ 16”: the distance therefore between the 
 parallels of these two places is 8° 31’ 11”. But it was found at the same time, by 
 measurement, that the distance between these parallels was 486058 toises, which 
 gives for a mean degree in the whole extent of France 57051 toises; and by correc- 
 tions made afterwards, this number was reduced to 57038. 
 
 During this operation care was also taken to determine the distance of the first 
 meridian, which in France is that of the observatory of Paris, from the principal places 
 between which it passes. 
 
 The meridian of France continued, enters Spain, leaving Giromne on the east, at the 
 distance of about 4 of a degree; passes two or three thousand toises to the east of 
 Barcelona, traverses very nearly the island of Majorca, to the east of that city, and 
 then enters Africa, about seven minutes of a degree west of Algiers. But we shall 
 not follow its course farther through unknown nations and countries, and shall only 
 observe that it issues from Africa in the kingdom of Ardra. The astronomers 
 of France have, since the above, repeated the measurement of the said arc through 
 the country, with no great difference ; from whence they have deduced the length 
 of the meridional quadrant, which has been assumed as the standard of the new uni- 
 versal measures. Also several degrees of the meridian through England are now 
 measuring by Colonel Mudge, and Colonel Colby, of the Royal Artillery, under 
 the auspices of the Master General and Board of Ordnance; and an arc exceeding 
 
 in amplitude all others that have yet been measured, is in course of execution in 
 India. 
 
 PROBLEM X. 
 Of the real figure of the Earth. 
 We have already said that the rotundity of the earth is proved-by various astrono- 
 
 mical and physical phenomena; but these phenomena do not prove that it is a perfect 
 
‘ 
 420 ASTRONOMY AND GEOGRAPHY. 
 
 sphere. Accurate methods for measuring it were no sooner employed, than doubts 
 began to be entertained respecting its perfect sphericity. In fact, it is now demon- 
 strated that our habitation is flattened or depressed towards the poles, and elevated 
 about the equator; that is to say, the section of it through its axis, instead of being 
 a circle, is a figure approaching very near to an ellipse, the less axis of which is the 
 axis of the earth, or the distance from the one pole to the other, and the greater the 
 diameter of the equator. Newton and Huygens first established this truth, on 
 physical reasoning deduced from the centrifugal force and rotation of the earth ; and 
 it has since been confirmed by astronomical observations. 
 
 The manner in which Newton and Huygens reasoned, was as follows. If we 
 suppose the earth originally spherical and motionless, it would be a globe, the greater 
 part of the surface of which would be covered with water. But it is at present 
 demonstrated, that the earth has a rotary motion around its axis, and every one knows 
 that the effect of circular motion is to make the revolving bodies recede from the 
 centre of motion: thus the waters under the equator will lose a part of their gravity, 
 and therefore they must rise to a greater height, to regain by that elevation the force 
 necessary to counterbalance the lateral columns, extended to other points of the earth, 
 where the centrifugal force, which counterbalances their gravity, is less, and acts in 
 a less direct manner. The waters of the ocean then must rise under the equator as 
 soon as the earth, supposed to be at first motionless, assumes a rotary motion round 
 its axis: the parts near the equator will rise a litfle less, and those in the neighbour- 
 hood of the poles will sink down; for the polar column, as it experiences no centrifu- 
 gal force, will be the heaviest of all. This reasoning cannot be weakened, but by 
 supposing that the nucleus of the earth is of an elongated form; or by supposing a 
 singular contexture in its interior parts, expressly adapted for producing that effect ; 
 but this is altogether improbable. 
 
 The philosophers however on the continent persisted a long time in refusing to 
 admit this truth. Their principal arguments against it were founded on the measure- 
 ment of the degrees of the meridian made in France; by which it appeared that a 
 degree was less in the northern part of the kingdom than in the southern, and hence 
 they concluded that the figure of the earth was a spheroid elongated at the poles. 
 If the earth, said they, were perfectly spherical, by advancing uniformly under the 
 same meridian, the elevation of the pole would be uniformly changed. Thus, in 
 advancing from Paris, for example, towards the north 57070 toises, the-elevation of 
 the pole would vary a degree ; and to make the elevation of the pole increase ano- 
 
 ther degree, it would be necessary to advance towards the north 57070 toises more; j 
 
 and so on throughout the whole circumference of a meridian. 
 
 If, in proportion as we proceed northwards, it is found necessary to travel farther 
 than the above number of toises before the latitude is changed one degree, there is 
 reason to conclude that the earth is not spherical, but that it is Jess curved or more 
 flattened towards the north, and that the curvature decreases the nearer we approach 
 the pole, which is the property of an ellipsis having its poles at the extremities of its 
 less axis. In the contrary case, it would be a proof that the curvature of the earth 
 decreased towards the equator ; which is the property of a body formed by the revo- 
 lution of an ellipsis around its greater axis. 
 
 But it was believed in France at first, that the degrees of the meridian were found 
 to increase the more they approached the south. Thedegree measured in the neigh- 
 bourhood of Collioure, the austral boundary of the meridian, appeared to be equal to 
 57192 toises, while that in the neighbourhood of Dunkirk, which was the most 
 northern, seemed to be only 56954. There was reason therefore to conclude that 
 the earth was an elongated spheroid, or formed by the revolution of an ellipsis around 
 its greater axis. 
 
 The partisans of the Newtonian philosophy, at that time too little known in France. 
 
FORM OF THE EARTH. 421 
 
 replied, that these observations proved nothing, because the above difference, being 
 so inconsiderable, could be ascribed only to the errors unavoidable in such operations. 
 As 19 toises correspond to about a second, the 238 toises of difference would amount 
 only to about 12 seconds; an error which might have arisen from various causes : 
 they even asserted that this difference might be on the opposite side. 
 
 - To decide the contest, it was then proposed to measure two degrees as far distant 
 from each other as possible, one under the equator, and the other as near the pole as 
 the cold of the polar regions would admit. For this purpose, Maupertius, Camus, 
 and Clairaut, were dispatched by the king, in the year 1735, to measure a degree of 
 the meridian at the bottom of the Gulf of Bothnia, under the arctic polar circle; 
 and Bouguer, Godin, and Condamine were sent to the neighbourhood of the equator, 
 where they measured, not only a degree of the meridian, but almost three. It 
 resulted from these operations, performed with the utmost care and attention, that a 
 degree near the polar circle was equal to 57422-toises, and that a degree near the 
 equator contained 56750, which gives a difference of 672 tolses, and therefore tuo 
 considerable to be ascribed to the errors unavoidable in the necessary observations. 
 Since that time it has been generally admitted that the earth is flattened towards the 
 poles, as Newton and Huygens asserted. We shall here add that the measurements 
 formerly made in France having been repeated, it was found that the degree goes 
 on increasing from south to north, as ought to be the case if the earth be an oblate 
 spheroid. 
 
 This truth has been since confirmed by other measurements of the meridian, made 
 in different parts of the earth. The Abbé de la Caille having measured a degree 
 at the Cape of Good Hope, that is under the latitude of abont 33° south, found it to 
 be 57037 toises; and in 1755, Fathers Mairé and Boscovich, two Jesuits, having mea- 
 sured a degree in Italy, in latitude 43°, found it to be 56979: it is therefore certain 
 that the degrees of the terrestrial meridian go on increasing from the equator towards 
 the poles, and that the earth has the form of an oblate spheroid. 
 
 Other operations of the same kind for measuring a degree of the terrestrial me- 
 ridian have been since undertaken at different times, as by the Abbé Liesganig in 
 Germany, near Vienna; by Father Beccaria in Lombardy; and by Messrs. Mason 
 and Dixon, members of the Royal Society of London, in North America; and again 
 more lately by Mechain and De Lambre in France. They all confirm the diminution 
 of the terrestrial degrees as they approach the equator, though with inequalities 
 difficult to be reconciled with a regular figure. But it may here be asked, why should 
 the earth have a figure perfectly regular ? 
 
 It is, indeed, impossible to determine with perfect accuracy the proportion be- 
 tween the axis of the earth and its diameter at the equator: it has been proved that 
 the former is shorter, but to find their exact ratio would require observations which 
 can be made only at the pole. However the most probable ratio is that of 177 to 
 
 78. 
 
 Consequently, if this ratio be admitted, the axis of the earth, from the one pole to 
 the other, will be 6525376 toises, and the diameter of the equator 6562242. 
 
 In the last place, the difference between the distance of any point of the equator 
 
 ‘on a level with the sea, to the centre of the earth, and the distance of the pole from 
 
 the same centre, will be 18433 toises, or about 22 English miles. 
 
 Since Montucla wrote the above, however, the French astronomers Mechain and 
 De Lambre, in 1799, completed their measurement of the meridian, from Dunkirk in 
 France, to near Barcclona in Spain, an extent of almost 10 degrees; "from whence 
 it has been more accurately deduced, that the flattening of the earth at the poles is 
 only the 334th part, the ratio of the axes being that of 334 to 333; that the polar 
 axis is 78993 English miles, the equatorial diameter 79234 miles, their half difference 
 only 118 miles, which is the height of the equator more than at the pole, from the 
 
422 ASTRONOMY AND GEOGRAPHY. 
 
 centre ; the mean diameter 79113 miles, the mean circumference 24873 miles, the 
 greatest or equatorial circumference 24892? miles, the least or meridional circle 24855 
 miles, and the difference of the two 373 miles. 
 
 Corollaries.—I. From what has been said, several curious truths may be deduced. 
 The first is, that all bodies, except those placed under the equator and the poles, do 
 not tend to the centre of the earth; for a circle is the only figure in which all the 
 lines perpendicular to its circumference tend to the same point. In other figures, 
 the curves of which are continually varying, as is the case with the meridians of the 
 earth, the lines perpendicular to the circumference all pass through different points 
 of the axis. 
 
 II. The elevation of the waters under the equator, and their depression under the 
 poles, being the effect of the earth’s rotation around its axis, it may be readily con- 
 ceived that if this rotary motion should be accelerated, the elevation of the waters 
 under the equator would increase; and as the solid part of the earth has assumed, 
 since its creation, a consistence which will not suffer it to give way to such an ele- 
 vation, the rising of the waters might become so great, that all the countries lying 
 under the equator would be inundated ; and in that case the polar seas, if not very 
 deep, would be converted into dry land. 
 
 On the other hand, if the diurnal motion of the earth should be annihilated, or 
 become slower, the waters accumulated, and now sustained under the equator, by 
 the centrifugal force, would fall back towards the poles, and overwhelm all the 
 polar parts of the earth; new islands and new continents would be formed in the 
 torrid zone by the sinking down of the waters, which would leave new tracts of Jand 
 uncovered. 
 
 Remark.—We caunot help here remarking one advantage which France, and all 
 countries near the mean latitude of about 45 degrees, would in this case enjoy. If 
 such a catastrophe should take place, these countries would be sheltered from the 
 inundation, because the spheroid, which is the real figure of the earth at present, 
 and the globe or less oblate spheroid into which it would be changed, would have 
 their intersection about the 45th degree; consequently the sea would not be altered 
 in that latitude. i 
 
 PROBLEM XI. 
 To determine the length of a degree on any given parallel of latitude. 
 
 As the difference between the greater and less diameter of the earth does not 
 amount to the 800th part, in this and the following problems we shall consider it as 
 absolutely spherical; especially as the solution of these problems, if we supposed 
 the earth to be a spheroid, would be attended with difficulties inconsistent with the 
 plan of this work. 
 
 Let it be proposed then to determine how many miles or yards are equal to a de- 
 gree on the parallel passing through London ; that is to say under the latitude of 51 
 degrees 31 minutes This problem may be solved either geometrically or by calcu- 
 lation, according to the following methods, 
 
 Ist. Draw any straight line as (Fig. 4.), and divide it into 
 23 equal parts, because a degree of the equator contains 69°14 
 miles, or about 23 leagues. Then from the point a as a centre, 
 with the distance as, describe the arc Bc, equal to 51° 31 ; 
 and from the point c draw cp perpendicular to as: the part 
 AD will indicate the number of leagues contained in a degree 
 on the parallel of 51°31. 
 
 2d. This however may be found much more correctly by trigonometrical calcula- 
 tion; for which purpose nothing is necessary but to make use of the following pro= 
 portions: viz. 
 
 ‘Fig. 4. 
 
 gue 
 
 B D A 
 
- 
 
 ———  - 
 
 LENGTHS OF DEGREES.—DISTANCES OF PLACES, 493 
 
 Rad. ; cosine lat. :; the miles in a degree of the equator : to the miles in a de- 
 gree of the parallel. 
 The above is worked by logarithms in the following manner: 
 
 A stating >:.eee nets ed hee cole eee 10°0000000 
 is to the cosine of the latitude 51° 31’.. 9°7939907 
 So is the number of miles contends in 
 a degree of the meridian, viz. 69:14.... 1:8397294 
 * toa fourth term. tate Ditto t ststecg ats Shes 1°6337201 
 
 which in the table of Acypurithens will be fori answering to 43°025 miles, as before. 
 A degree therefore on the parallel of London contains nearly 43 miles, or about 
 75643 yards. 
 
 The demonstration of this rule is easy, if it be recollected that the circumferences 
 of two circles, or degrees of these circles, are to each other in the ratio of their 
 radii. But the radius of the parallel of London is the cosine of the latitude; whereas 
 the radius of the earth, or of the equator, is the real radius or sine of 90°, and hence 
 the above rule. 
 
 3d. If the circumference of the earth at the given parallel be required, nothing is 
 necessary but to multiply the degree found as above by 360: thus as a degree on the 
 varallel of London is equal to 43 miles, if this number be multiplied by 360, we shall 
 have 15480 miles, for the whole circumference of the circle of that parallel. ’ 
 
 The following table, which shews the number of miles contained in a degree on 
 
 . every parallel, from the equator to the pole, is computed of the supposition that the 
 
 length of the degrees of the equator are equal to those of the meridian, at the 
 medium latitude of 45°, which length is nearly 69), English miles. 
 
 se 
 = &, 
 
 —5| ————s——— | | i | 
 
 COnNTOoaUhWwWNeH © 
 
 PROBLEM XII. 
 
 Given the latitude and longitude of any two places on the earth, to find the distance 
 between them. 
 
 We must here observe, that the distance of any two places on the surface of the 
 earth, ought to be the arc of the great circle intercepted between them. The distance 
 therefore of any two places, lying under the same parallel, is not the are of that pa- 
 rallel intercepted between them, but an arc of a great circle, having the same 
 extremities as that arc ; for on the surface of a sphere, it is the shortest way from one 
 point to another, as a,straight line is upon a plane surface. 
 
 This being premised, it may be readily seen that this problem is susceptible 
 
424 ASTRONOMY AND GROGRAPHY. 
 
 of several cases; for the two places proposed may lie under the same meridian; that 
 is to say have the same longitude, but different latitudes; or they may have the 
 same latitude ; that i is, lie under the equator or under the same parallel; or in the last 
 place, their longitudes and latitudes may be both different: there is also a subdivision 
 into two cases, viz. one where the two places are in the ’same hemisphere, and another 
 where one isin the northern and the other in the southern hemisphere. But we shall 
 confine ourselves to the solution of the only case which is attended with any 
 difficulty. a 
 For it is evident that if the two places are under the same meridian, the are which 
 measures their distance is their difference of latitude, provided they are in the same 
 hemisphere, or the sum of these latitudes if they are in different. hemispheres. 
 Nothing then is necessary but to reduce this arc into leagues, miles, or yards, and 
 the result will be the distance of the two places in similar parts. 
 If the places lie under the equator, the amplitude of the are which separates them 
 may be determined with equal ease ; and can then be reduced into leagues, miles, &c. 
 Let us suppose then, which is the only case attended with difficulty, that the 
 places differ both in longitude and latitude, as London and Constantinople, the for- 
 mer of which is 28° 53’ farther west than the latter, and 10° 31’ farther north. If we 
 conceive a great circle passing through these two cities, the arc comprehended he- 
 tween them will be found by the following construction. 
 From a asa centre (Fig. 5.), with any opening of the com- 
 Fig. 5. passes taken at pleasure, describe the semicircle Bc DE, 
 representing the meridian of London. Take the are BF 
 
 find its place in F, and draw the radius a F. 
 
 In the same semi-circle, if the arcs B c and E vp be taken 
 each equal to 41°, the latitude of Constantinople, the line c p 
 will be the parallel of Constantinople, the place of which 
 must be found in the following manner. 
 
 On cD as a diameter, describe the semi-circle c Gp; and in the circumference of it 
 take the are c G equal to the difference of longitude between London and Constan- 
 tinople, that is 28° 53’; then from the point G draw Gc u, perpendicular to c p, to have 
 in H the projection of the place of Constantinople ; and from the point u draw H1, 
 perpendicular to A F and terminated at 1 by the are BCD E: if the arc F 1 be 
 
 S| 
 | 
 
 equal to 51° 31’, which is the latitude of London, in order to. 
 
 measured, it will give the distance required in degrees and minutes. In this case it _ 
 
 is about 22 degrees. * 
 
 If one of the places be on the other side of the equator, | 
 
 as the city of Fernambouc in Brazil is in regard to London, 
 being in 7° 30’ of south latitude, the are Bc must be assumed 
 on the other side of the diameter BE, (Fig. 6.), equal! to 
 the latitude of the second place given, which is here 7° 30’; 
 and as the difference of longitude between Loudon and Fer- 
 nambouc is 35° 5’, it will be necessary to make the areca 
 = 35° 5’. By these means the arc F 1 will be found to 
 
 be equal to about 66° +, which reduced into miles of 69°07 to a degree, gives 4558 — 
 
 miles for the distance between London and the above city of Brazil. 
 This problem may be solved trigonometrically thus: If the latitudes are both 
 north or both south, either add them both to 90°, or subtract them both from 90°; if 
 
 one is north and the other south, add either to 90° and subtract the other. Call the 
 
 results 7 and 1’, and the difference long. L. 
 Then call half the sum of Zand /’ are ], and add together (in jopeataen twice 
 
 * Calculation by spherical trigonometry gives 22 deg. 23 min. 
 + Trigonometrical calculation gives 66 deg. 15 min. 
 
rf . 
 ' 
 4 
 ; - 
 es 
 i 
 
 ITINERARY MEASURES, 425 
 
 the cosine of half 1, and the sines of 7 and/, and half the sum, rejecting tens, will be 
 the sine of arc 2: add together the sines of the sum and difference of arcs] and 2, 
 and half the sum will be the sine of half the required distance. 
 
 Remark.—When the distance between the two places is not very considerable, as 
 
 is the case with Lyons and Geneva, the latter being only 36’ farther north than the 
 - former, and more to the east by 6 minutes of time, which is equal to 1° 30’ under 
 the equator, the calculation may be greatly shortened. 
 
 For this purpcse, take the mean latitude of the two places, which in this instance 
 is 46° 4’, and find by the preceding problem the extent of a degree on the parallel 
 passing through that latitude, which will be = 47-922 miles. The difference of longi- 
 tude between these places is 1° 30’, which on that parallel, allowing 47°922 miles to 
 a degree, gives 71°88 miles, and the miles corresponding to the difference of latitude 
 are 41°44. 
 
 If we therefore suppose a right angled triangle, one of the sides of which adjacent 
 to the right angle is 4144 miles, and the other 71°88, by squaring these two numbers, 
 adding them together, and extracting the square root of the sum, we shall have the 
 hypothenuse equal to 82°97 miles; which will be the distance, in a straight line, 
 between Lyons and Geneva. 
 
 As this is the proper place for making known the measures employed by different 
 nations, in measuring itinerary distances, it will doubtless be gratifying to our readers 
 to find here a table of them, especially as it is difficult to collect them: for the same 
 reason we have added some of the itinerary measures of the ancients, the whole 
 expressed in English feet. 
 
 TABLE OF ITINERARY MEASURES, 
 
 ANCIENT AND MODERN. 
 
 Ancient Greece. Feet. 
 Feet. | The mile 123 to a degree ...... 28995 
 The Olympic stadium.......... 604 |_The same 15 to adegree ....... 24292 
 A smallerstadium ..ccccseoses 482 : 
 fire. least stadium ....':....-.. 322 Arabia. 
 Heimer dee ess acter cs woesees 6929 
 Egypt 
 Me schenus 2... eccessscscess 19421 Ar ance, 
 The mile of 1000 French toises.. 6392 
 Persia. The small league of 30 to a degree 12159 
 The parasang or farsang ......-- 14499 | The mean league of 25toadegree 14594 
 The great or marine league of 20 
 Roman Empire. EOUAMIO ET CCl einrs'o als evis c's gate eats 18238 
 The mile (milliare) .....-+.0¢+ . 4833 nein 
 Judea. — (he: mile 7.3% 32)>.3 se ua we sag eters 35050 
 The rast or stadium .......se0 486 Decnark 
 The berath or mile .......--.. ea aD hes tnilecren: cc's acute ae 25123 
 Ancient Gaul. England. 
 The league (Jeug) .....++++++- 4249 V The mile stas ean teen ot ace 5280 
 Germany. — Scotland. 
 The rast or league ....... ones L400 | Lhe mile wn. dees. ate erste etay ss 7332 
 
426 ASTRONOMY AND GEOGRAPHY. 
 
 Treland. Russia. 
 Feet. : Feet. 
 The pile s.ssceks helt BRET 6724 | The ancient werst ..ee--seeese 4193 
 The modern werst ..ssccsseses 3497 
 Spain. 
 
 The league (legale of 5000 vares) 13724 
 The common league 174 to a de- 
 gree wooo eceoereevreceeersece 20846 India. 
 
 The.little. cosa s/05:. sas seceecene 8579 
 The great COSS,...sesccveerces DOE 
 The Roman mile.,....+2+++++- 4909 | The gau of the Malabar coast .. 38356 
 The Lombard mile ......-+ee+e+ 5425 | The nari or nali of the same .-e. 5753 
 
 Tur key. 
 The agash-......s0:. saeense~ sae OG 
 
 Italy. 
 
 The Venetian mile .....+...+-- 634] 
 China 
 Poland. The present li ....... «4:60 
 The league .¢;....:+.-- seee.ee- 18223 | The pu, equal to 10 lis ....-... 18857 
 
 These values are extracted from a work by Danville, entitled Traité des Mesures 
 ttinéraires anciennes et modernes, Paris, 1768, 8vo., in which this subject is treated 
 with great erudition and sagacity ; so that, amidst the uncertainty which prevails in 
 regard to the precise relation bet ween these measures and ours, the evaluations given 
 by Danville may be considered as the most probable, and the best founded. We 
 have deviated therefore in many points from those given by Christiani, in his book 
 
 Delle Misure d’ogne genere antiche é moderne. This work is valuable in some respects ; 
 
 but the subject is far from being éxamined there in so profound a manner as it has 
 been by Danville. 
 
 PROBLEM XIII. 
 To represent the terrestrial Globe in plano. 
 
 A map, which represents the whole superficies of the terrestrial globe on a flat 
 
 surface, is called a planisphere, or general map of the world. 
 A map of this kind is generally represented in two hemispheres ; because the arti- 
 ficial globe, which represents the globe of the earth, cannot be all seen at one view: 
 
 hence, when delineated in plano, it is necessary to divide it into two halves, each of 
 
 which is called ahemisphere. It may be thus represented in three ways. 
 
 The first is to represent it as divided by the plane of the meridian into two hemis- 
 pheres, one eastern the other western. This method is that generally used for a map 
 of the world, because it exhibits the old continent in the one hemisphere, and the 
 whole of the new in the other. 
 
 The’ second is to represent it as divided by the equator into two hemispheres, the 
 one northern and the other southern. This representation is in some cases attended 
 with advantage, because the disposition of the most northern and most southern 
 countries are better seen. Some maps of this kind have been published, in which 
 the tracts pursued by our modern navigators, and all the discoveries made by them 
 in the South Seas, are accurately delineated. 
 
 The third method is to exhibit the globe of the earth as divided: by the horizon 
 into two hemispheres, the upper and lower, according to the position of each. 
 
 Under certain circumstances, this form has its advantages also. The disposition 
 
 of the different parts of the earth, in regard to the proposed place, are better seen, — 
 
 and a great many geographical problems can be solved by it with much greater 
 facility. 
 
 Father Chrysologue of Gy, in Franche-Comté, published some years ago two he- 
 
— 
 7 
 ‘#* 
 4 
 i 
 
 PLANS OF MAPS. 427 
 
 mispheres of this kind, the centre of one of which was occupied by Paris ; and he 
 added an explanation of the different uses to which they might be applied. 
 
 Two methods may be employed in these representations. 
 
 According to one of them, the globe is supposed to be seen by the eye placed 
 without it; and such as it would appear at an infinite distance. 
 
 According to the other, each hemisphere is supposed to be viewed on the concave 
 side; as if the eye were placed at the end of the central diameter, or at the pole 
 of the opposite hemisphere; and it is conceived to be projected on the plane of its 
 ‘base. Hence arise the different properties of these representations, which we shall 
 here describe. 
 
 I. When the globe is represented as seen on the convex side, and divided into 
 two hemispheres by the plane of the first meridian, the eye is supposed to be at an 
 infinite distance, opposite to the point where the equator is intersected by the 90th 
 meridian. All the meridians are then represented by ellipses, the first excepted, 
 which is represented by a circle, and the 90th which becomes a straight line: the 
 parallels of latitude also are represented by straight lines. This representation is 
 
 attended with one great fault, viz., that the parts near the first meridian are very 
 much contracted, on account of the obliquity under which they present them- 
 selves. 
 
 When the hemispheres are represented by the second method, that is to say as seen 
 on the concave side, and projected on the plane of the meridian, the contrary is the 
 case. It is supposed, in regard to the eastern hemisphere, that the eye is placed at 
 the extremity of the diameter which passes through the place where the equator and 
 the 90th meridian intersect each other. In this case there is more of equality be- 
 tween the distances of the meridians; and even the parts of the earth represented 
 in the middle of the map lie somewhat closer than those towards the edges. Be- 
 sides, all the meridians and parallels are represented by arcs of a circle, which is very 
 convenient in constructing the map. It is attended however with this inconvenience, 
 that the parts of the earth have an appearance different from what they have when 
 seen from without. Asia for example is seen on the left, and Europe on the right ; 
 but this may be easily remedied by a counter-impression. 
 
 II. If a projection of the earth on the plane of the equator be required, the eye 
 according to the first method may be supposed at an infinite distance in the axis 
 produced: the pole will then occupy the centre of the map; the parallels will be 
 concentric circles, and the meridians straight lines. But it is attended with this 
 inconvenience, that the parts of the earth near the equator will be very much con- 
 tracted. 
 
 For this reason it will be better to have recourse to the second method, which 
 supposes the northern hemisphere to be seen by an eye placed at the south pole, and 
 vice versa: as there is here an inversion of the relative position of the places, it may 
 be remedied in like manner by a counter-impression. 
 
 III. If the eye be supposed in the zenith of any determinate place, as of London 
 for example, and at an infinite distance, we shall have on the plane of the horizon a 
 representation of the terrestrial hemisphere, the pole of which is occupied by Lon- 
 don, and which is of the third kind. But this representation will still be attended 
 with the inconvenience of the places near the horizon being too much crowded. 
 
 This defect however may be remedied by employing the second method, or by 
 supposing the above hemisphere to be seen through the horizon by an eye placed in 
 the pole of the lower hemisphere: the different meridians will then be represented 
 by arcs of a circle, as will also the parallels: the circles representing the distance 
 
 ee 
 
; & 
 " 
 be 
 ig 
 “ 
 
 428 ASTRONOMY AND GEOGRAPHY. 
 
 from the proposed place, to all other places of the earth, will be straight lines. The 
 inversion of position may be remedied as in the preceding cases. 
 
 The numerous uses to which this particular kind of projection can be applied, may 
 be seen in a work published by Father Chrysologue in 1774, and which was intended - 
 as an explanation of his double map of the world, already mentioned. | 
 
 Various other projections of the globe might be conceived: and by supposing the 
 eye in some other point than the pole of the hemisphere, more equality might be 
 preserved between the parts lying near to the centre and the edges of the projection ; 
 but this would be attended with other inconveniences, viz., that the circles on the 
 surface of the sphere or globe would not be represented by circles or straight lines, 
 which would render a description of them difficult. It is therefore better to adhere 
 to the projection where the eye is supposed to be in the pole of the hemisphere 
 opposite to that intended to be represented; whether the terrestrial globe, as in 
 common maps, is to be projected on the plane of the first meridian, or whether 
 it be required to project it on the plane of the equator, or on that of the horizon 
 of any determinate place. 
 
 A series of six maps of the stars, prepared under the direction of J. W. Lubbock, 
 Esq., has recently been published by the Society for the diffusion of Useful Know- 
 ledge. If we conceive the celestial sphere to be inscribed in a cube, touching at 
 the poles, and at four equi-distant points on the equinoctial; then if lines be drawn 
 from the centre of the sphere through the stars as depicted on its surface, these lines 
 continued will, on the faces of the squares, give the projected places of the stars, 
 This is the construction which Mr. Lubbock has adopted: and it is certainly attended 
 with many and great advantages. 
 
 PROBLEM XIV. 5 
 
 The latitude and longitude of two places, London and Cayenne for example, being 
 given ; to find with what point of the horizon the line drawn from the one to the 
 other corresponds ; or what angle the azimuth circle, drawn from the former of these 
 places through the other, makes with the meridian. 
 
 The solution of this problem is attended with very little difficulty, if spherical 
 trigonometry be employed, as it is reduced to the following: the two: sides of a 
 spherical triangle and the included angle being given, to find one of the other two 
 angles. But for want of trigonometrical tables, which I had lost with all my bag- 
 gage in consequence of shipwreck, I found myself obliged, on a certain occasion, to 
 solve this problem by a simple geometrical construction, which I shall here de- 
 scribe. I cannot however help mentioning the singular circumstance which con- 
 ducted me to it. . 
 
 Being at the island of Socotora, near Madagascar, on board a vessel belonging to 
 the East India Company, which had touched there, I formed an acquaintance with a 
 devout Mussulman, one of the richest and most respectable inhabitants of the island. 
 As he soon learned, by the astronomical observations which he saw me make, that I 
 was an astronomer, he requested me to determine, in his chamber, the exact direction 
 of Mecca; that he might turn himself towards that venerable place when he re- 
 peated his prayers. I at first hesitated on-account of the object; but the good 
 Tahia (that was his name) begged with so much earnestness, that I was not able to 
 refuse. Having neither charts nor globes, and knowing only the latitude and longi- 
 tude of the two places, I had recourse to a graphic construction on a pretty large 
 scale. I determined the angle of position which Mecca formed with the above 
 island ; and traced out, on the floor of his oratory, the line in the direction of which 
 he ought to look, in order to be turned towards Mecca. Words can hard y express 
 how much the good Iahia was gratified by my compliance with his wishes; and I 
 have no doubt, if still alive, that he offers up grateful prayers to his prophet for my 
 
a 
 / 
 
 al 
 REFRACTION. 429 
 
 conversion. But let us return to our problem, in which we shall take, by way 
 of example, London and Cayenne. 
 Fig. 7. To resolve it by a geometrical construction, de- 
 scribe a circle to represent the horizon of London, 
 which we shall suppose to be in the centre P: the 
 larger this circle is, the more correct will the -ope- 
 ration be. Draw the two diameters A Band cD, 
 cutting each other at right angles: and having 
 assumed DN, equal to the distance of Londen from 
 the pole, draw the radius np, and P E perpendicular 
 to it, which will represent a radius of the equator: 
 make the arc EK equal to the distance of the se- 
 cond place from the equator, which, in regard to Cayenne, is 4° 56’; draw also K F 
 and KG perpendicular to the radii pB and PN; and from the point G draw ¢G @ per- 
 
 - pendicular to the diameter a B, and continue it on both sides: if from 0 as a centre, 
 with the radius GK, a semicircle RH Q be then described on the line ro Q, the points 
 R and Q will necessarily fall within the circle; because P G being greater than Po, we 
 shall have, on the other hand, cx or oR less than os. 
 
 Having described the semicircle Ra Q, assume the arc HI equal to the difference 
 of the longitudes of the two places, that is towards the side c, which we here 
 suppose to represent the west, and towards the south if the second place lies to the 
 west of London and farther south, which is the case in the proposed example; for 
 Cayenne is situated to the west of London, and lies much nearer the equator. Hence 
 it may be readily seen what ought to be done, if the second place lay farther north, 
 or to the east, &c. The arc u1 then having been taken equal to 52° 1)’, draw 1h 
 perpendicular to the diameter RQ; and draw Hf, till it meet, in m, that diameter 
 continued: if mF be then drawn, which will cut L1in 7, the point T will represent 
 the projection of Cayenne on the horizon of London; and consequently, by drawing 
 the line pt, the angle TPA will be that formed by the azimuth of London passing 
 through Cayenne. 
 
 It will be found, by this operation, that the line of position of Cayenne, in regard 
 to London, makes with the meridian an angle of 61°48’, consequently Cayenne bears 
 
 from London south-west by west 4 west nearly. 
 
 It must however be allowed that this problem can be solved mechanically, by 
 means of a globe, with much more ease and convenience; for nothing more is 
 
 _ Necessary than to rectify the globe for the latitude of London, to screw fast the 
 quadrant of altitude to that point, and then to turn it till the edge of it corresponds 
 _ with Cayenne: if the number of degrees intercepted between it and the meridian 
 be then counted on the horizon, you will have the angle it forms with the meridian. 
 But as a globe may not always be at hand, nor tables of sines and tangents to solve 
 _ it trigonometrically, this want may be supplied by the graphic construction above 
 described. 
 
 THEOREM. 
 
 The heavenly bodies are never seen in the place where they really are: thus, for 
 example, the whole face of the sun is seen above the horizon, after he is actually 
 set. 
 
 Though this has the appearance of a paradox, it is a truth acknowledged by all 
 
 | astronomers, and which philosophers explain in the following manner. 
 The earth is surrounded by a stratum of a fluid much denser than that which fills 
 the expanse of the celestial regions. A smal] portion of the terrestrial globe en- 
 
 _ veloped by this stratum, commonly called the atmosphere, is seen represented Fig. 8. 
 
 If the sun then be ins, a central ray sr, when it reaches the atmosphere, instead 
 
Ae a 
 
 430 ASTRONOMY AND GEOGRAPHY. | : 
 
 of continuing its course in a straight line, is refracted 
 towards the perpendicular, and assumes the direction 
 EF. A spectator at F, must consequently see the 
 sun in the line rE; and as we always judge the ob- 
 pe ots ject to be in the direct continuation of the ray by | 
 cai FI. which the eye is affected, the spectator at F sees the 
 i va ia centre of the sun at s, a little nearer the zenith than | 
 he really is; and this deviation is greater, the nearer | 
 the body is to the horizon, because the ray then falls : 
 with more obliquity on the surface of the atmospheric fluid. | 
 
 Astronomers have found that when the body is on the horizon, this refraction is 
 about 33 minutes; therefore when the upper limb of the sun is in the horizontal 
 line, so that if there were no atmosphere he would seem only beginning to peep. 
 over the horizon, he appears to be elevated 33 minutes; and as the apparent diame. 
 ter of the sun is less than 33 minutes, his lower limb will appear to touch the 
 horizon. Thus the sun is risen in appearance, though he is not really so, and even 
 when he is entirely below the horizon. Hence follow several curious consequences, 
 which deserve to be remarked. 
 
 I. More than one half of the celestial sphere is always seen ; though in every trea- 
 tise on the globes it is supposed that we see only the half; for besides the upper 
 hemisphere, we see also a band round the horizon of about 33 minutes in breadth, 
 which belongs to the lower hemisphere. 
 
 If. The days are every where longer, and the nights shorter, than they ought to 
 be, according to the latitude of the place; for the apparent rising of the sun precedes 
 the real rising, and the apparent setting follows the real setting; therefore, though 
 the quantity of day and night ought to be equally balanced at the end of the year, the 
 former exceeds the latter in a considerable degree. 
 
 Ill. The effect of refraction, above described, serves also to account for another 
 astronomical paradox, which is as follows. 
 
 The moon may be seen eclipsed even totally and centrally, when the sun is above 
 the horizon. 
 
 A total and central eclipse of the moon cannot take place but when the sun anil 
 moon are directly opposite to each other. We here suppose that the reader is 
 acquainted with the causes of these phenomena, an explanation of which may be 
 found in every elementary work on astronomy. When the centre of the moon there- 
 fore, at the time of a total eclipse, is in the rational horizon, the centre of the sun 
 ought to be in the opposite point; but by the effect of refraction these points are 
 raised 33 minutes above the horizon. The apparent semi-diameter of the sun and 
 moon being only about 15 minutes, the lower limbs of both will appear elevated 
 about 18 minutes.—Such is the explanation of a phenomenon which must take place 
 at every central eclipse of the moon; for there is always some place of the earth 
 where the moon is on the horizon at the middle of the eclipse. 
 
 IV. Refraction enables us to explain also a very common phenomenon, viz., the 
 apparent elliptical form of the sun and moon, when on the horizon; for the lower 
 limb of the sun corresponding, we shall suppose, with the rational horizon, is elevated 
 33 minutes by the effect of refraction ; but the upper limb being really elevated 30 
 minutes, (which is nearly the apparent diameter of that luminary at its mean dis- 
 tances,) is elevated in appearance by refraction no more than 28 minutes above its 
 real altitude ; the vertical diameter therefore will appear shortened by the difference 
 between 33 and 28, that is to say 5 minutes; for if the refraction of the upper limb 
 were equal to that of the lower, the vertical diameter would be neither lengthened 
 
 nor shortened. The apparent vertical diameter will thus be reduced to about 28 
 minutes. 
 
= 
 
 ECLIPSES. 431 
 
 But there ought to be no sensible decrease in the horizontal diameter; for the 
 extremities of this diameter are carried only a little higher in the two vertical 
 circles passing through them, and which, as they meet in the zenith, are sensibly 
 parallel. The vertical diameter then being contracted, while the horizontal diameter 
 remains the same, the result must be, that the discs of the sun and moon will appa- 
 rently have an elliptical form, or appear shorter in the vertical direction than in the 
 horizontal. 
 
 V. There is always more than one half of the earth enlightened by a central illumi- 
 nation ; that is to say by an illumination, the centre of which is visible; for if there 
 were no refraction, the centre of the sun would not be seen till it corresponded with 
 the plane of the rational horizon ; but as the refraction raises it about 33 minutes, it 
 will begin to appear when it is in the plane of a circle parallel to the rational horizon, 
 and 33 minutes below it. 
 
 There is therefore a central illumination for the whole hemisphere, plus the zone 
 - comprehended between that hemisphere and a parallel distant from it 33 minutes ; 
 and there is a complete illumination from the whole disc of the sun to the same 
 hemisphere, and the zone comprehended between the border of it, and a parallel 
 about 16 minutes farther below the horizon. 
 
 What Ozanam therefore, or his continuator, endeavours to demonstrate after De- 
 schales with so much labour and tediousness, (see ‘‘ Recreations Mathematiques” 
 vol. 11. p. 277, edit. of 1750,) is absolutely false :* because no allowance is made for 
 refraction. 
 
 PROBLEM XV. 
 To determine, without astronomical tables, whether there will be an Eclipse at any new 
 or full moon given. 
 
 Though the calculation of eclipses, and particularly those of the sun, is exceedingly 
 laborious ; those which took place in any given year of the 18th century, that is 
 between 1700 and 1801, may be found, without much difficulty, by the following 
 operation. The method of finding those of the present or 19th century, will be shewn 
 in the additional remark to this problem. 
 
 For the New Moons. 
 
 Find the complete number of lunatious between the new moon proposed, and the 
 _ 8th of January 1701, according to the Gregorian calendar, and multiply that number 
 _ by 7361; to the product add 33890, and divide the sum by 43200, without paying 
 | any regard to the quotient. If the remainder after the division, or the difference 
 between that remainder and the divisor, be less than 4060, there will be an eclipse, 
 and consequently an eclipse of the sun. 
 Ezxample.—It is required to find whether there was an eclipse of the sun on 
 the first of April 1764. Between the 8th of January 1701, and the Ist of April 
 _ 1764, there were 782 complete lunations ; if this number then be multiplied by 7361, 
 | the product will be 5756302; to which adding 33890, we shall have 5790192; and 
 ' this sum divided by 43200 will leave for remainder 1392: this number being less 
 _ than 4060, shews that on the Ist of April 1764 there was an eclipse of the sun, 
 which was indeed the case; and this eclipse was annular to a part of Europe. 
 
 Bost the) Ball ooes: 
 
 _ Find the number of complete lunations between that which began on the 8th of 
 January 1701, and the conjunction which precedes the full moon proposed : multiply 
 this number by 7361; and having added to the product 37326, divide the sum by 
 43200: if the remainder after the division, or the difference between the remainder 
 _and the divisor, be less than 2800, it will shew that an eclipse of the moon took place 
 | at that time. 
 
 ¢ 
 
432 ASTRONOMY AND GEOGRAPHY. 
 
 Example.-—Let it be required to find whether there was an eclipse at the full moon _ 
 which took place on the 13th of December 1769. Between the 8th of January 1701, | 
 and the 28th of November 1769, the day of the new moon preceding the 13th | 
 of December, there were 852 complete lunations: the product of this number by 
 7361 is 6271572; to which if we add 37326, the sum will be 6308898. But this — 
 sum divided by 43200, leaves for remainder 1698, which being less than 2800, shews 
 that there was an eclipse of the mooh on the 13th of December 1769, as indeed may | 
 be seen by the almanacs for that year. | 
 
 Remark.—To determine the number of lunations, which have elapsed between the | 
 8th of January 170], and any proposed day, the following method, which is attended 
 with very little difficulty, may be employed. Diminish by unity the number of years _ 
 above 1700, and multiply the remainder by 365; to the product add the number of bis- | 
 sextiles between 1700 and the given year, and the result will be the number of days 
 from the 8th of January 1701 to the 8th of January of the proposed year. Thenadd the | 
 number of days from the 8th of January of the given year to the day of the new moon | 
 proposed, or to that which precedes the full moon proposed ; and having doubled the | 
 sum, divide it by 59, the quotient will be the number of lunations required. | 
 
 Let us propose, by way of example, the 13th of December 1769, the day of full — 
 moon. The preceding new moon fell on the 28thof November. If 69 be diminished 
 by unity, the remainder is 68; which, multiplied by 365, gives 24820. As in that 
 interval there were 17 bissextiles, we must add 17, which will give 24837. Lastly, — 
 the number of days from January 8th to November 28th 1769 was 309, which added to | 
 the above sum make 25146. This number doubled is 50292; which divided by 
 59 gives for quotient 852. The number of complete lunations therefore, between 
 the 8th of January 1701 and the full moon December 13th 1769, was 852. 
 
 Additional Remarks.—This easy method of finding eclipses was invented by M. de 
 la Hire, a celebrated French astronomer; but as it will require some alteration to 
 make it answer for the present century, we shall first explain the principles on which 
 it is founded, and then shew how this alteration is to be made. 
 
 Ist. In regard to the full moons, we shall suppose that the sun is at present in the 
 ascending node, and the moon in the descending: the former during the period of 
 a lunation will move from his node 80 degrees 40 minutes 15 seconds; which expressed 
 in quarters of a minute are equal to 736]. Hence M. de la Hire multiplies this 
 number by that of the complete lunations, between the new moon on the 8th of 
 January 1701, and the full moon proposed ; and the product necessarily gives all the 
 movements which the sun has made during that time, to recede from the one node 
 and to approach the other, 
 
 2d. The sun at the time of the full moon,in the month of January 1701, was 
 distant from his node 155. degrees 31 minutes 30 seconds, which, expressed in quarters 
 of a minute, give 37326: hence, according to M. dela Hire, this number must be added 
 to the product of 7361 multiplied by the lunations, 
 
 3d: The two nodes of the lunar orbit are distant from each other 180 degrees, or 
 10800 minutes; which multiplied by 4, give 45200: the distance therefore of the 
 one node from the other is represented by 43200. 
 
 4th. To obtain the true distance of the sun from the node 43200 must be sub-_ 
 tracted from the sum mentioned in the example, viz. 6308898, as many times as pos- 
 sible; and hence, according to M. de la Hire, this sum must be divided by 43200, — 
 neglecting the quotient. 
 
 5th. The remainder after the last division gives the true distance of the sun from 
 his node, which we have hitherto supposed to be the ascending node; that is to say, 
 the node by which the moon passes from the southern to the northern side of the 
 
 a 
 
> . ee ta 
 . ‘ 
 
 ECLIPSES. 433 
 
 ecliptic. If this remainder does not exceed 2800, there will be an eclipse, or at 
 least it will be possible ; because the sun will not be distant from his node 11 
 degrees 40 minutes. For 11 degrees 40 minutes are equal to 700 minutes; and 700 
 minutes multiplied by 4, give 2800 quarters of a minute. 
 
 6th. There may be an eclipse though the remainder after the last division exceeds 
 2800; but in that case the difference between this remainder and the divisor will be 
 less than 2800. The reason of this is, that the sun is necessarily distant from one of 
 the two nodes less than 1] degrees 40 minutes. The one node indeed being distant 
 from the other only 43200 quarters of a minute, andas the sun cannot recede from 
 the one node without approaching the other, if the difference between the remainder 
 after the division, and the divisor 43200, does not exceed 2800, there will necessarily 
 be one of the nodes from which the sun will not be distant 11 degrees 40 minutes. 
 
 But it may here be objected, as the sun during the time of a lunation does not pass 
 over 30 degrees of the ecliptic from west to east, why have we asserted that if he be 
 at present in the ascending node, he will remove from it, in the course ofa Junation, 
 30 degrees 40 minutes 15 seconds? 
 
 This objection will not appear of much consequence, but to those who imagine 
 that the nodes which the lunar orbit forms with the solar are fixed and immoveable. 
 This is not the case; these nodes havea periodical motion ; thatis, they pass through 
 the 12 signs of the zodiac in the course of almost 19 years, not from west to east, as 
 the sun, but from east to west: at the end of a lunation then the sun must be 30 
 degrees 40 minutes 15 seconds distant from the node he has quitted; because he not 
 only moves from his node, but bis node moves from him. 
 
 In regard to new moons, the only difference in the operation is, that 33890 is added 
 to the product of the lunations by 7361, instead of 37326. At the time of the new 
 moon in January 1701, the sun was distant from his node 141 degrees 12 minutes 
 30 seconds; which expressed in quarters of a minute are equal to 33890. For an 
 eclipse of the sun, therefore, 33890 must be added to the product of the lunations 
 by 7361. 
 
 It is to be observed also, that for solar eclipses, the remainder must be less than 
 4060 ; which represents the quarters of a minute contained in 16 degrees 55 minutes. 
 A solar eclipse indeed is not impossible but when the sun and moon are at a greater 
 distance from their nodes than 16 degrees 55 minutes: the remainder and divisor 
 therefore must not be compared with 2800, as for eclipses of the moon, but with 4060. 
 
 To apply the above rules to the present century. 
 
 It is evident, from what has been said, that to find, by the above method, the 
 eclipses of the sun and moon in the present century, nothing will be necessary but to 
 substitute, for the sun’s distance from the node at the time of the new and full moon in 
 the month of January 1701, the same distance at the time of the new and full moon in 
 the month of January 1801, and to count the lunations between the new moon in 
 January 1801, that is the 14th, and the time proposed. _ But the sun’s distance from 
 the node at the time of the new moon on the 14th day of January 1801, was 
 280° 56’ 44”, and his distance from the node at the time of the full moon on the 29th 
 of January 1801 was 297° 15'11”. The former of these reduced to quarters of a 
 minute gives 67427, and the latter reduced in the same manner gives 71341. 
 
 Example \st.—Let it be required to find whether there will bean eclipse at the full 
 moon on the 18th of March 1802. Between the 14th of January 1801 and the 3d of 
 March 1802, the day of the new moon preceding the 18th of March, there will be 
 14 complete lunations. The product of this number by 7361 is 103054 ; to which if 
 we add 71341, the sum will be 174395. But this sum divided by 43200, leaves for 
 remainder 1595; which, being less than 2800, shews that there will be an eclipse of 
 the moon on the 18th of March 1802 
 
 2 F 
 
4 a 
 } 
 | q 
 
 434 ASTRONOMY AND GEOGRAPHY. 
 
 Example 2d.—It is required to find whether there will be an eclipse of the sun on 
 the 3d of March 1802. Between the 14th of January 1801, and the 3d of March 
 1802, there will be 14 complete lunations; if this number be multiplied by 7361, the 
 product will be 103054, to which adding 67427, we shall have 170481 ; and this sum 
 divided by 43200, will leave for remainder 40881: this number is not less than 
 4060, but its difference from 43200, which is 2319, is less than 4060 ; we may con- 
 clude therefore that there will be an eclipse of the sun on the 3d of March 1802. 
 
 Eclipses of the Sun and Moon during the nineteenth century. 
 
 To gratify the curiosity of the reader, we shall here give a table of the eclipses, 
 both of the sun and moon, which will take place in the course of the present century ; 
 with the different circumstances attending them, such as the time of the middle of 
 the eclipse, and its extent ; and, in regard to eclipses of the moon, how many digits 
 will be eclipsed, &c. 
 
 We must however observe, that as this table is extracted from an immense labour,* 
 undertaken for another purpose, perfect exactness must not be expected, either in 
 extent or time, and particularly in regard to the eclipses of the sun, since it is well 
 known that a solar eclipse, on account of the moon’s parallax, varies in quantity 
 according to the place of the earth; that an eclipse, for example, which is central 
 and total to the regions of the southern hemisphere, may be only partial and small 
 to the northern. The author therefore, to whom we allude, was satisfied with 
 indicating, rather than calculating, these eclipses; and left the more exact determi- 
 nations to astronomers. 
 
 To render this table however more generally useful, we shall add the following 
 explanation. The hour marked indicates the middle of the eclipse in true time; 3 
 signifies one half, } one fourth of an hour, morn. morning, aft. afternoon. The 
 quantity of the eclipse is expressed in digits and divisions of a digit. A digit is one 
 twelfth part of the diameter of the luminary eclipsed. Six digits are equal to one 
 half of the disc; four digits to one third, &c. When an eclipse is marked 0 digits, 
 the meaning is that it is less than a quarter, or + of a digit,~ When the moon is within 
 a minute of a degree or less of the centre, the eclipse is marked central; when 
 within two minutes, almost central. The duration of eclipses is nearly proportioned 
 to their greatness; a total lunar eclipse will continue at least 3} hours, and at most 
 4 hours and some minutes; a partial eclipse, which exceeds six*digits, may continue 
 24 or 31 hours; eclipses of between three and six digits, are of 2 or 3 hours’ dura- 
 tion ; those of two digits will last about 14 hour; those of one digit about 1 hour; 
 and those of half a digit about 2 of an hour. The time therefore of the middle 
 of an eclipse, and its duration, being given, its beginning and end may be nearly 
 ascertained by the following rule: viz., subtract its semi-duration from the time 
 given, and the remainder will be the hour of the beginning; add the same quantity, 
 and the sum will be the time of the end. A lunar eclipse must begin and end every 
 where at the same time; with this difference, that so many hours must be added or 
 subtracted as the one place is to the eastward or westward of the other. Thus, an 
 eclipse that begins about 44 hours p. m. at Greenwich observatory, will begin about 
 12 p.m. at Pekin, as the latter is 7 hours 46 minutes eastward of the former. 
 
 _ In regard to solar eclipses, they are dated from the time of the conjunction of the — 
 sun and moon. Though this date be sensibly different from that of the middle 
 of the eclipse; yet this difference will never amount to two hours, and may be 
 nearly found by the following rules:—Ist. In the morning a solar eclipse must 
 always happen sooner, and in the evening later, than the time of the conjunction. 
 
 * This labour is a table of the solar and lunar eclipses since the commencement of the Christian 
 sera, to the year 1900, inserted in ‘* L’Art de verifier les Dates,” by the Abbe Pingere, a celebrated 
 ' astronomer, and member of the Royal Academy of Sciences. 
 
 ea 
 
ECLIPSES. 435 
 
 2d. The nearer the sun is to the horizon, the more sensible will be the difference. 
 3d. The acceleration in the morning will be great in proportion to the elevation 
 of the sun at mid-day, three months before, and the retardation in the evening 
 will be great in proportion to the sun’s elevation three months after, the time proposed. 
 It thence follows, Ist, That the difference must be greatest in the torrid zone; and 
 2d. That the greatest difference in the other latitudes must happen in the evening of 
 the vernal, and in the morning of the autumnal, equinox; for the greatest meridian 
 altitudes are observed three months before and after these seasons. 
 
 - The parts of the world where the eclipse is visible, are marked. If there be no 
 limitation, the whole or the greater part of Europe or Asia must be understood. 
 Particular divisions of these quarters are denoted by the letters E. W. N. and S., 
 that is, East, West, &c. When an eclipse is said to be visible in E. or W. of 
 Europe, &c., the meaning is, that it is visible in all the parts of the region specified, 
 where the sun is sufficiently elevated above the horizon at the time of conjunction. 
 When it is marked as visible N. or S. of any particular region, all places in every 
 other direction are excluded. The terms small and great for the most part refer to 
 the eclipses, and not to the places where they are visible. The latitude of those 
 places is marked in which an eclipse is central. South latitude is indicated by the 
 letter S., and North latitude by N., which is frequently omitted. An 0, or cipher, 
 denotes North latitude. 
 
 The course of a central eclipse is ofttimes pointed out by three numbers. The 
 first and third shew the latitude in which the eclipse is central in the planes of the 
 5th and 155th meridians; the second, included in crotchets, gives the latitude in 
 which it is central at mid-day. The place where an eclipse is central at mid-day, 
 may be easily found, when the time of the true conjunction at Paris is known. The 
 interval between the true conjunction as given, and mid-day, nearly shews how many 
 hours and minutes the required place is east or west of the meridian of Paris. 
 
 It is to be observed also, that the limits of eclipses are fixed to be the tropic 
 of Cancer in Africa, and the northern extremity of Lapland; and from 5° to 6° N. 
 lat. in Asia to the Polar circle. In longitude, the limits are the 5th and the 155th - 
 meridians, supposing the 20th to pass through Paris. 
 
 The first and third numbers above mentioned, do not always express the latitude, 
 under the 5th and 155th meridians. Sometimes an eclipse begins before the sun has 
 risen upon the former, and ends after it has gone down on the latter meridian. - In 
 these cases, the first number denotes the latitude in which the eclipse is central at 
 sun-rising ; and the next the latitude in which it is central at sun-set. The number 
 included in crotchets is omitted when there is no meridian within the limits pre- 
 - scribed, under which the time of mid-day coincides with the middle of the eclipse. 
 It is to be observed also, that a number is sometimes added to point out the increase 
 or decrease of an eclipse. 
 
 A. single character or number indicates the latitude in which an eclipse is central 
 in Europe or Africa at sun-set ; and towards the eastern extremity of Asia at sun- 
 rising. An asterisk * denotes that the course of a central eclipse extends many 
 degrees beyond the equator. A dagger + indicates that its course is beyond the pole ; 
 and the excess is sometimes added to 90. Thus 94 intimates that the eclipse re- 
 ferred to is central 4° beyond the pole. The sign + affixed to pen. is used to ex- 
 press that the penumbra is deep or strong. 
 
 An eclipse is visible from 32° to 64° north; and as far south of the place where it 
 is central. 
 
 or 2 
 
436 ASTRONOMY AND GEOGRAPHY. 
 
 LIST OF ECLIPSES, 
 FROM THE BEGINNING TO THE END OF THE PRESENT CENTURY. 
 
 1801.—Eclipse of the moon, total, March 30th, 5} morn. cent. Of the sun, April 
 13th, 44 morn. Europe N.E. Asia N. dim. from W. to E. Of the sun, September 
 8th, 6 morn. Asia N.E. small. Of the moon, total, September 22d, 7} morn. 
 
 1802.—Of the moon, March 19th, 114 morn. 5 dig. Of the sun, August 28th, 74 
 morn. Eur. Afr. Asia, cent. 69 (59) 23 an. Of the moon, partial, September 
 llth, 11 aft. 9 digits. 
 
 1803.—Of the sun, August 17th, 8} morn. great part of Eur. S. Afr. Asia, S. cent. 
 26 (12) * an. 
 
 1804.—Of the moon, partial, January 26th, 9 aft. Of the sun, February 11th, 1) 
 morn. Eur. Afr. Asia, W. cent. 25 (32) 64. Of the moon, partial, July 22d, 
 54 aft. 102 dig. 
 
 1805.—Of the moon, total, January 15th,9 morn. Of the sun, June 26th, 11 aft, 
 part of Asia, N.E. Of the moon, total, July llth, 9 aft. 
 
 1806.—Of the moon, partial, January 5th, 0 morn. 9 dig. Of the sun, June 16th, 4 
 aft. Eur. Afr. W. cent. 31—16 tot. Of the moon, partial, June 30th, 10 aft. pen, 
 Of the sun, December 10th, 24 morn. small, Asia, S.E. 
 
 1807.—Of the moon, partial, May 2st, 5} aft. 1} dig. Of the sun, June 6th, 5} 
 morn. small, Asia S.E. Of the moon, partial, November 15th, 8} morn. 3 dig. Of 
 the sun, November 29th, merid. all Eur. Afr. Asia, W. cent. 18 (13) 9 — 25. 
 
 1808.—Of the moon, total, May 10th, 8 morn. Of the moon, total, November 3d, 9 
 morn. Of the sun, November 18th, 3 morn. great part of Asia N. incr. from’ 
 W. to E. 
 
 1809.—Of the moon, partial, April 30th, 1 morn. 10 dig. Of the moon, partial, 
 October 23d, 94 morn. 94 dig. 
 
 1810.—Of the sun, April4th, 2 morn. Asia, S.E. cent. * 10 an. 
 
 1811.—Of the moon, partial, March 10th, 64 morn. 5 dig. Of the moon, partial, Sep- 
 tember 2d, 11 aft. 7 dig. ; 
 
 1812.—Of the moon, total, February 27th, 6 morn. almost cent. Of the moon, total, — 
 August 22d, 3 aft. 
 
 1813.—Of the sun, February Ist, 9 morn. Eur. Afr. Asia, cent. 32 — 24 (26) 
 55 an. Of the moon, partial, February 15th, 9 morn. 7} dig. Of the moon, partial, 
 August 12th, 33 morn. 4} dig. 
 
 1814.—Of the sun, January 2lst, 2) aft. Eur. S.E. Afr. cent.* 10 an. Of the sun, 
 July 17th, 7 morn. Eur. S. Afr. E. Asia, S. cent. 14 — 33 (31) 5 tot. Of the 
 moon, partial, December 26th, 114 aft. 6 dig. 
 
 1815.—Of the moon, total, June 21st, 6} aft. 121 dig. Of the sun, July 7th, 0 morn, 
 Eur. and Asia, N. cent. 62 ¢ tot. Of the moon, partial, December 16th, 1d aft. 
 
 1816.—Of the moon, total, June 10th, 1} morn. Of the sun, November 19th, 104 morn. 
 Eur. Afr. Asia, W. cent.59 (38) 33 — 37 tot. Of the moon, partial, December 
 4th, 9 aft. 73 dig. 
 
 1817.—Of the sun, May 16th, 7 morn. Asia, S. cent. * (7) 12—7an. Of the 
 moon, partial, May 3d. 3) aft. pen. +. Of the sun, November 9th, 2! morn. Asia, 
 E. cent. 26 — 5S. tot. 
 
 1818.—Of the moon, partial, April 21st, 0} morn. 53 dig. Of the sun, May 5th, 
 7 morn. Eur. Afr. Asia, cent. 13 (51) 60 — 53 an. Of the moon, partial, October 
 14th, 6 morn. 2 dig. 
 
 1819.—Of the moon, total, April 10th, 14 aft. Of the sun, April 24th, merid. N. of 
 Eur. and of Asia, dim. from W. to E. Of the sun, September 19th, 1 aft. Eur. 
 N.E. small. Of the moon, total, October 3d, 34 aft. 
 
LIST OF ECLIPSES, 437 
 
 1820.—Of the moon. partial, March 29th, 7 aft. 6 dig. Of the sun, September 7th, 
 2 aft. Eur. Afr. Asia, W. cent. 62— 29 an. Of the moon, partial, September 
 22d, 7 morn. 10 dig. 
 
 1821.—Of the sun, March 4th, 6 morn. Asia, S.E. cent. * (7 S.) 24 tot. 
 
 1822.—Of the moon, partial, February 6th, 5 morn. 43 dig. Of the moon partial, 
 Aug. 3d, 0} morn. 9 dig. 
 
 1823.—Of the moon total, January 26th, 5! aft. Of thesun, February 11th, 3 morn. 
 great part of Asia N. small. Of the sun, July 8th, 64 morn. Eur. and Asia, N. 
 Of the moon, total, July 23d, 31 morn. 
 
 1824.—Of the moon partial, January 16th, 9morn. 9 dig. Of the sun, June 26th, 
 114 aft. Asia, E. cent. 27 —41 tot. Of the moon, partial, July 11th, 45 morn. 1 
 dig. Of the sun, December 20th, 11 morn. Indies, S. small. 
 
 1825.—Of the moon partial, June Ist, 0} morn. Of the sun, June 16th, O} aft. Afr. 
 small cent. * (0)*. Of the moon partial, November 25th, 44 aft. 2} dig. 
 
 1826.—Of the moon, total, May 2Ist, 3faft. Of the moon, total, November 14th, 44 aft. 
 Of the sun, November 29th, 114 morn. Eur. Afr. Asia, W. 
 
 1827.—Of the sun, April 26, 31 morn. Eur. N.E. Asia, N. cent. 49 (81) 84 an. Of 
 the moon partial, May 11th, 83 morn. 11; dig. Of the moon sara November 
 3d, 5 aft. 10 dig. 
 
 1828.—Of the sun, April 14th, 93 morn. small part of Eur. §.E. Afr, Asia, cent. 
 2S. (18) 29— 26. Of the sun, October 9th, 0} morn. Asia S.E. cent. 7 * an. 
 
 1829.— Of the moon, partial, March 20th, 2 aft. 4 dig. Of the moon, partial, Sep- 
 tember 13th, 7 morn..53 dig. Of the sun, September 28th, 2) morn. Asia, E. 
 cent, 59 — 40 an. 
 
 1830.—Of the sun, February 23d, 5 morn. Asia, N. dim. from W. toE. Of the moon, 
 total, March 9th, 2 aft. Of the moon, total, September 2d, 11 aft. cent. 
 
 1831.—Of the moon, partial, February 26th, 5 aft. 8 dig. Of the moon, partial, 
 August 23d, 10} morn. 6 dig. 
 
 1832.—Of the sun, July 27th, 24 aft. Eur. S. Afr. Asia, S.E. cent. 23 N. 3 S. tot. 
 
 1833.—-Of the moon, partial, January 6th, 8 morn. 53 dig. Of the moon, partial, 
 July 2d, 1 morn. 103 dig. Of the sun, July 17th, 7 morn. Eur. Afr. E. Asia, 
 N. cent. 83 (80) 73 tot. Of the moon, total, December 26th, 10 aft. 
 
 1834.—Of the moon, total, June 21st, 8{ morn. Of the moon, partial, December 
 16th, 53 morn. 8 dig. 
 
 1835. _Of the sun, May 27th, 14 aft. small a of Eur. Afr. Asia, S: W. cent. 
 7—8—3S. an. Of the moon partial, June 10th, 11 aft. 0§ dig. Of the sun, 
 November 20th, 11 morn. small part of Eur. S.W. Afr. small part of Asia, 8. W. 
 cent. 4 (11 S.) * tot. 
 
 1836.—Of the moon, partial, May Ist, 84 morn. 44 dig. Of the sun, May 15th, 2} 
 aft. Eur. Afr. Asia, W. cent. 53 — 54— 44 an. Of the moon partial, October 24th, 
 13 aft. 14 dig. 
 
 1837.—Of the moon, total, April 20th, 9 aft. Of the sun, May 4th, 74 aft. small 
 part of Eur. N. great part of Asia, N.E. Of the moon total, October 13th, 11) aft. 
 
 1838.—Of the moon, partial, April 10th, 2} morn. 7 dig. Of the moon, partial, April 
 10th, 2! morn.7 dig. Of the moon, partial, October 3d, 3 aft. 03 dig. 
 
 1839.—Of the sun, March 15th, 2) aft. Eur. S. Afr. Asia, 8.W. cent 17 — 26 tot. 
 Of the sun, September 7th, 10} aft. extrem. of Asia, E. cent. 37. an. 
 
 -1840.—Of the moon, partial, February 17th, 2 aft. 44 dig. Of the sun, March 4th, 
 
 4 morn. cent. 16 (37) 48. -Of the moon, partial, August 13th, 7} morn. 7} dig. 
 
 -1841.—Of the moon total, February 6th, 2} morn. Of the sun, February 2lst, 
 
 1] morn. almost all Eur. N. Asia, N.W. dim. from W.to E. Of the sun, July 
 18th, 2 aft. great partof Eur. N.E. Asia, N.W. incr. from W. to E. Of the moon, 
 total, August 2d, 10 morn. 
 
x 
 438 ; _ ASTRONOMY AND GEOGRAPHY. 
 
 1842.—Of the moon, partial, January 26th, 6 aft. 9 dig. Of the sun, July 8th, 
 7 morn. Eur. Afr. Asia, cent. 35 — 50 (49) 21 tot. Ofthe moon, partial, July 
 22d, 11 morn. 3 dig. 
 
 1843—-.Of the moon, partial, June 12th, 8 morn. pen. Of the moon, partial, De- — 
 cember 7th, 0} morn. 21 dig. Of the sun, December 21st, 5, morn. Asia, cent. 25 
 (8) 21 tot. 
 
 1844.Of the moon, total, May 31st, 114 aft. Of the moon, total, November 25th, 04 
 morn. 
 
 1845.—Of the sun, May 6th, 10} morn. almost all Eur. N.W. Asia, N.W. cent. 90 
 (98) fan. Of the moon, total, May 2Ist, 43 aft. 123 dig. Of the moon, partial, 
 November 14th, 1 morn. 10} dig. 
 
 1846.—Of the sun, April 25th, 5} aft. Eur. and Afr. W. cent. 28 — 26. Of the 
 sun, October 20th, 84 morn. Europe S.W. Afr. Asia, S.W. cent. (18 S.)* an. 
 
 1847.—Of the moon, partial, March 31st, 91 aft. 23 dig. Of the sun, September 
 24th, Saft. 44 dig. Of the sun, October 9th, 94 morn. Eur. Afr. Asia, cent. 58 
 (31) 16—17 an. 
 
 1848.—Of the moon total, March 19, 9} aft. Of the moon total, September 13th, 
 64 morn. Of the sun, September 27th, 10 morn. Eur. N.E. Asia, N. 
 
 1849.—Of the sun,February 23d, 14 morn. Asia, E. cent. 3] —28—32an. Of the 
 moon partial, March 9th, 1 morn, 84 dig. Of the moon partial, September 2d, 5} 
 aft. dig. 
 
 1850.—Of the sun, February 12th, 61 morn. Asia, S.E. cent. * (11 8.) 17 N.an. Of 
 
 » the sun, August 7th, 10 aft. extrem. of Asia, E. cent. 14 tot. 
 
 1851.—Of the moon partial, January 17th, 5 aft. 51 dig. Of the moon, partial, July — 
 13th, 7; morn. 8} dig. Of the sun, July 28th, 23 aft. Eur. Afr. Asia, W. cent. 
 70 — 39 tot. 
 
 1852.—Of the moon, total, January 7th, 64 morn. Of the moon total, July Ist, 32 
 aft. Of the sun, December I 1th, 4 morn. Asia, E. cent. 59 (36) 35 tot. Of the 
 moon, partial, December 26th, 1 aft. 8 dig. 
 
 1853.—Of the moon, partial, June 21st, 6 morn. 23 dig. 
 
 1854. Of the moon, partial, May 12th, 4 aft. 3 dig. Of the moon, partial, November 
 4th, 9} aft. 1 dig. 
 
 1855.—Of the moon, total, May 2d, 44 morn. Of the sun, May 16th, 2} morn. 
 great part of Asia, N. dim. from W. to E. Of the moon, total, October 25th, 
 8 morn. 
 
 1856.— Of the moon, partial, April 20th, 9} morn. 83 dig. Of the sun, September 
 29th, 4 morn. Asia, N. cent. 84 (67) 66 an. Of the moon, partial, October 13th 
 114 aft. 114 dig. E 
 
 1857.— Of the sun, September 18th, 6 morn. Eur. and Afr. E. Asia, S. cent. 40 (12) 
 12 S. an. ; 
 
 1858.—Of the mceon partial, February 27th, 103 aft. 4 dig. Of the sun, March 15th, 
 Of aft. Eur. Afr. Asia, W. cent. (40) 68. Of the moon, partial, August 24th, 24 
 aft. 54 dig. 
 
 1859.— Of the moon, total, February 17th, 11 morn. Of the sun, July 29th 9 aft. 
 small, Asia, N.E. Of the moon total, August 13th, 45 aft. 
 
 1860.— Of the moon, partial, February 7th, 24 morn. 9f dig. Of the sun, July 18th, 
 2 aft. Eur. Afr. Asia, W. cent. 49—16 tot. Of the moon, partial, August Ist, 
 5b aft. 43 dig. 
 
 1861.—Of the sun, January 11th, 31 morn. small, Asia S.W. Of the sun, July 8th, 
 2 morn. Asia, S.E. cent.* 9 an. Of the moon, partial, December 17th, 8i morn. 
 2 dig.—Of the sun, December 3lst, 2} aft. all Eur. Afr. cent. 17 — 36 tot. 
 
 1862.—Of the moon total, June 12th, 63 morn. Of the moon, total, December 6th, 
 8 morn. Of the sun, December 21st, 5! morn. great part of Asia, N. 
 
LIST OF ECLIPSES. 439 
 
  1863.—Of the sun, May 17th, 5 aft. great part of Eur. N. Of the moon, total, June 
 2d,0 morn. Of the moon, partial, November 25th, 9 morn. 11 dig. 
 
 1864.— Of the sun, May 6th, 03 morn. Asia, S.E. cent. 6 — 23. 
 
 1865.—Of the moon, partial, April 11th, 5 morn. 14 dig. Of the moon, partial, Oc- 
 tober 4th, 11 aft. 33 dig. Of the sun, October 19th, 5 aft. extrem. of Eur. and 
 of Afr. W. cent. 16 an. 
 
 1866.— Of the sun, March 16th, 10 aft. small, Asia, N.E. Of the moon, total, March 
 3lst,5 morn. Of the moon, total, September 24th, 2h aft. Of the sun, October 
 8th, 53 aft. Eur. W. dim. from N. to S. 
 
 1867.—Of the sun, March 6th, 10 morn. Eur. Afr. Asia, cent. 31 (45) 69 an. Of 
 the moon, partial, March 20th, 9 morn. 93 dig. Of the moon, partial, September 
 14th, 1 morn. 8 dig. 
 
 1868.— Of the sun, February 23d, 24 aft. Eur. S. Afr. Asia, S.W. cent. 9— 2] an. 
 Of the sun, August 18th, 5$ morn. Eur. §.E. Afr. Asia, S. cent. 14 — 18 
 (11) 0 tot. 
 
 1869.—Of the moon, partial, January 28th, 13.morn. 5$ dig. Of the moon, partial, 
 July 23d, 2 aft. 63 dig. Of thesun, August 7th, 10 aft. Asia, N.E. cent. 46 tot. 
 
 1870.—Of the moon, total, January 17th, 3 aft. Of the moon, total, July 12th, 11 
 aft. Of the sun, December 22d, 03 aft. Europe, Africa, Asia, W. cent. (386) 49, 
 total. 
 
 1871.—Of the moon, partial, January 6th, O aft. 8 dig. Of the sun, June 18th, 
 21 morn. Asia, S.E. small. Of the moon, partial, July 2d, 14 aft. 4 dig. Of the 
 sun, December 12th, 44 morn. Asia, S. cent. 17 * total. 
 
 1872.—Of the moon, partial, May 22d, 114 aft. 14 dig. Of the sun, June 6th, 34 
 morn. Asia, cent. 8 (42) 43 an. Of the moon, partial, November 15th, 53 morn. 
 0 dig. 
 
 1873.—Of the moon, total, May 12th, 114 morn. Of the sun, May 26th, 9 morn. 
 great part of Europe N.W. Africa W. Asia N. dim. from W.to E. Of the moon, 
 total, November 4th, 44 aft. 
 
 1874.—Of the moon, partial, May Ist, 44 aft. 93 dig. Of the sun, Oct. 10th, 114 
 morn. Europe, Africa, Asia, W. cent. 82 (74) 55 an. Of the moon, partial, Oc- 
 tober 25th, 8 morn. 12 dig. 
 
 1875.—Of the sun, April 6th, 7 morn. Asia, S.E. cent. * (1) 21 total. Of the sun, 
 September 29th, 14 aft. small part of Europe S.W. Africa, Asia, 8.W. cent. 13 
 (10) 13S. an. 
 
 1876.—Of the moon, partial, March 10th, 64 morn. 34 dig. Of the moon, partial, 
 September 3d, 94 aft. 4 dig. 
 
 _ 1877.—Of the moon, total, February 27th, 74 aft. Qfthe sun, March 15th, 3 morn. 
 great part of Asia N. ann, from W.to E. Of the sun, August 9th, 5 morn. Asia, 
 N.E. small. Of the moon, total, August 23d, 114 aft. almost cent. 
 
 1878.—Of the moon, partial, February 17th, 114 morn. 9} dig.. Of the sun, July 
 29th, 94 aft. extremity of Asia, E. cent. 52 total. Of the ae partial, August 
 13th, 04 morn. 63 dig. 
 
 1879. i; the sun, January 22d, merid. small, Asia, S.W. cent. * 7 an, Of the 
 sun, July 19th, 9 morn. Europe S. Africa, Asia, S.W. cent. 8 — 16 (12) * an, 
 Of the moon, partial, December 28th, 45 aft. 1} dig. 
 
 1880.—Of the sun, January 11th, 11 aft. Asia, E. cent. 16 total. Of the moon, 
 total, June 22d, 2 aft. 123 dig. Of the moon, total, December 16th, 4 aft. Of the 
 sun, December 3Ist, 2 aft. Europe, Africa, dim. from N. to S. 
 
 1881.—Of the sun, May 28th, 0 morn. Asia, N. dim. from W. to E. Of the moon, 
 total, June 12th, 71 morn. Of the moon, partial, December 5th, 5} aft. 114 dig. 
 
 1882.—Of the sun, May 17th, 8 morn. Europe, S.E. Africa, Asia, cent. 10 (88) 42 
 — 26 total. Of the sun November I]th, 0 morn. Asia, S.E. cent. 2 * an 
 
440 ASTRONOMY AND GEOGRAPHY. 
 
 1883.— Of the moon, partial, April 22d, merid. 01 dig. Of the moon, partial, October 
 16th, 74 morn. 3 dig. Of the sun, October 31st, 0} morn. Asia, E. cent. 46 an. 
 1884.—Of the sun, March 27th, 6 morn. small, great part of Europe N.E. dim. in 
 
 Asia, from W.to E. Of the moon, total, April 10th, merid. Of the moon, 
 total, October 4th, 104 aft. Of the sun, October 19th, 1 morn. Asia, N. : 
 1885.—Of the moon, partial, March 30th, 5 aft. 10 dig. Of the moon, partial, Sep- 
 tember 24th, 8! morn. 9 dig. 
 1886.—Of the sun, August 29th, 14 aft. extremity of Europe, S.W. Africa, cent. 
 6 (4) * total. 
 1887.—Of the moon, partial, February 8th, 10! morn. 53 dig. Of the moon, par- 
 tial, August 3d, 9 aft.5 dig. Of the sun, August 19th, 6 morn. Europe and Africa, 
 E. Asia, cent. 54 — 62 (54) 29 total. 
 1888.—Of the moon, total, January 28th, 114 aft. Of the moon, total, July 23d, 6 
 morn. almost central. 
 1889.—Of the moon, partial, January 17th, 5$ morn, 81 dig. Of the moon, partial, 
 July 12th, 9 aft. 5§ dig. Of the sun, December 22d, 1 aft. Asia, S.W. cent. * 5 
 total. 
 1890.—Of the moon, partial, June 3d, 6 morn. 03 dig. Of the sun, June 17th, 10 
 morn. Europe, Africa, Asia, cent. 25 (38) 19 an. Of the moon, partial, November 
 26th, 2 aft. 01 dig. 
 1891.—Of the moon, total, May 23d, 7 aft. Of the sun, June 6th, 4 aft. great part 
 of Europe, N. cent. | Of the moon, total, November 16th, 02 morn. 
 1892.—Of the moon, partial, May 11th, 11) aft. 11} dig. Of the moon, total, No- 
 vember 4th, 44 aft. 12% dig. 
 1893.—Of the sun, April 16th, 3 aft Europe, S. Africa, cent. 20 — 18 total. 
 1894.—Of the moon, partial, March 2Ist, 24 aft. 3 dig. Of the sun, April 6th, 44 
 morn. Europe, N.E. Asia, cent. 10 (43) 8. Of the moon, partial, September 
 15th, 43 morn. 25 dig. Of the sun, September 29th, 5$ morn. Africa, E. small. 
 1895.—Of the moon, total, March 11th,4 morn. Of the sun, March 26th, 10 morn. 
 almost all Europe, N.W. Asia, N.dim. from W. to E. Of the sun, August 20th, 
 OL aft. Asia, N. small. Of the moon, total, September 4th, 6 morn. 
 1896.—Of the moon, partial, February 28th, 8 aft. 10 dig. Of the sun, August 9th, 
 41 morn. Europe, E. Asia, cent. 60 — 68 (59) 49 total. Of the moon, partial, Au- 
 gust 23d, 7 morn. 8 dig. 
 1897.—No eclipse. 
 1898.—Of the moon, partial, January 8th, 0 morn. 1} dig. Of the sun, January 22d, ~ 
 8 morn. Europe, E. Africa, E.all Asia,cent. 1] —5(10)44 total. Of the moon, 
 partial, July 3d, 9§ aft, 11 digs Of the moon, total, December 27th, 12 aft. 
 1899.— Of the sun, January 11th, 11 aft. extremity of Asia, E.dim, from N. to 8S. 
 Of the sun, June 8th,7 morn. Europe, W. and N. Asia, N. Of the moon, total, 
 June 23d, 2Laft. Of the moon, partial, December 17th, 14morn. 11} dig. 
 1900.—Of the sun, May 28th, 33 aft. Europe, Africa, cent. 45 — 26 total. Of the 
 moon, partial, June 13th, 4 morn. pen.+. Of the sun, November 22d, 8 morn. 
 small eclipse, Africa, cent. 3 S. * an. 
 
 PROBLEM XVI. 
 To observe -an Eclipse of the Moon. 
 
 To observe an eclipse of the moon, in such a manner as to be useful to geography 
 and astronomy, it will be necessary, in the first place, to have a clock or watch that 
 indicates seconds, and which you are certain is so well constructed as to go in an 
 uniform manner: it ought to be regulated some days before by means of a meridian, 
 if you have one traced out, or by some of the methods employed for that purpose by 
 
. yi 
 : 
 1g 
 
 x. LUNAR ECLIPSES. 441 
 
 astronomers; and you must ascertain how much it goes fast or slow in 24 hours; that 
 ‘the difference may be taken into account at the time of the observation. 
 
 You ought to be provided also with a refracting or reflecting telescope, some feet 
 in length; for the longer it is, the more certain you will be of discerning exactly the 
 moment of the phases of the eclipse; andif you are desirous of observing the quantity 
 of the eclipse, it should be furnished with a micrometer. 
 
 When you find the moment of the eclipse approaching, which may be always koe 
 either by a common almanac, or the Ephemerides published by the astronomers in 
 different parts of Europe, you must carefully remark the instant when the shadow 
 of the earth touches the moon’s disc. It is necessary here to mention, that there will 
 always be some uncertainty on account of the penumbra; because it is not a thick 
 
 _ black shadow which first covers the moon’s disc, but an imperfect one that thickens 
 by degrees. This arises from the sun’s dise being gradually occulted from the moon ; 
 and hence it is difficult to fix with exactness the real limits of the shadow, and the 
 penumbra. Here, as in many other cases, observers are enabled by habit to distin- 
 guish this boundary; or at least prevented from falling into any great error. 
 
 When you are certain that the real shadow has touched the moon’s disc, the time 
 must be noted down; that is to say, the hour, minute, and second, at which it 
 happened. 
 
 In this manner you must follow the shadow on the moon’s disc, and remark at 
 what hour, minute, and second the shadow reaches the most remarkable spots: all this 
 likewise must be noted down. 
 
 If the eclipse is not total, the shadow, after having covered part of the lunar disc, 
 will decrease. You must therefore observe in like manner the moment when the _ 
 shadow leaves the different spots it before covered, and the time when the disc 
 of the moon ceases to be touched by the shadow, which will be the end of the 
 eclipse. 
 
 If the eclipse is total, so that the moon’s disc remains some time in the shadow, you 
 must note down the time when it is totally eclipsed, as well as that when it begins to 
 be illuminated, and the moments when the shadow leaves the different spots. 
 
 When this is done, if the time of the commencement of the eclipse be subtracted 
 from that of the end, the remainder will be its duration; and if half the duration 
 be added to the time of commencement, the result will be the middle. 
 
 To facilitate these operations, astronomers have given certain names to most of 
 the spots with which the moon’s disc is covered. The usual denominations are those 
 of Langrenus, who distinguished the greater part of them by the names of astronomers 
 and philosophers who were his contemporaries, or who had flourished before his time. 
 Some others have been since added; but there was no room for the most celebrated 
 of the moderns, such as Huygens, Descartes, Newton, and Cassini. Hevelius, far more 
 judicious in our opinion, gave to these spots names taken from the most remarkable 
 places of the earth: in this manner he calls the highest mountain of the moon, Mount 
 Sinai, &c. This however is a matter of indifference, provided there be no confusion, 
 We have here subjoined a representation of the moon, by means of which and the 
 following catalogue they can be easily known, on comparing the numbers in the 
 latter with those in the former. 
 
442 ASTRONOMY AND GEOGRAPHY. 
 
 ui] \ S 
 So 
 
 V) 
 
 i 
 
 Ing 
 All 
 
 Un 
 
 ——$—— 
 
 \ 
 
 MTT 
 
 \ li 
 
 => 
 
 1 Grimaldi. 
 
 2 Gallileo.. 
 
 8 Aristarchus. 
 4 Kepler.: 
 
 5 Gassendi. 
 
 6 Schikard. 
 7 Harpalus. 
 8 Heraclides. 
 9 Lansberg. 
 10 Rheinhold. 
 1] Copernicus. 
 12 Helicon. 
 13 Capuanus. 
 14 Bullialdi.. 
 
 A Sea of humours. 
 
 B Sea of clouds. 
 C Sea of rain. 
 
 15 Eratosthenes. 
 
 16 Timocharis. 
 
 17 Plato. 
 
 18 Archimedes. 
 
 19 Isle of the middle 
 Bay. 
 
 20 Pittacus. 
 
 21 Tycho. 
 
 22 Eudoxus. 
 
 23 Aristotle. 
 
 24 Manilius. 
 
 25 Menelaus. 
 
 26 Hermes. 
 
 27 Posidonius. 
 
 D Sea of nectar. 
 E Sea of tranquillity. 
 F Sea of serenity. 
 
 28 Dionysius. 
 
 29 Pliny. 
 
 30 Catharina, Cyrillus, 
 Theopilus. 
 
 3] Fracastorius. 3 
 
 32 The acute promontory. 
 
 83 Messala. 
 
 34 Promontory of dreams. 
 
 35 Proclus. 
 
 36 Cleomedes. 
 
 37 Snellius and Furnerius. 
 
 38 Petau. 
 
 39 Langrenus. 
 
 40 Taruntius. 
 
 G Sea of fecundity 
 H Sea of crises. 
 
SOLAR ECLIPSES. 443 
 
 PROBLEM XVII. 
 | To observe an Eclipse of the Sun. 
 Ist. The same precautions, in regard to the measuring of time, must be employed 
 in this case, as in that of lunar eclipses ; that is to say, care must be taken to regulate a 
 good clock by the sun on the day before, or even onthe day of the eclipse. 
 
 2d. A good telescope must be provided, of at least three or four feet in length ; 
 which must be directed towards the sun ona convenient supporter. If you are 
 then desirous to look at the sun without the telescope, you must employ a piece 
 of smoked glass, or rather two pieces, the smoked sides of which are turned towards 
 Bach other ; but are prevented from coming into contact by means of a small dia- 
 phragm cut from a card placed between them. These two bits of glass may be then 
 cemented at the edges, so as to make them adhere. By means of these glasses inter- 
 posed between the eye and the telescope, you may then view the sun without any 
 ‘danger to the sight. 
 _ About the time when the eclipse ought to commence, you must carefully observe 
 the moment when the solar disc begins to be touched by the dise of the moon: 
 his period will be the commencement of the eclipse. If there are any spots on 
 the solar disc, you must observe the time when the moon’s disc reaches them, and 
 also when it again permits them to appear; in the last place, you must observe, with 
 all the attention possible, the instant when the moon’s dise ceases to touch the 
 
 : 
 E 
 
 solar disc, which will be the end of the eclipse. 
 
 But if, instead of observing in this manner, you are desirous of making an ob- 
 
 servation susceptible of being seen by a great number of persons at the same time, 
 
 affix to your telescope, on the side of the eye-glass, an apparatus to support a piece 
 of very straight pasteboard at the distance of some feet. This pasteboard ought to 
 be perpendicular to the axis of the telescope, and if it be not sufficently white, you 
 must paste to it a sheet of white paper. Make the end of the telescope, which 
 contains the object glass, to pass through the window-shutter of a darkened room, 
 or one rendered considerably obscure ; and if the axis of the telescope be directed 
 to the sun, the image of that luminary will be painted on the paper, and of a larger 
 size according as the paper is at a greater distance. It is necessary here to remark, 
 that before you begin to observe, a circle of a convenient size must be delineated on 
 it; so that, by moving it nearer to or farther from the telescope, the image of the 
 sun may be exactly comprehended within it. The space contained within this 
 circle must be .divided by twelve other concentric circles, equally distant from 
 each other; so that the diameter of the largest may be divided into 24 equal parts, 
 each of which will represent a semi-digit. 
 
 It may now be readily conceived, that if a little before the commencement of the 
 eclipse you look with attention at the image of the sun, you will see the moment 
 when it begins to be obscured by the entrance of the moon’s body; and that you 
 may in like manner observe the end of it, and also its extent. 
 
 It must not however be expected that the same exactness can be attained by em- 
 ploying this method, as by the former; especially if you are furnished with a long 
 telescope, and a good micrometer, 
 
 In observing the great solar eclipse of May 15th, 1836, two gentlemen residing 
 at Greenwich, one observing the sun’s image on paper in a dark chamber, and the 
 other looking directly at the sun, found that the time at which they observed the 
 contact of the moon with the spots on the sun, were in several instances identical. 
 
 E 
 
 Remarks.—There are partial eclipses of the sun, that is to say, eclipses in which 
 only a part of the solar disc seems to be covered, and these are most common. 
 Others are total and annular. 
 
444 ASTRONCMY AND GEOGRAPHY. 
 
 Total eclipses take place when the centre of the moon passes over that of the 
 sun, or nearly so; and when the apparent diameter of the moon is equal to that 
 of the sun, or greater. In the latter case, the total eclipse may be what is called 
 cum mord; that is to say, with duration of darkness: of this kind was the famous 
 eclipse of 1706. 7 
 
 During eclipses which are total and cum mord, so great darkness prevails, that 
 the stars are seen in the same manner as at night, and particularly Mercury and 
 Venus. But what excites a sort of terror, is the dismal appearance which all 
 nature assumes during the last moments of the light. Animals, struck with fear, 
 retire therefore to their habitations, sending forth loud cries; the nocturnal birds 
 issue from their holes; the flowers contract their leaves; a coldness is felt, and the 
 dew falls; but as the moon has suffered a few rays of the solar light to escape, all 
 is again illumination; day instantly returns, and with more brightness than when the 
 weather is cloudy. 
 
 Some eclipses, as already said, are really annular: they take place when the 
 eclipse is very near being central, while the apparent diameter of the moon is less 
 than that of the sun; which may be the case if the moon at the time of the eclipse 
 is at her greatest distance from the earth, and the sun at his nearest distance to it. 
 The eclipse of the sun, on the Ist of April 1764, was of this kind to a part of 
 Europe, and also that of May 1836. 
 
 During eclipses of this kind, when the sun is entirely eclipsed, a luminous circle of 
 a silver colour, and as broad as the twelfth part of the diameter of the sun or moon, 
 is often observed around the former ; it is effaced as soon as the smallest part of the 
 sun begins to shine: it appears more lively towards the sun’s limb, and decreases in 
 brilliancy the further it is distant. Some are inclined to believe that this circle is 
 formed by the luminous atmosphere with which the sun is surrounded; others have 
 conjectured that it is produced by the refraction of his rays in the atmosphere of the 
 moon; and some have ascribed it to the diffraction of the light. Those who are 
 desirous of farther information on this subject, may consult the Memoirs of the 
 Academy of Sciences, for the years 1715 and 1748. 
 
 On various occasions, persons who have witnessed the formation and dissolution 
 of the annulus in annular eclipses, have recorded certain singular appearances as 
 having been noticed by them; and the eclipse above referred to of May 15th, 1836, 
 being annular in the north of England and south of Scotland, Francis Bality, 
 of London, well known for his devotion to astronomical science, went to Jedburgh, 
 in Roxburgshire, over which the central line of the moon’s umbra passed on that ~ 
 occasion, to observe the rare phenomenon, and to see what truth there might be in 
 these reported singular appearances. 
 
 It is needless to say that he was provided with all needful instruments; and 
 he got the error of his chronometers from the neighbouring observatory of Sir T. 
 Brisbane, at Makerston. He was fortunate also in meeting at Jedburgh with an 
 able assistant in Mr. Veitch, of that place, a most ingenious mechanic, a self-taught 
 maker of telescopes, and enthusiastically attached to scientific pursuits. The day 
 was uncommonly favourable. 
 
 Mr. Baily, in his account of the eclipse, published in the 10th volume of the Me- 
 moirs of the Royal Astronomical Society, says, ‘‘ The diminution of light, during 
 the existence of the annulus, was not so great as was generally expected; being 
 little more than what might be caused by a temporary cloud passing over the sun; 
 the light however was of a peculiar kind, somewhat resembling that produced by the 
 sun shining through a mist. The thermometer in the shade fell only 3 or 4 degrees; 
 it was 61° during the time of the annulus. About twenty minutes before the formation 
 of the annulus, Venus was seen with the naked eye; and a few minutes afterwards 
 I found it impossible to fire gunpowder with the concentrated rays of the sun 
 
SOLAR ECLIPSES. 445 
 
 through a lens three inches in diameter. The same lens had no effect on the bulb 
 of a thermometer during the existence of the annulus.” ; 
 
 After some further remarks, Mr. Baily goes on to say :—‘ I shall now proceed 
 
 to detail the singular appearances which occurred at the formation and dissolution 
 of the annulus.’’—‘* When the last portion of the moon’s disc was about to enter on 
 the face of the sun, I prepared myself to observe the formation of the annulus.’ I 
 was in expectation of meeting with something extraordinary ; but imagined that it 
 would be momentary only, and consequently that it would not interrupt the noting of 
 the time of its occurrence. Inthis however I was deceived, as the following facts will 
 shew. For when the cusps of the sun were about 40° asunder, a row of lucid points, 
 like a string of bright beads, irregular in size and distance from each other, suddenly 
 formed round that part of the circumference of the moon that was about to enter, 
 or which might be considered as having just entered, on the sun’s disc. Its formation 
 iudeed was so rapid, that it presented the appearance of having been caused by the 
 ‘ignition of a fine train of gunpowder. This I intended to note as the correct time 
 of the formation of the annulus; expecting every moment to see the thread of light 
 completed round the moon; and attributing this serrated appearance (as others had 
 done before me) to the lunar mountains, although the remaining part of the moon’s 
 circumference was comparatively smooth and circular, as seen through a telescope. 
 My surprise however was great on finding that these luminous points, as well as the 
 dark intervening spaces, increased in magnitude, some of the contiguous ones ap- 
 pearing to run into each other like drops of water; for the rapidity of the change 
 was so great, and the singularity of the appearance so fascinating and attractive, that 
 the mind was for the moment distracted, and lost in the contemplation of the scene, 
 so as to be unable to attend to every minute occurrence. Finally, as the moon pur- 
 sued. her course, these dark intervening spaces (which, at their origin, had the ap- 
 pearance of lunar mountains in high relief, and which still continued attached to the 
 sun’s border) were stretched out into long, black, thick, parallel lines, joining the 
 limbs of the sun and moon; when, all at once, they suddenly gave way, and left the 
 circumference of the sun and moon in those points, as in the rest, comparatively 
 smooth and circular; and the moon perceptively advanced on the face of the sun, 
 
 ‘«* The appearances here recorded passed off in less time than it has taken me now to 
 describe them, but they were so extraordinary and so rapid that all idea of time was lost, 
 except by the recollection afterwards of what had-passed; for I was so riveted to 
 
 the scene, that I could not take my eye away from the telescope, to note down any 
 thing during the progress of the phenomenon. I estimate, however, that the whole 
 lasted about six or eight seconds, or perhaps ten at the utmost. 
 - « After the formation of the annulus thus described, the moon preserved its usual 
 circular outline during its progress across the sun’s disk, till its opposite limb again 
 approached the border of the sun, and the annulus was about to be dissolved. When, 
 all at once (the limb of the moon being at some distance from the edge of the 
 _ sun) a number of long, black, thick, parallel lines, exactly similar in appearance to 
 | the former ones above mentioned, suddenly darted forward from the moon, and 
 joined the two limbs as before: and the same phenomena were repeated, but in 
 inverse order. For, as these dark lines got shorter, the intervening bright parts 
 assumed a more circular and irregular shape, and at length terminated in a fine curved 
 line of bright heads (as at the commencement) till they ultimately vanished, and 
 _ the annulus consequently became wholly dissolved.” 
 Mr. Baily says that he shall not attempt to account for this phenomenon: but 
 he says the lines *‘ were as plain, as distinct, and as well detined, as the open fingers 
 of the human hand held up to the light.” 
 It appears that, in a total eclipse April 22, 1715, Dr, Halley noticed similar pheno- 
 
ce 
 
 446 "ASTRONOMY AND GEOGRAPHY. ' 
 
 mena, as did M. Cassini May 22, 1724; ea Ellicot, June 16th, 1810, at total cclipal 
 which they observed. 
 
 An annular eclipse of the sun was observed by Mr. S. Webber, April 3, 179], 
 and precisely similar phenomena were noticed; and like phenomena had been noticed 
 by M. Nicolai and Professor Moll of Leyden. It would appear also that similar 
 phenomena have been noticed in the transits of Venus over the sun’s disc. 
 
 PROBLEM XVIII. 
 To measure the Height of Mountains. 
 
 The height of a mountain may be measured by the common rules of geometry: 
 for if we suppose cs D (Fig. 10), to be a mountain, the perpen- 
 dicular height of which is required, the following method 
 can be employed. If the nature of the adjacent ground will 
 admit, measure a horizontal line a B, in the same vertical plane 
 as the summit.s of the mountain. The greater the extent of 
 this line, the more correct will be the result. At the two 
 stations a and B, measure the angles sa E and s BB, which are 
 the apparent heights of the summit s, above the horizon, when 
 seen from a and B. It will then be easy, by means of plane trigonometry, to find, in 
 the right-angled triangle s £ A, the side E A, as well as the perpendicular s 8, or the 
 elevation of the summit s above A & continued. 
 
 Now let us suppose the vertical line sr to be drawn, intersecting BE in¥F. As, 
 in dimensions of this kind, the angle Es Fr, formed by the vertical line and the per- 
 pendicular s £, will for the most part be exceedingly small, and much below one 
 degree, the lines s E and s F may be considered as equal.* On the other hand, the 
 line F H, comprehended between the line a © and the spherical surface c A, is evidently 
 the quantity by which the real level is lower than the apparent level, in an extent 
 such as A F, or more correctly in a mean length between AF and BF: for this reason 
 take the mean length between aE and BE, which differ very little from a F and B F; 
 and in the table of differences between the apparent and real levels, find the height 
 corresponding to that mean distance: if this height be then added to the height sz 
 ors F, already found, you will have s u for the corrected height of the mountain, 
 above the spherical surface, where the points a and B are situated. 
 
 If it be known how much this surface is higher than the level of the sea, it 
 will be known also how much the summit s of the mountain is elevated above the 
 
 same level. 
 
 Another Method. 
 
 As it may be difficult to establish a horizontal line, so as to be in the same vertical © 
 plane with the summit of the mountain, it will perhaps be better to proceed in the 
 following manner : 
 
 Trace out your base in the most convenient manner, so as to be horizontal: we 
 
 ; shall here suppose that it is represented by a6 (Fig. 11.); let s ¢ be | 
 ig M1. the perpendicular from the summit s to the horizontal plane passing 
 through a b; and let ¢ be the point where this plane is met by the 
 perpendicular: if the lines a c and 6 c be drawn to that point, we — 
 shall have the triangles sacands bc, right-angled at c; and the 
 angles sac and s be may be found by measuring, from the points 
 a and b, the apparent height of the mountain above the horizon: the 
 angles sa 6 ands ba, in the triangle as 6, must also be measured. 
 Now, since in the triangle s ab, the angless a 6 and sbaare known, 
 
 * For even in the case of this angle being a degree, they would not differ'a ten thousandth part, 
 which would suppuse the distance of the stations from the mountain to be more than 100000 yards. 
 
 4 
 
| HEIGHT OF MOUNTAINS. 447 
 
 and also the side ab; any one of the other sides, such for example as sa, may be 
 _ easily determined by plane trigonometry. In the triangle ac s, right-angled at c, as 
 _ the angle s ac is known, the side ac and the perpendicular sc may be found in the 
 _ same manner. When this is done, the method pointed out in the preceding ope- 
 _ ration must be employed: that is to say, find the depression of the level below the 
 
 apparent level for the number of feet or yards comprehended in the line a c, and 
 
 add it to the height s c: the sum will be the height of the point s, above the 
 real level of the points a and 3. 
 
 Ezxample.—Let the horizontal length a b be 2000 yards, or 6000 feet: the angle 
 sab 80° 30’; and the angle s b a 85° 10’; consequently the angle bd s a will be 14° 20’. 
 By means of these data, the side sa of the triangle as b will be found to be 8050 
 yards. On the other hand, if we suppose the angle sac to have been measured, © 
 and to be 18°, the side a ¢ will be found, by trigonometrical calculation, to be 7656 
 yards; and sc, perpendicular to the horizontal plane passing through a b, will be 
 
 ~ found equal to 2488. Now, as the depression of the real level below the apparent 
 level at the distance of 7656 yards, is 121 feet, or 4 yards 6 inches,* if this quantity 
 be added to the height sc, we shall have 2492 yards 6 inches for the real height 
 of the mountain. 
 
 Remark.—When either of these methods is employed, if the mountain to be 
 ‘measured is at a considerable distance, such as 20000 or 40000 yards, as its summit 
 in that case will be very little elevated above the horizon, the apparent height must 
 be corrected by making an allowance for refraction, otherwise there may be a very 
 considerable error in the result. The necessity of this correction may be easily con- 
 
 ceived by observing, that the summit c of the mountain B c 
 “ts Fig. 4. (Fig. 12), is seen by a ray of light = ca, which is not rectili- 
 neal, but bent; so that the summit cis judged to be in p, 
 according to The direction of the line a D, a tangent to the 
 curve A C E, which in the small space a c may be considered 
 as the arc of acircle. The angle p a B therefore, of the ap- 
 parent height of the mountain, exceeds the height at which the summit would appear 
 without refraction, by the quantity of the angle c aD; which must be determined. But 
 it will be found that this angle c a D is nearly equal to half the refraction which 
 would belong to the apparent height D a B. You must therefore find, in the tables, 
 the refraction corresponding to the apparent height p a B of the mountain, and sub- 
 tract the half of it from that height: the remainder will be that of the summit of the 
 mountain, such as it would be seen without refraction. 
 
 Let us suppose, for example, that the summit of the mountain seen at the distance 
 of 20000 yards appears to be elevated above the horizon 5 degrees: the refraction 
 corresponding to 5 degrees is 9’ 54”, the half of which is 4’ 57”; if 4’ 57” therefore 
 be subtracted from 5°, the remainder will be 4° 55’ 3” which must be employed as 
 the real elevation.t 
 
 It may thence be seen, that, to proceed with certainty in such operations, it will 
 be necessary to make choice of stations at a moderate distance from the mountain ; 
 so that its summit may appear to be elevated several degrees above the horizon ; 
 otherwise the difference of the refraction, which is very variable near the horizon, 
 will occasion great uncertainty in the measurement. 2 
 
 We shall give hereafter another method for measuring the height of mountains, by 
 means of the barometer ; but in this case it is supposed that it is possible to ascend 
 
 * See the table in the additional remark. 
 
 + Montucla here employs the common tables of refraction used for nautical and astronomical 
 purposes, such as that given in ‘‘ Riddle’s Navigation,” and other works of the kind. In regard to 
 mda refraction, and the allowance wade for it, see the additional remark at the end of this 
 article. 
 
q 
 
 «I 
 
 . 
 
 ae - ‘ 
 +7 
 f 
 t 
 
 448 ASTRONOMY AND GEOGRAPHY. 
 
 to the summit of them. We shall also give a table of the heights of the principat 
 mountains of the earth above the level of the sea. We shall here only observe that 
 the highest mountains in the world, at least in that part of it which has hitherto been 
 accessible to scientific men, are situated in the neighbourhood of the equator; and it 
 is with justice that an historian of Peru says, that when compared with our Alps 
 and our Pyrenees, they are like the towers and steeples of the churches in our cities, 
 compared with common edifices. The highest yet known is one in the Himalayan 
 range in India, which rises nearly 27000 feet in a perpendicular direction above the 
 level of the sea. 
 
 As all the known mountains in Europe are scarcely two-thirds of the ieee of 
 those enormous masses, the falsity of what the ancients, and some of the moderns, 
 such as Kircher, have published respecting the height of mountains, will readily 
 appear. According to these authors, tna is 4000 geometrical paces in height; the 
 mountains of Norway 6000; Mount Heemus and the Peak of Teneriff 10000; Mount 
 Atlas and the Mountains of the Moon in Africa 15000 ; Mount Athos 20000; Mount 
 Cassius 28000. It is asserted that these heights were found by means of their sha- 
 dows; but nothing is more destitute of truth, and if ever any observer ascends to the 
 summit of these mountains, or measures their height geometrically, they will be found 
 very inferior to the mountains of Peru; as is the case with the Peak of Teneriff, 
 which when measured geometrically was found to be only about 7000 feet. 
 
 Hence it appears that the elevation of the highest mountains is very little, when 
 compared with the diameter of the earth, and that its regular form is not sensibly 
 altered by them; for the mean diameter of the earth is about 79573 miles; thereforé 
 if we suppose the height of a mountain to be 34 miles, it will be only the 2273d part 
 of the diameter of the earth, which i is less than the elevation of half a line ona gion 
 six feet in diameter. 
 
 Additional Remark.— As Montucla has not here explained the method of finding 
 the difference between the apparent and true level, we think it necessary to add a | 
 few observations on the subject. ‘Two or more places are said to be ona true level, 
 when they are equally distant from the centre of the earth. One place also is higher 
 than another, or out of level with it, when it is farther from the centre of the 
 earth; anda line equally distant from that centre in all its parts, is called the line 
 of true level. Hence, because the earth is round, that line must be a curve, or at 
 least parallel or concentric to it. But the line of sight, given by operations of level- 
 ling, which is a tangent, or a right line perpendicular to the semi-diameter of the 
 earth at the point of coutact, always rising higher above the true curve line of level 
 the farther the distance, is called the apparent line of level; and the difference 
 between the line of true level and the apparent, is always equal to the excess of the 
 secant of the arch of distance above the radius of the earth. Hence it will be found 
 that this difference is equal to the square of the distance between the places, divided 
 by the diameter of the earth ; and consequently it is always proportional to the square 
 of the distance. 
 
 From these principles is obtained the following table, which shews the height of 
 the apparent above the true level for every 100 yards of distance on the one hand, 
 aud for every mile on the other. 
 
 The common methods of levelling are sufficient for laying pavements of walks, or 
 for conveying water to small distances, &c.; but in more extensive operations, asin _ 
 levelling the bottoms of long canals, which are to convey water to the distance of 
 many miles, and such like, the difference between the true and apparent level must 
 be taken into account. 
 
HEIGHT OF MOUNTAINS. 449 
 ® 
 
 BD PTY of 3 Diff. of eee Hie. of Oe a OF 
 inst. | Level. piek: Level. eats Level. Dist, | Level. 
 Yards. | Inches. Yards Inches. Miles. } Ft. In. Miles. Ft. Ins 
 100 | 0°026 1000 | 2°570 4 0 of 7 32 6 
 200 | 0:103 1100 | 3°110 4 0 2 8 42 6 
 300 | 0:231 1200 | 3°701 3 o 44 9 53. 9 
 400 | 0°411 1300 | 4°344 1 02-8 10 66 4 
 500 | 0°'643 1400 | 5°038 2 2 8 11 80 3 
 600 | 0'925 1500 | 5°'784 3 6 0 12 95 7 
 700 | 1°260 1600 | 6°580 4 Lowey 13 112 2 
 800 | 1°645 1700 | 7°425 | 5 16 7 14 180.1 
 900 | 2:081 $ ' (OCG 23 11 
 
 By means of these tables of reductions, the difference between the true and appa- 
 
 rent level can be found by one operation; whereas the ancients were obliged to 
 employ a great many; for being unacquainted with the correction answering to any 
 distance, they levelled only from one 20 yards to another, as they had occasion to 
 continue the work to some considerable extent. 
 
 These tables will answer several useful purposes: First, to find the height of the 
 apparent level above the true, at any distance. If the given distance be contained 
 in the table, the correction of the level will be found in the same line with it. For 
 example, at the distance of 1000 yards the correction is 2°57, or nearly two inches 
 and a half; and at the distance of ten miles, it is 66 feet 4 inches. But if the 
 exact distance be not found in the table, multiply the square of the distance in yards 
 by 2°57, and divide by 1000000, or cut off six places on the right for decimals, the 
 rest will be inches ; or multiply the square of the distance in miles by 66 feet 4 inches, 
 and divide by 100. 
 
 2d. To find the extent of the visible horizon, or how far can be seen from any given 
 height on a horizontal plane, as at sea, &c. Let us suppose the eye of an observer 
 on the top of aship’s mast at sea, to be at the height of 130 feet above the water, 
 it will then see about 14 miles all around; or from the top of a cliff by the sea 
 side, the height of which is 66 feet, a person may see to the distance of nearly 10 
 miles on the surface of the sea. Also, when the top of a hill, or the light in a 
 light-house, the height of which is 130 feet, first comes into the view of an eye on 
 
 board a ship, the table shews that the distance of the ship from it is ]4 miles, if 
 
 the eye be at the surface of the water; but if the height of the eye in the ship 
 be 80 feet, the distance will be increased by nearly 11 miles, making in all about 
 25 miles. 
 
 3d. Suppose a spring to be on the one side of a hill, anda house on an opposite 
 
 hill, with a valley between them, and that the spring seen from the house appears, 
 by a levelling instrument, to be on a level with the foundation of the house, which 
 we shall suppose to be at the distance of a mile from it: this spring will be 8 inches 
 | above the true level of the house; and that difference would be barely sufficient for 
 the water to be brought in pipes from the spring to the house, the pipes being 
 laid all the way under ground. 
 
 4th. If the height or distance exceed the limits of this table: Then first, if the 
 distance be given, divide it by 2, or by 3, or by 4, &c., till the quotient come withic, 
 _the distances in the table; then take out the height answering to the quotient, and 
 multiply it by the square of the divisor, that is by 4, or by 9, or by 16, &c., which 
 will give the height required. Thus, if the top of a hill be just seen at the distance 
 of 40 miles; then 40 divided by 4, is 10, and opposite to 10 in the table will be 
 found 663 feet, which multiplied by 16, pies of 4, gives 106]4 for the height 
 
450 ASTRONOMY AND GEOGRAPHY. 
 
 of the hill. But when the height is given, divide it by one of these square numbers, 
 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 distance sought. Thus, when the top of the peak of Teneriff, said to be about 
 3 miles or 15840 feet high, just comes into view at sea, divide 15840 by 225, or the 
 square of 15, and the quotient is 70 nearly, to which in the table corresponds by pro- 
 portion nearly 10? miles; which multiplied by 15, will give 154 miles and 3 for the 
 distance of the mountain. 
 
 In regard to the terrestrial refraction, which in measuring heights is to be taken into 
 account also, as it makes objects to appear higher than they really are, it is estimated 
 by Dr. Maskelyne at 4, of the distance observed, expressed in degrees of a great 
 circle. Thus if the distance be 10000 fathoms, its 10th part 1000 fathoms is the 60th 
 part of a degree on the earth, or 1’, which is therefore the refraction in the altitude 
 of the object at that distance. 
 
 Le Gendre, however, says he is induced by several experiments to allow only 7th 
 part of the distance for refraction in altitude. So that upon the distance of 10000 
 fathoms, the 14th part of which is 714 fathoms, he allows only 44” of terrestrial re- 
 fraction, so many being contained in the 714 fathoms. 
 
 Delambre, an ingenious French astronomer, makes the quantity of terrestrial re- 
 fraction to be the 11th part of the arch of distance. But the English measurers, 
 Col. Ed. Williams, Capt. Mudge, and Mr. Dalby, from a multitude of exact observa- 
 tions made by them, determine the quantity of refraction to be the 12th part of the 
 said distance. The quantity of this refraction however is found to vary, with the 
 different states of the weather and atmosphere, from the 15th part of the distance 
 to the 9th part; the medium of which is the 12th, as above mentioned. 
 
 PROBLEM XIX. 
 
 Method of knowing the Constellations. 
 
 To learn to know the heavens, you must first provide yourself with some good 
 celestial charts, or a planisphere of such a size, that stars of the first and second mag- 
 nitude can be easily distinguished. At the end of the present article we shall point 
 out the best works on this subject. 
 
 Fig. 13. Having placed before you one of these 
 HC yguus charts, that containing the north pole, turn 
 “1 your face towards the north, and first 
 +t find out the great bear, commonly called 
 Arcturus + tae Charles’s wain (Fig. 13.) It may be easily 
 Bear known, as it forms one of the most remark- 
 Gp *, | able groups in the heavens, consisting of 
 Bear 8" E* Cassi i seven stars of the second magnitude, four 
 + “Star peta : ; 
 +7 of which are arranged in such a manner as 
 pa. to represent an irregular square, and the 
 erseus: * . : 
 Copella 4 other three a prolongation in the form of 
 
 a very obtuse scalene triangle. Besides, 
 by examining the figure of these seven stars, as exhibited in the chart, you will easily 
 distinguish those in the heavens which correspond to them. When you have made 
 yourself acquainted with these seven principal stars, examine on the chart the con- 
 figuration of the neighbouring ones, which belong to the Great Bear; ard you will 
 thence learn to distinguish the other less considerable stars which compose that 
 constellation, 
 
 After knowing the Great Bear, you may easily proceed to the Lesser Bear; for 
 nothing will be necessary but to draw, as seen in the annexed figure, a straight line 
 through two anterior stars of the square of the Great Bear, or the two farthest’ 
 
| 
 THE CONSTELLATIONS. 451 
 
 distant from the tail: this line will pass very near the polar star, a star of the second 
 ‘magnitude, and the only one of that size in a pretty large space. At a little distance | 
 from it, there are two other stars of the second and third magnitude, which, with four 
 ‘more of a less size, form a figure, somewhat similar to that of the Great Bear; but 
 smaller, This is what is called the Lesser Bear; and yéu may learn, in the same 
 manner as before, to distinguish the stars which compose it. 
 
 Now, if a straight line be drawn through those stars of the Great Bear, nearest to 
 the tail, and through the polar star, it will conduct you toa very remarkable group 
 of five stars arranged nearly in this form 4: these are the constellations of Cassi- 
 opeia, in which a very brilliant new star appeared in 1572; though soon after it 
 became fainter, and at length disappeared. . 
 
 If a line, perpendicular to the above line, be next drawn, through this constella- 
 tion, it will conduct, on the one side, to a very beautiful star called Algenib, which 
 is in the back of Perseus; and, on the other, to the constellation of the Swan, re- 
 ‘markable by a star of the first magnitude. Near Perseus is the brilliant star of the 
 Goat, called Capella, which is of the first magnitude, and forms part of the constel- 
 lation of Auriga. 
 
 After this, if a straight line be drawn through the two last stars of the tail of the 
 Great Bear, you will come to the neighbourhood of Arcturus, one of the most 
 brilliant stars in the heavens, which forms part of the constellation of Bootes. 
 
 In this manner you may successively employ the knowledge you have obtained 
 of the stars of one constellation, to enable you to find out the neighbouring ones. 
 We shall not enlarge farther on this method ; for it may be easily conceived, that we 
 cannot proceed in this manner through the whole heavens: but any person of inge- 
 nuity, in the course of a few nights, may learn by these means to know a great part 
 of the heavens ; or at any rate the principal stars. 
 
 The ancients were not acquainted with, or rather did not insert into their cata- 
 logue, more than 1022 fixed stars, which they divided into 48 constellations; but 
 their number is much greater, even if we confine ourselves to those which can be 
 distinguished by the naked eye. The Abbé de la Caille observed 1492 in the small 
 space comprehended between the tropic of Capricorn and the south pole; a part 
 of which he formed into new constellations. But this space is to the whole sphere, 
 as 3 to 10 nearly; so that in our opinion, the whole number of the stars visible to 
 the naked eye may be estimated at about 6500. It is a mere illusion that makes us 
 conclude, on the first view, that they are innumerable ; for if you take a space come 
 prebended between four, five, or six stars of the second and third magnitude, and try 
 to count those it contains, you will find that it can be done without much difficulty ; 
 and some idea may be thence formed of their total number, which will not much 
 exceed that above stated. 
 
 The stars are divided into different classes, viz., stars of the first, second, third, 
 _&c. magnitude, as far as the sixth, which are the smallest perceptible to the naked 
 eye. There are 20 of the first maguitude, 76 of the second, 223 of the third, 512 
 of the fourth, &c. 
 
 __ In regard to the constellatious, the number of those commonly admitted is 90; 
 of which 33 belong to the northern hemisphere, 12 to the Zodiac, and the remaining 
 45 to the austral or southern hemisphere. We shall here give a catalogue of them, 
 with the number of the principal stars of which each is composed, together with the 
 names of some of the most remarkable stars: the constellations which have this 
 mark * against them, are modern ones, the others ancient. The figures placed against 
 the principal stars, denote their magnitudes. 
 
 2a2 
 
452: ASTRONOMY AND GEOGRAPHY. F 
 
 I. PRINCIPAL CONSTELLATIONS NORTH OF THE ZODIAC. 
 
 No. Constellations. |N°-of] Chief Stars. No. | Constellations. |N9-°f] Chief Stars. 
 Stars Stars 
 1 | Ursa Minor 24 | Pole Star 2] 19 | Serpens -| 50 
 2 | Ursa Major 105 | Dubhe 1 | 20 | ScutumSobieski] 76 
 3 | Perseus 72 | Algenib 2] 21 | Hercules cum 
 4 | Auriga 66 | Capella 1 Ramo et Cer- 
 5 |*Bootes 54 | Arcturus 1 bero 117 | Ras Algiatha3 
 6 | Draco 84 | Rastaber 3 | 22 |*Serpentarius 
 7 |*Cepheus 51 | Alderamin 3 siveOphiucus/142 } Ras Alhagus 3 
 8 |*Canes Venatici] | 23 |*Taurus Ponia- 
 scil. Asterian towski 7 
 et Chara 63 24 | Lyra 24 | Vega 1 
 9 '*Cor Caroli 3 25 *Vulpecula et 
 10 |*Triangulum 10 Anser- 36 
 1] | Triangulum 26 | Sagitta 12 
 minus 5 27 | Aquila 40 | Altair 1 
 12 |*Musca 6 28 | Delphinus all b 
 13 |*Lynx 48 29 |*Cygnus 82 | Deneb Adige 1 
 14 |*Leo Minor 59 30 |*Equuleus 10 
 15 *ComaBerenices| 45 31 |*Lacerta 16 
 16 |*Camelopardalus| 78 32 |* Pegasus 88 | Markab 2 
 17 |*Mons Menelaus} 11 33 |*Andromeda 71 | Almaac 2 
 
 18 | CoronaBorealis, 21 
 
 II. CONSTELLATIONS IN THE ZODIAC. 
 
 No. Constellations. f oes Chief Stars. No.}| Constellations. ee Chief Stars. 
 
 1 eA vies 67 7 | Libra 55 |Zubenich Mali2 
 2 | Taurus 143 |Aldebaran 1 8 | Scorpio 37 |Antares 1 
 3 | Gemini 87 |Castor and 9 | Sagittarius 73 
 
 Pollux 1.2 | 10 | Capricornus 54 
 4 | Cancer 87 11 | Aquarius — 119 |Scheat 3 
 5 | Leo 101 |Regulus 1 | 12 | Pisces 115 
 6 | Virgo 117 |SpicaVirginis 1 : 
 
 III. PRINCIPAL CONSTELLATIONS SOUTH OF THE ZODIAC. 
 
 No. | Constellations. R ie Chief Stars. No. | Constellations. iecee Chief Stars. 
 1 |* Phoenix 13 Canis Major | 31 | Sirius 1 
 2 |\*Officina Sculp- *Equuleus Pic- 
 toria 12 torius 8 
 3 | Eridanus 76 | Achernar *Monoceros 31 
 4 |*Hydrus 31 Canis Minor |-18 | Procyon MM 
 5 |*Cetus 70 | Menkar *Chameleon 10 
 6 |*FornaxChemica] 14 *Pyxis Nautica | 4 
 7 \*Horologium 12 *Piscis Volans 8 
 8 |*Reticulus Hydra 43 | Cor Hydre 1. 
 Rhomboidalis} 10 *Sextans 43 4 
 9 )*Xiphias - 7 *Robur Caroli- ! 
 10 |*Celapraxitellis | 16 num 12 
 11 |*Lepus 18 *Machina Pneu- ; 
 12 |*Columba Noa. matica 3 : 
 chi 10 *Crater 11] Alkes 3. 
 13 | Orion 70 | Betelguese 1 }| 27 |*Corvus 9 | Algorab | 
 14 |] Argo Navis 43 | Canopus 1 | 28 |*Crosiers 6 im 
 
CELESTIAL CHARTS. 453 © 
 
 No.| Constellations. og Chief Stars. No. Constellations. ye off Chief Stars. 
 % tars 4 Stars 
 
 29 |*Musca 4 38 |*Corona Aus- 
 
 30 |* Apis Indica 11 tralis 12 
 
 31 |*Circinus 124, 39 |*Pavo 14 
 
 32 | Centaurus 36 40 |*Indus 12 
 
 33 |*Lupus 24 41 |*Microscopium | 10 
 
 34 *QuadraEuclidis| ‘ 42 |*Octans Hadlei- 
 
 35 |*Trangulum Aus- | anus 43 
 : trale 43 |*Grus 14 
 
 36 | Ara 9 44 |*Toucan 9 
 37 |* Telescopium 9 45 | Piscus Austra- 
 lis | 20 | Fomalhaut 1 
 
 IV. NUMBER OF STARS OF EACH MAGNITUDE. 
 
 — 
 = Magnitudes. Total 
 Constellations, a | Number 
 US I, | If. iil. Iv. Vie Vii of Stars. 
 In the Zodiac 12 16 44 | 120 | 183 646 1014 
 
 5 
 Ir. the N. Hemisphere | 33 6 | 24 95 | 200 | 291 635 1251 
 In the S. Hemisphere 45 9 | 36 | 84] 190 | 221 | 323 865 
 
 90 | 20 | 76 | 223 | 512 | 695 | 1604 3130 
 
 > 
 
 We shall not here enter into any physical details respecting the stars; as we re- 
 serve these for another place, where we shall speak of their distances, magnitudes, 
 notion, and various other things relating to this subject; such as new stars, change- 
 ible or periodical stars, &c. 
 
 The best celestial charts were for a long time those of Bayer’s Uranometria, a work 
 
 n folio, published in 1603,,and which has gone through a great many editions. But 
 these charts have given place to the magnificent Celestial Atlas of Flamsteed, pub- 
 ished in folio at London, in 1729; a work indispensably necessary to every practical 
 istronomer. Ofthe other charts or planispheres, those of Pardies, published in 1673, 
 n six sheets, magnificently engraved by Duchange, are esteemed. We have also the 
 wo planispheres of De la Hire, in twosheets. Senex, an English engraver, published 
 ikewise two new planispheres, according to the observations of Flamsteed ; one of 
 hem in two sheets, where the two hemispheres are projected on the plane of the 
 ‘quator ; and the other where they are projected on the plane of the ecliptic. Those 
 vho have not the Celestial Atlas of Flamsteed must provide themselves with either 
 if these planispheres. ‘The modern astronomers, and particularly La Caille, having 
 dded a great number of new constellations to the old ones in the southern hemis- 
 jhere, two new planispheres have on that account been formed. One of them, by 
 ML Robert, consists of two sheets, where the ground of the heavens is coloured blue ; 
 o that the constellations are very distinctly seen. It is constructed according to the 
 -ewest observations; and it is accompanied with useful instructions respects ths 
 aethod of knowing the heavens. 
 
 As it is of the greatest importance to astronomers, to be acquainted with the con- 
 tellations and stars of the Zodiac, because the planets move in that circular band, 
 jenex, before mentioned, published, about half a century ago, The Starry Zodiac, 
 rom Flamsteed’s Observations ; and as it was difficult to be procured at Paris, the 
 eur Dheuland, engraver, gave, in 1755, a new edition of it; with such corrections 
 s the interval between that period and the time when Senex published his edition, 
 ad rendered necessary. He was directed in this undertaking by M. de Seligny, a 
 
. 454 ; ASTRONOMY AND GEOGRAPHY. 
 
 young officer in the service of the East-India Company. To the Zodiac of Dheuland 
 is annexed a minute catalogue of the Zodiacal stars, with their longitudes and latitudes, 
 reduced to the year 1755. This catalogue comprehends 924 stars; but the author, 
 to render his work more useful in nautical observations, gives to his Zodiac ten 
 degrees of latitude, on each side of the ecliptic. It may be readily seen, from what 
 has been here said, that those who are not possessed of the Celestial Atlas of 
 Flamsteed, must procure the Zodiac and Catalogue of Dheuland, or rather of Seligny, 
 and that even possessing the former work does not supersede the necessity of the 
 latter. 
 
 A new edition of Flamsteed’s Atlas, reduced to a third of its original size, has 
 since been published, with a planisphere of the austral stars observed by La Caille. 
 M. Fortin, the author, reduced all the stars to the year 1780; and added a chart 
 of the stars representing the different figures which they form, together with their 
 relative positions. 
 
 To the above list we may add the large Celestial Atlas lately published by Pro- 
 fessor Bode, ot Berlin, consisting of twenty sheets. 
 
 Remark.—Since the period when mankind began to observe the stars, various 
 astronomers, at different times, have undertaken to exhibit, in charts, their places, 
 relative distances, and magnitudes. To the works of this kind before mentioned, we 
 may add also the Calum Stellatum of Julius Schiller, 1627; the Firmamentum Sobes- 
 cianum of Hevelius, 1690, in 54 sheets; and Doppelmayer’s Celestial Atlas, Nurem- 
 berg 1742. In the year 1729 Flamsteed’s Celestial Atlas was published in 28 sheets, 
 containing 2919 stars, observed by that astronomer at Greenwich, and divided into 
 56 constellations. In the year 1776, an edition of it, reduced to the quarto form, 
 was published at Paris by Fortin, in 30 sheets; in the year 1796 La Lande and 
 Mechain published the same plates, considerably improved and enlarged with seven 
 new constellations. In the year 1782 M. Bode published the same Atlas in 34 sheets, 
 small folio; but he added, besides the old observations, a great many new ones, and 
 above 2100 fixed stars and nebule. In the year 1748, a new Uranographia, of the 
 same kind as that of Bayer, to consist of 50 sheets, was announced to be published 
 by subscription in England. Dr. Bevis, a noted astronomer, was at the head of 
 this undertaking, and some of the sheets were engraved; but the work was never 
 completed.* The Atlas now published by professor Bode, in 20 sheets, is constructed 
 according to an entirely new projection, Flamsteed’s charts were each 21 inches in 
 breadth and 28 inches in length; those of Bode’s Atlas are 26 inches in breadth and 
 38 in length. Flamsteed’s Atlas contains only 56 constellations on 28 sheets; that 
 of Bode contains 106 on 18 sheets, together with the stars around the south pole, 
 and two hemispheres. Of late years, by the continued assiduity of astronomers, 
 the number of stars observed has been much increased. Dr. Herschel, with his excellent | 
 telescopes, has discovered above 2500 nebule, groups of stars, and double stars. 
 Baron von Zach of Gotha constructed a new and complete catalogue of the fixed 
 stars, from his own observations ; but Professor Bode for the greatest number of his | 
 improvements was indebted to La Lande. This meritorious astronomer supplied him 
 at different times with new stars, amounting altogether to about 6000, which were 
 observed by himself and his nephew Le Francois, at the Military school, with a mural 
 quadrant by Bird. But the first manuscripts transmitted by La Lande, contained the 
 right ascensions only to minutes of time; and consequently were not accurately 
 enough defined for the large scale on which these charts are constructed. Professor 
 
 _* Another little known Celestial Atlas, which at least is not mentioned by La Lande, is that of Cor- 
 binianus Thomas, a Benedictine and professor of mathematics at Augsburg. It is entitled “ Fir 
 mamentum Firmianum,”’ in honour of the then bishop of the house of Firmian, and was published 
 a Augsburg in small folio, in the year 1731. In this Atlas the northern crown is called “ Corona 
 
 irmiana. 
 
 | 
 P| 
 
CELESTIAL CHARTS. 455 
 
 Bode therefore inserted only some of these stars into his charts, being obliged to 
 leave out the greater part of them. La Lande sent afterwards more correct positions ; 
 and though the professor encountered many difficulties in reducing them, in conse- 
 quence of errors in the transcribing or calculation, he was enabled to add to his charts 
 some thousands of new stars, furnished by the above astronomer. The professor 
 however found several vacuities, and being desirous that the improvement introduced 
 into his work shonld be uniform, he resolved to supply these deficiencies from his 
 own observations. He began therefore in the month of December 1796, at the royal 
 observatory of Berlin, to search for and observe new stars, with a mural quadrant by 
 Bird; and by these means was enabled to enrich his Atlas with some hundreds of 
 stars, of the 6th and 7th magnitudes, not to be found in any of the catalogues. 
 
 Flamsteed, for his charts, made choice of a kind of projection by which, especially 
 under great declinations, no proper idea is given of the real figure of the circles of 
 the sphere. In these charts the parallels to the equator are straight lines, which 
 intersect the meridians, where the cosines of their distance from the mean meridian 
 falls. They appear therefore as crooked lines; the meridians or great circles appear 
 alse crooked, and the parallels or less circles straight lines, entirely contrary to the 
 real form which these circles of the sphere exhibit. Professor Bode therefore made 
 choice of another kind of projection, namely, that conical projection described by 
 Kastner in his Geometrical Treatises, and in which the semi-diameter of the mean 
 parallel is the cotangent of its declination. The mean meridian, on the other 
 hand, is lengthened where these cotangents fall; and from this point as a centre 
 are drawn the parallel circles at every 5 degrees. At this centre the vaiue of the 
 angle of right ascension, for example 10 degrees, is made = sin. decl. 10°; and the 
 meridians are drawn as straight lines. By this construction the degrees of ascension 
 are kept in the proper proportion to those of declination, in the mean zones lying be- 
 tween the parallels, as far as they extend east or west; and the principal stars which 
 each sheet exhibits, fall in these mean zones, Each sheet generally contains about 75°, 
 on the equator, of right ascension, and 54° in declination. When the equator falls in 
 the middle of the chart, the parallels and meridians are straight lines, placed at equal 
 distances, and intersecting each other at right angles. ‘The polar regions are deli- 
 neated according the sterecgraphic projection. The scale of these charts, the two 
 polar ones excepted, is 10° declination to 4 inches English. 
 
 The names of all the constellations are given in Latin, according to the general 
 practice; the original constellations, when they form the principal figures in the 
 chart, are completely shaded; but in such a manner that the smallest stars and the 
 nebulous spots are apparent. The names are given in large Roman shaded characters. 
 - The constellations introduced in modern times are shadedin the punctured manner ; 
 and the names are adged in large open Roman characters. Besides the Arabic and 
 Latin names already known, the old Arabian names are also added to many of the stars. 
 The epoch of the right ascension of these stars is fixed at the lst of January, 1801. 
 
 The ‘Society for the Diffusion of Useful Knowledge have published two sets of 
 celestial maps on the gnomonic projection. 'The smaller is of a quarto size, and the 
 larger of 25 inches square, and forms one of the most useful celestial Atlases hitherto 
 published. The late Professor Harding, well known to astronomers as the discoverer 
 of the planet Juno, published an Atlas of the heavens, which is considered exceedingly 
 accurate—especially that part of it where planets may be expected toappear. At the 
 death of the. Professor many copies of this valuable Atlas were in possession of the 
 family, and several copies were purchased on the occasion by English astronomers. 
 
 An Atlas has for some time been in progress of construction from actual observations 
 made by several astronomers in Germany and one or two in England, each taking 
 a separate part of the heavens and filling up from his observations skeleton forms with 
 which he is furnished. This work when finished will doubtless beof standard character. 
 
456 ASTRONOMY AND GEOGRAPHY. 
 
 CHAPTER II. 
 
 A SHORT VIEW OF THE PRINCIPAL FACTS IN REGARD TO PHYSICAL ASTRONOMY, 
 OR THE SYSTEM OF THE UNIVERSE. 
 
 THERE is no difference of opinion at present among enlightened philosophers, in 
 regard to the position of the planets and of thesun. All those capable of estimating 
 the proofs deduced from astronomy and physics, admit that the sun occupies the 
 centre of an immense space, in which the following planets revolve around him at 
 different distances, viz., Mercury and Venus; the Earth, always accompanied by 
 the moon; Mars; Pallas and Vesta, discovered by Dr. Olbers; Ceres, discovered by 
 M. Piazzi; Juno, by Harding ; ; and Jupiter, followed by his four moons or satellites ; 
 Saturn, surrounded by his ring, and accompanied by seven satellites; the Georgian 
 planet, discovered by Dr. Herschel, together with its satellites; and lastly a great 
 number of comets, which have been shewn to be nothing else but planets having 
 orbits very much elongated. 
 
 The path in which each of the planets moves around the sunis not a circle, but an 
 ellipsis more or less elongated, in one of the foci of which that luminary is placed ; 
 so that when the planet is at the extremity of the axis, beyond the centre, it is at 
 its greatest distance from the sun; and when at the other extremity of that axis, it is 
 at its nearest distance. This ellipsis however is not very much elongated: that de- 
 scribed by Mercury is the most of all of the ancient planets; for the distance of its 
 focus from the centre is equal to a fifth part of its semi-axis. That of Venus is 
 nearly acircle. In the orbit of the earth, the distance from the focus to the centre 
 is only about a 57th part of the semi-axis. The last discovered planet, Pallas, it is 
 said has its orbit the most elongated of any, its Ps being about one third 
 of its mean distance from the sun. 
 
 The motion of all these bodies around the sun is regulated by two celebrated laws, 
 the discovery of which has rendered the name of Kepler immortal. The first of 
 these laws, which relates to the motion of a planet in the different points of its orbit, 
 is, that it always movesin such a manner, that the area described by the radius vector, 
 
 Fig. 14. or the straight line drawn from the planet to the sun, increases 
 uniformly in equal times, or is always proportional to the time; 
 ~ so that if a planet, for example, employs 30 days in moving from 
 
 A to x (Fig. 14.) and 26 in moving from x to p, the mixtilineal 
 
 area A S x will be to the mixtilineal area x s p, as 30 to 20; or © 
 Asm is to asp, as 30 to 50, or as 3to 5. In double the time 
 therefore this area is double, and so on; whence it follows, that 
 when the planet is at its greatest distamee, it moves with the 
 least velocity in its orbit. The ancients laboured under 
 
 a mistake, when they imagined that the retardation which 
 they observed in the motion of any of the heavenly bodies, such as the sun for 
 example, was a mere abe illusion: this retardation is partly real, and partly 
 apparent. 
 
 The second law, discovered by Kepler, is that which regulates the distances 
 of the planets from the sun, and their periodical times, or the times of their revo- 
 lutions. According to this law, the cubes of the mean distances of two planets 
 from the sun, around which they perform their revolutions, are always in proportion © 
 to each other as the squares of their periodical times; thus, if the mean distances 
 of two planets from the sun, be the one double of the other, since the cubes of these 
 distances will be as 1 to 8, the squares of the periodical times will be as 1 to 8; con- 
 sequently the times themselves will be to each other as 1 to the square root of 8, 
 which is 23 nearly. 
 
a 
 
 OF THE SUN. 457 | 
 
 This rule holds good, not only in regard to the principal planets, those which 
 revolve about the sun, but also in regard to the secondary planets, which revolve 
 around a primary planet, as the four satellites of Jupiter and the seven satellites 
 of Saturn. If the earth had two moons, they also would observe this law in regard 
 to each other by a mechanical necessity. 
 
 These two laws, first discovered by Kepler, from his observations and those 
 of Tycho Brahe, were afterwards confirmed and proved by Newton, from the princi- 
 ples and laws of motion; so that those who deny truths so well established, must be 
 incapable of feeling the force of a demonstration. 
 
 We know of no secondary cause that could have any influence in regulating the 
 distances of the planets from the sun, yet there appears a relation between the dis- 
 tances which is too remarkable to be considered accidental. This was first remarked 
 by Bode of Berlin, who remarked that a planet was wanting to complete the rela- 
 tion; and that want has since been supplied by the discovery of the four new planets 
 
 - at almost the precise distances from the sun which Bode suggested that a planet 
 
 ought to be. According to Bode (and it is nearly true), the distances of the planets 
 may be expressed as follows :—that of the Earth being 10. 
 
 MEGHCULY Cis vas ow eicts we engl | 
 Venus he stiecdee: 2p eas OF 
 Barts, Poeeee 0 2? - B2h ==97 10 
 Mara cine vadeuecg Bart Hoa 2 a1G 
 New planets ....000. 2? + 3:23 = 28 
 Jupiter..ccceseseeeee 22+ 3:2! = 52 
 Saturn .......e-+.0.. 2? + 3:2) = 100 
 Georgium Sidus..... - 27-4 3°28 = 196 
 
 We shall now lay before our reader every thing most remarkable in regard to those 
 celestial bodies of which we have any knowledge, beginning with the sun. They 
 who can behold this sublime picture without emotion, ought to be classed among 
 those stupid beings whose minds are insensible to the most magnificent works 
 of the Deity. 
 
 I.—Of the Sun. 
 
 The sun, as we have already said, is placed in the centre of our system, as a source 
 of light and heat, to illuminate and vivify all the planets subordinate to it. Without 
 
 _ his benign influence the earth would be a mere block, which in hardness would sur- 
 _ pass marble and the most compact substances with which we are acquainted; no 
 
 vegetation, no motion would be possible: in a word, it would be the abode of dark- 
 ness, inactivity, and death. The first rank therefore among inanimate beings cannot 
 be refused to the sun; and if the error of addressing to a created object that adora- 
 tion which is due to the Creator alone, could admit of excuse, we might be tempted 
 to excuse the homage paid to the sun by the ancient Persians, as is still the case 
 
 _ among the Guebres, their successors, and some savage tribes in America. 
 
 The sun is, or seems to be, a globe of fire, the diameter of which is equal almost 
 
 to 111 times that of the earth, being about 883217 English miles ; its surface there- 
 
 fore is 12321 times greater than that of the earth; and its mass 1367631 times. 
 Its distance from the earth, according to the latest observations, is about 95 millions 
 
 _ of miles. ; 
 
 This enormous mass is not absolutely at rest: for modern astronomers have found 
 that it revolves round its axis, in 25 days 12 hours. This motion takes place, on an 
 axis inclined to the plane of the ecliptic about 73°; so that the equator of the sun 
 has the same inclination to the earth’s orbit. 
 
 This phenomenon was discovered by means of the spots, with which the surface 
 of the sun is covered at certain periods: with the assistance of a telescope, these 
 
458 ASTRONOMY AND GEOGRAPHY. 
 
 spots, which are dark, and generally of a very irregular form, and which often remain 
 some months, may be observed on the disk of this luminary. They were first dis- 
 covered by Galileo, who thus gave a mortal blow to the opinion of the philosophers 
 of that time, some of whom, treading in the steps of Aristotle, considered the ce- 
 lestial bodies as unalterable. He repeatedly observed, at different periods, large 
 spots on the sun’s disk ; saw them always approach in the same direction, and almost 
 in a straight line to one of the edges; then disappear, and re-appear afterwards at 
 the other edge; whence he concluded that the sun had a rotary motion about his 
 axis. It is remarked that these spots employ 27 days 12 hours to return to the same 
 point of the disk where they began to be observed; hence it follows that they require 
 25 days 12 hours, to perform a complete revolution ;*, and consequently the sun. em- 
 ploys that time in revolving about his axis. 
 
 It thence follows, also, that a point in the sun’s equator moves about four times 
 and a third as fast as a point of the terrestrial equator, during its diurnal motion; 
 for, the circumference of a solar great circle being 111 times as great, these points 
 would move with the same velocity if the period of the sun’s revolution were 111 
 days. But being only 25 days and some hours, it is about four times and a third as 
 rapid, 
 
 Astronomers have also had the curiosity to measure the extent of some of these solar 
 spots; and have found that they are sometimes much larger than the whole earth. 
 
 In regard to the nature of these spots ; some philosophers have conjectured, that 
 they can be nothing else than parts of the nucleus of the sun which remain unco- 
 vered, in consequence of the irregular movements of a fluid violently agitated. An 
 English astronomer, Professor Wilson of Glasgow, revived this idea in the Philoso- 
 phical Transactions for 1773, with this difference, that according to his theory the 
 luminous matter of the sun is not fluid, but of such a consistence that, under par- 
 ticular circumstances, there may be sometimes formed in it considerable excavations, 
 which discover a portion of the nucleus. The sloping sides of these excavations, 
 according to his opinion, form the faculz, or that border less luminous, without being 
 black, with which these spots are generally surrounded. This theory he endeavours 
 to establish, by examining the phenomena that ought to be exhibited by such exca- 
 vations, according to the manner in which they might present themselves to an ob- 
 server. 
 
 Other philosophers have supposed these spots to be only clouds of fuliginous 
 vapours, which remain suspended over the surface of the sun, in the same manner as 
 the smoke that rises from Vesuvius at the time of an eruption; and which to an eye 
 placed in the atmosphere would appear to cover a Jarge tract of country. Some also 
 have imagined them to consist of a kind of scum produced by the combustion 
 of heterogeneous matters, which have fallen on the sun’s surface. But, in all pro- 
 bability, nothing certain will ever be known on the subject. For considerable pe- 
 riods no large spots are seen on the sun’s disk, and sometimes a great many are ob- 
 served. In 1637 it is said they were so numerous, that both the heat and splendour 
 of that luminary were in some measure diminished by them.t If the opinion 
 of Descartes, respecting the incrustation of the stars, and their conversion into opake 
 planets, had been then known, some apprehensions might have been entertained 
 of seeing the sun, to the great misfortune of the human species, undergo this strange 
 metamorphosis. . 
 
 We shall here remark that a certain figure of the sun, given on the authority of 
 Kircher, and copied in various maps of the world, ought to be considered merely as 
 
 * The reason of this difference is, that while the sun performs a complete revolution on its axis, 
 the earth, moving in its orbit, advances about 25 degrees towards the same side: on which account 
 
 the spot must still pass over about 25 degrees, before it can be in the same point of view in regard 
 to the earth. 
 
 + In September 1839, they were very numerous, and many of them large. 
 
 & 
 
a 
 
 vo 
 <7 ie 
 
 OF MERCURY. 459 
 
 au imaginary production.. No observations have ever been made by any astronomer, 
 
 that can serve as the least foundation for it. 
 
 In 1683, Cassini discovered that the sun not only has a proper light of his own, 
 but that he is accompanied by a kind of luminous atmosphere, which extends to an 
 immense distance, since it sometimes reaches the earth. But this atmosphere is not 
 of a form nearly spherical, like that of the earth: it is lenticular, and situated in such 
 a manner that its greatest breadth coincides almost with the prolongation of the solar 
 equator. We indeed often see, during very serene weather, and a little after sun- 
 set, a light somewhat inclined to the ecliptic, several degrees broad at the horizon, 
 and decreasing to a point, which rises to the height of 45°. It is principally towards 
 the equinoxes that this phenomenon is observed; and asit has been since seen, and in 
 various places, by a great number of astronomers, these appearances cannot perhaps 
 be accounted for, but by supposing around the sun an atmosphere such as that above 
 mentioned. ; 
 
 It has been observed that this Zodiacal light is most distinct about the first of 
 March, at about 7 o'clock in the evening; but it has been seen in January, and 
 according to M. Toulquier it is always seen at Guadaloupe in fine weather. ; 
 
 Doctor Herschel has two ingenious papers in the Philosophical Transactions for 
 1795 and 1802, containing many new and curious speculations on the nature and con- 
 stitution of the sun, his light, &c. Dissatisfied with the old terms used to denote 
 certain appearances on the surface of the sun, Dr. Herschel rejects them ; and instead 
 of the words, spots, nuclei, penumbra, luculi, &c., he substitutes openings, shal- 
 lows, ridges, nodules, corrugations, indentations, pores, &c. He imagines that the 
 body of the sun is an opake habitable planet, surrounded and shining by a luminous 
 atmosphere, which-being at times intercepted and broken, gives us a view of the sun’s 
 body itself, which are the spots, &c. He conceives that the sun has a very exten- 
 sive atmosphere. consisting of elastic fluids, that are more or lesa lucid and tran- 
 sparent, and of which the lucid ones furnish us with light. ‘‘ ‘This atmosphere, 
 he thinks, is not less than 1843, nor more than 2765, miles in height: and he sup- 
 poses that the density of the luminous solar clouds need not be much more than 
 that of our aurora borealis, in order to produce the effects with which we are ac. 
 quainted. The sun then, if this hypothesis be admitted, is similar to the other globes 
 of the solar system, with regard to its solidity—its atmosphere—its surface diversi- 
 
 . fied with mountains and valleys—the rotation on its axis—and the fall of heavy bodies 
 
 on its surface; it therefore appears to be a very eminent, large, and lucid planet, the 
 principal one in our system, disseminating its light and heat to all the bodies with 
 which it is connected. 
 
 = 
 
 Il.—Of Mercury. 
 
 Mercury is the smallest of all the ancient planets, and the nearest the sun: its 
 distance from that luminary is about 3% of that of the earth: Mercury therefore re- 
 volves about the sun at the distance of about 37 millions of miles. On account of 
 this position, it is never more than 28° 20’ from the sun, and on this account it 
 is very difficult to be seen. When at about its greatest elongation from the sun 
 it appears through a good telescope as a crescent like the moon towards her 
 quadratures. 
 
 It has not yet been ascertained from any observations whether Mercury has a 
 motion round its axis, which however is very probably the case. 
 
 This planet completes its revolution round the sun in 87 days 23 hours 15 minutes; 
 and its diameter is to that of the earth as 2 to 5; so that its bulk is to that of the 
 earth as 8 to 125. 
 
 The distance of Mercury from the sun being no more than 3% of that of the earth; 
 and as heat increases in the inverse ratio of the squares of He distance ; it ieee 
 
460 ASTRONOMY AND GEOGRAPHY. 
 
 follows that, ceteris paribus, it is nearly seven times as hot in that planet as on our 
 eaith, This heat even far exceeds that of boiling water. If Mercury therefore 
 has the same conformation as our earth, and is inhabited, the beings by which it is 
 peopled must be of a nature very different from those of the latter. In this there is 
 nothing repugnant to reason; for who will dare to confine the power of the Deity 
 to beings almost similar to those with which we are acquainted on the earth? We 
 shall shew hereafter that the conformation of the surface of Mercury, and the 
 nature of the circumambient fluid, may be such as to make it not impossible for 
 such beings as ourselves to exist in it. 
 
 III.—Of Venus. 
 
 Venus is the most brilliant of all the planets in the heavens. This planet, as is well 
 known, sometimes precedes the sun; and on that account is called Lucifer, or the 
 morning star: sometimes it follows him, appearing the first after he is set; and on 
 that account is distinguished also by the name of Vesper, or the evening star. 
 
 This planet revolves about the sun at a distance from him, which is to that of 
 the earth from the sun, as 68 to 95; consequently its distance from the sun is 
 about 68 millions of miles: its greatest elongation from the sun, in regard to us, 
 is about 48°, and it exhibits the same phases as the moon. 
 
 The revolution of Venus around the sun is performed in 224 days 16 hours 49 
 minutes: its diameter, according to the latest and most correct observations, is nearly 
 the same as that of the earth, and consequently it is of equal bulk also. Changeable 
 spots have been discovered on the surface of Venus, which serve to prove the revo- 
 lution of that planet about its axis; but the period of this revolution is not yet fully 
 ascertained. M. Bianchini makes it to be 24 days, and M. Cassini 23 hours 20 mi- ~ 
 nutes. For our part we are inclined to adopt the latter opinion; but unfortunately 
 these spots, seen by Maraldi and Cassini, are no longer visible, even with the help 
 of the best telescopes, at least in Europe: at present not a single spot can be ob- 
 served in this planet; and therefore the question must remain undetermined till new 
 ones are seen. 
 
 Venus may sometimes pass between the earth and the sun, in such a manner as to 
 be seen on the disk of the latter, where it appears as a black spot, of about a minute 
 apparent diameter. It was seen for the first time passing over the sun’s disk in Nov. 
 1631; it was again observed under the like circumstances on the 6th of June, 1761; 
 and the same observation was made on the 3d of June, 1769. It will not be again 
 seen passing over the sun’s disk, till the 9th of December, 1874. Inthe observation. - 
 of this phenomenon, all the states of Europe interested themselves, as the founda- 
 tion of the best method of finding the sun’s parallax, from which his distance from 
 the earth may be computed, and thence his distance from any.other planet whose 
 time of revolution is known. 
 
 IV.—Of the Earth. 
 
 The Earth, which we inhabit, is the third in the order of the planets hitherto 
 known. Its orbit, the semi-diameter of which is about 95 millions of miles, com- 
 prehends within it those of Venus and Mercury. It performs its revolution about 
 the sun in 365 days 6 hours 11 minutes; for it is necessary that a distinction should 
 be made between the real and complete revolution of the earth, and the tropical revo- 
 lution or what is called-the solar year. The latter consists of 865 days 5 hours 49 
 minutes ; because it-represents only the time which the sun employs in returning to 
 the same point of the equinoctial; but as the equinoctial points go back every year 
 50”, which makes the stars seem to advance the same quantity, in the same period ; 
 when the earth has returned to the point of the vernal equinox, it must still pass over 
 i)” before it can attain to the point of the fixed sphere, where the equinox was the 
 
— *y 
 r 4 x r 
 Fe * 
 
 Bs 
 
 OF THE MOON. 461 
 
 preceding year. But as it employs for this purpose about 20 minutes, these added to 
 the tropical year will give, as the time of the complete revolution, from a point of 
 
 “the fixed sphere to the same point again, 365 days 6 hours 11 minutes, as mentioned 
 
 above. 
 
 During a revolution of this kind, the earth, in consequence of the laws of motion, 
 always maintains its axis parallel to itself; and it performs its revolution around this 
 axis, with respect to the fixed stars, in 23 hours 56 minutes ; for it is in regard to the 
 fixed stars that this revolution ought to be measured, and not in regard to the sun, 
 which has apparently advanced in the same direction about a degree per day, This 
 parallelism of the earth’s axis produces the variation of the seasons ; as it exposes 
 sometimes the northern and sometimes the southern part to the direct influence of the 
 sun’s rays. 
 
 This parallelism however is not absolutely invariable. In consequence of certain 
 physical causes, it has a small motion, by which it deviates from it, at each revolution, 
 about 50 seconds; asif it had a conical motion, exceedingly slow, around the move- 
 
 able and supposed axis of the ecliptic. On account of this motion, the apparent pole 
 
 of the world, among the fixed stars, is not fixed; but revolves about the pole of the 
 ecliptic, and approaches certain stars, while it recedes from others. The polar star 
 has not always been the nearest the arctic pole ; nor is it yet at its greatest degree of 
 proximity: it will attain to this situation about the year 2100 of our era, and its 
 distance from the pole at that period will be 28’ or 29 ; the arctic pole will then recede 
 more and more from it, so that in the course of ages there will be another polar star, 
 and even others after that in succession. 
 
 The axis of the earth is inclined to the plane of the ecliptic, at present, in an angle 
 of nearly 23° 28, which causes the inclination of the ecliptic to the equator, and pro- 
 duces the different changes of the seasons. This inclination is also variable, and, 
 according to modern observations, decreases about a minute every century: the 
 ecliptic therefore slowly approaches towards the equator, or rather the equator towards 
 the ecliptic ; and.if this motion takes place with the same velocity, and in the same 
 direction, the equator will coincide with the ecliptic in about 140,000 years; and then 
 a perpetual spring, as well as an equality of the days and nights, will prevail all over 
 the earth. 
 
 But it has been shewn by Laplace, that.all variable planetary phenomena are periodi- 
 cal, and restricted with respect to their amounts within certain and comparatively 
 narrow limits; one of which being attained, they recede again towards the opposite 
 one; but the times of periodical variation are many of them of great extent. 
 
 V.— Of the Moon. 
 
 Of all the celestial bodies which surround us, and by which we are illuminated, the 
 most interesting, next to the sun, is the moon. Being the faithful companion of our 
 
 ‘globe during its immense revolution, she often supplies the place of the sun, and by 
 
 her faint light consoles us for the loss we sustain when the rays of that luminary are 
 withdrawn. It is the moon which raising, twice every day, the waters of the ocean, 
 produces in them that reciprocal motion, known under the name of the flux and 
 reflux ; a motion which is perhaps necessary in the economy of the globe. 
 
 The mean distance of the moon from the earth is about 60} semi-diameters of the 
 latter, or 240,000 miles. Her diameter is in proportion to that of the earth, as 20 to 
 73, or nearly as to 3 to 11; so that her mass, or rather bulk, is to that of the earth, 
 nearly as ] to 482. 
 
 The moon is an opake body ; but we do not think it necessary to adduce here any 
 proof of this assertion. She is not a polished body, like a mirror; for if that were 
 the case, it would scarcely transmit to us any light, as a convex mirror disperses the 
 rays in such a manner that an eye, at any considerable distance, sees only one point 
 
462 ASTRONOMY AND GEOGRAPHY. 
 
 on the surface illuminated ; whereas the moon transmits to us from her whole disk a 
 light sensibly uniform. 
 
 To this we may add, that observation shews in the body of the moon asperities 
 still greater, considering her magnitude, than those with which the earth is covered, 
 If the moon indeed be attentively viewed, some days after her conjunction, the boun- 
 dary of the shaded part will be seen as it were indented; which can arise only from 
 the effect of its inequalities. Besides, ata little distance from that boundary, in the 
 part not yet illuminated, there are observed luminous points, which, increasing gra- 
 dually as the luminous part approaches them, are at length confounded with it, and 
 form the indentations above mentioned: in short, the shadows of those parts, when 
 they are entirely illuminated, are seen to project themselves to a greater or less dis- 
 tance, and to change their position, according as they are illuminated on the one side 
 or the other, and ina direction more or less oblique. It is in this manner that the 
 summits of the mountains on our earth are illuminated, while the neighbouring 
 valleys and plains are still in obscurity; and that their shadows are projected toa. 
 greater or less distance, on the right or the left, according to the elevation and _ posi- 
 tion of the sun. Galileo, the author of this discovery, measured the height of one of 
 these lunar mountains geometrically ; and found it to be about 3 leagues, which is 
 nearly double the height of the most elevated peaks of the highest mountains known 
 on the earth. But later astronomers, by more accurate measurements, have 
 found that few of the lunar mountains rise above a mile in height, and that the 
 majority do not reach above half that height. 
 
 _ The best time to observe the shadows is about half moon, when the separation 
 of light from the dark part is a straight line; the shadows are then seen of their 
 full extent, not being foreshortened. 
 
 We have already spoken of the names given by astronomers to these spots, and 
 of their use in astronomy. We shall therefore not repeat them here, but proceed to 
 something more interesting. On the surface of the moon there are spots of different 
 kinds, some luminous, and others in some measure obscure. It was long considered 
 as fully established that the most luminous parts were land, and the obscure parts 
 sea; for it was said, as water absorbs a part of the light, it must transmit a weaker 
 splendour than the land, which reflects it very strongly. But this reasoning is not 
 well founded; for if these spots, which are obscure in regard to the rest of the 
 moon, consisted of water, when illuminated obliquely, as they are in respect to us 
 during the first days after the conjunction, they ought to transmit to us a very lively 
 light; as a mirror which seems black to those not placed in the point to which it- 
 reflects the solar rays, appears on the other hand exceedingly bright to an eye situ- 
 ated in that point. 
 
 Others have hence been induced to believe that these obscure parts are immense 
 forests; and this indeed may be more probable. We have no doubt that if the vast 
 forests still in Europe, and those of America, were seen at a great distance, they 
 would appear darker than the rest of the earth’s surface.* 
 
 But is this observation sufficient to make us conclude that these spots are really 
 forests? We do not think it is; and the reasons are as follow: 
 
 It is in a manner proved, that the moon has no atmosphere; for, if she had, it 
 would produce the same effects as ours. A star, on the moon approaching it, would 
 change its colour: and its rays, broken by that atmosphere, would give it a very — 
 irregular motion, even at a considerable distance from the moon. But nothing of this 
 kind is observed. A star covered by the dark edge of the moon suddenly disappears, 
 
 # About the eighth day of the moon’s age, towards the upper part of the moon, and not far from 
 the line which divides the light from the dark part, is seen a straight and deep excavation running 
 through aa extensive range of hills. If we were to meet with such an excavation on the globe we 
 inhabit, we should certainly consider it a monument of ancient art. 
 
. ; 3 is 
 
 OF THE MOON. 463 
 
 without changing its colour, or experiencing any sensible refraction. Some astro- 
 nomers indeed have imagined that they saw lightning in the moon, during total 
 eclipses of the sun; but this no doubt was an illusion, owing to their eyes being 
 fatigued by looking too attentively at the sun. Besides, if there were clouds and 
 vapours in the moon, they would sometimes be seen to conceal certain known parts 
 of her surface; as an observer placed in the moon would certainly see certain pretty 
 large portions of the earth, such as whole provinces, concealed sometimes for days, 
 and even weeks, by those clouds, which frequently cover them, during as long a 
 period. M. de la Hire has shewn that an extent as large as Paris would be percepti- 
 ble to an observer in the moon, if viewed through a telescope which magnified 
 objects about 100 times. 
 
 But if there be no dense atmosphere, no elevation of vapours, on the surface of the 
 moon, it is difficult to conceive how there can be any kind of vegetation in it; and 
 if this be the case, it can produce neither plants, trees, nor forests, and conse- 
 quently noanimals. Itis therefore probable that the moon is not inhabited; besides, 
 if it were inhabited by animals nearly similar to man, or endowed with some kind 
 of reason, it is hardly to be supposed that they would not make some changes in 
 the surface of that globe. But since the invention of the telescope, to the present 
 time, no alteration has been observed in its surface. 
 
 The moon always presents to the earth very nearly the same face; and therefore 
 she must have a rotary motion about an axis, nearly perpendicular to the eckiptic, the 
 duration of which forms the lunar month; or in one of its hemispheres there must 
 be some cause, which makes it incline towards the earth. ‘The latter conjecture is 
 more probable ; for why should this revolution of the moon around its axis be per- 
 formed exactly in the period of its rotation about the earth. However, as the moon 
 always presents the same face to the earth, it thence follows, that her whole surface 
 is illuminated by the sun, in the course of a lunar month; the days therefore in the 
 moon are equal to about 15 of ours, and the nights of the same duration. 
 
 But if we suppose, notwithstanding what has been said, that there are inhabitants 
 in the moon, they will enjoy a very singular spectacle: an observer placed towards 
 the middle of the lunar disk, for example, will always see the earth motionless to- 
 
 ‘wards his zenith, or having only a motion of nutation, in consequence of reasons 
 
 which we shall explain hereafter. In a word, each inhabitant of that hemisphere 
 will always see the earth in the same point of the horizon; while the sun will appear 
 to perform his revolution ina month. On the contrary, the inhabitants of the other 
 hemisphere will never see the earth; and if there are astronomers in it, some of them 
 no doubt will undertake a voyage to the hemisphere which is turned towards us, for 
 
 the purpose of observing this sort of motionless moon, suspended in the heavens like 
 
 a lamp,and the more remarkable as it must appear to the lunar inhabitants of a diameter 
 four times as large as that of the moon appears to us; with a great variety of spots 
 performing their revolutions in the interval of 24 hours: for there can be no doubt 
 that our earth, intersected by vast seas, large continents, and immense forests, such 
 as those of America, must exhibit to the moon a disk variegated with a great many 
 spots, more or less luminous. 
 
 We have said that the moon always presents the same disk to the earth; but 
 strictly speaking this is not exactly the case; for it has been found that the moon 
 has a certain motion, called libration, in consequence of which the parts nearest the 
 edge alternately approach to or recede from that edge by a kind of vibration. Two 
 kinds of libration are in particular distinguished; one called a libration in latitude, 
 by which the parts near the austral or the boreal poles of the moon, seem to vibrate 
 from north to south, and from south to north, through an are which may comprehend 
 about 5 degrees. This however is a mere optical effect, produced by the parallelism 
 of the moon’s axis of rotation, which is inclined 24 degrees to the ecliptic. 
 
464 ASTRONOMY AND GEOGRAPHY. 
 
 The other libration is that in longitude ; which takes place around the above axis, 
 
 at an angle of nearly 74 degrees; and as both are combined, it needs excite no wonder 
 that this phenomenon should have long been an object of research to philosophers. 
 However, it is evident that the inhabitants of the moon, if there really be any, who 
 are situated near the edge of the disc turned towards the earth, must see our globe 
 alternately rise and set, describing an arc of only afew degrees. 
 
 The moon is a little depressed at the poles in consequence of her rotation, but 
 because she presents always the same face to the earth she must be elongated in the 
 direction of that axis which points towards the earth. According to the theory this 
 excess ought to be about the 3000th part of the least axis. It is remarkable that the 
 satellites of Jupiter also present always the same face to the planet, and there can be 
 no doubt that it isa general consequence of the combined laws of gravitation and 
 revolution. 
 
 The planes of the earth’s equator and the moon’s orbit, and the plane drawn 
 
 through the moon’s centre parallel to the ecliptic, have always very nearly the same 
 
 intersection. 
 
 VI.—Of Mars. 
 
 Mars, which may be easily distinguished by its reddish splendour, is the fourth in 
 the order of the primary planets. Its orbit incloses that of Mercury, Venus, and the 
 earth; consequently the motions of these planets must exhibit to the inhabitants of 
 Mars the same phenomena, as are presented by Mercury and Venus to the inhabitants 
 of our globe. 
 
 The revolution of Mars around the sun is performed in 686 days 23 hours 30 minutes, 
 or nearly two years. Its mean distance from the sun is more than 1} that of the 
 earth, or about 144 millions of miles. 
 
 Spots are observed sometimes on the disc of Mars, by which it is proved that it 
 revolves on an axis almost perpendicular to its orbit; and that this revolution is 
 completed in 24 hours 39 minutes. The days therefore, to the inhabitants of Mars, 
 if there are any, must be nearly equal to ours; and the days and nights in this planet 
 must be of the same length, since its equatcr coincides with its orbit. 
 
 Mars has a very dense atmosphere, and when either of the poles emerges from dark-_ 
 
 ness into the rays of the sun, itis found to be decidedly brighter than the other parts 
 
 of the surface; and the extent of the bright spot gradually becomes less during — 
 
 the time that the pole remains in the light. This appearance has been conceived to 
 
 be caused by snow deposited during the polar winter, and the gradual melting of it 
 
 during the summer. The diameter of Mars is about 4100 miles. 
 
 VIL.—Of Jupiter. 
 
 The next planet to Mars, of the ancient ones, is Jupiter. Its distance from the 
 sun is above 5 times that of the earth, being 490 millions of miles. ‘The period of its 
 revolution around the sun is 11 years 317 days 12 hours 20 minutes. Its diameter, 
 compared with that of the earth, is as 11 to 1; so that its bulk is 1331 timesas great 
 as that of our globe. 
 
 This bulk does not prevent Jupiter from revolving around his axis with much 
 
 more rapidity than our earth. The spots observed on the disc of this planet have 
 indeed shewn that this revolution is performed in 9h. 56m.; so that it is more than 
 twice as quick, and as any point in the equator of Jupiter is eleven times as far dis- 
 tant from the axis asa point of the earth’s equator is from the terrestrial axis, it thence 
 follows that this point in Jupiter moves with a velocity about twenty-four times as 
 great. 
 
 It has therefore been observed that the body of Jupiter is not perfectly spherical : | 
 
 it is an oblate spheroid, flattened at the poles, and the diameter of its equator is to 
 
THE FOUR NEW PLANETS. 465 
 
 that passing from the one pole to the other, according to the latest observations 
 made with the most perfect instruments, as 14 to 13. 
 
 On the four new Planets, commonly, from their smallness, called Astroids. 
 
 These planets CerEs, Patuas, JuNo, and Vesta, are situated between the orbits 
 of Mars and Jupiter, and are nearly at the same distance from the sun. 
 
 Ceres was discovered by Piazzi, on January 1, 1801. It is of a ruddy though not 
 very deep colour, and, being surrounded by an extensive and dense atmosphere, it 
 exhibits a distinct dise when viewed with a magnifying power of 200. Like the 
 atmosphere of the earth, that of Ceres is very dense near the planet, and becomes rarer 
 at a greater distance. As this planet approaches the earth its apparent size increases 
 much more rapidly than it ought to do from the diminution of the distance; an effect 
 which, Schroeter observes, arises from the finer exterior strata becoming visible as it 
 approaches. 
 
 It performs its revolution round the sun in about four years seven months and ten 
 days. Its mean distance from the sun is about 2°669 times that of the earth; and 
 its diameter has been variously estimated at from 163 to 1630 miles. 
 
 Pallas was discovered March 28th, 1802, by Dr. Olben, of Bremen. It is 
 nearly of the same apparent magnitude as Ceres, but of a less ruddy colour. It is 
 surrounded with a nebulosity, but of less extent than that of Ceres. But Pallas is 
 
 distinguished from al! other planets by the great inclination of its orbit to the plane 
 
 of the ecliptic. While the other planets revolve in orbits which are nearly circular, 
 and deviating only a few-degrees from the plane of the ecliptic, that of Pallas is 
 inclined to this plane about 35 degrees ; and while the mean distance of this planet from 
 the sun is nearly the same as that of Ceres, from the greater eccentricity of the orbit 
 of Pallas, the orbits of these two planets intersect each other, a phenomenon quite 
 anomalous in the solar system. The diameter of this planet, like that of Ceres, has 
 not been satisfactorily determined: it has been estimated at from 80 to upwards of 
 2000 miles. 
 
 Juno was discovered by Mr. Harding, at Lilienthal, on September Ist, 1804. It 
 is of a reddish colour, and is free from the nebulosity which surrounds Ceres and 
 Pallas. It is probably the smallest of the new planets; but it is distinguished by the 
 great eccentricity of its orbit, its greatest distance from the sun being double its deast 
 distance. Its time of revolution is about 4 years and 123 days; and its mean apparent 
 diameter, as seen from the earth, is, according to Schroeter, 3°057.” 
 
 Vesta.—From the regularity observed in the distances of the old planets from the 
 sun, some astronomers supposed that a planet existed between the orbits of Mars and. 
 
 ‘Jupiter. The discovery of Ceres seemed to confirm this conjecture, which, 
 
 however, was overturned by the discovery of Pallas and Juno. Dr. Olben, how- 
 ever, imagined that these small celestial bodies might be the fragments of a large 
 planet, which had been burst. by some internal convulsion, and that several 
 more might be discovered between the orbits of Mars and Jupiter. And he 
 
 | conceived that if they originated in this manner, though their orbits might be 
 _ differently inclined to the ecliptic, yet as they must all have diverged from the same 
 point, they must have two common points of reunion in opposite regions of the 
 
 heavens. ‘These nodes he found, from observation on the planets that had been 
 
 _ discovered, to be in Virgo and the Whale. With the hope therefore of detecting 
 _ other fragments of the supposed planet, Dr. Olben examined thrice every year al] the 
 
 i _ little stars in these two opposite constellations, and he discovered Juno in the Whale; 
 
 and on March 29th, 1807, he discovered Vesta in. Virgo. 
 The appearance of this planet is similar to that of a star of the 5th or 6th mag- 
 
 nitude, and it may be seen on a clear night by the naked eye. It is not surrounaed 
 
 _ iby any nebulosity; and, with a power of 636, Dr, Herschel saw no appearance of a 
 2H 
 
466 ASTRONOMY AND GEOGRAPHY. 
 
 planetary disc. Its orbit intersects that of Pallas, but not in the same place where 
 it is cut by Ceres. Its time of revolution is about 3 years and 66 days. 
 
 Sir David Brewster has some ingenious speculations on the origin of these small 
 planets, and he is decidedly of opinion that the phenomena which they exhibit lead 
 to the conclusion that they are the dispersed fragments of a single planet. See his 
 edition of Ferguson’s Astronomy, and the article Astronomy in the Edinburgh En. 
 cyclopeedia. 
 
 The axis of Jupiter is almost perpendicular to the plane of its orbit; for its 
 inclination is only 3 degrees: the days and nights therefore in this planet must be 
 nearly equal at all seasons. 
 
 The surface of Jupiter is for the most part interspersed with spots, in the form of 
 bands; some of them obscure, and others luminous; at certain periods they are 
 scarcely visible; nor are uniformly marked throughout their whole extent ; so that 
 they are, as it were, interrupted : their number also varies, and they can be very well 
 
 seen by the assistance of a telescope magnifying 50 times, but best when Jupiter is 
 at his least distance from the earth. The year 1773 was exceedingly favourable for 
 
 these observations; because Jupiter was then as near to the orbit of the earthas possible, 
 
 The distance of Jupiter from the sun being above 5 times that of the earth, it is 
 evident that the sun’s diameter must appear five times less, or about 6 minutes 
 only; consequently the splendour of the sun at Jupiter will be 25 times less than 
 itis tothe earth. But a light 25 times less than that of the sunis still pretty strong, 
 and more than sufficient to produce a very clear day: the inhabitants therefore of 
 Jupiter, forit is probable that there are some in this planet, will have no great cause 
 to complain. 
 
 ‘But if they are treated less favourably in this respect than the inhabitants of the 
 earth, they possess advantages in others; for while the earth has only one moon, 
 to make up for the absence of the-sun, Jupiter has four. These moons, or satellites, 
 were first discovered by Galileo; and they enabled him to reply to those who 
 objected, in opposition to the earth’s motion, the impossibility of conceiving how 
 the moon could accompany the earth during its revolution; Galileo’s discovery 
 reduced them to silence. 
 
 The satellites of Jupiter revolve around him in the periods, and at the distances 
 
 indicated in the following table. 
 
 | 
 
 Order of the Dist. in semi-diame- | Periodical Times 
 
 Satellites. || ters of Jupiter. 
 D 6H. OM. 
 I. 6°048 I 1S es 
 HH, 9°623 3 13 14 
 . Hil. - 15 350 yg Wena 3 
 (Sg hey 8373 
 
 LV 0 Sm 27°998 1 | 
 
 The inhabitants of Jupiter then, in this respect, enjoy much greater advantages 
 than those of the earth; for having four moons, some of them must be always above 
 the horizon which is not illuminated by the sun; they will even sometimes see the 
 whole four, one as a crescent, another full, and a third half full; they will see them 
 eclipsed, as we see the moon deprived of her light from time to time, when she enters 
 the shadow projected by the earth, but with this difference, that, being much nearer 
 to Jupiter, considering his bulk, they cannot pass behind him, in regard to the sun, 
 without suffering an eclipse. 
 
 When the brilliancy of the satellites of Jupiter is examined at different times, 
 it is found to undergo considerable change. By comparing the mutual positions of 
 
 the satellites with the times when they attain the maximums of light, Sir Wm.. 
 
 Herschel concluded, that, like our moon they all turned round their axes in the same 
 time that they performed their revolution round Jupiter, 
 
 >: ae 
 
OF SATURN. 467 
 
 From the theory of reciprocal attraction, La Place discovered two remarkable 
 theorems concerning the motions of these satellites. 
 
 First. The mean-motion of the first satellite added to twice the mean motion of 
 the third, is rigorously equal to thrice the mean motion of the second satellite. 
 
 Second. The mean longitude of the first satellite minus three times that of the 
 second, plus twice that of the third, is exactly equal to a semi-circle or 180 degrees. 
 
 By following out these laws, we find, lst. When the second and third satellites of 
 Jupiter are simultaneously eclipsed, the first is always in conjunction with Jupiter. 
 2d. When the first and third satellites are simultaneously eclipsed, the difference 
 between either of their longitudes and that of the second is 60°. 3d. When the 
 first and second are simultaneously eclipsed, the difference between either of their 
 longitudes and that of the third is 90°. 
 
 It is obvious from these results, that a wonderful provision is made in the system 
 of Jupiter to secure to the planet the benefit of his satellites ; for it 2s ¢mpossible for 
 - the planet to be deprived of the light of all the satellites at the same time. 
 
 Astronomers, however, not contented with establishing the existence of these 
 moons attached to Jupiter, have done more ; for they have calculated their eclipses 
 with as much correctness, at least, as those of our moon. The Nautical Almanac, 
 
 and other astronomical Ephemerides, exhibit for each day of the month, the aspects 
 of the satellites of Jupiter, and announce the hour, minute, and second at which 
 their eclipses will commence, and whether they will be visible or not on the horizon 
 of the place; they give also the time when any of these satellites will be hid behind 
 the dise of Jupiter, or disappear by passing before it. ‘These predictions are not 
 matters of mere curiosity, since they are of great utility in determining the longitude. 
 
 VILl.—Of Saturn. 
 
 Saturn, which is still farther from the sun than Jupiter, exhibits a most singular 
 spectacle, on account of his seven moons, and the ring by which he is surrounded. 
 He performs his revolution around the sun in 29 years 174 days 6 hotirs 36 minutes; 
 and his mean distance from that luminary is about 93 times as Gs as that of the 
 earth, or 900 millions of miles. 
 
 At such an immense distance the apparent diameter of the sun, to a spectator in 
 Saturn, is no more than # of what is to us; and its light as well as heat must be 
 90 times less. An inhabitant of Saturn transported to Lapland, or even to the polar 
 regions, coyered with perpetual ice, would experience there an insupportable heat; 
 and would no doubt perish sooner than a man immersed in boiling water; while an 
 inhabitant of Mercury would freeze in the most scorching climates of our torrid zone. 
 
 By noticing carefully the changes of certain dark spots on the disc of Saturn, Sir 
 Wm. Herschel found that he revolved on his axis in 10h. 16m., and that.the axis of 
 rotation is perpendicular to the plane of the ring. 
 
 Nature seems to have been desirous to indemnify Saturn for his great distance from 
 _ thesun, by giving him seven moons, whichare called his satellites. Their distances 
 | from the centre of Saturn, in’ semi-diameters of that planet, and the periods of their 
 revolution, are as expressed in the following table. 
 
 - Satellites. | Distances. Revolutions. 
 Dan hue NE 
 I. 5'284 RPPOR ARIS 
 II. 6°819 274] 
 II. 9°524 4.12 95 
 IV. 22°081 15 ' 23°41 
 Vv. 64°359 79° 7 48 
 Vi. 4°300 tii Sib 
 
 VIL. 3°351 0 22 40 
 
 2 2 
 
468 ASTRONOMY AND GEOGRAPHY. 
 
 Of these satellites, five were discovered by Cassini and Huygens, before the year | 
 1685; and it was imagined there were no more, till two were discovered by Dr. | 
 Herschel in 1787 and 1788. These are nearer to Saturn than any of the other five; 
 but to prevent confusion in the numbers, with regard to former observations, the 
 are called the 6th and 7th satellites. 
 
 The inclination of the first four satellites to the ecliptic: is from 30 to 31 degrees, 
 The fifth describes an orbit inclined in an angle of from 17 to 18 degrees to the | 
 orbit of Saturn. Dr. Herschel observes that this satellite turns once round its | 
 axis exactly in the time in which it revolves about Saturn; and in this respect it | 
 resembles our moon. 
 
 We shall not here enlarge on the advantages which this planet must derive from. 
 so many moons; what we have said in regard to Jupiter is applicable in a greater de- | 
 gree to Saturn also. | 
 
 But something still more singular than these seven moons, is the ring by which | 
 Saturn is surrounded. Let the reader conceive a globe placed in the middle of a flaé | 
 thin circular body, with a concentric vacuity: and that the eye is placed at the ex- 
 tremity of a line oblique to the plane of this circular ring. Such is the aspect ex- 
 hibited by Saturn, when viewed through an excellent telescope; and such is the 
 position of a spectator on the earth. The diameter of Saturn is to that of the | 
 vacuity of the ring, as. 3 to 5; and the breadth of the ring is nearly equal to the 
 interval between the ring and Saturn. It is fully proved that this interval it a va- | 
 cuity; for a fixed star has been seen between the ring and the body of the planet; 
 this ring therefore maintains itself around Saturn as a bridge would do concentric to 
 the earth, and having every where a uniform gravity. ‘ | 
 
 What is called tre ring of Saturn certainly consists of at least two rings, concentric 
 with the planet and each other, both lying in one plane, and separated from each 
 other by.a very narrow interval. It has been surmised by some observers that the 
 outer ring consists of several narrow ones. In the fourth volume of the Memoirs 
 of the Royal Astronomical Society, there is a very interesting account by Captain 
 Kater, of some observations which he and some other persons made on the ring; and 
 the conclusion which all the observers came to was, that the outer ring was certainly 
 composed of several separate ones. 
 
 Any tolerable telescope will shew the division of the ring which divides the outer 
 from the inner part; and Professor Struve has given the following dimensions cof the 
 planet and the rings from micrometrical measurements with the magnificent Fraun- 
 hofer achromatic telescope, at the Dorpat observatory. 
 
 Miles. 
 Exterior diameter of exterior ring......... Aver s ee eee . 176418 
 Interior ditto Cito RUT SE BD Ree Lee ee sea wes OOLTS 
 Exterior ditto intemor dittO st. ea ete ake ee ops LOLOOU 
 Interior ditto CELLO TM Noes alerts a te ee eet toe - 117339 
 Equatorial diameter of the planet .......... die is 8 66min eee ee ee 
 Interval between the planet and interior ring .............. 19090 
 Interval between the rings ......... Sei Peder PSE ee te ue 1791 ; 
 Thickness of the ring, not exceeding ...........000008 44 100 
 
 (See Mem. Astron. Soc., vol. 3.) 
 
 That the ring is an opake substance is shewn by its casting a deep shadow on the 
 body of the planet, on the side next the sun; and receiving the shadow of the planet 
 on the other side. The time of revolution has been found to be 10h. 29m. 17s. 
 
 Recent micrometrical measurements seem to indicate that the rings, though nearly, 
 are not accurately, concentric with the planet. And it has been stated that this want 
 of perfect concentricity is essential to the stability of the system of which the rings 
 
¥ 
 
 ae 
 
 OF SATURN. ; 469 
 
 form so conspicuons a part. Sir J. Herschel, in his Astronomy, has the following 
 striking remarks on the phenomena of Saturn: 
 
 ‘The rings of Saturn must present a magnificent spectacle from those regions 
 of the planet which lie above their enlightened sides, as vast arches spanning the sky 
 from horizon to horizon, and holding an invariable situation among the stars. On 
 the other hand, in the regions beneath the dark side, a solar eclipse of 15 years’ dura- 
 tion must afford (to our ideas) an inhospitable asylum to animated beings, ill com- 
 pensated by the faint light of the satellites. But we shall do wrong to judge of the 
 fitness or unfitness of their condition from what we see around us, when perbaps 
 the very combinations which convey to our minds images of horror, may be in reality 
 theatres of the most striking and glorious displays of beneficent contrivance.”’ 
 
 This body, of a conformation so singular, is alternately illuminated on each side 
 by the sun; for it makes, with the plane of Saturn’s orbit, an invariable angle of 
 about 31° 20’; always remaiving parallel to itself; in consequence of which it pre- 
 sents to the sun, sometimes the one face, and sometimes the opposite one; the inha- 
 bitants, therefore, of the two hemispheres of Saturn enjoy the benefit of it alter- 
 nately. 
 
 Saturn is seen sometimes from the earth without his ring; but this phenomenon 
 may be easily explained. 
 
 Saturn’s ring may disappear in consequence of three causes. Ist. It disappears 
 when the continuation of its plane passes through the earth and the sun; for in that 
 case its surface is in the shade, or too weakly illuminated by the sun to be visible at 
 so great a distance; and its edge is too thin, even though illuminated, to be seen 
 from the earth. This phenomenon is observed when Saturn’s place is about 19° 45’ 
 of Virgo and Pisces. 
 
 2d. The ring of Saturn must disappear also, when the continuation of its plane 
 passes between the earth and the sun; for the flat part of the ring, which is then 
 turned towards the earth, is not that illuminated by the sun. It cannot therefore 
 
 be seen from the earth; but its shadow may te scen projected on the disk of 
 Saturn. 
 
 The nature of this singular ring affords much matter for conjecture. Some have 
 supposed that it may be a multitude of moons, all circulating so near each other, that 
 the distance between them is not perceptible from the earth, which gives them tbe 
 appearance of one continued body. But this is very improbable. @ 
 
 Others have imagined that it is the tail of a comet, which, passing very near 
 Saturn, has been stopped by it. But such an arrangement of a circulating fluid 
 
 would be something very extraordinary. In our opinion, while we admire this work 
 
 of the sovereign Artist, the Creator of the universe, we must suspend our conjec- 
 tures respecting the nature of it, till a farther improvement in telescopes shall enable 
 us to obtain new facts to support them. 
 
 The distance of Saturn from the sun is so great, that, if it be inhabited by intelli- 
 gent beings, it is very doubtful whether they have any knowledge of our existence, 
 and much less of that of Mercury and Venus; for in regard to them, Mercury will 
 never be farther from the sun than 2° 25’, Venus than 4° 15’, and the Earth than 6°; 
 Mars will be distant from the sun only about 9°, and Jupiter 28° 40’: it will there- 
 fore be much more difficult for the Saturnians to see the first three or four of these 
 planets, than it is for us to observe Mercury ; which can scarcely ever be seen, as it 
 is almost always concealed among the rays of the sun. 
 
 It is however true that the light of the sun is, on the other hand, very weak; and 
 that the constitution of Saturn’s atmosphere, if it has one, may be of such a nature, 
 that these planets are visible as soon as the sun has set. 
 
470 ASTRONOMY AND GEOGRAPHY. : 
 
 1X.—Of the Georgian Planet. 
 
 pags 
 
 It was long supposed that Saturn was the remotest planet in our system; but P 
 
 from inequalities in the motions of Jupiter and Saturn, astronomers at length 
 began to suspect that another more distant planet existed ; and this conjecture was 
 confirmed in 1781, when Dr. Herschel discovered a new planet, which he called the 
 
 Georgium Sidus, in honour of the then reigning king of England, George III. The — 
 
 French call it Herschel, in honour of the discoverer ; and Professor Bode, of Berlin, 
 gave it the name of Uranus, who was the father of Saturn, as Saturn was of Jupiter. 
 An interesting history of the discovery was presented to the Academy of Sciences 
 at Brussels, in May 1785, by Baron von Zach of Gotha, and is inserted in the first 
 volume of the Memoirs of that Academy. 
 
 The distance of this planet from the sun is immense; being about 1800 millions 
 of miles, which is double that of Saturn. It performs its annual revolution in 83 
 years 150 days and 18 hours of our time; and its motion in its orbit must conse- 
 quently be above 7000 miles an hotr. To a good eye, unassisted by a telescope, it 
 appears like a faint star of the fifth magnitude ; and it cannot readily be distinguished 
 from a fixed star with a less magnifying power than 200. Its apparent diameter, to 
 an observer on the earth, subtends an angle of no more than 4 seconds; but its real 
 diameter is about 35000 miles, and therefore it must be about 80 times as big as the 
 earth. Hence we may infer, as the earth cannot be seen under an angle of quite a 
 second by the inhabitants of the Georgian planet, that it has never yet been disco- 
 vered by them, unless their eyes and instruments are considerably better than ours. 
 The orbit of this planet is inclined to the ecliptic at an angle of 46 minutes 26 se- 
 
 conds; but as no spots have been discovered on its surface, the position of its axis, 
 
 and the length of its day and night, are not known. 
 
 On account of the immense distance of the Georgian planet from the sun, it was 
 highly probable that it was accompanied with several satellites or moons; and the 
 high powers of Dr. Herschel’s telescopes indeed enabled him to discover six; but 
 there may be some others which he has not yet seen. The first and nearest the 
 
 ' planet, revolves at the distance from it of 123 of its semi-diameters ; and performs — 
 its revolution in 5 days 21 hours 25 minutes; the second revolves at the distance ~ 
 
 from the primary of 163 of its semi-diameters, and completes its revolution in 18 
 days 17 hours | minute : the third, at the distance of 19 semi-diameters, in 10 days 
 23 honrs 4 minutes; the fourth, at 22 semi-diameters, in 13 days ll Wests 5 mi- 
 nutes; the fifth, at 44 semi-diameters, in 38 days 1 hour 49 minutes; and the sixth, 
 at 88 semi-diameters, in 107 days 16 hours 40 minutes. It is remarkable that the 
 orbits of these satellites are almost all at right angles, to the plane of the ecliptic; 
 
 and that the motion of every one of them, in their own orbits, is retrograde, or con- 
 
 trary to that of all the other known planets. 
 
 X.-- Of Comets. 
 
 ‘ Comets are not now considered, as they were formerly, to be signs of celestial 
 vengeance; the forerunners of war, famine, or pestilence. Mankind in those ages 
 must have been exceedingly credulous to imagine that scourges confined to a very 
 small portion of the globe, which itself is but a point in the universe, should be an- 
 nounced by a derangement of the natural and immutable order of the heavens. WNei- 
 ther are comets, as supposed by the greater part of the ancient philosophers, and 
 those who trod in their footsteps, meteors accumulated in the middle of the air. 
 Astronomical observations made at the same time, in different parts of the earth, 
 
 have shewn that they are always at a distance much greater than that even of the — 
 
 moon ; and consequently that they have nothing in common with the meteors formed 
 in our atmosphere. 
 
i 
 
 ‘ 
 } 
 bs 
 
 i 
 § 
 
 OF COMETS. 471 
 
 The opinions entertained by some ancient philosophers, such as Apollonius the 
 Myndian, and particularly Seneca, have been since confirmed. According to these 
 philosophers, comets are as old and as durable as the planets themselves; their revo- 
 lutions are regulated in the same manner; and if they are seldem seen, it is because 
 they perform their course in such a manner, that in a part of their orbits they are so 
 far distant from the earth as to become invisible; so that they never appear but when 
 in the lower part of them. 
 
 Newton and Halley, who pursued the same path, have proved by the observations 
 of different comets, which appeared in their time, that they describe elliptical orbits 
 around the sun, which is placed in one of the foci; and that the only difference be- 
 tween these orbits and those of the planets is, that the orbits of the latter are nearly 
 circular, whereas those of comets are very much elongated ; in consequence of which, 
 during a part of their course, they approach near enough to our earth to become 
 visible ; but during the rest they recede so far from us, as to be lost in the immensity 
 of space. ‘These two philosophers have taught us also, by the help of a small num- 
 ber of observations, made in regard to the motion of a comet, how to determine the 
 distance at which it has passed, or will pass, the sun; as well as the period when it 
 is at its least distance, and its place in the heavens for any given time. Calculations 
 made “according to these principles agree in a surprising manner with observa- 
 tions. 
 
 _ The modern philosophers have even done more: they have determined the periods 
 of the return of some of these comets. The celebrated. Dr. Halley, considering 
 that comets, if they move in ellipses, ought to have periodical revolutions, because 
 these curves return into themselves, examined with great care the observations 
 of three comets, which appeared in 1531 and 1532, 1607, and 1682; and having cal- 
 culated the position and dimensions of their orbits, found them to be nearly the 
 same, and consequently that these comets were only one, the revolution of whick 
 was completed in about 75 years: he therefore ventured to predict that this comet 
 would re-appear in 1758 or 1759 at latest. It is well known that this prediction was 
 verified at the time announced ; hence it is certain that this comet has a periodical 
 revolution around the sun, in 75 years anda half.* According to the dimensions 
 of its orbit, determined by observations, its least distance from the sun is #83 of the 
 semi-diameter of the earth’s orbit ; it afterwards recedes to a distance which is equal 
 to 354 of these semi-diameters ; so that its greatest elongation from the sun is about 
 four times as great as that of Saturn. The inclination of its orbit to the ecliptic, is 
 17° 40’, in a line proceeding from 23° 45’ of Taurus to 23° 45’ of Scorpio. 
 _ There are still two comets, the return of which is expected with some sort 
 of foundation; viz., that of 1556, expected in 1848; and that of 1680 and 1681, 
 which it is supposed, though with Tess confidence, will re-appear about 2256. The 
 latter, by the circumstances which attended its apparition, seems to be the same as 
 that seen, according to history, 44 years before the Christian era, also in 531, 
 and in 1106; for between all these periods there is an interval of 575 years. There 
 is reason therefore to suppose that this comet has an orbit exceedingly elongated, and 
 that it recedes from the sun about 135 times the distance of the earth, 
 
 What is very remarkable also in this comet is, that in the lower part of its orbit 
 it passed very near the sun; that is, at a distance from its surface which scarcely 
 exceeded a sixth part of the solar diameter; hence Newton concludes, that at the 
 time of its passage it was exposed to a heat 2000 times greater than that of red-hot 
 
 __ * Its re-appearance in 1835, in the very place in which observers had been instructed to look for 
 it, is one of the greatest triumphs of modern science. A very interesting series of observations on 
 this comet were made at the Cape of Good Hope, by Sir J. Herschel, and Mr. Maclear, in 1836. 
 
 ore is a very elaborate paper on this comet published as a supplement to the Nautical Almanac 
 or 1839. 
 
472 ASTRONOMY AND GEOGRAPHY. 
 
 iron. This body therefore must be exceedingly compact, to be able to resist so pro- 
 digious a heat, which there is reason to think would volatilize all the terrestrial 
 bodies with which we are acquainted. - 
 
 Within these few years two small comets have been discovered, which revolve 
 round the sun in comparatively short periods, The first, which is called Encke’s 
 comet (from the name of the astronomer who first predicted its return), completes 
 its revolution inabout 3 yearsand4 months. Thesecond, called Biela’s comet, which 
 was first observed by an Austrian officer whcse name was Biela, revolves in about 6 
 years and 8 months. Both these comets have been repeatedly observed ; they have 
 no tails; and Biela’s is so transparent that Sir J. Herschel saw some small stars 
 through the centre of it. It appears indeed to consist of extremely attenuated - 
 vapour; and it is wonderful that a thing so light should continue its course through — 
 the regions of space, obeying, with the most exact regularity, the laws of gravi- 
 tation. 
 
 At present there are more than 100 comets, the orbits of which have been calcu- 
 lated; so that their position, and the least distance at which they must pass the sun, 
 are known. When a new comet therefore shall appear, and describe the same, or 
 nearly the same path, we may be assured that it isa comet which has appeared 
 before: we shall then know the period of it revolution, and the extent of its axis, 
 
 -which will determine the orbit entirely: in short we shall be enabled to calculate 
 the times of its return, and other circumstances of its motion, in the same manner 
 as those of the other planets. : “: 
 
 Comets have this in particular, that they are often accompanied by a train or tail. 
 These tails or trains are transparent, and of greater or less extent; some have been. 
 seen which were 45, 50, 60, and even 100 degrees in length, as was the case with 
 those of the comets which appeared in 1618 and 1680. Sometimes however the tail 
 consists merely of a sort of luminous nebula, of very little extent, which surrounds 
 the comet in the form of a ring, as was observed in the comet of 1585: it frequently 
 happens that these tails cannot be seen unless the heavens be exceedingly serene, and 
 free from vapours. The celebrated comet, which returned about ‘the end of the year 
 1758, seemed at Paris to have a tail scarcely 4 degrees in length; whereas some ob- 
 servers at Montpelier found it to be 25°; and it appeared still longer to others 
 at the Isle of Bourbon.* 
 
 In regard to the cause which produces the tails of comets, there are only two opi- 
 _hions which seem to be founded on probability. According to Newton, they are - 
 vapours raised by the heat of the sun, when the comet descends into the inferior 
 regions of our system. It is therefore observed that the tail of a comet is longest 
 when it has passed its perihelion ; and it always appears longer the nearer it approaches 
 to the sun. But this opinion is attended with considerable difficulties: According 
 to M. de Mairan, these tails are a train of the zodiacal light, with which comets 
 become charged in passing between the earth and the sun. It is remarked that 
 comets which do not reach the earth’s orbit, have no sensible tail; or are at most 
 surrounded by aring. Of this kind was the comet of 1585, which passed the sun at 
 a distance 7, greater than that of the earth; the comet of 17)8, which passed at a dis- 
 tance almost equal to that of 1729, that is, at a distance nearly quadruple; and that 
 of 1747, which passed at a distance more than double. It is indeed true, that the 
 comet of 1664, which passed at a greater distance from the sun than that of the earth, 
 appeared with a tail, but it was of a moderate size; and as the distance of its peri- 
 helion was’ very little more than that of the earth from the sun, and as the solar 
 
 * The tail of Halley’s comet underwent some remarkable chances during the appearance of the 
 comet im 1855—6. A beautiful set of drawings, by Mr. P. Smith, of the appearance of the tail at 
 the Cape of Good Hope, have been engraved and published in vol. 10, Mem. Royal Ast. Soc. 
 
OF COMETS. 473 
 
 atmosphere extends sometimes beyond the earth’s orbit, no objection of any great 
 weight cau thence be made, in opposition to the opinion of M. de Mairan. 
 
 We shall remark, in the last place, that while the other planets perform their revo- 
 lutions in orbits very little inclined to the ecliptic, and proceed in the same direction, 
 comets on the other hand move in orbits, the inclination of which to the ecliptic 
 amounts even to a right angle. Besides, some move according to the order of the 
 signs, and are called direct ; others move in a contrary direction, and are called retro- 
 grade. These motions being combined with that of the earth, give them an appear- 
 ance of irregularity, which may serve to excuse the ancients for having been in an 
 error respecting the nature of these bodies. : 
 
 , It has been already said that there are some comets which pass very near the 
 earth; and hence a catastrophe fatal to our globe might some day take place, had not 
 the Deity, by particular circumstances, provided against any accident of the kind. 
 
 A comet, indeed, like that of 1744, which passed at a distance from the sun only 
 
 ' greater by about a 50th than the radius of the earth’s orbit, should it experience any 
 derangement in its course, might fall against the earth or the moon, and perhaps carry ~ 
 away from us the latter. Asa multitude of comets descend into the lower regions 
 of our system, some of them, in their course towards the sun, might pass so near the 
 orbit of our earth, as to threaten us with asimilar misfortune. But the inclination of 
 the orbits of comets to the ecliptic, which is exceedingly. varied, seems to have been 
 established by the Deity to prevent that effect. It would be a curious calculation to 
 determine the least distances at which some of these comets pass the earth; we 
 should by these means be enabled to know those from which we have any thing to 
 apprehend: that is, if it could be of any utility to be acquainted with the period of 
 such a catastrophe ; for where is the advantage of foreknowing a danger which can 
 neither be retarded nor prevented ? 
 
 It would seem however, from the extreme tenuity of the matter of which comets 
 are generally formed, that except from actual contact, the more weighty bodies of the 
 system have little to apprehend from them. One comet passed near if not among 
 
 the satellites of Jupiter; and the orbit of the comet was altogether deranged by the 
 attraction of the planet, but the motions of the satellites were not found to be 
 affected in the slightest degree. 
 
 An English astronomer, who possessed more imagination and learning than sound- 
 
 ness of judgment, the clebrated Whiston, entertained an opinion that the deluge was 
 occasioned by the earth’s meeting with the tail of a comet, which fell down upon it 
 in the form of vapours and rain: he advanced also a conjecture, that the general con- 
 _ flagration, which according to the sacred Scriptures is to precede the final judgment, 
 "will be occasioned by a comet like that of 1681; which returning from the sun, with 
 a heat two or three thousand times greater than of red-hot iron, will approach so 
 near the earth as to burt even its interior parts. Such assertions are bold; but they 
 rest on a very weak foundation: and in regard to a general deluge, occasioned by the 
 
 | tail of a comet, we need be under very little apprehension on that head; for if we 
 consider the extreme tenuity of the ether in which the comets float, it may be readily 
 
 conceived that the whole tail of a comet, even if condensed, could not produce a 
 quantity of water sufficient for the effect ascribed to it by Whiston. 
 
 Cassini thought he observed that comets pursue their course ina kind of Zodiac, 
 
 which he even denoted by the following verses :— 
 
 | Antinous, Pegasusque, Andromeda, Taurus, Orion, 
 
 Procyon, atque Hydrus, Ceutaurus, Scorpio, Arcus. 
 
 But the observations of a great number of comets have shewn that this supposed 
 Zodiac of comets has no reality. 
 
474 ASTRONOMY AND GEOGRAPHY. - 'G 
 
 X1.—Of the Fixed Stars. 
 
 As it now remains for us to speak of the fixed stars, we shall here collect every 
 thing most curious in the modern astronomy on this subject. 
 
 The fixed stars may be easily distinguished-from the planets. The former, at least 
 in-our climates, and when they are of a certain magnitude, have a splendour accom- 
 panied with a twinkling called scintillation. But one thing by which they are parti- 
 cularly distinguished is, that they do not change their place in regard to each other, 
 at least in a sensible manner: they are therefore a kind of fixed points in the heavens, 
 to which astronomers have always referred the positions of the moving bodies, such 
 as the moon, the planets, and the comets. 
 
 We have said that the fixed stars in our climates exhibit a sort of twinkling. This 
 phenomenon seems to depend on the atmosphere ; for we are assured that in certain 
 parts of Asia, where the air is exceedingly pure and dry, as at Bender-Abassi, the 
 stars havea light absolutely fixed; and that the scintillation is never observed, except 
 when the air is charged with moisture, as is the-case in winter. This observation of 
 M. Garcin, which was published in the History of the Academy of Sciences for 1743, 
 deserves to be farther examined. 
 
 The distance between the fixed stars and the earth, is so immense, that the diame- 
 ter of the earth’s orbit, which is 190 millions of miles, is in comparison of it only @ 
 point; for in whatever part of its orbit the earth may be, the observations of the 
 same star shew no difference in its aspect; so that it has no sensible annual parallax, 
 Some astronomers however assert that they discovered, in certain fixed stars, an 
 annual parallax of a few seconds. Cassini, in a memoir on this parallax, says he 
 observed in Arcturus an annual parallax of seven seconds, and in the star ealled 
 Capella one of eight.* This would make the distance’ of the sun from the former 
 of these stars equal to about 20250 times the radius of the earth’s orbit, which, 
 being 95 millions of miles, would give for that distance 19237500000000 miles. Be- 
 tween the fixed stars and the Georgian planet, which is the most distant of our 
 system, there would therefore remain a space equal to more than 10000 times the 
 distance of that planet from the sun. : 
 
 Placed at such an immense distance from us, what can the fixed stars be but im- 
 mense bodies, which shine by their own light; in short, suns, similar to that which 
 affords us heat, and around which our earth performs its revolutions? It is very 
 probable also that these suns, accumulated as we may say on each other, have the same’ 
 destination as ours; and are the centres of so many planetary systems, which they 
 vivify and illumine. It would however be ridiculous to form conjectures respect- 
 ing the nature of the beings by which these distant bodies are peopled; but of 
 whatever kind they may be, who can believe that our earth, or our system, is the 
 only one inhabited by beings capable of enjoying the pleasure which arises from the 
 contemplation of such noble works? Who ean believe that an immense whole, a 
 creation almost without bounds, should have been formed for an imperceptible point, 
 a quantity infinitely small ? 
 
 The apparent diameter of the fixed stars is in no manner magnified by the best 
 telescopes; on the contrary, these instruments, while they increase their splendour, 
 seem to diminish their magnitude so much, that they appear only as luminous points; — 
 but they shew in the heavens a multitude of other stars, which cannot be observed © 
 without their assistance. -Galileo, by means of his telescope, which was far inferior 
 to those now employed, counted in the Pleiades 36 stars, invisible to the naked 
 eye; in the sword and belt of Orion 80; in the nebula of Orion’s head 21, and in — 
 
 * It may be safely affirmed that this estimate (for it can have been nothing more) of Cassini is 
 greatly in excess. It is doubtful whether the parallax of any fixed star amounts to 1”. 
 
 e 
 
THE FIXED STARS. 475 
 . 
 
 that of Cancer 36. Father de Rheita says he counted 2000 in Orion, and 188 in 
 : the Pleiades. In that part of the Austral hemisphere, comprehended between the pole 
 and the tropic, the Abbé de la Caille observed more than 6000 of the 7th magni- 
 tude, that is to say perceptible with a good telescope, of a foot in length; a longer 
 telescope shews others apparently more distant, so in progression perhaps without 
 end. What immensity in the works of the Creator! And how much reason to 
 exclaim with the Psalmist: ‘‘ The heavens declare the glory of God, and the firma- 
 ment sheweth his handy work !” 
 The fixed stars seem to have a common and general motion, by which they revolve 
 _arouud the pole of the ecliptic, at the rate of a degree in 72 years. It is in conse- 
 quence of this motion that the constellations of the zodiac have all changed their 
 positions, Aries occupies the place of Taurus, the latter that of Gemini, and so of 
 the rest; so that the constellations or signs have advanced about 30 degrees beyond 
 the divisions of the zodiac to which they gave names. But this motion is only ap- 
 parent, and not real; and arises from the equincctial points going back every year 
 about 51 seconds on the ecliptic. The explanation of this phenomenon however is 
 _of such a nature as not to come within the object of this work. 
 It has always been believed that the fixed stars have no real motion, or at least- 
 no other than that by which they change their longitude. But it has been disco- 
 vered, by the very accurate observatious of modern astronomers, that some of them 
 have a small motion peculiar to themselves, by which they slowly change their 
 places. ‘Thus Arcturus, for example, has a motion by which it approaches the ecliptic 
 about 4 minutes every 100 years. The distance between this star and another very 
 small one, in its neighbourhood, has been sensibly changed in the course of the last 
 century. Sirius also seems to have a motion in latitude, of more than 2 minutes per 
 century, by which it recedes from the ecliptic. A similar motion has been observed 
 in Aldebaran or the Bull’s Eye, in Rigel, in the eastern shoulder of Orion, in the 
 Goat, the Eagle. &c. Some others seem to have a peculiar motion in a direction 
 parallel to the equator, as is the case with the brilliant star in the Eagle; for in the 
 ‘course of 48 years it has approached one star in its neighbourhod 73”, and receded 
 from another 48”. All thestars perhaps are subject to a similar motion; so that ina 
 series of ages the heavens will afford a spectacle very different from what they do at 
 present. So true it is that nothing in the universe is permanent! In regard to the 
 cause of this motion, however astonishing it may at first seem, it will appear 
 less so, if it be recollected that it has been demonstrated by Newton, that a whole 
 planetary system may have a progressive and uniform motion in space, without the 
 particular motion of the different parts being thereby disturbed. It needs therefore 
 excite no surprise that suns, as the fixed stars are, should have a motion of their 
 own. ‘The state of rest being of one kind only, and that of motion in any direction 
 being infinitely varied, we ought rather to be astonished to see them absolutely at 
 rest, than to discover in them any movement. 
 | But these are not the only phenomena exhibited to us by the fixed stars; for’ 
 ‘some have appeared suddenly, and afterwards disappeared. The year 1572 is cele- 
 brated for a phenomenon of this kind. In the month of November of that year, an 
 exceedingly bright star suddenly appeared in the constellation of Cassiopeia: its 
 ‘splendour at first was equal to that of Venus when in its perigeum, and then to that 
 of Jupiter when he exhibits the greatest brightness ; three months after its appear- 
 ance it was only like a fixed star of the first magnitude: its splendour gradually 
 decreased till the month of March 1574, at which time it entirely disappeared. 
 There are other stars which appear and disappear regularly at certain periods; of 
 this kind is that in the neck.of the whale. When in its state of greatest brightness it 
 is nearly equal to a star of the second magnitude ; it retains this splendour for about 
 
 | , 
 
 “oe 
 
476 ASTRONOMY AND GEOGRAPHY. 
 
 fifteen days, after which it becomes fainter, and at length disappears; it then re- 
 appears, and attains. to its greatest splendour, after a period of about 330 days. 
 
 The constellation of the Swan exhibits two phenomena of the same kind ; for in 
 the breast of the Swan there is a star which has a period of 15 years, during 10 of 
 which it is invisible; it then appears for 5 years, varying in its magnitude and splen- 
 dour. Another, which is situated in the neck near the bill, has a period of about 
 13 months. In the same constellation a star was observed in 1670 and 1671, which 
 disappeared in 1672, and has never since been seen. 
 
 Hydra also has a star of the same kind, which is attended with this remarkable 
 circumstance, that it appears only 4 months; after which it remains invisible for 20, 
 so that its period is about two years. 
 
 Some stars seem to have become extinct since the time of Ptolemy; for he 
 enumerates some in his catalogue which are not now to be seen : others have changed 
 their magnitude; this diminution of size is proved in regard to several of the fixed 
 stars; among this number may be classed the star B in the Eagle, which at the 
 beginuing of the last century was the second in splendour, but which at present is 
 scarcely of the third magnitude. Of this kind also is a star in the left leg of Serpen- 
 tarius or Ophiuchus. 
 
 It uow remains that we should say a few words respecting those stars called 
 nebule. They are distinguished by this name, because, when seen by the naked 
 sight, they appear only like a small luminous cloud. There are three kinds of-them. 
 Some consist of an accumulation of a great number of stars crowded together, 
 and as it were heaped upon each other; but when viewed through a telescope, 
 they are seen distinct, and without any nebulous appearance. Among these is the 
 famous nebula of Cancer, or the presepe Cancri, forming a collection of 25 or 30 
 stars, which may be counted by means of a telescope. ‘Similar groups may be seen 
 in various parts of the heavens. 
 
 Other nebulz consist of one or more distinct stars, but accompanied or 
 surrounded by a whitish spot, through which they seem to shine. ‘There are 
 two of this kind in Andromeda; one in the girdle, and another smaller about a 
 degree farther south than the former. Of this kind also is that in the head 
 of Sagittarius ; that between Sirius and Procyon; that in the tail of the Swan; and 
 three in Cassiopeia. It is probable that our sun appears under this form, when seen 
 from the neighbourhood of those fixed stars which are situated towards the prolonga- 
 tion of his axis; for he has around hima lenticular and luminous atmosphere, which 
 extends nearly to the earth. The Abbé de la Caille counted, in the Austral hemi- 
 sphere, fourteen stars surrounded in this manner with nebulosities; but the most 
 remarkable appearance of this kind, is that of the nebula in the sword of Orion; for, 
 when viewed through a telescope, it is found to be formed of a whitish spot, nearly tri- 
 angular, and containing seven stars, one of which is itself surrounded by a small cloud, 
 brighter than the rest of the spot. One is almost inclined to believe that this 
 spot has experienced some alteration since the time of Huygens, by whom it was 
 discovered. 
 
 The third kind of nebule is composed of a white spot, in which no stars are 
 seen when viewed with the telescope. Fourteen of this kind are found in the 
 Austral hemisphere, among which the celebrated spots, near the South pole, called — 
 by sailors the Magellanic clouds, hold the first rank. They are like small detached 
 portions of the Milky Way. But it may be thought an error to ascribe the splen- 
 dour of that part.of the heavens to small stars accumulated there in a greater mul- 
 titude than any where else; for it does not contain a number, visible by common > 
 telescopes, sufficient to produce that effect; and there are portions of the Milky 
 Way no less brilliant than the rest, though no stars are observed in them, unless with 
 the yery highest improved instruments. 
 
SYSTEM OF THE UNIVERSE. _ 477 
 
 Respecting the milky way, nothing certain is known; but we may conjecture, 
 not without probability, that it consists of some matter similar to that of the solar 
 atmosphere, and which is diffused throughout that celestial space.* If our whole 
 system indeed were filled with a similar matter, it would exhibit to the neighbouring 
 fixed stars the same appearance as the milky way. But why are all these systems, 
 with which that part of the heavens is interspersed, filled with this luminous matter ? 
 To this question no answer certainly can be given. 
 
 We shall here remark, that the famous new star in Cassiopeia had its origin in 
 the milky way, and was perhaps formed by a prodigious quantity of this luminous 
 matter being precipitated on some centre. But it is more difficult to explain why, 
 and in what manner, the star disappeared. This origin of the new star may acquire 
 some probability, if it be true that in the part of the milky way where it was seen, 
 
 there is a vacuity similar to the other parts of the heavens. 
 
 Many of the fixed stars, when examined with telescopes, are found to be double, or 
 ~to consist of two stars. Sir Wm. Herschel has enumerated more than 500 of such 
 stars within 30” of each other, and Struve and other modern observers have in-. 
 creased the list to at least five times the number. Some of the stars which form 
 
 these double ones are within less than one second of each other. In catalogues they 
 are divided into classes, the closest forming the first class. 
 
 On observing the direction of the line joining the centre of these stars with respect 
 to the meridian, Sir Wm. Herschel found that some of them formed dynamic systems, 
 actually revolving round each other, or round a common centre, in circular or 
 
 elliptical orbits of different degrees of eccentricity ; and by continuing to watch their 
 motions, the times of revolution of several are now kuown with very considerable 
 accuracy. 
 
 The stars forming y Virg?nis revolve in about 629 years, Castor in about 253 
 years, 70 Ophiuchi in about 80 years, and € Ursz in about 58 years. 
 
 The following list of a few of the larger class of double stars may be useful for the 
 
 trial of telescopes :— 
 
 Onject. = | Haat pram aa 
 
 y Arietis 1H 44M pale ae ey 9” 
 
 Castor Taues 57 45 5 
 e Piscium 1°. Sa 88 5 5 
 ¢ Aquarii 22° 20 90 55 4 
 » Draconis AFee as ston, eS: 4 
 
 Andron. 37 932 51 ty laa 3 4 
 
 Bodtis 44 14 58 41 38 3 
 — Urse Le 67) 30 2 
 ~ Bootis 14 33 76. 31 1 
 y Virginis | Vase ‘90 29 id 
 
 Besides these double stars, others are found to present more complicated com- 
 | binations, consisting of three, of four, or sometimes of a greater number of separate 
 stars; and it not unfrequently happens that the separate stars are of different 
 colours. 
 
 _ XIi.—Recapitulation of what has been said respecting the System of the Universe. 
 | We shall terminate this chapter with a familiar comparison, calculated to shew, by 
 _known and common measures, the small space which our planetary system occup es 
 
 in the immensity of the universe; and the poor figure, if we may be allowed the 
 
 expression, which our earth makes in it. This consideration will no doubt serve to 
 
 * Unless, with Dr. Herschel, we suppose it isa far extended stratum of stars, by us seen edgeways, 
 
478 ASTRONOMY. rs 
 
 humble those proud beings, who, though they occupy but an infinitely small portion 
 of this atom, have the vanity to think that the universe was created for them. 
 
 ‘To form an idea of our system as compared with the universe, let us suppose the 
 sun to be in Hyde Park, as a globe of 9 feet 3 inches diameter: the planet Mercury 
 will be represented by a globule of about 4 of a line in diameter, placed at the 
 distance of 37 feet. Venus will bea globe of a little more than a line in diameter, 
 circulating at the distance of 68 feet from the same centre: if another globule, a 
 line in diameter, be placed at the distance of 95 feet, it willrepresent the earth, that 
 theatre of so many passions, and so much agitation; on the surface of which the 
 greatest potentate scarcely possesses a point, and where a space often imperceptible 
 excites, among the animalcula that cover it, so many disputes, and occasions so much 
 bloodshed.. Mars, which in magnitude is somewhat inferior to the earth, will be 
 represented by a globule of a little less than a line in diameter, and placed at the 
 distance of 144 feet; Jupiter, by globe 10 lines in diameter, 490 feet from the cen- 
 tral globe; Saturn, about 7 lines in diameter, at the distance of about 900 feet; and 
 the Georgian planet, 4 lines in diameter, at. the distance of 1800 feet. 
 
 But the distance from the Georgian planet to the nearest fixed stars, is immense. 
 The reader may perhaps imagine that, according to the supposition here made, the 
 first star ought to be placed at the distance of two or three leagues. This is the idea 
 which one might form before calculation has been employed; but it is very erroneous, 
 for the first, that is to say the nearest star, ought to be placed at the same distance as 
 that between London and Edinburgh, which is more than 300 miles. Such then is 
 the idea which we ought to have of the distance between the sun and the nearest of 
 the fixed stars ; and there is reason even to think that it is much greater, for we have 
 supposed, in this calculation, that the parallax of the earth’s orbit is the same as the 
 horizontal parallax of the sun, that is to say 8:5”. But certainly this parallax is much 
 less ; for it can hardly be believed that it could have escaped astronomers had it been 
 so great. 
 
 Our solar system then, that is the system of our primary and secondary planets, 
 which circulate around the sun, is to the distance of the nearest fixed stars almost as 
 a circle of 1800 feet radius would be to a concentric one of 300 miles radius; and in 
 the first circle our earth would occupy a space a line in diameter, appearing like a 
 grain of mustard seed. - 
 
 Another comparison, proper to convey some idea of the immense distance between 
 the sun, which is the centre of our system, and the nearest of the neighbouring 
 bodies of the same nature, is as follows: It is well known that the velocity of 
 light is so great, that it passes over the distance between the sun and the earth in” 
 about half a quarter of an hour: in a second and a half it would go to the moon and 
 return, or rather it would go fifteen times round the earth inasecond. What time would 
 light then employ in coming to us from the nearest of the stars ?—Not less than 108 days; 
 or if the annual parallax be only two or three seconds, it would require a year and more. — 
 
 What immense distance then between this inhabited point and the nearest of its 
 neighbours ! ‘Is it not probable that in this vast interval there are planets which will 
 remain for ever unknown to the human species ? 
 
 Modern astronomy indeed has discovered that this space is not entirely desert: it is 
 now known that about a hundred comets move in it, at greater or less distances, but 
 do not penetrate to a very great depth. Those of 1531, 1607, 1682, and 1759, the — 
 only ones the periods and orbits of which are known, do not immerge farther than 
 about 374 times the radius of the earth’s orbit, or four times the distance of Saturn 
 from the sun. If that of 1681 has a revolution of 575 years, as supposed, it must 
 recede from us about 130 times the distance of the earth from the sun, or about 14 
 times that of Saturn from the same body ; which is only a point when compared with 
 the nearest of the fixed stars. But there are comets perhaps which perform their reve- 
 
 . 
 
 a 
 
CHRONOLOGY. | 479 
 
 » Jution only in 10000 years, and which scarcely approach so near the sun as Saturn: in 
 _ that case these would penetrate into the immense space, which separates us trom the 
 first of the fixed stars, as far as a fifteenth part of its depth. 
 
 Those desirous of seeing a great many curious conjectures respecting the system of 
 _the universe, the habitation of the planets, the number of the comets, &c., may con- 
 sult a work by M. Lambert, member of the Royal Academy of Berlin, entitled « Sys- 
 
 téme du Monde,” Bouillon 1770, 8vo. Every one almost is acquainted with the 
 _“ Pluralité des Mondes,” of Fontenelle; the ‘ Cosmotheoros,” of Huygens, the 
 _“Somnium,” of Kepler, and the ‘Iter extaticum,” of Kircher. ‘The first of these, 
 _the ‘“ Pluralité des Mondes,” is an ingenious and pleasing work, but alittle too affected. 
 _ The second. is learned and profound, and, like Kepler’s “‘ Somnium,” will please none 
 
 but astronomers. In regard to the last, however much we may esteem the memory 
 
 of Kircher, it can be considered in no other light than as a production altogether 
 _ pedantic and ridiculous. 
 
 CHAPTER III. 
 
 OF CHRONOLOGY, AND VARIOUS QUESTIONS RELATING TO THAT SUBJECT. 
 
 Aut polished nations keep an account of the time which has elapsed, and of that 
 which is to come, by means of periods that depend on the motions of the heavenly 
 bodies ; and this is even one of those things which distinguish man ina state of civili- 
 zation, from man in the animal and savage state: for, while the former is enabled at 
 every moment to count that part of the duration of his existence which has elapsed ; 
 to foresee, at an assigned period, the recurrence of certain events, labours or duties; 
 the latter, though in some measure happier, since he enjoys the present without recol- 
 lecting the past, or anticipating the future, cannot tell his age, nor foresee the period 
 of the renovation of his most common occupations: the most striking events of which 
 he has been a witness, or in which he has had a share, exist in his mind only as past; 
 while the civilized man connects them with precise periods and dates, by which they 
 are arranged in their proper order. Without this invention, every thing hitherto 
 done by mankind would have been lost to us; there would be rio historical records ; 
 and men, whose existence in the social state requires the united efforts of its different 
 members in certain circumstances, could not employ that concurrence of action, which 
 is necessary. No real civilized society therefore can exist without ‘an agreement 
 to count time in a regular manner; and hence the origin of chronology, and the 
 various computations of time employed by different nations. 
 
 But, before we proceed farther, it will be proper to present the reader with some 
 definitions, and a few historical facts, necessary for comprehending the questions 
 which will be proposed in the course of this article. 
 
 There are two kinds of year employed by different nations ; one of which is regu- 
 jlated by the course of the sun, and the other by that of themoon. The first is called 
 ithe solar, and the second the lunar year. ‘The solar year is measured by a revolution 
 of the sun through the ecliptic, from one point of the equinoctial, that of the vernal 
 equinox for example, to the same point again; and, as already said, consists of 365 
 days 5 hours 49 minutes. 
 
 The lunar year consists of twelve lunations ; and its duration is 354 days 8 hours 
 44 minutes 3 seconds. Hence it follows that the lunar year is about 1) days 
 shorter than the solar; consequently, if a lunar and a solar year commence on the 
 same day, at the end of three years the commencement of the former will have 
 advanced 33 days before that of the latter. The commencement therefore of the 
 lunar year passes successively through all the months of the solar year, in a retro- 
 grade direction. The Arabians and Mussulmans in general, count only by lunar 
 lyears ; and the Hebrews and Jews never employed any other. 
 
480 CHRONOLOGY. 
 
 But the most polished and enlightened nations have always endeavoured to com- 
 bine these two kinds of year together. This the Athenians accomplished by means | 
 of the famous golden cycle, invented by Meto, the celebrated mathematician whom 
 Aristophanes made the object of his satirical wit; and the same thing is done at 
 present by the Europeans, or the Christians in general, who have borrowed from the. 
 Romans the solar year for civil uses; and from the Hebrews their lunar year for 
 their ecclesiastical purposes. 
 
 Before Julius Cesar, the Roman calendar was in the utmost confusion; but it is 
 here needless to enter into any details on the subject: it will be sufficient to ob-. 
 serve, that Julius Cesar, being desirous to reform it, supposed, according to the sug-. 
 gestiou of Sosigines his astronomer, that the duration of the year was exactly 365. 
 days 6 hours. He therefore ordered that, in future, there should he three successive | 
 years of 365 days, and a fourth of 366. This last year was afterwards distinguished | 
 by the name of bissextile, because the day added every fourth year followed the sixth | 
 of the calends which was counted twice; and because, to avoid any derangement in 
 the denomination of the following days, it was thence called bissexto calendas. Among 
 us it is added to the end of February, which has then 29 days instead of 28, which | 
 is the number it contains in common years. This form of year is called the Julian 
 year, and the calendar in which it is employed is called the Julian calendar. 
 
 But Julius Cesar was mistaken, when he considered the year as consisting exactly 
 of 365 days 6 hours; as it contains only 365 days 5 hours 49 minutes; and hence 
 it follows that the equinox always retrogrades in the Julian year 1] minutes _ 
 annually; which gives precisely three days in 400 years. Hence it happened 
 that the equinox, which at the time of the council of Nice corresponded to: 
 the 21st of March, after the lapse of about 1200 years, that is to say, in the year 
 1500, fell about the 11th. Pope Gregory XIII. being desirous to reform this error, | 
 suppressed, in 1582, ten consecutive days; counting, after the 1]th of October, the: 
 21st: and by these means brought back the vernal equinox following to the 2Ist 
 of March; and, in order that it might never deviate any more, he proposed that 
 three bissextiles should be suppressed in the course of 400 years. For this reason 
 the years 1700 and 1800 were not bissextile, though they ought to have been so” 
 according to the Julian calendar; the case will be the same with the year 1900, but 
 the year 2000 will be bissextile; in like manner the years 2100, 2200, and 2300 will 
 not be bissextile ; but 2400 will; and so of the rest. | 
 
 All this is sufficient, and more than sufficient, for the solar year. But the great 
 difficulty of our calendar arose from the lunar year, which it was necessary to com- 
 bine with it ; for as the Christians had their origin among the Jews, they were de- 
 sirous of connecting their most solemn festival, that of Easter, with the lunar year; 
 because the Jews celebrated their Passover at a certain lunation, viz. on the day of 
 the full moon which immediately followed the vernal equinox. But the council-of Nice, 
 that the Easter of the Christians might not concur with the Passover of the Jews, | 
 ordained, that the former should celebrate their festival on the Sunday after the full | 
 moon which should take place on the day of the vernal equinox, or which should 
 immediately follow it. Hence has arisen the necessity of forming periods of luna- | 
 tions, that the day of the new or full moon may be found with more facility, in order 
 to determine the paschal moon. ; 
 
 The council of Nice supposed the cycle of .Meto, or the golden number, according 
 to which 235 lunations are precisely equal to 19 solar years, to be perfectly exact. | 
 After the period of 19 years, therefore, the new and full moons ought to take place 
 on the same days of the month. It was thence easy to determine, in each of these 
 years, the place of the lunations; and this was what was actually Gone by means 
 of the epacts, as shall be her baker explained. | 
 
 But in reality 235 lunations are less, by an hour and a half, than 19 solar Ju> 
 
: 
 
 BISSEXTILE, 481 
 
 Jian years; whence it happens, that in 304 years the new moons retrograde a day 
 towards the commencement of the year: and consequently four days in 1216 years. 
 On this account, about the middle of the 16th century, the new and full moons had 
 anticipated, by four days, their ancient places: so.that Easter was frequently cele- 
 brated contrary to the disposition of the council of Nice. 
 
 Gregory XIII. undertook to remedy this irregularity by an invariable rule, and 
 proposed the problem to all the mathematicians of Europe: but it was an Italian 
 physician and mathematician who succeeded best in solving it, by a new disposition 
 of the epacts, and which the church adopted. This new arrangement is called the 
 Gregorian calendar. It began to be used in Italy, France, Spain, and other Catholic 
 countries, in 1582. It wassoon adopted, at least in what concerns the solar year, 
 even by the Protestant states of Germany: but they rejected it in regard to the lunar, 
 and preferred finding the day of the paschal full moon by astronomical calenlation: 
 the Roman Catholics therefore do not always celebrate Easter at the same time as 
 the Protestants in Germany. The English were the most obstinate in rejecting the 
 Gregorian year, and almost for the same reason which made them long exclude 
 Peruvian bark from their pharmacopeia: that is to say, because they were indebted 
 for it to the Jesuits: but they at length became sensible that whatever is good in | 
 itself, and useful, ought to be received were it even from enemies: and they con- 
 formed to the method of computing time employed in the rest of Europe. This 
 change did not take place till the year 1752. Before that period, when the French 
 counted the 21st of the month, the English counted only the 10th. In the course 
 of ages they would therefore have had the vernal equinox at Christmas, and the winter 
 at Midsummer. The Russians are the only people of Europe who still adhere to the 
 Julian calendar. 
 
 After this short historical sketch, we shall now proceed to the principal problems 
 of chronology. 
 
 PROBLEM I, 
 To find whether a given year be Bissextile or not ; that is to say, whether it consists of 
 366 days. 
 
 Divide the number which indicates the given year by 4, and if nothing remains the 
 year is bissextile: if there be a remainder, it shews the number of the year current 
 after bissextile. We shall here propose, as an example, the year 1774. As 1774 
 divided by 4 leaves 2 for remainder, we may conclude that the year 1774 was the 
 second after bissextile. 
 
 To this rule however there are some limitations. Ist. If the year is one of the 
 
 “centenaries posterior to the reformation of the calendar by Gregory XIII., that is 
 _to say 1582, it will not be bissextile unless the number of the ceuturies which it 
 
 denotes be divisible by 4; thus 1600, 2000, 2400, 2800 have been or will be bissex- 
 tiles; but the years 1700, 1800, 1960, 2100, 2200, 2300, 2500, 2600, 2700, were 
 not, or will not be, bissextiles, for the reason already mentioned. 
 
 2d. If the year be centenary, and anterior to 1582, but without being below 474, it 
 has been bissextile. 
 
 3d. Between 459 and 474 there was no bissextile. 
 
 4th. There was none among the first six years of the Christian era. 
 
 5th. As the first bissextile after the Christian era was the seventh year, and as 
 the bissextiles regularly followed each other every four years till 459; when the given 
 year is between the 7th and the 459th, first subtract 7 from it, and then divide it 
 by 4; if nothing remains, the year has been bissextile ; but if there be any remainder, 
 it will shew what year after bissextile the proposed year was. Let the proposed year, 
 for example, be 148: if 7 be subtracted, the remainder is 141, which divided by 4, 
 
 pint 
 
482 CHRONOLOGY. 
 
 leaves 1 for remainder ; consequently the year 148 of the Christian ra was the | 
 
 first after bissextile. 
 
 Of the Golden Number and Lunar Cycle. 
 
 The golden number, or lunar cycle, is a revolution of 19 solar years, at the end of 
 which the sun and moon return very nearly to the same position. The origin of it is 
 as follows. 
 
 Since the solar Julian year, as already said, consists of 365 days 6 hours; and as 
 the duration of one lunation is 29days 12 hours 49 minutes; it has been found, by 
 combining these two periods, that 235 lunations make nearly 19 solar years; the 
 difference being only 1h.3lm. It is therefore plain that after 19solar years the new 
 moons ought to take place on the same days of the month, and almost at the same 
 hour. In the first of these solar years, if the new moon happen on the 4th of Janu- 
 
 ary, the 2d of February, &c., at the end of 19 years the new moons will take place | 
 
 also on the 4th of January, the 2d of February, &c.; and this will .be the case eter- 
 nally, if we suppose that 235 lunations are exactly equal to 19 solar revolutions, 
 Hence is sufficient to have once determined, during 19 solar years, the days of the 
 month on which the new moons happen; and when it is known what rank a given 
 year holds in this period, we can immediately tell on what days of each month the 
 new moons fall. 
 
 The invention of this cycle appeared to the Athenians to be so ingenious, that, 
 when proposed by the astronomer Meto, it was received with acclamations, and in- 
 scribed in the public square in golden letters: hence the name of the golden number. 
 It is distinguished also by the less pompous denomination of the lunar cycle, or cycle 
 of Meto, from the name of its inventor. 
 
 PROBLEM II. 
 To find the Golden Number of any given year ; or the rank which it holds m the 
 Lunar Cycle. 
 To the given year add 1, and divide the sum by 19: if nothing remains, the golden 
 number of the given year will be 19; but if there be a remainder, which must neces- 
 
 sarily be less than 19, it will be the golden number required. 
 Let the given year, for example, be 1802. If 1 be added to 1802, and if the sum 
 
 1803 be divided by 19, the remainder will be 17; which indicates that 17 is the 
 
 golden number of 1802, or that this year is the 17th of the lunar cycle of 19 years, 
 
 If the year 1728 be proposed, it will be found, by a similar operation, that the re- 
 
 mainder is nothing: which shews that the golden number of that year was 19. 
 
 The reason of adding 1 to the given year, is because the first year of the Christian 
 
 era was the second of the lunar cycle, or had 2 for its golden number. 
 If any year before the Christian wra be proposed, such as the 25th for example, 
 substract 2 from that number, and divide 23 the remainder by 19; if 4 the remainder 
 
 be then taken from 19, the result will be the golden number of the year 25 before ! 
 
 Jesus Christ ; which in this case is 15. 
 
 Remark.—It may be readily seen, that when the golden number of any year has 
 been found, the golden number of the following year may be obtained by adding 1 to 
 the former. The golden number of the preceding year may be obtained also by sub- 
 tracting 1 from the golden number already found. Thus, having found the golden 
 
 number of the year 1802, which is 17, by adding 1 to it, we shall have 18 for that of — 
 the year 1803; and 1 subtracted from it will give 16 for the golden number of © 
 
 1801. 
 Of the Epact. 
 
 The epact is nothing else than the number of days denoting the moon’s age at the . 
 
THE EPACT. 483 
 
 end of a given year. The formation of it may be easily conceived by considering that 
 the lunar year, which consists of 12 lunations, is less than a Julian year by about 11 
 days ; therefore if we suppose that a lunar and a solar year begin together on the Ist 
 of January, the moon at the end of the year will be 11 days old; for 12 complete 
 lunations, and 11 days ofa thirteenth, will have elapsed; and therefore the moon, at 
 the end of the second year, will be 22 days old, and at the end of the third 33. But 
 as 33 days exceed a lunation, one of 30 days is intercalated, by which means that year 
 -has 13 lunations; and consequently the moon is only 3 days old at the end of the 
 third year. 
 
 Such then is the progress of the epacts. That of the first year of the lunar 
 cycle is 11; this number is afterwards continually added, and when the sum exceeds 
 
 - 80, if 30 be subtracted, the remainder will be the epact; except in the last year 
 of the cycle, where the product of the addition being only 29, the same number is 
 deducted to have O for epact:: this announces that the new moon happens at the end 
 of that year, which is also the beginning of the next one. The order of the epacts 
 therefore is 11, 22, 3, 14, 25, 6, 17, 28, 9, 20, 1, 12, 23, 4, 15, 26, 7, 18, 29. 
 
 This arrangement would have been perfect and perpetual, if 19 solar years, of 365 
 days 6 hours, had been exactly equal to 235 lunations, as supposed by the ancient 
 astronomers; but unfortunately this is not the case. On the one hand, the solar 
 year consists only of 365 days 5 hours 49 minutes; and besides, 235 lunations are 
 less than 19 Julian years by one hour and a half; so that in 304 years the real new 
 moons are anterior, by one day, to the new moons calculated in this manner. Hence it 
 happened that in the middle of the 16th ceutury, they preceded by four days those 
 found by calculation; as four- revolutions of 304 years had elapsed between that 
 period and the council of Nice, at which the use of the lunar cycle had been adopted 
 for computing the time of Easter, it was therefore found necessary to correct the 
 calendar, that this festival might not be celebrated, as was often the case, contrary 
 to the intention of that council ; and with this view some changes were made in the 
 calculation of the epacts, which form two cases. One of them is that when the 
 proposed year is prior to the reformation of the calendar, or to 1582: the second is 
 when the years are posterior to that epoch. We shall illustrate both cases in the 
 following problem. 
 
 PROBLEM III. 
 Any year being given, to find its Epact. 
 
 I. If the proposed year be anterior to 1582, though posterior to the Christian 
 wera, which forms the first case; find by the preceding problem the golden number 
 for the given year, and having multiplied it by 11, subtract 30 from the product as 
 many times as possible: the remainder will be the epact required. 
 
 Let the given year, for example, be 1489. Its golden number, by the preceding 
 problem, is 8, which multiplied by 11 gives 88; and this product divided by 30 leaves 
 
 | for remainder 28; the epact of the above year therefore was 28. 
 
 ' - In like manner, if 1796 be considered as a Julian year, that is to say, if those wha 
 
 have not adopted the new style of reformation in the calendar wished to know the 
 epact of that year, it would be necessary first to find the golden number, which is 
 
 11; this multiplied by 11 gives 121; and the latter divided by 30, leaves 1 for re- 
 
 _™ainder. Hence it appears that the epact of 1796, considered as a Julian year, 
 
 was i. 
 
 | Il. We shall now suppose that the given year is posterior to the reformation 
 of the calendar, or to the year 1582; which forms the second case. In this case, 
 multiply the golden number by 11, and from the product subtract the number of days 
 cut off by the reformation of Gregory XIII., that is to say 10, if the year is between 
 
 212 
 
484. CHRONOLOGY. 
 
 1582 and 1700; 11 between 1700 and 1900; 12 between 1900 and 2200, &c.; divide 
 
 what remains after this deduction by 30, and the remainder will be the epact — 
 
 required.* 
 
 Let it be proposed, for example, to find the epact of the Gregorian year 1693, the 
 golden number of which was 3: multiply 3 by 11, and from 33, the product, subtract 
 10: as the number 23 cannot be divided by 30, that number was the epact of the 
 year 1693. ! 
 
 If the epact of the year 1796 were required, the golden number of which was 11; 
 multiply 11 by 11, and from the product 121] subtract 11, which will leave 110: this 
 number divided by 30, gives for remainder 20, which was the epact of the year 
 1796. : 
 
 If the epact of the year 1802 were required, the golden number of which is 17 ; 
 multiply 17 by 11, and from the product 187 subtract 11; the remainder, 176, divided 
 by 30, leaves for remainder 26, which therefore is the epact for the year 1802. 
 
 Remarks.—The epact, according to the Julian calendar, may be found without a 
 division, in the following manner: Assign to the upper extremity of the thumb 
 of the left hand the value of 10; to the middle joint 20, and to the last or root 30, 
 or rather 0. Count the golden number of the proposed year on the same thumb, 
 beginning to count 1 at the extremity, 2 on the middle joint, 3 on the root; then 4 
 at the extremity, 5 on the joint, 6 on the root; and so on, till you come to the 
 golden number found; to which, if it falls on the root, nothing is to be added, be- 
 cause the value assigned to it was 0: but if it falls on the extremity add 10 to it; 
 and if on the middle joint 20; because these were the values assigned to them. The 
 sum, if less than 30, will be the epact required ; if greater than 30, subtract 30 from 
 it and the remainder will be the epact. ; 
 
 Thus, if the epact of 1489 were required: as the golden number of that year was — 
 
 8, count 8 on the thumb, as above mentioned, beginning to count | on the extremity, 
 2 on the middle joint, 3 on the root; then 4 on the extremity, and so on. Because 
 8, in this case, falls on the middle joint, add to it 20, and the sum 28 will be the 
 epact of the above year 1489. In like manner, if the epact of 1726 be required, the 
 
 golden number of which is 17; count 1 on the extremity of the thumb, 2 on the | 
 
 middle joint, &c., till you complete 17, which will fall on the joint; and if 20, the 
 value assigned to that joint, be then added to the golden number, the sum will be 
 
 ' 
 ' 
 
 37; from which if 30 be subtracted, there will remain 7 for the epact of 1726, ac- | 
 
 cording to the Julian calendar. 
 
 By the same artifice the epact for any year of the 17th century might be found ; | 
 provided 20 be assigned to the extremity of the thumb, 10 to the joint, and 0 to the © 
 root; and that you begin to count 1 on the root, 2 on the joint, and so on. | 
 
 PROBLEM IV. 
 To find the day of the New Moon in any proposed Month of a given Year. 
 First find the epact of the given year, as taught in the two preceding problems ; and 
 
 add to it the number of months, reckoning from March inclusively: subtract the sum _ 
 
 from 30 if less, or from 60 if greater ; and the remainder will give the day of the 
 new moon. ' . 
 
 Let it be required, for example, to find on what day the new moon happened in 
 
 } 
 
 the month of May 1802. The golden number of 1802 was 17, which multiplied by | 
 1] gives 187; and if 11 be subtracted, according to the rule, we shall have for re- | 
 
 mainder 176: this divided by 30 leaves 26 = the epact of that year, as before found. 
 
 When the golden number is 1, if the year be posterior to 1900 3 i ‘ i 
 by 1}, and then procerd as. above Girecuon : ‘ PEL et hehe ea 
 
THE MOON’S AGE. 485 
 
 Now the number of months from March, including May, is 2; and 2 added to the 
 epact makes 28, which subtracted from 30 leaves 2: new moon therefore took place 
 on the 2d of May 1802. Accordingly the Almanacs shew it was new moon on the 
 2d at lh. 43m. in the morning. 
 
 Remark.—In calculations of this nature, great exactness must not be expected. 
 The irregular arrangement of the months which have 31 days, the mean numbers 
 necessary to be assumed in the formation of the periods from which these calculations 
 are deduced, and the inequality of the lunar: revolution, may occasion an error of 
 nearly 48 hours. 
 
 More correctness may perhaps be obtained by employing the following table; 
 which indicates what ought to be added to the epact for each commencing month. 
 
 January ecccee 1 
 
 May secccessecee 3B September.... 8 
 
 February...... 2 | JUNE eee eeeee. coe 4 October tae 8 
 Marthicss s4s2%6 J ‘ UR Va sterecsi erties oa eee | November.... 10 
 Aptilevs.cenaau, 2 AUP USC Hs silas ie 94 6 December .... 10 
 
 PROBLEM Y. 
 To find the Moon’s Age on any given day. 
 
 To the epact of the year add, according to the above table, the number belonging 
 to the month in which the proposed day is ; and to this sum add the number which 
 indicates the day: if the result be less than 30, it will be the moon’s age on the 
 given day; if it be 30, it shews that new moon took place on that day; but if it 
 exceeds 30, subtract 30 from it, and the remainder will be the age of the moon. 
 
 Let it be required, for example, to find what was the age of the moon on the 
 20th of March 1802. The epact of 1802 was 26, and the number to be added 
 for the month of March, according to the preceding table, is 1: this added to 26 
 makes 27, and 20, the number of the proposed day, added to 27, makes 47; from 
 which if 30 be subtracted, the remainder is 17 = the moon’s age on the 20th of 
 March ; and this indeed is agreeable to what is indicated by the Almanacs. 
 
 The moon’s age, during the present century, may be found with sufficient exactness 
 by the following rule. : 
 
 From the given year subtract 1800, multiply the remainder by 109, and divide the 
 product by 295. othe tenth part of the remainder add the day of the month, and 
 the number from the subjoined table; and the sum, if it does not exceed 30, is the 
 
 _ moon’s age,—if it exceed 30, the excess is the moon’s age. 
 
 TABLE. 
 
 Jan. Feb. Mar. Ap. May. June. July. Aug. Sep. Oct. Nov. Dee. 
 Common Year 95 6 5 6 7 8 9 10 °s511D:0212 Y sapere 
 
 BR ears Oy oe, Gon Zion SHY HME 104 alk ulB OLB welSeees 
 
 Thus taking the example proposed above, 1800 deducted from 1802 leaves 2; and 
 2 multiplied by 109 gives 218 for a product, which divided by- 295 gives 0 for a 
 quotient, and 218 for a remainder; the tenth parts of which is 21'8 or 22 nearly. 
 
 Remainder ........ A ae se? 25 
 Day of month.......... wi) (20 
 Tab. NO.acs oats isides sas 5 
 
 47 
 Dediicts waded stds Tees oS 
 
 Moon’s age ...... Fecleecee 17 days. 
 
486 CHRONOLOGY. 
 
 Of the Solar Cycle and Dominical Letter. 
 
 The solar cycle is a perpetual revolution of 28 years, the origin of which is as 
 follows: 
 
 Ist. The seven first letters of the alphabet A Bc D EF G are arranged in the calen- 
 dar in such a manner, that a corresponds to the Ist of January, B to the 2d, c 
 to the 3d, p to the 4th, £ to the 5th, F to the 6th, c to the 7th, a to the 8th, B to 
 the 9th, and so on through several revolutions of seven. The seyen days of the 
 week, called also feria, are represented by these seven letters, 
 
 2d. Because a year of 365 days contains 52 weeks and 1 day, and as that remaining 
 day is the first of a 53d revolution, a common year of 365 days ought to begin and 
 end with the same day of the week. 
 
 3d. According to this disposition, the same letter of the alphabet corresponds to 
 the same day of the week, throughout the course of a common year of 365 days. 
 
 4th. As these letters all serve alternately to indicate Sunday, during a series 
 
 of several years, they have on that account been called dominical letters. 
 
 5th. 1t hence follows that if a common year begins by a Sunday, it will end. 
 
 by a Sunday; the Ist of January therefore of the following year will be a Monday, 
 which will correspond to the letter a; and the 7th will be a Sunday, which will 
 correspond to the letter G, which will be the dominical letter of that year. For the 
 same reason the dominical letter of the following year will be F, that of the next 
 one £, and so on, circulating in an order retrograde to that of the alphabet. From 
 
 this circulation of the letters has arisen the name of solar cycle; because Sunday 
 
 among the pagans was called dies solis, the day of the sun. 
 
 6th. If there were no days to be added for bissextile years, all the different 
 changes of the dominical letters would take place in the course of seven years. But 
 this order being interrupted by the bissextile years, in which the 24th of February 
 corresponds to two different feriz of the week: the letter ¥, for example, which 
 would have indicated a Saturday in a common year, will indicate a Sunday in a 
 bissextile year: or if it indicated a Sunday in a common year, it will indicate a Sun- 
 day and a Monday in a bissextile, &c. Hence it follows that in a bissextile year, 
 the dominical letter changes, and that the letter which marked a Sunday in the com- 
 mencement of the year, will mark a Monday after the addition of the bissextile. This 
 is the reason why two dominical letters are assigned to each bissextile year; one 
 
 which serves from the Ist of January to the 24th of February, and the other from - 
 
 the 24th of February to the end of the year; so that the second dominical letter 
 would naturally be that of the following year, if a day had not been added for the 
 bissextile. 
 
 7th. All the possible varieties to which the dominical letters are subject, both in 
 common and in bissextile years, take place in the course of 4 times 7, or 28 years; 
 for after 7 bissextiles the dominical letters return and circulate as before. This 
 revolution of 28 years has been called the solar cycle, or the cycle of the dominical 
 letter. 
 
 PROBLEM VI. 
 To find the Dominical Letter of any proposed year. 
 
 I. To find the dominical letter of any given year, according to the Gregorian 
 Calendar, add to the number of the year its fourth part, or, if it cannot be exactly 
 divided by 4, the least nearest to it; from the sum subtract 5 for 1600, 6 for the 
 following century 1700, 7 for 1800, and 8 for 1900 and 2000; because the years 1700, 
 1800, and 1900, are not bissextiles ; 9 for 2100, 10 for 2200, and 11 for 2300 and 2400, 
 because the three years 2100, 2200, and 2300 will not be bissextiles; divide what 
 
 remains by 7, and the remainder will be the dominical letter required, counting from — 
 
7 
 _—.. 
 
 = 
 
 DOMINICAL LETTER. 487 
 
 the last letter c towards a the first; so that, if nothing remains, the dominical letter 
 willbe a; if 1 remains, the dominical letter will bea; if 2 remains, it will be F; and 
 so of the rest. 
 
 Thus, to find the dominical letter of the year 1802: add its fourth part 450, which 
 makes 2252,and from this sum subtract 7; if the rexainder 2245 be divided by 7, the 
 remainder 5 will shew that the dominical letter is c, since it is the fifth, counting in a 
 retrograde order, from the last letter c. 
 
 We must here observe, that to find with more certainty, by this operation, the. 
 dominical letter of a bissextile year, it will be necessary to find first the dominical 
 letter of the preceding year, which will serve till the 24th of February of the bissextile 
 year; after which the next letter in the retrograde order must be used for the remain- 
 ing part of the year. Thus, if it be required to find the dominical letter of the 
 year 1724; first find that of 1723, by adding to it its nearest less fourth part, 430; 
 subtracting 6 from the sum 2153, and dividing the difference 2147 by 7: the remainder 
 
 5 shews that the dominical letter of the year 1723 was c; which is the fifth of the 
 
 first seven letters of the alphabet, counting in the retrograde order. Since it is 
 known that c was the dominical letter of 1723, it may be readily seen that B was the 
 dominical letter of the following year 1724. But as 1724 was bissextile, B could be 
 used only till the 24th of February, after which a, the letter preceding B, was em- 
 ployed to the end of the year; hence it is seen that a and B were the two dominical 
 letters of the year 1724. In like manner the dominical letters of any future bissextile 
 year may be found. . 
 
 2d. To find the solar cycle, or rather the current year of the solar cycle, corre- 
 sponding to a given year; add 9 to the proposed year, and divide the sum by 28: if 
 nothing remains, the solar cycle of that year is 28; but if there be any remainder it 
 indicates the number of the solar cycle required. 
 
 Thus, if the solar cycle of 1802 be required; add 9, which makes 1811, and divide 
 this sum by 28; the remainder, being 19, shews that 19 is the solar cycle of 1802. 
 
 The reason of this rule is, that the first year of the Christian era was the 10th of 
 the solar cycle; or in other words, that at the commencement of this era 9 years of 
 
 the solar cycle were elapsed. 
 
 Remarks.—The solar cycle of any year whatever may be found with great ease, and 
 without division, by means of the subjoined table. 
 
 7 
 
 a 
 | Years, A se Years. eae Centuries. She Centuries. Orci, 
 
 1 1 10 10 100 16 1000 20 
 
 2 2 20 20 200 25 2000 12 
 
 3 3 30 2 300 20 3000 + 
 
 4 4 40 12 400 8 4000 24 
 
 5 Z 50 22 500 24 5000 16 
 
 6 6 60 4 600 12 6000 8 
 
 7 7 70 14 700 0 7000 0 
 
 8 8 80 24 800 16 8000 20 
 
 9 9 90 6 900 yg 9000 12 
 
 fe NEE ISL eae 1 Bgl Coe ie Le BE Oe SR Se i Sees SAREE MESES 
 
 The method of constructing this table is as follows :—— 
 
 Having placed opposite to the first ten years the same numbers as the solar cycles 
 of these years, and 20 for the solar cycle of the 20th ; instead of putting down 30, for 
 the 30th year, set down only 2, which is the excess of 30 above 28, or above the 
 
 _ period of the solar cycle, For the 40th year inscribe the numbers which correspond 
 
 | 
 
 to 30 and to 10, that is 2 and 10; and so of the rest, always subtracting 28 from the 
 
488 - CHRONOLOGY. 
 
 sum when it is greater. Having thus shewn the method of constructing this table, 
 we shall now explain the use of it. 
 
 In the first place, if the proposed year, the solar cycle of which is required, be in 
 the above table, look for the number opposite to it in the column on the right, 
 marked solar cycle at the top, and add 9 to it; the sum will be the solar cycle 
 required: thus if 9 be added to 12, which stands opposite to the year 2000, we 
 shall have 2] for the solar cycle of that year. 
 
 But, if the given year cannot be found exactly in the above table, it must be divided 
 into such parts as are contained in it. If the numbers corresponding to these parts 
 be then added, their sum increased by 9 will give the solar cycle of the required 
 year ; provided this sum is less than 28; if greater, 28 must be subtracted from it as 
 many times as possible. | 
 
 Let it be required, for example, to find by the above table the solar cycle of the | 
 year 1802. Divide 1802 into the three following parts 1000, 800, 2, and find the. 
 numbers corresponding to them in the right hand columns, which are 20, 16, 2; the 
 sum of these is 38, and 9 added makes 47; from which if 28 be subtracted we shall 
 have for remainder 19, the solar cycle of 1802. 
 
 II. The reason of adding 9 to the sum of all these numbers, is because the. 
 solar cycle, before the first year of the Christian era, was 9; consequently this 
 cycle had begun 10 years before the birth of Christ, which may be ascertained in this’ 
 manner :— 
 
 Knowing the solar cycle of any year, either by tradition or in any other manner, — 
 that of the year 1693 for example, which was 22; subtract 22 from 1693, and divide 
 the remainder 1671 by 28; then subtract 19, which remains, from 28, and the re-_ 
 mainder 9 will be the solar cycle before the first year of the Christian era. 
 
 III. A table to shew the golden number of any proposed year might be con- | 
 structed in the same manner; with this difference, that instead of subtracting 28 
 it would be necessary to subtract 19, because the period of that cycle is 19; and that 
 instead of adding 9, it would be necessary to add only 1; because the golden number, | 
 
 before the first year of the Christian era, was 1: consequently this cycle began two | 
 ‘years before the birth of Christ ; that is to say, the golden number for the first year | 
 of the Christian era, was 2, &c. | 
 
 IV. The dominical letter of any proposed year may be found by another method; | 
 and when this letter is known, it will serve to shew the letter which corresponds to | 
 every day throughout the whole of the same year.* 
 
 Divide by 7 the number of days which have elapsed between the first of January | 
 and the proposed day inclusively; and if nothing remains the required letter will be 
 G; if there be any remainder, it will indicate the number of the required letter, | 
 reckoning according to the order of the alphabet, A 1, B 2, &c. 
 
 Thus, to find the dominical letter of the year 1802; take any Sunday, the 28th 
 of February for example, and find how many days have elapsed between it inclusively, | 
 and the lst of January: as the number is 59, divide this number by 7, and the 
 remainder 3 will shew that C, the third letter of the alphabet, is the dominical 
 letter required. 
 
 _ The days which have elapsed between the first of January and any given period © 
 of the year, may be readily found by means of the following table; but it is to be 
 observed that after February, in bissextiles, the number of days must be increased 
 by unity. y | 
 
 * It is here to be observed, that when you wish to find the dominical letter, the proposed day 
 must be a Sunday ; otherwise you will find only the letter which belongs to some other day. | 
 
WEEK DAYS.—EASTER-DAY. 489 
 
 Days. Days. 
 
 From Jan. to Feb.ccccscscssse 31 From Jan. to August ........ 212 
 Jan. to, March ....4s2.0.: 9 59 Jan. to Sept. ....cecees 243 
 PantOuA PE l iiss 042 AeO Jats tds Octal > vt acens tee 
 Jan. to May...... soeeee 120 _ sat. to Nov, «2.015... 304 
 Matis FO SUNG | «ci sessces (LOL —~Jan. to Dec. weccccsess dot 
 
 Jan. to July.....-.-+-- 18) Jan. to Jan....... mate att « 365 
 
 : PROBLEM VII. 
 To find what day of the Week corresponds to any given day of the Year. 
 
 To the given year add its fourth part, or, when it cannot be found exactly, its 
 nearest least fourth part; and to the sum add the number of days elapsed since the 
 first of January, the proposed day included: from the last sum subtract 14, for the 
 - present century, and divide what remains by 7: the remainder will indicate the day 
 of the week, counting Sunday 1, Monday 2, Tuesday 3, and so on: if nothing re- . 
 mains, the required day is a Saturday. 
 
 Thus, if it be required to know what day of the week corresponded to the 27th 
 of April 1802; add to 1802 its nearest least fourth part 450, and to the sum 2252, 
 add 117, the number of days elapsed between that day inclusive and the Ist of 
 January. If 14 be subtracted from the last sum, which is 2369, and if 2355 which 
 
 remains be divided by 7, the remainder will be 3: consequently the 27th of April 
 1802 was a Tuesday. 
 
 Remark.—If the proposed year be between 1582 and 1700, it will be necessary to 
 deduct only 12 from the sum formed as above. 
 
 If the “year be anterior to 1582, it will be necessary to deduct only 2; because in 
 1582 ten days were suppressed from the calendar. Asa bissextile was suppressed in 
 1700, which makes an eleventh day suppressed, 13 must be subtracted if the given 
 year be in the last century. 
 
 For the same reason 14 must be subtracted in the present century, 15 in the 
 twentieth and twenty-first, and so on. 
 
 PROBLEM VIII. 
 To find Easter-Day and the other Moveable Feasts. 
 By the reformation of the calendar, the 14th day of the paschal moon was brought ° 
 back to the same season in which it was found at the time of the council of Nice, 
 and from which it had removed more than four days, According to the decree of 
 that council, Easter ought to be celebrated on the first Sunday after the 14th day of 
 the moon, if this 14th day should happen on or after the 21st of March. Hence it is 
 obvious that Easter cannot happen sooner than the 22nd of that month, nor later 
 than the 25th of April: which on that account have been called the paschal limits. 
 The following is a table of these limits from the year 1700 to 1900. 
 
 Lunar Paschal Lunar Paschal | Lunar Paschal 
 
 Cycle. Limits. Cycle. Limits. i Cycle. Limits. 
 1 April 13 8 March 27 14 March 21 
 2 April 2 9 April 15 15 April 9 
 3 March 22 10 April 4 16 March 29 
 4 April 10 11 March 24 17 April 17 
 5 March 30 12 April 12 18 April 6 
 6 April 18 13 April 1 19 March 26 
 7 April. 17 
 
490 
 
 CHRONOLOGY. 
 
 By means of this table Easter may be found in the following manner. First find 
 the golden number or lunar cycle of the year, and opposite to it, in the above table, | 
 
 will be found the day of the month on which the paschal full moon happens in that 
 year. The Sunday immediately following is Easter-day, according to the Gregorian 
 calendar. If the full moon happens on a Sunday, Easter-day will be the Sunday 
 
 following. 
 
 | 
 
 Thus, if Easter-day 1802 were required, as the golden number of that is 17, op- : 
 posite to it will be found April 17th; and as the following day, or the 18th, is a 
 Sunday, Easter-day happens on the 18th of April. | 
 
 columns, each divided into seven parts. 
 
 falls. 
 
 Second Method. 
 
 Easter may be found also by means of the following table, which consists of nine 
 The first column contains the dominical 
 letters, the seven following the epacts, and the ninth the day on which Easter 
 
 TABLE FOR FINDING EASTER. 
 
 am ne | ns | | | ns | | es | | 
 
 HSM 291 2) 
 1851173) e216 
 A 11 | 10 9 
 4 3 2 
 27 | 26 | 25 
 23 | 22 | 21 
 17 eelOmieo. 
 B 10 9 8 
 3 2 1 
 | 26 | 25 | 24 
 93%) 29.21 
 16,4. 15 714 
 C 9 8 7 
 2 1 * 
 25 | 24 
 23 
 OMT 20 
 D Tome 4 ies 
 8 a 6 
 1* |.29 | 28 
 23 | 22 
 21 | 20 | 19 
 E TAs iS) 612 
 7 6 5 
 *1.29.4.28 
 pa ak sj ones Re Nw or ee 
 931922 1 OF 
 20 | 19 | 18 
 F Bay 19411 
 6 5 4 
 29 | 28 | 27 
 93 | 22 | 21 
 19 | 18 | 17 
 G E27 ULI ELO 
 5 4 3 
 28 127. 4826 
 
 —— 
 
 20 | 19 26 March 
 15] 14 | 13) 12 2 April 
 8 aad gt Gains 9 April 
 
 Lity © 4129.1023, 1G April 
 24 | 23 April 
 20 | 19 | 18 7 March 
 LAM PES 1a) es 3 April 
 
 Tivwaye Br lite Baek 10 April 
 B29 28 plat el eA eid 
 24 April 
 
 20°41 1997. 18 a7 28 March 
 L3eiy12.P TIANi0 4 April 
 CIS ile 45 8a Ali 
 29 | -298. | 27.44.26 18 April 
 
 25 April 
 22 March 
 29 March 
 L2HE1aH) YOUNGS 5 April 
 
 19° 18 tLe 1G 
 
 Bir Beale 33st  1seApnl 
 27 |. 26 |. 25 [24 | 19-Apria 
 
 23 March 
 YSr TT 116s "15" 30 Naren 
 
 BL?) TOM O18 6 April 
 
 ASH yCQA} 13 April 
 27. 1.261325) 24 |. 20) April 
 
 24 March 
 17 16°) T5414 31 March 
 
 TOM) POTTS RIPE, 7 April 
 
 Sp 2) (1 ae ee a te pr 
 26 1 25) )-04 21 April 
 20 25 March 
 16 | 15 | 14 | 13 1 April 
 
 9 8 ¥i 6 8 April } 
 
 15 April 
 25 | 24 22 April 
 
’ 
 
 WEEK DAYS, &e. 491 
 
 To use this table, the epact and dominical letter for the given year must be found. 
 Thus if 1802 were proposed, the dominical letter of which is C, and the epact 26; 
 look in one of the cells, opposite to that inscribed C, for the epact 26, and opposite 
 to it will be found, in the last column on the right, the 18th of April, which is 
 Easter-day. aa 
 
 Third Method. 
 
 If the epact of the proposed year does not exceed 23, subtract it from 44; and 
 the remainder, if less than 31, will give the paschal limits in March; if greater than 
 $1, the surplus will be the paschal limits in April. 
 
 But if the epact is greater than 23, subtract it from 43, or from 42, when it-is 24 
 or 25; the remainder will be the day of the paschal limits in April, and the Sunday 
 following will be Easter. 
 
 Remark.—Since all the other moveable feasts are regulated by Easter, when the 
 day on which it falls is known, it will be easy to find the rest. Septuagesima Sun- 
 day is nine weeks or 64 days before it, both the Sundays included. Ash-Wednesday 
 is the 47th day preceding Easter, and the Sunday following Ash-Wednesday is the 
 first Sunday in Lent. Ascension-day is 40 days, Pentecost. or Whit-Sunday is 50 
 days, and Trinity Sunday is 57 days, after Easter. 
 
 PROBLEM IX. 
 To find on what day of the week each month of the year begins. 
 
 As it has been usual in the calendars to mark the seven days of the week with 
 the first seven letters of the alphabet, always calling the lst of January A, the 
 2d B, the 3d C, the 4th D, the 5th E, the 6th F, the 7th G, and so on throughout 
 the year; the letters answering to the first day of every month in the year, according 
 to this disposition, may be known by the following Latin verses: 
 
 Astra Dabit Domiaous, Gratisque Beabit Egenos, 
 _ Gratia Christicole Feret Aurea Dona Fideli. 
 
 Or by these French verses :— 
 
 Au Dieu De Gloire Vien Espere ; 
 Grand Coeur, Faveur Aime De Faire. 
 
 Or by the well known English ones :— 
 
 At Dover Dwells George Brown lisquire, 
 
 Good Caleb Finch, And David Frier. 
 Where the first letter of each word is that belonging to the first day of each month, 
 in the order from January to December. 
 
 Now, as these letters, when the dominical letter is A, indicate the day of the 
 week by the rank which they hold in the alphabet, it is evident in that case that 
 January begins on a Sunday, February on a Wednesday, March on a Wednesday, 
 
 _ April on a Saturday, and so on. But when the dominical letter is not A, count 
 either backwards or forwards from the letter of the proposed month, till you come 
 to the dominical letter of the year, and see how many days are between them; for, 
 as the dominical letter indicates Sunday, it will be easy, by reckoning back, to find 
 the day of the week corresponding to the letter of the proposed month. 
 
 Thus, if it were required to find on what day of the week February 1802 began ; 
 as the dominical letter of 1802 is C, and as the letter corresponding to February is D, 
 _ which is the one immediately following C, in the order of the alphabet, it is evident 
 
 that February began on a Monday. In like manner if April 1802 were proposed, as 
 the letter G which belongs to that month is the third from C, the dominical letter, it 
 may be readily seen that April 1802 began on a Thursday. 
 
492 ' CHRONOLOGY. 
 The day of the week on which any proposed month begins, may be found also by 
 means of the following table :— 
 MONTHS. A B C D E F G 
 
 January | Sunday} Satur. | Friday | Thurs. | Wedn. | Tues. | Mond. 
 
 February | Wedn. | Tues. Mond. | Sunday ; Satur. | Friday | Thurs, 
 
 —_—— 
 
 March Wedn. | Tues. | Mond. | Sunday | Satur. | Friday | Thurs. 
 
 —__ 
 ——_ 
 
 April Satur. | Friday | Thurs. | Wedn. | Tues. | Mond. | Sunday 
 
 et me eee 
 
 May Mond. | Sunday | Satur. | Friday | Thurs. | Wedn. | Tues. 
 | June Thurs. | Wedn. | Tues. | Mond. | Sunday | Satur. | Friday 
 July Satur. | Friday | Thurs. | Wedn. | Tues. | Mond. | Sunday 
 
 —_ 
 
 August Tues. | Mond. | Sunday | Satur. | Friday | Thurs. | Wedn. 
 
 = 
 
 —- | ————__ --____ 
 
 September, Frida Thurs. | Wedn. | Tues. | Mond. | Sunday | Satur. 
 P sf 
 
 ——— —— 
 
 October, | Sund. | Satur. | Friday | Thurs. | Wedn. eats Mond. 
 
 | 
 
 —. 
 
 November| Wedn. | Tues. Mond. 
 
 —_—_— 
 
 Sunday | Satur.. | Friday | Thurs. 
 
 December | Friday | Thurs. | Wedn. | Tues. | Mona. Sunday | Satur. 
 
 To use this table, look for the dominical letter of the given year at the top, and in 
 the column below it, and opposite to each month will be found the day on which it 
 begins. Thus, as the dominical letter for 1802 is C, it will be seen, by inspecting the 
 table, that January began on a Friday, February on a Monday, March on a Monday, 
 April on a Thursday, and so of the rest. 
 
 PROBLEM X. 
 To find what Months of the year have 31 days, and those which have only 30. 
 
 Raise up the thumb « (Fig. 15.) the middle finger c, and the little, 
 finger E, of the left hand; and keep down the other two, viz. the. 
 fore finger B, which is next to the thumb, and the ring-finger p, which 
 is between the middle finger and the little finger. Then begin 
 to count March on the thumb a, April on the fore finger p, May on 
 the middle finger c, June on the ring-finger p, July on the little finger 
 FE, and continue to count August on the thumb, September on the fore- | 
 finger, October on the middle finger, November on the ring-finger, 
 and December on the little finger; then beginning again continue’ 
 to count January on the thumb and February on the fore-finger: all 
 those months which fall on the fingers raised up A, c, E, will have 31 days ; and those 
 which fall on the fingers kept down, viz. 8 and p, will have only 30, except February, 
 “which in common years has 28 days, and in bissextiles 29. | 
 
 The number of the days in each month may be known also by the following, 
 lines :— 
 
 Thirty days hath September, 
 April, June and November; 
 All the rest have thirty-one, 
 Except February alone. 
 
LUNAR YEARS, &¢. 493 
 
 PROBLEM XI. 
 To find to what Month of the Year any lunation belongs. 
 
 In the Roman calendar, each lunation is considered as belonging to that month in 
 which it terminates, according to this ancient maxiw-or the computists, 
 
 In quo completur, mensi lunatio detur. 
 
 Hence, to determine whether a lunation belongs to a certain month of any given year, 
 as the month of May 1693 for example; having found, by Prob. 5, that the moon’s 
 age on the last day of May was 27; this age 27 shews that the lunation ends in the 
 next month, that is to say in June, and consequently that it belongs to that month. 
 It indicates also that the preceding lunation ended in the month of May, and there- 
 fore belonged to that month. 
 
 PROBLEM XII. 
 
 To determine the Lunar Years which are common, and those which are embolismic. 
 
 This problem may be readily solved by means of the preceding, from which we 
 easily know that the same solar month may have two lunations. For two moons 
 may end in the same month, which has 30 or 31 days, as November, which has 30; 
 or one moon may end the first of that month, and the fellowing moon on the last or 
 30th of the same month: this year then will have had 13 lunations; and consequently 
 will be embolismic. We shall here give an example. 
 
 In the year 1712, the first moon having ended on the 8th of January, the second 
 on the 6th of February, the third on the 8th of March, the fourth on the 6th of April, 
 the fifth on the 6th of May, the 6th on the 4th of June, the seventh on the 4th of 
 July, the eighth on the 2d of August, the ninth on the Ist of September, the tenth on 
 the Ist of October, the eleventh also on the 380th of the same month, the twelfth on 
 the 29th of November, and the thirteenth on the 28th of December; we know that 
 this year, as it had thirteen moons, was embolismic. 
 
 We know that all the civil lunar years of the new calendar, which begin on the 
 first of January, are embolismic, when they have for epact 29, 28, 27, 26, 25, 24, 
 23, 22, 21, 19; and also 18, when the golden number is 19. 
 
 Thus we know, that in the year 1693, the epact of which was 8, the lunar civil 
 year was embolismic; that is to say, had thirteen moons: this happened because the 
 
 “month of August bad two lunations, one of which ended on the first, and the follow- 
 ing one on the thirtieth of the same. 
 
 PROBLEM XIII. 
 
 An easy method of finding the Calends, Nones, and Ides, of any month in the year. 
 The denomination of Calends, Nones, and Ides, was a singularity in the Roman 
 
 Calendar; and as these terms frequently occur in classical authors, it may be useful 
 
 to know how to reduce them to our method of computation. This may be easily 
 done by means of the three following Latin verses. 
 
 Principium mensis cujusque vocato calendas: 
 Sex Maius nonas, October, Julius et Mars ; 
 Quator at reliqui : dabit idus quidlibet octo. 
 
 Which have been thus translated into French: 
 
 A Mars, Juillet, Octobre et Mai 
 Six Nones les gens ont donné ; 
 Aux autres mois quatre guardé; 
 Huit Ides 4 tous accordé. 
 
 The meaning of these verses is, that the first day of each month is always called 
 the calends ; 
 
raves ARS 
 
 494 CHRONOLOGY. 
 
 That in the months of March, May, July and October the nones are on the seventh 
 day, and in all the other months on the fifth ; 
 
 Lastly, that the ides are eight days after the nones, viz. on the fifteenth of March, — 
 
 May, July and October; and on the thirteenth of the other months. 
 
 It must now be observed that the Romans counted the other days backwards; 
 always decreasing, and that they gave the name of nones to those days of the month 
 which were between the calends and nones of that month; that of ides to those 
 
 days which were between the nones and ides of that month ; and the name of calends | 
 to those days which remained between the ides and the end of the preceding 
 
 month. 
 Thus in the four months of March, May, July and October, where the nones 
 
 had six days, the second day of the month was called sexto nonas ; that is to say the | 
 sixth day before the nones, the preposition ante being here understood. In like ~ 
 manner the third day was called quinto nonas ; that is to say the fifth day of the nones, — 
 or before the nones; and so of the rest. But instead of calling the sixth day of the 
 month secundo nonas, they said pridie nonas ; that is, the day preceding the nones, — 
 
 They said also postridie calendas, the day after the calends ; postridie nonas, the day 
 after the nones ; postridie idus, the day after the ides. 
 
 PROBLEM XIV. 
 
 To find what day of the Calends, Nones, or Ides, corresponds to a certain day of any 
 given month. 
 
 To solve this problem, attention must be paid to the remark already made, that 
 all the days between the calends and the nones belong to the nones; that those be- 
 tween the nones and the ides bear the name of ides; and that those between the 
 ides and calends of the following month, have the name of the calends of that month. 
 This being premised, the following method must be pursued. 
 
 Ist. If the day of the month belongs to the calends, add 2 to the number of the 
 days in the month, and from the sum subtract the given number; the remainder will 
 be the day of the calends. 
 
 Thus, for example, to find to what day of the Roman calendar the 25th of May 
 corresponds, it is first to be observed that it belongs to the calends, since it is be- 
 tween the ides of May and the calends cf June. As the month of May has 31 days, 
 add 2 to this number, which will make 33; andif 25 be subtracted from the sum, the 
 remainder 8 will shew that the 25th of May corresponds to the 8th of the calends of 
 June; that is to say, the 25th of May among the Romans was called octavo calendas 
 Junit. 
 
 2d. If the day of the month belongs to the ides or the nones, add 1 to the number 
 of days elapsed between the first of the month and the ides or nones inclusively ; from 
 this sum subtract the given number, which is the day of the month, and the remainder 
 will be exactly the day of the nones or ides. 
 
 We shall suppose, for example, that the given day is the 9th of May, which be- 
 longs to the ides; as it is between the seventh day of the nones and the fifteenth day 
 of the ides. If 1 be added to 15, and 9 be subtracted from the sum 16, the 
 remainder 7 willshew that the 9th of May corresponds to the 7th of the ides of that 
 month ; that is, the 9th of May among the Romans was called septime idus Maiti. 
 
 In like manner, if the proposed day be the 5th of May, which belongs to the 
 nones, because it is between the Ist and 7th; add 1 to 7, and from the sum 8, sub- 
 tract 5, or the given day of the month; the remainder 3, shews that the 5th of May 
 
 corresponds to the 3d of the nones; or that the Romans called the 5th of May, tertio 
 nonas Maiti. 
 
ROMAN INDICTION, 495 
 
 PROBLEM Xv. 
 
 The day of the Calends, Ides, or Nones, being given ; to find the corresponding day of 
 the month. 
 
 This problem may be solved by a method similarto that employed in the preced- 
 ing, but with this difference, that instead -of subtracting the day of the month, to 
 obtain that of the calends, &c., the latter is subtracted to obtain the day of the 
 month. 
 
 Let it be required, for example, to find what day of the month corresponds to 
 the 6th of the calends of June, which the Romans expressed by sexto calendas Junit. 
 As the calends are counted in a retrograde order from the Ist of June towards the 
 ides of May, it is evident that the 6th of the calends of June corresponds to some 
 day in the month of May; and as that month has 31 days add 2 to 31, and from the 
 
 sum 33 subtract 6, or the given day of the calends, the remainder 27 shews that the 
 
 o 
 
 9th of the calends of June corresponds to the 27th of May. 
 
 The same operation must be employed, in regard to the nones and the ides. 
 _ Remark.—The above two questions may be easily solved also by means of a table 
 of the Calends, Nones, and Ides, which will be found with other tables at the end 
 of this part. 
 
 Of the Cyele of Indiction. 
 
 The cycle of indiction is a period of fifteen years, distinguished by that name, 
 according to some authors, because it served to indicate the year in which a certain 
 tribute was paid to the Roman republic; and hence it is called the Roman In- 
 diction. 
 
 It is called also the Pontifical Indiction, because employed by the court of Rome 
 in its bulls, and in all its decrees. The following, it is said, is the origin of tbis 
 custom. In the year 312, Constantine issued an edict, by which he authorised the 
 exercise of the Christian religion throughout the whole empire. Some years after, 
 the council of Nice was assembled, which in 328 condemned the heresy of Arius: 
 in the space therefore of fifteen years, Christianity triumphed over persecution and 
 heresy ; and on that account it was considered as a memorable period. To preserve 
 the remembrance of it, the cycle of indiction was established; the commencement 
 of which was fixed at the Ist of January 313, to make it begin with the solar year ; 
 
 though the epoch of this cycle, according to the institution of Constantine, had 
 
 been fixed at the month of September 312, the date of his edict in favour of the 
 Christians. It was the emperor Justinian however who first ordered, that the 
 
 - method of computing by the indiction should be introduced into the public acts. 
 
 But, whatever may have been its origin, which Petau considers as very doubtful, 
 
 it is certain that the first year of the indiction was the year 313 of the Christian 
 ra. The year 312 therefore must have corresponded to 15 of the indiction, had 
 
 _ this method of computation been then in use; and if 312 be divided by 15, the 
 
 remainder will be 12; which shews that the 12th year of the Christian zra was the 
 _ 15th of the indiction: consequently this cycle must have begun three years before 
 
 the birth of Christ: or, in other words, the first year of the Christian era corres- 
 
 _ ponded to the fourth of the indiction, and hence we have a solution of the following 
 | problem. 
 
 PKOBLEM XVI. 
 To find the number of the Roman Indiction which corresponds to any given year. 
 Add 3 to the given year, and divide the sum by 15: the remainder will indicate 
 
 the current year of the indiction. | 
 
‘ee 
 
 496 CHRONOLOGY. 
 
 Let it be required, for example, to find the indiction of the year 1802. If 3 be 
 added to 1802, we shall have 1805, and if this sum be divided by 15, the remainder 
 will be 5. Hence it appears that the indiction for 1802 is 5. 
 
 Of the Julian Period ; and some other periods of the like kind. 
 
 The Julian period is formed by combining together the lunar cycle of 19 years, 
 the solar of 28, and the cycle of indiction of 15. The first year of this period is 
 
 supposed to be that which corresponded to 1 of the lunar cycle, | of the solar cycle, 
 
 and 1 of the cycle of indiction. 
 If the numbers 19, 28, and 15, be multiplied together, the product, 7980, will be 
 the number of years comprehended in the Julian period; and we are assured by the 
 
 laws of combination, that there cannot be in one revolution two of these years 
 
 which have at the same time the same numbers. 
 
 This period is merely an artificial one, invented by Julius Scaliger; but it is 
 convenient on account of its extent, as we can refer to it the commencement of all 
 known eras, and even the creation of the world, were that epoch certain; for, 
 according to the common chronology, it was only 3950 years before the Christian 
 
 ! 
 
 wera. But the commencement of the Julian period goes 4714 years beyond that era; | 
 and hence it follows that the creation of the world corresponds to the year 764 of the 
 
 Julian period. 
 
 The method by which it is found that the year of the birth of Jesus Christ was 
 
 the 4714th of the Julian period, is as follows. It is shewn, by a retrograde calcula- 
 
 tion, that if the three cycles, viz., the sclar, lunar, and that of indiction, had been in 
 use at the birth of Christ, the year in which he was born would have been the 2d ) 
 
 of the lunar cycle, the 10th of the solar, and the fourth of the cycle of indiction. 
 But these characters belong to the year 4714 of the above period, as will be seen in 
 the following problem. That year therefore must be adapted to the year of the 
 
 birth of Christ; from which if we proceed backwards, calculating the intervals | 
 of anterior events, from the profane historians and sacred Scriptures, it will be found | 
 that there were 3950 years between that period and the creation of Adam. If 3950 | 
 then be subtracted from 4714, the remainder will be 764; so thatthe Julian period | 
 
 is anterior to the creation of the world by 764 years. 
 
 PROBLEM XVII. 
 
 Any year of the Julian period being given; to find the corresponding year of the 
 
 lunar cycle, the solar cycle, and the cycle of indiction. 
 
 Let the given year of the Julian period be 6522. Divide this number by 19, and » 
 the remainder 5, neglecting the quotient, will be the golden number; divide the | 
 
 same number by 28, and the remainder 26 will be the year of the solar cycle; if 
 6522 be then divided by 15, the remainder 12 will indicate the indiction. If nothing 
 
 remains, when the given year has been divided by the number belonging to one 
 of these cycles, that number itself is the number of the cycle. Thus, if the year 
 6525 were proposed; when divided by 15, nothing remains, and therefore the indic- | 
 
 tion is 15. 
 
 But if itwere required to find what year of the Christian era corresponds to any — 
 given year of the Julian period; such for example as 6522, nothing is necessary but 
 to subtract from it 4714; the remainder 1808 will be the number of years elapsed » 
 
 since the commencement of the Christian era. 
 All this is so plain that it requires no farther illustration. 
 
 a 
 
THE JULIAN PERIOD. . 497 
 
 PROBLEM XVIII. 
 The Lunar and Solar Cycles and the Cycle of Indiction corresponding to any year being 
 given; to find its place in tivee’uJian period. 
 
 Multiply the number of the lunar cyele by 4200, that of the solar cycle by 4845, 
 and that of the indiction by 6916. 
 
 Add together all these products, and divide the sum by 7980; the number which 
 remains will indicate the year of the Julian period.* 
 
 Let the lunar cycle be 2, the solar 10, and the indiction 4; which is the character 
 of the first year of the Christian era. In this case 4200 x 2— 8400; 4845 x 10 
 = 48450; and 6916 x 4= 27664; the sum of these products is 84514, which di- 
 vided by 7980, leaves for remainder 4714. The year therefore in the Julian period, 
 to which the above characters correspond, is the 4714th, or the origin of the Julian 
 
 _ period is 4713 years anterior to the Christian era. 
 
 Remarks.—I. There is another period, called the Dionysian, which is the product 
 
 _ of the lunar cycle 19, and the solar cycle 28; consequently it comprehends 532 years, 
 
 It was invented by Dionysius Exiguus, about the time of the council of Nice, to in- — 
 clude all the varieties of the new moons and of the dominical letters; so that, after 
 532 years, they were to recur in the same order, which would have been very con- 
 venient for finding Easter and the moveable feasts; butas it supposed the lunar cycle 
 to be perfectly correct, which is not the case, this period is no longer used. 
 
 II. As among the cycles of the Julian period there is one, viz. that of indiction, 
 which is merely a political institution—that is to say, which has no relation to the 
 motions of the heavenly bodies—it would have been of more utility perhaps, to substi- 
 tute inits place that of the epacts, which is astronomical, and contains 30 years: the 
 
 _ number of years of the Julian period would, in this case, have been 15960. This 
 . period of 15960 years was called by the inventor of it, Father John Louis d’ Amiens, a 
 
 Capuchin friar, the period of Louis the Great. But it does not appear that it met 
 
 with that reception from chronologists, which the author expected. 
 
 Of some Epochs or Periods celebrated in History. 
 The first of these epochs is that of the Olympiads. It takes its name from the 
 
 _ Olympic games, which, as is well known, were celebrated with great solemnity every 
 
 four years, about the winter solstice, throughout all Greece. These games were 
 
 “instituted by Hercules; but having fallen into disuse, they were revived by Iphitus, 
 
 one of the Heraclide, or descendants of that hero, in the year 776 before Jesus Christ ; 
 and after that time they continued to be celebrated with great regularity; till the | 
 
 conquest of Greece by the Romans put an end tothem. The era or epoch of the 
 
 i 
 
 Olympiads, begins therefore at the summer solstice of the year 776 before Christ. 
 
 * The year of the Julian period may be found also by the following general rule: Multiply the 
 golden number. by 3780, and the indiction by 1064; subtract the sum of these products from the 
 product of 4845 multiplied by the solar cycle; divide the difference, if it can be done, by 7980, and 
 the remainder will be the year of the Julian period. 
 
 The reason of this rule may be found in the solution of the following algebraic problem: To find a 
 number which, divided by 28, shall leave for remainder @ ; divide by 19, shall leave 6; and by 15, 
 shall leave c. 
 
 Call the three quotients, arising from the division of the required number according to the terms 
 
 of the problem, z, y, z Then the number will be == 287 +@=—-19y+ 6 =15z%+c. From 
 
 : 92+a—b ; 9x+a—b 
 the first equation 28 2 + a.==19y + b, we havey— a@ + qo: Now since Tr is an 
 Ae i 92+a—b . m—a+b ; 
 integer number, let us suppose it == m, then m == ener and 7 =~ 2 m+——79-— ? or making 
 m—at+b 
 9. = 7, or m = 19 nm + a — b, we have, by substitution, a2 == 19” + 2a—2b. There- 
 
 fore 28 r+ a—532n + 57a—56b—U52+ Ce by the third quotient; and by resolving this 
 
 equation in the same manner, putting p and g to denote the successive fractions, we shall find the 
 ' Qumber sought to be 15z + c == 7980qg + 4845 a — 37806 b — 1064 c. 
 
 2K 
 
498 CHRONOLOGY. 
 
 PROBLEM XIX. 
 
 To convert years of the Olympiads into years of the Christian era, and vice versd. 
 
 Ist. To solve this problem, subtract unity from the number of the olympiads, and 
 multiply the remainder by 4; then add to the product the number of years of the olym- 
 piad which have been completed, and from the last sum subtract 775; or, if the 
 sum be less, subtract it from 776: in the first case, the result will be the current 
 year of the Christian era, and in the second, the year before that zra. 
 
 Let the proposed year, for example, be the third of the seventy-sixth olympiad. 
 Unity subtracted from 76 leaves 75, which multiplied by 4 gives for product 300, 
 The complete years of an olympiad, while the third is current, are 2: if 2 therefore 
 be added to 300, we shall have 302. But as 302 is less than 775, we must subtract 
 the former from 776, and the remainder 474, will be the current year before Jesus 
 Christ. 
 
 As a second example we shall take the 2d year of the 201st olympiad. If 1 be 
 subtracted from 201, the remainder is 200; which multiplied by 4 gives 800, and 1 
 complete year being added makes 801. But 775 subtracted from 801 leaves 26; 
 which is the year of the Christian era, corresponding to the 2d year of the 201st 
 olympiad. 
 
 2d. To convert years of the Christian wera into years of the olympiads; the number 
 of years, if anterior to the birth of Christ, must be subtracted from 776; or, if pos- 
 terior to that period, 775 must be added to them: if the result be divided by 4, the 
 
 quotient increased by unity will be the number of tne olympiad; and the remainder, 
 
 also increased by unity, will be the current year of that olympiad. 
 Let the proposed year, for example, be 1715. By adding 775, the sum is 2490; 
 and this number divided by 4, gives for quotient 622, with a remainder of 2. The 
 
 year 1715 therefore was the 3d year of the 623d olympiad; or more correctly, the » 
 last six months of the year 1715, with the first six months of 1716, corresponded to | 
 
 the 3d year of the 623d olympiad. 
 
 II. The era of the Hegira is that used by the greater part of the followers 
 of Mahomet: it is employed by the Arabs, the Turks, the various nations in Africa, 
 &c. ; consequently, it is necessary that those who study their history, should be 
 able to convert the years of the hegira into those of the Christian era, and vice 
 versa. 
 
 For this purpose, it must be first observed, that the years of the hegira are nearly 
 lunar; and as the lunar year, or twelve complete lunations, forms 354 days 8 hours — 
 
 48 tdinupene if the year were always made to consist of 354 or 355 days, the new — 
 moon would soon sensibly deviate from the commencement of the year. To prevent | 
 this inconvenience, a period of 30 years has been invented, in which there are ten — 
 common years, that is to say of 354 days; and 11 embolismic, or of 355 days. The | 
 
 latter are the 2d, 5th, 7th, 10th, 13th, 15th, 18th, 21st, 24th, 26th, and 29th. 
 
 It is to be observed also, that the first year of the hegira began on the 15th of 
 
 July 622, of the Christian ra, 
 
 PROBLEM XX. 
 To find the year of Hegira which corresponds to a given Julian year. 
 
 To resolve this problem, it must first be observed that 288 Julian years form | 
 
 nearly 235 years of the Hegira. 
 
 This being supposed, let us take, as example, the year 1770 of the Christian era. 
 Now as 621 years complete of our era had elapsed when the hegira began, we must — 
 
 first subtract these from 1770, and the remainder will be 1149. "We must then 
 employ this proportion: if 228 Julian years give 235 years of the hegira, how many 
 
REMARKABLE EVENTS. 499 
 
 will 1149 give. the answer will be 1184, with a remaiy/der of 99 days. The year 
 1770 therefore, of the Christian era, corresponded, at lgast in part, to the year 1184 
 of the hegira. | 
 
 On the other hand, if it be required to find the year of the Christian era which 
 corresponds to a given year of the hegira, the reverse of this operation must be em- 
 ployed: the number thence resulting will be that of the Julian years elapsed since the 
 commencement of the hegira; and by adding 621, we shall have the current year 
 after the birth of Christ. 
 
 We shall say nothing further on this subject, but terminate the present article 
 with a few useful tables. The first contains the dates of the principal events 
 recorded in history, and of the commencement of the most celebrated zras; the 
 second is a table of the golden numbers for every year from the birth of Christ to 
 5600; the third a table of the dominical letters from 1700 to 5600; the fourth a 
 
 table of the index letters for the same period ; the fifth a table of the epacts; and 
 the sixth a table of the calends, nones, and ides. 
 
 A TABLE 
 
 OF THE YEARS OF THE MOST REMARKABLE EPOCHS OR ZRAS AND EVENTS. 
 
 Years , Years 
 
 Julian 
 Remarkable Events. Pariah: oe neon 
 Memorentign Of the World “Seessescceceses.ce- ans» s07 9 0.0 706 0 | 4007 
 mene Deluge, or Noah’s flood *....cccccccoccecs Oy cre ooo 2362 | 1656 | 2351 
 _ Assyrian monarchy founded by Nimrod ...... Oe Ee 2537 | 1831 | 2176 
 Bemccecor Abraham, waar 8 Se sles el Te Vee oe ed 2714 | 2008 | 1999 
 Kingdom of Athens founded by Cecrops .........-..++5 - | 3157 | 2451 | 1556 
 mutrance of the Israelites into Canaan ...ccs.-seceseeees 3262 | 2556 | 1451 
 BCRtTUCtION Of LTOVS von cc eedseis eres) aun eslece cle aiatee 3529 | 2823 | 1184 
 Solomon’s temple founded ........ Tetras as poet ss see ses {3701 }-2995 |. 1012 
 me Arronautic expedition’ vesccossscecctasase cats e ses. 3776 | 3070 | 937 
 mumeurous. formed his laws )\.,.iisees occ stcnvaveccevess us 3829 | 3103 884: 
 meeeeees 1st king of ithe Medes 2 as o's c.crs--0 vk amis ae old o's 3838 | 3132 875 
 Olympiads of the Greeks began ....s.ceecsccnnceeessces | 3938 | 3232 | 775 
 BEECHES OF TLOMAI) TA y sc cc ossicles s'o0 6.0 5's 6.) aps aioe 40 o's 3961 | 3255 752 
 Aira of Nabonassar ......... RS Serie tae x cree 3967 | 3261 | 746 
 _ First Babylonish captivity by Nebuchadnezzar ........... . 4107 | 3401 606 
 The 2d ditto and birth of Cyrus ........ eee Sic goete toe . | 4114 | 3408 | 599 
 fametomon's temple destroyed. 12... ese sclesuccscece soee | 4125 | 3419) 588 
 eros began to reign in Babylon. ....csserccsecseesees | 4177 | 3471 536 
 Bereponnesian war began .....seessesesesees aieea pues Mut eSea| SO204) 4 ook 
 @uexander the Great died .......ceceecsee: eas Sere Sees 4390 | 3684 | 323 
 Captivity of 100,000 Jews by Ptolemy ..... ‘Ar iiabeshe sess | 4893 |"3687 | 320 
 @eemedes:killed at Syracuse 9.20... cee eet erwin 4506 | 3800 | 207 
 Julias Cesar invaded Britain ............- Be Sibes Sie oes. | 4659 | 3953 54 
 | He corrected the calendar ..... PVE RETE CCE er Sark oe 4667 ; 3961 46 
 pamee rue year of Christ’s birth .........0..seecsssccces 4709 | 4003 4 
 Christian Era begins here. 
 \ : Years ; Years 
 Remarkable Events. Ee ioe shoe Christ 
 Dionysian or vulgar era of Christ’s birth ....-eeeesenceee | 4713 | 4007 0 
 Christ crucified, Friday April 3d ...4--+-+--- cucccccese |:4746:| 4040 33 
 Jerusalem destroyed ...........+.0005 Be etek ie Re 4783 | 4077 70 
 Adrian’s wall built in Britain ............. Ay Lipa ecaenue 4833 | 4127 120 
 ‘Dioclesian epoch, or that of Martyrs ...... ceccsecceeccee | 4997 | 4291 | 284 
 MeO of Nice 2. 0. oo. eect toss cece tena. 5038 | 4332 | 325 
 
 ‘Constantine the Great died .......0...-.e0008 A es 5050 | 4344 } 337 
 
500 CHRONOLOGY. 
 ulian se 
 Remarkable Events. clits 4: cae 
 The Saxons invited into Britain ...ceccesccceccccesseees 5158 | 4452 
 Hegira or flight of Mohammed... eeeee...es sap idle Hike 5335 | 4629 
 Death of Mohammed ....+e...eeeeee AC TRE ioe 5343 | 4637 
 The Persian yesdegird ........ee+ees pe sisid aw auiets Sage 5344 | 4638 
 Sun, moon, and planets seen from the ble oe Be tse soles! € acta 5899 | 5193 
 Art of printing discovered ae oastyce she eh ots Veis see caret ae 6153 | 5447 
 Constantinople taken by the Turks .......... went ee visti 6166 | 5460 
 Reformation begun by Martin Luther ........sece-see.. 6230 | 5524 
 The calendar corrected by Pope Gregory  ....ee+eee-s sere 6295 | 5589 
 Sir Isaac Newton born .... .«%csie'swieee  piedirieinis oterel tints are 6355 | 5649 
 Made president of the Royal Society ........-- esis ytes t 6416 | 5710 
 Died; March 2OUi se <0. as cc NG Se = nae ose Gee Ron eee 6440 | 5734 
 
 Years - 
 since 
 
 Christ. 
 
 445 + 
 662 
 630. 
 631 
 1186 
 1440 
 1453 | 
 1517 
 1582. 
 
 1642 
 1703 
 1727 
 
 TABLE OF SOME OTHER REMARKABLE EVENTS, RELATING CHIEFLY TO THE 
 
 ARTS AND SCIENCES. 
 
 Use of bells introduced into churches ........cesseececeses Ah Ore it 
 Alexandrian library destroyed, and Egypt conquered by the Saracens ........ 
 Organs first used in churches .......... nce bicteiniacarets yr i ee AN bioawiae . 
 Glass invented by a bishop, and brought to Bagi by a Reece monk.. 
 Arabic ciphers introduced into Europe by the Saracens ......eeeesseeseee 
 Astronomy and Geography brought to Europe by the Moors .........+.-. 
 
 Silk manufacture introduced at Venice from Greece .....+esceccceeccecece 
 Spectacles invented by a monk of Pisa........ © 4¢ia\w Rips hem bie e oy4 epee gierale 
 The mariner’s compass invented or improved by asic se eccecceceerscves 
 Gunpowder invented by a monk of Cologne ........ 4 iol ard Boas ue as eile ees 
 
 The art of weaving cloth brought from Flanders to England ....-seseeees0 
 
 Cannon first used in the English service by the Governor of Calais .....ses 
 First company of linen-weavers settled in England ....sescsseesseeees ay 
 Cards invented for the amusement of the French king .....escccecereceres 
 Algebra brought to Europe from Arabia ..... s vie'etee Wetnela isis ete gf sleet ates ale 
 Great guns first used in England at the siege of Berwick ............00.. 
 Paper made of linen rags invented ........ een ess 06 nice Cs sees guess ene 
 Printing invented in Germany ....ceccvccceccsseee a swiss him bibs 015 sey else 
 Engraving and etching invented ....... eee e tee cerrccecseecerescensens 
 Cape of Good Hope discovered ...... Coc ee cc eceececceessctoceeenesocess 
 Geographical maps and sea charts brought to England .......-sceereseeess 
 America discovered by Columbus ....... AGE ha eae A cece ec onencccecs 
 «Algebra taught by a Friar at Venice ...000 eoecesees ER i Aes itech sen ea 
 First voyage round the world by Magellan ....csceccesvcccevssces=vestes 
 Variation of the compass discovered by Cabot .c.cecesecrseeccoteecececs 
 Iron cannon and mortars made in England ......cecaseccvccee socvsscece 
 Glass first manufactured in England ......... WARS Fo SUIT ce asee ee eweeh sam 
 First proposal of settling a colony in America .......scevccsccesesorcccues 
 Bomb-shells invented at Venloo ........c.eeeeeeecs th elena 4 wine ea ae 1 seem 
 Telescopes invented by Jansen, a spectacle-maker of Holland ............ 
 Art of weaving stockings invented by Lee in Cambridge ...... Pe 
 Watches brought to England from Germany ........+0. etese eas yh see 
 Thermometers invented by. Drebbel, a Dutchman ......+.esee-eeee- cesece 
 Galileo first observed three of Jupiter’s satellites, January 7th ........ 
 Logarithms invented by Lord Napier of Scotland...... Piri te 
 Circulation of the blood discovered by Hervey ..-sec..0+---see:- so cee 
 Gazettes first published at Venice......... Bre nh oh ata ee APT ROL 
 
ry sa * , “ 
 “8 
 7 
 =» 
 a 
 my 
 # 
 
 BRITISH PHILOSOPHERS AND MATHEMATICIANS, 
 
 Transit of Mercury over the sun’s dise first observed by Gassendi, Nov. 17th. 
 Galileo condemned by the inquisition .... 
 French academy established, January 
 Transit of Mercury observed by Cassini, Nees 11th 
 Polemoscope invented by Hevelius .......ccecescceccece 
 Transit of Venus observed by Horrox, November 24th 
 Barometers invented by Torricelli ....cseees++s-: : ° 
 Royal academy of painting founded by Louis XIV. ............ etiiatene eo. 
 Galileo first applied the pendulum to clocks ...csecsscccseccsoes seseses 
 Air-pump invented by Otto Gueric of Magdeburg ........ gradal'o pista? besita we ae 
 Huygens first discovered a satellite of Saturn, March 25th ........0+5 cose 
 Royal Society of London established, July 15th ......... venansepesdcewue - 
 Royal Academy of inscriptions and belles-lettres founded..........sceeee8- 
 . Academy for sculpture established in France ..........0.ceseeeeees Pile nage 
 The observatory of Paris founded ........+....55 aelalghetete Misia! sietalele « hehere 
 Magic lantern invented by Kircher ..... Be mphmerihe’d y 64 «ot ae eaecclelele ol te ciety 
 _ Academy of sciences established in France .......... solace wows vate ola Mi 
 Cassini discovered 4 of Saturn’s satellites in the course of a few years ...+.... 
 The royal observatory at Greenwich built .......eee.ees seeees 
 The anatomy of plants made known by Grew........ We gr etteste cay ers es see 
 The Newtonian philosophy was published ...... Same ocekee nar: cece 
 The academy of sciences founded at Berlin ...ccecccesserseves 
 Academy of sciences established at Petersburgh ....-coccevccersoecececs 
 _ Aberration of the fixed stars discovered and accounted for by Bradley ...... 
 Transit of Mercury observed by Cassini, Nov. L]th .........0+ cece veee 3S 
 ’ Academy of sciences founded at Stockholm .cecesesseeereceneceseeecsas 
 New style introduced into Great Britain, Sept. 3d, being Fsehoned Sept. 14th 
 
 eeeeseeGeor *t ses eeaeee eer es oe veee 
 
 @erereece eee evoesetseaeane 
 
 eooee emerere 
 
 British Museum established at Montague-House, by an act of parliament . 
 Transit of Venus over the sun, June 6th..... sna eis sane waar eatels ote ccs es Pe 
 ' Royal academy of arts erepuhed Ab LONCONT vane cio we oes ame es veld allne bier 
 
 Transit of Venus over the sun’s disk, June 34d, CO IRRR A SRM Dire ii 
 
 EMINENT BRITISH PHILOSOPHERS AND MATHEMATICIANS. 
 
 | Arbuthnot, John, M.D. .ooe..oe.. sevens Oo ceo c ee nen seers enanaveccsess 
 famecon, Hoger, philosopher 6. vccvcsssesectcececcsssesesncsnccconcerecs 
 | Bacon, Lord, ditto ...... Snot ia atte’ o Cake ee wikis 8H oe € Mhk bite che Wels btafe eisGievemts ce ¢ 
 | Barrow, Isaac, mathematician ......... seid Ad wabe sid telsia Wie didi pian h eie's cieisl's 8 ae 
 | Boyle, Robert, phil....seeseeessssee vee wieeve cts eo wd Cee eecsw sans oe baccoe 
 trerewood, Edward, phil. and math, ...cccccecscccccnscnvcyeusce cocececs 
 | Briggs, Henry, math. ....s.seesseseccceseesecee sc ececcees ccc cece evesece 
 | Cheyne, George, phys. and phil. .......... 6 0000s oieigs ticle v1 sccccercsocces 
 | Clark, Samuel, phil. and math. .....0..sescerccccees pi aterisreis © seach: aoe 
 Cook, James, navigator... csicccescsccessecs estaia delete eat stein es ese cen ne 
 Derham, William, philosopher ......... AE TALE Fe ht ae Seon cee 
 Dudley, Sir Robert, phil. and math. .....seseesseseesees HSER CAIN ioxBt : 
 _ Evelyn, John, phil. .....+....00. SR APy here eR Dee adids MRE ONO Foe, 
 Ferguson, James, phil. and mech. ....seeeecsssserees Prrrrrrer er 
 Graham, George, math. and mech. ...........++- seeees owlel$ «sie aUnoE tae ose 
 Gregory, James, prof. St. Andrew's ...+.seseeeseeeerseeerenes edevees A 
 Gregory, David, prof. Oxford, astronOMy’....--++.eeeesevece waaisie at ae 
 Gemeiter, Edmund, astron. .. 92.202 sccccccsvesssreoene eatin a diehe Carats a OSs ce 
 | Hales, Stephen, phil, ...ccesececceerererereeece re one bie dia eieis o%s 
 Hailey, Edmund, astron. ....-... sahaka hack (VaNe kn: Cer Dodeeetorks ee 
 
 | 
 
 ae 
 
x A 3 
 ‘ 
 4 
 ' > = 
 Fs 
 
 es 
 
 q 
 
 502 GOLDEN NUMBERS, 4 
 Died, 
 
 Harriot, Thomas, math, 2.2... cecssvercccensesecccsssanensooevences steel fame 
 Harrison, John, inventor of the time-keeper .......6..-.-. bade bw els ola apatite? 
 Harvey, William, phys. discoverer of the circulation of the blood ...... cece.’ 160 
 Horrox, Jeremiah, astron.®... - ..)eiclse sa sls'as viele se sitio die ci celalscty sleleidy 6 seal een 
 Keil, John, mathxand astron, «2... .ccccecvccesccssedsectecscccoscscsscw Liam 
 Locke, John, phil. ........ o eit vide uate niece weep itee sid baat dalore carat ccceccee LOE 
 Long, Robert, astron. Decal nes ecw ale Als oyna: dWaiTeie seule cules glatoels #6 otelel an 
 Lyons, Israel, math. ....:...+scccesseseeees sels a dic Wie o(a'd bea vee etter eenans kane mn 
 Maclaurin, Colin, math.......... arok Kk shalt Saale wa aldla’e @ Qelalate d'etat Lie Sie teens 1746 
 Newton, Sir Isaac, atk and Sphild: cvs lsayeiers 2 2!e5:dte bwtaleoee © i s.alelé atolls] alevate iene mma 
 Pell, John, math. ........ awk wie slbisie « d'b\e eip's o Nie d'a ela.» s¥ ale Ieneilt's = oi aisles tia aes een 
 Pemberton, Henry, phil. ........ pe secitla we sh «eee 4 di ote ia weg tae 1771 
 Ray, John, phil. ..ccessseeseetecccccrersccsvcccerscscesssccessccaceees 1705 
 Simpson, Thomas, math. .......... son cle 'e fits Lathe ach eta 'e'oieeth fat aaa sesus aera a 
 Watts, Isaac, phil. and math. ............ saene s dUGW Se oe ca ce Sle cules aan 
 Whiston, William, astron: /...-.e5ss0. ass dn eS wdialiw aw ohahe Wiese algal ame oes ree 
 Wilkins, John, phil. ...... eer care tae saPElTib ele $1 0.0.8 be wheels rem eh wleratenent ge nn 
 Wren, Sir Christopher, math... cccssessesesccssecccscccccsevcscsscetseees Ling 
 
 TABLE OF THE GOLDEN NUMBERS, 
 
 FOR EVERY YEAR SINCE THE BIRTH OF CHRIST TO THE YEAR 5600. 
 
 = 
 
 The centenary years ; 
 that is, the last years 
 of each century. 
 
 ——— ———— er 
 
 GOLDEN NUMBERS. 
 Intermediate years. 
 
 1 6 11116 Ph if 
 
 12 17. 31 8 13 18] 4 9 14/19 5 10 15 
 fee ee a . 
 1 20 39/58 77 96| 2 7 1217 3 813 18 4! 9 14 19] 5 10 15! 1 611 16 
 2 21 40159 78 97| 3 8 13/18 4 9/14 19 5/1015 1|'61116/2 71217 
 3 22 41/60 79 98} 4 9 14/19 5 10115 1 6/11 16 2! 7 1217/3 813 18 
 4 23 42161 80 99| 5 10 15/1 611116 2 712 17 3| 81318] 4 9 14 19 
 5 24 43/62 81 6 1116/2 712117 3 81318 4| 91419151015 1 
 6 25 44163 82 712 1413 8 13118: 4 9114 19 °5/10°15 1) 6 71 Teme 
 7 26 45/64 83 8 13 18| 4-9 14/19 5 10115 °1° 6111 16 9| 7 19°070Ne 
 8 27 46/65 84 9 14 19] 5°10 15] 1 611116 2 4/12 17 3] 8 13 18) 
 9 28 47/66 85 10 15 116.11 16/2 7 12117 .3°.8113.128 4) 9 142 90mm 
 10 29 48/67 86 11 16 21712171 3 8 13118 .4- 9114 19. 5iLO 15, Slee 
 11 30 49/68 87 12 17 -3| 8 13 18} 4 9 14:79 -5 10}115 1 6|1) 16° 03m 
 12 31 50/69 88 13 18 4) 9 14 19] 5 10 15] 1 6 11/16 °2 7\12 17° "See 
 13 32 51/70 89 1419 5/1015 2) 6 11 16/2 7 12117 3. 8113 18° 4a 
 14 33 52/71 90 15.1 6/1116 2) 7.12 17| 3 8 13118 4 9114 19 (Sec 
 15 34 53/72. 91 16 2 712,17. 3] 8 13 18! 4 9 14119 5 10115 1 Gomme 
 16 35 54/73 92 17 3 S13 18° 4) 9 14 39) 5.10.15¢1 611/16 2 7 12) 
 17 36 55|74 93 18 4 9/1419 5/10 15 1/611 16} 2 7 1217 3°\ Samm | 
 18 37 56/75 94 19 6 1015 1 6/1116 2|-7-12 17/3 8 13/18 4 Sin 
 19 38 57|76 95 1 611/116 2 7112 17 31.8 138 18] 4 9 14/19 5 LOR 
 
DOMINICAL LETTERS. 
 
 TABLE OF THE DOMINICAL LETTERS, 
 
 rrRom 1700 To 5600. 
 
 —— 
 
 2300 
 
 2900 2700 3100 | 2800 3200 
 3300 3700 | 3400 3800 | 3500 3900 | 3600 4000 
 4100 4500 | 4200 4600 |} 4300 4700 | 4400 4800 
 5500 
 
 Jentenary years; 
 that is, the last 
 years of each 
 century. 
 
 Q 
 to 
 q2 
 > 
 
 Intermediate 
 years. 
 
 19295 75.85 
 2 30 58 86 
 eee bo) 9 87 
 4. 32 60 88 
 
 ee ee ee | ee en 
 
 ky 
 yo 
 Q 
 
 ie 
 Q 
 
 FHOaD 
 
 | 
 | 
 
 mart] Uiaa 
 
 eoes 
 
 Qprns 
 iss} ' 
 
 Bi Bey Meal ee 
 6 34 62 90 
 (em OS ot 
 8 36 64 92 
 9 37 65 93 
 10 38 66 94 
 PSI 671495 
 12 40 68 96 
 134) 69 497 
 14 42 70 98 
 152 643% 72, 99 
 16 44 72 
 
 rHWao 
 Q 
 EQ 
 o 
 
 te 
 Emap | 
 
 aria 
 | 
 PWad 
 C2 
 
 Hauer 
 > 
 
 BOQ 
 2b 
 Q | 
 ETAL 
 
 | 
 | 
 | 
 
 17 45 73 
 18 46 74 
 19 47. 75 
 20 48 76 
 
 oo 
 
 Bate 
 - 
 Seam 
 Q 
 Ey dg 
 9 
 
 21 49 77 
 
 Se wo 
 Qt: 
 
 ky 
 
 ye 
 
 — 
 
 mp te | 
 ic > | 
 > 
 
 qeaU 
 
 603 
 
504 
 
 1700 
 1800 
 1900 
 
 2100 
 2200 
 2300 
 2400 
 2500 
 2600 
 2700 
 2800 
 2900 
 3000 
 3100 
 3200 
 3300 
 3400 
 3500 
 3600 
 
 Hgets sr HH ett ES SS Pir DWAO 
 
 2000 © 
 
 CHRONOLOGY; 
 
 TABLE OF THE INDEX LETTERS, 
 From 1700 To 5600. 
 
 Metemptosis.* p Met. 
 
 Met. & proemptosis.f]} n Met. 
 
 Met. n Met. and proem. 
 Bissextile. n Bissextile. 
 
 Met, and proem. m Met. 
 
 Met. 1 Met. 
 
 Met. ] Met, and proem. 
 Bissext. and proem. || 1 Bissextile. 
 
 Met. k Met. 
 
 Met. k Met. and proem. 
 Met. and proem. i Met. 
 
 Bissextile. i Bissextile. 
 
 Met. i Met. and proem. 
 Met. and proem. h Met. 
 
 Met. g Met. 
 
 Bissextile. h Bissext. and proem. 
 Met. and proem. g Met. 
 
 Met. f Met. 
 
 Met. f Met. and proem. 
 Bissext. and proem. || f Bissextile. 
 
 * Metemptosis, or the solar equation, is the suppression of a day. There was a metemptosis m 
 the year 1800, because that year, which ought naturally to have been bissextile, was not so. Since 
 the reformation of the calendar it takes place three times in 400 years. 
 
 + Proemptosis, or the lunar equation, is the anticipation of the new moon. There is a proemptosis 
 in about every 300 years, because the new moon takes place then a day sooner than it ought to do. 
 
 ny 
 
 “ 
 
 ne . 
 as 
 = 
 
505 
 
 Ww} = =61xx x| xtxx| tax! MA| IAxxX AX AI} xx 1x 1 xx xt IMAXX| ITAX TA; AXX AIx] J 
 TI} = TIxxX 1X «| xtx| UIA) MAXX TAX Aj} AIxXxX THX 1 xx X| XIXX] ITIIAX WA) TAXX Ax| 3 
 AI} HIXxX/ TIX | Xx) — NPY Pree TAR A ARE) AIS) OS | oe #|  XIK) MIA) WAxx| TAX) 
 A| AIXX] TIX | xx x| XIXX| IIAx TA] TAXX AX At| 11xx 11x I xx XI] WIAXX| TAX] Tf 
 TA) AXX| AIX] OT) Txx| IX «| XIX} WTA) Axx) TAX A) AIXX| IIx WM) ree} Xx) Xixx| WAX) ¥ 
 
 HA} TAXX| AX| = At] WIXX| = TIX Tie we XX |S CSXT) Asx | TARP ee emeARe || PAI OH eee | @axe) Se 
 
 THA | MAXX} TAX A] Atxx| HTX | xx KX} XIXx| MAX] 1A] TAXX| = AX/ \ AT] WIXX] IX XX} wi) & 
 
 Ae Ne BES tA AXX| ant A se at S * Xie eA A oe RCA es a Txx| U| & 
 
 X| XIXX] MIAX WA| TAXX AX AL} TIXx TIX I xx XI|MWIAXX | TAX TA] AXX| AIX tr} xx} b] - 
 
 1x «| XIX] WIA] Axx] [Ax A} AIXX| TTX Tt) = 1xx x| xixx| max} oa} taxx] Ax] © at] Mixx] d| & 
 
 ux I xx XI|IMAXX| ILAX TA] AXX| AIX W| 1xx 1x é xix| THA] WAXx} = [AX A| Arxx| 1| § 
 
 TIX i) ‘1XX x| XIxx| TITAX WA | TAXX AX AI| T1XxX 11x 1 xXx XI] UWIAXX] TAX TA] AXX) 8] © 
 ; AIX Ut] 1xx 1X »| XIX} A] WAxXx) [AX A] AIXxX| IITX 1} mx x] XIxx] TIAX TA] TAXx}] 4 
 Ay Al} WIXxX| Tx 1 xXx X1| MAXX} TAX TA] AXX| AIX Tr} = TIxx Ix «| XIX} tA! MAXX] f 
 = 1AX A} AIxXx] IIx IL oe xx x| XIxXx| TAX] TIA; [AXX AX At} xx 11x I xx XI} WAXx| W 
 pu TAX AXX| AIX lt} xx Aj AIxx| IIx | xx x} Xixx| @ 
 SQ ojrmax| oma; taxx] ax| AT) Mixx 1) 
 
 *SLOVda 
 
 eS Ree De eet ae a ee ines Pale Sa ee 
 xix | wax] Tax | TAX, [OAS =|) Ake fe | 11x 1x | x | xI 
 
 "SHAAMAN NACTOO 
 
 | ITA | 1A 
 
 009¢ Ob OOLI UVAA AHL WOU ‘SLOVdA AHL dO ATAVL 
 gk oe ee 
 
 OT OSs SP 
 
506 CHRONOLOGY. 
 
 A TABLE OF THE CALENDS, NONES, AND IDES. 
 
 De of April, June, Sept., January, August, March, May, July, 
 
 fee November. December. October. February. 
 1 Calendee. Calende. Calende. Calende. 
 2 IV IV Vi 1V 
 3 111 Ill V III 
 4 Prid. Non Prid. Non IV Prid. Non 
 5 None. None. Ill None 
 6 VIII VIII Prid. Non VIII 
 7 VII VII None VII 
 8 VI VI VIII VI 
 9 V V VII V 
 10 IV IV VI IV 
 11 III III V III 
 12 Prid. Id Prid. Id. IV Prid. Id 
 13 Idus Idus. Ill Idus. 
 14 XVIII XIX Prid. Id XVI 
 15 XVII XVIII Idus XV 
 16 XVI XVII XVII XIV 
 17 XV XVI xXVI XIII 
 18 XIV XV XV XII 
 19 XIII XIV XIV XI 
 20 XII XIII XIII x 
 21 XI XII XII IX 
 22 x XI XI Vill 
 23 Ix xX xX VII 
 24 VIII IX IX VI 
 25 VII VIII VIII V 
 26 VI Vil VII IV 
 27 V ny Pa VI III 
 28 IV V V Prid. Cal. 
 29 III IV 1V Martii. 
 30 Prid. Cal. III III 
 31 Mensis sequen-| Prid. Cal. Prid. Cal. 
 tis. Mens. seq. Mens. seq. 
 
 USE OF THE FOREGOING TABLES. 
 Ist. Table of the Golden Numbers. 
 
 This table contains the centenary years, that is to say, the last years of each 
 century, arranged in cells at the top, and the intermediate years in the ten cells on 
 the left hand. The centenary years which have the same golden number, are placed 
 in different cells, but below each other in a line, as 1800, 3700, 5600. The golden — 
 numbers belong, some to the centenary years, and others to the intermediate years. 
 The former are placed in arow by themselves below the centenary years, and are 
 as follow: 1, 6, 1], 16, 2, 7, 12, 17, 3, 8, 18, 18, 4, 9, 14, 19, 5, 10, and 15. The 
 latter will be found ina line with the intermediate years distributed in 80 different 
 cells. 
 
DOMINICAL LETTERS—-INDEX LETTERS—EPAOCTS, 507 
 
 I. Now to find the golden number of a centenary year, for example 1800; first 
 look for the centenary year in the cell to which it belongs, and immediately below 
 it, in the row at the bottom standing by itself, will be found 15, which was the 
 golden number of that year. 
 
 II. To find the golden number of an intermediate year, 1802 for example. Find 
 the centenary year 1800 in its proper cell, and the intermediate year 2 in the cells on 
 
 the left hand; then on a line with 2, and exactly below 1800, will be found 17, the 
 golden number of 1802. 
 
 2d. Tuble of the Dominical Letters. 
 
 The centenary years are arranged in this table, as in the preceding, in the four 
 “cells at the top, and the intermediate years in the seven cells on the left. All the 
 centenary years which have the same dominical letter are arranged together in one 
 cell, Those which have C for dominical letter in the first, those which have E in 
 the second, those which have G in the third, and those which have B A in the 
 fourth. As in 40 centenary years, the number comprehended in this table, there are 
 10 bissextiles, these 10 years have been placed in the fourth cell, and the other 30 in 
 the first three. The intermediate years placed horizontally in the same cell, differ 
 by 28 years, because the solar cycle contains only that number. Thus the difference 
 between 1] and 29 in the first cell, is 28, and the case is the same with 29 and 57, &c. 
 Each collateral cell contains four perpendicular rows, consisting each of four num- 
 bers, because a bissextile recurs every four years. The four first dominical letters, 
 in the four upper cells, viz., B, D, F, G, correspond to the numbers 1, 29, 57, 85, 
 in the first cell of intermediate years; the case is the same with the dominical letters 
 in the next row, A, C, E, F, in regard to the numbers 2, 30, 58, 86; and so on 
 throughout the table. 
 
 I. To find the dominical letter of a centenary year, 1800 for example. Look for 
 1800, which stands in the second cell at the top, and immediately below it will be 
 found the letter E. . 
 
 II. To find the dominical letter of an intermediate year, as 1802. First find the 
 centenary year 1800 in its proper cell; then look for 2 among the intermediate years, 
 on a line with which, and below the cell containing 1800, will be found the letter C. 
 
 3. Table of the Index Letters, and Table of the Epacts. 
 
 The use of the first of these tables will readily appear, when we have explained 
 _ the nature of the second. The table of the epacts contains the golden numbers in 
 the horizontal column at the top; the index letters are arranged in the first perpendi- 
 cular column, and the epacts in columns parallel to it. Nowif the epact of any year 
 be required; first find the golden number of the proposed year, and in the table 
 of index letters, the letter corresponding to the century; then look for the same 
 letter in the table of the epacts, and also for the golden number at the top; and on 
 a line with the index letter, and directly below the golden number, will be found the 
 epact required. 
 
 Let it be proposed, for example, to find the epact of 1802, the golden number 
 of which is 17. Look in the table of the index numbers, and it will be found that 
 the letter corresponding to 1800 is C; then find C in the first column on the left 
 of the table of epacts; and ona line with it, and directly below xvii among the 
 golden numbers, will be found xxvi, the epact of the year 1802. The epact of any 
 other year, till the year 5600, may be found in like manner. 
 
 * 
 
es 
 
 4th. Table of the Calends, Nones, and Ides. o 
 
 _ This table requires little explanation: look for the given month at the top, an 
 in the column below it, and opposite to the proposed day, will be found the corres. 
 ponding day of the Roman calendar. The day of our calendar, corresponding to : 
 given day of the Roman calendar, may be found with the same ease. sae 
 
DIALLING. 509 
 
 PART SEVENTH. 
 
 CONTAINING THE MOST USEFUL AND INTERESTING PROBLEMS IN 
 GNOMONICS OR DIALLING. 
 
 Gnomonics, or Dialling, is the art of tracing out on a plane, or even on any surface 
 whatever, a sun-dial; that is, a figure, the different lines of which, when the sun 
 shines, indicate, by the shadow of a style, the different hours of the day. This 
 science depends therefore on Geometry and Astronomy, or at least on a knowledge 
 of the sphere. 
 
 As many people construct sun-dials without having a clear idea of the principle 
 
 which serves as a basis to this part of the Mathematics, it may not be improper to 
 begin with an explanation of it. 
 
 The General Principle of Sun-dials, 
 
 Conceive a sphere, with its twelve horary circles or meridians, which divide the 
 equator, and consequently al] its parallels, into 24 equal parts. Let this sphere be 
 placed in a situation suited to the position of the dial; that is, let its axis be directed 
 to the pole of the place for which the dial is constructed, or elevated at an angle 
 equal to the latitude. Now if we suppose a horizontal plane cutting the sphere 
 through its centre, the axis of the sphere will represent the style, and the different 
 intersections of the horary circles with that plane will be the hour-lines; for it is 
 evident, that if the planes of these circles were infinitely produeed, they would form 
 in the celestial sphere the horary circles, which divide the solar revolution into 
 twenty-four equal parts. When the sun 
 therefore has arrived at one of these cir- 
 cles, that of three in the afternoon for 
 example, he will be in the plane of the si- 
 milar circle of the sphere above mentioned ; 
 and the .shadow of the axis or style will 
 fall upon the line of intersection which 
 that circle forms with the horizontal plane: 
 this line then will be the line of 3 o’clock ; 
 and so of the rest. 
 
 All this is illustrated in Fig. 1, which 
 represents a part of the sphere, with six 
 of the horary circles. Pp is the axis, in 
 which all these circles intersect each other ; 
 AH Bh the horizontal plane, or horizon 
 of the sphere, indefinitely continued; a B 
 the meridian; p«£ the diameter of the 
 equator, which is in the meridian; DHEA 
 the circumference of the equator, of which 
 pHeEis a half, and paw a quarter. This 
 quarter of the equator is divided into six 
 
 Fig. 1. 
 
510 DIALLING. 
 
 equal parts, p 1, 1 2,2 3,3 4,4 5,5 6, and through these pass the horary circles, the 
 planes of which evidently cut the horizon in the lines c 1, c 2, c 3, c 4, c 5, c 6: 
 these are the hour-lines , and if we suppose them continued to a F, which is perpen- 
 dicular to the meridian c a, they will give the hour-lines c 1, © 11, C IIT, C IV, C V, C VI. 
 The style will be a portion cs of the axis of the sphere; which consequently ought 
 to form with the meridian, and in its plane, an angle s c A, equal to the height of the 
 pole or PCA. 
 
 Should this reasoning appear too dry and tedious, another method may be em- 
 ployed to acquire a clear idea of the principles of dialling. Construct a solid sphere, 
 divided by its twelve horary circles, and cut it in such a manner that one of its poles 
 shall form with the plane of the section an angle equal to the height of the pole 
 of the given place. 
 
 If the sphere, cut in this manner, be then made to rest on a horizontal plane, with 
 its pole directed towards the pole of the world, the points where the horary circles 
 intersect the horizontal plane, will be readily seen ; and the common section of all 
 the circles, which is the axis, will shew the position of the style. 
 
 For the sake of illustration, we have bere supposed the section of the sphere to be 
 formed by a horizontal plane; but if the plane were vertical, the case would be 
 similar, and the lines of intersection would be the hour-lines of a vertical dial. If the 
 plane be declining or inclining, we shall have a declining or inclining dial: it may 
 even be easily seen that this holds good in regard to every surface, whatever be its 
 form, convex, concave or irregular, and whatever may be its position. 
 
 The style is aniron rod, generally placed in an inclined direction, the shadow 
 of which serves to point out the hours: as before said, it is a portion cs of the 
 axis of the sphere; and in that case it shews the hour by the shadow of its whole 
 length. 
 
 An upright style, however, such as s Q, is sometimes given to dials; but in that 
 case it is only the shadow of the summit s that indicates the hour, because this 
 summit is a point of the axis of the sphere. 
 
 The centre of the dial is the point c, where all the hour-lines meet. It sometimes 
 happens however that these do not meet. This is the case in dials which have 
 their plane parallel to the axis of the sphere; for it is, evident that in such dials 
 the intersections of the horary circles must be parallel lines. These dials are called 
 dials without a centre. Vertical east and west dials, and dials turned directly towards 
 the south, and inclined to the horizon at an angle equal to the latitude, or which - 
 if pr edeed would pass through the pole, are of this number. 
 
 The meridian line, as is well known, is the intersection of the plane of the meri- 
 dian with the plane of the dial; when the plane of the dial is vertical, it is always 
 perpendicular to the horizon. 
 
 The substylar line is that marked out by the plane perpendicular to the plane 
 of the dial, passing through the style. As this line is of great importance in de- 
 clining dials, it is necessary to have a very distinct idea of it: For this purpose, 
 conceive a perpendicular let fall on the plane of the dial, from any point in the style; 
 and that a plane is made to pass through the style and the perpendicular: this plane, © 
 which will necessarily be perpendicular to that of the dial, will cut it ina line passing 
 through the centre, and through the bottom of the perpendicular, and this line will 
 be the substylar line ! 
 
 This line is the meridian of the plane; that is, it shews the moment at which the 
 elevation of the sun above that plane is greatest. Care however must be taken not 
 to confound this meridian with the meridian of the place, or the south line of the 
 dial; for the latter is the intersection of the plane of the dial with the meridian 
 of the place, which is the plane passing through the zenith of the place and the pole; 
 whereas the meridian of the plane of the dial, is the intersection of that plane 
 
MERIDIAN LINE—SUBSTYLAR LINE. 511 
 
 with the meridian, or the horary circle passing through the pole and the zenith of the 
 lane. 
 f In the horizontal plane, or any other which has no declination, the substyle and 
 ‘the meridian of the place coincide; but in every plane not turned directly towards 
 the south or the north, these lines form greater or less angles. 
 Lastly, the equinoctial is the intersection of the plane of the equator with the 
 dial: it may easily be seen that this line is always perpendicular to the substyle. 
 
 PROBLEM I. 
 To find the Meridian Line on a horizontal plane. 
 To find the meridian line is the basis of the whole art of constructing sun-dials ; 
 but as it is at the same time the basis of all astronomical operations, and as we have 
 already treated of it in that part of this work which relates to astronomy, it would 
 
 be needless to repeat here what has been already said on the subject. We shall 
 
 therefore confine ourselves to one ingenious and little-known operation. 
 We shall give also hereafter a method of determining the position of the meridian 
 line at all times, and in all places, provided the latitude be known. 
 
 PROBLEM II. 
 To find the Meridian by the observation of three unequal shadows. 
 
 The meridian line on a horizontal plane is found generally by means of two equal 
 shadows of a perpendicular style; the one observed in the forenoon and the other in 
 the afternoon. For this purpose, several concentric circles are described from the 
 bottom of the style; but notwithstanding this precaution, it may happen that it will 
 be impossible to have two shadows equal to each other. This inconvenience how- 
 ever may be remedied by three observations instead of two. For this ingenious 
 method, we are indebted to a very old author on Gnomonics, named Muzio oddi da 
 Urbino, who published it in a treatise entitled ‘Gli Orologi solari nelle superficie 
 plane.” ‘Chis author was exceedingly devout; for he piously thanks our Lady of 
 Loretto for having communicated to him, by inspiration, the precepts he has taught 
 in his work. The operation is as follows. 
 
 Let p (Fig. 2.) be the bottom of the style, and 
 Ps its height; and let three shadows projected 
 by it be pa, PB, and Pc; which suppose to be 
 tmequal, and let pc be the shortest of them. 
 From the point P draw PD, Pz, and PF, per- 
 pendicular to PA, PB, and Pc, and all equal to 
 each other, as well as to ps. Draw also the 
 lines DA, EB, and Fc, on the two largest of 
 which, viz. Da and EB, assume DG and EH 
 equal to Fc; then from cg and # draw ci and 
 HK, perpendiculars to p A and PB, and join the: 
 points 1 and kK by an indefinite line: make IM 
 
 and Kt perpendicular to 1x, and equal to@1 and xu; and draw ML, which will 
 
 meet 1K in the point N: if through n and c the line cn be drawn, it will be perpen- 
 
 | dicular to the meridian ; consequently by. drawing, from p, the line Po perpendicular 
 _ to cn, it will be the meridian required. 
 
 As the demonstration of this problem would be too long, we must refer the 
 
 _Teader to the fifth book of a work by Schootten, entitled ‘ Exercitationes Ma- 
 
 thematice,”’ 
 
512. DIALLING. 
 
 PROBLEM IIl. 
 To find the Meridian on a plane, or the substylar line. 
 
 After what has been already said, in regard to thé substylar line, this operation 
 will be easy; for since this line is the meridian of the plane, nothing is necessary but 
 to consider it as if it were horizontal, and to trace out on it the meridian by the same 
 method: the line resulting will be the substyle, the determination of which is very 
 necessary for constructing inclined or declining dials, and those which are both at the 
 same time. 
 
 PROBLEM Iv. 
 To describe an Equinoctial Dial. 
 
 From any point c (Fig. 3.) asa centre, describe a 
 circle AEDB; and having drawn the two diameters 
 intersecting each other at right angles in the centre 
 c, divide each quadrant into six equal parts; and 
 draw the radii c 1, c 2, c3, and so on, as seen in the 
 figure. ‘These radii will shew the hours by means 
 of a style perpendicular to the plane of the dial, 
 which must be placed in the plane of the equator; 
 that is, in such a manner as to form with the horizon 
 an angle equal to the complement of the latitude. 
 The line aD must coincide with the plane of the 
 meridian, and in north latitude the point a must be 
 directed towards the south. 
 
 Remarks.—I. When this equinoctial dial is erected, if the hour-lines look towards 
 the heavens, it is called a superior dial, but if they are turned towards the earth, an 
 inferior. 
 
 II. A superior equinoctial dial shews the hours of the day only in the spring and 
 summer; and an inferior one only during the autumn and winter; but at the equi- 
 noxes, when the sun is in the equator, or very near it, equinoctial dials are of no 
 use, as at those periods they are never illuminated by the sun. 
 
 ILI. At London the elevation of the plane of the equator is 38° 29’, which is the 
 complement of the elevation of the pole: the angle therefore which the plane of an 
 equinoctial dial at London should form with the horizon, ought to be 38° 29”. 
 
 IV. It hence appears that it is easy to construct an universal equinoctial dial, which 
 may be adjusted to any elevation of the pole whatever. For this purpose, join 
 together two pieces of ivory, or copper, or any other matter, a B cD and cD EF, 
 
 Fig. 4: (Fig. 4), by means of a hinge at c p: then describe on 
 the two surfaces of the piece a Bc D, two equinoctial dials; 
 and in the centre x place a style extending both ways in 
 a direction perpendicular to ancpD. At, inthe middle 
 of the piece c p EF, fix a magnetic needle, covered with 
 a plate of glass, and towards the edge of the same piece 
 apply a quadrant uu divided into degrees, and passing 
 through an aperture H, made to receive it in the upper 
 piece ABCD. The degrees and minutes must begin to 
 bé counted from the point L. 
 
 When this dial is to be used, place the needle in the meridian, making a proper 
 allowance for the variation; and cause the two pieces A BC Dand c DEF to form an 
 angle BCF, equal to the elevation of the equator at the given place; thatis, equal 
 
POLAR AND VERTICAL DIALS. 513 
 
 to the complement of the latitude. If care be then taken to turn the quadrant 
 towards the south, either of these equinoctial dials will shew the hours at that place, 
 except on the day of the equinox. 
 
 PROBLEM V, 
 Construction of the most important of the other regular Dials. 
 
 Regular dials are those which have the hour lines, forming equal angles on each 
 side of the meridian: these dials therefore are, the equinoctial, the horizontal, the 
 north and south vertical, and the polar. Having already spoken of the equinoctial 
 and horizontal, we shall now proceed to the north and south vertical dials. 
 
 Of the South Vertical Dial. 
 
 If the vertical dial be turned directly towards the south; then make the angle EcK 
 or the arc E kK (Fig. 5.) equal to the height of the pole; if cx v be then 
 made a right angle, the point v will be the centre of the dial; and the 
 angle ¢ vK, which will then be equal to the complement of the latitude or 
 
 of the elevation of the pole, will denote the angle which the ptyle, in the 
 _ plane of the meridian, ought to form with the plane of the dial. 
 
 Of the North Vertical Dial. 
 
 If the vertical dial be north, make, as before, the angle oc k (Fig. 5.) 
 equal to the height of the pole, and the angleckua right angle: the point 
 H will be the centre of the dial; and the angle c Hf will be that which 
 the style forms with the meridian. The style, instead of being inclined downwards, 
 must be turned in a contrary direction, as may be readily conceived when we consider 
 che position of the pole in regard to a vertical plane turned directly towards the 
 iorth. 
 
 EX 
 
 PROBLEM VI. 
 Of Vertical East and West Dials. 
 
 “Next to the dials already described, the simplest are those which Chests front the 
 ast or the west. The method of constructing them is as follows :— 
 Fig. 6 Draw the horizontal line oR, (Fig. 6.) 
 coe and assume it in any point P, for the bottom 
 of the style, the upper extremity of which 
 is intended to shew the hours. At the 
 point p, make, towards the left for an east 
 dial, and towards the right for a west one, 
 the angle H P E, equal to the comple- 
 ment of the latitude of the pole above 
 the horizon; and continue ePpto N. The 
 line E N will be the equinoctial. Then 
 through the point p draw the line c a, 
 in such a manner as to form with the 
 line H R the angle a P H, equal to the 
 elevation of the pole; then a c, which 
 will intersect the equinoctial E N at right 
 angles, will be the hour-line of VI in the 
 morning, and also the substylar line. 
 “When these lines have been traced out, the hour lines may be drawn in the 
 lowing manner. In the substylar line A c, assume a point a, at any distance from 
 te point P according to the intended size of the dial; and from a, as a centre, describe 
 semicircle of any radius at pleasure, Divide this semicircle into twelve equal parts» 
 21 
 
 =A. re 
 
o4° Gy 
 
 514 ) DIALLING, 
 
 beginning at the point p, and then from the centre a draw dotted lines through each of the 
 points of division in the semicircle, till they meet the equinoctial £ N: if lines parallel 
 to the substylar line be then drawn through the points where these dotted lines eut 
 the equinoctial, they will be the hour lines required, the substylar line being that of 
 vt in the morning. The parallels above the substylar line, in the east dial, will cor- 
 respond to 1v and v in the morning; those below it to vit, vitt, &c. in the afternoon, 
 
 Fig. 7. The style, the figure of which is seen in Fig. 7. is placed 
 parallel to the line of v1, on two supports raised perpendicular 
 to the plane of the dial, and at a distance above it equal to that 
 of VI. hours from mz or from 1x. It is here evident that a 
 west is exactly the same as an east dial, only ina contrary situ- 
 Fig. 8. ation (Fig. 8.) ; but instead of marking on 
 it the morning hours, as Iv, V, VI, &c., you. 
 must inscribe on it those of the afternoon, 
 as I, I, W1,1v,&c. Ifan east dial be traced 
 out on a piece of oiled paper, and if the 
 paper be then inverted, but not turned 
 upside down, on holding it between you 
 and the light, you will see a west dial. 
 
 It may be easily seen that these dials 
 cannot shew the hour of noon: for the 
 sun does not begin to illuminate the latter 
 till that hour, and the former ceases to be 
 illuminated at the same period. 
 
 PROBLEM VII. 
 
 To describe a horizontal or a vertical South Dial, without having occasion to find tht 
 horary points on the equinoctial. | 
 
 Fig. 9. Let the line a 8 (Fig. 9.) be the meri, 
 
 dian of thesdial, which we suppose a hori, 
 zontal one; and let c be its centre: mak« 
 the angle H cB equal to the elevation oO 
 the pole, in order to find the position} 
 the style, and from the point B, assumed ai 
 pleasure, but in such a manner that ct) 
 shall be of a proper length, draw 8 F per- 
 pendicular to cu. If we conceive the 
 triangle BrFc raised vertically above the 
 plane of the dial, it will represent the 
 style. | 
 From the point c, with the radius cB 
 describe a circle BD a £; and from the 
 same centre, with the radius B F, describe 
 another circle MQNP. P| 
 
 | 
 
 Divide the whole circumference of the first circle into 24 equal parts, B 0,0 0,00 
 &c., aud then divide the second circle into the same number of equal parts, N R, R Ri, 
 &e.: from the points of division 0, of the great circle, draw lines perpendicular tc 
 the meridian; and from the corresponding points R of the less circle, draw lines 
 parallel to that meridian. ‘These parallels and perpendiculars will meet in certall 
 points, which will serve to determine the hour-lines. For example, the lines 03, 
 R38, which proceed from the third of the corresponding points of divison, will mee! 
 
 ; 
 |) 
 | 
 
| TO DRAW ANY DIAL, | ey De 
 
 in the point 3; through which if c 3 be drawn, it will be the position of the line of 3 
 o’clock ; and so of the rest, 
 It is evident that the larger the circles, the more distinct will be the intersections 
 formed by the lines drawn through the points of division o and rR. 
 _ Itis remarkable that all these points of intersection are found inthe circumference 
 of an ellipse, the greater axis of which is equal to twice cs; and the less rp @ to 
 ‘twice CN, or twice BF. 
 | The reason of this construction will be easily discovered by geometricians. 
 } 
 \. PROBLEM VIII. 
 To trace out a Dial on any plane whatever, either vertical or inclined, declining or not, 
 on any surface whatever, and even without the sun shining. 
 / 
 
 | This problem, as may be seen, comprehends the whole of Gnomonics; ; and the 
 “operation may be practised by any person who knows how to find the meridian, and 
 ‘to construct an equinoctial dial. The solution of it is as follows. 
 
 Having made the necessary 
 preparation (Fig. 10.), trace 
 out a meridian line ona table, 
 according to themethod taught 
 in the first problem; and by 
 means of this meridian, place 
 an equinoctial dial in such a 
 situation, that the plane of it 
 Shall be raised at the proper 
 angle; that is, at an angle 
 equal to the elevation of the 
 equator, or complement of the 
 latitude, and that its south 
 line shall coincide with the 
 
 above meridian. Adjust along 
 the axis a piece of packthread, which being stretched shall meet the plane on which 
 the dial is to be described: the point where it meets this plane is that where the 
 style or axis ought to be placed, so as to form one straight line with the packthread 
 and the style of the equinoctial dial. 
 
 When this is done, and when the axis of the dial has been fixed, hold a candle or 
 taper before the equinoctial dial, in such a manner, that the style shall shew noon; 
 the shadow projected, at the same time, by the packthread, or the axis of the dial 
 about to be constructed, will be the south line. You must therefore assume a point 
 which, together with the centre, will determine that line. If you then change the 
 position of the taper, so that the equinoctial dial shall shew one o’clock, the shadow 
 projected by the packthread, or the axis of the proposed dial, will be the hour-line 
 of 1; and so of the rest. 
 
 Fig. 10. 
 
 } 
 4 
 ) 
 af 
 
 Remarks.—I. If the plane, on which the dial is to be described, be situated in 
 ‘such a manner that it cannot be met by the axis continued, according to the pre- 
 ceding method, two supporters must be affixed to the plane, for the purpose of re- 
 ceiving a rod of iron, so as to make one line with the packthread; and the operation 
 may then be performed as above described. 
 
 II. Instead of an equinoctial dial, a horizontal one may be employed ; provided it 
 be placed in such a manner, that the south line corresponds with the meridian which 
 has been traced out. 
 
 III. This operation may be performed in the day time when the sun shines. In 
 
 2L 2 
 
4 
 
 3 
 
 516 DIALLING, 
 
 this case you must employ a mirror, the reflection of which will produce the sume 
 effect as the taper or candle. 
 
 PROBLEM IX. 
 To describe aVertical Dial on u pane of glass, which will shew the hours without a style 
 by means of the solar rays. 
 
 Ozanam relates that he once constructed a vertical declining dial on a pane of glass 
 in a window, which had no style; and by which the hours could be known when the 
 sun shone. 
 
 I detached, says he, from the window frame on the outside a pane of glass, and 
 described upon it a vertical dial, according to the declination of the window and the 
 height of the pole above the horizon; taking as the height of the style the thickness 
 of the window frame. I then fixed the pane of glass against the frame in the inside; 
 having given to the meridian line a situation perpendicular to the horizon, as it 
 ought to have in vertical dials. I then cemented to the window frame on the out= 
 side, opposite to the dial, a piece of strong paper, not oiled, in order that the surface | 
 of the dial might be more obscure. And thatI might be able to know the hours’ 
 without the shadow of a style, I made a small hole in the paper with a pin, opposite | 
 to the bottom of the style, which I had marked out. As this hole represented the 
 extremity of the style, the rays of the sun passing through it formed on the glass a 
 luminous point ; which, while the rest of the dial was obscure, indicated the hours , 
 in an agreeable manner. | 
 
 PROBLEM X. 
 
 In any latitude, to find the Meridian by one observation of the sun, and at any hour of 
 the day. 
 _ Provide an exact cube, each side of which is about 8 inches; and describe on the | 
 upper face a horizontal dial, adapted to the latitude of the place. On the vertical 
 face, which stands at right angles to the meridian of this dial, describe a vertical one; 
 on the adjacent face to the left an east dial, and on the opposite one a west dial, | 
 of which must be furnished with the proper style. | 
 When you are desirous of finding the meridian on a horizontal plane, place this. 
 quadruple dial on it, so that the vertical one shall nearly face the south; and gra-| 
 dually turn it till three of these dials all shew the same hour: when this takes place, | 
 you may be assured that the three dials are in their proper position. If a line be! 
 then drawn with a pencil, or other instrument, along one of the lateral sides of the 
 cube, it will be in the true direction of the meridian. f 
 It is indeed evident that these three dials cannot shew the same hour, unless they 
 are all placed in a proper position in regard to the meridian ; their concurrence there. || 
 fore will shew that they are properly placed; and that their common meridian is the 
 meridian of the place. 
 
 PROBLEM XI. 
 To construct a Dial on the convex surface of a globe. | 
 This dial, which is the simplest and most natural of all, is formed by dividing the. 
 equatorial circle into 24 parts. If a globe be placed on a pedestal, in such a man- 
 ner that its axis shall be in the plane of the meridian, and exactly elevated according | 
 to the height of the pole of the place, nothing then will be necessary to complete the | 
 dial, but to divide its equator into 24 equal parts. 
 The globe (Fig. 11.) in this state, may be used without any farther apparatus ; for” 
 one half of it being enlightened by the sun, the boundary of the illumination will 
 exactly follow on the equator, the-motion of the sun from east to west. At*noon, 
 
ARMILLARY SPHERE. 517 
 
 it will fall on those points of the equator turned directly 
 to the east and west. At one o’clock, it will have 
 advanced 15°; and so on. To render this globe then 
 fit for being employed as a dial; vr must be inscribed at 
 the division which corresponds with the meridian; vit 
 at the following one, and so of the rest; so that the 
 twelfth will be exactly in the point turned towards 
 the west; then 1, 1, 11, &£c. will be under the hori- 
 zon. Nothing then will be necessary but to observe 
 what division corresponds with the boundary of the 
 light and shadow; for the number belonging to that 
 division will be the hour. 
 
 This dial Eagerst is attended with a very great inconvenience : as the boundary 
 ~ between the light and shadow is always badly defined, it cannot be precisely known 
 | where it terminates ; it will therefore be better to employ this dial in the following 
 manner. 
 
 Adapt to this globe a half meridian, made of a piece of flat wire, 7 or 8 lines in 
 breadth, and half a line in thickness, and moveable at pleasure arsine its axis, which 
 must be the same as that of the globe. Then, when you wish to know the hour, 
 _tnove the half meridian in such a manner, that it shall project the least shadow pos- 
 
 sible, and this shadow will shew the hour on the equator. In this case however it is 
 evident that the numbers naturally belonging to the points of division in the meri- 
 dian, should be inscribed on them; that is, x11 at the meridian, 1 at the following 
 
 division, towards the west, and so on. 
 
 i 
 
 PROBLEM XII. 
 Another kind of Dial, in an Armillary Sphere. 
 Fig. 12. This dial is equally simple as the preceding, and is 
 py attended with this advantage, that it may serve by 
 7 way of ornament in a garden. 
 x Conceive an armillary sphere (Fig. 12.) consisting 
 \ only of its two Ctolures, its equator, and zodiac, and 
 \ 
 
 7 \ 
 / 
 
 furnished with an axis passing through it. If we sup- 
 s  —— | pose this sphere to be placed ona pedestal, in such 
 fee. Det, / a manner that one of its colures shall supply the 
 \ place of a meridian, and that its axis shall be directed 
 towards the pole of the place, it is evident that the 
 shadow of this axis, by its uniform motion, will shew 
 the hours on the equator. Ifthe equator therefore 
 be divided into 24 equal parts, and if the numbers 
 belonging to the hours be inscribed at these divisions, 
 the dial will be constructed. 
 But as the equator, in general, is not of suf- 
 Saale ficient thickness, the hours must be marked on 
 ‘the inside of the zone which represents the zodiac, and which, on that aceount, 
 should be painted white. But in this case, care must be taken not to divide 
 “each quarter of the zodiac into equal parts ; for the shadow of the axis, which passes 
 “over equal arcs on the equator, will pass over unequal ones on the zodiac: these 
 divisions will be narrower towards the points of the greatest declination of that 
 circle; so that the division in the zodiac nearest to the solstitial colures, instead 
 of 15°, which are equal to the interval of an hour on the equator, ought to compre- 
 hend Be 13° 45’; the second 14°15’; the third 15°20’; the fourth 15° 25’; the tifth 
 at 55’; and the rut or nearest the equinoxes, 16°20’, It is in this manner that 
 
518 DIALLING. 
 
 4 
 mo 
 | 
 
 the zodiacal band, on which the hours are marked, must be divided; otherwise there | 
 will be several minutes of error; but each interval may be divided into four equal 
 parts for quarters, without any sensible error. Transversal lines may then be drawn 
 through the breadth of the zodiac, taking care to make them concur in the pole. | 
 We have seen dials of this kind constructed by ignorant artists, who paid no atten- | 
 
 tion to the above remark, and which therefore were very incorrect. 
 
 PROBLEM XIII. 
 
 To construct a Solar Dial, by means of which a blind person may know the hours. 
 
 This may appear a paradox; but we shall shew that a sun-dial might be erected — 
 near an hospital for the blind, by which its inhabitants could tell the hours of the 
 day. 
 
 If a glass globe, 18 inches in diameter, be filled with water, it will have its focus 
 at the distance of 9 inches from its surface; and the heat produced in this focus - 
 will be so considerable, as to be sensible to the hand placed in it. This focus also | 
 will follow the course of the sun, since it will always be diametrically opposite to it; | 
 and therefore to construct the proposed dial, we may proceed as follows, 
 
 Let the globe be surrounded by a portion of a concentric sphere, 9 inches distant | 
 from its surface, and comprehending only the two tropics, with the equator, and the 
 two meridians or colures ; and let the whole be exposed to the sun in a proper posis 
 tion; that is, with the axis of the globe parallel to that of the earth. 
 
 Let each of the tropics and the equator-be divided into 24 equal parts; and let the 
 corresponding parts be connected by a small bar, representing a portion of ihe hour 
 circle comprehended between the two tropics. By these means all the horary circles | 
 will be represented in such a manner, that a blind person can count them, beginning . 
 at that which corresponds to noon, and which may be easily distinguished by some | 
 particular form. 
 
 When a blind person then wishes to know the hour by this dial, he will first put 
 his hand on the meridian, and count the hour circles on the bars which represent them; 
 when he comes to the bar on which the focus of the solar rays fall, he will readily - 
 perceive it by the heat, and consequently will know how many hours have elapsed | | 
 since noon; or how many must elapse before it be noon. ) 
 
 Each interval between the principal bars, that indicate the poe may be easily | 
 divided by smaller ones, in order to have the half-hours and quarters. | 
 
 PROBLEM XIV. | 
 Method of arranging a Horizontal Dial, constructed for any particular latitude, in such | 
 amanner as to make it shew the hours in any place of the earth. 
 
 Every dial, for whatever latitude constructed, may be disposed in such a manner 
 as to shew the hour exactly in any given place; but we shall here confine ourselves | 
 to a horizontal dial, and shew how it may be employed in any place whatever. 
 
 Ist. If the latitude of the place be less or greater than that of the place for which | 
 the dial has been constructed, after exposing it in a proper manner, that is, with its 
 meridian in the meridian of the place, and its axis turned towards the north, nothing — 
 will be necessary but to incline it till its axis forms with the horizon an angle equal | 
 to the latitude of the place in which it is to be used. Thus, for example, if it has | 
 been constructed for the latitude of Paris, | 
 which is 49° 50’, and you wish to employ | 
 it at London, in latitude 51° 31’; as the - 
 difference of these two places is 1°41’, the — 
 plane of the dial must make with the ho-— 
 rizon an angle of 1°41’, as seen in the 
 figure (Fig. 13.), where sN is the meridian, — 
 
USEFUL TABLES. 519 
 
 ABCD the plane of the dial, and ane, or abe, the angle of the inclination of that 
 _plane to the horizon. If the latitude of the primitive place of the dial be less than 
 that of the place for which it is used, it must be inclined in a contrary direction. 
 2d. When the second method of rendering a horizontal dial universal is em ployed, 
 
 the hour-lines must not be described on it, but only the points of division in the 
 
 equinoctial line. In regard to the style, it must be moveable, in the following man- 
 | ner. Let ase (Fig. 14.) represent 
 | Fig. 14. the triangle in the plane of the meri- 
 | dian, where n B cis the axis or oblique 
 style, and a8 the radius of the equa- 
 tor. The style must be moveable, 
 though it always remain in the plane 
 of the meridian, so that the radius 
 A B of the equator, having a joint in 
 the point a, may form the angle Bac 
 equal to a given angle; that is, equal 
 to the complement of the latitude. 
 For this reason a groove must be formed in the meridian, so as to admit this triangle 
 _ to be raised up or lowered, always remaining in the plane of the meridian. 
 
 When every thing has been thus arranged, to adapt the dial to any given latitude, 
 
 such as that of 51°31’, for example, take the complement of 51°31’, which is 38¢ 
 | 29’, and inake the angle B a c = 38°29, The style then will be in the proper posi- 
 tion, and the dial being exposed to the sun, with its meridian corresponding to the 
 “meridian of the place, the shadow of the style, which ought to be pretty long, will 
 shew the hour at the place where it intersects the equinoctial. 
 
 ) 
 
 PROBLEM XV. 
 Method of constructing some Tables necessary in the following problems. 
 
 There are three tables frequently employed in Gnomonics, and which we shall have 
 occasion to make use of hereafter. These are: ‘ 
 . Ist. A table of the angles which the hour-lines form with the meridian on an 
 
 horizontal dial, according to the different latitudes. 
 
 _ 2d. A table of the angles which the azimuth circles, passing through the sun at 
 different hours of the day, form with the meridian, according to the different lati- 
 tudes, and the sun’s place in the ecliptic. 
 _ 3d. A table of the sun’s altitude at different hours, ona given day, and ina place the 
 latitude of which is given. 
 
 From the latter is deduced the sun’s zenith distance, at different hours of the day, 
 in a given place, and on a given day; for the sun’s zenith distance is always the 
 complement of his altitude. 
 
 _ The first of these-tables may be easily calculated by means of the following pro- 
 portion : 
 
 _. As radius is to the sine of the latitude of the given place, so is the tangent of the 
 angle which measures the sun’s distance from the meridian, at a given hour, to the 
 tangent of the angle which the hour-line forms with the meridian. 
 
 By means of this analogy, we have calculated the following table, which we conceive 
 will be sufficient : as it comprehends the whole extent of Great Britain, and particu. 
 larly the latitude of London. 
 
 | 
 
520 DIALLING. 
 
 A TABLE 
 
 OF THE ANGLES WHICH THE HOUR-LINES FORM WITH THE MERIDIAN ON A HORI- 
 ZONTAL DIAL, FOR EVERY HALF DEGREE OF LATITUDE, FROM 50° To 59" 30’. 
 
 A. M. A. M. A. M. A. M. AeiM. | AseN 
 
 Latitude. 
 be De BS TT. XX. | fT. 1X. |TV. VEII.| V. VIEW| Vie VI. 
 5O°. Well? BS) 239.51) 37°275) 63°50 70> 434) a0 ee 
 Bo 30 1.11.41.) 24° 2. 187 404168117 1770051) SnD 
 51 | 11 46 | 24 10 | 37 51] 53 24 | 70 58!.90 0O 
 5131 ive E 19 | 38 4 163 36 | 91-64 90 0 
 52 11 55 | 24 27 | 38 14 | 53 46 | 71 13-| 90 0O 
 BF 3011.12 1011924.86 |-38.95 153 SS 1991-204 oo ao 
 53 12 5 | 24 45 | 38 371 54 8 | 71°27) 90.0 
 53.30 | 12 9 |.24 54 | 38 48 | 54 19 | 71 341 90 O 
 54 12 14 | 25 2{| 3858 | 54 29! 71 40 | 90 Oo 
 54 30 | 12 18 | 25 10 | 39 8 | 54 39 | 71.47 | 90 O 
 55 12 23 | 25 19 | 39 19 | 54 49 | 71 53 | 90 O 
 55 30 | 12 28 | 25 27 | 39 29 | 54 59 |.71 59 | 90. O 
 56 12 32 | 25 3 39740). 55-58 1 70.08 1 eOO gee 
 56 30 | 12 36 | 25 43 | 39 50 | 55 18 |] 72 12 | 90 O 
 57 12 40,).95 51 | .39 59 | 55 27 | 72.17 | 90° 6 
 57 30 | 12 44 | 95°58 | 40 -9°) 56 37 | 72 22°) 90° 0 
 58 12 48 | 26 5 | 4018] 55.45 | 72 27 | 90 0O 
 58 30 | 12 52 | 26 13 | 40 27 | 55 54 | 72 33 | 90 O 
 59 12 56 | 26 20 | 40 36 | 56 21 72 39 | 90 O 
 59°30 13) 0°) 26 27 0 
 
 ! 
 
 We have not marked, iv this table, the angles formed by the lines v hours in the 
 morning and vi hours in the evening, Iv hours in the morning and vir in the evening, 
 because these lines are only a continuation of others; for example, that of rv hours in | 
 the morning, is the continuation of rv in the evening; that of vim hours in the 
 evening is the continuation of viz in the morning; and so of the rest. | 
 
 The use of this table may be easily comprehended. If the place for which a hori- | 
 
 | 
 
 | 
 
 zontal dial is required, corresponds with any latitude of the table, such as 52 for 
 example, it may be seenat one view, that the hour-lines of x1 and 1 must fori, with 
 the meridian, an angle of 11° 55’, at the centre of the dial; that of x and 11 an 
 angle of 24° 27’; and so of the rest. | 
 
 If the latitude be not contained in the table, the proportional parts may be taken | 
 without any sensible error. Thus, if it were required to find the angle which the | 
 hour-line of 1 or x1 forms with the meridian, on a dial for the latitude of 54° 15’; as’ 
 the difference of the horary angles, for 54° and 54° 30, is 4’, take the half of 4, and 
 add it to 12° 14’, which will give 12° 16’ for the horary angle between the hours of 1 
 or xr and the meridian, on a dial for the latitude of 54° 15’. The same operation | 
 may be employed for the other horary angles. | 
 
 It is necessary to observe that this table, though constructed for horizontal dials, 
 may be used also for vertical south or north dials; for it is evident that a south verti- | 
 eal dial, for any particular place, is the same as a horizontal dial for another, the lati- | 
 tude of which is the complement of the former.. Thus a south vertical dial for the | 
 latitude of London 51° 3)’, is the same as a horizontal dial for the latitude of 38° 2%, 
 and vice vérsa. 
 
 It is in the construction of these vertical dials that the utility of such tables will 
 be most apparent; for as these dials are in general very large, the commen rules of | 
 Gnomonics cannot easily be applied to them. To remedy this inconvenience, when | 
 
1 
 
 , 
 
 4 
 
 USEFUL TABLES, 521) 
 
 the centre and equinoctial of the dial have been fixed, assume, as radius, that part 
 of the meridian comprehended between the equinoctial and the centre, and divide it 
 into 1000 parts; then find in some table, or by calculation as above shewn, for the 
 
 - given latitude—that is, for its complement if a vertical dial is to be constructed, the 
 
 tangents of the angles which the hour-lines form with the meridian, at 1, 1, 
 mi, tv, &c., and lay them off on both sides on the equinoctial: the points 
 where they terminate will be the horary points of 1 and x1 hours, m and x 
 hours, &c. 
 
 Let us suppose, for example, that a south vertical dial is to be constructed for the 
 latitude of 51° 31’, the complement of which is 38° 29. A vertical south dial for 
 lat. 51° 31’, may be considered as a horizontal dial for the latitude of 38° 2%. 
 But the angles which the hour-lines form with the meridian on a horizontal dial, 
 for that latitude, are 9° 28’; 199 46’; 319 53’; 47° 9’; 66° 42’; 90° 0’; the tangents 
 
 of which, radius being divided into 1000 parts, are 166, 359, 622, 1078, 2321, 
 
 infinite. If the portion of the meridian therefore, comprehended between the centre 
 and the equinoctial, be divided into 1000 parts, and if 166 of these parts be set off on 
 each side of the meridian, we shall have the points of xr and 1 hours; if 359 parts 
 be then laid off in the same manner, we shall have the points of x and 1 hours; 
 and so of the rest. Straight lines drawn from the centre, to each of these points, 
 will be the hour-lines. 
 The last tangent, which corresponds to vr hours, being infinite, indicates that the 
 bour-line corresponding to it must be parallel to the equinoctial. 
 In order to give an idea of the construction of 
 Fig. 15. the second table, let the circle MBND (Fig. 15.), 
 represent the horizon of the place; z its zenith, p 
 the pole, z B the azimuth circle passing through the 
 sun, and psa the horary circle in which the sun is 
 at any proposed time of the day; it is here evi- 
 dent, that if the hour be given, the angle z Ps is 
 known; that the day of the year being given, the 
 sun’s distance from the equator is known, and 
 consequently the arc Ps, which in our ay Te 
 is the fourth part of a great circle, minus the sun’s 
 declination, if it be north, or plus that declination, if it be south; and lastly, 
 that if the elevation of the pole be given, the arc Pz which is its complement, is 
 also known. In the spherical triangle z ps, we have therefore given the arcs z P 
 and p38, with the included angle z ps; and hence we may find the angle p z s, which 
 subtracted from 180 degrees, will leave the angle MZB or MCB, the sun’s azimuth 
 from the south. 
 In the same triangle, we can find the side ZS, the complement of the sun’s altitude 
 at the same time; and consequently the altitude itself. 
 By these means, the following tables have been constructed, for the latitude of 
 London 51°31’. Those who are tolerably versed in spherical trigonometry, may 
 easily construct similar tables for any other latitude. 
 
x 
 
 %* = 
 onsen 
 
 522 DIALLING. 
 ~~ 4 
 A TABLE OF THE SUN’S AZIMUTH FROM THE SOUTH, AT HIS ENTRANCE INTO | 
 THE 
 
 EACH OF THE TWELVE SIGNS, AND AT EACH HOUR OF THE DAY, FOR 
 LATITUDE OF LONDON, 51° 31’, 
 
 | 
 Hours. oS TE $25) Wee Vee EN il ae Ww 
 
 XI. I | 28° 2’| 26° 9’| 22°18’| 19°13’| 16°19’ | 14°46’| 14° 9‘ 
 X. II | 50 50 | 48 7 | 42 9 | 36 25 | 31 49 | 28 53 | 27 49 
 TXS:) TID) +68) 31 4°65"22) 58)48+) 5157.46 3/242" 791740539 
 VIIT. IV | 82 2179 27 | 72 55 | 65 41 | 59 O |} 54 24 
 Vitz = V 4793-547) °91 25-185. 289078 100 Fi) 38 
 VI. VI 1/105 7 1102 54/97 8] 90 0 
 VoL VELS TGs 6 TS 
 IV. VIII |127 23 
 
 A TABLE OF THE SUN’S ALTITUDE AT HIS ENTRANCE INTO EACH OF THE TWELVE 
 SIGNS, AND AT EACH HOUR OF THE DAY, FOR THE LATITUDE OF LONDON 51° 3)’, 
 
 Hours. oS IL + 2-1). wel oe lee) se any We 
 XII. 61°57’ | 58°41") 49° 617) 38° 297] (967437) "18" 19744 15 
 XI. I | 59 40 | 56 34/148 2! 36 571] 25 30 | 17 32 | 13 54 
 X. II | 53 44 | 50 56 | 43 4 | 32 37] 21 42 | 13 40 | 10 32 
 TX TEE 945041 1743509 485-59 26 77 15 6D 8 15 ee We 
 VIII. IV }°36, 40 | 34 14.4 97°91 | 1878 8 18 1 16 
 WITS OV 91927 -9951724 1567108819" O17 147 
 
 PNT VL 118 10 | 1541 8 53 
 Vue VET 9 26 6 50 
 IV. VIII 1 31 
 
 PROBLEM XVI. 
 
 The sun’s altitude, the day of the month, and the elevation of the pole, being given; to 
 find the hour by a geometrical construction. 
 We give this construction merely as a geometrical curiosity; for it is certain that 
 the same thing can be performed with much greater accuracy by calculation. How- 
 ever, as the solution of this problem forms a very ingenious example of the graphic 
 solution of one of the most complex cases of spherical trigonometry, we have no 
 doubt that it will afford gratification to our readers; or at least to such of them as 
 are sufficiently versed in geometry to comprehend it. 
 Let us return then to Fig. 15, in which pz represents the complement of the lati- 
 tude or elevation of the pole; zs the complement of the sun’s altitude, which is 
 
 known, being given by the supposition; and ps the sun’s distance from the pole, 
 
 | 
 
 which is also given, since the declination of the sun, or his distance from the eguator | 
 
 each day, is known. In the triangle z Ps therefore, there are given the three sides, 
 to find the angle zps, the hour angle, or angle which the horary circle, passing 
 through the sun, forms with the meridian. This case then is one of those in sphe- 
 rical trigonometry, where the three sides of an oblique triangle being given, it is re- 
 quired to find the angles ; and which may be solved geometrically in the following 
 manner, 
 
 In the circumference of a circle, which must be sufficiently large to give quarters 
 
 of degrees, (Fig. 15 and 16), assume an arc equal to pz, and draw the two radii cP — 
 
 and cz. On the one side of this arc make p s equal to the arc P s, and on the other 
 
GNOMONICAL PARADOX, 523 
 
 » Fig. 16. z R equal to the arc zs: from the points R and s let 
 noey fall, on the radii pc, cz, two perpendiculars s 7 and 
 ve a vals R v, which will intersect each other in some point x: 
 / ae he then, if svT be radius, we shall have Tx for the cosine 
 WA es 5 of the required angle, which may be constructed in 
 q) a /  _ the following manner : 
 
 / F th From the centre T, with the radius rs orTs, 
 | f which is equal to it, describe a quadrant, compre- 
 hended between TP and Tx continued; if x y be 
 then drawn parallel to T p, the are ys will be the one 
 required, or the measure of the hour angle SP 2; 
 
 Ss arwae therefore y TX will be equal to that angle. 
 SS ee By a similar construction we might find the angle 
 
 z, the complement of which is the sun’s azimuth; 
 
 but this is sufficient in regard to an operation which is rather curious than useful. 
 
 This construction is much simpler and far more elegant, than that given by Oza- 
 nam, for the solution of the same problem. 
 
 GNOMONICAL PARADOX. 
 
 Every sun-dial, however accurately constructed, is false, and even sensibly so, in regard 
 to the hours near sun-set. 
 
 The truth of what is here asserted, will be readily perceived by astronomers, who 
 are acquainted with the effects of refraction. ‘The following observations will make 
 it sensible to our readers. ‘ 
 
 It is a fact now well known to all philosophers, that the heavenly bodies always 
 appear more elevated than they really are, except when they are in the zenith. This 
 phenomenon is produced by the refraction, which the rays of light, proceeding from 
 
 them, experience in the atmosphere; and the effect of it is very considerable in the 
 
 neighbourhood of the horizon ; for when the centre of the sun is really on the horizon, 
 he still appears to be elevated more than half a degree, or 33 minutes, which in our 
 latitudes is the quantity of the horizontal refraction. The centre of the sun then is 
 really on the horizon, and astronomically set, when his lower limb does not touch the 
 horizon, but is still distant from it an apparent semi-diameter of the sun. 
 
 Let us suppose then, that on the day of the equinox, for example, the hour indi- 
 cated by a vertical west-dial, near the time of sun-setting, has been observed at the 
 
 -moment when a well regulated clock strikes six: the shadow of the style ought to 
 
 be on the hour of six, and it would indeed be soif the sun were on the horizon; but 
 being elevated 33 minutes above the horizon, the shadow of the style will be within 
 six hours, for it is by the apparent image of the sun that this shadow is formed: it 
 will even not reach that line till the sun has still descended 33’, for which he will em- 
 ploy, in the latitude of London, about 3m, 28s. of time. But, in a sun-dial, an error 
 of 3m. 28s. is more than sensible. 
 
 If the sun be at the summer solstice; as he employs in the latitude of London 
 more than 4’ to descend vertically 33 minutes on the horizon, on account of the 
 obliquity with which the tropic cuts that circle, the difference will be more sensible 
 as the space passed over by the shadow between the hours of seven and eight, is suffi- 
 ciently great to suffer an error of a twelfth or a fifteenth to be very perceptible. We 
 have seen, on a dial of this kind, the point of the shadow, which ought to have fallen 
 on the line of seven o’clock, more than an inch distant from it; though at all the 
 other hours of the day the dial was very exact, and corresponded with an excellent 
 watch which was compared with it. 
 
524 DIALLING. 
 
 PROBLEM XVII. 
 To describe a Portable Dial on a Quadrant. 
 
 As the construction of this dial depends also on the sun’s titude at each hour of 
 the day, in a determinate latitude, according to his place in te zodiac, the tables 
 before mentioned must be employed here also. 
 
 : Let a Bc then, (Fig. 17.), be 
 
 Fig. 17. 
 
 scribe, at pleasure, seven qua- 
 drants equally distant from each 
 other, to represent the com- 
 mencement of the signs of the 
 
 assumed as the tropics, and that 
 in the middle as the equater. 
 Mark on each of these parallels 
 of the signs, the points of the 
 hours, according to the altitude 
 which the sun ought to have at 
 these hours, which may be found 
 in the table above mentioned, 
 To determine, for example, the 
 point of m in the afternoon, or x in the morning, for the latitude of London, when 
 the sun enters Leo; as the table shews that the sun’s altitude is at that time 50° 56,, 
 
 make in the proposed quadrant the angle 8 a o equal to 50° 56’, and the place where 
 
 the parallel of the commencement of Leo is intersected by the line a o, will be the 
 required point of 11 in the afternoon and x in the morning. 
 
 Having made a similar construction for all the other hours, on the day of the sun’s 
 entrance into each sign, nothing will be necessary but to join, by curved lines, all the 
 points belonging to the same hour, and the dial will be completed. Then fix a small 
 perpendicular style in the centre a, or place on the radius ac, or any other line pa- 
 rallel to it, two sights, the holes of which exactly correspond; and from the centre 4 
 suspend a small plummet by means of a silk thread. “a 
 
 When you use this instrument, place the plane of it in such a manner as to be in 
 the shade; and give such a direction to the radius that the shadow of the small style 
 shall fall on the line ac, or that the sun’s rays shall pass through the two holes of 
 ' the sights: the thread from which the plummet is suspended will then shew the hour, 
 
 by the point where it intersects the sun’s parallel. 
 
 To find the hour with more convenience, a small bead is put on the thread, but in 
 such a manner as not to move too freely. If this bead be shifted to the degree and 
 sign of the sun’s place, marked on the line A‘c, and if the instrument be then directed 
 towards the sun, as above mentioned, the bead will indicate the hour on the hour- 
 line which it touches. . 
 
 Remark.—To render this dial more commodious, it will be better, instead of the signs, 
 to mark the days of the month on which the sun enters them. For example, instead of 
 marking the small circle with the sign ve, mark December 21; close tothe second place on 
 one side January 21, instead of w, the sign of Aquarius; and on the other November 
 21, instead of 7, the sign of Sagittarius, &c.; for if we suppose the equinoxes inva- 
 riably fixed at the 21st of March and the 2st of September, the days on which 
 the sun enters the different signs of the zodiac will be nearly the 21st of each month: 
 to use the dial, nothing will then be necessary but to know the day of the month. 
 
 a quadrant, the centre of which — 
 is A. From the centre a de- 
 
 : 
 . 
 
 zodiac; the first and last being | 
 
“PORTABLE DIAL. . - O25 
 
 PROBLEM XVIII. 
 To describe a Portable Dial on a card. 
 
 This dial is generally called the Capuchin, because it resembles the head of a Capu- 
 chin friar with the cowl inverted. It may be described on a small piece of pasteboard, 
 or even a card, in the following manner, 
 
 Having described a circle, Fig. 18, 
 at pleasure, the centre of which is a, 
 and the diameter s 12, divide the cir- 
 cumference into 24 equal parts, or at. - 
 every 15 degrees, beginning at the di- 
 ameter B 12. If each two points of 
 division, equally distant from the di- 
 ameter B 12, be then joined by parallel 
 lines, these parallels will be the hour- 
 lines; and that passing through the 
 “3p centre a, will be the line of six 
 oS o’clock. 
 
 Then at the point 12, make the 
 angle B 12 Y equal to the elevation of 
 the pole, and having drawn through 
 } the point Y, where the line 12 Y inter- 
 ony sects the line of 6 o’clock, the inde- 
 ee” finite line 95 vw, perpendicular to the 
 line 12 y, draw from the extremities 
 of the line 95 vr, the lines 12 95, and 
 12 we, which will each make with the 
 line 12 ‘, an angle of 231 degrees, 
 which is the sun’s greatest declination. 
 
 The points of the other signs may be found on this perpendicular 95 vp, by de- 
 scribing from the point 7, as a centre, through the points %, vr, the circumference of 
 a circle, and dividing it into 12 equal parts, or at every 30 degrees, to mark the com- 
 mencement of the 12 signs. Join every two opposite points of division, equally dis- 
 tant from the points 5, w, by lines parallel to each other, and perpendicular to the 
 diameter 95, v7: these lines will determine, on this diameter, the commencement of 
 
 Se al 
 
 _ the signs ; from which, as centres, if circular ares be described through the point 12, 
 
 j 
 p 
 
 | 
 1s 
 
 | 
 
 they will represent the parallels of the signs; and therefore must be marked with the 
 appropriate characters as seen in the figure. 
 
 A slit must be made along the line 95 w, to admit a thread furnished with a small - 
 weight, sufficient to stretch it; and in which it must glide, but not too freely; so 
 that its point of suspension can be shifted toany point of the line 25 vw at pleasure. 
 
 T hese ares of the signs will serve to indicate the hours when the sun shines, in the 
 following manner: Having drawn at pleasure the line c vw, parallel to the diameter B 
 12, fix at its extremity ¢ a small style in a perpendicular direction, and turn the plane 
 of the dial to the sun, so that the shadow of the style shall cover the line c we: the 
 thread and plummet being then freely suspended from the sun’s place, marked on the 
 line 95 vr, will indicate the hour on the arc of the same sign at the bottom. 
 
 The thread may be furnished with a small bead to be used as in the preceding 
 problem. 
 
 Remark.—This dial originated from an universal rectilineal dial constructed by 
 Father de Saint-Rigaud, a jesuit, and professor of mathematics in the college of 
 Lyons, under the name of Analemma Novum. But though Ozanam has given a con- 
 
526 DIALLING. 
 
 spicuous place to it in his Recreations, as well as to another «universal rectilineal 
 analemma, it appeared to us that his description of them was too complex to be ad- 
 mitted into a work of this kind. 
 
 PROBLEM XIX. 
 Method of constructing a Ring-dial. 
 
 Portable ring-dials are sold by the common instrument-makers; but they are very 
 defective. ‘The hours are marked in the inside on one line, and a small moveable 
 band, with a hole in it, is shifted till the hole correspond with the degree and sign of 
 the sun’s place marked on the outside. Such dials however, as already said, are 
 defective ; for as the hole is made common to all the signs of the zodiac, marked on 
 the circumference of the ring, it indicates justly none of the hours but noon: all the 
 rest will be false. Instead of this arrangement, therefore, it will be necessary to 
 describe, on the concave surface of the ring, seven distinct circles, to represent as 
 many parallels of the sun’s entrance into the signs; and on each of these must be 
 marked the sun’s altitude on his entrance into the sign belonging to the parallel 
 to which the circle corresponds. When these points are marked, they must be 
 joined by curved lines, which will be the real hour-lines, as has been remarked by 
 Deschales. 
 Fig. 19. Having provided a ring, Fig. 19, or 
 
 rather described a circle of the size of the 
 
 ring which is to be divided; and having 
 fixed on B as the point of suspension, make 
 
 B A and BO, on each side of zB, equal to 
 
 51° 31’, for the latitude of the place, sup- 
 
 pose London, that is, equal to the distance 
 of the zenith from the equator: then 
 through the points a and o draw the chord 
 
 A Oo, and a D perpendicular to it: if the 
 
 line a 12 be then draw through a and the 
 
 centre of the circle, the point 12 will 
 be the hour of noon on the day of: the 
 equinox. 
 To find the other hour-points for the 
 same day, at the commencement of Aries 
 Libra; from the centre a describe the 
 quadrant o D; and from the point 0, set off 
 toward p the sun’s altitude at the different 
 
 hours of the day, as at 1 and 11, 2 and 10, 
 
 &c.; the lines drawn from the centre A 
 
 through these points of division, if con- 
 
 tinued to the circumference of the circle B 
 
 12 a, will give the hour-points for the day 
 
 of the equinox. P 
 
 To obtain the hour-divisions on the — 
 circles corresponding to the other signs, 
 
 first set off, on both sides of the point A 
 
 Fig. 20, the sun’s declination when he 
 
 enters each of the signs, viz. the ares A Band 
 
 A1of 23 degrees, for the commencement 
 
 of Taurus or Virgo; of Scorpio or Pisces; 
 
 A F of 40° 26’ for the commencement of 
 Gemini and Leo; a xk equal to it for the 
 
—s 
 As 
 
 RING=-DIAL. 527 
 
 commencement of Sagittaris and Aquarius; and a c and a Lof 47° for the commence. 
 ment of Cancer and Capricorn. 
 
 Now to find the hour-points on the circle, (that corresponding to the commencement 
 of Aquarius, for example, ) through the point K, which corresponds to the sun’s entrance 
 into that sign, draw & P parallel to a o, and also the line x 12: from the same point 
 K describe, between x 12 and the horizontal line K P, the arce@ R; on which set off, 
 from R towards Q, the sun’s altitude at the differeut hours of the day, when he 
 enters Sagittarius and Aquarius, as seen in the figure; and if lines be then drawn from 
 k to these points of division, you will have the hour-points of the two circles corre- 
 sponding to the commencement of Sagittarius and Aquarius. By proceeding in the 
 same manner for the sun’s entrance into the other signs, you will have the hour- 
 points of the circles which correspond to them, : 
 
 Then trace out, on | 
 the concave surface of 
 the circle, seven pa- 
 rallel circles (Fig. 
 21.), that in the mid- 
 dle for the equinoxes ; 
 the two next on each 
 side for the~ com- 
 mencement of the 
 signs Taurus and Vir- 
 go,Scorpioand Pisces; 
 the following two on 
 the right and left for 
 Gemini and Leo, Sa- 
 gittarius and Aqua- 
 rius ; and the last two 
 for Cancer and Capri- 
 corn: if the similar | 
 hour-points be then 
 joined bya curved line, 
 the ring-dial will be 
 completed. 
 
 The next thing to be done, is to adjust properly the hole which admits the solar 
 rays; for it ought to be moveable, so that on the day of the equinox it may be at the 
 “point a; on the day of the summer solstice at Gc; on the other days of the year in 
 the intermediate positions. For this purpose the exterior part of the ring cB D must 
 have in the middle of it a groove, to receive a small moveable ring or hoop, with a 
 hole in it. The divisions L, K, 1, A, E, F, G, must be marked on the outside of this 
 part of the ring by parallel lines, inscribing on one side the ascending signs, and on 
 the other the descending: when this construction has been made, it will be easy to 
 place the hole of the moveable part a on the proper division, or at some intermediate 
 point; for if the ring be pretty large, each sign may be divided into two or three 
 parts. 
 
 To know the hour, move the hole a to the proper division, according to the 
 sign and degree of the sun’s place; then turn the instrument in such a manner 
 that the sun’s rays, passing through the hole, may fall on the circle corresponding to. 
 the signin which the sun is: the division on which it falls willshew tke hour. 
 
 Remark.—I. To render the use of this instrument easier, instead of the divisions 
 of the signs, the days corresponding to the commencement of the signs might be 
 
528 ; DIALLING. 
 
 marked out on it: for example, June 21 instead of 95; April 20, August 20, instead 
 6 and m, and so on. 
 
 ll. The hole a might be fixed, and the most proper position for it would be that 
 which we origially assigned to the day of the equinox; but in this case, the hour of 
 noon, instead of being found on a horizontal line, for all the circles of the signs, 
 according to the preceding method, would be a curved line; and all the other hour- 
 iines would be curved lines also. As this would be attended with a considerable 
 degree of embarrassment and difficulty, it will be better, in our opinion, that the hole 
 A should be moveable. f 
 
 PROBLEM XX. 
 
 How the shadow of a style, on a Sun-dial, might go backwards, without a miracle. 
 
 This phenomenon, which on the first view may appear physically impossible, is 
 however very natural, as we shall here shew. It was first remarked by Nonius or 
 Nugnez, a Portuguese mathematician, who lived about the end of the sixteenth cen- 
 tury. It is founded on the following theorem. 
 
 In all countries, the zenith of which is situated between the equator and the tropic, 
 as long as the sun passes beyond the zenith, towards the apparent or elevated pole, 
 he arrives twice before noon at the same azimuth, and the same thing takes place in 
 the afternoon. 
 
 Fig. 22. Let z (Fig. 22.) 
 be the zenith of any — 
 TZ 
 
 place situated be- 
 tween E the equa- 
 tor, and T the point 
 through which the 
 sun passes on the 
 day of the summer 
 solstice ; let the 
 circle HAQBKH 
 represent the hori- 
 zon; RE Q one 
 half of the equator; © 
 T F the eastern part 
 of the tropic above. 
 the horizon, and G 
 7 the western part, 
 It is here evident, 
 that from the zenith 
 z there may be 
 
 drawn an azimuth circle, such as z1, which shall touch the tropic in a point 0, for 
 example: and which shall fall on the horizon in a point 1, situated between the 
 points @ and F, which are those where the horizon is intersected by the equator and 
 the tropic; and, for the same reason, there may be drawn another azimuth, as Z H, 
 which shall touch in o the other part of the tropic. eg 4 
 Let us now suppose that the sun is in the tropic, and consequently rising in the . 
 point F; and let a vertical style, of an indefinite length, be erected in c. Draw also 
 the lines 1c K, and FCN; itis evident that at the moment of sun-rise the shadow 
 of the style will be projected inc nN; and that when the sun has arrived at the point 
 of contact o, the shadow. will be projected inc x. While the sun is passing over 
 F 0, it will move from c nN toc K, but when the sun has reached the meridian, the 
 shadow will be in the line c B; it will therefore have gone back from c kK to C Bt 
 
d 
 
 from sunrising to noon then it will have gone fromc Nn toc x, and from cx toc pb: 
 consequently it will have moved in a contrary or retrograde direction, since it 
 first moved from the south towards the west, and then trom the west towarde 
 the south. 
 
 Let us next suppose that the sun rises between the points F and 1. In this case 
 the parallel he describes before noon will evidently cut the azimuth z1 in two points ; 
 and therefore, in the course of a day, the shadow will first fall within the angle KCL; 
 it will then proceed towards c x, and even pass beyond it, going out of the angle; 
 but it will again enter it, and, advancing towards the meridian, will proceed thence 
 towards the east, even beyond the line cx, from which it will return to disappear 
 with the setting of the sun within the angle Los. 
 
 It is found by calculation, that in the latitude of 12 degrees, when the sun is in 
 the tropic on the same ade, the two lines c nN and c kK form an angle of 9° 48’; to 
 pass over which the shadow requires 2 hours 7 minutes, 
 
 SHADOW GOING BACKWARDS. 529 
 
 PROBLEM XxXI. 
 
 To construct a Dial, for any latitude, on which the shadow shall retrograde, or move 
 backwards. 
 
 For this purpose incline a plane, turned directly south, in such a manner, that its 
 zenith shall fall between the tropic and the equator, and nearly about the middle 
 of the distance between these two circles: in the latitude of London, for example, 
 which is 51°31’, the plane must make an angle of about 38°. In the middle of the 
 plane, fix an upright style of such a length, that its shadow shall go beyond the 
 plane; and if several angular lines be then drawn from the bottom of the style to- 
 wards the south, about the time of the solstice, the shadow will retrograde twice in 
 the course of the day, as above-mentioned. 
 
 This is evident, since the plane is parallel to the horizontal plane, having its zenith 
 under the same meridian, at the distance of 12 degrees from the equator towards the 
 north: the shadows of the two styles must consequently move in the same manner 
 in both. 
 
 Remark.—Some may here say, that this isa natural explanation of the miracle, 
 which, as we are told in the Sacred Scriptures, was performed in favour of Hezekiah, 
 king of Jerusalem; but God forbid that we should entertain any idea of lessening 
 
 _the credibility of this miracle. Besides, it is very improbable, if the retrogradation 
 which took place on the dial of that prince had been a natural effect, that it should 
 not have been observed till the prophet announced it to him, as a sign of his cure ; 
 for in that case it must have always occurred when the sun was between the tropic 
 and the zenith: the miracle therefore, recorded in the Scriptures, remains unim- 
 peached, 
 
 PROBLEM XXII. 
 | To determine the Line traced out, on the plane of a Dial, by the summit of the style 
 We here suppose that the sun, in the course of a diurnal revolution, does not 
 sensibly change his declination ; for if he did, the curve in question would be of two 
 complex a nature, and very difficult to determine. 
 
 Let the sun then be in any parallel whatever. It may be easily seen that the central 
 solar ray, drawn to the point of the style, describes a conical surface, unless the 
 sun be in the equator ; consequently the shadow projected by that point, which is 
 always directly opposite to it, passes over, in its revolution, the surface of the oppo- 
 site cone, which is united to it by its summit. Nothing then is necessary but to 
 know the position of the plane which cuts the two cones; for its intersection with 
 the conical surface, described by the shadow, will be the curve required. 
 
 Those therefore who have the least knowledge of conic sections will be able to 
 2M 
 
530 DIALLING. 
 
 solve the problem. For, Ist, If the proposed place be under the equator, and the 
 
 plane horizontal, it is evident that this 
 
 Fig. 23. plane intersects the two opposite cones 
 
 / at the summit: consequently the track 
 
 of the shadow will be an hyperbola zc D” 
 
 (Fig. 23.), having its summit turned to- 
 wards the bottom of the style. 
 
 But it may be easily seen, that as the 
 sun approaches the equator, this hyper- 
 bolic line becomes flatter and flatter; 
 and at length, on the day of the equinox, 
 is changed into a straight line; that it 
 afterwards passes to the other side, and always becomes more and more curved, 
 till the sun reaches the tropic, &c. : , 
 
 We shall here add, that the sun rises every day in one of the asymptotes of an 
 hyperbola, and sets in the other. 
 
 2nd. In all places situated between the equator and the polar circles, the track 
 of the shadow, on a horizontal plane, is still an hyperbola; for it may be easily seen 
 that this plane cuts the two opposite cones, united at their summits, which are de- 
 scribed by the solar ray that passes over the point of the style; since in all these 
 latitudes the two tropics are intersected by the horizon. 
 
 3d. In all places situated under the polar circle, the line described by the shadow 
 or a horizontal plane, when the sun is in the tropic, is a parabolic line: but that de-. 
 scribed on other days is hyperbolic. | 
 
 4th. In places situated between the polar circle and the pole, as long as the sun’ 
 rises and sets, the tract described by the shadow of the summit of the style is an. 
 hyperbola: when the sun has attained to such a high latitude that he only touches 
 the horizon, instead of setting, the track is a parabola; and when the sun remains 
 the whole day above the horizon, it is an ellipsis, more or less elongated. 
 
 Sth. Lastly, it may be easily seen that under the pole the track of the shadow. 
 of the summit of the style is always a circle ; since the sun, during the whole dey, | 
 - remains at the same altitude. 
 
 | 
 
 Corollary.—As the ares of the signs are nothing else than the track of the shadow) 
 of the summit of the style, when the sun in his diurnal motion passes over the 
 parallel belonging to the commencement of each sign, it follows that these arcs are all, 
 conic sections, having their axis in the meridian or substylar line. In horizontal, 
 dials, constructed for places between the equator and the polar circles, and in all 
 vertical dials, whether south, north, east, or west, constructed for places in the tem- 
 perate zone, they are hyperbolas. This may be easily perceived, on the first view. 
 in most of the dials in our latitude. : 
 
 These observations, which perhaps may be considered by common gnomonists as 
 of little importance, appeared to us worthy the consideration of those more verse¢ 
 in geometry; especially as seme of them may not have attended to them. For this 
 reason we resolved to give them a place in this work. 
 
 PROBLEM XXIII. 
 To describe the Arcs of the Signs on a Sun-dial. 
 Of the appendages added to sun-dials, the ares of the signs may be classed among, 
 the most agreeable ; for by their means we can know the sun’s place in the differen| 
 signs, and as we may say can follow his progress through the zodiac. We therefor: 
 ATTY it our duty not to omit, in this work, the method of describing them. 
 - For the sake of brevity, we shall suppose that the plane is horizontal. First de. 
 
ARCS OF THE SIGNS,—DIFFERENT KINDS OF HOURS. 531 
 
 scribe a dial such as the position of the plane requires, (that is, a horizontal one,) and: 
 fixin itan upright style, terminated by aspherical button, or by a circular plate, having 
 in its centre a hole, of a line or two in diameter, according to the size of the dial. 
 Then proceed as follows :— 
 
 Let it be required, for example, to trace out the are corresponding to the com- 
 mencement of Scorpio or Pisces. First find, by the table of the sun’s altitude, at each 
 hour of the day in the latitude of London, for which we suppose the dial to be con- 
 structed, the altitude when he enters these two signs. As this altitude is 26° 43’, 
 make the triangle s T £, Fig. 24, in 
 which s T is the height of the style, 
 and such that the angle se T shall be 
 equal to 26°43’: the point = will be 
 the first point of the are of these two 
 signs. 
 
 Then find, in the same table, the 
 sun’s altitude at one in the afternoon 
 of the same day, which will be found 
 equal to 25° 30’; and construct the tri- 
 es angle sT F, in such a manner that the 
 2. angle F shall be 25°30’; then from the 
 
 bottom of the style s, asa centre, with 
 
 the radius s F, describe an are of a 
 circle, intersecting the lines of 1 and x1 hours in the two points @ and 4; these will 
 be the points of the are of those signs on the lines of 1 and xt. 
 
 If the same operation be repeated for all the other hours, you will have as many ~ 
 points, through which if a curved line be drawn, by means of a very flexible ruler, 
 - you will obtain the arc of the signs Scorpio and Pisces. 
 
 By employing the like construction, the ares belonging to the other signs may be 
 obtained. 
 
 Of the different kinds of Hours. 
 
 Every thing hitherto said has related only to the equinoctial and equal hours; 
 such as those by which time is reckoned in England, the day being supposed to begin 
 at midnight, and the hours being counted to the following midnight, to the number 
 of 24, or twice twelve. This is the most common method of computing the hours in 
 Europe. The astronomical hours are almost the same; the only difference is, that 
 the latter are counted, to the number of 24, from the noon of one day to the noon of 
 the day following. 
 
 But there are some other kinds of hours, which it is proper we should here explain ; 
 because they are sometimes traced out on sun-dials: such are the natural or Jewish 
 hours, the Babylonian, the modern Italian, and those of Nuremburg. 
 
 ' The natural or Jewish hours begin at sun-rise; and there are reckoned to be 12 
 
 between that period and sun-set: hence it is evident that they are not of equal 
 length, except on the day of the equinox: at every other time of the year they are 
 unequal. Those of the day, in our hemisphere, are longer from the vernal to the 
 autumnal equinox: those of the night are, on the other hand, longer while the sun is 
 passing through the other half of the zodiac. 
 
 _ The Babylonian hours were of equal length, and began at sun- rise; they were 
 counted, to the number of 24, to sun-rise of the day following. 
 
 The modern Italian hours, for the ancient Romans counted nearly as we do from 
 midnight to midnight, are reckoned tc the number of 24, from sun-set to sun-set of 
 » the day following; so that on the days of the equinox noon takes place at the 18th 
 ‘hour, and then, as the days lengthen, the astronomical noon happens at 174 hours, 
 
 2M2 
 
532 DIALLING. 
 
 then at 17 hours, &c.; and vice versd. This singular and inconvenient method has 
 had its defenders, and that even among the French; who have found that witha 
 pencil, and a little astronomical calculation, one may fix the hour of dinner with ver 
 little embarrassment. 
 
 However, as these hours are still used throughout almost the whole of Italy, we 
 think it our duty to shew here the method of describing them, by way of a Gnomo- 
 nical curiosity. 
 
 PROBLEM XXIV. 
 To trace out, on a Dial, the Italian Hours. 
 
 Describe first on the proposed plane, which we here suppose to be a horizontal one, 
 a common horizontal dial, with the astronomical or European hours: delineate on it 
 also the arcs of the solsticial signs, Cancer and Capricorn; as well as the equinoctial 
 line, which is the arc of the equinoctial signs. 
 
 Then observe that, on the days of the equinox, noon, for a dial constructed at 
 London, takes place at the end of the 18th Italian hour; and on the day of the summer 
 solstice at 17 minutes after the 16th hour. Noon, therefore, or 12 hours, counted 
 according to the astronomical hours, corresponds, on the day of the equinox, to the 
 18th Italian hour; and on the day of the solstice to 17 minutes after the 16th ; con- 
 sequently the 18th Italian hour, on the day of the summer solstice, will correspond to 
 17 minutes past 2, counted astronomically. Join therefore, (Fig. 25), by a straight 
 
 Fig. 25. 
 
 line, the point of noon marked on the equinoctial line, and that of 2 hours 17 minutes 
 on the tropic or arcof the sign Cancer, and inscribe there 18 hours. Join also by 
 transversal lines 1 hour on the equinoctial and 3h. 17m. on the arc of Cancer; then 2h. 
 and 4h.17m., &c. ; and before noon I] 1h. and lh. 17m. ; 10h. and 12h. 17m.; 9h. and 1h. 
 17m. &c. ; efface then the astronomical hours, which we suppose ought not to appear 
 and continue the above transversal lines till they meet the parallel of Capricorn, 
 inscribing at their extremities the proper numbers; by which means you will have 
 your dial traced out as seen, 
 
 Remark.—It may be easily seen, by the ahove example, what calculation will be 
 necessary for a latitude different from that of London, where the length of the day,, 
 at the summer solstice, is 16 hours 34 minutes, and at the winter solstice only 7 hours 
 44 minutes. In another latitude, where the longest day is only 14 hours and the 
 shortest 10, noon at the summer solstice will take place at the end of the 17th Italiar 
 hour. Noon therefore, or 12 hours, counted astronomically, will on the day of the 
 solstice correspond to the 17th Italian hour ; and consequently the 18th Italian hour 
 
THE NATURAL HOURS, £33 
 
 ‘at the same period, will correspond to 1 in the afternoon counted astronomically.. 
 ‘Tohave the hour-line of the 17th Italian hour, therefore, nothing will be necessary, 
 
 but to join the point of 1 in the afternoon, on the are of Cancer, and the point of 
 noon on the equinoctial- And the case will be the same with the other hours. 
 
 PROBLEM XXV. 
 To trace out on a Dial the lines of the natural or Jewish hours. 
 We have already said, that the equal hours which can be counted from sun-rise to 
 
 ‘sun-set, to the number of twelve, are called the natural hours ; for it is this interval 
 of time which really forms the day. 
 
 This kind of hours may be easily traced out ona dial, which we shall here suppose 
 to be horizontal. For this purpose, it will be first necessary to draw the equinoctial, 
 
 ‘and the two tropics, by the preceding methods. 
 
 Now it must be observed, that as, in the latitude of London, the sun, on the day 
 
 ‘of the summer solstice, rises at 3h. 43m., and sets at Sh. 17m., the interval between 
 
 these periods is equal to 17h. 34m.; consequently, if we divide this duration into 12 
 parts, each of these will be about 1} hour: for this reason, draw lines from the centre 
 of the dial to the points of division on the equinoctial, corresponding to 54 hours, to 
 7 hours, to 84 hours, to 10 hours, to 114 hours, to ] hour, and so on; but marking 
 only, on the tropic of Cancer, the points of intersection which these hours form 
 with it. 
 
 In like manner, as the sun at the winter solstice, in the latitude of London, rises 
 at 8h. 8m., and sets at 3h. 52m., the duration of the day is only 7 hours 44 minutes; 
 
 which being divided into 12 parts, gives for each about 40 minutes, or 2 of an astro- 
 
 nomical hour. Draw therefore the hour-lines corresponding to 82 hours, to 94 hours, 
 to 10 hours, and so on; marking only the points where they intersect the tropic 
 of Capricorn; then, if the corresponding points of division, on the two tropics and 
 he eauinoctial, be joined by a curved line, the dial will be described, as seen Fig. 26, 
 
 Fig. 26. 
 
 If more exactness be required, it will be necessary to trace out two more parallels 
 f the signs, viz. those of Taurus and Scorpio, and to find on each, bya similar process, 
 he points corresponding to the natural hours: the natural hour-lines may then be 
 aade to pass through five points, by which means they will be obtained with much 
 aore exactness. 
 
534 _ DIALLING. 
 
 APPENDIX. 
 
 We shall conclude this pubyer? by giving a general method of describing sun- cals, 
 whatever be the declination or inclination of the plane. 
 
 This method is founded on the consideration that any plane whatever is always a a 
 horizontal plane to some plane on the earth; for a plane being given, it is evident 
 that there is some point of the earth the Peet or horizontal plane of which is‘ 
 parallel to it. It is evident, also, that two such parallel planes will shew the same 
 hours at the same time. Thus, for example, if we suppose at London a plane in- 
 clining and declining in such a manner, as to be parallel to the horizontal plane) 
 of Ispahan; then a dial traced out on that plane, as if it were horizontal, will give 
 the hours of Ispahan ; so that when the shadow falls on the substyle, we may say that: 
 it is noon at Ispahan, &c. 
 
 But as the hours of Ispahan-are not those wanted at London, it is necessary that. 
 we should find out the means of delineating those of London, which will not be 
 attended with much difficulty, when the difference of longitude between these two 
 cities is known. Let us suppose then that it is exactly 45 degrees, or 3 hours: when 
 it is noon at London then, it will be 3 in the afternoon at Ispahan ; and when it is ii) 
 in the forenoon at the former, it will be 2 in the afternoon at the latter, &c. Con- 
 sequently, on this dial, which we suppose to be horizontal, if we assume the line 
 
 . of 3 o’clock as that of noon, and mark it 12; and if we assume the other hour-lines 
 in the same proportion, we shall have at London the horizontal dial of Ispahan, which 
 will indicate, not the hours of Ispahan, but those of London, as required. 
 
 We flatter ourselves that we have here explained the principle of this method in 
 a manner sufficiently clear, to make it plain to such of our readers as have a slight 
 knowledge of geometry or astronomy ; but to render the application of it more fa- 
 miliar, we shall illustrate it by an example. | 
 
 Let us suppose then, at London, a plane forming with the horizon an angle of 12 
 degrees, and declining towards the west 221 degrees. 
 
 The first operation here is, to find the tonende and latitude of that place of the 
 earth where the horizontal plane is parallel to the given plane. 
 
 For this purpose, let us conceive an azimuth ar per- 
 pendicular to the given plane (Fig. 27.), and in this 
 azimuth, which we suppose to be traced out on the sur- 
 face of the earth, let us assume, on that side which is 
 towards the upper part of the plane, an arc a H, equal to 
 the inclination of that plane to the horizon: the extre- 
 mity of this are, that is the point u, will be that point 
 of the earth where the horizon is parallel to the given 
 plane. This is so easy to be comprehended that it re- 
 quires no demonstration. Let us next conceive a meri- 
 dian pH, drawn from the pole Pp to the point H: it is 
 evident that this will be the meridian of the given plane;, 
 and that the angle Apu, formed by this meridian and 
 that of London, will give the difference of longitude of the two places. We must 
 therefore determine this triangle, and to find it we have three things given, viz., Ist, 
 
TO DESCRIBE ANY DIAL. 535 
 
 -a pthe complement of the latitude of London, which is 38°29 ; 2d, a H the distance. 
 
 of London from the place, the horizontal plane of which is parallel to the given 
 plane, and which is 12°; 3d, the angle p a 4, comprehended between these two sides, 
 which is equal to the right angle w AL plus PAL, or that which the plane forms with 
 the meridian. 
 
 By resolving this spherical triangle, it will be found, that the angle ‘at the pole 
 aPH, or that formed by the two meridians, is 5° 59’ ; which is the difference of longi- 
 tude between the two places a and H. 
 
 The latitude of the place will be found also by the solution of the same triangle ; 
 
 for it is measured by the complement of the arc PH, of the triangle PA H: according 
 
 to calculation it is 40° 15’.* 
 Thus, a plane inclining 12° at London, and declining to the west 224 degrees, is 
 parallel to the horizontal plane of a place which has 5° 59’ of longitude west 
 
 from London, and 40° 15’ of latitude. The latter also is the angle which the style 
 
 _ A let fall a perpendicular hi, on the radiusca. On hi 
 
 _ plement of the angle pa H: draw KZ perpendicular to 
 _ hi, and from the point /, draw /m perpendicular to the 
 
 ought to form with the sub-style; for the angle which the axis of the earth forms with 
 
 the horizontal plane is always equal to the latitude. 
 
 It is here evident that when it is noon atthe place H, it will be 23m. 56s. after 
 noon at the place a; for 5° 59’ in longitude correspond to 23m. 56s. in time. Con- 
 sequently, at the place a, when the shadow of the style falls on the sub-style, which 
 is the meridian of the plane, it will be 23m. 56s. after twelve at noon. To find there- 
 fore the hour of noon, it will be necessary to draw, on the west side of the sub-style, 
 an hour-line corresponding to 11h. 386m. 4s., or 1lh. 36m. By the like reasoning, . 
 it will be found that 11 in the morning, at the place a, will correspond to 10h. 36m. 
 at the place u, &c. In the same manner, | in the afternoon, at the place a, will cor- 
 respond to 12h. 36m., or 36m. after 12, at the place w: 2 o’clock will correspond to 
 lh. 36m. ; 3 o’clock to 2h. 36m., and so of the rest. ; 
 
 Thus, if we suppose the sub-style of the plane, on which the dial ought to be de- 
 scribed, to be the meridian, it will be necessary to describe a dial which shall indicate 
 in the forenoon, 11h. 36m.; 10h. 36m. ; 9h. 36m.; 8h. 36m., &c.; and in the after- 
 
 f noon 12h. 36m.; lh. 36m.; 2h. 36m. ; 
 
 Fig. 29. 3h. 36m.; 4h. 36m., &c. 
 
 Pp When these calculations have been 
 made, the dial may be easily constructed. 
 For this purpose, first find, by Prob. 3, 
 the: sub-style, which is the meridian of 
 the plane. We shall suppose that it is 
 p E (Fig. 29.), and that Pp is the centre 
 of the dial. Having assumed PB of a 
 
 Fig. 28. 
 
 _* Trigonometrical calculation may be avoided by means 
 of a graphic operation exceedingly simple. In a circle of a 
 convenient size (Fig. 28.), assume an arc pa@ equal to PA 
 (Viz. 27.); make ah equal to'aH3 and from the pomt 
 
 describe a quadrant, or make h & equal to the arc which 
 measures the declination of the plane, or equal to the sup- 
 
 Tadiiis cp, and let / m be continued till it meet the circle 
 in 2; the arc pm will be equal to Pp H, and if an arcof a 
 
 circle be described on mo, and if 7 be drawn perpen- 
 
 dicular from the point 2, so as to meet this arc in z, the ; 
 
 angle «mJ will be equal to the required angle P of the ; - 
 
 triangle ar H. sie 
 
: 
 
 536 DIALLING. | 
 
 | 
 convenient length, draw, through the point B, the line a 8 c, perpendicular to P E; 
 
 if a be the western side, the line Pp d which corresponds to 11 hours 36 minutes, | 
 
 or which is distant from the meridian 24 minutes in time, may be found by making 
 
 use of the following analogy :: 
 
 As radius is to the cosine of the latitude, which is 40° 15’; so is the tangent of the 
 hour-angle corresponding to 24m. in time, or the tangent of 6°, to a fourth term, 
 which will be the tangent of the angle B P d. 
 
 By this analogy, it will be found equal to 80 parts, of which P D contains 1000: 
 if 80 of these parts therefore, taken from a scale, be set off from 38 towards d, 
 and if p d be then drawn, we shall have the hour-line of 11 hours 36 minutes for the 
 plane of the dial, or of the place H. 
 
 The line P e, of 10 hours 36 minutes, will be found in like manner, by this 
 analogy : 
 
 As radius is to the cosine of 40° 15’; so is the tangent of the hour-angle corre 
 sponding to 10h. 36m., or the tangent of 21° to the tangent of the angle B P e. 
 
 This tangent will be found equal to 293 of the above parts: if this number of 
 parts therefore, taken from the same scale, be laid off from B to e, we shall have the 
 hour-line P e corresponding to 10 hours 36 minutes. 
 
 The lines of the other hours before noon may be found in the like manner: the 
 two first terms of the analogy are the same, and the third is always the tangent of 
 an angle successively increased by 15°: these tangents therefore will be those of 
 6°, 21°, 36°, 51°, 66°, the logarithms of which must be added to the cosine 40° 15’; 
 and if the logarithm of radius be subtracted, the remainders will be the logarithms B 
 the tangents of the hour-lines: these tangents themselves will be for B d, Be, ‘&e. 
 80, 293, 554, 942, 1732, 4814, &c. in parts of which the radius or P D contains 
 1000. 
 
 A similar operation must be performed for the hours in the afternoon. As 36m. 
 in time correspond to 9°, the first hour-angle will be 9° ; the second, by adding 159, 
 will be 249; the third 39°; the fourth 54°, &c. The following proportions then 
 must be employed: As radius is tothe cosine of 40° 15’; so is the tangent of 9°, or 
 24°, or 399, &c. to a fourth term, which will be the tangent of the angles pl, or BP m, 
 orBpPn, &c. 
 
 Hence, if the logarithm of the sine of 49° 45’ be successively added to the loga- 
 rithmic tangent, of 9°, 24°, 39°, 54°, &c., and if radius be subtracted from the 
 different sums, we shall have the logarithms of the tangents of the angles which the 
 hour-lines Pp J, Pp m, Pn, &c. form with the sub-style; and these tangents themselves 
 will respectively be 121, 339, 618, 1050, 1988, 7268, parts of which Pp B contains 
 1000. If these numbers therefore, taken from the same scale as before, by means of 
 a pair of compasses, be set off from 8 to lJ, from B to m, from B to n, &c., and if the 
 lines PZ, P m, Pn, P Oo, &c. be then drawn, the dial will be nearly completed; as 
 nothing will be necessary but to mark the point d with x11, because P d is the meri- 
 dian of the place a; and to mark the other hour-points with the numbers which 
 belong to them, as seen in the figure. 
 
 To avoid the trouble of tracing out more hour-lines than are necessary, it will be 
 proper first to determine at what hour the sun rises and sets on the given plane, at 
 the time of the longest day; which may be easily done by means of the following. 
 consideration. 
 
 It may be readily seen that if we suppose two parallel planes, in two different 
 places of the earth, the sun will begin to illuminate both of them at the same mo- 
 ment; and that he will also set to both at the same time. The plane of the- 
 dial in question, being parallel to the horizontal plane of a place which has 409 i 
 of north latitude, nothing is necessary but to know at what hour the sun will rise 
 
TO DESCRIBE ANY DIAL. 637 
 
 in regard to that plane on the longest day. But it will be found that in the latitude 
 of 40° 15’ the longest day is 15 hours 24 minutes; or that the sun rises on that 
 day 7 hours 42 minutes before noon, and sets at 42 minutes past 7 in the evening. 
 It will be sufficient then, on the dial in question, to make the first hour-line in 
 
 the morning that of 4 hours 15 minutes, and the last in the evening 7 hours 30 
 minutes, 
 
538 NAVIGATION. 
 
 PART EIGHTH. 
 
 CONTAINING SOME OF THE MOST CURIOUS PROBLEMS [IN 
 NAVIGATION. 
 
 NAVIGATION may be classed among those arts which do the greatest honour to the 
 human invention; for in no department of science is the ingenuity of man displayed 
 to more advantage than in this art, by which he conducts himself through the wide 
 expanse of the ocean, without any other guide than the heavenly bodies and a 
 compass; by which he subdues the winds, and even employs them to enable him to 
 brave the fury of the ocean, which they excite against him: in short, an art which 
 connects in social intercourse the two worlds; forms the principal source of the 
 industry, commerce, and opulence of nations. Hence one of our poets very justly 
 says, / 
 Le trident de Neptune est le sceptre du monde. 
 
 But this is not a proper place for entering into a dissertation on the utility of navi- 
 gation. We shall only observe, that navigation may be considered under two points 
 of view. According to the first, it is a science which depends on astronomy and 
 geography: considered in this manner it is called Piloting, which is the art of deter- 
 mining the course that ought to be pursued in order to go from one place to another, 
 and of knowing at all times that point of the earth at which a ship hasarrived. <Ac- 
 cording to the other, it is an art founded on mechanics and the moving powers of 
 the vessel: considered under this point of view, it is called manceuvring, and teaches 
 how to give to that ponderous mass, which cleaves the billows, the feces direc- 
 tion by means of the sails and the rudder. 
 
 We shall here present the reader with every thing most curious in both these parts 
 of navigation. 
 
 PROBLEM I. 
 
 Of the curve which a vessel describes on the surface of the sea, when she sails on the 
 same point of the compass. 
 
 When a ship is about to set sail, it is necessary to find out the proper course: that 
 is, to determine the direction in which she ought to proceed, in order to arrive, in 
 the shortest time and with the greatest safety, at the place of her destination. 
 When this direction, or the angle it forms with the meridian, has been determined, 
 it is always pursued, unless particular circumstances prevent it. A vessel, by 
 thus steering for several days, on the same point of the compass, describes @ 
 line which always forms the same angle with the meridians: this is what is called 
 the lorodromic line, or oblique course ; and there hence results, on the surface of the 
 earth, a peculiar curve, the nature and properties of which have excited the attention 
 of mathematicians. On these properties the practical rules of navigation have been 
 founded ; and, as they are very remarkable, they deserve to be explained. » 
 
 We presume that the reader is acquainted with the nature of the compass, the 
 different points, &c.; and with the elements of navigation ; for it is impossible that 
 we should here enter into details merely elementary. 
 
hm 
 
 NAVIGATION, 539 
 
 Let us suppose that the sector acB (Fig. 1.) re- 
 presents a portion of the spherical surface of the ~ 
 earth, of which c is the pole, and as the equator ; 
 or only the are of a parallel comprehended between 
 two meridians, as Ac and Bc; and that cp, cE, and 
 CF, represent so many meridional arcs, very near to 
 each other. 
 
 Let a vessel depart from the point a of the arc 
 A B, the meridian of which is ac; and proceed ona 
 course forming with that meridian the angle ca Hu, 
 less than a right angle, for example an angle of 60 
 degrees; the vessel will describe the line a H, by 
 which means she will always change her meridian. 
 When she arrives at H, under the meridian c p, let her 
 continue in the same course, making with the meridian the angle c H1, equal to the 
 former; and so on, describing the lines AH, H1, IK, &c., always making the same | 
 angle (60°) with the meridians c a, cH, cI, cK, &c. As her course is continually. 
 inclined to the meridian at an angle of 60 degrees, it may be readily seen that the 
 line A HI will not be the arc of a great circle on the surface of the sphere ; for it _ 
 is demonstrated in spherics, that if aux were a circle of this kind, the angle cut 
 would be greater than c AH, and c1k greater than cu1,and soon. The case would 
 be the same if the curve A HIK were an arc of a lesser circle of the sphere; hence 
 there is reason to conclude, that the curve described by a ship, when she always 
 proceeds on the same course, is a peculiar curve, which constantly approaches the 
 pole. 
 
 Remarks.—I. It is here evident, that when the loxodromic angle vanishes; that 
 is, when the vessel steers directly north or south, the loxodromic line is an arc of the 
 meridian. 
 
 But, if the angle be a right angle, and if the vessel be under the equator, she will 
 describe an arc of the equator. In the last place, if out of the equator, she will 
 describe a parallel. 
 
 II. If the loxodromic line a KL, be divided into several parts, so small that they 
 may be considered as straight lines, and if as many parallels or circles of latitude be 
 made to pass through the points of division u, I, K, &c., all these circles will be equal 
 and equally distant from each other; so that, by making meridional arcs to pass 
 through the same points of division, the portions of these meridians, such as DH, 
 MI, NK, &c., will be equal, as well as the corresponding arcs a D, H M,1N, &e. 
 This equality however will not be in degrees, but in miles, as may be easily demon- 
 strated; for the triangles ADH, HMI, INK, &c., are evidently similar, because the 
 hypothenuses, aH, HI, 1, &c., being equal in length, the other sides will be re- 
 ‘spectively equal also. On the other hand, it is evident that if ap, which is part 
 of a great circle, be equa] in length, or in miles, to HM, which is part of a lesser 
 circle, the latter must contain a greater number of minutes or degrees than the 
 former. 
 
 III. When a very small portion of the loxodromic line, such as a H, has been 
 passed over, always pursuing the same course, on the vessel’s arrival at 4, if the dif- 
 ference of latitude, or the arc pu, be determined by observation, it will be easy to. 
 find the distance sailed a H; since p H is to A H, as the sine of the angle H a b, 
 which is known, is to radius. Ifthe angle c a u, for example, be 60 degrees, and 
 ‘consequently H ap 30 degrees; and if p be equal to half a degree, or 30 nautical 
 miles, the distance a H will be 60 nautical miles; for the sine of 80 degrees is 
 exactly equal to half the radius. 
 
540 NAVIGATION. 
 
 IV. If the course and distance sailed be known, the difference of latitude may be 
 found in like manner. 
 
 V. The loxodromic angle c a 4, or HAD, being known, as well as the difference of 
 latitude p H, the value of the arc aD may be found; for pH is to aD asthe sine of the 
 angle HaDis to its cosine. But when the length of the arc of a parallel, or the number 
 of miles it contains, is known, the degrees and minutes it contains may be determined 
 also. In this manner, the difference of longitude produced by the vessel’s change of 
 position, while passing over the small loxodromicare a H, is obtained ; and if the same 
 operation be performed in regard to all the other small arcs H M, 1 N, &c., we shall 
 have the whole difference of longitude, produced by the vessel’s passing over any 
 loxodromic are aK. The difficulty of this operation arises from these ares being 
 dissimilar, though equal in length. But geometricians have found means to avoid 
 these calculations, by ingenious tables or other operations, the explanation of which 
 does not fall within the plan of this work. 
 
 VI. This curved line has one property which is very singular, that it always 
 approaches the pole without ever reaching it. This evidently follows from the 
 nature of it; for if we suppose it to arrive at the pole, it will intersect all the meri- 
 dians in that point; consequently, since it cuts each meridian under the same angle, 
 it will cut them all at the pole under the same inclination, which is absurd; since 
 they are all inclined in that point to each other. It will therefore approach the pole 
 more and more, making an infinite number of circumvolutions around it, but without 
 ever reaching it. Hence, according to mathematical rigour, a ship which continually 
 pursues the same course, the cardinal points excepted, will always approach the pole 
 without ever arriving at it. . 
 
 VII. Though the loxodromic line, when it forms an acute angle with the meridians, 
 must make an infinite number of circumvolutions around the pole before it reaches — 
 it, its length is however finite; for it can be demonstrated, that the length of a loxo-— 
 dromic line, such as A KL, is to the length of the arc of the meridian that indicates — 
 the difference of latitude, as radius to the cosine, or sine complement, of the angle 
 which the loxodromic line forms with the meridian; consequently the difference 
 of latitude is to the loxodromic distance sailed, as the cosine of the above angle — 
 is to radius. 
 
 The above remark is principally intended for geometricians ; and exhibits a kind 
 of paradox which must astonish those to whom truths of this kind are not familiar: . 
 those, however, who comprehend the preceding demonstrations, can entertain no 
 doubt of it. But, for the sake of farther illustration, let us suppose a loxodromic 
 line inclined to the meridian at an angle of 60 degrees, with its infinite cireumyolu- 
 tions around the pole; if we employ the following proportion, As the cosine of 60 
 degrees, or the sine of 30°, is to radius, so is 90 degrees difference of latitude toa 
 fourth term, this fourth term will be the absolute length of the loxodromic line. 
 But the sine of 30 degrees is equal to half the radius; and hence it follows that the 
 fourth part of the circle is the half of the above loxodromic line; or this line, not- 
 withstanding the infinite number of its circumvolutions, is exactly equal to a semi- 
 circle of the sphere. 
 
 PROBLEM II. 
 How a Vessel may sail against the Wind. 
 
 What is here proposed will no doubt seem a paradox to those unacquainted with the 
 principles of mechanics. Nothing however is more common in navigation, as this is 
 always done when a vessel, according to the nautical term, is beating up on different — 
 tacks, or keeping as near to the wind as possible. But when we say a vessel can | 
 sail against the wind, we do not mean that she can proceed on a course directly oppo- | 
 site to the point from which the wind blows; it is only by making an acute angle | 
 
TO SAIL AGAINST THE WIND. 541 
 
 with the rhumb line passing through that point, which is sufficient; for by several 
 tacks she can then advance ina direction contrary to that of the wind. 
 
 Let us suppose a vessel, (Fig. 2.), the keel of which 
 is A B, and let one of the sails c p be set in sucha 
 manner, as to form with the keel an angle B £ p of 
 40 degrees: if the direction of the wind be £ F, 
 making with the same keel an angle of 60 degrees, for 
 example, it is evident that the angle D E F will be 20 
 degrees; consequently the sail will be impelled by a - 
 wind falling onit at an angle of 20 degrees. But ac- 
 cording to the principles of mechanics, the action of a 
 power falling obliquely on any surface, is exercised in 
 a direction perpendicular to that surface, and therefore 
 if = c be drawn perpendicular to c D, the line EG will be the direction according to 
 which the effort of the wind is exercised on the sail c D, but with a diminished force 
 on account of the obliquity of the stroke. 
 
 If the vessel were round, it would proceed in that direction; but as, in consequence 
 of its length, it can move with much greater facility in the direction of its keel = u, 
 than according to any other, it will assume a direction E K, somewhere between EG 
 and £ H, but much nearer to the latter than to the former, almost in the ratio of its 
 facility to move according to eu andrG. The angle kx E F therefore, which the 
 ship’s course forms with the direction of the wind, may be an acute angle. If the 
 angle K E H, for example, be 10 degrees, the angle K EF will be 703 degrees, conse- 
 quently the vessel will lie almost two points nearer to the wind. But it is shewn 
 by experience, that a vessel may be made to go on acourse still nearer to the direc- 
 tion of the wind, or to lie closer to it by about one point more; for if the vessel be 
 well constructed, there are 22 of the 32 points comprehended in the compass, which 
 may serve to make her proceed to the same place. 
 
 It is indeed true, that the nearer a ship lies to the wind, or to speak in common 
 terms, the sharper the angle of the wind’s incidence on the sail, the less will be its 
 force to push the vessel forwards ; but this is compensated by the quantity of sail 
 that may be set, for in this case none of the sails hurt each other, and a vessel can 
 absolutely carry all her sails. What therefore is lost in consequence of the weakness 
 of the force exerted on each, is gained by the quantity of surface exposed to the wind. 
 
 It may be easily conceived how advantageous this property of 
 Fig. 3. of vessels isto navigation; for whatever be the wind, it may be 
 employed to convey a ship to any determinate place, even if it 
 should blow directly from that quarter. For let us suppose 
 (Fig. 3.) that the direct course is from £ to F, and that the 
 wind blows in the direction Fs; the vessel must be kept as near 
 the wind as possible to describe the line # c, making with E F 
 the acute angle F EG; having proceeded some time in the direc- 
 tion E G, the vessel must then tack about to run down Gu; 
 then H1; then 1 K; and so on; by which means she will. 
 always approach nearer to.the place of her destination. 
 
 Fig. 2. 
 
 PROBLEM III. 
 Of the force of the Rudder, and the manner in which tt acts. 
 
 The force by which the rudder of a ship makes her move in any direction at plea- 
 sure, excites no small degree of astonishment; especially when we consider the 
 weak action of the enormous rudders with which some of the barges that navigate 
 our rivers and canals are furnished. The cause of this phenomenon we shall here 
 endeavour to explain and illustrate. 
 
le bw 
 ah 
 
 * “a Fe 
 
 542 NAVIGATION. 
 
 The rudder of a barge or vessel has no action unless impelled by the water. It is 
 the force resulting from this impulse, which being applied in a direction transversal to 
 the poop, tends to make the vessel turn around a point of its mass, called the spon- 
 taneous centre of rotation. The prow of the vessel describes around this point an 
 are of a circle, in a direction opposite to that described by the poop; hence it follows 
 that the prow of the vessel turns towards that side to which the rudder is turned, 
 consequently opposite to that side towards which the tiller or lever of the rudder is 
 moved. Hence, when the tiller is moved to the starboard side, the vessel turns 
 towards the larboard, and vice versa. 
 
 A force, and even a certain degree of intensity, must therefore be applied to the 
 rudder to make the vessel turn; and on this account the construction of the vessel 
 is so contrived, as to increase this force as much as possible; for while the barges 
 which navigate our rivers are in general very broad behind, and screen as we may say 
 the rudder, so that the water flowing along their sides can scarcely touch it, the stern 
 of vessels intended for sea are made narrow and slender, so that the water flowing 
 along their sides must necessarily strike against the rudder, if in the least moved from 
 the direction of the keel. Let us therefore endeavour to estimate nearly the force 
 which results from this impulse. 
 
 A vessel of 900 tons, when fully laden, draws 13 or 14 feet of water, and its rudder. 
 is about 2 feet in breadth. Let us now suppose that the vessel moves with the velo- 
 city of 2 leagues per hour, which makes 176 yards per minute, or about 9 feet per 
 second; if the rudder be turned in such a manner as to make with the keel conti- 
 nued an angle of 30 degrees, the water flowing along the sides of the vessel will impel 
 the rudder under the same angle, that is, 30 degrees. The part of the rudder under 
 water being 14 feet in length and 2 in breadth, presents a surface of 28 square feet, 
 impelled at an angle of 30 degrees, by a body of water flowing with the velocity of 9 
 feet per second. But the action of such a current, if it impelled a similar surface in 
 
 a perpendicular direction, would be 2205 pounds, which must be reduced in the ratio. 
 of the square of the sine of incidence to that of radius, or in the ratio to 1, since 
 
 the sine of 30 degrees is 4, radius being 1. The effort therefore of the water 
 will be 551 pounds. Such is the force exercised perpendicularly on the rudder; 
 and to find the quantity of this force that acts in a direction perpendicular to the 
 
 keel, and which makes the vessel turn, nothing is necessary but to multiply the pre- | 
 
 ceding effort by the cosine of the angle of inclination of the rudder to the keel, which 
 
 in this case is ,/ 3 or 0:866, which will give 477 pounds. a 
 
 The above computation is made on the old supposition, that the force of the water 
 
 is diminished in proportion as the square of the sine of the incident angle is less than — 
 the square of the radius. But, by more accurate experiments it is found (Dr. ° 
 
 Hutton’s Math. and Philos. Dictionary, Tab. 3, Resistance), that at an angle of 30. 
 
 degrees the absolute force is diminished only in the ratio of 840 to 278; hence 
 
 then, the whole force 2205 pounds, reduced in this ratio, comes out 730 pounds, for 
 the effective or perpendicular force on the rudder, to turn it or indeed the ship about, 
 supposing the rudder held or fixed firm in that position. 
 
 But there is one cause which renders this effort more considerable: the water 
 which flows along the sides of the vessel does not move ina direction parallel to the — 
 
 keel, but nearly parallel to the sides themselves which terminate in a sort of angle at 
 the stern-post, or piece of timber which supports the hinges of the rudder; so that 
 this water bears more directly on the rudder by an angle of about 30 degrees : hence, in 
 the above case, the angle under which the water impels the rudder will be nearly 60 de- 
 grees; we must, therefore, make this proportion: As the square of the radius is 
 
 { 
 
 to the square of the sine of 60 degrees, or as lis to 3; sois 2205 to 1653. The force ' 
 
 therefore which acts in a direction perpendicular to the keel, is 1653 pounds, Or, 
 
a. - 
 
 OF THE RUDDER. 543 
 
 by the table in the dictionary above quoted, as 840 isto 729 (for 60°), so is 2205 
 to 1913 pounds, the perpendicular force. 
 
 This effort will no doubt appear very inconsiderable when compared with the 
 effect it produces, which is to turn a mass of 900 tons; but it must be observed that 
 this effort is applied at a very great distance from the point of rotation and from 
 the vessel’s centre of gravity ; for this centre is a little beyond the middle of the 
 vessel towards-the prow, as the anterior part swells out, while the posterior tapers 
 towards the lower works, in order that the action of the rudder may not be inter- 
 rupted. On the other hand, it can be shewn that what is called the spontaneous 
 centre of rotation, the point round which the vessel turns, is also a little beyond the 
 middle and towards the prow; hence it follows, that the effort applied at the extre- 
 mity of the keel, towards the stern, acts to move the vessel’s centre of gravity, 
 by an arm of a lever 12 or 15 times as long as that by which this centre of gravity, 
 
 - where the weight of the vessel is supposed to be united, exerts its action, And 
 
 lastly, there is no comparison between the action exercised by this weight when 
 floating in water, and that which it would exert if it were required to raise it only 
 one line. It needs therefore excite no surprise, that the weight of one ton, applied 
 with this advantage, should make the vessel’s centre of gravity revolve around its 
 ceutre of rotation. 
 
 If the ship, instead of going at the rate of two leagues per hour, sails at the rate 
 of three, the force applied to the rndder will be to that applied in the former case, 
 in the ratio of 9 to 4; consequently, if the position of the rudder be as above sup- 
 posed, the actual force will be 3719 pounds, or rather 4304 pounds: if the velocity 
 of the vessel were 4 leagues per hour, this force in the same position of the rudder 
 would be 4 times as much as at first, or 6612 pounds, or rather 7652 pounds. 
 
 Hence it is evident why a vessel, when moving with rapidity, is more sensible to 
 the action of the helm; for when the velocity is double, the action is quadrupled: 
 this action then follows the square or duplicate ratio of the velocity. 
 
 PROBLEM IV. 
 
 What angle ought the rudder to make, in order to turn the vessel with the greatest 
 force ? 
 
 If the water moves in a direction parallel to the keel when it impels the rudder, 
 it will be found that this angle ought to be 54 degrees 44 minutes; but, as already 
 observed, the water is carried along in an angular manner towards the direction of the 
 keel continued; which renders the problem more difficult. If we suppose this angle 
 to be 15 degrees, which Bouguer considers as near the truth, it will be found that 
 the angle in question ought to be 46 degrees 40 minutes. 
 
 Ships do not receive the whole benefit of this force; for the length of the tiller 
 does not permit the helm to form with the keel an angle of more than 30 degrees. 
 
 PROBLEM V, 
 Can a vessel acquire a velocity equal to or greater than that of the wind? 
 
 This can never take place in a direct course, or when the ship sails before the 
 wind; for besides that in this case a part of the sails hurt or intercept the rest, it is 
 Prident that if the vessel should by any means acquire a velocity equal to that of the 
 wind, it would no longer receive from it any impulse ; its velocity then would begin 
 to slacken in consequence of the resistance of the water, until the wind should make 
 an impression on the sails equal to that resistance, and then the vessel would con- 
 tinue to move in an uniform manner, without any acceleration, with a velocity less 
 than that of the wind. 
 . But, when the course of the vessel is in a direction oblique to that of the wind, 
 
544 NAVIGATION. 
 
 this is not the case. Whatever may be its velocity, the sail is then continually 
 
 receiving an impulse from the wind, which still approaches more to equality, as the 
 
 course approaches a direction perpendicular to that of the wind: therefore, however 
 
 fast the vessel advances, it may continually receive from the wind a new impulse to” 
 motion, capable of increasing its velocity to a degree superior to that even of the 
 
 wind itself. . 
 
 But for this purpose it is necessary that the construction of the vessel should be 
 of such a nature, that, with the same quantity of sail, it can assume a velocity equal 
 to or 3 that of the wind. This is not impossible, if all the canvass which a 
 vessel can spread to the wind, in an oblique course, were exposed in one sail, ina 
 direct course. This then being supposed, Bouguer shews, that if the sails be set in 
 such a manner as to make with the keel an angle of about 15 degrees, and if they 
 receive the wind in a perpendicular direction, the vessel will continually acquire a new 
 acceleration, in the direction of the keel, until her velocity be superior to that of the - 
 wind, and that in the ratio of about 4 to 3. 
 
 It is indeed true, that, as the masts of vessels are placed at present, it is not 
 possible that the yards can form with the keel an angle less than 40 degrees; but 
 some navigators assert, that by means of a small change this angle might be reduced 
 to 30 degrees. In this case, and supposing that the vessel could acquire in the direct 
 line a velocity equal to 3 that of the wind, the velocity which it would acquire by 
 receiving the wind on the sails at right angles, might extend to 1°034 that of the 
 wind, which is a little more than unity, and therefore somewhat more than the velocity 
 of the wind. 
 
 If we suppose the same velocity possible in the direct course, and that the sail 
 forms with the keel an angle of 40 degrees, it will be found that the velocity 
 acquired by the vessel in an oblique course, will be nearly 48 the velocity of the 
 wind. 
 
 This at least will be the case, if in this position of the sails, in regard to the wind, 
 they do not hurt or obstruct each other. If all these circumstances therefore be 
 combined, it appears that though it is possible, speaking mathematically, that a vessel 
 can move with the same velocity as the wind, or even with a greater, it will be very 
 difficult to produce this effect in practice. 
 
 PROBLEM VI. 
 
 Given the direction of the wind, and the course which a vessel must pursue in order to. 
 reach a proposed place ; what position of the sails will be most advantageous for 
 that purpose ? 
 
 Let us suppose that the wind blows from the north, and that the ship’s course is 
 due east. If the ship, when her head is directed to that point, has her yards parallel 
 to the keel, her progress will be = 0; as she will receive no impulse but in a direc- 
 _ tion perpendicular to the keel. On the other hand, if the yards be perpendicular to 
 
 the keel, as the sails will not catch the wind, the vessel in this case again will not 
 move. Thus, from the first position to the latter, the impulse in the direction of the 
 keel, and consequently the velocity, goes on first increasing, and then decreasing. 
 
 There is some position therefore at which this impulse is strongest, or what is called 
 
 a maximum, and which will make the vessel move with the greatest velocity. The 
 
 question is to determine it. 
 
 Geometricians have solved the problem, and have found, that to determine this 
 angle, that between the wind and the proposed course must be divided in sucha 
 manner, that the tangent of the apparent angle, which the wind forms with the yard, 
 shall be double to that which the yard forms with the course, or with the keel. In 
 this case, therefore, the sail at first must be placed in such a situation, as to make 
 
BEST FORM OF THE SHIP, 545 
 
 with the keel an angle of 35 degrees 16 minutes, and consequently with the wind an 
 angle of 54 degrees 44 minutes. 
 
 We say the sail at first must be set in this manner; for as soon as the vessel has” 
 acquired a greater velocity, this angle will cease to be the most favourable, and will 
 become less so, the more the velocity is accelerated, as must be the case, till the 
 impulse of the wind be in equilibrio with the resistance which the vessel suffers from 
 the water ; but in proportion as the velocity is accelerated, the wind strikes the sail 
 more obliquely, and loses its force: for this reason the sail must be disposed in such 
 ‘manner, as to form with the keel an angle always more acute, and this angle may be 
 ‘reduced to 30 degrees and less; so that the wind shall make with the sail an angle of 
 60 degrees and more. 
 
 We have here considered the question independently of lee-way ; but if this be 
 ‘taken into account, supposing it for example in the present case to be one point, it 
 will be necessary to make the vessel’s head lie a point nearer to the wind: the angle 
 then which the wind forms with the course will be from 78 to 79 degrees; and it will 
 be found that on the outset, the angle formed by the wind and the sail ought to be 
 48° 45’; and that of the yard with the keel 29° 45’, which must gradually be reduced 
 to 24 or 25 degrees. By then steering wnwiw, the vessel will really proceed 
 east with the greatest velocity possible, or nearly so; and asin the neighbourhood of 
 those points which give a mazimum the progressive increase is insensible, this 
 greatest velocity will always be nearly obtained, even when the above angles are not 
 very exact. 
 
 PROBLEM VII. 
 
 In what manner must a vessel at sea be directed, so as to proceed from any given place 
 to another by the shortest course possible ? 
 
 As the loxodromic line, which navigators generally follow at sea, is not the 
 shortest way from one place to another, it is natural to ask whether there be not 
 
 some means by which the shortest course can be pursued; for it is evident, ceteris 
 paribus, that the way being shorter, the voyage would be sooner ended. 
 
 As this is no doubt possible, we shall first shew how it may be done, and then 
 examine with what advantage it is attended. 
 
 Every one knows that the shortest way from one place to another, on the surface 
 of the earth, is the are of a great circle drawn from the one to the other. Nothing 
 
 then is necessary but to keep the vessel continually on the’ arc of a great circle, or 
 
 at least to deviate very little from it. 
 
 _ Let us suppose, then, that a vessel is bound from London to the island of Trinidad, ~ 
 Tt will be found by trigonometrical calculation, that the arc of a great circle drawn 
 from London to Trinidad, makes at London with the meridian an angle of 69° 44’, 
 and at Trinidad: of 37° 30’; while that of the loxodromic line with the meridian is at 
 London 50° 40’. The angle formed by the course with the meridian, at the time of 
 ‘departure, ought therefore to be 69° 44’. 
 
 But to keep the vessel in this great circle, it will be necessary to change the angle 
 every day ; and strictly speaking every hour and every moment ; otherwise the vessel 
 will describe small loxodromic lines, and not the arc of a great circle. The following 
 method, which if not perfectly exact, approaches very near the truth, may be ems 
 ployed to effect this change. ; 
 
 _ As the angle at Trinidad is 37° 30’, it may be easily seen, that from the time of the 
 
 vessel’s departure, till that of her arrival at the place of destination, the angle of the 
 
 sourse must be gradually diminished, from 69° 44’ to 37° 30’. Let us divide the - 
 
 lifference, which is 32° 14’, into 10 equal portions, which will each be 3° 13’. Every. 
 
 ‘ime then that the difference of longitude is one tenth of the whole, or about 
 
 i® 37’, that is, when the vessel has made about 111 leagues of departure towards the 
 2N 
 
546 NAVIGATION. 
 
 west, it will be necessary to keep 3° 13’ more to the south. By these means the 
 vessel will be kept nearly on the arc of a great circle, passing through London and 
 Trinidad. 
 
 These angles might be more exactly determined by means of trigonometry ; that i ig; 
 by drawing a meridian at about every four degrees of longitude, and successively. 
 solving the spherical triangles thence resulting; but if we examine what advantage! 
 would arise from this operation, it will be found of very little importance. The dis-' 
 taice from Plymouth to Trinidad, measured on a great circle drawn from the one to: 
 the other, is about 1212 leagues; and if the loxodromic line drawn from the one to 
 the other be measured, it will be found to be about 1254. It-istherefore not worth 
 
 while to seek for the shortest course to save about 40 leagues; especially as in sea 
 voyages the principal object is not to pursue the shortest route, but to take advantage 
 of the wind whatever it may be, in order to complete the voyage. 
 
 | 
 
 PROBLEM VIII. 
 
 What is the most advantageous form of construction for the prow of the vessel, in | 
 - order that she may sai] better, or be easier steered ? | 
 
 If one only of these objects were to be attained, that for example of cleaving the 
 water with the greatest facility, the problem might be easily solved. The sharpera 
 vessel is at the prow, the easier she can cut the water, and consequently will be 
 better calculated for moving with rapidity. 
 
 But an object still more important than velocity, is that of being easily worked: 
 without this property a vessel, like a refractory borse, would render useless the 
 whole art of the navigator. But it is shewn, both by experience and reason, that a 
 vessel, to be manageable, must be narrow towards the prow, in the part immersed, 
 in order that the water which runs along her sides may strike the rudder with more! 
 facility. She will also be managed with more ease, the farther the centre of gravity 
 is from the stern; and for this reason the most obtuse and the widest part of the 
 vessel must be towards the head. This is actually the case in regard to all vessels, 
 destined for voyages. 
 
 Nature, in regard to this point, seems to have provided man with a model in the 
 form of fishes; for it may be readily seen that the thickest part of the fish is towards) 
 the head, which in general.is even pretty obtuse. Like our ships, they have much. 
 more need of being able to turn and direct themselves with ease, than to move with’ 
 
 rapidity. The best vessel perhaps would be that constructed according to the exact 
 
 dimensions of a migrating fish, such as the salmon; which seems to enjoy, in @ 
 greater degree than any other, the two properties of moving quick and directing it. 
 self with ease. 
 
 M. Camus, a gentleman of Lorraine, gives an account, in his Mechanics, of several 
 experiments, from which he endeavours to shew, that the model of a vessel will move 
 faster with the thick end foremost, than when cleaving the waves with the other, 
 which is sharper: he even assigns reasons for this idea, but they are certainly ill 
 founded. These experiments are in absolute contradiction to sound theory; and 
 if ships have that form, it is not that they may move faster, but in consequence 
 of the necessity which has been found, of sacrificing the advantage of velocity to 
 that of being easily manceuvred. 
 
 M. Montucla here, rather injudiciously, opposes theory to experiment, and censures 
 Camus improperly, whose experiments and reasonings have been confirmed by the 
 more accurate and extensive ones made, in the years 1793—1798, by the English So- 
 ciety for the improvement of Naval Architecture, as may be seen at large in the 
 Report of their Committee, printed in the year 1800. 
 
r x: 
 hs 
 
 be 
 
 BEST COURSE IN CHASING. 547 
 
 ' 
 PROBLEM IX. 
 
 What is the most expeditious method of coming up with a vessel which is jehneg. and 
 which is to the leeward ? 
 
 When a vessel is descried at sea, and you are desirous of coming up with her, you 
 would be much mistaken if you directed the head of your own vessels towards the 
 ‘one you are pursuing; for unless the chase were proceeding on the same course 
 exactly, you would either be obliged to change your direction every moment, or you 
 would lose the advantage of the wind by falling to the leeward. 
 
 e If a body a (Fig.4.) moves in the line a b cd, and if it 
 
 Fig. 4. be proposed that another body a should come up with it, 
 
 | the body a ought not to be impelled in the direction aa; 
 
 t\ |e for in a few moments a will have advanced on the line in 
 
 which it moves, and will have reached the point 8, for 
 
 example. Hence if we suppose that the body a always 
 
 changes its course, directing itself towards the one it 
 
 pursues, it will describe a curve such as ABCDBE, and 
 
 will at length reach the body a by going faster, but not 
 
 by the shortest way. If it does not change its direction 
 
 every moment, it will arrive at a point in the line ad, 
 
 which the body a has already left, and will pass it, unless 
 
 it set out to pursue it along the line ad, which would 
 still make it lose time. 
 
 . Fig. 5. To cause the body a Sider to come up with a, in 
 : Eye the least time possible, A must be directed to a point in 
 | the line ae (Fig. 5.), so situated, that 4 and ae shall be 
 to each other in the ratio of their respective velocities. 
 = But. these lines will be in this ratio, if the body a, at 
 
 D d ee in hs 
 c le every moment in its course, has that which it pursues 
 ae 8 similarly situated, in a direction parallel to the direction 
 
 Aa; that is, aa being directed to the south, if the body 
 a, when it reaches 8, is to the south of the body a when 
 it arrives at B; for it is evident that the lines ak, ae, will then be proportional to the 
 velocities of the two bodies, and they will arrive at the same time at E or e. 
 Navigators are sensible of this, both from practice and reason ; for if a vessel at a 
 espies another at a, the course of the latter ae, may be ascertained nearly without 
 much difficulty, and the ship in chase, instead of directing her, head towards a, will 
 follow a course such as a8, inclined from a, and at the same time the bearing of the 
 Vessel in the direction « a will be taken by means of a compass; when A has proceeded 
 some time, and reached B, for example, while a has reached 3, the bearing of the vessel 
 a in the direction 8b will be again taken: if it be still the same, it is a sign that a is 
 gaining ground, for a a and Bb are parallel. If the chace falls a little behind, it shews 
 that she may be pursued ina line making with the direction of her course, a less acute 
 angle ; but if she has got a-head, a line more inclined must be pursued to reach her ; 
 and if the line be as much inclined as possible, and approaches to parallelism, there 
 is reason to conclude that the chase is a better sailer, and that all hope of reaching 
 er must be given up. 
 
 _ It is here supposed that the chasing vessel has the advantage, or is to windward ; 
 for if she be to leeward, the manceuvring must be different, unless she has a great 
 _ idvantage in being able to lie near the wind. But this is not the proper place for 
  miarging on these manceuyres of the most ingenious of all arts. 
 
 DERN 
 
 ~* 
 
Wk 
 
 548 - NAVIGATION. 
 
 PROBLEM X. 
 
 On determining the Longitude at sea. | 
 
 To find the longitude at sea is a problem which has long engaged the attention of. 
 mathematicians ; and they have been stimulated in their efforts by munificent rewards. 
 which have been offered by some of the maritime states of Europe. The British par- 
 liament offered £20000. for the discovery of a practical method which could be relied. 
 on within certain specified limits; and the reward was paid to Mr. Harrison for the 
 improvements which he made in chronometers; one which he sent from England to 
 Jamaica being found less than two minutes in error on its return. 
 
 Since the days of Harrison chronometers have been greatly improved; they are 
 made of a more portable size, and are sold at a comparatively cheap rate. They form 
 an essential part of the outfit of every respectable navigator, and have conferred great 
 benefits on maritime science. i 
 
 Formerly the mariner had no guide to his longitude but the dubious one depending 
 on the compass and the log; and at the conclusion of along voyage errors amounting 
 to several degrees were not unfrequently found in the reckoning. ) 
 
 To determine the longitude by celestial observations it is necessary to find by inde- 
 pendent processes the time at the place where you are; and also at the same instant, 
 the corresponding time at some assigned or known meridian, as that of the Royal Ob-: 
 servatory at Greenwich, which is the meridian from which English seamen compute 
 their longitude. 
 
 Now, with respect to the time at the place of observation, we may observe, that for 
 every individual instant, every celestial object has a specific place in the heavens—. 
 changing its place as the time changes. At every instant its altitude, or its distance 
 from the horizon, varies ; the change of altitude being most rapid when the bearing of 
 the object is east or west, and slowest when the bearing is north or south, that is, 
 when the object is on the meridian. 
 
 If then the altitude of a celestial object be observed with a sextant or other 
 like instrument, when the altitude is varying quickly, the true mean time at the 
 place of observation may be found at the instant of taking the altitude. And if, when’ 
 such an altitude is observed, the time by a chronometer be noted, whose error for. 
 Greenwich mean time on agiven day, and daily gain or loss, are known, the true mean. 
 time at Greenwich may readily be inferred, and the difference between these local times 
 (one for the meridian of the ship, and the other for that of Greenwich) is the longitude 
 of the ship in time; and it may be converted into degrees by allowing 15 degrees 
 for every hour of time. 
 
 This method of finding the local time at sea, which is that depending on observed al- 
 titudes ina known latitude, may be considered as the only one generally practicable at 
 sea; on land many other methods may be used with advantage. We have adverted to one 
 method of finding the Greenwich time, viz. that by chronometers whose errors and rates. 
 are settled before going to sea. We have now to give an account of another method: 
 of determining this important element, which within the last 80 years has been brought 
 to a high degree of perfection—we mean that by lunar observations. By the succes- 
 sive efforts of men of science, building on the foundation laid by NEwTov, it is now 
 found practicable to predict the moon’s place in the heavens to a very minute degree of 
 accuracy ; and in fact, in the Nautical Almanac, her distance from the sun, nine fixed' 
 stars, and the four brightest planets, are given for every third hour of mean Green-' 
 wich time throughout the year, except when the moon is too near the sun to be 
 visible. At must not however be understood that her distance is given from all of 
 these objects for every day; a few conveniently situated on each side of her are 
 
 payer for each day, and they are changed for others as the moon changes her 
 place, 
 
OF THE LONGITUDE AT SEA. 549 
 
 _ Now if witha sextant we observe the distance of the moon from one of the objects 
 whose distances from her on the day of observation are given in the Nautical Almanac, 
 we must first find by computation what the observed distance would have been if 
 the observer had been at the centre of the earth. The methods which have been_ 
 devised for making this computation are numberless, and many are of nearly equal 
 merit. But for the details of the methods we must refer to works on practical na- 
 vigation. 
 
 Having found the distance of the moon from the sun, star, or planet, as seen 
 from the centre of the earth, we have only to turn to the Nautical Almanae for 
 the given day, and find to what Greenwich mean time the distance corresponds; and 
 comparing that time with the local mean time at the ship, the longitude is deduced 
 as before. 
 
 In deducing the distance seen at the centre from that observed on the surface of the 
 
 earth, the altitudes of both objects are necessary elements; and if either of the 
 
 objects be at a suitable distance from the meridian, the mean time at the place may be 
 deduced from its altitude, and the longitude found accordingly. 
 The longitude may also be found by comparing the mean time at the ship, of the 
 
 - immersion of one of Jupiter’s satellites in his shadow, or the emersion of the satellite 
 ’ from the shadow, with the Greenwich time of the immersion or emersion as given 
 for the day in the Nautical Almanac. Much difficulty has however been experienced 
 
 in holding a telescope, of sufficient magnifying power for observing these eclipses, 
 
 steady at sea; but we are assured that some persons have succeeded in doing so. 
 
 | Eclipses of the moon have also been used for finding the Greenwich time ; but the 
 
 times of beginning and end, or the contact of the border of the earth’s shadow with 
 
 the spots on the moon, can seldom be noted with requisite precision. 
 
 Eclipses of the sun and the kindred phenomena of occultations of stars by the moon — 
 have occasionally been made available for the same object. They admit of great pre- 
 cision in the results, but the necessary calculations are long and intricate, and the phe- 
 
 ' nomena occur too seldom to be of much use in the practice of navigation. 
 
 - Jt is tothe moon, however, that astronomers have in general looked for the solu- 
 - tion of this problem; the comparative quickness of her angular motion rendering her 
 decidedly the most eligible of all celestial objects for determining by her motions 
 ' small intervals of time. 
 
 _ Some persons have recently proposed her meridian altitudes as observed ina latitude 
 determined by other means, for finding the longitude ; and there is no doubt that the 
 longitude is involved as an element in the observed altitude. But no latitude can be 
 determined at sea with the precision requisite to render this method of any value, 
 
 and it is much to be wished that seamen were put on their guard against trusting to 
 
 auy such method of finding their longitude. 
 
 Before closing this article, we shall give an account of a method which has been 
 extensively used within the last few years, for finding differences of longitude on 
 land: the observations being the differences between the intervals of transit over 
 the meridian of the moon’s bright limb, and a star, as observed by transit instuments 
 at two different places ; availing ourselves of the form under which Mr. Riddle bas 
 
 _ put the problem in his ‘‘ Treatise on Navigation and Nautical Astronomy.” 
 
 It is evident (Mr. Riddle observes), that if the moon had no motion, and her 
 
 _semi-diameter did not change, the interval between the times of transit would be the 
 
 same at both places, and that the difference of the intervals arises from and is equal — 
 to the increase of the moon’s right ascension in the time of the limb’s passing from one 
 meridian to the other; or the westerly meridian must revolve in that time through 
 an angle equal to the sum of the difference of longitude, and the increase of the moon’s 
 
 Tight ascension. 
 
 Therefore, if p= the different longitude 1 = the increase, in time of the moon’s 
 
550 NAVIGATION. 
 
 right ascension in a sidereal hour, and Y’ the observed increase of her right ascension, — 
 
 (or the difference of the observed intervals) between the passing the two meridians, 
 ]h —I , 
 I 
 
 I 
 
 then r ; 1h33 7 : p-+1,’ whence p= 
 
 Now from the Nautical Almanac, the increase of the moon’s right ascension in an 
 
 hour may be taken out at once, and hence its increase in a sidereal hour may readily — 
 
 be found, whence the longitude may readily be computed. 
 
 : ]h — 
 Mr. Riddle gives a table of three pages, which contains the name of ie 
 
 for | 
 
 every possible value of 1, the argument of the table being 1 for an hour of mean time, but | 
 the number from the table corresponds to 1 for an hour of sidereal time; so that there — 
 is no necessity for reducing the mean time interval into a sidereal one. With the aid 
 
 of the table the longitude may be found by adding two logarithms together. 
 
 PROBLEM XI. 
 
 Ifa vessel should be able to reach either of the poles, what method ought the commander — 
 
 to pursue, in order to steer in the direction of a determinate Meridian ? 
 The difficulty which this problem seems, on the first view, to present, arises from 
 
 this circumstance, that if a vessel were at either of the poles, to whichever side she 
 
 might turn, her head would be directed towards the south or north. Every line drawn 
 from that point to any point whatever in the horizon, is a meridian; aud consequently 
 at the pole there is neither east nor west. But if there is neither east nor west, how 
 
 would she steer, or how would it be possible, all the meridians being similar, to find | 
 
 that in the direction of which it would be necessary to proceed, in order to reach the 
 proposed place ? 
 
 This however is not all: if a vessel should reach one of the poles, it is probable | 
 that the compass would become useless, or as the sailors say run entirely mad ;* and | 
 there are only two ways of navigating a vessel, either by the magnetic needle, or by | 
 
 observing the stars, or rather by both these methods combined. 
 Such is the problem which the astronomer who accompanied the Hon. Capt. 
 
 Phipps, afterwards Lord Mulgrave, sent out to attempt a passage through the northern. | 
 ocean, would have had to solve, had the expedition succeeded. If the progress of | 
 
 the vessel had not been stopped by the ice, he would have proceeded to the 90th 
 
 degree of latitude in order to arrive by the shortest passage at the strait which sepa- | 
 
 rates Asia from America—a strait, the existence of which is now confirmed by the» 
 
 expeditions of the Russians, and by the researches of Captain Cook, and which lies in 
 
 about the 176th degree of longitude. I proposed this problem to myself, in conse- 
 
 quence of a new attempt which was about to be undertaken in France, by M. de 
 
 Bougainville. I have heard that it was proposed to a celebrated astronomer, a 
 member of the Royal Academy of Sciences: I do not know what answer he returned; — 
 
 but my solution is as follows :— 
 
 Had I been the navigator entrusted with the expedition, that I might not be taken | 
 
 by surprise, I should have provided myself with two or three good time-keepers, all | 
 
 exactly set to the time at the port of departure, which I suppose to be Brest. 
 
 Let us now suppose that the sea was found open, and that [had arrived at the north - 
 
 pole. I,shall suppose also that my compass had become entirely useless; but that I 
 
 had the sun on the horizon, which is the case in summer, and therefore such an expe- 
 
 dition ought never to be undertaken but at that period, during which the sun is visible 
 in those regions for several months. It is evident that by consulting my time-keepers, 
 _ the moment when they indicated noon would be that when the sun was on the meri- 
 dian of Brest; consequeutly had I been desirous of returning thither, nothing would 
 
 * There is no reason to believe that the compass would have no directive power at the terres- 
 trial pole, as the magnetic pole and that of the earth are quite distinct. 
 
HOW TO SAIL FROM THE POLES, 551 
 
 have been necessary but to turn the ship’s head towards the sun, and to steer on that 
 course in sucha manner, as to have the sun at the end of an hour 15 degrees to the 
 starboard ; at the end of two hours 30 degrees, &c. It may be readily conceived that 
 by these means, though destitute of a compass, I should have kept my vessel pretty 
 exactly on the line of the determinate meridian. 
 
 Now, if the meridian, on which it was necessary I should steer, had been distant 
 from that of the place of departure 176°, as seems to be the case with that of the 
 strait which separates Asia from America, it may be easily seen that I should have 
 had nothing to do but to direct the ship’s head within about 4 degrees of the point 
 diametrically opposite to the sun, when the time-keepers indicated noon; or 
 towards the sun itself when they indicated 16 minutes after midnight, and then 
 to keep.on this course by the method above described; changing every hour the 
 angle formed by the ship’s course with the azimuth passing through the sun. If we 
 
 suppose the mouth of the strait in question to be, in regard to Brest; in the longi- 
 tude already mentioned, it is evident that I should not have failed to enter it. 
 
 But, it is to be observed, that this expedient would be necessary only when very 
 near to the pole: at the distance of ten degrees from it, other means of directing the 
 ship’s course might be employed. We shall not however enlarge farther on this sub- 
 ject ; for it would be of very little use to point out these means, since the latest 
 voyages seem to prove that the arctic pole, at the most favourable seasons, that is to 
 say during the summer of our hemisphere, is surrounded by a covering of ice ten de- 
 
 grees at least in diameter, and which even extends farther towards Asia and 
 America; or, in all probability, adheres to these two continents, except perhaps 
 
 during some excessively hot summers. In a word, I am fully persuaded that the 
 idea of traversing the frozen ocean, in order to proceed to the seas of China and 
 Japan, is a mere chimera; and that if a vessel should even be able to get thither, by 
 steering close along the shores of Asia or America, to the strait above mentioned, 
 
 the voyage would be attended with so many dangers, and require circumstances so 
 
 favourable, that it would be madness to attempt it. What indeed would become of 
 a ship if, retarded by any of the accidents so common in those seas, she should be 
 
 obliged to winter, nearly a whole year, in any port of the almost uninhabited 
 
 northern coast of Asia? What assistance could she expect from the Samoeides, or 
 
 any other of these nations still more barbarous? If the crew remained there, how 
 
 could they secure themselves from the intense cold of these climates? If they quitted 
 
 _ their vessel, to take up their lodging in a close hut, after carrying thither tbeir pro- 
 
 , 
 
 ‘visions, would not the vessel be exposed to the danger of being plundered or burnt ? 
 Such an enterprise would require, that the commercial nation which undertook it, 
 
 should have a port belonging to it in some advantageous situation, that ships obliged 
 to winter in those cold regions might have a convenient place of shelter. But what 
 appearance is there that Russia, the sole mistress of these countries, will ever con- 
 sent to such a measure; especially as the Russian government so long concealed the 
 
 - information it’had obtained in regard to the strait above mentioned ? 
 
NRG 
 
 552 ARCHITECTURE. 
 
 vee ps 
 
 PART NINTH. | 
 
 CONTAINING SOME CURIOUS PARTICULARS IN REGARD TO 
 ARCHITECTURE. 
 
 ARCHITECTURE may be considered under two points of view. According to the - 
 first, it is an art, the object of which is to unite utility and grace; to give to an | 
 edifice that form fittest for the purpose to which it is destined, and at the same time 
 the most agreeable by its proportions; to strike the beholder by magnitude or ex- | 
 tent, and to please by the harmony of the different parts and their relation to each 
 other: the more an architect succeeds in uniting all these requisites, the more he | 
 will be entitled to rank among the eminent men who have distinguished themselves 
 in this art. 
 
 But it is not under this point of view that we here consider it; we shall confine 
 ourselves to the geometrical and mechanical part of Architecture, as it presents us 
 with several curious and useful questions, which we shall lay before the reader. 
 
 PROBLEM I. 
 To cut a Tree into a Beam capable of the greatest possible resistance. 
 
 This problem belongs properly to Mechanics; but on account of its use in Archi- | 
 tecture, we thought it might be proper to give it a place here, and to discuss it both | 
 geometrically and philosophically. ‘We shall first examine it under the former point 
 of view. ! 
 
 Galileo, who first undertook to apply geometry to the resistance of solids, has | 
 determined on a very ingenious train of reasoning, that when a body is placed hori- 
 zontally, and fixed by one of its extremities, as is the case with a quadrangular beam | 
 projecting from a wall, if a weight be suspended from the other extremity, in order ' 
 to break it, the resistance which it opposes is in the compound ratio of the horizontal ' 
 dimension and the square of the vertical dimension. But this would be more 
 correctly true, if the matter of the body were of a homogeneous and inflexible 
 texture. | 
 _ It has been shewn also, that if a beam is supported at both extremities, and if a 
 weight, tending to break it, be suspended from the middle, the resistance it opposes | 
 is in the ratio of the product of the breadth and square of the depth, divided by half | 
 the length. 
 
 To solve therefore the proposed problem, we must cut from the trans of the 
 tree a beam of such dimensions, that the product of the — 
 square of the one by the other shall be the greatest possible. © 
 
 Let a B then (Fig. 1.), be the diameter of the circle, which 
 is the section of the trunk; the question is, to inscribe in 
 this circle a rectangle, as AEBF, of such a nature, that the 
 square of one of its sides a F, multiplied. by the other side 
 A&E, shall give the greatest product. But it can be proved — 
 that, for this purpose, we must first take, in the diameter AB, — 
 the part aD equal to a third of it, and raise the perpendicular 
 
 Fig. 1. 
 
THE STRONGEST BEAMS, 553 
 
 DE, till it meet the circumference in £: if BE and Ea be then drawn, and also aF 
 and fr B parallel to them, we shall haye the rectangle AEBF, of such a nature, that 
 the product of the square of a F by BF, will be greater than that given by any other 
 
 rectangle inscribed in the same circle. If a beam of these dimensions, cut from the 
 
 proposed trunk, be placed in such a manner, that its greatest breadth aF shall be, 
 
 perpendicular to the horizon, it will present more resistance than any other that . 
 
 could be cut from the same trunk ; and even than a square beam cut from it, though 
 the latter would contain more matter. 
 
 Remark.—Such would be the solution of the problem, if the suppositions from 
 which Galileo deduced his principles, in regard to the resistance of solids, were 
 altogether correct. He indeed supposes that the matter of the body to be broken 
 
 Is perfectly homogeneous, or composed of parallel fibres, equally distributed around 
 . the axis, and presenting an equal resistance to rupture; but this is not entirely the 
 _ ease with a beam cut from the trunk of a tree which has been squared. 
 
 By examining the manner in which vegetation takes place, it has been found, that 
 the ligneous coats of a tree, formed by its annual growth, are almost concentric; and 
 that they are like so many hollow cylinders, thrust into each other, and united by a 
 
 _ kind of medullary substance, which presents little resistance: it is therefore these 
 
 ligneous cylinders chiefly, and almost wholly, which oppose resistance to the force 
 that tends to break them. x 
 But, what takes place when the trunk of a tree is squared, in order that it may be 
 
 _ converted intoa beam? It is evident—and it will be rendered more sensible by 
 
 inspecting Fig. 2—that all the ligneous cylinders, greater than 
 the circle inscribed in the square, which is the section of the 
 beam, are cut off on the sides; and therefore the whole resist- 
 ance almost arises from the cylindric trunk inscribed in the solid 
 part of the beam. The portions of the cylindric coats which 
 are towards the angles, add indeed a little strength to that ecy- 
 linder, for they cannot fail of opposing some resistance to the 
 breaking force ; but it is much less than if the ligneous cylinder 
 
 were entire. In the state in which they are they oppose only a moderate effort to 
 
 flection, and even to rupture. For this reason, there is no comparison between the 
 strength of a joist made of a small tree, and that of another which has been sawn, 
 
 or cut with several others from the same beam or block. The latter is generally 
 _ Weak, and so liable to break, that joists, and other timber of this kind, ought to be 
 
 carefully rejected from all wooden work which has to support any considerable 
 
 , Weight. 
 
 We shall here add, that these ligneous and concentric cylinders are not all of equal 
 strength. The coats nearest the centre, being the oldest, are also the hardest; while, 
 according-to theory, the absolute resistance is supposed to be uniform throughout. ~ 
 
 It needs therefore excite no surprise, that experience should not entirely confirm, 
 
 _and even that it should sometimes oppose, the result of theory. Hence we are under 
 
 considerable obligations to Duhamel and Buffon, for having subjected the resistance 
 of timber to experiments; as it is of great importance in architecture to know the 
 
 strength of the beams employed, in order that larger and more timber than is neces- 
 | Sary may not be used. 
 
 * a 
 
 But notwithstanding what has been said, it is very probable that the beam capable 
 
 of the greatest resistance, which can be cut from the trunk of a tree, is not the — 
 
 Square beam; for the following experiments made by Duhamel seem to prove, the 
 »size being the same, that the beam which has more depth in proportion than breadth, 
 yhen the depth is placed vertically, presents so much more resistance ; and even 
 
 “ahr 
 
554 ARCHITECTURE. 
 
 without deviating very much from the law proposed by Galileo, viz., the compound 
 ratio of the square of the vertical dimension and that of the breadth. 
 
 Duhamel indeed caused to be broken twenty square bars of the same volume, to 
 determine what form of dressing would render them capable of the greatest resist- 
 ance. They all had 100 square lines of base, and four of each sort were employed of 
 the different dimensions, to compose the same area. 
 
 The first four, which were 10 lines in every direction, sustained a weight of 13] 
 pounds. 
 
 Four others, which were 12 lines in one direction and 8} in another, sustained each 
 154 pounds. ‘The above law would give 157 pounds. 
 
 The next four, which were 14 lines in height and 74 in breadth, supported each 
 164 pounds. Calculation would give 183 pounds. 
 
 Four more, which were 16 lines in height and 6! in breadth, sustained each 180 
 pounds. According to calculation they ought to have supported 209 pounds. 
 
 The last four, which were 18 lines in height and 53 in breadth, sustained each 243 
 pounds. Calculation would have given only 233 pounds. It is very singular that in 
 this case calculation should give less than experience; while in the other cases the 
 result was contrary. 
 
 Buffon began experiments on a larger scale, in regard to the resistance of timber, an 
 account of which may be seen inthe Memoirs of the Academy of Sciences for the — 
 year 1741. It is to be regretted that he did not pursue this subject, on which no one 
 could have thrown more light. It appears to result from these experiments, that the 
 resistance increases less than in the square of the vertical dimension, and decreases I in 
 a ratio somewhat greater than the inverse of the length. 
 
 In short, the result of the whole is, that to solve the proposed problem, it would 
 be necessary to have physical data of which we are not yet in possession; that the. 
 beam capable of the greatest resistance, that can be cut from the trunk of a tree, is 
 not a square beam; and that in general many researches are to be made respecting | 
 the lightening of carpenters’ work, which often contains forests of timber in a great. 
 part useless. 
 
 Our readers are referred to Barlow on the Strength of Timber, for a series of valu- | 
 able experiments and important deductions on this subject. | 
 
 PROBLEM II. ; 
 
 Of the most perfect form of an Arch. Properties of the Catenarian Curve, and their 
 application to the solution of this problem. 
 The most perfect arch, no doubt, would be that, the voussoirs of which being. 
 exceedingly thin, and even smooth on the sides in contact, should maintain theme , 
 selves in complete equilibrium. It may easily be perceived that, in consequence of. 
 this form, very light materials might be employed; and we shall shew also that its 
 push or thrust on the piers would be much less than that of any other arch of the 
 same height, constructed on the same piers. 
 This property and this advantage are found ina curve well known to geometri- | 
 cians under the name of the Catenarian, and called by the French la Chainette. This 
 name has been given to it because it represents the curve as-. 
 
 Fig. 3. sumed by achain acpz, Fig. 3, composed of an indefinite number | 
 A B of infinitely small and perfectly equal links, or by a rope per- 
 \ fectly uniform and exceedingly flexible, when suspended freely 
 
 by its two extremities. 
 
 The determination of this curve was one of those problems 
 which Leibnitz and Bernoulli proposed towards the end of the 
 17th century, in order to shew the superiority of their caleu- | 
 
OF ARCHES. 555 
 
 lation over the common analysis ; which indeed is hardly sufficient to solve a problem 
 of this nature. But we must here confine ourselves to a few of the properties of the 
 curve in question. 
 
 If the curve a Bc, (Figs. 3., 4.), be disposed in such a manner, 
 
 Fig .4. that its summit shall be uppermost; andif a multitude of globes 
 
 be so arranged, that their centre shall be in the circumference 
 of this curve, they will all remain motionless and in equilibrium : 
 much more will this equilibrium subsist, if, instead of balls, we 
 substitute thin voussoirs, the joints of which will pass through 
 the points in contact, as they will touch each other in a surface 
 ‘far more extensive than the points in which we suppose the balls to touch each 
 
 other. 
 
 Now to describe a curve of this kind is attended with no 
 Fig. 5. difficulty ; for let us suppose that the space a B, comprehended 
 between the two piers a and B of Fig. 5., is to be covered 
 with an arch, and that the elevation of this arch is to be sc. 
 _ Trace out on a walla horizontal linea db, Fig. 6, equal to aB; 
 then from the middle of a 6 draw e perpendicular to it, and 
 equal to sc; and having fixed to the points, a and 5, the 
 two ends of a very flexible rope or chain, formed of small links 
 perfectly equal and very moveable, so that when suspended 
 freely it shall pass through the point c, mark out onthe wall a 
 sufficient number of the points or eyes of these links, without 
 deranging them: the curve described through these points 
 will be the one required; and nothing will be easier than to 
 Fig. 6. trace out the plan of it on the wall as represented by a cB, 
 a i Fig. 5. 
 
 a, Then trace out at an equal distance, both without and within 
 \ A cB, two curves, which will represent the extrados and intra- 
 dos of the arch to be constructed. Divide the curve a c into 
 
 t 
 
 any number of equal parts at pleasure; and through these 
 
 points of division draw lines perpendicular to the curve, which 
 may be done mechanically with sufficient exactness for practice : 
 these perpendiculars will divide the arch into voussoirs; and 
 you willthus havea plan of thearch described on the wall. From 
 this plan it will be easy to construct the pannel or model boards for cutting the stones 
 according to the proper form. If these operations are accurately performed, were 
 ‘the line az a hundred feet, and the height s c still more, the voussoirs of this arch 
 
 e 
 
 would maintain themselves in equilibrium, however small the part in contact might . 
 
 ‘be: for, mathematically speaking, they ought to maintain themselves in equilibrium 
 ) even if the surfaces in contact were highly polished and slippery: consequently the 
 equilibrium will subsist much more when cut in the usual manner. 
 Now to find the force with which an arch of this kind pushes against its piers, or 
 tends to overturn them, draw a tangent to the point a the commencement of the 
 curve, Fig. 6., which may be done mechanically by assuming two points very near the 
 curve, and drawing through these points a line which will meet in ¢, the axis s ¢ con- 
 tinued.* This tangent being given, it can be demonstrated in mechanics that the 
 whole weight of the semi-arch a c, is to the weight or force with which it pushes the 
 
 * This tangent may be drawn geometrically in the following manner : make use of this proportion : 
 as2scistouc + $c,soisac—sc toa fourth term, which we shall calle w, if you then say: 
 as ¢ uw is to a c, sois as to s t, the point ¢ will be that where the tangent to the point a will meet 
 
 ) the axis. 
 
556 ARCHITECTURE. 
 
 pier in a horizontal direction, asst is tosa. On the other hand, we must add to the 
 weight of the pier the force with which the semi-arch presses upon it perpendicu- 
 larly; that is to say, the absolute weight of the semi-arch: in this manner the thick- 
 ness of the pier may be found, by the following arithmetical operation, which we shall 
 here substitute for a geometrical construction, as the latter might appear too complex 
 to the generality of our readers. 
 
 We shall suppose the span a B to be 60 feet, Fig. 5. and 6., and consequently 
 A 8s will be 30 feet; we suppose s c to be 30 feet also, in order that we may compare 
 the push or thrust of this arch with that of a semi-circular one. Let the length ae 
 be 45 feet 1 inch 8 lines,* and the breadth of the arch 1 foot; for, on account of the 
 reasons above mentioned, it may be constructed with safety in this light manner. If 
 the height of the pier then be 40 feet, required the thickness it ought to have in 
 order to overcome the thrust of the arch. 
 
 It will be found, on this supposition, that the tangent of the point a, the com- 
 mencement of the catenarian curve or arch, will meet its axis s ¢ produced, in a point 
 ¢ so situated, that s ¢ will be 71), feet. Ifsa be then divided by s t, we shall have 
 the number 399, which must be reserved, and which we shall call n. 
 
 Now take a fourth proportional to the height of the pier, the length a ¢ of the 
 semi-arch, and to its thickness, and let the half of this fourth proportional, which in 
 this is ~, be called p. 
 
 Then multiply a c by the thickness 1, and the product by double the reserved 
 number N, which will give 372; to this number add the square of p, and extract 
 the square root of the sum, which will be 61: if the above number p be taken from 
 this root, we shall have 5 feet 7 inches, for the breadth of the pier.t The pier being 
 constructed of materials homogeneous to the arch, it is certain that it will resist the 
 force with which the latter tends to throw it down; for, to simplify the calculation, 
 we have made a supposition which is not altogether exact, but which increases in some 
 measure the breadth of the pier. ‘This observation we think necessary, that we may 
 not be accused of committing a wilful error. | 
 
 If this breadth be compared with that necessary to support a circular arch forming — 
 a complete semi-circle, the latter will be found to be much greater, for it ought to be 
 near 8 feet. ye 
 
 The push of an arch constructed on a circular foundation, such as the arch of a | 
 dome, being only about one half that exerted on its piers by a vault arch of the same | 
 thickness, it thence follows that, on the above supposition, the side of such a dome 
 would require only 334 inchesin thickness. But it can bedemonstrated, even by the figure 
 of the catenarian curve, that the arch may be but about a foot in thickness. Hence 
 we may see howill founded was the objection made to the architect of the chureh of 
 Saint Genevieve, of its being impossible to construct, on the base he employed, the ; 
 dome which he projected ; for he could have done it even if we suppose his construc- 
 tion to be such as the author of the objection traced out. to him, according to the 
 precepts of Fontana, or rather according to the mode which that architect followed in 
 the construction of his domes. What then would have been the case, had the 
 architect alluded to, instead of first constructing a cylinder of 36 feet, which it how- 
 ever appears was never his design, made his arch rise immediately in a catenarian 
 curve, above the circular cornice, which crowned his pendentives, or above a socle | 
 of small height? It is evident that the push of this arch would have been mueh : 
 less; and it would not be surprising if it should be found by calculation that his | 
 piers would have been capable of sustaining the arch raised above them, even suppos- © 
 
 * It is found by calculation that this ought to be the length. { 
 + This determination of the breadth or thickness of the pier, if not mathematically correct, may | 
 at least be considered as sufficiently near for practical purposes. 
 
 ee cdi 
 ii 
 — 
 
, 
 
 DOME WITHOUT PUSH. 557 
 
 ing them insulated, and not allowing them any support from the re-entering angles 
 of the church, which might have been made to rest against them. 
 
 We shall conclude with observing, that if it were required to find, by prin- 
 ciples similar to those which gave rise to the discovery of the catenarian curve, 
 the most advantageous form for a dome, the problem would be exceedingly diffi- 
 cult; for if we suppose this arch divided into small sectors, it will be evident that 
 the weight of the voussoirs is not equal, and that their relation depends even 
 ou the form given to the arch. What has been here said, ought therefore to be 
 considered only as an approximation of the most advantageous figure which the 
 arch in that case ought to have. 
 
 PROBLEM III. 
 How to construct a hemispherical arch, or what the French Architects call an arch 
 en cul-de-four, which shall have no thrust on the piers. 
 
 The dispute carried on, some years ago, with a considerable degree of warmth, 
 respecting the possibility of executing the cupola of the new church of Saint Gene- 
 vieve, gave me an opportunity of examining whether, even on the supposition that 
 the supporters would be necessarily too weak to resist the thrust of an arch of 63 
 
 feet in diameter, there might not be found some resources to render the construction 
 
 of the cupola possible. I soon found that it was possible, by means of a very simple 
 
 artifice, to construct a hemispherical arch, or an arch in the form of a semi-spheroid, 
 _ which should have no sort of thrust on its piers, or on the cylindric tower by which 
 
 _it is supported. This will be readily conceived from the following reasoning and 
 _ illustration. 
 
 It is evident that a hemispherical arch would exert no thrust on its support, if the 
 first row were of one piece. But though this is impossible, the deficiency may be 
 supplied, and such an arrangement may be made, that not only the first row, but that 
 
 several of those above it, shall be disposed in such a manner, that their voussoirs 
 
 can have no movement capable of disjoining them, as we shall here shew. The he- 
 
 _mispherical arch will then exert no kind of thrust on its supporters; so that it may 
 not only be sustained by the lightest cylindric pier, but even by simple columns, 
 
 which would furnish the means of rearing a work very remarkable on account of its 
 
 construction. Let us see, then, how the voussoirs of any row can be connected in 
 such a manner as to have no motion tending to make them recede from the centre. 
 
 There are several methods of accomplishing this object. 
 Ist. Let a and zg (Fig. 7. No.2.) be two conti- 
 Fig. 7. No.1. guous voussoirs, which we shall suppose to be three 
 feet in length, and a foot anda half in breadth. 
 Cut out on the contiguous sides two cavities in the 
 form of a dove-tail, four inches in depth, with an 
 oe ios aperture of the same extent at ab; five or six 
 LS At inches in length, and as much in breadth at cd. 
 No. 2. This cavity will serve to receive a double key ot 
 cast iron, as appears In the same Fig. No. ] ; or even 
 of common forged iron, which will be still more secure, as the latter is not so brittle 
 as the former. These two voussoirs will thus be connected together in such a man- 
 
 "ner, that they cannot be separated without breaking the dove-tail at its re-entering 
 angle; but as each of its dimensions in this place will be four inches, it may be easily 
 
 seen that an immense force would be required to produce that effect; for we are 
 taught, by well known experiments on the strength of iron, that it requires a force 
 of 4500 pounds to break a bar of forged iron an inch square by the arm of a lever 
 
 _of 6 inches: consequently 288000 would be necessary to break a bar of 16 square 
 _inehes, like that in question. Hence there is reason to conclude that these youssoirs 
 
Se 
 
 558 ARCHITECTURE. « 
 
 will be connected together by a force of 288000 pounds, and as they will never ex- 
 
 perience an effort to disjoin them nearly so great, as might easily be proved by cal- 
 
 culation, it follows that they may be considered as one piece. 
 
 They might even be still farther strengthened, in a very considerable degree: for 
 the height of these dove-tails might be made double, and a cavity might be cut in— 
 the middle of the bed of the upper voussoir, fit to receive it entirely: the dove-tail | 
 could not then be broken without breaking the upper voussoir also. But it may 
 
 easily seen that, to produce this effect, an immense force would be required. 
 
 538 
 
 ' 
 
 | 
 
 2d. But as some persons may condemn the use of iron in works of this kind, we 
 shall propose another method, not attended with the same inconvenience, if it really 
 
 be one ;* and in which nothing is employed but stone combined with stone. 
 Let a and B (Fig. 8.) be two contiguous voussoirs 
 
 Fig. 8. No.1. of the first row, and c the inverted voussoir of the 
 
 upper next row, which ought to cover the joining. © 
 Each of the two former voussoirs being divided into | 
 two, cut out in the middle of each half a hemisphe- 
 rical cavity, half a foot in diameter ; then take with 
 
 great exactness the distance of the centres of the 
 cavities a and c, which are in two contiguous vous- | 
 No. 2. soirs, and by these means cut out two similar cavi-, 
 
 ties in the lower bed of the voussoir, which is to be. 
 placed in connection on the preceding. Then fill the cavities a andc with two globes 
 of very hard marble, and place the upper voussvir in such a manner that these two. 
 globes shall fit exactly into the cavities of its lower bed. If this operation be dex. , 
 terously performed throughout the whole range of the first, second, and third rows, | 
 it may be easily perceived, that all these voussoirs will form together one solid body, | 
 
 the parts of which cannot be separated; for the two voussoirs a and B cannot be 
 disunited without breaking either the balls'of marble which connect them with the 
 
 upper voussoir, or breaking the upper voussoir through the middle. But even if we | 
 
 suppose this effect, which could not be produced withont a force almost inconceiy- 
 able, or at least far superior to the action of the arch, the two halves of the broken 
 
 voussoir being themselves sustained in a similar manner by the superior voussoirs, no | 
 
 tendency to separate from each other could thence result: the three rows therefore 
 
 of the arch wouJd form only one piece, and there would be no thrust. It will be 
 
 sufficient if the base of this arch have such a thickness as to prevent it from being 
 crushed by its absolute weight; and a very moderate thickness, if the materials be 
 gocd, will answer this purpose. 
 
 ) 
 
 } 
 | 
 
 We think we have proved therefore, by these two methods, that a hemispherical 
 
 arch might be constructed without any thrust on its supporters; consequently if we 
 even suppose that the architect of Saint Genevieve had adopted the form of Fon- 
 
 tana’s domes, and had begun by raising on his pendentives a tower of about 36 feet 
 
 in height, to be crowned by a hemispherical dome, it would not have been impos- 
 sible to give it a solid construction. 
 
 PROBLEM IV. 
 In what manner the thrust of arches may be considerably diminished. 
 
 Architects, in our opinion, have not considered with sufficient attention the re- | 
 
 . * All architects, indeed, are not so nice in their choice of materials ; but it appears to us that the 
 
 frequent use of iron for strengthening buildings is subject to much inconvenience and danger. We . 
 
 at least wish that public monuments were constructed without it; for if they can support themselves 
 without iron, it is needless: if iron is essential to strength, it will certainly be consumed in the 
 course of time by rust, and the edifice will then tumble to pieces, or be greatly injured. ‘The use 
 of iron then in this case is attended with bad consequences. 
 
TO DIMINISH THE PUSH.—-DOUBLE STAIRCASE. 559° 
 
 sources afforded by mechanics, for diminishing, on many occasions, the thrust of 
 arches. We shall therefore present the reader with some observations on that subject. 
 
 When the manner in which an arch tends to overturn its piers is analyzed, it ap- 
 pears that the arch necessarily divides itself somewhere in its flanks, and that the 
 upper part acts in the form cf a wedge or a Jever on the remainder of the arch, and 
 on the pier, which are supposed to form one body. This consideration then suggests, - 
 that to diminish the thrust of the arch, or increase the stability of the pier, the 
 commencement of the flanks ought to be loaded; and that the thickness of the 
 youssoirs near the key ought to be considerably lessened: in short, to make the arch, 
 instead of having a uniform thickness throughout its whole extent, ta be very thick 
 at its origin, and at the key to be no thicker than what is necessary to resist the 
 pressure of the flanks, It may be easily perceived, that as by this method a part 
 of the force which acts to overturn it, is thrown upon that which resists being over- 
 
 turned, the latter will gain a great advantage over the former. 
 
 It is to arches in the form of a dome, in particular, that this consideration is appli- 
 cable; and not only might this method be employed, but also heterogeneity of ma- 
 terials. For this purpose, let us suppose ourselves in the place of the architect 
 of Saint Genevieve, and that it is necessary to construct his dome by first raising 
 
 a round tower 36 feet in height, to be afterwards crowned by an arch, which we 
 
 shall suppose to be hemispherical, though he was allowed to make it a little more 
 
 elevated than that form, in order that it might appear hemispherical when seen 
 at a moderate distance. It is found that giving to this arch the uniform thickness 
 of a foot and a half, the tower ought to be 43 feet in thickness at the utmost, which 
 added to some necessary enlargement at the foundation, for the sake of solidity, 
 
 exceeds the breadth of the basis which might be given to it in a part of its cir- 
 
 -cumference. But, according to the above considerations, what would prevent this 
 
 tower, and the first rows, even as far as towards the middle of the flanks of the 
 arch, from being constructed of materials much more ponderous than the rest of the 
 arch? For we are acquainted with some stones, such as hard and coarse marble, 
 which weigh 230 pounds the cubic foot; while the Saint Leu, in the neighbourhood 
 of Paris, weighs only 132, and brick much less. Instead of giving to the arch the 
 uniform thickness of a foot and a half» why might it not be made three feet at 
 
 the spring, and only eight inches towards the summit? But by making the fol- 
 
 lowing suppositions, namely that the tower and the first rows of the arch, as far 
 
 | as the middle of the flanks, are of the hard stone in the neighbourhood of Paris, 
 which weighs 170 pounds the cubic foot, and the rest of brick which weighs only 
 
 130; and that the arch at its spring, as far as the middle, is 2} feet in thickness, 
 
 ' and only 8 inches towards the summit; we have found that the tower in question 
 - ought to be only 1 foot 83 inches in thickness, to be in equilibrium with the thrust 
 ' of the arch. If this tower therefore were made 3 feet in thickness, it is evident 
 
 to the most timid architect, that it would be more than sufficient to counteract 
 
 | every effect of lateral pressure ; and it would be still more so were it made 34 feet in 
 
 thickness to a certain height, such as that of 9 feet, for example, and then 3 feet or 
 2 feet 9 inches to the commencement of the arch; asa pier is strengthened by 
 throwing to its lower part a portion of its thickness, instead of making it equally 
 
 _ thick throughout; since the point on which it ought to turn, in order to be thrown 
 _ down, is remoyed farther back. 
 
 But this is enough on a subject which we have introduced here occasionally. 
 
 PROBLEM V. 
 
 Two persons, who are neighbours, have each a small piece of ground, on which they 
 intend to build ; but, in order to gain as much room as possible, they agree to con- 
 struct a stair common to both houses, and of such a nature, that the inhabitants shall 
 
a4 i 
 SS 
 =| 
 
 560: ARCHITECTURE. 
 
 have nothing in common except the entrance and the vestibule. What method must 
 the architect pursue to carry this plan into execution 2? 
 
 The stairs here proposed may be constructed in the following manner, of whiclh 
 
 there are some examples. 
 
 Let Fig. 9. No. 1. be a plan of the stairs, the form. 
 
 Fig. 9. No. 1. of which is of such a nature as to ascend, without 
 
 j being too steep, from the lower to the first story in| 
 
 2 {ll one revolution, or somewhat less. In a common vese 
 tibule a, the entrance to which is through a common | 
 door P, construct on the right at B the commencement | 
 of the ramp intended for the house on the right ; and | 
 make it circulate from right to left, as far asa landing: 
 place, which must be constructed above the landing 
 place B: the stairs may then be continued in the same 
 
 manner to a second or third story. 
 
 The commencement of the other stair-case must. 
 be on the side diametrically opposite at c; and must. 
 circulate in the same direction, in order to arrive, 
 after one revolution, at a landing place forming the 
 entrance to the first story of the house on the left ; | 
 so that if the inside railing of these stair-cases be 
 open, as it may be easily made, those who ascend or 
 descend, by one of them, can see those who are on 
 the other, without having any communication but 
 by the common vestibule a, and the door of entrance. | 
 A ‘section of this double stair-case is seen Fig. 9, | 
 No. 2. 
 
 At the castle of Chambord there is a stair-case | 
 nearly of this form, which serves for the whole | 
 building. For, as this edifice consists of four grand | 
 vestibules or immense saloons, placed opposite to 
 each other; in the form of a Greek cross, and into | 
 which all the apartments open, Serlio, the architect, constructed the stair-case in 
 the centre of this cross; and, by means’of a double ramp, those who enter from the | 
 south vestibule on the ground floor, and who front the stair-case before them, arrive’ 
 after one revolution at the southern vestibule or saloon of the first Ree and vice . 
 versd. | 
 
 But though the form of this stair-case is very ingenious, it has some very great | 
 defects, which might have been easily avoided. Ist. The entrance of the stair-case, | 
 instead of being directly opposite to the middle of each saloon, is a little on one. 
 side. 2d. There is no landing place before the door which forms the entrance into | 
 this story. 3d. The interior railing, which might have been light, and almost en- 
 tirely open, has only a very small number of apertures. 
 
 If the ground would admit, the same artifice might be employed to construct a 
 stair-case with four ramps, all separate from each other, in order to ascend to four 
 different apartments. The plan of a stair of this kind, which is said to have been 
 constructed at Chambord, may be seen in Palladio. That of Serlio, on account | 
 of the four galleries to be entered, would no doubt have been much more beautiful, 
 had it been built on the same plan; but we can assert that the stair of Chambord _ | 
 has only two ramps as above described. 
 
 Remark.—Some stairs are distinguished by another peculiarity, namely, the bold-, 
 ness of their construction. Such are those stairs in the form of a screw, the helix of | 
 which formsa spiral entirely suspended ; so that there remains in the middle a vacuity | 
 
 Wf = (Qu 
 
 i: 
 hia...” 
 
FLOOR WITH SHORT JOISTS. 561 
 
 of greater or less extent. This bold construction depends on the manner in which 
 the steps are cut, and their being fixed by one end in the wall which on one side sup- 
 plies the place of a rail. A full account of the mechanism of them may be seen in 
 most works on architecture, 
 
 PROBLEM VI. 
 
 To construct a floor with joists, the length of which is little more than the half of that 
 necessary to reach from the one wall to the other. 
 
 Let the square a B c D (Fig. 10.), be the frame of the floor which is to be covered 
 with joists a little more in length than the half of one of the sides a 8. On the sides 
 of this square assume the parts a G, BI, C L, and pk, equal to the given length of the 
 
 joists, which must be arranged as seen in the figure ; that is, first 
 
 Fig. 10. place E F, and introduce below it Gc H, with its end H resting on 
 
 G B 1K; and let kK, the end of 1 k, rest upon L M, the end of which 
 
 Alialaily ids) M must be made to rest upon the first joist EF. It may be easily 
 
 proved, that in this position these joists will mutually support 
 each other, without falling. 
 
 It is almost needless to observe that the end of each joist 
 must be cut in such a manner as to enter a notch made for it in 
 the joist on which it rests, and into which it ought to be well 
 fitted. However, as a notch cut into a joist must lessen its 
 strength, it would perhaps be better to make the end of each joist rest on an iron 
 stirrup of a sufficient size, and affixed to them in a secure manner. 
 
 But it is not necessary that the joists should be a little longer than half the breadth of 
 the frame to be covered: a floor may be constructed with pieces of wood much 
 shorter, if they be cut and arranged in a proper manner. 
 
 Let us suppose, for example, that an area of 12 feet square is to be covered, and 
 that the pieces of wood, intended to support the floor, are only 2 feet in length. Cut 
 the extremities of one of these pieces of wood in an 
 oblique form, or into a bevel, as represented by the 
 section ac DorBEF (Fig. 11.); and in the middle 
 of the same piece, form on each side a notch, for re- 
 ceiving the end of another piece cut in like manner. 
 If the same operation be performed on all the rest, 
 they may then be arranged as seen in the figure; a 
 bare view of which will give a better idea of the arti- 
 fice here employed, than a long description. The 
 oblong spaces, which remain along the walls, may be 
 filled up with pieces of wood half the length of the 
 former. The scaffolding may then be removed with 
 great safety, for these pieces of wood willform a solid - 
 
 floor, and will mutually support each other, provided none of them is destroyed ; for 
 - it is to be observed that the breaking of one would make the whole fall to pieces. 
 
 Dr. Wallis, at the end of the third volume of his works, gives a great variety of 
 
 _ these combinations, and he says that this invention was employed in some parts of Eng- 
 _ land. But on account of the reasons already mentioned, it is to be considered rather as 
 
 ingenious than useful, and fit only to be adopted when there is a great scarcity of 
 _ timber, and for floors which have very little weight to support. 
 
 Remark.—Instead of pieces of wood, if we suppose stones to be cut in the same 
 manner, it is evident that they would form a flat arch ; but in this case, to avoid the 
 danger of breaking, it would be necessary that they should be at most 2 feet in length, 
 
 _ and of a suitable width and thickness. Anarch of this kind is generally called the 
 
 | 
 
 flat arch of M. Abeille; because it was proposed by that engineer to the Academy of 
 i 2 O . 
 
562 ARCHITECTURE. 
 
 Sciences, in 1699. It is attended with this advantage, that its whole thrust is exerted 
 on the four walls, which serve to support it ; whereas a flat arch, constructed accord-", 
 ing to the usual method, exerts its thrust or push only against two. But this advan. | 
 tage is more than overbalanced by the danger of the whole tumbling to pieces, if. 
 one stone ouly should be deficient. Frezier has treated on this subject at some | 
 length, in his work on cutting stones ; and has shewn how to vary the compartments | 
 of the intrados, or lower part, as well as of the extrados, or upper part, which might | 
 be formed with these arches. But we must here repeat that these things are more 
 curious than useful, or rather that this construction is very dangerous. 
 
 PROBLEM VII. 
 Of suspended Arches, called by the French ‘* Trompesdans l’ Angle.” 
 
 One of the boldest works in masonry, is that kind of arch called, by the French, _ 
 Trompe dans I’ Angle.* Let us suppose a conical arch, as s a F Bs (Fig. 12.) raised | 
 on the plane ofatriangle as B; if from the middle of the base there be drawn two lines | 
 E cand £ D, which in general are parallel to the respective sides s p and sc; and if | 
 upon these be raised two planes D E F andc E F, pers 
 pendicular to the base; these two planes will cut off 
 towards the summits, a part of the arch, as FD SCF, | 
 the half of which c F pc willbe suspended, or pro- , 
 ject beyond the foundation. This truncated part of | 
 the conical arch F D8 c ¥, is.what is called a trompe | 
 dans l’Angle ; because in general it is constructed ina | 
 re-entering angle to support some projecting part of 
 an edifice. For this purpose, on the curvilinear planes 
 c F and DF, there are raised walls which, though 
 suspended, have miieeinnt strength, provided the voussoirs be exactly cut; are long | 
 enough to be inserted in the half which is not suspended ; and provided also that this _ 
 part is properly loaded. i 
 
 Works of this kind are common ;. but the most singular is one at Lyons, which - 
 supports a considerable part of a house situated on the stone-bridge. One cannot 
 see, without some uneasiness, the corner of this edifice, which is three or four stories | 
 in height, project several yards above the river. It is said to be the work of Desar- | 
 gues, a gentleman of the Lyonnese, and an able geometrician, who lived in the time 
 of Descartes. In that case, this work must have stood about 150 years; which 
 seems to prove that this kind of construction has a real and greater solidity than is | 
 commonly supposed. i 
 
 Remark.—If the suspended arch be a right arch, that is to say a portion of a right - 
 cone AS BF; and if the section planes F Ep and F EC be respectively parallel to 
 s cand sv, the curves F c and F D, as is well known, will be parabolas, having their - 
 summit in p, and c & or DE for their axis. We must here take notice of a geome- | 
 trical curiosity, which is, that the conical surface Fr cs p F, though a curve, and ' 
 terminated in part by curved lines, is equal to a rectilineal figure; for if p G be | 
 drawn parallel to the axis s £, it can be demonstrated that the conical surface in ques- | 
 - tion, is equal to one and a third of the rectangle of s B, ors Fby EG. 
 
 PROBLEM VIII. = | 
 
 A gentleman has a quadrangular irregular piece of ground, as ABCD, in which he is | 
 desirous of planting a quincunx, in such a manner, that all the rows of trees, whether 
 transversal or diagonal, set be right lines. How must he proceed to carry this plan 
 mnto execution ? 
 
 * These arches are called tvompes, because they have a resemblance to the mouth of a trumpet, 
 
- X 
 i eae 
 , . 
 
 ve 
 ROWS OF TREES. 563 
 
 We shall suppose this quadrilateral to be so irregular, that the opposite sides a B 
 and D c, (Fig. 13.), meet in a point F, and the sides a p and cB in another point E. 
 Continue these sides two and two, to the points of meeting, E and F, which must be 
 joined by a straight line r F; then through the point p draw a line parallel to EF; 
 continue B c and B a till they meet that parallel in m and a, and divide c paud p H 
 into the same number of equal parts, which we shall suppose to be four: if through 
 
 Fig. 13. the points of division in G D, as many 
 
 : j : straight lines be drawn to the point F; 
 
 = <p ai and if straight lines be drawn, in like 
 S LP 
 
 manner, through the points of division 
 in D Hto..the point £, these lines wilk 
 intersect the sides of the quadrilateral, 
 and each other, in points, which will be 
 those where the trees must be planted, 
 in order so solve the problem: 
 
 For the demonstration we might refer 
 to Prob. 24 of Optics, where we have 
 shewn how a quadrilateral, such as 
 
 ABC D, may be the perspective representation of a parallelogram. We shall however 
 here repeat it. 
 
 Through the points p and H draw the lines p a and u 3, inclined to ca at an angle 
 of 45 degrees from right to left ; and through the points c and p two other lines, 
 ‘Gand ve, inclined also 45 degrees to G u, but in a contrary direction: these four 
 lines will necessarily cut each other at right angles, and form a rectangle a bc D,. 
 of which, according to the rules of perspective, the quadrilateral a B c p would be the 
 representation, to an eye situated in the point 1, which divides & F into two equal 
 ‘parts, and is at a distance from the plane of the picture equal to 1 E or 1 F. 
 Let us suppose then that the oblong a6 ¢ D is divided into similar oblongs, by four 
 
 lines parallel to its sides: these lines, if continued till they meet c p and p u, will divide 
 them into the same number of equal parts; and as D c and G aB are the perspective 
 ‘representations of p c and ca J, the lines proceeding from the equal divisions of 
 GD, and ending at the point r, will be the perspective representations of lines 
 ‘parallel to a b or pec. The case will be the same with the lines parallel to the two 
 ‘sides D a and c 6. The small quadrilaterals then formed by these lines cutting 
 ‘each other in the quadrilateral a B c D, will be the perspective pictures of the oblongs 
 ‘into which a bcp is divided. But all the points which are in a straight line in the 
 object, will be in a straight line in the picture; consequently, as the rows of 
 trees planted at the angles of the divisions of the oblong a 6 e p necessarily form 
 
 ‘straight lines, both transversely and diagonally, their places in the quadrilateral aBcD, 
 
 which are the pictures of these angles in the oblong, will also form straight lines in’ 
 (ne same direction ; for, in perspective representations, the pictures of straight lines 
 
 fl 
 i] 
 
 are always straight lines. 
 
 If the opposite sides a b and c D, of the given quadrilateral be very unequal, 
 they must not be divided into the same number of parts; for in that case they would 
 be too unequal, since in a plantation of this kind the quadrilaterals ought to be nearly 
 perfect squares. For example, if one side a 6 be 100 yards, and the other 40, by 
 dividing each of them into 20, the divisions on one side would be 5, and on the other 
 2 yards, which would form figures too oblong. Onthis supposition, it would be much 
 better to divide the first into 16, and the second into 6, whichwould give divisions almost 
 square, namely of 63 yards in one direction, and 62 in the other, but in this case there 
 would be no diagonal row of trees, either in the oblong a bc b, or in the proposed 
 quadrilateral aco. In short, by dividing one of the lines c p or p H into 16 parts, 
 ind the other into 6, all the rows of trees in the irregular figure will be straight lines. 
 
564 ARCHITECTURE. 
 
 To have a real quincunx,* it will be sufficient, after this operation, to draw, in 
 each small quadrilateral of the plantation, two diagonals, and to plant a tree in the. 
 point where they intersect each other: all these new trees will form straight lines. 
 also. 
 
 PROBLEM Ix. 
 
 To construct the frame of a roof, which, without tie-beams, shall have no lateral thrust 
 on the walls on which tt rests. | 
 
 We have seen at Paris, in a garden of the faubourg Saint-Honoré, a small building, 
 in the form of a tent, the walls of which were only a few inches in thickness, and | 
 which were covered by a roof without tie-beams; the whole being lined in the 
 inside, it had the real appearance of a tent. It was used as a summer apartment in, 
 the day-time, and formed a retreat truly delightful. 
 
 What surprised those who had any knowledge of architecture, was, how the roof | 
 of this small edifice would be constructed without tie-beams: for however light it 
 might be, the walls were.so thin, that any common roof must have overturned them. 
 The artifice, said to have been the invention of M. Arnoult, superintendant of the 
 theatres des Menus-Plaisirs, was as follows. ! 
 
 Two rafters, cp and Ep (Fig. 14.), resting on. 
 the two beams AB and ab, were strongly joined’ 
 together at the summit p. From the angles,’ 
 which these two rafters formed at c and £, pro-. 
 ceeded two other pieces of timber, which were 
 well united to the beams at F and g, to the rafters. 
 at H and 1, and also to each other at x, by means’ 
 of a double notch. For the sake of greater se- 
 curity, the pieces cD and ru, and Ep and GI, 
 were bound together by two cross pieces at L and. 
 M. It is evident that these four inclined pieces can have no tendency to separate, or. 
 to exert any lateral thrust on the walls upon which the beams a B and a6 are placed: 
 for they cannot separate without rendering the angle p more obtuse. For this purpose, 
 it would be necessary that ‘the angle K should become more obtuse also; but the 
 junctions at 1 and " oppose any movement of this kind: consequently this frame| 
 work will rest on the beams a B and ab, without separating in any manner, and will 
 exert no lateral pressure against the walls. | 
 
 It is hence evident that this artifice might be of great use in architecture, especially’ 
 when it is required to cover an extensive building, the walls of which are thin, and 
 
 to avoid the disagreeable effect produced by tie-beams, when not concealed from the 
 sight. 
 
 PROBLEM X. 
 
 On measuring arches en cul-de-four, surhaussé, and surbaissé. | 
 The apellation of cul-de-four is applied to vaults on a plan commonly circular, a 
 section of which through the axis is an ellipsis, or 
 as the French architects call it an anse de panier. 
 They differ from hemispherical arches in this, that 
 in the latter the height of the summit above the 
 plane of the base is equal to the radius of that: 
 base ; while in the former this height is greater or 
 less: if greater, the arch is called cul-de-four sur- 
 
 Fig. 16. 
 
 quare ; for the Mad 
 
 * A real quincunx is that where there is a tree in the middle of each s 
 guincunx means five trees in a square, which cannot be arranged otherwise. 
 
F- x 
 e 
 
 ’ 
 a. 
 
 MEASUREMENT OF VAULTS. 565 
 
 baissé. Both these arches are represented Figs. 15. and 16. The first is an arch en 
 ceul-de-four surhaussé ; the second arch en cul-de-four surbaissé. In the language 
 of geometry, the one is an elongated semispheroid, or an arch formed by the circum. 
 -yolution of a semi-ellipse around its greater semi-axis; the other a semi-spheroid 
 formed by the circumvolution of the same semi-ellipsis about its less semi-axis. 
 
 Books of architecture contain, in general, rules so false for measuring the super- 
 ficial content of these arches, that we think it necessary to give here methods more 
 
 correct. Bullet and Savot, for example, say that nothing is necessary but to multiply 
 the circumference of the base by the height; as if the arch to be measured were 
 hemispherical. This is an egregious error, and it is surprising those authors did not 
 observe that if this rule were correct, the superficial content of some arches en 
 cul-de-four surbaissé, would be less than the circle covered by them, which is 
 absurd. 
 
 For let us suppose, by way of example, an arch of a foot in height, ona circle 
 of seven feet diameter: the area of this circle, according to the approximation 
 of Archimedes, will be equal to 383 square feet: but if the circumference, 22 feet, 
 _be multiplied by one foot in height, we should have only 22 square feet; which is not 
 
 two-thirds of the base. In this case, the builder would be cheated of more than 
 two-thirds of what he ought to receive. We shall therefore give rules for measnring 
 such arches, sufficiently correct to be employed in the common purposes of archi- 
 tecture. 
 
 # 
 
 I.—For arches en cul-de-four surhaussé, or the Oblong Spheroid. 
 
 The radius of the base and the height of the arch being given, first make this 
 proportion: As the height is to the radius of the base, so is the latter to a fourth 
 _term, the third of which must be added to two-thirds of the radius of the base. 
 
 _ Then find the circumference corresponding to a radius equal to that sum, and mul- 
 
 _tiply this circumference by the height: the product will be the superficial content or 
 
 ) curve surface nearly. 
 
 _ Example.—Let the height be 10 feet, and the radius of the base 8. Then say, as 
 
 10 is to 8, so is 8 to 62, the third of which is 2%: two-thirds of 8 are 5}, which added 
 
 to 2%, make 7.7, feet, or 7 feet 5 inches 7 lines. j 
 
 » But the Se anterones corresponding to 77 feet radius, or 1433 feet diameter, is 
 
 | 4614 feet, which multiplied by 10 feet, the height of the arch, gives for product 4694 
 square feet, or 52 yards 14 foot. 
 
 | By Bullet’s rule, the superficial content would have been 55 yards 7 feet; the 
 ' difference of which in excess is 3 yards 6 feet, or about a 14th of the whole; and 
 | this in an arch which does not deviate much from a hemisphere: if it deviated more, 
 the error would be considerably greater. 
 
 Il.—For arches en cul-de-four surbaissé, or the Oblate Spheroid. 
 
 | The rule for these arches is nearly the same as the preceding. Find a third pro- 
 portional to the height and the radius of the base; and add two-thirds of it to the 
 radius of the base; then find the circumference corresponding to the sum as a radius, 
 -and multiply it by the height: the product will be the superficial content nearly 
 _of the given arch. 
 Let the radius of the base of an arch en cul-de-four surbaissé be 10 feet, and the 
 height be 8. As 8 is to 10, so is 10 to 123, two-thirds of which are 83; on the 
 other hand, the third of 10 is 34, which added to the former, gives 112 fet 
 But the circumference corresponding to 112 feet radius, or 233 diaretey is 734, 
 which multiplied by the height, that is 8 feet, gives for product 5862 feet, or 65 
 yards 12 foot =the superficial content of the arch. 
 
* 
 ' 
 
 | 
 
 | 
 
 4 
 
 ; 4 
 566 ARCHITECTURE. 
 
 “According to Bullet’s rule the superficial content would have been 55 yards 7 feet | 
 which makes an error in defect of 9 yards 32 feet, or above a6th part of the whole! 
 
 surface. 
 
 Remark.—It would ‘be easy to give, for those who are geometricians, rules still) 
 more exact; as it is well known that the dimensions of prolate spheroids depend 
 on the measurement ofa truncated elliptical or circular segment ; and that of the sur= 
 face of an oblate spheroid, on the measurement of an hyperbolical space; conse- 
 quently the former may be determined by means of a table of sines and circular a | 
 and the other by employing a table of logarithms. 
 
 In regard to the method above given, it is deduced from the same principles ; but 
 by considering a segment of a circle or hyperbola of a moderate extent, asa parabolie 
 area, which when this segment forms but a small part of the space to be measured, is 
 liable only to a very small error: in many cases this consideration supplies practical 
 rules exceedingly convenient. 
 
 Some architects may perhaps ask: Of what advantage is it to be able to ascertain| 
 with precision the superficial content of these domes, as a few yards more or less’ 
 can be of little importance? But it may be said in reply, that for the same reason, 
 accurate measurement in general is of little utility. To such persons it is of no 
 consequence that Archimedes has demonstrated that the surface of a hemisphere is’ 
 equal to that of a cylinder of the same base and height; or, to speak according to 
 their own terms, that the surface of a hemispherical arch is equal to the product of 
 the circumference of the base by the height. If they employ, in regard to the arches 
 in question, rules so erroneous, it is because they consider them as exact, and be, 
 cause they have been taught them by people so ignorant of geometry, as not to be. 
 able to give them Dane ones. 
 
 "PROBLEM XI. 
 To measure Gothic or Cloister Arches, and Arches d’aréte, or Groin Arches. 
 
 It frequently happens that on a square, an oblong, or polygonal space or edifice, an 
 arch vault is raised, consisting of several berceaur or vaults, which commencing at 
 the sides of the base, unite in a common point as a summit, and form in the inal 
 as many re-entering angles or groins, as there are angles in, 
 the figure which serves as a base. These arches are called 
 ares de cloitre, cloister arches. A representation of them is) 
 seen Fig. 17. 
 
 But if the space or edifice, a square for example, be covered 
 with two berceauz or vaults (Fig. 18.) which seem to penetrate. 
 each other, and which form two ridges or re-entering angles, 
 intersecting each other at the summit of the vault, such an arch’ 
 is called an arch d’aréte, or a groin arch. The most remark- 
 able properties of these arches are as follow. | 
 
 Ist. The superficial content of every circular cloister arch, 
 on any base, whether square or polygonal, is exactly double. 
 that of the base, in the same manner as the superficial content of a hemispherical. ’ 
 arch, or arch en cul-de-four or en plein ceintre, is double that of the circular base. 
 
 It may indeed be said that a hemispherical arch is only a cloister arch on a polygon 
 of an infinite number of sides. 
 
 When the superficial content therefore of such an arch is to be measured, nothing: 
 will be necessary but to double the surface of the base, provided the berceauzx be en 
 plein ceintre, or a complete semi-circle ; for if they are greater or less, they will have 
 to the base, the same ratio that an arch en cul-de-four surhaussé, or surbaissé, has to” 
 the circle of its base. 4 
 
WOODEN BRIDGES OF SMALL PIECES. 567 
 
 9d. A cloister arch, and a groin arch on a square, form together the two com- 
 plete berceaux or vaults, raised upon that square. 
 This may be readily seen in Fig. 19. Therefore if from two berceaux or vaults, 
 
 the cloister arch be deducted, there will remain the groin arch, 
 which in this case gives a simple method for measuring groin 
 arches; for if the superficial content of the cloister arch be sub- 
 tracted from the superficial content of the two vaults, the 
 remainder will be that of the groin. 
 
 If the base, for example, be 14 feet in both directions, the 
 circumference of the semi-circle of each will be 22 feet, and the 
 
 ‘superficial content will be 22 by 14, or 308 square feet ; consequently the superficial 
 content of both the berceaux will be 616 square feet. But the interior surface of the 
 Gothic arch is twice the base, or twice 196, that is 392; and if this number be sub- 
 ‘tracted from 616, we shall have 224 square feet, for the superficial content of the 
 
 groin arch. 
 
 3d. If the solid content of the interior of such an arch be required; multiply the 
 base by two thirds of the height. 
 
 This is evident from the reason already given in regard to the superficial content; 
 for arches of this kind, both in regard to their solidity and superficial content, are to 
 a prism of thesame base and height, inthe same ratio as the hemisphere to the cir- 
 cumscribed cylinder. 
 
 4th. The solidity of the space contained by a groin arch ona square or oblong plane 
 is 19 of the solid having the same base and height, supposing the approximate ratio 
 of the diameter of the circle to the circumference to be as 7 to 22. 
 
 This may be easily demonstrated also by observing, that the interior solid of such 
 an arch is equal to the sum of the two vaults or demi-cylinders, minus once the 
 solidity of the cloister arch, which is twice comprehended in this double, and con- 
 sequently ought to be deducted. 
 
 PROBLEM XII. 
 
 - How to construct a wooden bridge of 100 feet and more in length, and of one arch, with 
 
 pieces of timber, none of which shall be more thana few feet in length. 
 
 __ We shall here suppose that the pieces of timber intended for a bridge of this kind 
 _are 12 or 14 inches square, and only about 12 feet in length ; or that particular cir- 
 _ cumstances have prevented rows of piles from being sunk in the bed of the river, to 
 ‘support the beams employed in constructing the work. In what manner must the 
 architect. proceed to build the bridge, notwithstanding these difficulties ? 
 
 | The execution of this plan is not impossible: for it might be accomplished in the 
 
 following manner. 
 
 First trace out, on a large wall, a plan of the projected bridge, 
 
 _ by describing two concentric arches at such a distance from each other, as the length 
 _ of the pieces of timber to be employed will admit; which we shall suppose, for 
 example, to be 10 feet, giving them the form of an arc of 90 degrees from one pier 
 
 to another : 
 
 then divide this are into a certain number of equal parts, in such a man- 
 
 ‘ner that the arc of each shall not exceed 5 or 6 feet. 
 
 On the supposition here made, of the distance of 100 feet 
 between the two piers, an arc of 90 degrees which covers it 
 would be 110 feet in length, and its radius would be 70 feet. 
 Divide then this are into 22 equal parts, of 5 feet each, and 
 with the above pieces of timber, joined together, form a kind 
 of voussoirs, 8 or 10 feet in height, 5 feet in breadth at the in- 
 trados, and 5 feet 8 inches 6 lines at the extrados; for such are 
 the proportions of these arcs, according to the above dimen- 
 sions. Fig. 20 represents one of these voussoirs, which, as it 
 
568 ARCHITECTURE. 
 
 is evident, consists of four principal pieces of strong timber, at least 10 inches square, 
 which meet two and two at the centre of their respective arcs; of three principal, 
 cross bands at each face, as ac, BD, EF, ac, bd, ef, which saat be exceedingly 
 strong, and therefore ought to be 12 or 14 inches in height, and 10 inches in breadth; 
 and lastly, of several lateral bands, between the two faces, to bind them together in 
 different directions, and to prevent them from giving way. A voussoir of this kind. 
 may be about 6 feet in length, that is between the two faces AEFB and ae fb, i) 
 
 An arch must then be formed of these voussoirs, exactly in the same manner as 
 if they were stone, and when they are all arranged in their proper places, the different. 
 pieces may be bound together according to the rules of art, either with pins or 
 braces. Several arches or ribs of this kind must be formed, close to each other,’ 
 according to the intended breadth of the bridge ; and the pieces may be bound to-. 
 gether in the same manner as the first, so as to render the whole firm and secure. 
 By these means we shall have a wooden bridge of one arch, which it would be very 
 difficult to construct in any other manner. 
 
 It now remains to be examined whether these voussoirs will have sufficient’ 
 strength to resist the pressure which they will exert on each other. The following 
 calculation will shew that there can be no doubt of it. 
 
 It appears, from the experiments of Muschenbroeck,* and the theory of the re- 
 sistance of bodies, that a piece of oak 12 inches square, and 5 feet in length, can, 
 sustain in an upright position, without breaking, 264 thousand pounds; hence it 
 follows that a cross band, as ac or EF, 5 feet in length, and 12 inches by 10, can’ 
 support 220000 ; but for the greater certainty we shall reduce this weight to 150000:| 
 therefore, as we have six bands of this length, a few inches more or less, in each 
 of these voussoirs, it is evident that the effort which one of these voussoirs is capa- 
 ble of sustaining, will be at least 900000 pounds. Let us now examine what is the! 
 real effort to be resisted. 
 
 We have found, by calculating, the absolute weight of such a voussoir, and even 
 supposing it to be considerably increased, that it will weigh at most between 7 and 8 
 thousand pounds, or 7500, The weight then resting on one of the piers, most loaded, 
 having 10 voussoirs to support, will be charged only with the weight of 75000 pounds: | 
 a weight however which, on account of the position of the voussoirs, will exert a/ 
 pressure of 115000 pounds; but we shall suppose it to be even 120000. There is; 
 reason therefore to conclude from this calculation, that such a bridge would not only’ 
 have strength to support itself, but also to bear, without any danger of breaking, the’ 
 most ponderous burthens: it even appears that it would not be necessary to make the 
 pieces of timber so strong. 
 
 If the expense of such a bridge be compared with that attending the common’ 
 method, it will perhaps be found to be much less; for one of these voussoirs would 
 contain no more than 140 or 150 square feet of timber, which at the rate of 2s. per. 
 foot, would be only £15; so that the 22 voussoirs, of one course or rib, would cost. 
 £330: consequently, if we suppose the breadth of the bridge to consist of four 
 courses or ribs, the whole would amount only to £1320. It must indeed be allowed, | 
 that to complete such a bridge, other expenses would be required; but the object’ 
 here proposed, was to shew the possibility of constructing it, and not to calculate’ 
 the expense. | 
 
 The idea of such a bridge first occurred to me in consequence of a dangerous’ 
 passage I met with in the province of Cusco, in Peru; where I was obliged to cross 
 a torrent, that flows between two rocks, about 125 feet distant from each other, and: 
 more than 150 feet in height. The inhabitants of the country have constructed” 
 
 * Essais de Physique, vol. i. chap. 11. 
 
} 
 \ FRAMES WITHOUT LATERAL THRUST. 569 
 ‘there a Travita,* where I was in danger of perishing. When I arrived at the next 
 vyillage, I began to reflect on the best means of constructing in this place a wooden 
 bridge, and I contrived the above expedient. I proposed my plan to the Corregidor, 
 Don Jayme Alonzo y Cuniga, a very intelligent man, who, being fond of the French, 
 received me with great politeness. He approved of my idea, and agreed that, at the 
 expense of a thousand piasters, a bridge of 12 feet in breadth, which all Peru would 
 come to see through curiosity, might be constructed in that place. But as I set out 
 jthree days after, I do not know whether this project, with which this worthy man 
 seemed highly pleased, was ever carried into execution. - 
 
 ' It may here be remarked, that it would be easy to arrange the voussoirs of a 
 ‘bridge of this kind, in such a manner that, in case of necessity, any one of them 
 might be taken out, in order to substitute another in its stead: which would afford 
 the means of making all the necessary repairs. 
 
 PROBLEM XIII. 
 
 Is it possible to construct a Plat-band or Frame which shall have no lateral thrust ? 
 
 | It would be of great advantage to be able to execute a work of this kind; for one 
 of the obstacles which architects experience, when they employ columns, arises from 
 
 the thrust of their architraves, which requires that the lateral columns should be 
 
 strengthened by different means. This embarrassment they are particularly liable 
 
 to, when they make detached porches to project before an edifice, like that of 
 
 Sainte-Genevieve: the two frames, that of the face and the side, exert such a push 
 on the angular column or columns, that it is very difficult to secure them; and it is 
 even sometimes necessary to renounce them, if stones cannot be found sufficiently large © 
 to make architraves of one piece, from column to column, at least in the spaces 
 
 mearest the angles. 
 
 These difficulties would be obviated, if frames could be made without any thrust. 
 This we do not think impossible; and we propose the problem to architects in the 
 hope that some of them will be able to solve it. 
 
 PROBLEM XIV. 
 
 Is it a perfection, in the Church of St. Peter at Rome, that those who see it, for the 
 | first time, do not think it so large as it really is; and that it appears of its real 
 magnitude after they have gone over it ? 
 Though we announced, in the beginning of this work, that we meant to exclude 
 from it whatever was mere matter of taste; as the above question is connected with 
 physical and metaphysical reasons, we are of opinion that it may be admitted. ; 
 The impression which the church of St. Peter at Rome makes, on the first view, 
 has been boasted of as a perfection. Every person, as far as we have heard or read, 
 who enters this edifice, for the first time, conceives the extent of it to be far less 
 than itis generally accounted to be by public report. 'To havea just idea of its gran- 
 eur, one must have seen, and in some measure studied, every part of it. 
 _ Before we venture to say any thing decisive on this subject, it may perhaps be of 
 some use to examine the causes of this first impression. In our opinion, it arises 
 from two sources. 
 The first is the small number of principal parts into which this immense edifice is 
 divided ; for, from the entrance to the middle, which constitutes the dome, there are 
 
 * This is an Indian bridge, the very idea of which is enough to make one shudder. A man is 
 placed in a large basket, fastened by a pulley to a rope which is extended from the one side of a torrent 
 to the other. The basket and rope are both constructed of those creeping plants, which the inhabi- 
 tants of America employ in almost all their works. As soon as the man has got into the machine, it 
 is drawn over to the opposite side, by means of a rope fastened tothe pulley. If the rope, used for 
 dragging over the machine, should break, the man must remain suspended for some hours, until 
 means have been found to relieve him from his painful situation. 
 
 . 
 = he 
 " Pe 
 
570 ARCHITECTURE. 
 
 only three lateral arcades. But, though dividing a large mass into many small par 
 tends, in general, to diminish its effect, there is still a medium to be observed ; al, 
 it appears to us that Michael Angelo kept too far below it. 
 
 The second cause of the impression which we here examine, is the excessive si: 
 of the figures and ornaments, which serve as appendages to the principal parts. W 
 can indeed judge of the size of objects beyond our reach, only by comparing the 
 with neighbouring objects, the dimensions of which are familiar to us. But if the 
 objects, the dimensions of which are known, or are nearly given by nature, accompa 
 others to which they have a ratio that approaches too near to equality, it must nece 
 sarily follow that the latter, in the imagination of the spectator, will lose a part: 
 their magnitude. Such isthe case with the church of St. Peter at Rome: the figuri 
 placed in niches, which decorate the spaces between the pillars of the arcades; thoi 
 between the pilasters, and those which ornament the tympana of the lateral arcade 
 are truly gigantic ; but they are human figures; they are besides, for the most par 
 raised very high; consequently they appear less, and make the principal parts whic 
 they accompany to appear less also. 1 
 
 By some people, this illusion is considered to be a master-piece of the art an 
 genius of the celebrated architect, the principal author of this monument. Shall u, 
 be permitted to differ from them? For what is the object which the constructors ‘ 
 this immense edifice had in view; and which will be the aim of all those who rais 
 edifices that exceed the usual measures ? Doubtless to excite astonishment and adm. 
 ration. We are convinced that Michael Angelo would have been much mortifiec 
 had he hearda stranger, just arrived at Rome, and entering St. Peter’s for the firs 
 time, say publicly: ‘‘ This is the church respecting the immensity of which we hay. 
 heard so much: it is a large building, but not so large as generally reported.’’ | 
 
 In our opinion, it would display much more ingenuity to construct an edifice whick 
 though of a moderate size, should immediately excite in the mind the idea of consider, 
 able extent; than to construct an immense one which, on the first view, shoul 
 appear of a moderate size. We do not think that on this subject there can be an 
 difference of opinion. Whatever then may be the perfection, which it must Db’ 
 allowed the church of St. Peter possesses, so far as harmony of proportion, beaut 
 and magnificence of architecture, are concerned, we are of opinion that Michae 
 Angelo missed his aim in regard to the object in question ; and itis probable that h| 
 would have approached much nearer to it, had he employed less gigantic appendages 
 If the children, for example, which support the bénitiers* had been of less size; if th. 
 figures which accompany the arcbivaults of his lateral arcades, as well as those whic) 
 decorate the niches between the pilasters, had been ona scale not so enor mous; a.com) 
 parison of the one with the other would have made the principal parts appear muel 
 greater. Those who turn their eyes from these gigantic objects, and direct then 
 towards a man near the middle, or at the extremity of the church, experience thi. 
 effect : it is then, by comparing their own size with that of the principal parts of thi 
 edifice in the neighbourhood, that they begin to form an idea of its extent, and ar 
 struck with astonishment: but this second impression is the effect of a sort of rea 
 soning ; and the sensation, when produced in this manner, has not the same energy, 
 as when it is the effect of a first view. 
 
 While we are on this subject, we shall take the liberty of offering a few observa: 
 tions on the means of enlarging, as we may say, any space by the help of the imagi: 
 nation. In our opinion, nothing contributes more to produce this effect, than insu- 
 lated columns; that is to say; columns not regularly connected; for, when coupleé 
 or grouped, they always produes this effect, more or less, thotieh it would doubtless 
 be much better to employ them single. The result is, that every time a spectatol| 
 
 * Benitiers are vases for holding holy-water. 
 
ST. PETER’S CHURCH. Bye e 
 
 tee his position, different openings occur ; and a variety of aspects which asto- 
 ush and deceive the imagination. ' 
 _ But when columns are employed, they ought to be large ; for in the same degree as 
 hey have a majestic appearance when constructed on a grand a scale, they are, inour 
 pinion, mean and diminutive when small, and particularly when supported on pedestals. 
 The court of the Louvre, though in other respects beautiful, would have a much more | 
 triking effect, if the columns, instead of being mounted on meagre pedestals, rose from 
 he ground supported merely bya socle, like those in some of the vestibules of that 
 “alace. One might almost say, and there is some reason to think, that pedestals were 
 vented to render fit for use, columns collected at hazard, and which have not the 
 " equisite dimensions. 
 
 \ If Michael Angelo then, instead of forming his lateral spaces of immense arcades 
 “‘upported by pillars, decorated with pilasters, had employed groups of columns ies 
 -nstead of placing only three rows of lateral arcades, between the entrance and.the 
 ‘art of the dome, he had placed a greater number, which this arrangement would 
 ‘ave allowed him to do; and if the figures employed amidst this decoration had not 
 ar exceeded the natural size, we entertain no doubt that the spectator would have 
 ‘een struck with astonishment on the first view, and that the edifice would have ap- 
 eared much larger. “ 
 
 But it is to be observed, at the same time, that the knowledge which we now pos- | 
 ess, in regard to the resistance of materials, and the philosophy or mechanical part 
 f architecture, was not known at the time when Michael Angelo lived. It is pro- 
 able that he durst not venture to load columns, even when grouped, with a weight 
 0 considerable as that which he had to raise upon these pillars. But it is proved, by 
 ite experiments in regard to stones, that there is no weight that an insulated 
 olumn, six feet in diameter, made of very hard stone, well chosen and prepared, is 
 ‘ot capable of supporting. Our ancient churches, called improperly Gothic, are a proof 
 : fit; for there are some of them, the whole mass of which rests on pillars scarcely 
 (x feet in diameter, and often less: they therefore in general convey an idea of extent 
 ‘hich the Greek architecture, employed in the same places, does not excite. 
 
 ' 
 
 } 
 
 nee eae noone 
 
572 PYROTECHNY, 
 
 PART TENTH. 
 
 CONTAINING THE MOST CURIGUS AND AMUSING OPERATIONS IN 
 REGARD TO PYROTECHNY. i 
 
 Wuy it has been usual to consider Pyrotechny as a branch of the mathematics, w 
 donot know. The least reflection will readily shew, that it is an art by no mear, 
 mathematical, though dimensions, proportions, &c., are employed in it. There are | 
 great number of other arts which havea much better claim to be included amon: 
 these sciences. 
 
 However, as we might be blamed for omitting an art which affords a considerable fiel 
 for amusement, and as it is connected, at least, with natural philosophy, we shall mak 
 it the subject of one of the divisions of this work. But as we do not intend to giv 
 a complete treatise of Pyrotechny, we shall confine ourselves to those parts whic| 
 are most common and most curious: we shall also avoid every thing that relates t 
 the fatal art of destroying men. We can seeno amusement in the motion of a bulle 
 which carries off files of soldiers, nor in the action of a bomb or shell that sets fire t. 
 a town. The preceding editors and continuators of Ozanam, seem to hay, 
 possessed a very military spirit, if they considered all these things as harmless re 
 creation, For our part, having imbibed other principles in that happy country 
 Pennsylvania, we shudder even at the idea of introducing such atrocities under th: 
 form of amusement. 
 
 Pyrotechny, as we consider it in this work, is the art of managing fire, and c 
 making, by means of gunpowder and other inflammable substances, various compo 
 sitions agreeable to the eye, both by their form and their splendour. Of this kin, 
 are rockets, serpents, sheaves of fire, fixed or revolving suns, and other piece 
 employed in decorations and fire-works. 
 
 I 
 Gunpowder being the most common ingredient in Pyrotechny, we shall begin wit, 
 an account of its composition. 
 
 y 
 
 ARTICLE I. 
 
 Of Gunpowder. 
 
 Gunpowder is a composition of sulphur, salt-petre, and pounded charcoal: thes: 
 three ingredients mixed together, in the proper quantities, form a substance exceed 
 ingly inflammable, and of such a nature, that the discovery of it could be owing onl, 
 to chance. A single spark is sufficient to inflame, in an instant, the largest mass 0 
 this composition. The expansion, suddenly communicated either to the air, lodge: 
 in the interstices of the grains of which it consists, or to the nitrous acid which is on. 
 of the elements of the saltpetre, produces an effort which nothing can resist ; ani 
 the most ponderous masses are driven before it with inconceivable velocity. W: 
 must however observe that this invention, to which the epithet of diabolical is fre. 
 quently applied, is not so destructive to the human race as it might at first appear 
 battles seem to have been attended with less slaughter since gunpowder began ti 
 be used; and, as is remarked by the celebrated Marshal Saxe, the noise and smoki 
 
 is ’ 
 < 
 
OF GUNPOWDER, : 573 
 
 roduced by fire arms, during a battle, are more terrible than the execution they 
 take. We must however except cannon when well directed. But let us return to 
 ur subject, and give an account of the process for making gunpowder. 
 | Sulphur is found ready formed, and almost in its last degree of purity, in voleanic 
 roductions. It is found also, and much more frequently, in the state of sulphuric 
 cid; that is to say, combined with oxygen: it isin this state that it is found in 
 rgil, gypsum, &c. It may be extracted likewise from vegetable substances, and 
 ‘nimal matters. 
 - To purify sulphur, melt it in an iron pan: by which means the earthy and metallic 
 arts will be precipitated ; and then pour it into a copper kettle, where it will form 
 nother deposit of the foreign matters, with which it is mixed. After keeping it in 
 usion some time, pour it into cylindric wooden moulds, in order that it may be 
 ormed into sticks. 
 
 Saltpetre, or, as it is called in the modern chemistry, nitrate of potash, exists in a 
 atural state, but in small quantities. It is found sometimes~at the surface of the 
 round, as in India, and sometimes on the surface of calcareous walls, the roofs 
 f cellars, under the arches of bridges, &c. 
 
 ’ To extract the saltpetre from the lime of walls, or other earths impregnated with 
 t, the earths are put into casks, placed on timbers, and water is poured over them 
 o the height of about three inches. When the water has remained in that state 
 ‘ive or six hours, it is suffered to run off by apertures made in the bottom of the 
 asks, from which it falls into a gutter that conveys it to a common reservoir sunk 
 ‘n the earth. When the sediment has been deposited, the clear liquor is drawn 
 ff into a proper vessel, in order to be evaporated. 
 
 ' When the liquor is in a state of ebullition, in proportion as it evaporates, there is 
 “yrecipitated calcareous earth, and then muriate of soda. To know when it is suffi- 
 ‘iently evaporated, put a drop of it on a piece of cold iron, and if it becomes fixed, 
 ind assumes a white solid globular form, it is time to slacken the fire. The liquor 
 nust then be left at rest for twenty-four hours, after which it is run off and set to 
 ‘rystallize. 
 
 It is needless to describe charcoal, as it is every where known. We shall only 
 dbserve, that the charcoal found by experience to be the fittest for the composition 
 of gunpowder, is that made from the alder, willow, or black dog-wood. 
 
 ' To make gunpowder, mix together 6 parts of pounded nitre, well purified, | part 
 of pounded sulphur, exceedingly pure, and 1 part of pounded charcoal, adding a 
 quantity of water sufficient to reduce them toa soft paste. Put the whole into a 
 ‘wooden or copper mortar, and with a pestle of the same materials, to prevent 
 nflammation, pound these ingredients for twenty-four hours, to mix them thoroughly ; 
 aking care to keep them always moderately moist. When they are well incorpo- 
 ‘ated, pour the mass upon a sieve pierced with small holes of the size which you 
 ntend to give to the grains of powder.~ If it be then pressed, shaking the sieve, it 
 will pass through in grains, which must be dried in the sun, or over a stove without 
 ire. When dry, it ought to be put into vessels capable of preserving it from 
 moisture. 
 _ Every one knows that, in consequence of the great consumption of gunpowder, 
 sertain machines, called powder-mills, have been invented. These machines consist 
 of a beam turned by means of a water-wheel, and furnished with a great number 
 of projecting arms, which raise up and let fall in succession a series of pestles or 
 stampers, below which are placed copper vessels or mortars containing the matter to 
 de pounded or incorporated. These mills however are exceedingly disagreeable 
 ieighbours ; for notwithstanding the precautions taken there are few of them which 
 Jo not some time or other blowup. On this account they ought always to be erected 
 it a distance from towns or dwellings. 
 
 1 ae 
 
574 PYROTECHNY. 
 
 As the enlarged state of chemistry has introduced some improvements in the 4 
 of making gunpowder, we shall here, in addition to what has been above said, gi’ 
 the following account of the process employed for this purpose in some of the Engli 
 mannufactories. ! 
 
 Gunpowder is made of three ingredients ; saltpetre, charcoal, and brimston 
 which are combined in the following proportions: for each 100 parts of gunpowde 
 saltpetre 75 parts, charcoal 15, and sulphur 10. 
 
 The saltpetre is: either that imported principally from the East Indies, or th 
 which has been extracted. from damaged gunpowder. It is refined by solution, filtr’ 
 tion, evaporation, and crystallization ; after which it is fused; taking care not tou 
 too much heat, that there may be no danger of decomposing the nitre. 
 
 The sulphur used is that which is imported from Sicily, and is refined by meltit 
 and skimming: the most impure is refined by sublimation. | 
 
 The charcoal formerly used in this manufactnre, was made by charring wood in #) 
 usual manner. This mode is called charring in pits. The wood is cut into piee! 
 of about three feet in length ; it is then piled on the ground, in a circular form, thre’ 
 four, or five cords of wood making what is called a pit, and then covered with stray 
 fern, &c. kept down by earth or sand: and vent holes are made, as may be necessar | 
 in order to give it air. As this method is uncertain and defective, the charcoal no 
 used in the manufacturing of gunpowder, is made in the following manner. Tr 
 wood to be charred is first cut into pieces of about nine inches in length, and py 
 into an iron cylinder placed horizontally. The front aperture of the cylinder is the 
 closely stopped: at the other end there are pipes connected with casks. Fire beir’ 
 made under the cylinder, the pyro-ligneous acid, attended with a large portic| 
 of hydrogen gas, comes over. The gas escapes, and the acid liquor is collected |) 
 the casks. The fire is kept up till no more gas or liquor comes over, and the carbe 
 remains in the cylinder. 
 
 The several ingredients, being thus prepared, are ready for manufacturing. The 
 are first ground separately to a fine powder; they are then mixed together in tl 
 proper proportions ; and the composition in this state is sent to the gunpowder mil. 
 which consists of two stones placed vertically, and running on a bed-stone. On th) 
 bed-stone the composition is spread ont, and moistened with as small a quantit 
 of water as will reduce it to a proper body, but not to a paste: after the stone rul| 
 ners have made the proper revolutions over it, it may then be taken off. 
 
 A powder mill is a slight wooden building, with a boarded roof. Only about 40 ¢ 
 50 Ibs. of composition is worked here at a time, as explosions may happen by th 
 
 | 
 
 runners and bed-stone coming into contact, and even from other causes. These mil 
 are worked either by water or by horses. 
 
 The composition, when taken from the mill, is sent to the corning house, tot 
 corned or grained. Here it is first formed into a hard and firm mass; it is then broke. 
 into small lumps, and afterwards grained, by these lumps being put into sieves, /| 
 each of which is a flat circular piece of lignum vite. The sieves are made of parcl | 
 ment skins, having round holes punched through them. Several of these sieves a!’ 
 fixed in a frame, which by proper machinery has such a motion given to it, as to mak’ 
 the lignum vite runner in each sieve go round with great velocity, so as to break th 
 lumps of powder, and by forcing it through the holes to form it into grains of severi) 
 
 4 
 
 sizes. The grains are then separated from the dust by sieves and reels made for the 
 purpose. 
 
 The grains are next hardened, and the rougher edges are taken off by shakin 
 them a sufficient time ina close reel, moved in a circular direction with a prope 
 velocity. 
 The powder for guns, mortars, and small arms, is generally made at one time, an’ 
 
OF GUNPOWDER. . 575 
 
 lways of the same composition. The only difference is in the size of the grains, 
 which are separated by sieves of different fineness. 
 
 i The gunpowder thus corned, dusted, and reeled, which is called glazing, as it 
 ‘ives it a small degree of gloss, is then sent tothe stove and dried; care being taken 
 ot to raise the heat so much as to decompose the sulphur. The heat is regulated 
 ya thermometer placed in the door of the stoves, if dried in a gloom-stove.* . 
 A gunpowder stove dries the powder either by steam or by the heat from an iron 
 loom, the powder being spread out on cases placed on proper supports around the 
 om. 
 If gunpowder is injured by damp in a small degree, it may be recovered by again 
 rying it in a stove; but if the ingredients are decomposed, the nitre must be ex- . 
 
 racted, and the gunpowder re-manufactured. 
 _ There are several methods of proving and trying the goodness and strength of gun- 
 pwder. The following is one by which a tolerably good idea may be formed of its 
 jurity, and also some conclusion as to its strength. 
 -Lay two or three small heaps, about a dram or two of the powder on separate pieces 
 f clean writing paper; fire one of them by ared hot wire; if the flame ascends ra- 
 idly, with a good report, leaving the paper free from white specks, and without burping 
 oles in it; and if sparks fly off and set fire to the adjoining heaps, the goodness 
 { the ingredients and proper manufacture of the powder may be safely inferred ; but 
 otherwise, itis either badly made, or the ingredients are impure. 
 _Dr. Hutton, the former editor of the English edition of the Recreations, was for- 
 mate enough to succeed in constructing the most convenient and most. accurate 
 yrouvette that has perhaps ever been contrived for accurately determining the 
 mparative strength of gunpowder. It consists of asmall cannon, or gun, suspended 
 eely, like a pendulum, with the axis of the gun horizontal. This being charged 
 ith the proper charge of powder and then fired, the gun swings, or recoils back- 
 ard, and the instrument itself shews the extent of the first or greatest vibration, 
 hich indicates the strength to the utmost nicety. 
 Having thus given an account of almost every thing necessary to be known in 
 gard to the process for making gunpowder, we shall now say a few words respect- 
 g the physical causes of its inflammation and exploding. 
 ,Gunpowder being composed of the above ingredients, when a spark, struck from 
 piece of flint and steel, falls on this mixture, it sets fire to a certain portion of the 
 tarcoal, and the inflamed charcoal causes the nitre with which it is mixed or in 
 mtact, to detonate, and also the sulphur, the combustibility of which is well known. . 
 -ortions of the charcoal contiguous to the former take fire in like manner, and pro- 
 ice the same effect in regard to the surrounding mass: thus the first portion 
 flamed, inflames a hundred others; these hundred communicate the inflammation 
 » ten thousand; the ten thousand toa million, and so on. It may be easily con- 
 ived that an inflammation, the progress of which is so rapid, cannot fail to extend 
 self in the course of a very short time, from the one extremity to the other of the 
 tgest mass..- 
 \We shall observe, in support of this inflammation, that granulated powder in- 
 ‘mes with much more rapidity than that which is not granulated. The latter 
 ily puffs away slowly, while the other takes fire almost instantaneously ; and of 
 je granulated kinds of gunpowder, that in round grains, like the Swiss powder, 
 
 * This kind of stove consists of a large cast-iron vessel, projecting into one side of a room, and 
 aes from the outside, tiJl it absolutely glows. From the construction it is hardly possible that 
 2 can be thrown from the gloom as it is called; but stoves heated by steam passing through steam- 
 ‘ht tubes, or otherwise, ought certainly to be preferred; for the most cautious workman may 
 famble, and if he has a case of powder in his hand, some of it may be thrown upon the gloom: 
 id it is not improbable that some of the accidents which have happened to powder mills may have 
 lan occasioned in this manner. 
 
 Bs 
 
576 PYROTECHNY. . 
 inflames sooner than that in oblong irregular grains, like the French. The reason’ 
 this is, that the former leaves to the flame of the grains first inflamed, larger and fire 
 interstices, which produced the inflammation with more rapidity. 
 
 In regard to the expansion of inflamed gunpowder, is it occasioned by the air inti 
 posed between its grains, or by the aqueous fluid which enters into the compositi| 
 of thenitre? We doubt much whether it be the air, as its expansibility does not see 
 sufficient to explain the phenomenon ; but we know that water, when converted in| 
 vapour by the.contact of heat, occupies a space 14000 times greater than its | 
 bulk, and that its force is very considerable. 
 
 In the foregoing account however Montucla seems to have missed the true cause. 
 the expansive force of fired gunpowder, the discovery of which is chiefly due to t, 
 English philosophers, and particularly to the learned and ingenious Mr. Robins. Tl 
 author apprehends that the force of fired gunpowder consists in the action of a pe 
 manently elastic fluid, suddenly disengaged from the powder by the combustic 
 similar in some respects to common atmospheric air, at least as to elasticity. J] 
 shewed, by satisfactory experiments, that a fluid of this kind is actually disengaged || 
 firing the powder; and that it is permanently elastic, or retains its elastici, 
 when cold, the force of which he‘measured in this state. He also measured 1. 
 force of it when inflamed, by a most ingenious method, and found its strength in th, 
 state to be about a thousand times the strength or elasticity of common atmosphei 
 air. This however is not its utmost degree of strength, as it is found to increase ini 
 force when fired in larger quantities than those employed by Mr. Robins; so mu) 
 so indeed, that, by more accurate and effectual experiments, we have found its for, 
 rise as high as 1500 or 1600 times the force of atmospheric air in its usu 
 state. Much beyond this it is not probable it can go, nor indeed possible, 
 there be any truth in the common and allowed physical principles of mechanic 
 With an elastic fluid, of a given force, we infallibly know, or compute the effects, 
 can produce, in impelling a given body; and on the other hand, from the effects 
 velocities with which given bodies are impelled by an elastic fluid, we as certain 
 know the force or strength of that fluid. And these effects we have found perfect 
 to accord with the forces above mentioned. If any gentleman therefore thinks heh. 
 discovered that fired gunpowder is 50 or 60 times as strong as above stated, he mu’ 
 have been deceived by mistaking or misapplying his own experiments; and we a 
 prehend it would not be difficult, if this were the proper place, to shew that tl 
 has actually been the case. | 
 
 Mr. Robins’s discovery and opinion have also been corroborated by others, amo. 
 the best chemists and philosophers. Lavoisier was of opinion that the force of fir: 
 gunpowder depends, in a great measure, on the expansive force of uncombined calori 
 supposed to be let loose, in a great abundance, during the combustion or deflagrati 
 of the powder; and Bouillon Lagrange, in his Course of , Chemistry, says, whi 
 gunpowder takes fire, there is a disengagement of azotic gas, which expands in ¢ 
 astonishing manner, when set at liberty ; and we are even still ignorant of the exte 
 of the dilatation occasioned by the heat arising from the combustion. A decompo: 
 tion of water also takes place, and hydrogen gas is disengaged with elasticity; al 
 by this decomposition of water there is formed carbonic acid gas, and even sulphw 
 ated hydrogen gas, which is the cause of the smell emitted by burnt powder. 
 
 Remarks.—I. It is ridiculous therefore to believe in the existence of white gw 
 powder ; that is, a kind of powder which impels a ball without any noise; for the 
 can be no force without sudden expansion, nor sudden expansion without a concussit 
 of the air, which produces sound. | 
 
 II. It was childish to give precepts, as in the preceding editions of this work, f 
 naking red, blue, green, &c. gunpowder; as they could answer no good purposes 
 
OF ROCKETS. Sie 677 
 
 | We shall now proceed to our principal object, the construction of the most common 
 and curious pieces of fire-works. 
 
 ARTICLE It. 
 “Construction of the Cartridges of Rockets. ; 
 A rocket is a cartridge or case made of stiff paper, which being filled in part with 
 sunpowder, saltpetre, and charcoal, rises of itself into the air, when fire is applied 
 oO it. 
 _ There are three sorts of rockets: small ones, the calibre of which does not exceed 
 - pound bullet; that is to say, the orifice of them is equal to the diameter of a leaden 
 vullet which weighs only a pound; for the calibres, or orifices of the moulds or 
 nodels used in making rockets, are measured by the diameters of leaden bullets :—— 
 niddle sized rockets, equal to the size of a ball of from one to three pounds; and 
 arge rockets, equal to a ball of from three to a hundred pounds. 
 | To give the cartridges the same length and thickness, in order that any number 
 if rockets may be prepared of the same size and force, they are put into a hollow 
 ylinder of strong wood, called a mould. This mould is sometimes of metal; but 
 t any rate it ought to be made of some very hard wood. fl 
 This mould must not be confounded with another piece of wood, called the former 
 t roller, around which is rolled the thick paper employed to make the cartridge. 
 f the calibre of the mould be divided into 8 equal parts, the diameter of the roller 
 aust be equal to 5 of these parts. See Fig. 1, where a is the mould, ands the roller. 
 “he vacuity between the roller and the interior surface of the mould, that is to say, 
 of the calibre of the mould, will be exactly filled by tbe cartridge. : 
 As rockets are made of different sizes, moulds of different lengths and diameters 
 aust be provided. The calibre of a cannon is nothing else than the diameter of its 
 routh; and we here apply the same term to the diameter of the aperture of the 
 —r0uld. . 
 | The size of the mould is measured by its calibre; but the length of the moulds 
 br different rockets does not always bear the same proportion to the calibre, the 
 mgth being diminished as the calibre is increased. The length of the mould for 
 _ mall rockets ought to be six times the calibre, but for rockets of the mean and larger 
 _ ze it will be sufficient if the length of the mould be five times or even four time 
 ie calibre of the mould. : 
 _ At the end of this section we shall give two tables, one of which contains the 
 _ 4libres of moulds below a pound bullet ; and the other the calibres from a pound to 
 ‘hundred pounds bullet. 
 . For making the cartridges large stiff paper is em- 
 ployed. This paper is wrapped round the roller zB, 
 (Fig. 1.), and then cemented by means of common 
 paste. The thickness of the paper when rolled up 
 in this manner, ought to be about one-eighth anda 
 half of the calibre of the mould, according to the 
 proportion given to the diameter of the roller. But 
 if the diameter of the roller be made equal to 3 the 
 calibre of the mould, the thickness of the cartridge 
 5 | @ must be a twelfth and a half of that calibre. — 
 _ When the cartridge is formed, the roller B is drawn out, by turning it round, until 
 is distant from the edge of the cartridge the length of its diameter. A piece 
 ‘cord is then made to pass twice round the cartridge at the extremity of the roller; and 
 to the vacuity left in the cartridge, another roller is introduced, so as to leave some 
 ace between the two. One end of the packthread must be fastened to something 
 _ sed, and the other to a stick conveyed between the legs, and placed in such a man. 
 2P 
 
* 
 
 R78 PYROTECHNY. 
 
 & 
 
 ner, as to be behind the person who choaks the cartridge. The cord is then to | 
 stretched by retiring backwards, and the cartridge must be pinched until there re., 
 mains only an aperture capable of admitting the piercer DE. The cord employer 
 for pinching it is then removed, and its place is supplied by a piece of pack-thread 
 which must be drawn very tight, passing it several times around the cartridge, afte, 
 which it is secured by means of running knots made one above the other. 
 
 Besides the roller 3, a rod c (Fig. 1.), is used, which being employed to load thy 
 cartridge, must be somewhat smaller than the roller, in order that it may be easil; 
 introduced into the cartridge. The rod c is pierced lengthwise, to a sufficient dept] 
 to receive the piercer DE, which must enter into the mould a, and unite withi) 
 exactly at its lower part. The piercer, which decreases in size, is introduced int 
 the cartridge through the part where it has been choaked, and serves to preserve) 
 -cavity within it. Its length, besides the nipple or button, must be equal to about two: 
 thirds of that of the mould. Lastly, If the thickness of the base be a fourth par; 
 of the calibre of the mould, the point must be made equal to a sixth of the calibre. 
 
 It is evident that there must be at least three rods, such as c, pierced in proportio- 
 to the diminution of the piercer, in order that the powder which is rammed in by mean, 
 of a mallet, may be uniformly packed throughout the whole length of the rocket, 
 Is may be easily perceived also, that these rods ought to be made of some very har 
 wood, to resist the strokes of the mallet. 
 
 In loading rockets, it is more convenient not to employ a piercer. When load! 
 on anipple, without a piercer, by means of one massy rod, they are pierced with , 
 bit, and a piercer fitted into the end of a bit-brace. Care however must be taken t) 
 make this hole suited to the proportion assigned for the diminution of the pierce, 
 That is to say, the extremity of the hole at the choaked part of the cartridge, ough 
 to be about a fourth of the calibre of the mould; and the extremity of the hol, 
 which is in the inside for about two-thirds of the length of the rocket, ought to b) 
 a sixth of the calibre. This bole must pass directly through the middle of th 
 rocket. In short, experience and ingenuity will suggest what is most convenien 
 and in what manner the method of loading rockets, which we shall here explain, ma' 
 be varied. 
 
 After the cartridge is placed in the mould, pour gradually into it the prepared con 
 position; taking care to pour only two spoonfuls ata time, and to ram it immediate) 
 down with the rod c, striking it ina perpendicular direction with a mallet of a props: 
 size, and giving an equal number of strokes, tor example 3 or 4, each time that a ne. 
 quantity of the composition is poured in. 
 
 When the cartridge is about half filled, separate with a bodkin the half of tL 
 folds of the paper which remains, and having turned them back on the compositio. 
 press them down with the rod and a few strokes of the mallet, in order to compre 
 
 the .paper on the composition, ; 
 
 Then pierce three or four holes in the folded paper, by means of 
 piercer, which must be made to penetrate to the compesition of the ro. 
 ket, as seen at a (Fig. 2.) These holes serve to forma communicati( 
 between the body of the rocket and the vacuity at the extremity of i 
 cartridge, or that part which has been left empty. 
 
 In small rockets this vacuity is filled with granulated powder, whit 
 serves to let them off: they are then covered with paper, and pinched | 
 the same manner as at the other extremity. But in other rockets, the p 
 containing stars, serpents, and running rockets, is adapted to it, as will) 
 shewn hereafter. 
 
 It may be sufficient however to make, with a bit or piercer, only one hol 
 which must be neither too large nor too small, such as a fourth part of t) 
 diameter of the rocket, to set fire to the powder! taking care that this he 
 
 | 
 
 | 
 
/ 
 . 
 
 OF ROCKETS. 579 
 
 _beas straight as pessible, and exactly in the middle of the composition. A little of the 
 
 composition of the rocket must be put into these holes, that the fire may not fail to 
 be communicated to it. 
 
 It now remains to affix the rocket to its rod, which is done in the following man- 
 ner. When the rocket has been constructed as above described, make fast to it a 
 rod of light wood, such as fir or willow, broad and flat at the end next the rocket, 
 and decreasing towards the other. It must be as straight and free from knots as 
 possible, and ought to be dressed, if necessary, with a plane. Its length and weight 
 ‘must be proportioned to the rocket ; that is to say, it ought to be six, seven, or eight 
 feet long, so as to remain in equilibrium with it, when suspended on the finger, within 
 ‘an inch or an inch and a half of the neck. Before it is fired, place it with the neck 
 ‘downwards, and let it rest on two nails, in a direction perpendicular to the horizon. 
 To make it ascend straighter and to a greater height, adapt to its summit a a pointed 
 
 ‘cap or top, as c, made of common paper, which will serve to facilitate its passage 
 ‘through the air. 
 
 These rockets, in general, are made in a more complex manner, several other 
 ‘things being added to them to render them more agreeable, such for example as a 
 ‘petard, which is a box of tin-plate, filled with fine gunpowder, placed on the summit. 
 ‘The petard is deposited on the composition, at the end where it has been filled ; and 
 
 the remaining paper of the cartridge is folded down over it to keep it firm. The 
 
 petard produces its effect when the rocket is in the air, and the composition is con- 
 
 sumed, 
 
 Stars, golden rain, serpents, saucissons, and several other amusing things, the 
 ‘composition of which we shall explain hereafter, are also added to them. This is 
 done by adjusting to the head of the rocket an empty pot or cartridge, much larger 
 
 ‘than the rocke+, in order that it may contain serpents, stars, and various other ap- 
 _'pendages, to render it more beautiful. 
 
 Rockets may be made to rise into the air without rods. For this 
 purpose four wings must be attached to them in the form of a cross, 
 and similar to those seen on arrows or darts, as represented at A (Fig. 
 3.) In length these wings must be equal to two-thirds that of the 
 rocket ; their breadth towards the bottom should be half their length, 
 and their thickness ought to be equal to that of a card. 
 
 But this method of making rockets ascend is less certain, and more 
 inconvenient than that where arod is used; and for this reason it is 
 rarely employed. 
 
 | We shall now shew the method of finding the diameters or calibre of rockets, 
 according to their weight; but we must first observe that a pound rocket is that 
 
 just capable of admitting a leaden bullet of a pound weight, and so of the rest. The - 
 
 calibre for the different sizes may be found by the two following tables, one of which 
 
 is calculated for rockets of a pound weight and below; and the other for those from 
 a pound weight to 50 pounds. 
 
 i 
 
580 PYROTECHNY. 
 
 I.—TABIE OF THE CALIBRE OF MOULDS OF A POUND WEIGHT AND BELOW. 
 
 Ounces. | Lines. Drams. Lines. 
 16 19} 14 77 
 12 17 12 7 
 
 8 15 10 62 
 7 143 8 6+ 
 6 144 6 52 
 5 13 4 4} 
 4 122 - 2 33 
 3 
 2 
 1 
 
 The use of this table will be understood merely by inspection; for it is evident 
 that a rocket of 12 ounces ought to be 17 lines in diameter; one of 8 ounces, 18 
 lines; one of 10 drams, 64 lines; and so of the rest. 
 
 On the other hand, if the diameter of the rocket be given, it will be easy to find 
 the weight of the ball corresponding to that calibre. For example, if the diameter’ 
 be 13 lines, it will be immediately seen, by looking for that number in the column 
 of lines, that it corresponds to a ball of 5 ounces. 
 
 Il.—TABLE OF THE CALIBRE OF MOULDS FROM 1 TO 50 POUNDS BALL. 
 
 Pounds.| Calibre. {Pounds.| Calibre. |Pounds.| Calibre. {Pounds.| Calibre. }Pounds.| Calibre, 
 
 _ eed 
 
 1 100 11 222 21 275 31 314 41 344 
 2 126 12 228 22 280 32 317 42 347 
 3 144 13 235 23 284 33 320 43 350 
 4 158 14 241 24 288 34 323 44 353 
 5 171 15 247 25 292 35 326 45 355 
 6 181 16 252 26 296 36 330 46 358 
 7 191 17 257 27 300 37 333 47 361 
 8 200 18 262 28 304 38 336 48 363 
 9 208 19 267 29 307 39 339 | 49 366 
 10 215 20 271 30 310 40 341 50 368 
 
 The use of the second table is as follows: If the weight of the ball be given 
 which we shall suppose to be 24 pounds, seek for that number in the columno. 
 pounds, and opposite to it, in the column of calibres, will be found the number 288 
 Then say, as 100 is to 193, so is 288 to‘a fourth term, which will be the number 0) 
 lines of the calibre required; or multiply the number found, that is 288, by 193, ant 
 from the product 5616, cut off the two last figures: the required calibre will there, 
 fore be 56°16 lines, or 4 inches 8 lines. : 
 
 On the other hand, the calibre being given in lines, the weight of the ball may bi 
 found with equal ease: if the calibre, for example, be 28 lines, say as 19 is to 28 
 so is 100 to a fourth term, which will be 143-5, or nearly 144. But in the abov: 
 table, opposite to 144, in the second column, will be found the number 8 in the first 
 
 which shews that a rocket, the diameter or calibre of which is 28 lines, is a rocke 
 of a 3 pounds ball. 
 
COMPOSITION FOR ROCKETS, 581 
 
 ARTICLE III. 
 Composition of the Powder for Rockets, and the manner of filling them. 
 The composition of the powder for rockets must be different, according to the 
 different sizes; as that proper for small rockets would be too str ong for large ones. 
 This is a fact respecting which almost all the makers of fire-works are agreed. The 
 
 quantities of the ingredients, which experience has shewn to be the best, are as 
 follow : 
 
 For rockets capable of containing one or two ounces of composition. 
 
 To one pound of gunpowder, add two ounces of soft charcoal; or to one pound 
 of gunpowder, a pound of the coarse powder used for cannon; or to nine ounces 
 
 of gunpowder, two ounces of charcoal ; or to a pound of gunpowder, an ounce and 
 ahalf of saltpetre, and as much charcoal. 
 
 For rockets of two or three ounces. 
 
 To four ounces of gunpowder, add an ounce of charcoal; or to nine ounces of gun- 
 powder, add two ounces of saltpetre. 
 
 For a rocket of four ounces. 
 
 To four pounds of gunpowder, add a pound of saltpetre, and four ounces of char- 
 soal: you may add also, if you choose, half an ounce of sulphur; or to one pound 
 iwo ounces and a half of gunpowder, add four ounces of saltpetre, and two ounces 
 of charcoal; or to a pound of powder, add four ounces of saltpetre, and one ounce 
 f charcoal; or to seventeen ounces of gunpowder, add four ounces of saltpetre, and 
 
 he same quantity of charcoal; or to three ounces and a half of gunpowder, add ten 
 yunces of saltpetre, and three ounces and a half of charcoal. But the composition 
 will be strongest, if to ten ounces of gunpowder, you add three ounces and a half of 
 ialtpetre, and three ounces of charcoal. 
 
 For rockets of five or six ounces. 
 To two pounds five ounces of gunpowder, add half a pound of saltpetre, two 
 yunces of sulphur, six ounces of charcoal, and two ounces of iron filings. 
 For rockets of seven or eight ounces. 
 
 To seventeen ounces of gunpowder, add four ounces of saltpetre, and three 
 unces of sulphur. 
 
 For rockets of from eight to ten ounces. 
 
 To two pounds and five ounces of gunpowder, add half a pound of saltpetre, two 
 vunces of sulphur, seven ounces of charcoal, and three ounces of iron filings. 
 
 For rockets of from ten to twelve ounces. 
 | To seventeeen ounces of gunpowder, add four ounces of saltpetre, three ounces 
 nd a half of sulphur, and one ounce of charcoal. 
 For rockets of from fourteen to fifteen ounces. 
 _To two pounds four ounces of gunpowder, add nine ounces of saltpetre, three 
 unces of sulphur, five ounces of charcoal, and three ounces of iron filings. 
 For rockets of one pound. 
 
 To one pound of gunpowder, add one ounce of sulphur, and three ounces of 
 harcoal, 
 
> 
 | 
 
 582 PYROTECHNY. S| 
 
 | 
 For a rocket of two pounds, 
 
 To one pound four ounces of gunpowder, add two ounces of saltpetre, one ounce, 
 of sulphur, three ounces of charcoal, and two ounces of iron filings, | 
 
 | 
 
 For a rocket of three pounds. | 
 To thirty ounces of saltpetre, add seven ounces and a half of sulphur, and eleven 
 ounces of charcoal. 
 
 For rockets of four, five, six, or seven pounds. 
 
 To thirty-one pounds of saltpetre, add four pounds and a half of sulphur, and ten 
 pounds of charcoal. 
 
 For rockets of eight, nine, or ten pounds. 
 
 To eight pounds of saltpetre, add one pound four ounces of sulphur, and two 
 pounds twelve ounces of charcoal. 
 
 We shall here observe that these ingredients must be each pounded separately, ane 
 sifted: they are then to be weighed and mixed together for the purpose of loading 
 the cartridges, which ought to be kept ready in the moulds. The cartridges must by. 
 made of strong paper, doubled, and cemented by means of strong paste, made 0) 
 fine flour and very pure water. as 
 
 Of Matches. 
 
 Before we proceed farther, it will be proper to describe the composition of thi 
 matches necessary for letting them off. Take linen, hemp, or cotton thread, ani 
 double it eight or ten times, if intended for large rockets ; or only four or five times 
 if to be employed for stars. When the match has been thus made as large as necessary 
 dip it in pure water, and press it between your hands, to free it from the moisture) 
 
 _ Mix some gunpowder with alittle water, to reduce it to a sort of paste, and immers| 
 the matchin it; turning and twisting it, till it has imbibed a sufficient quantity of th 
 powder ; then sprinkle over it a little dry powder, or strew some pulverised dr 
 powder upon a smooth board, and roll the match over it. By these means you wil, 
 have an excellent match, which, if dried in the sun, or ona rope in the shade, wil 
 be fit for use. 
 
 ARTICLE IY. 
 
 On the cause which makes rockets ascend into the air. 
 
 As this cause is nearly the same as that which produces recoil in fire-arms, i] 
 is necessary we should first explain the latter. | 
 When the powder is suddenly inflamed in the chamber, or at the bottom of th 
 barrel, it necessarily exercises an action two ways at the same time; that is to say, 
 against the breech of the piece, and against the bullet or wadding, which is place | 
 above it. Besides this, it acts also against the sides of the chamber which it ocew 
 pies; and as they oppose a resistance almost insurmountable, the whole effort of th 
 elastic fluid produced by the inflammation is exerted in the two directions aboy 
 mentioned. But the resistance opposed by the bullet being much less than that op 
 posed by the mass of the barrel or cannon, the bullet is forced out with great velocity 
 It is impossible, however, that the body of the piece itself should not experience |: 
 movement backwards; for if a spring is suddenly let loose, between two moveabl 
 obstacles, it will impel them both, and communicate to them velocities in th 
 inverse ratio of their masses: the piece therefore must acquire a velocity backwaré 
 nearly in the inverse ratio of its mass to that of the bullet. We make use of th, 
 term nearly, because there are various circumstances which give to this ratio certai) 
 
 t 
 ] 
 ¢ 
 
CHINESE AND BRILLIANT FIRE, 583 
 
 modifications; but it is always true that the body of the piece is driven backwards, 
 ‘and that if it weighs with its carriage a thousand times more than the bullet, it 
 ‘acquires a velocity which is a thousand times less, and which is soon annihilated by 
 the friction of the wheels against the ground, &c. ' 
 
 The cause of the ascent of a rocket is nearly the same. At the moment when the 
 powder begins to inflame, its expansion produces a torrent of elastic fluid, which 
 acts in every direction; that is, against the air which opposes its escape from the car- 
 tridge, and against the upper part of the rocket; but the resistance of the air is more 
 considerable than the weight of the rocket, on account of the extreme rapidity with 
 which the elastic fluid issues through the neck of the rocket to throw itself down- 
 wards, and therefore the rocket ascends by the excess of the one of these forces over 
 the other. 
 __ This however would not be the case, unless the rocket were pierced to a certain 
 depth. A sufficient quantity of elastic fluid would not be produced; for the compo- 
 ‘sition would inflame only in circular coats of a diameter equal to that of the rocket; 
 and experience shews that this is not sufficient. Recourse then is had to the very 
 ‘ingenious idea of piercing the rocket witha conical hole, which makes the composition 
 ‘burn in conical strata, which have much greater surface, and therefore produce a auch 
 
 greater quantity of inflamed matter and fluid. This expedient was certainly not the 
 work of a moment. 
 
 ARTICLE v. 
 Brilliant fire and Chinese fire. 
 
 As iron-filings, when thrown into the fire, inflame and emit a strong light, this 
 
 “property, discovered no doubt by chance, gave rise to the idea of rendering the fire 
 ‘of rockets much more brilliant than when gunpowder, or the substances of which it 
 is composed, are alone employed. Nothing is necessary but to take iron-filings, very 
 clean aud free from ‘rust, and to mix them with the composition of the rocket. It 
 ‘must however be observed, that rockets of this kind will not keep longer than a week ; 
 because the moisture contracted by the saltpetre rusts the iron-filings, and destroys 
 the effect they are intended to produce. 
 But the Chinese have long been in possession of a method of rendering this fire 
 ‘much more brilliant and variegated in its colours ; and we are indebted to Father 
 @Incarville, a jesuit, for having made it known. It consists in the use of a very 
 simple ingredient: namely, cast iron reduced to a powder more or less fine; the 
 Chinese give it a name, which is equivalent to that of iron sand. 
 
 To prepare this sand, take an old iron pot, and having broken it to pieces on an 
 anvil, pulverise the fragments till the grains are not larger than radish seed: then sift 
 them through six graduated sieves, to separate the different sizes, and preserve these 
 
 ‘six different kinds, in a very dry place, to secure them from rust, which would 
 ‘render this sand absolutely unfit for the proposed end. We must here remark, that 
 the grains which pass through the closest sieve, are called sand of the first order ; 
 _those which pass through the next in size, sand of the second order; and so on. 
 
 This sand, when it inflames, emits a light exceedingly vivid. It is very surprising 
 to see fragments of this matter no bigger than a poppy seed, form all of a sudden 
 luminous flowers or stars, 12 and 15 lines in diameter. These flowers are also of 
 different forms, according to that of the inflamed grain, and even of different colours 
 according to the matters with which the grains are mixed. But rockets into which 
 this composition enters cannot be long preserved, as those which contain the finest 
 sand will not keep longer than eight days, aud those which contain the coarsest, fif- 
 ‘teen. The following tables exhibit the proportions of the different ingredients for 
 Tockets of from 12 to 36 pounds. 
 
584 PYROTECHNY. 
 
 For red Chinese fire. 
 
 HK | 
 
 Calibres. | Saltpetre. | Sulphur. | Charcoal. Patel 
 
 st order. 
 Pounds. Pounds. Ounces. Ounces. Oz dr. 
 12 to 15 
 18 to 21 
 24 to 36 
 For white Chinese fire. a | 
 : Bruised 1 Sand of the 
 Calibres, | Saltpetre. Gunpowder: Charcoal. BA Geion: 
 Pounds. Pounds. Ounces. oz ar. oz dr.|— | 
 ; 12 to 15 1 12 7 8 11 O | 
 18 to 21 1 11 8 0 i) ees 
 24 to 36 1 11 Sano 1270 
 
 When these materials have been weighed, the saltpetre and charcoal must be three 
 times sifted through a hair sieve, in order that they may be well mixed: the iron sand, 
 is then to be moistened with good brandy, to make the sulphur adhere, and they must. 
 be thoroughly incorporated. The sand thus sulphured must be spread over the mix 
 ture of saltpetre and charcoal, and the whole must be mixed together by spreading it 
 over a table with a spatula. | 
 
 ARTICLE VI. 
 Of the Furniture of Rockets, 
 
 The upper part of rockets is generally furnished with some composition, which 
 taking fire when it has reached to its greatest height, emits a considerable blaze, or, 
 produces a loud report, and very often both these together. Of this kind are’ 
 saucissons, marroons, stars, showers of fire, &c. 
 
 To make room for this artifice, the rocket is crowned with a part of a greater 
 diameter called the pot, as seen in Fig. 4, The method of 
 making this pot, and connecting it with the body of the rocket,| 
 is as follows. ; =| 
 
 The mould for forming the pot, though of one piece, must 
 consist of two cylindric parts of different diameters. That on 
 which the pot is rolled up must be three diameters of the rocket 
 in length, and its diameter must be three fourths that of the 
 rocket ; the length of the other ought to be equal to two of 
 these diameters, and its diameter to Z that of the rocket. 
 
 Having rolled the thick paper intended for making the pot, and which ought to be’ 
 of the same kind as that used for the rocket, twice round the cylinder, a portion of 
 it must be pinched in that part of the cylinder which has the least diameter ; this. 
 part must be pared in such a manner as to leave only what isnecessary for making the 
 pot fast to the top of the rocket, and the ligature must be covered with paper. 
 
 To charge such a pot, attached to a rocket; having pierced three or four holes in 
 the double paper which covers the vacuity of the rocket, pour over it a small quantity 
 of the composition with which the rocket is filled, and by shaking it, make a part 
 enter these holes; then arrange in the pot the composition with which it is to be! 
 
 i 
 ] 
 i) 
 
 | 
 | 
 | 
 | 
 =| 
 | 
 
OF SERPENTS—MARROONS. 585 
 
 charged, taking care not to introduce introduce into it a quantity heavier 
 than the body of the rocket. 
 
 The whole must then be secured by means of a few small balls of 
 paper, to keep every thing inits place, and the pot must be covered with 
 paper cemented to its edges: if a pointed summit or cap be then added 
 to it, the rocket willbe ready for use. 
 
 We shall now give an account of the different artifices with which 
 such rockets are loaded. 
 
 I.—Of Serpents. 
 
 Serpents are small flying rockets, without rods, which instead of rising 
 
 in a perpendicular direction, mount obliquely, and descend in a zig-zag 
 
 : form without ascending toa great height. The composition of them is 
 
 nearly the same as that of rockets ; and therefore nothing more is neces- 
 
 sary than to determine the proportion and construction of the cartridge, which is a 
 follows. 
 
 The length a c (Fig. 6), of the cartridge may be about 4 inches ; 
 
 Fig. 6. it must be rolled round a stick somewhat larger than the barrel 
 
 saan 54 of a goose quill, and after being choaked at one of its ends, fill 
 
 =. -B A it with the composition a little beyond its middle, as to B; and 
 
 ; then pinch it so as to leave a small aperture. The remainder 
 
 BC, must be filled with grained powder, which will occasion a report when it bursts. 
 
 Lastly, choak the cartridge entirely towards the extremity c; and at the other ex- 
 
 tremity A place a train of moist powder, to which if fire be applied, it will be com- 
 
 municated to the composition in the part a B, and cause the whole to rise in the air. 
 
 The serpent, as it falls, will then make several small turns in a zig-zag direction, till 
 
 the fire is communicated to the grained powder in the part-Bc; on which the perpen 
 will burst with a loud report before it falls to the ground. 
 
 If the serpent be not choaked towards the middle, instead of moving in a zig-zag 
 direction, it will ascend and descend with an undulating motion, and then burst as 
 before. 
 
 ' The cartridges of serpents are generally made of playing cards. These cards 
 are rolled round a rod of iron or hard wood, a little larger, as already said, than 
 the barrel of a goose quill. To confine the card, a piece of strong paper is cemented 
 over it. 
 
 | The length of the mould must be proportioned to that of the cards employed, and 
 
 _the piercer of the nipple must be three or four lines in length. These serpents are 
 
 loaded with bruised powder, mixed only with a very small quantity of charcoal. To 
 introduce the composition into the cartridge, a quill, cutinto the form of a spoon, may 
 be employed: it must be rammed down by means of a small rod, to which a few strokes 
 are given with a small mallet. 
 
 - When the serpent is half loaded, instead of pinching it in that part, you may intro- 
 duce into it.a vetch seed, and place granulated powder above it to fill up-the remainder. 
 \Above this powder place asmall pellet of chewed paper, and then choak the other end 
 of the cartridge. If you are desirous of making larger serpents, cement two playing 
 cards together ; and, that they may be managed with more ease, moisten them a little 
 with water. The match consists of a paste made of bruised powder, anda small 
 quantity of water. 
 
 Il.—Marroons. 
 
 Marroons are small cubical boxes, filled with a composition proper for making them 
 burst, and may be constructed with great ease. 
 Cuta piece of pasteboard, according to the method taught in geometry to form the 
 
536 PYROTECHNY. 
 cube, as seen Fig. 7.; join these squares at the edges, leaving, 
 Fig. 7. only one to be cemented, and fill the cavity of the cube with 
 L grained powder ; then cement strong paper in various directions, 
 aa over this body, and wrap round it two rows of pack- thread, 
 oo dipped in strong glue: then make a hole in one of the corners, 
 
 and introduce into it a match. 
 
 If you are desirous to have luminous marroons, that is to say marroons which, 
 before they burst in the air, emita brilliant light, cover them with a paste, the com. 
 position of which will be given hereafter for stars; and roll them in pulverised gun- 
 powder, to serve as a match or communication. 
 
 Il1.— Saucissons. | 
 
 Marroons and saucissons differ from each other only in their form. The cartridges 
 of the latter are round, and must be only four times their exterior diameter in length. 
 They are choked at one end in the same manner as a rocket ; and a pellet of paper is) 
 driven into the aperture which has been left, in order to fill itup. They are then 
 charged with grained powder, above whicli is placed a bali of paper gently pressed. 
 down, to prevent the powder from being bruised; the second end of the saucisson 
 being afterwards choaked, the edges are pared on both sides, and the whole is covered 
 with several turns of pack-thread, dipped in strong glue, and then left to dry. | 
 
 When you are desirous of charging them, pierce a hole in one of the ends; aud 
 apply a match, in the same manner as to marroons. 
 
 IV.—Stars. | 
 
 Stars are small globes of a composition which emits a brilliant light, that may be 
 compared to the light of the stars in the heavens. These balls are not larger than a 
 nutmeg or musket bullet, and when put into the rockets must be wrapped up in towt 
 prepared for that purpose. The composition of these stars is as follows. 
 
 To a pound of fine gunpowder well pulverised, add four pounds of saltpetre, and’ 
 two pounds of sulphur. When these ingredients are thoroughly incorporated, take) 
 about the size of a nutmeg of this mixture, and having wrapt it up in a piece of linen-! 
 rag, or of paper, form it into a ball; then tie it closely round with a pack-thread, and! 
 pierce a hole through the middle of it, sufficiently large to receive a piece of prepared) 
 tow, which will serve asa match. This star, when lighted, will exhibit a most| 
 beautiful appearance ; because the fire as it issues from the two ends of the hole ir 
 the middle, will extend to a great distance, and make it appear much larger. 
 
 If you are desirous to employ a moist composition in the form of a paste, instead 01 
 a dry one, it will not be necessary to wrap up the star in any thing but prepared tow: 
 because, when made of such paste, it can retain its spherical figure. There will be. 
 no need also of piercing a hole in it, to receive the match ; because when newly made’ 
 and consequently moist, it may be rolled in pulverised gunpowder, which will adhere 
 to it. This powder, when kindled, will serve as a match, and inflame the compositior 
 of the star, which in falling will form itself into tears. 
 
 Another method of making Rockets with Stars. 
 
 Mix three ounces of saltpetre, with one ounce of sulphur, and two drams of pul 
 verized gunpowder ; or mix four ounces of sulphur with the same quantity of salt: 
 petre, and eight ounces of pulverized gunpowder. When these materials have beei 
 well sifted, besprinkle them with brandy, in which a little gum has been dissolved 
 and then make up the star in the following manner. 
 
 Take a rocket mould, eight or nine lines in diameter, and introduce into it a nip 
 ple, the piercer of which is of a uniform size throughout, and equal in length to th 
 height of the mould. Put into this mould a cartridge, and by means of a pierce: 
 
 | 
 ry 
 
' SHOWERS OF FIRE, AND SPARKS, 587 
 
 +. 
 ‘rod load it with one of the preceding compositions ; when loaded, take it from the 
 ‘mould, without removing the nipple, the piercer of which passes through the com- 
 position, and then cut the cartridge quite round into pieces of the thickness of three 
 or four lines. The cartridge being thus cut, draw out the piercer gently, and the 
 pieces, which resemble the men employed for playing at drafts, pierced through the 
 middle, will be stars, which must be filed on a match thread, which, if you choose, 
 ‘may be covered with tow. { 
 
 To give more brilliancy to stars of this kind, a cartridge thicker than the above 
 dimensions, and thinner than that of a flying-rocket of the same size, may be em- 
 ployed ; but, before it is cut into pieces, five or six holes must be pierced in the cir- 
 ‘cumference of each piece to be cut. When the cartridge is cut, and the pieces have 
 been filed, cement over the composition small bits of card, each having a hole in the 
 middle, so that these holes may correspond to the place where the composition is 
 pierced. 
 
 _ Remarks.—\. There are several other methods of making stars, which it would be 
 ‘too tedious to describe. We shall therefore only shew how to make étoiles a pet, or 
 ‘stars which give a report as loud as that of a pistol or musket. 
 
 Make small saucissons, as taught in the third section; only it will not be necessary 
 to cover them with pack-thread: it will be sufficient if they are pierced at one end, 
 ‘in order that you may tie to it a star constructed according to the first method, the 
 composition of which,is dry; for if the composition be in the form of a paste, there 
 will be no need to tie it. Nothing will‘be necessary in that case, but to leave a little 
 more of the paper hollow at the end of the saucisson which has been pierced, for the 
 \purpose of introducing the composition; and to place in the vacuity, towards the 
 neck of the saucisson, some grained powder, which will communicate fire to the 
 -saucisson when the composition is consumed. 
 
 If. As there are some stars which in the end become petards, others may be made 
 which shall conclude with becoming serpents. But this may be so easily conceived 
 ‘and carried into execution, that it would be losing time to enlarge further on the 
 subject. We shall only observe, that these stars are not in use, because it is difficult 
 for a rocket to carry them to a considerable height in the air : they diminish the effect 
 of the rocket or saucisson, and much time is required to make them. 
 
 V.—Shower of Fire. 
 
 To form a shower of fire, mould small paper cartridges on an iron rod, two lines 
 jand a half in diameter, and make them two inches and a half in length. They must 
 not be choaked, as it will be sufficient to twist the end of the cartridge, and having 
 ‘put the rod into it to beat it, in order to make it assume its form. When the car- 
 tridges are filled, which is done by immersing them in the composition, fold down the 
 ‘other end, and then applya match. This furniture will fill the air with an undulating 
 ‘fire. The following are some compositions proper for stars of this kind. 
 
 Chinese fire.—Pulverised gunpowder one pound, sulphur two ounces, iron sand 
 ‘of the first order five ounces. 
 
 Ancient fire.—Pulverised gunpowder one pound, charcoal two ounces, 
 
 Brilliant fire. —Pulverised gunpowder one pound, iron filings four ounces. 
 
 The Chinese fire is certainly the most beautiful. 
 
 VI.—Of Sparks. 
 Sparks differ from stars only in their size and duration; for they are made smaller 
 ‘than stars; and are consumed sooner. They are made in the following manner, 
 Having put into an earthern vessel an ounce of pulverised gunpowder, two ounces 
 of pulverised saltpetre, one ounce of liquid saltpetre, and four ounces of camphor 
 
- 
 ia 
 a 
 
 588 PYROTECHNY. 
 
 | 
 
 | 
 
 | 
 
 reduced to a sort of farina, pour. over this mixture some gum-water, or brandy in 
 which gum-adraganth or gum-arabic has been dissolved, till the composition acquire. 
 the consistence of thick soup. Then take some lint which has been boiled in brandy, 
 or in vinegar, or even in saltpetre, and then dried and _ unravelled, and throw into the: 
 mixture such a quantity of it as is sufficient to absorb it entirely, taking care to stir 
 it well. 
 Form this matter into small balls or globes of the size of a pea; and having dried. 
 them in the sun or the shade, besprinkle them with pulverised gunpowder, in order 
 that they may more readily catch fire, | 
 
 Another method of making Sparks. 
 
 Take the saw-dust of any kind of wood that burns readily, such as fir, elder-teaml| 
 poplar, laurel, &c., and boil it in water in which saltpetre has been dissolved. When 
 the water has pode some time, take it from the fire, and pour it off in such a man- 
 ner that the saw-dust may remain in the vessel. Then place the saw-dust on a. 
 table, and while moist besprinkle it with sulphur, sifted through a very fine sieve: 
 you may add to it also a little bruised gunpowder. Lastly, when the saw-dust has 
 been well mixed, leave it to dry, and make it into sparks as above described. | 
 
 VII._—Of Golden Rain. 
 
 There are some flying-rockets which, as they fall, make small undulations in the 
 air, like hair half frizzled. These are called fusées chevelues, bearded rockets; they 
 finish with a kind of shower of fire, which is called golden rain. The method 
 of constructing them is as follows. 
 
 ‘Fill the barrels of some goose quills with the composition of flying-rockets, ana | 
 place upon the mouth of each a little moist gunpowder, both to keep in the composi- | 
 tion, and to serve asa match. If a flying-rocket be then loaded with these quills, 
 they will produce, at the end, a very agreeable shower of fire, which on account of 
 its beauty has been called golden rain. 4 
 
 | 
 | 
 
 ARTICLE VII. 
 Of some Rockets different in their effect from common rockets. 
 
 Several very amusing and ingenious works are made by means of simple rockets, of 
 which it is necessary that we should here give the reader some idea. 
 
 I.—Of Courantins, or Rockets which fly along a rope. | 
 
 A common rocket, which however ought not to be very large, may be made to 
 run along an extended rope. For this purpose, affix to the rocket an empty cartridge, 
 and introduce into it the rope which is to carry it; placing the head of the rocket . 
 towards that side to which you intend it to move: if you then set fire to the rocket, | 
 adjusted in this manner, it will run along the rope without stopping, till the matter 
 it contains is entirely exhausted. 
 
 If you are desirous that the rocket should move in a retrograde direction; first. 
 fill one half of it with the composition, and cover it with a small round piece of 
 wood, to serve as a partition between it and that put inte the other half; then 
 make a hole below this partition, so as to correspond with a small canal filled with) 
 bruised powder, and terminating at the other end of the rocket: by these means. 
 the fire, when it ceases in the first half of the rocket, will be communicated through: 
 the hole into the small canal, which will convey it to the other end; and this end | 
 being then kindled, the rocket will move backwards, and return to the place from 
 which it set out. 
 
 Two rockets of equal size, bound together by means of a piece of strong pack) 
 thread, and disposed in such a manner that the head of the one shall be opposite 
 
WATER ROCKETS, 589 
 
 to the neck of the other, that when the fire has consumed the composition in the one 
 it may be communicated to that in the other, and oblige both of them to move ina 
 retrograde direction, may also be adjusted to the rope by means of a piece of hollow 
 reed. But to prevent the fire of the former from being communicated to the 
 second too soon, they ought to be covered with oil-cloth, or to be wrapped up in paper. 
 
 Remark.—Rockets of this kind are generally employed for setting fire to various 
 other pieces when large fire-works are exhibited ; and to render them more agreeable, 
 they are made in the form of different animals, such as serpents, dragons, &c.; on 
 which account they are called flying dragons. These dragons are very amusing, espe- 
 cially when filled with various compositions, such as golden rain, long hair, &c. 
 They might be made to discharge serpents from their mouths, which would produce 
 a very pleasing effect, and give them a greater resemblance to a dragon, 
 
 Il.—Rockets which fly along a rope, and turn round at the same time. 
 
 Nothing is easier than to give to a rocket of this kind a rotary motion around the 
 ‘ope along which it advances ; it will be sufficient for this purpose, to tie to it another 
 “ocket, placed in a transversal direction. But the aperture of the latter, instead of 
 yeing at the bottom, ought to be in the side, near one of the ends. If both rockets 
 ye fired at the same time, the latter will make the other revolve around the rope, 
 while it advances along it. 
 
 Ill.—Of rockets which burn in the water. 
 
 Though fire and water are two things of a very opposite nature, the rockets above 
 iescribed, when set on fire, will burn and produce their effect even in the water ; but 
 is they are then below the water, the pleasure of seeing them is lost; for this reason, 
 when it is required to cause rockets to burn as they float on the water, it will be 
 iecessary to make some change in the proportions of the moulds, and the materials 
 of which they are composed. 
 
 In regard to the mould, it may be eight or nine inches in length, and an inch in 
 liameter: the former, on which the cartridge is rolled up, may be nine lines in thick- 
 less, and the rod for loading the cartridge must as usual be somewhat less. For 
 oading the cartridge, there is no need of a piercer with a nipple. 
 | The composition may be made in two ways; for if it be required that the rocket, 
 while burning on the water, should appear as bright as a candle, it must be composed 
 f three materials mixed together, viz., three ounces of pulverised and sifted gun- 
 yowder, one pound of saltpetre, and eight ounces of sulphur. But if you are desirous 
 
 /hat it should appear on the water with a beautiful tail, the composition must consist 
 f eight ounces of gunpowder pulverised and sifted, one pound of saltpetre, eight 
 vunces of pounded and sifted sulphur, and two ounces of charcoal. 
 | When the composition has been prepared according to these proportions, and the 
 iocket has been filled in the manner above described, apply a saucisson to the end of 
 t; and having covered the rocket with wax, black pitch, rosin, or any other 
 ubstance capable of preventing the paper from being spoilt in the water, attach to 
 ba small rod of white willow, about two feet in length, that the rocket may con- 
 eniently float. 
 If it be required that these rockets should plunge down, and again rise up; a cer- 
 ain quantity of pulverised gunpowder, without any mixture, must be introduced into 
 hem, at certain distances, such for example, as two, three, or four lines, according 
 0 the size of the cartridge. 
 
 b 
 
 | Remarks.—I. Small rockets of this kind may be made, without changing the mould 
 r composition, in several different ways, which, for the sake of brevity, we are 
 bliged to omit. Such of our readers as are desirous of further information on this 
 
590 PYROTECHNY. | 
 sil gg may consult those authors who have written expressly on pyrotechny, some 
 of whom we shall mention at the end of the 12th section. 
 
 IJ. It is possible also to make a rocket which, after it has burnt some time on the 
 water, shall throw out sparks and stars; and these after they catch fire shall ascent, 
 into the air. This may be done by dividing the rocket into two parts, by means 0. 
 a round piece of wood, having a hole in the middle. The upper part must be filler 
 with the usual composition of rockets, and the lower with stars, which must be mixet 
 with grained and pulverised gunpowder, &c. 
 
 II. A rocket which takes fire in the water, and, after burning there half the tim 
 
 of its duration, mounts into the air with great velocity, may be constructed in | 
 following manner. 
 
 Take a flying rocket, furnished with its rod, and by means of a little glue attach} 
 to a water rocket, but only at the middle a (Fig. 8.) in such a mannner that th. 
 latter shall have its neck uppermost, and the other its neck downward. Adjustt 
 their extremity B asmall tube, to commuicate the fire from the one to the other 
 and cover both with a coating of pitch, wax, &c., that they ma. 
 v; not be damaged by the water. 
 
 So Then attach to the flying rocket, after it has been thy 
 cemented to the aquatic one, a rod of the kind describedi 
 the 2d article, as seen in the figure at p; and from F suspen| 
 a piece of packthread, to support a musket bullet x, mad 
 fast. to the rod by means of a needle or bit of iron wire, 
 When these arrangements have been made, set fire to the par 
 c after the rocket is in the water ; and when the compositio. 
 is consumed to B, the fire will be communicated through th 
 small tube to the other rocket: the latter will then rise an 
 leave the other, which will not be able to follow it on accoul 
 of the weight adhering to it. 
 
 E | 
 | 
 IV.—By means of rockets, to represent several figures ¢ in the ar. 
 If several small rockets be placed upon a large one, their rods being fixed around th 
 large cartridge, whichis usually attached to the head of the rocket, to contain what | 
 is destined to carry up into the air ; and if these small rockets be set on fire while th. 
 large one is ascending, they will represent, in a very agreeable manner, a tree, th 
 trunk of which will be the large rocket, and the branches the small ones. | | 
 If these small rockets take fire when the large one is half burned in the air, the 
 will represent a comet; and when the large one is entirely inverted, so that its hea’ 
 begins to point downwards, in order to fall, they will represent a kind of fier 
 fountain. I 
 If the barrels of several quills, filled with the composition of flying rockets, « 
 above described, be placed on_a large rocket; when these quills catch fire, they wi 
 represent, toan eye placed below them, a beautiful shower of fire, or of half frizzle 
 hair if the eye be placed on one side. 
 If several serpents be attached to the rocket witha piece of pack-thread, by the ent 
 that do not catch fire; and if the pack-thread be suffered te hang down two or thre 
 inches, between every two, this arrangement will produce a variety of agreeable an| 
 
 amusing figures. 
 
 V.—A rocket which ascends in the form of a screw. | 
 A straight rod, as experience shews, makes a rocket ascend perpendicularly, aim 
 
 a straight line: it may be compared to the rudder of a ship, or the tail of a bird, tl 
 
 effect of which is to make the vessel or bird turn downwards that side to which it 
 
 inclined: ifa bent rod therefore be attached to a rocket, its first effect will be to mal 
 
GLOBES AND BALLS. 591 . 
 
 the rocket incline towards that side to which it is bent; but its centre of gravity 
 bringing it afterwards into a vertical situation, the result of these two opposite 
 efforts will be that the rocket will ascend in a zig-zag or spiral form. In this case 
 
 indeed, as it displaces a greater volume of air, and'describes a longer line, it will not 
 ascend so high, as if it had been impelled in a straight direction; but, on account of the 
 singularity of this motion, it will produce an agreeable effect. 
 
 ARTICLE VIII. 
 Of Globes and Fire Balls. 
 
 We have hitherto spoken only of rockets, and the different kinds of Works Ss 
 ean be constructed by their means. But there are a great many other fireworks, the most 
 
 )remarkable of which we shall here describe. Among these are globes and fire balls ; 
 /some of which are intended to produce their effect in water ; others by rolling or leap- 
 
 | ing on the ground; and some, which are called bombs, do the same in the air. 
 
 I.— Globes which burn on the water. 
 
 | These globes, or fire balls, are made in three different forms; spherical, spheroidal, 
 
 \or cylindrical ; but we shall here confine ourselves to the spherical. 
 
 To make a spherical fire ball, construct a hollow wooden 
 globe of any size at pleasure, and very round both within and 
 without, so that its thickness a cor B D (Fig. 9.), may be equal 
 to about the ninth part of the diameter a B. Insert in the upper 
 part of it aright concave cylinder & ¥F GB, the breadth of which 
 E F may be equal to the fifth part of the diameter a B; and hay- 
 ing an aperture, L M or 0 N, equal to the thickness ac or Bp, 
 that is, tc the ninth part of the diameter az. It is through this 
 aperture that fire is communicated to the globe, when it has 
 been filled with the proper composition, through the lower aper- 
 turerx. A petard of metal, loaded with good powder, is to be introduced also through 
 the lower aperture, and to be placed horizontally as seen in the figure. 
 
 | When thisis done, close up the aperture 1 K, which is nearly equal to the thickness 
 EF or G u, of the cylinder FG H, by means of a wooden tompion dipped in 
 /warm pitch; and melt over it sucha quantity of lead that its weight may cause the 
 globe to sink in water, till nothing remain above it but the part ¢ a; which will 
 be the case if the weight of the lead, with that of the globe and the composition, be 
 | equal the weight of an equal volume of water. If the globe be then placed in the water, 
 
 | the lead by its gravity will make the aperture 1K tend directly downwards, and keep in 
 
 /€@ perpendicular direction the cylinder & F G H, to which fire must have been pre- 
 viously applied. 
 
 To ascertain whether the lead, which has been added to the globe, renders its 
 
 | weight equal to that of an equal volume of water, rub the globe over with pitch or 
 | grease, and make a trial, by placing it in the water. 
 _ The composition with which the globe must be loaded, is as follows: to a pound 
 | of grained powder, and 32 pounds of saltpetre reduced to fine flour, 8 pounds of sul- 
 )phur, ] ounce of scrapings of ivory, and 8 pounds of saw-dust previously boiled in 
 ‘asolution of saltpetre, and dried in the shade or in the sun. 
 
 Or, to 2 pounds of bruised gunpowder, add 12 pounds of saltpetre, 6 pounds of 
 sulphur, 4 pounds of iron filings, and 1 pound of Greek pitch. 
 
 It is not necessary that this composition should be beaten so fine as that intended 
 for rockets; it requires neither to be pulverised nor sifted; it is sufficient if it 
 
 be well mixed and incorporated. But to prevent it from becoming too dry, it 
 will be proper to besprinkle it with a little oil, or any other liquid susceptible of 
 inflammation. 
 
592 : PYROTECHNY. 
 
 T1.—Of Globes which leap or roll on the ground. 
 
 I. Having constructed a wooden globe a, (Fig. 10.) with | 
 cylinder c, similar to that above described, and having loaded i 
 with the same composition, introduce into it four petards, | 
 even more, loaded with good grained gunpowder to their orifice 
 as AB; which must be well stopped with paper or tow. If | 
 globe, prepared in this manner, be fired by means of a match a, 
 c, it will leap about, as it burns, on a smooth horizontal plane 
 according as the petards are set on fire. 
 
 Instead of placing these petards in the inside, they may b 
 affixed to the exterior surface of the globe; which they wil 
 make to roll and leap as they catch fire. They may be applied in any manner to th: 
 surface of the globe, as seen in the figure. i) 
 
 II. A similar globe may be made to roll about on a horizontal plane, with a ver! 
 rapid motion. Construct two equal hemispheres of pasteboard, and adjust in one o| 
 
 Fig. 11. them, as a B, (Fig. 11.), three common rockets c,p, £, fille! 
 and pierced like flying rockets which have no petard: thes 
 rockets must not exceed the interior breadth of the hemisphere) 
 and ought to be arranged in such a manner, that the head of thi 
 one shall correspond to the tail of the other. 
 
 The rockets being thus arranged, join the two hemispheres, by 
 cementing them together with strong paper, in such a manner, 
 that they shall not separate, while the globe is moving and turning, at the same time 
 that the rockets produce tleir effect. To set fire to the first, make a hole in the 
 globe opposite to the tail of it, and introduce intoit amatch. This match will com. 
 municate fire to the first rocket; which, when consumed, will set fire to the second! 
 by means of another match, and so on to the rest; so that the globe, if placed ona’ 
 smooth horizontal plane, will be kept in continual motion. ( 
 
 It is here to be observed, that a few more holes must be aa in the globe, other- 
 wise it will burst. | 
 
 The two hemispheres of pasteboard may be prepared in the following manner: 
 construct a very round globe of solid wood, and cover it with melted wax ; ‘tat 
 cement over it several bands of coarse paper, about two inches in breadth, giving it’ 
 several coats of this kind, to the thickness of about two lines. Or, what will be still 
 easier and better, having dissolved, in glue water, some of the pulp employed by the! 
 paper makers, cover with it the surface of the globe; then dry it gradually at a slow 
 fire, and cut it through in the middle; by which means you will have two strong 
 hemispheres. The wooden globe may be easily separated from the pasteboard by” 
 means of heat; for if the whole be applied to a strong fire the wax will dissolve, 
 so that the globe may be drawn out. Instead of melted wax, soap may be em- 
 ployed. 
 
 Il. _of Aérial Globes, called Bombs. 
 
 These globes are called aérials, because they are thrown into the air from a mortar, 
 which is a short thick piece of artillery of a large calibre. | 
 
 Though these globes are of wood, and have a suitable thickness, namely, equal to 
 the twelfth part of their diameters, if too much powder be put into the mortar, they 
 will not be able to resist its force; the charge of powder therefore must be propor-— 
 tioned to the globe to be ejected. The usual quantity is an ounce of powder fora | 
 globe of four pounds weight; two ounces for one of eight, and so on. | 
 
 As the chamber of the mortar may be too large to contain the exact quantity of 
 powder sufficient for the fire ball, which ought to be placed immediately above the 
 
AERIAL GLOBES. 593 
 
 yowder, in order that it may be expelled and set on fire at the same time, another 
 mortar may be constructed of wood, or of pasteboard with a wooden 
 bottom, as a B, (Fig. 12.) It ought to be put intoa large iron mortar, 
 and to be loaded with a quantity of powder proportioned to the 
 weight of the globe. 
 
 This small mortar must be of light wood, or of paper pasted 
 together, and rolled up in the form of a cylinder, or truncated cone, 
 the bottom excepted ; which, as already said, must be of wood. The 
 ) chamber for the powder a c must be pierced obliquely, with a small 
 imblet, as seen at BC; so that the aperture B, corresponding to the aperture of the 
 netal mortar, the fire applied to the latter may be communicated to the powder 
 vhich is at the bottom of the chamber ac, immediately below the globe. By these 
 neans the globe will catch fire, and make an agreeable noise as it rises into the air; 
 yut it would not succeed so well, if any vacuity were left between the powder and 
 she globe. 
 
 _ A profile or perpendicular section of such a globe is represented by the right-angled 
 parallelogram aBcop, (Fig. 13.), the breadth of which a B is 
 nearly equal to the height ap. ‘The thickness of the wood, 
 towards the two sides, L, mM, is equal, as above said, to the 
 twelfth part of the diameter of the globe; and the thickness, 
 E F, of the cover, is double the preceding, or equal to a sixth part 
 of the diameter. The height @ x or HI of the chamber, HI K, 
 where the match is applied, and which is terminated by the 
 semicircle L GH M, is equal to the fourtb part of the breadth ax; 
 and its breadth cH is equal to the sixth part of a B, 
 
 _ We must here observe that it is dangerous to put wooden covers, such ag EF, on 
 rial balloons or globes: for these covers may beso heavy, as to wound those on 
 vhom they happen to fall. It will be sufficient to place turf or hay above the 
 lobe, in order that the powder may experience some resistance. 
 
 The globe must be filled with several pieces of cane or common reed, equal in 
 ength to the interior height of the globe, and charged with a slow composition, 
 nade of three ounces of pounded gunpowder, an ounce of sulphur moistened with 
 , small quantity of petroleum oil, and two ounces of charcoal: and in order that 
 hese reeds or canes may catch fire sooner, and with more facility, they must be 
 harged at the lower ends, which rest on the bottom of the globe, with pulverised 
 junpowder moistened in the same manner with petroleum oil, or well besprinkled 
 ‘vith brandy, and then dried. 
 
 | The bottom of the globe ought to be covered with a little gunpowder half pulve- 
 ‘sed and half grained; which, when set on fire, by means of a match applied to the 
 nd of the chamber a u, will set fire to the lower part of the reed. But care 
 aust have been taken to fill the chamber with a composition similar to that in the 
 eds, or with another slow composition made of eight ounces of gunpowder, four 
 “unces of saltpetre, two ounces of sulphur, and one ounce of charcoal; the whale 
 ust be well pounded and mixed, 
 
 _ Instead of reeds, the globe may. be charged with running rockets, or paper petards, 
 id a quantity of fiery stars or sparks mixed with pulverised gunpowder, placed 
 _ ithout any order above these petards, which must be choaked at unequal heights, 
 _ tat they may perform their effect at different times. 
 
 These globes may be constructed in various other ways, which it would be tedious 
 2re to enumerate. We shall only observe that, when loaded, they must be well 
 »vered at the top; they must be segs in apiece of cloth dipped in glue, and 
 
 Se 
 
594 PYROTECHNY. 
 
 a piece of woollen cloth must be tied round them, so as to cover the hole which con 
 
 tains the match. , 
 ARTICLE IX. ! 
 Jets of Fire. 
 
 Jets of fire are a kind of fixed rockets, the effect of which is to throw up into the ai. 
 jets of fire, similar to jets of water. They serve also to represent cascades ; for if; 
 series of such rockets be placed horizontally on the same line, it may be easily seer 
 that the fire they emit will resemble a sheet of water. When arranged in a circula 
 form, like the radii of a circle, they form what is called a fixed sun. ! 
 
 To form jets of this kind, the cartridge for brilliant fires must, in thickness 
 be equal to a fourth part of the diameter; and for Chinese fire, only to a sixtl 
 
 art. 
 : The cartridge is loaded on anipple, having a point equal in length to the same diame 
 ter, and in thickness to a fourth part of it; butas it generally happens that the moutl 
 of the jet becomes larger than is necessary for the effect of the fire, you must begir 
 to charge the cartridge, as the Chinese do, by filling it to a height equal to a foutl 
 part of the diameter with clay, which must be rammed down as if it were gunpowder: 
 By these means the jet will ascend much higher. When the charge is completed witl, 
 the composition you have made choice of, the cartridge must be closed with a tompioi 
 of wood, above which it must be choaked. 
 
 The train or match must be of the same composition as that employed for loading | 
 otherwise the dilatation of the air contained in the hole made by the piercer, wouli| 
 cause the jet to burst. | 
 
 Clayed rockets may be pierced with two holes near the neck, in order to haw 
 three jets in the same plane. e 
 
 If a kind of top, pierced with a number of holes, be added to them, they will imitat 
 a bubbling fountain. 
 
 Jets intended for representing sheets of fire ought not to be choaked. They mus' 
 be placed in a horizontal position, or inclined a little downwards. | 
 
 It appears to us that they might be choaked so as to form a kind of slit, and bi! 
 pierced in the same manner; which would contribute to extend the sheet of fire stil 
 farther. A kind of long narrow mouths might even be provided for this particula’ 
 purpose, 
 
 PRINCIPAL COMPOSITIONS FOR JETS OF FIRE. 
 
 \ 
 
 Ist. For Jets of 5 lines or less, of interior diameter. =f 
 
 Chinese fire.—Saltpetre 1 pound, pulverised gunpowder 1 pound, sulphur € 
 ounces, charcoal 2 ounces. 
 
 White fire.—Saltpetre 1 pound, pulverised gunpowder 8 ounces, sulphur 3 ounces, 
 charcoal 2 ounces, iron sand of the first order 8 ounces. 
 
 2d. For Jets of from 10 to 12 lines in diameter. : 
 Brilliant fire.—Pulverised gunpowder 1 pound, iron-filings of a mean size, 5 ounces. 
 White fire.—Saltpetre ] pound, pulverised gunpowder 1 pound, sulphur 8 ounces, 
 charcoal 2 ounces. | 
 Chinese fire.—Saltpetre 1 pound 4 ounces, sulphur 5 ounces, charcoal 5 ounces, 
 sand of the third order 12 ounces. | 
 | 
 3d. For Jets of 15 or 18 lines in diameter. 
 
 Chinese fire.—Saltpetre 1 pound 4 ounces, sulphur 7 ounces, charcoal 5 ounces, of 
 the six different kinds of sand mixed 12 ounces. a 
 
FIRES OF DIFFERENT COLOURS. 595 
 
 _ Father d’Incarville, in his memoirs on this subject, givés various other proportions 
 for the composition of these jets; but we must confine ourselves to what has been 
 here said, and refer the reader to the autbor’s memoirs, which will be found in 
 the ‘‘ Manual de 1’ Artificier.” 
 
 The saltpetre, pulverised gunpowder, and charcoal, are three times sifted througha 
 hair sieve. The iron sand is besprinkled with sulphur, after being moistened witha 
 little brandy, that the sulphur may adhere to it; and they are then mixed together : 
 ithe sulphured sand is then spread over the first mixture, and the whole is mixed with 
 a ladle only; for if a sieve were employed, it would separate the sand from the 
 other materials. When sand larger than that of the second order is used, the 
 ‘composition is moistened with brandy, so that it forms itself into balls, and the jets 
 
 are then loaded: if there were too much moisture, the sand would not perform its 
 effect. 
 
 ARTICLE X. 
 
 Of Fires of different Colours. 
 
 It is much to be wished that, for the sake of variety, different colours could be 
 given to these fire-works at pleasure; but though we are acquainted with several 
 materials which communicate to flame various colours, it has hitherto been possible 
 to introduce only a very few colours into that of inflamed gunpowder. 
 
 To make white fire, the gunpowder must be mixed with iron or rather steel- 
 ilings, 
 
 _ To make red fire, iron sand of the first order must be em 
 
 ployed in the same 
 nanner. 2 
 
 _ As copper filings, when thrown into a flame, render it green, it might be concluded, 
 hat if mixed with gunpowder, it would produce a green flame; but this experiment 
 loes not succeed. It is supposed that the flame is too ardent, and consumes the 
 nflammable part of the copper too soon. But it is probable that a sufficient 
 iumber of trials have not yet been made; for is it not possible to lessen the force of 
 junpowder in a considerable degree, by increasing the dose of the charcoal ? 
 
 _ However, the following are a few of those materials which, in books on Pyro- 
 echny, are said to possess the property of communicating various colours to fire- 
 vorks. 
 
 Camphor mixed with the composition, makes the flame to appear of a pale white 
 olour. 
 
 _ Raspings of ivory give a clear flame of a silver colour, inclining a little to that of 
 ead; or rather a white dazzling flame. 
 Greek pitch produces a reddish flame, of a bronze colour. 
 
 Black pitch, a dusky flame, like a thick smoke, which obscures the atmosphere. 
 _ Sulphur, mixed in a moderate quantity, makes the flame appear bluish, 
 
 Sal ammoniac and verdigrise give a greenish flame. 
 
 Raspings of yellow amber communicate to the flamea lemon colour. 
 _ Crude antimony gives a russet colour. 
 | Borax ought to produce a blue flame ; for spirit of wine, in which sedative salt, one 
 f the component parts of borax, is dissolved by the means of heat, burns with a 
 eautiful green flame. 
 
 Much, however, still remains to be done in regard to this subject ; but it would 
 ld to the beauty of artificial fireworks, if they could be varied by giving them dif- 
 Tent colours: this would be creating for the eyes a new pleasure, 
 | ARTICLE XI. 
 
 Composition of a Paste proper for representing animals and other devices in fire. 
 
 It is to the Chinese also that we are indebted for this method of re 
 
 presenting 
 2 ¢. 2 
 
596 - PYROTECHNY, 
 
 ‘figures with fire. For this purpose, take sulphur reduced to'an impalpable powder, 
 and having formed it into a paste with starch, cover with it the figure you are desir.) 
 ous of representing on fire: it is here to be observed, that the figure must first be 
 coated over with clay, to prevent it from being burnt. 
 
 When the figure has been covered with this paste, besprinkle it while still moist, 
 with pulverised gunpowder; and when the whole is perfectly dry, arrange some smal], 
 matches on the principal parts of it, that the fire may be speedily communicated to it] 
 on all sides. ' 
 
 The same paste may be employed on figures of clay, to form devices and various, 
 designs. Thus, for example, festoons, garlands, and other ornaments, the flowers of 
 which might be imitated by fire of different colours, could be formed on the frieze o} 
 a piece of architecture covered with plaster. The Chinese imitate grapes exceedingly 
 well, -by mixing pounded sulphur with the ay of the jujube, instead of flour 
 
 paste. 
 ARTICLE XII. 
 
 Of Suns, both fixed and moveable. 
 
 None of the pyrotechnic inventions can be employed with so much success, in. 
 artificial fire-works, as suns; of which there are two kinds, fixed and revolving : the 
 method of constructing both is very simple. 
 
 For fixed suns cause to be constructed a round piece of wood, into the circumference’ 
 of which can be screwed twelve or fifteen pieces in the form of radii; and to these. 
 radii attach jets of fire, the composition of which has been already described; so that’ 
 they may appear as radii tending to the same centre, the mouth of the jet being’ 
 towards the circumference. Apply a match in such a manner, that the fire communi-, 
 cated at the centre may be conveyed, at the same time, to the mouth of each of the 
 jets, by which means, each throwing out its fire, there will be produced the appear- | 
 ance of aradiating sun. We here suppose that the wheel is placed in a position per-, 
 pendicular to the horizon. 
 
 These rockets or jets may be so arranged as to cross each other in an angular manner ; 
 in which case, instead of a sun, you will have a star, or a sort of cross resembling that’ 
 of Malta. Some of these suns are made also with several rows of jets: these are 
 called glories. : 
 
 Revolving suns may be constructed in this manner. Provide a wooden wheel, of) 
 any size at pleasure, and brought into perfect equilibrium around its centre, in order. 
 that the least effort may make it turn round. Attach to the circumference of it fire-, 
 jets placed in the direction of the circumference ; they must not be choaked at the 
 bottom, and ought to be arranged in such a manner that the mouth of the one shall) 
 be near the bottom of the other, so that when the fire of the one is ended, it may 
 immediately proceed to another. It may be easily perceived, that when fire is applied 
 to one of these jets, the recoil of the rocket will make the wheel turn round, unless 
 it be too large and ponderous: for this reason, when these suns are of a consider-' 
 able size, that is when they consist for example of 20 rockets, fire must be communi- 
 cated at the same time to the first, the sixth, the eleventh, and the sixteenth; from, 
 which it will proceed to the second, the seventh, the twelfth, the sevéntenmen and 
 soon. These four rockets will make the wheel turn round with rapidity. | 
 
 If two similar suns be placed one behind the other, and made to turn in a con-| 
 trary direction, they will produce a very pretty effect of cross-fire. 
 
 Three or four suns, with horizontal axes passing through them, might be im-| 
 planted in a vertical axis, moveable in the middle of a table. These suns, revolving 
 around the table, will seem to pursue each other. Jt may be easily perceived that; 
 to make them turn around the table, they must be fixed on their axes, and these 
 axes, at the place where they rest on the table, ought to be furnished witha very move: 
 able roller. 
 
 | 
 
 I 
 
¥ 
 
 { 
 H _ OPTICAL PYROTECHNY. 597 
 
 _ We shall say nothing farther on artificial fireworks; because it is not possible in 
 ‘this work to give a complete treatise of Pyrotechny. We shall therefore content 
 ourselves with pointing out, to those who are fgnd of this art, a few of the best 
 authors gn the subject. One is, ‘ 'Traité des Feux d’Artifice de M. Frezier,’’ anew 
 edition of which was published in 1745. We shall mention also the work of M. 
 Perrinet d’Orval, entitled “ Traité des Feux d’Artifice, pour le Spectacle et pour 
 Ja Guerre.” To these we may add ‘‘ Le Manuel de 1’Artificier,” Paris 1757, 12mo. 
 which contains, in a very small compass, the whole substance of the art of making 
 ‘artificial fireworks: it is an abridgment of the latter work, augmented with several 
 ‘new and curious compositions, in regard to the Chinese fire, by Father d’Incarville. 
 
 ARTICLE. XIII, 
 Of Ointment for Burns. 
 
 It is proper that we should terminate a treatise on pyrotechny by some remedy for 
 burns; as accidents must often take place in handling such a dangerous element as 
 fire. We shall therefore not hesitate to follow the example of Ozanam, who in this 
 respect is himself a follower of Siemienowitcz, and the greater part of those who 
 have written on this subject: we shall even confine ourselves to the remedy he 
 proposes. : 
 
 Boil fresh hog’s lard in common water, over a slow fire; skim it continually till 
 no more scum is left, and let the melted lard remain in the open air for three or four 
 ‘nights. Melt it again in an earthen vessel, over a slow and moderate fire, and strain 
 ‘it into cold water through a piece of linen cloth; then wash it well in pnre river or 
 ‘spring water, to free it from its salt, and to make it become white; then press it into 
 ‘a glazed earthern vessel and preserve it for use. 
 
 It generally happens, in cases of burning, that the skin rises in blisters, which 
 however must not be opened till the third or fourth day after the ointment has been 
 applied. : . 
 
 ARTICLE XIV. 
 Pyrotechny without fire, and merely Optical. 
 
 As the inventions which we have here described, are necessarily attended with 
 ‘considerable expense, and are besides dangerous, attempts have been made in modern 
 ames, and with a considerable degree of success, to imitate the different kinds of 
 ire-works by optical effects, and to give them the appearance of motion, though in 
 ‘eality fixed. By means of this invention, the spectacle of artificial fire-works may 
 be exhibited at a very small expense, and if the pieces employed are constructed with 
 ngenuity, if the rules of perspective are properly observed, and if, in viewing the 
 spectacle, glasses which magnify the objects and render them somewhat less distinct 
 ye employed, a very agreeable illusion will be produced. 
 | The artificial fire-works imitated with most success by this invention, are fixed suns, 
 serbes and jets of fire, cascades, globes, pyramids and columus moveable around their 
 es. To represent a gerbe of fire, take paper blackened on both sides, and very 
 
 , opake, and having delineated on a piece of white paper the 
 fig. 14. figure of a gerbe of fire, apply it to the black paper, and 
 tie with a point of avery sharp penknife make several slashes 
 Saieay (Fig. 14.) in it, as 3, 5 or 7, proceeding from the origin 
 eae of the gerbe: these lines must not be continued but cut through 
 
 at unequal intervals. Pierce these intervals with unequal holes 
 made with a pinking iron, (Fig. 14.), in order to represent the 
 sparks of such a gerbe. In short you must endeavour to paint, 
 by these lines and holes, the well known effect of the fire of 
 
” b 
 ‘ ¥ 
 
 ‘ 
 
 * 
 
 ' 
 * 
 
 598 . PYROTECHNY. 
 
 inflamed gunpowder, when it issues through a small aper- 
 Fig. 15. ture. 
 
 ; According to the same principles, you may delineate the case 
 cades (Fig. 15) and jets of fire which you are desirous pf intro- 
 ducing into this exhibition, which is purely optical; and those 
 jets of fire which proceed from the radii of suns, either fixed or 
 moveable. It may easily be conceived that in this operation 
 taste must be the guide. 
 
 If you are desirous of representing globes, pyramids, or revolv- 
 ing columns, draw the outlines of them on paper, and then cut 
 them out in a helical form; that is, cut out spirals with the 
 
 _ point of a penknife, and of a size proportioned to that of the 
 piece. 
 
 It is to be observed also, that as these different pieces have 
 different colours, they may be easily imitated by pasting on the 
 = back of the paper, cut as here described, very fine silk paper 
 coloured in the proper manner. As jets, for example, when 
 loaded with Chinese fire, give a reddish light, you must paste 
 to the back of these jets transparent paper, slightly tinged with 
 red; and proceed in the same manner in regard to the other 
 colcurs by which the different fire- works are distinguished. 
 
 When these preparations have been made, the next thing is to give motion, or the 
 appearance of motion, to this fire, which may be done two ways according to cir- 
 cumstances. : 
 
 If a jet of fire, for example, is to be represented, prick unequal holes, and at unequal 
 distances from each other, ina band of paper, (Fig. 17.), and then move 
 this band, making it ascend between a light and the above jet: the 
 rays of light which escape through the holes of the moveable paper 
 will exhibit the appearance of sparks rising into the air. It is to be 
 observed that one part of the paper must be whole, that another must 
 be pierced with holes thinly scattered; that in another place they must 
 be very close, and then moderately so: by these means it will represent 
 those sudden jets of fire observed in fire-works. 
 
 To represent a cascade, the paper pierced with holes, instead of 
 moving upwards, must be made to descend. . 
 
 This motion may be easily produced by means of two rollers, on one 
 of which the paper is rolled up while it is unrolled from the other. 
 
 Suns are attended with some more difficulty; because in these it is 
 necessary to represent fire proceeding from the centre to the circumfe- 
 rence. The artifice for this purpose is as follows. 
 
 On strong paper describe a circle, equal in diameter to the sun which you are desi- 
 rous to exhibit, or even somewhat larger; then trace out on this circle two spirals, at 
 the distauce of a line or half a line from each other, and open the interval between 
 them with a penknife, in such a manner, that the paper may be cut from the circum- 
 
 ference, decreasing in breadth toa certain distance from the 
 
 Fig. 18. centre, (Fig. [8.); cut the remainder of the circle into spirals 
 
 , of the same kind, open and close alternately, then cement the 
 paper circle to a small iron hoop, supported by two pieces of 
 iron, crossing each other in its centre, and adjust the whole toa” 
 small machine, which will suffer it to revolve round its centre. ' 
 If this moveable paper circle, cut in this manner, be placed 
 aes before the representation of your sun, with a light behind it, as 
 soon as if is made to move towards that side to which the convexity of the spirals is 
 
OPTICAL PYROTECHNY. 599 
 
 turned, the luminous spirals, or those which afford a passage to the light, will give, on 
 
 ‘the image of the radii or jets of fire of your sun, the appearance of fire in continual 
 
 ‘notion, as if undulating from the centre to the circumference. 
 
 _ The appearance of motion may be given to columns, pyramids, and globes, cut 
 
 vhrough in the manner above described, by moving upwards, in a vertical direction,.a 
 yand of paper cut through into apertures inclined at an angle somewhat different from 
 
 hat of the spirals. By these means the spectators will imagine that they see fire 
 continually circulating and ascending along these spirals; and the result will be a sort 
 of illusion, in consequence of which the columns or pyramids will seem to revolve 
 vith them. 
 
 | But we shall not enlarge farther on this subject; it is sufficient to have explained 
 he principle on which this cheap kind of pyrotechny can be exhibited; the taste of 
 he artist may suggest to him many things to give more reality to this representation, 
 
 and to render the deception stronger. 
 
 _ We shall however add a few words respecting illuminations which form a part of 
 oyrotechny. 
 
 | Take some prints respecting a castle, or palace, &c.; and having coloured them 
 
 yroperly, cement paper to the back of them, in such a manner that they shall be only 
 
 jemi-transparent; then, with pinking irons of different sizes, prick small holes in 
 he places and on the lines where lamps are generally placed, as along the sides of the 
 windows, on the cornices, or balustrades, &c. But care must be taken to make these 
 ioles smaller and closer, according to the perspective diminution of the figure. With 
 ther irons of a larger size, cut out, in other places, some stronger lights; soas to 
 epresent fire-pots, &c. Cut out also the panes in some of the windows, and cement 
 
 0 the back of them transparent paper of a green or red colour, to represent curtains 
 
 lrawn before them, and concealing an illuminated apartment. 
 
 _ When the print is cut in this manner, place it in the front of a sort of small 
 heatre, strongly illuminated from the back part, and look at it through a convex glass 
 fa pretty long focus, like that used in those small machines called optical boxes 
 f the rules of perspective have been properly observed in the prints, and if the lights 
 ind shades have been distributed with taste, this spectacle will be highly agreeable. 
 lt may be intermixed with some of the pyrotechnic artifices above described; as such 
 lluminations are in general accompanied with fire-works, 
 
600 PHILOSOPHY, ~ 
 
 PART ELEVENTH. | 
 
 CONTAINING EVERY THING MOST CURIOUS IN PHILOSOPHY IN 
 GENERAL, AND IN ITS VARIOUS BRANCHES. 
 
 Havina gone through the different parts of the Mathematics, and of the sciences or, 
 arts comprehended under that head, we now enter the field of Philosophy, which pre-. 
 sents as many objects worthy of exciting our curiosity as the mathematics; or, rather, 
 which is indeed still more fertile in that respect, and affording matter cull better. 
 adapted to the comprehension of the generality of readers. This matter is even so, 
 abundant, that we can scarcely establish divisions in it ; sothat this part of our work | 
 will be a kind of a miscellany, without much order, of every thing that belongs to 
 philosophy in general. We shall successively review in it the principal properties of 
 bodies ; the inventions, whether useful or amusing, to which these properties have 
 given birth’ various questions relating to the system of the world, to meteors, and 
 the origin of springs, with other objects, which it would be too tedious to en umerate,. 
 
 But before we enter this vast field, it is necessary that we should establish some. 
 -general principles, which we shall doi in the following account of what philosophers, 
 have called the four elements, viz., fire, air, water, and earth. 
 
 PRELIMINARY DISCOURSE, 
 On the Elements of Bodies. 
 
 In analyzing any material, when we have arrived at its last component parts, 
 and cannot decompose them farther, we ought to regard them as its elements. Now. 
 every one knows that all or most bodies, submitted to analysis, furnish a fixed matter; 
 also something that is inflammable; an invisible fluid, which manifests itself only 
 by its expansibility and its resistance; and lastly, another which heat raises into 
 vapours, and which afterward re-unites under a visible form. These four component; 
 parts have been named earth, fire, air,and water. 'These enter into the composition 
 of most bodies; but cannot themselves be decomposed. They ought therefore to 
 be considered as the elements of all other bodies; which justifies the common deno- 
 mination, which has been established almost from the first dawn of philosophy, accord- 
 ing to which there are in nature the four elements, fire, air, water, and earth. 
 
 I.—Of Fire, both elementary and material. 
 
 What is Fire? This is perhaps one of the most obscure questions in philosophy, and 
 the least susceptible of a satisfactory answer. The most probable account, however 
 which its known properties enable us to give, is the following. 
 
 Fire is a fluid universally diffused throughout nature; which penetrates all bodies 
 with more or less facility ; is susceptible of being accumulated in some of them, and 
 this accumulation produces, in regard to us, that sensation which we call heat. 
 When this accumulation is carried to a higher degree, it produces inflammation and 
 combustion, which are always accompanied with light. In every state, this fluid 
 
| , 
 | . OF FIRE. 601 
 
 dilates bodies in proportion to the greater or less quantity of it present; and it at 
 length separates their parts, which we call fusing, burning, calcining. 
 
 That fire is a fluid, there can be no reason to doubt: for if it were not, how could 
 it be diffused Pcavhcat the atmosphere, and throughout water, without forming an. 
 obstacle to the motion of bodies ? How could it penetrate the densest and most 
 compact bodies, such for example-as metals ? 
 
 Nay, fire is not only a fluid, but it is even the principle of all auiditye without its 
 influence, all the fluids with which we are acquainted, would be reduced to masses 
 absolutely solid. Metals become fixed at a degree of heat far superior to that of 
 boiling water. Water loses its fluidity as soon as the heat or quantity of fire has been 
 diminished to a certain degree. Spirit of wine, and even mercury, are congealed by 
 the progressive diminution of heat. There isa degree of cold, or privation of heat, 
 perhaps, which would convert air into a fluid-like water, and even into a solid body ; 
 but we are at a prodigious distance from that term. 
 | Fire penetrates all bodies with more or less facility. This follows from the commu- 
 nication of heat from a hot to a cold body. It is with greater or less, a moderate 
 facility, and not with extreme facility, that heat is communicated : for it is well known 
 that this communication is not instantaneous: if the point of a pretty long needle 
 be presented to the flame of a taper, both its ends do not become equally hot at the 
 same time. One body receives this heat more readily than another; or, as we may 
 say, has a greater affinity for heat. 
 
 The accumulation of the igneous fluid produces on our bodies that sensation which we 
 call heat. This requires no proof; but the sensation is only relative. As long as 
 the palm of the hand, for instance, is hotter than the body with which it is in contact, 
 the latter seems to us cold; but it will, on the contrary, seem warm, if the hand 
 be colder, or contains less of the igneous fluid; or if that fluid tends to pass 
 gradually, as it does, from that body into the hand. Every person almost is 
 acquainted with the following experiment: heat one hand in a very high degree, 
 and cool the other almost to the temperature of ice; if you then immerse both of 
 them into tepid water, the one will experiencea selieation of cold, and the other of heat. 
 
 This accumulation, carried to a considerable degree, produces inflammation, always 
 wecompanied with light. It results from some experiments made by Buffon, that 
 ‘ron exposed, without being in contact, to the action of another ‘body in a state of 
 nflammation, becomes itself inflamed and red. But an ignited body is nothing else 
 than a body in which the ingneous fluid is accumulated to such a degree as to become 
 ‘uminous. All light indeed is not accompanied with heat; but all heat, carried to a 
 on degree, produces light. 
 
 | Has fire ‘weight? It appears to us that there can be no doubt that fire is heavy: it 
 $8 matter, since it acts upon matter; and therefore it must possess weight. But the 
 juestion is, to know whether this weight is perceptible, and can be indicated by the 
 nstruments which we employ. S’Gravesande and Muschenbroeck made some expe- 
 ‘imeuts on this subject; but they found no difference between masses of ignited iron, 
 yt iron penetrated with fire, and the same masses when cold. They however con- 
 
 Hluded from them, that as ignited iron, which by its increasing in volume ought to 
 veigh somewhat less in air, weighed the same in that state as when cold, this equa- 
 ity must have arisen from the addition of the weight of the fire present init. But 
 heir experiments were not made with the necessary degree of care. 
 
 _ Buffon, who, by means of the forges belonging to him, was enabled to make a 
 hiteh greater number of experiments, and on a larger scale, always found that pieces 
 f forged’ iron, brought to a state of ignition, weighed a little more than when cold; 
 nd he fixed the diminution at a 600th part of the weight of the ignited body. But 
 > must be allowed, and Buffon was sensible of it himself, that this experiment could 
 ot be decisive : fok he has shewn, that iron kept for some time at a red heat, con- 
 
602 PHILOSOPHY. ; 
 = 
 tinually loses a part of its weight, because it gradually burns: on this account he 
 made further experiments on a substance very common in furnaces, namely slag. He 
 first assured himself that slag retains its weight, or loses only an insensible portion 
 of it, in consequence of being heated and cooled again. He then weighed some of 
 this slag cold, by a very delicate balance; he next brought it toa white heat, and then 
 weighed it a second and a third time after it had cooled. Five experiments of this 
 kind always gave an excess of weight in the ignited slag, above that which it had 
 when cold, both before and after. This difference amounted to a 580th or a 600th: 
 part of that of the piece of slag. | 
 
 But it may be said, if this be the case, fire is heavier than air; for the specific 
 gravity of slag is to that of water, as 24to 1; therefore this gravity is to that of air 
 as 2125 tol. But the fire contained in a piece of ignited slag, is about a 600th part 
 of its weight; consequently it is to the weight of an equal volume of air, as 34 to 1, 
 which is not credible. So great is the tenuity of fire, that we can hardly allow our. 
 selves to think that its specific gravity approaches even near to that of air. 
 
 But it must be observed, that in an ignited mass brought to a white heat, a large 
 quantity of fire is accumulated the weight therefore of fire, in its ordinary state, ' 
 and in bodies heated to the mean temperature of our atmosphere, may be utterly 
 insensible; but when five or six hundred times, or more than that quantity, has 
 been accumulated, and to such a degree as to produce ignition, its gravity may then 
 become sensible. Let us suppose, for example, that the fire diffused throughout air, 
 heated to 1 degree of the thermometer, weighs only the 300th part of the weight of 
 that air; when five or six hundred parts more have been introduced into it, to pro- 
 duce ignition, its weight may then equal, and even surpass, the weight of such air as 
 we breathe. We do not know whether this would have been Buffon’s answer ; ba 
 such, in our opinion, is that which might be given. 
 
 Those persons however are mistaken, who consider the increase of weight which 
 metals acquire by calcination, as a Ehooe of the heaviness of fire, which by this ope-' 
 ration they suppose to become fixed, and in some measure solidified along with’ 
 the metallic calees. It is now known that fire has no part in this augmentation of 
 weight. | 
 
 Fire dilates bodies: by dilating them, it separates their constituent parts, and at 
 length liquefies them. This phenomenon, in regard tothe effect, is well known. 
 That fire dilates bodies, is well known, as will be shewn hereafter by means of a 
 very ingenious machine, which serves to determine the degree and ratio of this’ 
 dilatation. But it cannot produce this effect without separating the consti. ' 
 tuent parts of these bodies; and this is the mechanism by which it is afterwards 
 able to liquefy, and even to volatilize them; for the solidity cf a body is the effect of 
 the mutual adhesion of its constituent parts; an adhesion which, in all probability, is 
 produced by the contact of these molecule in large surfaces. But when fire, intro- 
 duced between them, produces a separation, and causes them scarcely to touch each 
 other, the bod has then attained to such an extreme degree of fluidity, as to be. vo- 
 jected 
 
 These particles, having no longer any adhesion, can be carried away by the least 
 effort, such as that of heat, which continually exercises an action to extend itself in’ 
 every direction. 
 
 There are some bodies, bowever, which fire at first tends to contract: but this is 
 because they contain pridciples which the fire dissipates: of this kind is clay, which 
 at first shrinks in the fire; but, if exposed to a greater degree of heat, it dilates, © 
 liquefies, and is changed into glass. ‘ | 
 
 Il.—Of Air. 
 Air is an elastic, heavy fluid, susceptible of compression; which expands by heat, © 
 
| 
 i OF AIR. 603 
 and contracts by cold. It is necessary for maintaining life to all the animals with 
 which we are.acquainted ; it becomes charged with, and combines with, water, as 
 water combines withit. Such are the principal properties of air, of which we shall 
 nere give a general view, and which we shall prove hereafter by some curious experi- 
 nents. 
 
 _ Air is a heavy fluid. To discover this property in air, and to prove its existence, 
 equires only a slight knowledge of philosophy. It may be demonstrated by a very 
 nmple experiment. Take a glass globe, 6 inches in diameter, furnished with a tube 
 ‘hat can be opened and shut by a stop-cock ; exhaust it of air by means of a pneumatic 
 nachine, and then shut it, so as to exclude the external air; weigh the globe thus 
 ixhausted of air by a very nice balance; if you then admit the external air, by turn- 
 ng the cock, the equilibrium will be immediately destroyed, and that end of the 
 valance which supports the globe will preponderate. For a globe of the size 
 ‘ove mentioned, 45 or 50 grains must be added to the weight, to restore the equi- 
 ibrium. 
 
 , Ar ts an elastic fluid. This may be proved by the following very simple experi- 
 aent. Introduce air into a bladder, but in such a manner as not entirely to fill it. 
 f the bladder be then carried, in that state, to the summit of a mountain, it will 
 je more and more distended ; and by carrying it to the top of a very high mountain, 
 uch as the Cordilleras of Peru, it might be made to-distend so much as to 
 urst. 
 
 | The same effect will be produced if the bladder be placed under the receiver of an 
 ir-pump. For if the receiver be then exhausted of air, on the first stroke 
 f the piston, the bladder will swell out, even if it contains only an inch of air; 
 od when the external air is suffered to re-enter the receiver, it will resume its 
 xrmer state. 
 
 There can be no doubt that this effect is produced by the elastic force of the air; 
 thich, when the pressure of the external air is removed, increases in volume; and 
 hen the pressure is restored, it assumes its former state. It is like a spring, more 
 less compressed by a weight, and which extends itself in a greater or less degree, 
 s the weight is heavier or lighter. 
 
 Air isa fluid susceptible of compression. This is a consequence of its elasticity. 
 
 It is proved, by experience, that a double weight compresses it in such a manner 
 3 to occupy only one half of its former volume; a quadruple weight reduces it to a 
 surth part of that volume, and so on. So that it may be said in general that the 
 ime mass of air, the temperature remaining the same, occupies a volume which is 
 | the inverse ratio of the compressing weight. 
 
 Air expands by heat, and contracts by cold. This property of air may be proved 
 so by a very simple experiment. In an apartment, brought to a mean degree 
 ‘ heat, introduce air into a bladder, but in such a manner as not entirely to fill it. 
 ‘it be then brought near the fire, so as to be exposed to a degree of heat greater 
 ‘an the mean temperature, we see the bladder distend, and occupy a larger volume. 
 exposing it to colder air, a contrary effect is produced. 
 | dir is necessary for maintaining animal life. This truth is well known. It may 
 * proved in the most evident manner, by shutting up animals in the receiver of an 
 ‘-pump: for as soon as you begin to exhaust it of air, the animals shew every 
 30 of uneasiness; they pant for breath, and at length expire, when only a small 
 iantity of air remains. If the air be gradually re-admitted, before they are quite 
 (ad, they recover life and motion. 
 
 Air becomes charged with water, and combines with it ; as water, on the other hand, 
 icomes charged and combines with air. The first part of this proposition is suffi- 
 ‘ntly proved by facts, with which every one is acquainted. Air is sometimes more 
 ‘d sometimes less humid. Air charged with moisture deposits it in certair bodies, 
 
 | 
 
= ” . A 
 + ¥ 
 ; “a 
 . 4 : 
 ae * - « 4 
 ; . at 
 
 : a 
 
 604 PHILOSOPHY. : 
 - capable of attracting and absorbing it in a great degree ; such as salt of tartar, which’ 
 becomes so much impregnated with it, that it resolves itself into a liquid, merely 
 by the, contact of common air, though it has been dried by a violent heat. It i 
 water, disengaged from the air with which it was combined, that occasions thi 
 moisture which deposits itself on stones, marble, &c., and during weather distin’ 
 guished by the appellation of damp. The contact of the air alone gradually dimi, 
 nishes the water contained in any vessel, especially if the air be in motion; becausi 
 a new portion of air is every moment applied to the surface of the water. It is by 
 this mechanism that those winds which have passed over a large extent of sea, asi) 
 the case with our west and south-west winds, become charged with water, and ary 
 mostly attended with rain. 
 
 Water, in its turn, becomes charged with air. This is proved by a curious expe: 
 riment, made by Mariotte. Take a certain quantity of water, and having freed i 
 as much as possible from air, put it into a small bottle, leaving no vacuity in it bu’ 
 a space of the size of a pea; at the end of twenty-four hours the water will occup) 
 the whole capacity of the bottle. What can have become of the air, which wa: 
 in the vacuity, if it has not been absorbed by the water, which was in contac 
 with it ? | 
 
 This property, which air has of combining with water, of even becoming satu 
 rated with it, and of afterwards quitting it, is the cause of various physical effects 
 such as the production of clouds and rain, the rising or falling of the barome: 
 ter, &c. But these phenomena we shall explain more at length in another place. 
 
 II1.—Of Water. 
 
 The principal properties of this common and well-known fluid are as follows: 1 
 is transparent, insipid, and inodorous; it always tends to put itself in equilibric 
 that is, to assume a form the surface of which is concentric with the earth, a pro’ 
 perty it possesses in common with all the other heavy and non-elastic fluids; iti 
 incompressible : can be reduced to vapour by heat, carried to a certain degree, an: 
 in that state is endowed with a very great elastic force. When exposed to a certai. 
 . degree of cold, it is transformed into a solid and transparent body: it dissolves salts: 
 and a multitude of other substances; and by these means it becomes the vehicle of th’ 
 nourishing particles both of animals and vegetables, which renders it so essential] 
 necessary in the animal economy, that it is in some measure more difficult to liv 
 without water, or without some fluid of which it forms the basis, than without soli 
 aliment. | 
 
 Such are the properties of water, of which we shall here give a few proofs, ti! 
 we come to another part of this work, where we shall have an opportunity of en 
 larging farther on the same subject. 
 
 It is needless to adduce any proof that water is transparent, inodorous, and insipid 
 When this fluid possesses either taste or smell, it is because it holds in solution som 
 foreign bodies. People ought therefore to be suspicious of water which is said to b: 
 agreeable to the taste, as it is certain that it is not pure. 
 
 Water always arranges itself in such a form that its surface is concentric with th 
 earth. Every body is acquainted with this property of water, which it possesses 1) 
 common with all the other non-elastic fluids, and which is the basis of the art of level 
 ling. When two masses of water communicate with each other we may rest assure 
 that their surfaces are level, or at an equal distance from the centre of the earth 
 Those persons then are mistaken, who believe that the water of the Mediterranea’ 
 is more or less elevated than that of the Red Sea, at the bottom of the Gulf ¢! 
 Suez, which, as is said, caused the plan for cutting through the isthmus to be aban’ 
 doned, lest the Mediterranean should run into the Red Sea, or the latter into th 
 former. Nothing can be more absurd, since these two seas have a communicatio’ 
 
 : | 
 
OF WATER. 605 
 
 Se 
 
 with each other by the ocean. Had they been originally created on a different level, 
 they would not have failed soon to assume the same level. 
 
 Water is incompressible. The members of the Academy del Cimento, the first 
 who it appears. adopted the true method of philosophizing, namely, by subjecting 
 every thing to the test of experiments, made a very curious one, which proves this pro- 
 perty of water. They inclosed a quantity of water in a hollow ball of gold, of a cer- 
 tain considerable thickness, taking care to ascertain that the cavity was completely 
 filled; they then subjected the ball to the blows of a hammer, by which means its 
 capacity was diminished; but the water, instead of suffering itself to be com- 
 pressed, passed through the pores of the gold, though exceedingly small, This 
 experiment was repeated by Mr. Boyle and by Muschenbroeck, who both attest the 
 truth of it. 
 | Water by a certain degree of heat is reduced into highly elastic vapour. This truth 
 may be proved also by very simple experiments. If a small quantity of water be 
 . i upon a strong fire, you will immediately see it transformed into vapour. 
 
 - If water be kept in a state of violent ebullition in a close vessel, there arises from it 
 an elastic vapour of so great force, that unless a vent be opened for it, or if the 
 vessel be not sufficiently strong to resist its action, it will undoubtedly burst : 
 for this reason, in the boiler of the steam-engine there is a valve, which must be 
 opened when the steam has acquired a certain force, otherwise it would be shivered to 
 pieces.” 
 
 This vapour, according to the calculations made by philosophers, occupies a space 
 14000 times greater than the water which produced it. Hence arises the shen 
 force it acquires, when confined in a much less space. 
 | Water, when exposed to a certain degree of cold, is transformed into a solid trans- 
 parent body, which we callice. This fact is so well known, that it is needless to 
 prove it. We shall therefore confine ourselves to an explanation of this singular 
 effect. 
 
 It is fully proved, by the formation of ice, that the primitive state of water was 
 that of a solid body. It isa solid fused by a degree of heat far below that which, 
 according to our sensations, we call temperate; for it would be a strange error to 
 imagine, that what we call zero of the thermometer, is the absence of all heat. Since 
 spirit of wine, and various other liquors, remain fluid at degrees of cold much greater 
 than that which freezes water, it is evident that the degree called zero, which is 
 marked 0, is merely a relative term, the commencement of the division. 
 
 Water then is only a liquefied solid, which keeps itself in a liquid state at a degree 
 ‘of heat very little more than that marked 0, on our common thermometers, and which 
 in that of Fahrenheit is marked 32. The reason of this we shall explain when we 
 come to speak of thermometers. 
 | Let us now take a short view of water in its solid state. When heated to a certain 
 degree, the matter of fire, with which it is then impregnated, raises up and separates 
 from each other the molecule of which it is composed; for as these molecule do no 
 longer touch each other by so large surfaces, though still within the limits of adhesion, 
 they easily run one over the other. ‘Thus we have ice brought to a state of fusion, 
 as lead is by a heat of 226 degrees. The matter of fire escapes to diffuse itself in 
 equilibrium in other bodies, which have less, for it is in this manner that cooling is 
 effected ; these molecule approach each other; they come in contact by the small 
 facets which they reciprocally present, thus adhere and form a solid body. What is 
 here said,in regard to the small facets of the particles of water, seems to be proved 
 by the ramifications of ice, for these ramifications, both in ice and in snow, are always 
 formed under angles of 60 or 120 degrees; which indicates planes uniformly inclined. 
 We shall enlarge further, in another place, on this phenomenon, which depends on 
 crystallization. 
 
| 
 
 ; ist 
 
 606 : PHILOSOPHY. 
 
 iT 
 
 - It would be ridiculous, at present, to explain the formation of ice by the supposec 
 - frigorific particles, the existence of which seems to rest on no foundation. Wate) 
 freezes at a degree of heat which can no longer keep it in fusion: for the same reason 
 and by the same mechanism, that lead becomes fixed at a degree of heat less thar 
 226 of Reaumur. But thesame philosophers who, to explain the congelation of water’ 
 have recourse to the frigorific particles diffused throughout the atmosphere, do no’ 
 recur to them in the present case: they well know that the-fixation of lead arise 
 only from the particles, which the fire does not keep sufficiently separated, approach. 
 ing each other. Why then should we recur to any thing else in regard to the con 
 gelation of water ? | 
 It is indeed true, that in the congelation of water there is one phenomenon exceed. 
 ingly singular, which is, that water decreases in volume in proportion as it cools, at 
 least toa certain degree; but at the moment when ice is formed, this volume increases 
 very sensibly ; hence the philosophers above mentioned conclude, that some foreign 
 matters, or their supposed frigorific particles, have been introduced into it. But we 
 shall observe, Ist. That the case is the same with iron. 2d. That this is the effect 
 of crystallization ; for we must here repeat, that the congelation of water is merely a 
 crystallization, by which its molecule assume an arrangement which is determined by 
 their primitive form. But this arrangement cannot be effected without producing an 
 increase of volume, as happens in regard to iron when it becomes fixed, or loses its 
 fluidity, merely by the diminution of heat, which kept it in fusion. This will become 
 more evident when we have explained the phenomena of crystallization. | 
 Water dissolves salts and a variety of other substances.—Every one knows that all’ 
 saline bodies, whether acids, alkalis, or neutral salts, are soluble in water, in a greater 
 or less quantity ; and a very singular phenomenon in this respect is, that water which 
 holds in solution as much of a certain salt as it can contain, will still dissolve some’ 
 other salt. But, for the most part, itabandons one of them when it becomes charged’ 
 with the other, if it has a greater affinity for the latter. j 
 Of the other substances, which water dissolves, we shall mention in particular the 
 gummy or mucilaginous part of animals or vegetables, which forms the nourishment 
 of the former. It is in consequence of this property, that water is so useful to the, 
 animal economy; for the nutritive part of aliment must be dissolved and diluted in 
 water, or some other fluid of the same kind, before it is swallowed, or this solution! 
 must be effected in the stomach after deglutition, Hence it is that water, in some} 
 measure, is the first aliment of man and of animals. It is not an aliment itself, but’ 
 it is the vehicle of every thing that serves as aliment. : 
 Finally, water is the base of all the other aqueous fluids, such as spirits, oils, &e:3' 
 for water may be extracted from all of them by a very simple process, namely dis-! 
 tillation. Combustion produces the same effect by disengaging the matter purely — 
 aqueous. 
 
 IV.—Of Earths. 
 
 Earth is that part of compound bodies, which remains fixed after they are analyzed. 
 When by the action of fire, we have consumed or raised in exhalations the inflammable. 
 part, have expelled or driven the air into the atmosphere, have raised the water in 
 vapours, there remains a solid and fixed body, not farther alterable by fire, that 
 is the elementary earth, the different kinds of which it is that commonly constitutes 
 the nature of the mixture. ) 
 
 It must indeed be acknowledged, at least till we arrive at a decomposition beyond | 
 that of the fixed body, that this elementary earth is not all of the same kind; con- | 
 trary to which, it is found that all water, all respirable air, is homogeneous : for where, — 
 by calcination, for instance, we have reduced a metal to calx, which is vitrifiable, | 
 that calx or earth is not necessarily homogeneous, neither to another metallic calx, | 
 
 - | 
 
Ber 
 ; 
 
 ur 
 
 EARTHS.—AIR PUMP. | 607 
 
 nor to caput mortuum, or to the earth of another body, as the calx of stone, or the 
 earth of any vegetables or animal calces. The proof of this is simple; for metallic 
 calx being revivified by the addition of phlogiston, produces only the same metal 
 
 _ which had given the calx ; and, by whatever way we proceed, the earth of any other 
 
 compound will not yield a metal, however we may combine it. This property of 
 metallic calx, is the basis of the art of separating the metals from the earths and 
 stones with which they are mineralized; for as soon as their calcés, vitrified by the 
 violence of fire, comes in contact with the carbonic matter, those of metals regain 
 
 _ their metallic form, and disengage themsel ves by their weight from the vitrified calces 
 of the other heterogeneous bodies with which they were confounded. 
 
 It has been usual to distinguish earths into calcareous, vitrifiable, and refractory. 
 
 _ Calcareous earths are those which, burned in the fire, reduce into a calx. The pro- 
 
 perties of this calx are well known, the principal characteristic of which is that of 
 
 _ attracting and absorbing moisture violently, and of effervescing with water. But it 
 , isnot necessary to subject them to that test to know them: they are easily distin- 
 guished by exposing them to the action of any gentle acid. Calcareous earths dis- 
 _ solve with more or less effervescence ; whereas other earths suffer no dissolution. 
 
 Vitrifiable earths are those which, exposed to a fire more or less active, suffer a 
 
 _ fusion, and become more or less fluid. 
 
 The refractory earths, are those on which the most violent heat excited in our fur- 
 
 _ Maces produces no effect or alteration. 
 
 We say the most violent heat excited in our furnaces; for perhaps, if all the earths 
 
 are not found to be vitrifiable, this happens only because we have not been able to 
 , produce a sufficient degree of heat. In fact, in proportion as we have succeeded in 
 _ producing more considerable degrees of heat, we also are able to vitrify materials 
 
 t 
 
 which had resisted the former degrees of’ fire. But it isa remarkable circumstance 
 that some earths which separately are unfusible, on being mixed together become 
 fusible and vitrifiable. Thus, for example, calcareous earth, mixed with argil, runs 
 and becomes glass, Usually, metallic matters, mixed either with calcareous earths, 
 or with refractory, as pure argil, communicate to them also fusibility, which these 
 have not separately. 
 
 We shall limit ourselves here as to what might be said concerning the elements; 
 what has been now said being the most solid and best proved part of the subject. 
 We shall now pass on successively through all the branches of physics, in selecting 
 
 what they offer the most curious and interesting. We have already said we shall hardly 
 
 regard much order in these observations: from the bowels of the earth we shall some- 
 
 times suddenly raise ourselves to the upper regions of the atmosphere; from a 
 
 problem in the celestial. physics, we shall pass to a question in mineralogy. We 
 
 Shall treat apart on electricity, on magnetism, and on chemistry, because these branches 
 
 of philosophy are extremely fertile in curious experiments, and present each of them 
 materials enough for separate treatises. 
 
 PROBLEM I. 
 | Construction of the Pneumatic Machine, or Air-Pump, with an account of the prin- 
 cipal experiments in which it is employed. 
 
 Air being an elastic fluid, it may be easily conceived that if it be shut up in a 
 close vessel, and if to this vessel be adapted a pump, made to communicate with it, 
 when the piston is drawn up the air contained in the vessel will enter the body 
 of the pump. If the communication between the vessel and the body of the pump, 
 de then intercepted, and if that between the latter and the external air be opened, 
 2y pushing down the piston, the air contained in the body of the pump will be ex- 
 yelled. If the communication between the body of the pump and the external air 
 ye then shut, and that between the body of the vessel be opened, when the piston 
 
608 PHILOSOPHY. 
 
 is drawn up, the air in the vessel will again rush into the body of the pump; and | 
 
 by thus repeating the same operation as hefore. the whole air contained in the vessel _ | 
 will be evacuated. If the body of the pump be equal, for example, to the capacity | | 
 of the vessel with which it communicates, the first operation will reduce the ‘air 
 
 to one half of its density; the second to the half of that half, or to a fourth, and | 
 so on in succession: hence a very few strokes of the piston will reduce the air con- 
 tained in the vessel to a very great degree of tenuity. | 
 
 Such is the mechanism of the air-pump, of which the fol. 
 lowing is a more minute description. a B (Fig. 1.) isa cylin. | | 
 dric pump or barrel, in which the piston D is made to play by | 
 means of the branch or handle pc, having at its extremity a 
 stirrup for receiving the foot, by means of which it can be | 
 forced downwards. The body of the pump is fitted into a. 
 collar, from which proceed three or. four branches that form a | 
 sort of stand. From the top of the pump a there arises a, 
 tube, about an inch in diameter, to the upper part of which 
 is adapted a circular plate, with a small raised border, or rim 
 around it. On this plate is placed the receiver, in the form | 
 of a bell. The small tube above mentioned, which serves to 
 establish a communication between the vessel and the body 
 of the pump, generally passes through this plate, and has a_ 
 screw at the end, in order that the tube of another vessel, , 
 such as a bell or small balloon, from which it is required to 
 evacuate the air, may be screwed upon it. Beneath the plate, 
 and between it and the body of the pump, is a stop-cock 1, so constructed that, by 
 turning it to one side, a communication is established between the body of the pump | 
 and the receiver, while all communication is prevented between it and the exterior 
 air; and by turning it in a contrary direction a contrary effect is produced. Such is 
 the form of the pneumatic machine ; at least of certain simple kinds of it, for there. 
 are others more complex. One kind, for example, consists of two cylinders, the 
 pistons of which are alternately worked by means of a crank; so that one of them. 
 always becomes filled with air from the receiver, while the other throws out into. 
 the atmosphere the air it contained. But it is needless to enter into these details: | 
 those who are desirous of seeing the newest improvements in regard to air-pumps, 
 may have recourse to the different treatises on natural philosophy, where they will 
 find a description and figures of the different additions made to this machine by : 
 mechanics and philosophers, to render the use of it more convenient or more 
 general. 
 
 By combining this description with what has been said in regard to the air, it wait: 
 be easy to conceive in what manner this machine is employed. When a bell- formed , 
 receiver is used, a piece of oiled leather, with a hole in the middle of it, to afforda 
 passage to the tube u, is sometimes placed on the platera. This wet leather causes, 
 the contact of the edges of the receiver to be more exact, than if it rested on metal; / 
 for some aperture or cleft would often remain, through which the exterior air would | 
 introduce itself. The receiver is then placed upon the plate, with or without the 
 leather, and the cock is turned in such a manner as to open a communication be-. 
 tween the body of the pump and the receiver. The piston, which we suppose. 
 raised up to the top, is then forced down by pressing the foot on the stirrup, and. 
 when it is as low as possible, the cock is turned in such a manner as to intercept the 
 first communication, and establish that between the body of the pump and the 
 exterior air; the piston being then raised, the air in the body of the pump is ex-. 
 pelled, and the cock is turned in the contrary direction which shuts the second coni- ; 
 munication, and opens the former; the piston is then forced down again, and the 
 
AIR-PUMP. | 609 
 
 jame effect takes place. Or the pump is otherwise worked in the manner peculiar 
 ‘0 its form and construction. Every stroke of the pump expels a portion of the air 
 originally contained in the receiver, and in a decteasing geometrical progression. 
 (hus, for example, if the body of the pump is equal in capacity to the receiver, the 
 irst stroke of the piston will expel one half of the air contained in the receiver; 
 he second will expel the fourth part; the third the eighth part; the fourth the 
 ixteenth part, &c.; so that it may with truth be said that it can never be entirely 
 tvacuated ; but after fourteen or fifteen strokes of the piston it will be so rarefied, 
 hat there will remain only a part infinitely small; for, on the above supposition, 
 he: quantity of air remaining after the first stroke of the piston will be 4, after 
 he second +, after the third 4, and so on; after the fifteenth then it will be only 
 he 32768th part, which in ae is equivalent to a perfect vacuum, for experi- 
 nents such as those that are usually made. 
 
 | After these observations on the form and use of the pneumatic machine, we 
 hall proceed to a few of the most curious experiments. 
 
 Experiment 1. 
 
 | Place on the plate of the machine a receiver in the form of a bell; if you try to 
 emove it you will experience no resistance ; but if you give only one stroke with 
 he piston, it will adhere to the plate with considerable force: after 2, 3, or 4, it will 
 dhere with more force; and after 18 or 20, with the force of several hundred 
 ounds weight. If the base of the receiver be, for example, a circle a foot in dia- 
 heter, the adhesive force will be about 1617 pounds, 
 
 This experiment is a proof of the gravity of the air of the atmosphere; for the 
 
 ir is the only body which, by pressing on the receiver, can produce the adhesion 
 xperienced. When the air under the receiver is as dense as the external air, there 
 sno adhesion, the air within and without being then in equilibrio with each other; 
 ‘ut when that within is evacuated, either in whole or in part, the equilibrium is de- 
 troyed, and the external compresses the receiver against the plate on which it rests, 
 vith the excess of the weight it has over the force of the internal air. It will be 
 ound that this force is equal to that of a cylinder of water 33 feet in height, and having 
 ' base equal to that of the receiver. It was by these means that we found the result 
 f 1617 pounds; for a cylindric foot of water weighs 49 pounds; consequently 33 feet 
 reighs 1617. 
 | Experiment 2. 
 | Place under a receiver an apple, much shrivelled, or a very flaccid bladder, in 
 thich there remains but a small quantity of air. If the receiver be then exhausted, 
 ou will see the skin of the apple become distended, so that it will assume almost the 
 ame form, and have as fresh an appearance, as it had when plucked from the tree. The 
 ladder will, in like manner, swell up, and may even be distended to such a degree as 
 9 burst. When the airisre-admitted into the receiver, they will both resume their 
 ormer contracted state. 
 We have here an evident proof of the elasticity of the air. While the wrinkled 
 pple or flaccid bladder is immersed in atmospheric air, its weight counteracts the 
 lastic force of the air contained in both; but when the latter is freed from the 
 veight of the former, its elasticity begins to act, and by these means it distends the 
 ides of the vessel which contains it. When the air is re-admitted, the elasticity is 
 ounteracted as before, and the apple and bladder resume their former shape. 
 
 Experiment 3. 
 
 | Place under the receiver a small animal, such as a cat, or a mouse, &c. If you then 
 ump out the air, you will immediately see the animal become troubled, swell up, and 
 2R 
 
610 - PHILOSOPHY. 
 
 at length expire, distended and foaming at the mouth. These phenomena are th) 
 effect of the air contained in the animal’s body, which being no longer compressed bj 
 the external air, exercising its elasticity, it distends the membranes, and throws ou 
 the humours which it meets with in its way. | 
 
 Experiment 4. 
 
 If butterflies or common flies be placed under the receiver, you will see al 
 fly about as long as the air contained in the machine is similar 1o the externa 
 air; but as soon as you have given a few strokes with the piston, they will in vail 
 make efforts to rise, as the air has become too much rarefied to support them. 
 
 Experiment 5. 
 
 Adapt to a flat bottle a small tube, so constructed that it can be screwed upon th 
 end of the tube, which rises above the plate of the machine. On the second or evei 
 the first stroke of the piston, you will see the bottle burst; for this reason it ough: 
 to be covered with a piece of wire netting, to prevent the fragments of it from doin 
 mischief by flying about. 
 
 The same effect is not produced on a receiver, because its spherical form gives i 
 strength, in the same manner as an arch, to resist the pressure of the exterior air. 
 
 Experiment 6, 
 
 Provide a small machine consisting of a bell and hammer, the latter of whicl 
 can be put in motion by wheel work, so as to strike the bell and make it sound’ 
 Wind up this small machine, and having put it in motion, place it below the re’ 
 ceiver and exhaust the air. As the air is exhausted you will hear the sound 0. 
 the bell always become weaker ; and -if you continue to exhaust the air, the sount 
 will at length cease entirely, or be scarcely heard. On the other hand, if you 
 begin to re-admit the air, the sound will be revived, and will increase more ani 
 more. , 
 
 This experiment, which we have mentioned in another place, fully proves tha 
 air is absolutely necessary for the transmission of sound, and that it is the vehicli 
 of it. 
 
 Experiment 7. \ 
 
 Provide a receiver with a hole in the top, and through this aperture introduci 
 the tube of a barometer, so that the bulb shall be in the inside of the receiver ' 
 then close the remaining aperture with mastic, or with a metal plate, so as 
 exclude the external air. Place the receiver thus prepared on the plate of the instru. 
 ment, and begin to exhaust it of air. On the first stroke of the piston you will see’ 
 the mercury fall considerably: a second stroke will make it still fall, but a quantity 
 less than the former; and so on in a decreasing proportion. In short, as the air ir’ 
 the receiver becomes less, the mercury will descend more and more towards the 
 level of that in its bason. 
 
 Experiment 8. 
 
 Provide two hollow hemispheres of brass or copper, two feet in diameter, more) 
 or less, with very smooth edges, so that they can be fitted to each other in such 
 a manner as to form a hollow globe. To one of these hemispheres let there be 
 adapted a tube passing into the inside of it, furnished with a stop-cock, and con- 
 structed in such a manner that it can be screwed on the end of the tube u of the 
 pneumatic machine. Each of these hemispheres must have affixed to ita ring, OI. 
 handle, by one of which the globe can be suspended, while a weight is attache to 
 the other. | 
 
— 
 
 TO INVERT A GLASS OF WATER. 611 
 
 - When these arrangements are made, adapt the two concave hemispheres to 
 vach other, so that the edges may be in perfect contact. Screw upon the end of 
 he tube H of the pneumatic machine, the end of that which communicates with 
 he inside of the globe, and exhaust it of air as much as possible, by forty, fifty, or 
 nore strokes of the piston. Then shut the communication between the inside of the 
 lobe and the external air, by turning the stop-cock, and remove the globe from the 
 aachine. If you then suspend the globe by one of the rings, and attach a consider- 
 ble weight to the other, you will find that the weight will not be able to separate 
 he two hemispheres. If the globe indeed be two feet in diameter, and well ex- 
 usted of air, the force with which the edges are pressed together will be equal to 
 bout 6500 pounds. 
 
 | This is what is called the celebrated experiment of Magdeburgh, because first made 
 y Otto Guerik, a burgomaster of that town. He applied to the globe several pairs 
 ff horses, some dragging in one direction and some in another, without their being 
 ble to separate the two hemispheres: and in this there is nothing astonishing, for 
 ough six horses draw a waggon, loaded with a weight equal to several thousand 
 ounds, it is well known that, one with another, they do not exert a continued effort 
 reater than about 180 pounds, and, dragging by jerks, their exertion does not exceed 
 erhaps 4 or 500 pounds. The effort of six horses, therefore, is equal to no more 
 1an 3000 pounds. We shall even suppose it to be 4 or 5000 pounds ; but if the six 
 orses draw in different directions, they do not double that force ; they only oppose 
 » the first the resistance necessary to make it act, and do nothing more than what 
 ‘ould be done by a fixed obstacle to which the globe might be attached. It needs: 
 aerefore excite no surprise that, in the experiment of Magdeburgh, twelve horses 
 vere not able to disjoin the hemispheres ; for according to this disposition, these 
 welve horses were equivalent only to six; and it has been shewn that the effort of 
 
 tese six horses, according to the above calculation, was very inferior to that which 
 tey had to overcome. 
 
 | PROBLEM II. 
 
 | ; To invert a glass Sull of water, without spilling it. 
 
 Pour water or any liquor into a glass, till it is full to the edge, and place over ita 
 luare bit of pretty strong paper, so as to cover the mouth of it entirely; and above 
 '€ paper place any smooth body, such as the bottom of a plate, or a piece of glass, 
 ' even your hand. If you then invert the whole, and afterwards raise it up, you 
 jill see the paper adhere to the glass, and the water will not fall out. 
 
 'This effect is produced by the gravity of the air, for as the air presses on the paper, 
 hich covers the mouth of the glass, with a weight superior to that of the water, it 
 
 ‘ust necessarily support it. But as the paper becomes moist, and affords a passage 
 the air, it at length suddenly falls down. 
 
 | 
 
 Remark.—In consequence of the same principle, water or any other liquor may be 
 
 Fig. 9, drawn from a vessel, by means of a pipe open at both ends. For, let 
 a B, (Fig. 2.), be a tube, thick in the middle, and tapering towards both 
 A ends, which terminate in two pretty narrow apertures. Immerse in 
 any liquor with both ends open until it is full; and then place your finger 
 | on the upper end, so as to close the aperture ; if you then draw it from the 
 fluid, the liquor it contains will remain suspended in it, though the lower 
 end be open; and it will not flow out till you remove your finger from the 
 
 UB upper orifice. j 
 Instead of employing a tube like that above described, you may use a 
 ‘ssel, such as a B, (Fig. 3.), made like a bottle, and having its bottom pierced 
 \th a great number of small holes: If you immerse this vessel in water with the 
 
 2n 2 
 
612 PHILOSOPHY. 
 
 bottom downwards, the liquor will enter through the holes ang 
 fill it; and if you then place your finger on the mouth of it 
 and draw it from the fluid, the water will remain suspended ir 
 it as long as your finger continues in that situation ; but as soor 
 as it is removed, the water will run out. 
 
 This is what is called the clepsydra or watering pot of Aris 
 totle; but neither Aristotle, nor any of those philosophers wh 
 followed him, till the time of Torricelli, assigned a better rea 
 son for this effect, than the horror which nature, as they said 
 had of a vacuum. 
 
 PROBLEM III. 
 
 To draw off all the liquor contained in a vessel, by means of a syphon. 
 The name syphon is given to a tube or pipe, consisting of two branches, A B an 
 cp, (Fig. 4.), united by a crooked part Bc. Whether thi) 
 part be straight or bent is of no importance. It has sometime: 
 an aperture which serves for filling the two branches, or fo 
 sucking up the liquor in which the shorter branch is immersed 
 while the other isshut. It isemployed in the following manner! 
 for solving the problem: proposed. 
 
 Having filled the two branches of the syphon with liquor 
 close them with your fingers, and immerse the shorter one in th 
 vessel, so that the end of it shall almost touch the bottom ; ther 
 remove your finger from the end of the longer branch, which will be’lower than th 
 bottom of the vessel to be emptied, and you will see the liquor run out at the extre 
 mity p of this branch, till the vessel is entirely emptied. Or the syphon may b 
 filled and set a-running after it is placed in the liquid, by sucking the air out at th: i 
 lower end with your mouth. 
 
 This phenomenon is also an effect of the gravity of the air: for when the syphon i) 
 full of liquor, and placed as above described, the air by its weight exercises a pressur: 
 on the surface of the liquor to be emptied, and at the same time on the orifice of thi 
 lower branch. The latter pressure indeed is for this reason somewhat superior to thi 
 former; but as this branch is full of a liquor heavier than air, the advantage must bi 
 in its favour, and the column ought to fall down. At the same time the air pressing 
 on the surface of the fluid in the vessel, forces the liquor into the branch of the 
 syphon immersed in it; which furnishes a new supply to the longer one, and so or 
 in succession, till the whole liquor is exhausted. 
 
 Remarks.—I. All the wine contained in a cask might easily be drawn off in thi 
 manner by the bung-hole; and this indeed is the method employed in some place: 
 for transferring liquor from one cask to another, without disturbing the lees, whicl 
 are at the bottom. : 
 
 II. By the same means, water may be conveyed from any place to another oné 
 lower level, making it pass over an obstacle higher than either, provided the place 
 which the water has to surmount is not higher than 32 or 33 feet above the leve. 
 where it begins to ascend ; for it is well known that the gravity of the atmosphere 
 cannot support a column of water greater than 32 or 33 feet. It is even necessary 
 that the obstacle should be several feet less in height than 32 feet above the leve! 
 of the water to be raised; otherwise the water will move in a very slow manner, 
 unless the orifice of the longer branch be much lower than that level. 
 
 This is a very economical kind of pump, and might be employed to convey water 
 from one place to another, when it is impossible or inconvenient to pierce an inter- 
 vening obstacle, to establish a pipe of communication. As we have never made the. 
 
RECIPROCATING VESSELS. 613 
 experiment, we cannot venture to give this as a very certain method, on account 
 of the air which might lodge itself at the summit of the bending of the pipe. 
 It is on the property of the syphon also that thd following hydraulic amusements 
 depend. 
 
 PROBLEM IV. 
 
 To construct a vessel which, when filled to a certain height with any liquor, shall 
 
 retain tt ; and which shall suffer the whole to escape, when filled with the same liquor 
 to a height ever'so little greater. 
 
 __ Those who may be desirous of giving to this small hydraulic machine a more 
 mysterious air, add to it a small figure, which they call Tantalus, because in the 
 attitude of drinking; but as soon as the water has reached its lips, it suddenly runs 
 out. The construction of this machine is as follows. 
 
 Let there be a metallic vessel a 3c p, (Fig. 5.) divided 
 into two parts by a partition rf: in the middle of this 
 partition is a small round hole, to receive a tube ms, 
 about two lines in diameter, the lower orifice of which 
 must descend a little below the partition. This tube is 
 covered by another somewhat larger, closed at the top, 
 and having on one side, at the bottom, an aperture, so 
 that when water is poured into the vessel, it may force 
 itself between the two tubes, and rise to the upper ori- 
 fice s of the first. This mechanism must be concealed 
 by a small figure in the attitude of a man stooping to 
 drink, and having its lips a little above the orifice s. 
 | If water be poured into this vessel, as soon as it reaches the lips of the figure, 
 being above the orifice s, it will begin to run off; and a sort of sypbhonic motion will 
 take place, in consequence of which the whole of the water will run into the lower 
 ‘cavity, which ought to have in the side, towards the partition, an aperture to let the 
 ‘air escape at the same time. 
 | This hydraulic machine might be rendered still more agreeable, by constructing | 
 ‘the small figure in sucha manner, that when the water has attained to its utmost 
 height, it shall cause the figure to move its head, in order to approach it; which 
 
 would represent the gestures of Tantalus, endeavouring to catch the water to quench 
 this thirst. 
 
 } 
 
 PROBLEM V. 
 Construction of a vessel which while standing upright retains the liquor it contains ; 
 but which, if inclined, as for the purpose of drinking, immediately suffers it to 
 | escape. 
 
 ' Pierce a hole in the bottom or side of the vessel to which you are desirous of giving 
 this property, and insert in it the longer branch of a syphon, the other extremity 
 / of which must reach nearly to the bottom, as seen Fig. 6. 
 Then fill the vessel with any liquor, as far as the lower side 
 of the bent part of the syphon: it is evident that when in- 
 clined, and applied to the mouth, this movement will cause 
 the surface of the water to rise above the bending, and from 
 the nature of the syphon, the liquor will then begin to flow 
 off; and if the vessel is not restored to its former position, 
 
 ’ will continue doing so till it becomes empty. 
 This artifice might be concealed by means of a double cup, 
 is it appears Fig. 7; for the syphon abc, placed between the two sides, will pro- 
 luce the same effect. If the vessel be properly presented to the person whom you 
 
 ——— 
 
614 - PHILOSOPHY. 
 
 are desirous of deceiving, that is to say in such a manner ag 
 to make him apply his lips to the side b, the summit of th 
 syphon, the inclination of the liquor will cause it to rise aboye 
 that summit, and it will immediately escape at c. Those 
 persons however who are acquainted with the artifice wil) 
 apply their lips to the other side, and not meet with the same 
 disappointment. 
 
 PROBLEM VI. : 
 Method of constructing a fountain, which flows and stops alternately. 
 This fountain, the invention of M. Shermius, is exceedingly ingenious, and affords 
 a very amusing spectacle, because it seems to flow and stop 
 at command. It depends on the operation of a syphon, 
 which, by the peculiar mechanism of this machine, is some- 
 times obstructed and sometimes left free, as will appear by the 
 following description. 
 aB (Fig. 8.) is a vessel shaped like 4 drum, and close on 
 all sides, except a hole in the middle of the bottom Ff, into 
 which is soldered a tube cD, open at both its extremities, o 
 and D ; but the upper one cought not to touch the top of the 
 cylinder, in order that the water may have a free passage. 
 When this vessel is to be filled, it must be inverted, and the 
 water is then introduced through the aperture D, till it is 
 nearly full. 
 From the centre of the bottom of a cylindric vessel G H, somewhat larger, rises a 
 tube D £, a little narrower, so that it can be fitted exactly into the former. Its 
 height also ought to be somewhat less ; and its summit = must beopen. These two’ 
 tubes, c p and zp, have two corresponding holes, 1, 7, at an equal height above the 
 bottom of the lower vessel, so that when the one tube is inserted into the other, the 
 holes may be made to correspond, and establish a communication between the exter- 
 - nal air and that in the upper vessel. Lastly, there must be two or four holes, as K L, 
 in the bottom of the vessel a 3, through which the water may flow into the lower! 
 vessel @ H; and in this vessel also there ought to be one or two holes, m.N, of @ 
 smaller size, through which the water may escape into another large vessel placed: 
 below the whole apparatus, 
 To make the machine play, pour water into the vessel 4 8, till it is almost entirely : 
 full ; having then stopped the pipes K and, introduce the tube p £ into c D, so that 
 the vessel cu shall serve as a base, and make the two holes 1 and 7 correspond with each’ 
 other ; if the holes or pipes k and x be then unstopped, as the external air will have: 
 a@ communication, by the apertures 1, 7, with that which is above the water in the: 
 vessel a B, the water will flow readily into the vessel c #: but the quantity which 
 escapes from G H being less than that which falls from the upper vessel A B, it will. 
 soon rise above the apertures 1, 7, and intercept the communication between the 
 external air and at that of vessel a B; consequently the water will soon after cease 
 to flow. But as the water will continue to flow from the lower vessel, while no. 
 more falls into it from the upper one, the apertures 1, ¢, will soon be uncovered, and 
 the above communication will be re-established : the water therefore will again begin 
 to flow through the pipes K and 1, and, rising above the apertures 1, 2, will soon’ 
 after begin to escape again ; and this play will take place alternately, till no more ' 
 water is left in the vessel A B. ; 
 The time when the air is about to be introduced through the apertures, 1, i, into’ 
 the top of the vessel a B, will be known by a small gurgling noise; and at that | 
 moment you must command the fountain to flow. When you see the water begin to 
 
 Fig. 8. 
 
| ot 
 LA 
 fe 
 f. 
 a 
 } 
 
 - 
 
 : 
 
 WATER CLOCKS.—TOWER OF BABEL. 615 
 
 rise above the same apertures 1, i, you must command it to stop. Hence the name 
 given to this machine, the fountain of command. 
 
 | PROBLEM VII. . 
 How to construct a clepsydra, which indicates the hours by the uniform efflux of water. 
 
 We have shewn, in the Mechanics, that if a vessel has a hole in its bottom, the 
 water flows out faster at first than it does afterwards, so that if we wished to employ 
 the efflux of water to indicate the hours, as the ancients did, it would be necessary 
 tomake the divisions very unequal; because, if the whole height were divided into 
 (144 equal parts, the highest, if the vessel were cylindrical, ought to contain 23, the 
 ssecond 21, &c., and the last only 1. 
 _ Are there any means then of causing the water to flow off in a uniform manner ? 
 ‘This is a problem which naturally presents itself in consequence of the preceding 
 jobservation. We have already solved it in Mechanics, by shewing what form ought 
 ito be given to'the vessel, that the efflux of the water through a hole in its bottom 
 ‘may be uniform. But we shall here give a more perfect solution, as it is equally 
 ‘exact whatever may be the law of the retardation of the water. 
 
 This solution is founded on the property of the syphon, and is very old, since it 
 was described by Hero of Alexandria. It is as follows. 
 , Provide a syphon a 8B c (Fig. 9.) and affix to the shorter 
 
 Fig. 9. branch a B a piece of cork, capable of keeping the whole 
 
 syphon in a vertical situation, as seen in the figure. When 
 this apparatus is made to play, and the water begins to flow off 
 through the longer branch, it will continue to escape with the 
 same velocity, whatever may be the height of the water: for, 
 in this machine, the efflux takes place in consequence of the 
 inequality of the force with which the atmosphere presses on 
 the surface of the liquid, and on the orifice of the longer 
 branch ; since the syphon then sinks down as the surface of the 
 liquid falls, it is evident that the velocity of its efflux will be 
 ie uniform. 
 | If the height of the vessel p & be therefore divided into equal parts, these divi- 
 sions will indicate equal intervals of time. To render this elepsydra more curious, 
 the branch a B might be concealed by a small light figure made to float on the 
 surface of the water in the vessel, and indicating the hour with a rod, or its finger, on 
 A small dial-plate. 
 _ On the other hand, the water might be made to flow from any vessel whatever, 
 through a similar syphon, into another vessel of a prismatic or cylindric form, from 
 
 which might arise a small figure floating on the water, to indicate the hour as above 
 described. 
 
 PROBLEM VIII, 
 
 What is the greatest height to which the Tower of Babel could have been raised, before 
 | the materials carried to the summit lost all their gravity. 
 
 To answer this mathematical pleasantry, which belongs as much to the physical part 
 f astronomy as to mechanics, we must observe : 
 _ Ist. That the gravity of bodies decreases in the inverse ratio of the square of their 
 listarice from the centre of the earth. A body, for example, raised to the distance of 
 , Semi-diameter of the earth above its surface, being then at the distance of twice the 
 adius, will weigh only 3 of what it weighed at the surface. 
 
 2d. If we suppose that this body partakes with the rest of the earth in the ro- 
 ary motion which it has around its axis, this gravity will be still diminished by the 
 entrifugal force; which on the supposition that unequal circles are described in the 
 
616 PHILOSOPHY. 
 
 same time, will be as their radii. Hence at a double distance from the earth thiv 
 force will be double, and will deduct twice as much from the gravity as at the surfacy 
 of the earth. But it has been found, that under the equator the centrifugal fore 
 lessens the natural gravity of bodies 3!,th part. | 
 3d. In all places, on either side of the equator, the centrifugal force being less, 
 and acting against the gravity in an oblique direction, destroys a less portion, in the 
 ratio of the square of the cosine of the latitude to the square of the radius. . 
 These things being premised, we may determine at what height above the surface 
 of the earth a body, participating in its diurnal motion in any given latitude, ought 
 to be to have no gravity. 
 But it is found by analysis that under the equator, where the diminution of gravity 
 at the surface of the earth, occasioned by the centrifugal force, is about sh5, the, 
 _ required height, counting from the centre of the earth, ought to be 3,/289, or € 
 semi-diameters of our globe plus #%,, or 5 semi-diameters and /§, above the surface. 
 Under the latitude of 30 degrees, which is nearly that of the plains of Mesopo. 
 tamia, where the descendants of Noah first assembled, and vainly attempted, as we’ 
 learn from the Scriptures, to raise a monument of their folly, it will be found that! 
 the height above the surface of the earth ought to have been 63% semi-diameters 
 of the earth. i 
 Under the latitude of 60 degrees, this height above the surface of the earth ought. 
 to have been 9,3, semi-diameters of the earth. } 
 Under the pole this distance might be infinite; because in that part of the earth 
 there is no centrifugal force, since bodies at the pole only turn round themselves. 
 
 PROBLEM IX. 
 If we suppose a hole bored to the centre of the earth, how long time would a heavy, 
 ball require to reach the centre, neglecting the resistance of the air ? ; 
 
 As the diameter of the earth is about 7930 miles, the semi-diameter will be 3965 
 miles, or 20935200 feet. If the acceleration were uniform, the solution of the 
 problem would be attended with no difficulty; for nothing would be necessary 
 but to say, according to Galileo’s rule, As 164, feet are to 20935200 feet, so is the’ 
 square of ] second, which is the time employed by a heavy body in falling 164, feet, 
 to a fourth term, which will be the square of the number of seconds employed in, 
 falling 20935200 feet. But this fourth term will be found to be 1301940; and if we 
 extract the square root of it, we shall have the required number, that is 1140 seconds, 
 or 19 minutes. Such, according to this hypothesis, would be the time employed by 
 a heavy body in falling to the centre of the earth. 
 
 But it is much more probable that a body, proceeding along the radius of the 
 earth, would lose its gravity, as it approached the centre ; for at the centre it would 
 have no gravity at all; and it can be demonstrated, supposing the density of the earth 
 to be uniform, and that attraction is in the inverse ratio of the squares of the dis- 
 tances, that the gravity would decrease in the same proportion as the distance from 
 the centre. The problem therefore must be solved in another manner, founded on 
 the following proposition demonstrated by Newton: 
 
 If a quadrant be described with a radius equal to that of the earth, an are which’ 
 has 164, feet for versed sine, will be to the quadrant, as 1 second employed.-to pass | 
 Over in falling these 164, feet, is to the time employed to fall through the whole’ 
 semi-diameter of the earth. 
 
 But an arc of the earth corresponding to 164, feet of fall, or versed sine, is 4’ 16”. 
 5”; and this are is to the quadrant, as 1 to 1265-2. Consequently we have this 
 proportion, as 4’ 16” 5” are to 90°, or as 1 to 1265°2, so is 1 second, employed in 
 falling 164, feet at the surface of the earth, to 126353 12 , or 21m 5s 12th, This will 
 be the time employed by a heavy body in falling from the surface of the earth to the 
 
 | 
 ee | 
 
TIME OF THE. MOON FALLING TO THE EARTH. 617 
 
 centre, according to the second supposition, which is more consistent with the 
 principles of philosophy than the former. 
 
 PROBLEM X. 
 
 What would be the consequence, should the moon be suddenly stopped in her circular 
 motion ; and in what time would she fall to the earth 2 
 As the moon is maintained in the orbit which she describes around the earth, only 
 
 »by the effect of the centrifugal foree which arises from her circular motion, and 
 which counterbalances her gravitation towards the earth, it is evident that, if the 
 circular motion were annihilated, the centrifugal force would be annihilated also ; 
 ‘the moon then would be abandoned to a tendency towards the earth, and would fall 
 jupon it with accelerated velocity. 
 
 | But this motion would not be accelerated according to the law discovered by 
 Galileo; for this law supposes that the force of gravity is uniform, or always the 
 same. In the present case, the gravity of the moon towards the earth would vary, 
 vand be increased in the inverse ratio of the square of the distance, according as she 
 ‘approached the centre; which renders the problem much more difficult. 
 
 | Newton however has taught us the method of solving it: this philosopher has 
 shewn, that this time is equal to the half of that which the same planet would employ 
 ‘to make a revolution around the same central body, but at half her present distance 
 from it. Now, it is well known that the lunar orbit is nearly a circle, the radius 
 of which is equal to 60 semi-diameters of the earth, and her revolution is 27 days 7 
 hours 43 minutes ;* hence it is found, by the celebrated rule of Kepler, that if she 
 were distant from the earth only 30 of its semi-diameters, she would employ in her 
 revolution around it no more than 9 days 15 hours 51 minutes, Consequently her 
 semi-revolution would be 4 days 19 hours 553 minutes, which is therefore the time 
 the moon would employ in falling to the centre of the earth. 
 
 | &Remark.—If we examine, by the same method, in what time each of the cireum- 
 solar planets, under the like circumstances, would fall into the sun, it.will be found 
 
 that ; D. 4H. 
 Mercurywould fall in J. saeewcseied sartiees :e lh ail 
 Venus BOOT 9 SO 2 (8! 80 16 8:60 0 0 O08 600 o's 6.004 66-8) eere ee 39 17 
 
 Pet E AM eres nents ost is wade sais tie aC TES 
 LENT ae AS A ec eccceee seccccsccecee 121 10 
 ero Eerste sis tis gts acln.cls bose cine Gale i «ee 297 6 
 BAe Tess oaid'n cremtitels cs ciiuts | coder siet dss 5 S01 4 
 | ALES ee ate Fi sibs hats eee ees eceweseeeesseceses 354 19 
 ) Vesta oe cnn ones sia ctee Sevsie hecceeet weds a Son. 405 on 0 
 DUDILER Es arise Maar sie asifbicls niles aetivas tis os tok TE TS 
 ARUP Wines leticisisk thin ed eeetaveaet adeeb LOO! 0 
 Creorgium Sidings. ss. seanids osts oc bedevede 5425. 0 
 i 
 PROBLEM XI. 
 What would be the gravity of a body transported to the surface of the sun, or any 
 
 other planet than the earth, in comparison of that which it has at the surface of our 
 globe ? 
 
 It can be demonstrated, to all those capable of comprehending the proofs, that the 
 
 | * We make the revolution of the moon 27 days 7 hours 43 minutes, 
 ninutes ; for the revolution here meant, is from any point of the heavens to the same point again, 
 tnd not a synodical revolution, which is longer ; because when the moon has described a complete 
 arcle, she has still to come up with the sun, which in the course of 27 days has advanced in ap- 
 earance 27 degrees, or nearly. 
 
 and not 29 days 12 hours 44 
 
618 PHILOSOPHY. 
 
 gravity of a body at the surface of the earth, is nothing else than the tendency of tha, 
 body towards every part of the earth; the result of which must be a compound ten, 
 dency passing through the centre, provided the earth be a perfect globe, which w 
 here suppose, on account of the small difference between its figure and that of | 
 sphere. It can be demonstrated also, that as attraction takes place in the direc, 
 ratio of the masses, and the inverse ratio of the square of the distances, a partie] 
 of matter placed on the surface of a sphere, which exercises on it a power c 
 attraction, will tend towards it with the same force as if its whole mass were unite 
 in its centre. 
 
 It thence follows, that if we suppose two aphbran unequal both in their diamee 
 and masses, the gravity of the particle on the one, will be to that of the same partic), 
 on the other, in the compound ratio of their masses taken directly, and of the square 
 of their semi-diameters taken inversely. 
 
 But it has been demonstrated by astronomical observations, that the sun’s semi. 
 diameter is equal to about 111 of the earth’s semi-diameters, and that his mass is t 
 that of the earth, as 341908 to 1: the gravity then of a body at the surface of thi 
 sun will be to that of the same body at the surface of the earth, in the compound rativ, 
 -of 341908 to 1, and of the inverse of the square of 11] to 1, that is of 12321 tol. | 
 
 If the number 341908 be therefore divided by 12321, we shall have 273 nearly ; con. 
 sequently a body of a pound weight, transported to the surface of the sun, woult 
 weigh 273 pounds. 
 
 But we shall endeavour to illustrate this subject by a reasoning still simpler. Ii 
 the whole mass of the sun, which is 341908 times as great as that of the earth, were 
 compressed into a globe equal in size to the earth, the body in question, instead oj 
 weighing one pound, would weigh 341908. But as the surface of the sun is 111 times 
 as far from his centre as that of the earth is from its centre, it thence follows that the 
 above weight must be diminished in the ratio of 12321, or of the square of 111, to the 
 square of unity 5 that is, we must take only the 12321st part of the weight above 
 found, which gives the preceding result, viz. 273. 
 
 By a similar reasoning it will be found, that a body of a pound weight carried to, 
 the surface of Jupiter, would weigh 27, pounds; to that of Saturn 1,4,, and to that of 
 the moon only 23 ounces. : 
 
 The masses of Mercury and the other planets which have no satellities can only. 
 be guessed at from the effect which they produce in disturbing the motions of the 
 other planets. 
 
 The mass of the Moon as compared with that of the sun may be deduced by com-. 
 paring their influence in producing the tides, and the precession of the equinoxes. 
 
 Tabular view of the comparative light and heat, volumes, mass, density, and gravity at 
 the surface of the sun and principal planets. 
 
 Volume. Mass. Density. Gravity. | Light & Heat.! 
 Mercury ...e-e- 0:063 TOWsTS | A 1:03 6°680 
 Venuis.t.een ee es 0°927 BieTT Ole ee 0:98 1-91 ta) 
 Earth). ny 1-000 351936 3-9326 1:00 1-000 |! 
 Mara’. ee niece 0°139 354630 gta) 0°33 431° |} 
 Jupiter: ..eees 1280900 10THS “9926 ype 037 
 Saturn ...... oy! 995-000 3542 “5500 1:01 ‘Oll | 
 Georgium Sidus.. 80-490 T7518 1:1000 Api 003. | 
 SU eoraasie sap 1384472:000 l 10000 27°90 | . 
 
 O16 | a 
 
 Moon sireieed.4 | 020 SBBI0IUT 
 
 2°4815 
 
AIR FOUNTAIN—WATER CHANGED TO WINE. 619 
 
 PROBLEM XII. 
 | ~ To construct a fountain which shall throw up water by the compression of the air. 
 
 Let there be a vessel, a section of which is represented 
 (Fig. 10.), namely, composed of a cylindric pedestal or paral- 
 lelopipedon, crowned with a kind of cup FAED. This 
 pedestal is divided, by a partition n 0, into two cavities, 
 the lower one of which must be somewhat smaller than the 
 other. 
 
 A tube Gu, passing through the partition, reaches nearly 
 to the bottom c Bs; while another tube t mu has its upper 
 orifice near the bottom of the cup, and its lower m near the par- 
 tition no, A third tube 1 x, which, like the first, passes 
 through the bottom of the cup, tapers to a point at the upper 
 md, and with the other reaches nearly to the partition. 
 
 ' When the vessel has been thus constructed, pour water into the upper cavity, 
 ‘hrough a lateral hole, till it reaches nearly to the orifice L of the tube mL; then 
 arefully stop the lateral hole, and pour water into the cup; this water, flowing into 
 he cavity N 8, will compress the air in it, and will force it, in part, to pass through 
 L into the space above the water in the upper cavity, where it will be more and 
 10re condensed, and force the water to spout out through the orifice 1, especially if 
 ; be some time confined, either by keeping the finger on the orifice 1, or by means of 
 ‘small stopcock, which can be opened when necessary. 
 | 
 | Remarks.—I. This small fountain may be varied different ways. Thus, for ex- 
 mple, if the weight of the water which flows through c u into the lower cavity NB, 
 e not sufficient to give the necessary force to the water which issues through 1, water 
 ught be introduced by means of a syringe, or even air by means of a pair of bellows 
 lapted to the orifice c, and furnished at the nozzle with a stopcock. ; 
 | Quicksilver might also be poured into it: this fluid would enter it notwithstanding 
 ae resistance of the air, and force it to exercise a powerful action on the fluid con- 
 tined in the upper cavity. 
 II. This fountain might be constructed in a manner still 
 Fig. 11. simpler. Provide a bottle, such as a 3B (Fig. 11.), and intro- 
 duce into it, through the cork, a tube c p, the lower orifice of 
 which reaches nearly to the bottom, while the upper one termi- 
 nates in a narrow aperture. The communication between the 
 external air and that in the bottle ought to be completely inter- 
 cepted at a. Let us now suppose that this bottle is three- 
 fourths filled with water: if you breathe with all your force 
 into the tube through the orifice c, the air in the space AEF 
 will be condensed to such a degree as to press on the surface 
 } of the water © F, which will make it issue with impetuosity 
 rough the orifice c, and even force it to rise to a considerable height. When 
 e play of the machine has ceased, if any water remains in it, to make it recom. 
 ence its play, nothing will be necessary but to blow into it again. 
 
 _ Fig. 10. 
 
 | PROBLEM XIII. 
 
 9 construct a vessel, into which if water he poured, the same quantity of wine shall 
 issue from it. 
 
 The solution of this problem is a consequence, or rather a simple variation of the 
 
 jeceding. Wet us suppose that the small tube 1 x (Fig. 10.), is suppressed, and 
 
 at the cavity a o is filled with wine; if a small cock nx be inserted into the 
 
620 ? PHILOSOPHY. / 
 machine near the bottom N 0, it is evident that when water is poured into the o 
 F AE D, the air, being forced into the upper cavity, will press on the surface of tl 
 wine, and oblige it to flow through the cock, until it be in equilibrium with the weig] 
 of the atmosphere: if more water be then poured into the cup F D, nearly as muy 
 wine will issue through the cock: so that the water will appear to be convert¢ 
 into wine. 
 
 Hence, if it be allowable to make allusion to a celebrated event recorded in tl 
 Sacred Scriptures, were this machine constructed in the form of a wine-jar, it might} 
 called the pitcher of Cana. 
 
 PROBLEM XIV. | 
 Method of constructing an hydraulic machine, where a bird drinks up all the water th 
 spouts up through a pipe and falls into a bason. 
 
 Let aBnpc (Fig. 12.), be a vessel, divided int 
 two parts by a horizontal partition E F; and let th 
 upper cavity be divided into two parts aie by a vel 
 tical partition G u. A communication is forme 
 between the upper cavity B F and the lower one E¢ 
 by a tube um, which proceeds from the lower parti 
 tion, and descends almost to the bottom pc. 4 
 similar communication is formed between the lowe 
 cavity ec and the upper one AG, by the tube 18 
 which rising from the horizontal partition & F, proceed. 
 nearly to the top azB. A third tube, terminating a 
 the upper extremity in a very small aperture, descend. 
 nearly to the partition © F, and passes through th 
 centre of a bason R s, intended to receive the wate. 
 which issues from it. Near the edge of this bason is a bird with its bill im 
 mersed in it; and through the body of the- bird passes a bent syphon q P, the aper 
 ture of which P is much lower than the aperture @. Such is the construction of thi 
 machine, the use of which is as follows. j 
 
 Fill the two upper cavities with water through two holes, made for the purpose i 
 the sides of the vessel, and which must be afterwards shut. It may be easily see 
 that the water in the cavity a G ought not to rise above the orifice x of the pip’ 
 KI. Ifthe cock adapted to the pipe L m be then opened, the water of the uppe 
 cavity BG will flow into the lower cavity, where it will compress the air, and maki, 
 it pass through the pipe K 1 into the cavity AG; in this cavity it will compress th: : 
 air which is above it, and the air, pressing upon it, will force it to spout up throug] 
 the pipe n 0, from whence it will fall down into the bason. 
 
 But at the same time that the water flows from the cavity B c into the lower one 
 the air will become rarefied in the upper part of that cavity : hence, as the weight o| 
 the atmosphere will act on the water already poured into the bason through the orifice 
 o of the ascending pipe n 0, the water will flow through the bent pipe qs P, int¢ 
 the same cavity B G; and this motion, when once established, will continue as long, 
 as there is any water in the cavity a c. 
 
 3 
 4 
 PROBLEM XV. 
 
 To construct a fountain, which shall throw up water, in consequenee of the rarefaction’ 
 of air dilated by heat. 
 
 Construct a cylindric or prismatic vessel, a section of which is represented (Fig. 
 13.) raised a little on four feet, that a chaffing dish with coals may be place(’ 
 beneath it. The cavity of this vessel must be divided into two parts by a partitior 
 z F, having in it a round hole about an inch in diameter : into this hole is inserte¢ 
 a pipe GH, which rises nearly to the top, and over the top is placed a vessel in 
 
 | 
 
THERMOMETERS, 621 
 
 : the form of a bason, to receive the water furnished 
 by the jet. Another piper x passing through the 
 top, into which it is soldered or cemented, descends 
 nearly to the partition zr: this pipe may be made a 
 little wider at the lower extremity, but the upper 
 end ought to be somewhat narrow, that the water 
 may spout up to a greater height. It will be proper to 
 adapt to the upper part of this pipe a small stop- 
 cock x, by means of which the water can be confined 
 till the air is sufficiently rarefied to produce the 
 jet. 
 
 When the machine is thus constructed, pour water 
 into the upper cavity till it reaches the orifice u of 
 f the tube cH; then place a chafing-dish of burning 
 soals, ora lamp with several wicks, below the bottom of the vessel. By these means 
 ‘heair contained in the lower cavity will be immediately rarefied, and passing through 
 he pipe c u, into the space above the water contained in the upper cavity, will 
 orce it to rise through the orifice 1 of the pipe 1k, and to spout up through the aper- 
 ure K. 
 
 _ To render the effect more sensible and certain, it will be proper to put a small 
 juantity of water into the lower cavity; for when this water begins to boil, the 
 lastie vapour produced by it, passing into the upper cavity, will exert a much 
 reater pressure on the water, and force it to rise to a more considerable height. 
 | Care however must be taken, if the steam of boiling water be employed, not to 
 eat the machine too much; otherwise the violent expansion of the water might 
 urst it. 
 
 PROBLEM XVI. 
 
 To measure the degree of the heat of the atmosphere, and of other fluids. History 
 of thermometers, and the method of constructing them. 
 
 One of the most ingenious inventions, by which the revival of sound philosophy 
 as distinguished in the beginning of the 17th century, was that of the instrument 
 nown under the name of the thermometer, so called because it serves to measure 
 ae temperature of bodies, and particularly that of the atmosphere, :and of other 
 uids, into which it can be immersed. This invention is generally ascribed to the 
 \eademy del Cimento, which flourished at Florence, under the protection of the 
 ‘and-dukes of the house of Medici, and which was the first in Europe that applied 
 > experimental philosophy. It is asserted also, that Cornelius Drebbel of Alemaer 
 (north Holland, who lived at the court of James I. king of England, had a share 
 | this invention. But we shall not here enter into a discussion of this point in the 
 story of Natural Philosophy, as it is foreign to our design.* 
 
 The invention of the thermometer is founded on the property which all bodies, 
 ad particularly fluids, have of dilating by the heat which pervades them. As spirit 
 ' wine possesses this property in an eminent degree, this liquid was employed in 
 eference to any other. A very narrow glass tube, terminating in a bulb of about 
 iinch in diameter, was filled with this liquor, after it had been coloured red by 
 eans of tincture of turnsol, or orchilla weed, in order to render it more visible, 
 
 ” The first description of a thermometer ever published, is that of Solemon de Caux, a French 
 ‘gmeer, in his book ‘* Des Forces Mouvantes,”’ printed in 1624, in folio, but written, as appears, 
 ‘or to that period, for the dedication to Louis XIII. is dated 1615, and the privilege granted by 
 at monarch is of 1614. The thermometer here alluded to, acts by the dilatation of air confined in 
 ‘20x, which, pressing against water, forces it to rise in a tube. As Drebbel’s thermometer was 
 (the same kind, it may be asked whether his invention was prior to that of Solomon de Caux ? 
 is is a question which seems difficult to be determined.— Note of the French Censor. 
 
§22 ‘ PHILOSOPHY. 
 
 i 
 
 It may be easily conceived, that the size of the bulb being considerable, compare¢ 
 with that of the tube, as soon as the liquor became in the least dilated, it would bi 
 in part forced to pass into the tube: the liquor therefore would be obliged to ascend 
 On the other hand, when condensed by cold, it would of course descend. It wa’ 
 only necessary to take care, that during very cold weather the liquor should no 
 entirely descend into the bulb; and that during the greatest degree of heat to bi 
 measured, it should not patrely escape from it. Towards the lower part, som 
 degrees of temperature were inscribed by estimation: such as cold, and a little lowe’ 
 great cold ; towards the middle temperate, and at the top heat, and great heat. 
 
 Such is the construction of that thermometer called the Florentine, which wai 
 used for almost a century; and such are those still sold in many country places by 
 itinerant venders, and which are purchased with confidence by the ignorant. 
 
 This thermometer, though its form and the greater part of its construction hal 
 been retained, is attended with this fault, that it indicates the variations of heat 
 only in a very vague and uncertain manner. By its means we may indeed know 
 that one day has been hotter or colder than another; but that degree of heat oi 
 cold cannot be compared with another degree, nor with the heat or cold in anothel: 
 place: besides, the words heat and cold are merely relative. An inhabitant of the 
 planet Mercury would probably find one of our hottest summers exceedingly cool, 
 and perhaps very cold; while an inhabitant of Saturn, if transported to the frigi¢ 
 zone of our earth during winter, would perhaps find it intolerably hot. We our 
 selves at the close of a fine day in summer, experience a sensation of cold, wher 
 removed into air much less bot, and vice versd. all 
 
 On this account, attempts have been made to construct thermometers, by which’ 
 the degrees of heat and cold could be compared to a degree of heat or cold invariable 
 in nature; so that all thermometers constructed according to this principle, though by: 
 different artists, in different places and at different times, should correspond with: 
 each other, and indicate the same degree when exposed to the same temperature.’ 
 This was the only method of making experiments that could be of utility. 
 
 This was at length accomplished by means of the two following principles, which’ 
 were discovered by experience. 
 
 The first is, that the degree of the temperature of pounded ice, beginning to’ 
 melt, or of water beginning to freeze, is constantly the same, at all times and in . 
 places, 
 
 The second is, that the degree of the temperature of boiling water is ib 
 constant. We here speak of fresh water; and we suppose also that the height 
 of the atmosphere does not vary ; for we know that when water is pressed with a 
 greater weight, it requires a degree of heat somewhat greater than when it is less’ 
 pressed. This is proved by the pneumatic machine, from which if a part of the alr’ 
 be exhausted, water boils at a less degree of heat than when exposed to the open 
 air. Hence arises a sort of paradox, that at the summit of a mountain, the same 
 quantity of heat is not required to boil water as at the bottom of it. But when the: 
 gravity of the air is the same, and when the water holds no salt in solution, it begins: 
 to boil at the same degree of heat; and when it once attains to that state, it never | 
 acquires a greater degree, hdasevat long it may be boiled. : 
 
 These two constant degrees of heat and cold, so easy to be obtained, have there- | 
 fore appeared to philosophers very proper for being employed in the construction of 
 thermometers. The simplest methcd for this purpose is as follows :— 
 
 Provide a tube, one of the ends of which is blown into a bulb of about an inch in | 
 diameter; if the tube be a capillary one, the bulb may be smaller. By a process” 
 which we shall describe hereafter, pour quicksilver into the tube, till it rises to the | 
 height of a few inches above the bulb; and then immerse the bulb into pounded ice | 
 putintoa bason. When the mercury ceases to fall, make a mark on the tube, in’ 
 
{ THERMOMETERS. 623 
 
 wrder that this point may be known; then immerse the thermometer into boiling 
 vater, and mark the point where the mercury ceases to rise, which will be that of 
 voiling water. Nothing then will be necessary but to divide that interval, between 
 hese two marks, into any equal number of parts at pleasure, such as 100, for instance, 
 vhich appear to us to be the most convenient. For this purpose affix the tube to a 
 mall piece of board, having a piece of paper cemented to it, and divide the interval 
 etween the marks into the number of parts you have chosen; if 8 be inscribed at 
 he point of freezing, and if a few degrees be marked below it, your thermometer will 
 
 e constructed. 
 
 , Care however must be taken, to ascertain whether the diameter of the tube is uni- 
 orm throughout: for it may be easily seen that a tube of unequal calibre would 
 ender the motion of the mercury irregular. For this purpose, introduce a small drop - 
 f mercury into the tube, and make it pass from the one end to the other; if it every 
 yhere occupies the same length, it is evident that no part of the tube is narrower 
 aan another ; if the drop becomes lengthened or shortened, the tube must be rejected 
 3 faulty. 
 
 | Several of the modern philosophers, with a view to improve the construction of 
 aermometers, have entered into minute details in regard to the increase of volume 
 ‘hich mercury and spirit of wine acquire, when they pass from the degree of freezing 
 ) that of boiling water; but it appears to us, that as these two terms have been 
 mund to be invariable, they might have saved themselves the trouble of entering 
 ito these considerations, which tend only to render their processes more difficult. 
 
 It now remains that we should describe the method of filling the bulb and tube 
 ‘ith the fluid intended to form the thermometer ; and which, for reasons to be men- 
 oned hereafter, we shall suppose to be mercury ; for this operation is attended with 
 me difficulty, especially when the tube is very small. i 
 |The first thing to be attended to, is to clean very well the inside of the tube; 
 hich, if it be not a capillary one, may be done by means of a very dry plug fixed to 
 1e end of a wire, and then drawn up and down the tube. If the tube be capillary, 
 | must first be heated, and then the bulb: the air issuing from the latter will expel 
 ly dirt that may be adhering to it. 
 |The mercury ought to be exceedingly pure, or revived by means of cinnabar; it 
 just also be boiled to expel the air which may be diffused through it. 
 
 When these preparations have been made, attach to the summit of the tube a small 
 sper funnel; apply the tube to a chafing-dish in such a manner as to heat it gra- 
 aally, and then heat the bulb in the same manner, till it cannot be held in the hand 
 ithout a thick glove. When the thermometer has acquired this degree of heat, if 
 ie small funnel above mentioned be filled with heated mercury, in proportion as the 
 ass cools the air will become rarefied, and afford a passage to the mercury into the 
 ilb, till it be in equilibrium with it. Repeat the same operation to introduce a new 
 aantity of mercury, and so on till the tube is full ; and then graduate the thermo- 
 eter, expelling from it, by the means of heat, what is more than necessary to make 
 jreach the highest point marked towards the upper extremity of the tube. when 
 mersed in boiling water. When this point has been fixed mark it by means of a 
 read, or by notching the tube with a file, and having suffered the thermometer to 
 
 ‘ol, immerse it in melting ice, which will give the freezing point. 
 
 It may be readily conceived, that if the whole of the mercury, during this opera- 
 ‘mn, should enter the bulb, it will be necessary to introduce a little more, in order to 
 ary the point of boiling water somewhat higher. 
 
 Melt and draw out the upper end of the tube, by applying it to an enameller’s 
 Inp, and heat the mercury to such a degree, as to make it ascend near to the 
 ‘mmit ; then seal it hermetically at the lamp, and by these means nothing will 
 
624 PHILOSOPHY. 
 
 remain in the upper end of the tube but a quantity of air imperceptible, or exceei 
 ingly small. 
 
 Then affix the tube toa Speed made of some wood which has the property of expan 
 ing but a very little in length by heat: fir has this property as well as that of ligh 
 ness; the bulb must be insulated from the board that the air may surround it mo} 
 freely) and that it may not be affected by the heat which the wood may acquire. | 
 
 A question here naturally arises: What kind of liquor is the best, and mo 
 convenient for constructing accurate and durable thermometers—Spirit of wine ( 
 mercury ? 
 
 In our opinion, this question is attended with no difficulty ; for all philosophe 
 must agree that mercury is the fittest fluid for constructing thermometers. N 
 doubt of its superiority over spirit of wine can be entertained, by those wh 
 consider :— 
 
 Ist. That spirit of wine, unless well dephlegmated, is not always the same; ar 
 who can assert that, in its different states, its progress is always the same, or thati 
 dilatation is not different at the same degree of heat? This point has been determine 
 by experience ; and therefore no certain comparison can be made between differei 
 spirit of wine thermometers. 
 
 2d. If spirit of wine be well dephlegmated, as it then becomes highly spirituoy 
 and volatile, is it not to be apprehended that its volume may be gradually diminished 
 It is indeed true, that to prevent this inconvenience, the tube is hermetically close 
 at the top; but this precaution will not prevent the most volatile part from beir 
 exhaled in the upper part of the tube; and in that case the spirit of wine, becomir 
 less expansible, will remain below the degree at which it ought to be; and the sam 
 thing will take place i in every state of the spirit of wine, whether it be employed mm 
 water, as is usual, in order to moderate its dilatibility. 
 
 3d. Spirit of wine boils at a degree of heat less than that of boiling water ; cor 
 sequently it is not proper for examining degrees of heat which are greater ; for beyor 
 ebullition the progress of the dilatation of any liquor does not follow the same laws 
 because after that term it becomes volatilized, or is suddenly reduced into vapour ¢ 
 a volume a thousand times greater. 
 
 On the other hand, spirit of wine, when united with water, is susceptible of freezin 
 at a degree of cold not much less than that at which water congeals; and therefor 
 it is very improper for measuring degrees of cold much below that term. 
 
 Mercury is attended with none of these defects. This substance, as far as chemis 
 have been able to ascertain, is of an uniform nature when pure; to make it boil re 
 quires a degree of heat six times as far distant from zero, or the term 0, as that ¢ 
 which water boils ; and it does not freeze but at a degree of cold very far indeed belo 
 that of the pragcuayion of water. 
 
 Another advantage of mercury, whether employed in thermometers or barome 
 ters, is, that while in the act of rising, the small column assumes a convex form ¢ 
 the top, and when it falls a concave form; for this reason, when the summit assume 
 a convex form, we can say that the mercury is in the act of rising; and when : 
 becomes concave, we may conclude that it begins to fall; which is very convenien 
 for prognosticating heat, and for ascertaining whether it increases or has become sta 
 tionary, or has begun to decrease. 
 
 B 
 
 PBOBLEM XVII. 
 Description of the most celebrated thermometers, or those chiefly used : Method of i 
 reducing the degrees of one to corresponding degrees of another. : 
 Several thermometers, different in the division of their scale, though constructe 
 on the same principle, are employed in Europe. As the division of the scale i 
 
THERMOMETERS. 625 
 
 “altogether arbitrary, it is necessary that we should point out the method of reducing 
 degrees of the one to corresponding degrees of another. 
 
 | These thermometers are that of Fahrenheit, that of Reaumur, that of Celsius, and 
 that of Delisle. 
 
 The first of these thermometers is constructed with mercury, and is graduated in a 
 manner which, on the first view, may appear rather whimsical. The freezing point 
 corresponds to the 32d degree; and between this point and that of boiling water 
 there are 180 degrees; so that the heat of boiling water corresponds to the 212th 
 degree. The reason of this division is that Fahrenheit assumed, as the Zero of his 
 \thermometer, the greatest degree of cold which he could produce by a mixture of 
 snow and spirit of nitre; having then immersed his instrument in melting ice, and 
 afterwards i in boiling water, he divided the interval between these two points into 180 
 degrees, which gave him 32 between the above artificial cold and that of common 
 freezing. Experience has since shewn that it is possible to produce an artificial cold 
 much more intense than that produced by Fahrenheit. 
 
 , Thisthermometer is that generally used in England ; butit appears that the scale 
 is not the most commodious. It might however be Figanaes by transposing zero to 
 the place of the 32d degree, in which case there would be 180 degrees between the 
 freezing point and that of boiling water; and the degree now marked 0 in this ther- 
 mometer, would be — 32, denoting the degrees below freezing by the negative sigu 
 minus. Fahrenheit, it appears, was the first person who employed mercury in the 
 construction of this instrument. 
 
 | Reaumur’s thermometer is generally made with spirit of wine, and is graduated in 
 such a manner, that the degree of melting ice is marked 0, and that which corre- 
 sponds to boiling water is marked 80; consequently there are 80 degrees between 
 these two points. The scale below O is marked 1, 2, 3, 4, &c.; and when these 
 degrees are used, the words below freezing are added, or for the sake of brevity the 
 sign —. 
 
 | Delisle’s thermometer is much used in the North; and for this reason it is neces- 
 sary we should make known the mannerin which it is divided. Delisle begins his 
 scale at the point of boiling water, and proceeds downwards to the freezing point; 
 setween which and that of boiling water there are 150 degrees: 150 degrees of his 
 hermometer correspond therefore to 80 of Reaumur, or 180 of Fahrenheit. 
 
 Celsius of Upsal, and Christin of Lyons, sensible of the defects of spirit of wine, 
 md finding the division into 80 degrees inconvenient, endeavoured to remedy these 
 faults by constructing a thermometer with mercury, and dividing the interval be- 
 sween the freezing point and that of boiling water into 100 degrees. The only 
 ‘lifference between this thermometer and that of Reaumur, is, that mercury is 
 ised instead of spirit of wine, and that 100 divisions are employed in the 
 fame space in which Reaumur~employs 80: one degree of the thermometer of 
 elsius is therefore equal to 4 of a degree of that of Reaumur; consequently, to con- 
 yert the degrees of Celsius’s thermometer to corresponding degrees of Reaumur, 
 Peer ly by 4 and divide by 5: to convert Reaumur’s degrees to those of Celsius, a 
 ontrary operation must be employed. To convert the degrees of Celsius’s thermo- 
 aeter to those of Fabreuheit’s, multiply by 9, then divide by 5, and to the product 
 dd 32. To convert Fahrenheit’s degrees to those of Celsius, subtract 32 from the 
 umber of degrees proposed, and having multiplied the remainder by 5, divide the 
 roduct by 9. 
 
 _ To convert degrees of Fahrenheit into degrees of Reaumur, the following process 
 hust be employed : if the degrees of Fahrenheit are above 32, subtract 32 from them, 
 hen multiply the remainder by 4, and divide the product by 9, the quotient will be the 
 
 1, ‘orresponding degree of Reaumur’s division. Let the proposed degree of Fahrenheit, 
 28 
 
626 PHILOSOPHY. 
 
 for example, be 149: if 32 be subtracted from this number, the remainder will he 117, : 
 which multiplied by 4, gives for product 468; and if this product be divided by 9, 
 we shall have for quotient 52, which is the corresponding degree of Reaumur’s “ 
 mometer. 
 
 If the degree of Fahrenheit be between 0 and 32, it must be subtracted from 32; | 
 then multiply the remainder by 4, and divide the product by 9: the quotient will be 
 the corresponding degree of Reaumur’s thermometer. In this manner it will be 
 found, that 12 degrees of Fahrenheit correspond to 8 degrees of Reaumur, below 
 freezing. 
 
 Lastly, when the proposed degree is below 0, add it to 32, and then proceed as 
 above directed: the quotient will be the corresponding degree of Reaumur’s ther- 
 mometer. Thus, it will be found that the 45th degree of Fahrenheit, below 0, cor- 
 responds to 342 degrees below 0 of Reaumur. 
 
 It may here be readily seen, that to convert degrees of Reaumur’s scale to the 
 corresponding degree of Fahrenheit’s, the reverse of this operation must be Per 
 formed. 
 
 In regard to Delisle’s thermometer; it is evident, from 1ts construction, that the 
 150th degree in its scale, corresponds to the zero of Reaumur’s scale. If the pro- 
 posed degree of Delisle’s thermometer then be less than 150, it must be subtracted 
 from 150: if you then multiply by 8, and divide by 15, the quotient will be the cor- 
 responding degree of Reaumur, above freezing. 
 
 Let the proposed degree of Delisle’s thermometer, for example, be 120: if this 
 number be subtracted from 150, the remainder will be 30; then say, as 150 to 80, or 
 as 15 to 8, so is 30 to a fourth term, which will be 16= the degree of Reaumuag 
 thermometer, above 0, or the freezing point. 
 
 If the degree of Delisle’s thermometer exceeds 150, as if it be 190, for example 
 subtract 150 from it, which will leave for remainder 40; then make use of this pro- 
 portion: As 15 is to 8, so is 40 to 214, which will be the degree of Reaumur’s ther- 
 mometer, below 0, corresponding to the 190th degree of Delisle’s thermometer. 
 
 As it will be easy to perform the reverse of this operation, in order to convert 
 the degrees of Reaumur’s thermometer into those of Delisle’s, more examples ve 
 needless. 
 
 It is certainly much to be wished that all philosophers would agree to employ only 
 one kind of thermometer, that is to say, constructed in the same manner, with mer- 
 cury, and having the same scale. In regard to the latter, there can be no doubt that 
 the division of 100 parts, between the freezing point and that of boiling water, 
 would be preferable to any other, as decimal divisions are attended with many ad- 
 vantages in regard to facility of calculation; and this mode of division has since 
 been adopted in France. The thermometer so divided is called the centigrade ther- 
 mometer. | 
 
 ON REGISTER THERMOMETERS. : 
 
 Many contrivances have been proposed for ems the thermometer register the 
 variations of temperature. One invented by Mr. Six of Colchester, and named after the 
 inventor, is represented in Fig. 14. It is nothing but a spirit-of-wine thermometer, 
 with a long cylindrical bulb; and its tube bent in the form of a syphon with parallel 
 legs, and ending in a small cavity. A part of both legs, as from a to J, is filled with 
 mercury ; and the ramainder of the legs, and a small portion of the cayity, are filled 
 with highly rectified spirit of wine. The double column of mercury gives motion 
 to two indices, ce and d. Each index consists of a bit of iron wire inclosed in a glass 
 tube, capped at each end with a button of enamel. They are of such size that they, 
 would move freely in the tube, were it not for a thread of glass drawn from the 
 
 | 
 | 
 | 
 4 
 | 
 
 4 
 
REGISTER THERMOMETERS. 627 
 
 upper cap of each, and inclined so as to press against one side 
 of the tube with sufficient force to retain the index attached at 
 any part of the tube to which it is raised by the mercurial column. 
 
 The instrument, then, effects its object in the following man- 
 ner. When the spirit in the bulb is expanded by heat, it de- 
 presses the mercury in the limb a, and raises it in the limb 4, 
 of the syphon. When the spirit d in the bulb contracts by cold, 
 the mercury in the limb 6 descends, and causes a proportional 
 rise of the column ina. Now it will be seen that when the 
 mercury in either column rises, it will carry the index in that 
 column with it; and when it begins to fall, it will leave the 
 index attached to the side of the tube, by the small glass spring 
 above adverted to; the lower part of the index marking the 
 highest point to which the mercury in that tube had risen. 
 
 In this way the highest and lowest temperatures are seen 
 that have occurred between any two times of observation. To 
 prepare the instrument for a new observation, both indices are 
 brought to the surface of the mercury by the attraction of a 
 magnet. 
 
 It is obvious, from this description, that there must be an as- 
 cending scale attached to 6 to measure the expansion of the 
 . spirit by heat, and a descending scale to a to mark the con- 
 traction of the spirit by cold, 
 
 ' Some makers, instead of the glass spring, insert a bristle into the cap of the in- 
 ex; but a fine wire of silver or platina would be preferable. 
 
 ' Another invention of Dr. John Rutherford’s, called the day and night thermome- 
 er, has, from its cheapness, and the simplicity of its construction, in some measure 
 uperseded that of Six. 
 
 Fig. 15. 
 
 _ This instrument is represented in Fig. 15, where a represents a spirit, and B a 
 -ercurial, thermometer ; both placed horizontally on the same piece of wood or 
 ory. 
 
 The index of 8 is a piece of steel wire, which is pushed before the mercury, and 
 ft where the mercury had attained its greatest expansion ; and marking therefrom 
 ‘e highest temperature. The index of a isa piece of glass about half an inch 
 lng, with a small knob at each end. It lies in the spirit, which passes freely beyond 
 i when expanded by heat; and when contracted by cold, the last film of the 
 dumn of spirit is enabled, by the attraction between the spirit and the glass, to 
 ry the index back towards the bulb ; leaving it at the point which marks the great- 
 G cold that has taken place since the setting of the index. 
 
 From the position ef the thermometers, it is obvious that, to bring both the in- 
 es to the surface of the respective fluids, it is only necessary to incline the instru- 
 int, making the end towards c the lowest. 
 
 282 
 
 | 
 
628 PHILOSOPHY. 
 
 For ordinary purposes this instrument is very convenient; it is not easily deranged, 
 and it can be adjusted in a moment. 
 
 For a description of other instruments of the kind, see the treatise on the Ther- 
 mometer and Pyrometer, in the ‘‘ Library of Useful Knowledge.” 
 
 PROBLEM XVIII. 
 
 Construction of another kind of thermometer, which measures heat by the dilatation 
 of a bar of metal. 
 
 The property which all metals have of dilating by heat, serves as a principle for 
 the construction of another thermometer, exceedingly useful, as much greater 
 degrees of heat can be measured by it than by other thermometers ; for a spirit-of- 
 wine thermometer cannot measure a degree of heat greater than that acquired by spirit 
 of wine in a state of ebullition; and a mercurial thermometer cannot measure any 
 degree of heat greater than that of boiling mercury. It was perhaps for this reason 
 that Newton employed, in his thermometer, linseed oil; for it is well known that 
 fat oils, before they are brought to ebullition, require a degree of heat much greater 
 than that which fuses the greater part of the metals and semi-metals, such as lead, 
 tin, bismuth, &c. 
 
 Muschenbroeck is the inventor of this new kind of thermometer, called also 
 Pyrometer. Its construction is as follows. 
 
 If we supposea small bar of metal, 12 or 15 inches in length, made fast at one of 
 its extremities, it is evident, that if it be dilated by heat, it will become lengthened, 
 and its other extremity will be pushed forwards. If this extremity then be affixed te 
 the end of a lever, the other end of which is furnished with a pinion adapted toa 
 wheel, and if this wheel move a second pinion, the latter a third, and so on, it will 
 be evident that, by multiplying wheels and pinions in this manner, the last one will 
 have a very sensible motion; so that the moveable extremity of the small bar cannot 
 pass over the hundredth or thousandth part of a line, without a point of the circum. 
 ference of the last wheel passing over several inches. Ifthis circumference then have 
 teeth fitted into a pinion, to which an index is affixed, this index will make several 
 revolutions, when the dilatation of the bar amounts only to a quantity altogether 
 insensible. The portions of this revolution then may be measured on a dial-plate, 
 divided into equal parts; and by means of the ratio which the wheels bear to the 
 pinions, the absolute quantity which a certain degree of heat may have made the small 
 bar to dilate, can be ascertained; or, by the dilatation of the bar, the degree of heat 
 which has been applied to it may be determined. 
 
 Such is the construction of Muschenbroeck’s pyrometer. It is necessary to obsess 
 that a small cup is adapted to the machine, in order to receive the liquid or fused 
 matters, subjected to experiment, and in which the bar to be tried is immersed. | 
 
 When it is required to measure, by this instrument, a considerable degree of heat, 
 such as that of boiling oil or fused metal, fill the cup with the matter to be tried, and 
 immerse the bar of iron into it. The dilatation of the bar, indicated by the turning 
 of the index, will point out the degree of heat it has assumed, and which must neces- 
 sarily be equal to that of the matter into which it is immersed. 
 
 This machine serves to determine the ratio of the dilatation of metals; for by sub- 
 stituting, in the room of the pyrometric bar, other metallic bars of the same length, 
 and then exposing them to’an equal degree of heat, the ratios of their dilatation will 
 be shewn by the motion of the index. | 
 
THERMOMETERS, - 629 
 
 A Taste of the different degrees of heat at which different matters begin to melt, to 
 
 Jreeze, or to-enter into ebullition, according to the thermometers of Fahrenheit, 
 Reaumur, and Celsius. 
 
 Degrees Degrees Degrees 
 Names of the Matters. Fe ; 
 
 re) of of 
 Fahrenheit. | Reaumur.| Celsius. 
 
 Mercury congeals  ... 11. 0ss see one vee] —= 39 | — 31d | —. 392 
 EVE OTSA AR cl hector \ Feed! Lenk: lens 708 300 375 
 MRTERRTCCZES Biers ter, Uncen. -doouts teh ale’. Sade 32 0 0 
 Water boils ... ... ah oe a ar 212 80 100 
 Rectified spirit of wine 2 freezes ete jee Hess} —— 33 | — 29 jee 
 fihe'same boils .... ... woes (tee 175 634 79 
 Brandy consisting of equal parts spirit. and 
 | 
 water freezes ... ... dee ter eae eee eel me 7 | IR | — 212 
 | The same boils ...... ee 190 70 874 
 | Water saturated with marine salt boils . He 218 822 1032 
 : Lixivium of wood ashes boils... 0... w..sasj 240 924 114 
 Burgundy, Bourdeaux, &c., wine freezes ...)J —- 20 | — 53 | — 7} 
 : PERT OPSUMEEVAFEOZES: NS. ch fit oh Sew fetes esef nt 400 | ea} og 
 eee cate: DOL Se Seems brea! i BOS UL RSE ats 242 935 116 
 | Peskuraehiarcscokis) Ii. seel tye ti ..2 lay well Measity ce 142 49% 624 
 _*| Butter melts... vee eee = eas} 80 to 90 121 to 26/26 to 32] 
 —_{ Oil of turpentine begins fo holy tee ee 560 234 292 
 | Olive oil becomes fixed ... 0 00. ue ee aes 43 5 6} 
 el Rape-seed oil boils, and is ready to inflame ... 714 298 372 
 | Peon fusedirsr: At ese 05,08. ON RHUL, Was 408 | 167 309 
 ee HIGGS 1.4) Ess yc oG fers Le ok. ubceh Rieder Goes ses 540 226 282 
 | Bismuth ditto... comitatus til hots 460 190 238 
 "| Regulus of anatomy ditto... 0. sue ses eee 805 344 430 
 
 Tasue of the different degrees of heat or cold observed in various paris of the earth, 
 | or in certain circumstances, or in consequence of certain operations, according to 
 Reaumur’s thermometer. 
 
 Degrees, 
 Constant heat of the vaults below the observatory of Paris ...... 94 
 men amovnich chickens aré hatched... ..<seee sec nsrise Kies ces : 35D 
 . at which silk-worms are hatched ...........--cseccsseccce 19 
 Sites OFAN POL Var «sie gin diols ai ais 5p, <vioiei sy sietv eat ¢ tel gb. « Sinieole «ia ¢ 15 
 BOF PIN Gad Clients owas x, sfoly: ol 'h'o!d, esse. ei¢jvieimsso.4\sie\s s7e0 os sie cigs 18 
 -—— for the chamber of a sick person «<..0.cececsvseucesecoes ive 
 BEA RAL OY. C Belg Sane s ccel sir esata a ole eka Nee ae airs a hens eters uate 12 
 | —— of the human skin ....... Silene Sedtisldiele tues + 5 shy oo ee ce s.n\CotOoU 
 -—— of the interior of the human body .........-.... hein ese 4.47520 3] 
 AREA tri Misric dc cis Weel feeb cy dcisid she Minded s Sterec e's. smie de ak DOTOAOUEE 
 gieat observed at Paris. in 1753 2.2 eseega.seccomesersesinecces 304 
 ——— BLM CNE PAE 6 oe s'a'viele s\ataan attcloie aos « sieielap clecescdas 37 
 2 ADJ OUT IBs EN GL LAO sins sen edi idictereliein cuales coe sioceoce 35 
 PEP OIBTEINICO ss ow sctslerele tie ere ersutd sislanielda seine eiis 32 
 MeaTNSELVCR Gtrkraris 11) { OUisc s aco e'seis.e ale nce hnace a pate c-00/c come gel mae 93 
 In 1740. ceccceccerccscsreeeeeescceses —= IO 
 = ATL GOA feo ob nina o.0 eM ai) wip ide ch init craic oie aL 
 a ATU S# RR EE « Ssceig Je ,8 Reisen, (ad paigheneletabrecer ih Led 
 1M 1768 peeseissceeasssecoveonersee +» —— 144 
 ct ectecrtreeeeentig ATV LAID) oben: wien 9 © a0e'd 6'8 6. 8. bki os Peace me 15} 
 
630 PHILOSOPHY. 
 
 R Degrees. 
 Cold observed at Paris in 1776 ..ccceccsccssccceccccccscesces <= 162 
 ooh hs, a at Petersburgh December 1759 ...... ceccccesees ——= Sat 
 a attra a ———- Decemben 17722 ccesces ccc sen subeltce) \—— ane 
 —_——— at Torneo in 1737'......-: cules cliete maar eet ae — 3s 
 a at Quebec «aces uecestt ode en e's sale usaldisla be eden eee 
 ———— at Epsal in 1733 .. sccccecerecsecccrsscsesecess —— 40 
 ———— at Kiringa in Siberia in 1738 (See Flora Siberica) — 70 
 Artificial cold with spirit of nitre and snow cooled to 33 degrees — 170 
 
 Table of the ratio of the dilatation of metals by he&t, according to Mr. Ellicot. 
 
 Names of Respective Names of Respective 
 Metals. Dilatation. Metals. Dilatation, 
 Gold vac ers ’ 73 Tron eI IE a. A: 60 
 Silver) vets oia's a 103 — Steel vs oye waar 56 
 Copper’ iccss bs 89 © Lead Pais y 149° 
 Similorapes-.. se <s 95 Tin ee eager +a 148 
 
 OBSERVATIONS ON THE PRECEDING TABLES. 
 
 I. The first observation we shall here make, is on the congelation of mercury by | 
 au extraordinary degree of cold. This singular experiment was made, for the first ) 
 time, at Petersburgh in the month of December 1759, and deserves that we should | 
 here give a particular account of it. 
 
 The cold having become very intense in that city, in the month of December 1759, 
 Mr. Braun embraced that opportunity of makiig some experiments on the artificial 
 cold that could be produced assisted by its means. He put into a glass vessel snow 
 already cooled to 208 degrees of Delisle’s thermometer, or 31 degrees of Reaumur, 
 and having cooled to the same degree good fuming spirit of nitre, he poured it upon 
 thesnow. He then immersed in the mixture the bulb of a thermometer, so con- 
 structed that the scale of it extended about 6000 degrees, both above and below 
 zero, which in Delisle’s thermometer is the point of boiling water, and saw with 
 astonishment the mercury rapidly descend to the 470th degree below that term. 
 The mercury having then stopped, Mr. Braun shook the thermometer, and found that 
 the mercury had no motion. He broke the bulb, and found that the mercury was 
 completely frozen. This experiment was repeated either the same day, or on the 26th 
 of December, when the natural cold was still more intense, and the mercury fell to 
 the 212th degree of Delisle’s thermometer, or the 33d of Reaumur. Several of the 
 Academicians of Petersburgh were present at the latter experiment, and confirmed the — 
 truth of it. The small ball of congealed mercury was hammered, and it appeared to 
 have the ductility of lead. 
 
 One thing very singular, and which Mr. Braun remarks with astonishment, is, that 
 in several of these experiments, the mercury fell with moderate velocity from the — 
 poiut of the temperature of the air to that of 470 degrees below zero; but when it 
 reached that term it fell at once below the 600th degree, without the bulb of the 
 thermometer being broken. 
 
 This phenomenon, in our opinion, is nearly the inverse of that which takes place 
 in the congelation of water. It is well known that in proportion as water cools, it~ 
 diminishes in volume; but when it reaches the degree of congelation, it suddenly — 
 increases in volume, so that if athermometer were constructed with pure water, it is 
 probable the water would first fall, and then burst the ball of the thermometer. 
 This is an effect of the new arrangement of the parts which takes place, with a force © 
 almost irresistible, at the moment when they are all in contact. 
 
 But there is reason to think, that in mercury the contrary is the case; that is to 
 
 ‘ 
 
 , 
 
CONGELATION OF MERCURY. 631 
 
 say, when cooled to such a degree that its component particles are almost in con- 
 tact, they suddenly arrange themselves in a certain form by their mutual attraction ; 
 aud this form is apparently such, that in this disposition they must occupy less 
 yolume, as those of water occupy more. 
 
 But however this may be, it is confirmed by the experiment of Mr. Braun, that 
 mercury is only a metal kept in a state of fusion by a degree of heat much less than 
 that which freezes water, and a multitude of other liquors. We must even remove 
 it from the class of semi-metals, and rank it among the number of real metals, 
 
 We find also, in this experiment, the reason why mercury is the most volatile 
 of the metals. Since the degree of heat necessary to keep it in fusion, is so far 
 ‘below that which melts i ice, it needs excite no astonishment that at the 300th degree 
 of Reaumur’s thermometer it begins to be volatilised ; for this degree is about the 
 same as the 600th would be to lead, or the 1200th to copper, &c. 
 
 IJ. Another remark is, that the searee of water beginning to freeze is indeed fixed ; 
 ‘but the case is not entirely the same with that of boiling water. It has been found 
 that the more water is charged with the weight of the atmosphere, the greater is the 
 degree of heat necessary tomake it boil. This was remarked by M.le Mounier, who 
 found at the summit of the Canignou that boiling water raised the thermometer only 
 to the 78th degree. This has been since proved by other philosophers, such as M. 
 de Secondat, the son of the celebrated Montesquieu, on the Pic du Midi, one of 
 the highest mountains of the Pyrenees, and by Mr. Deluc on a mountain still 
 higher. . Water has also been made to boil under the receiver of an air pump, at 
 a degree much below the 80th of the thermometer: this effect may be produced 
 by partly evacuating the air. 
 
 It is therefore necessary that this degree of the thermometer should be fixed, 
 taking into consideration the height of the barometer ; and in rectified and compara- 
 tive thermometers, of which we have heard, the 80th degree is that which indicates 
 boiling water when the height of the barometer is 27 inches Paris measure, or 
 28°8 English inches: this is what we ought to understand by the degree of boiling 
 water. 
 
 It has been found also that the thinnest liquors boil at a degree of heat less 
 than water; but that fat oils require a much greater degree. 
 III. We have rectified, according to the observations of Deluc, or obser- 
 vations made at his request, the temperature of the vaults of the observatory at 
 Paris, which is not 10 as commonly said, but 91 at most. We have rectified also, by 
 the observations of M. Braun, the degree of boiling mercury, which is generally 
 placed at the 600th degree of Fahrenheit, but which, according to that philosopher, 
 is the 708th or 709th. 
 IV. In the ¢able of the dilatation of metals, it is seen that steel is that which 
 dilates the least by heat; the next is iron, and the next to that is gold. Lead and 
 tin dilate the most. It appears also by this table, that the dilatability does not fol- 
 low the ratio of the specific gravities, nor that of the ductility, nor of the strength 
 fthese metals: there are even irregularities in their dilatations, on which account 
 ‘tis to be wished that a greater number of experiments were made on this subject, 
 ind in a more correct manner. 
 
 Remark.—It is rather matter of surprise, that M. Montucla has taken no notice 
 of the accounts of the freezing and fixing of mercury, in the Philos. Trans. for the year 
 (783, especially as the errors of M. Braun concerning this matter are there corrected, 
 ind the degree of cold at which it freezes is ascertained by many different persons, 
 doth by natural cold, and by artificial mixtures, with perfect satisfaction. It is there 
 roved that the degree of cold at which mercury freezes, is — 39 of Fahrenheit, or 
 39° below 0 in that scale. It isalso shewn that the extraordinary degree of depression 
 € thermometers accompanying frozen mercury, which deceived M. Braun and some 
 
632 PHILOSOPHY. 
 
 other persons, is owing to the sudden contraction of mercury in the act of fr eezing, , 
 and after it; contrary to the nature of water, which expands and enlarges in the 
 same circumstances. Hence it happens that congealed mercury, becoming more. 
 dense and compact, sinks in fluid mercury; while common ice, or congealed water, | 
 
 floats in that fluid. | 
 
 PROBLEM XIX. 
 What is the cause of the intense and almost continual cold experienced on the tops of 
 high mountains, and even of those situated in the torrid zone ; while it is hot in the 
 
 neighbouring plains or valleys ? 
 
 The cold experienced on high mountains, while the neighbouring plains are exposed | 
 to the most violent heat, is a phenomenon which has long excited the attention of 
 philosophers. It is now known that one of the hottest climates in the world is the, 
 coast of Peru, and yet those who gradually ascend the Cordilleras from it, observe. 
 that the heat progressively decreases; so that when they have got to the valley of - 
 Quito, at the height of about 1400 toises above the level of the sea, the thermometer, , 
 in the course of the whole year, scarcely rises 13 or 14 degrees above zero. If they 
 ascend still higher, this temperature is sueceeded by a severe winter, and when they 
 get to the perpendicular height of about 2400 toises, they meet with nothing, even 
 under*the equinoctial line, but eternal ice. : 
 
 Some philosophers have asked how this is possible. In proportion as they rise | 
 above the surface of the earth, they approach the sun, consequently his rays ought to 
 be warmer; and yet they experience the contrary. Some have thence concluded 
 that the rays of the sun are not the principle of the heat which we experience ; ; for if: 
 they are, say they, how comes it that they have less activity exactly in the place 
 where they ought to have more? This paradox we shall endeavour to explain. 
 
 It must first be observed, that when people ascend to the height of some thousands - 
 of yards above the surface of the earth, it is wrong for them to conclude that the rays | 
 of the sun ought to have more activity there than at the surface. This difference 
 would be insensible even if they should ascend to a height equal to the earth’s semi- ' 
 diameter, or some thousands of miles, for the sun oe at the distance of 22000 semi- 
 diameters of the earth, and as the heat of the sun’s rays increases in the inverse ratio 
 of the squares of the distances, the direct heat of the sun at the height ofa semi- | 
 diameter of the earth will be to that experienced at the surface, as the square of 
 21999 to the square of 21998; a ratio which will be found to be that of 10999 to 
 10998 or of 11000 to 10999; so that the heat would be only one 11000th part less at: 
 the surface than at the distance of the earth’s semi-diameter above it, a difference | 
 quite insensible. What then can be the difference at the height of four or five thou- 
 sand feet above the surface of the earth? certainly nothing, and therefore no atten 
 tion ought to be paid to it. 
 
 But there are very sensible physical causes, in consequence of which bodies are less » 
 susceptible of heat, and while on those elevated parts of the earth retain it a shorter | 
 time than when they are nearer the surface. It is certain that the heat which we ex- 
 perience at the surface of the earth is not merely the effect of the direct heat of the 
 sun, but of several causes united. 
 
 These causes are, Ist. The mass of the heated bodies, which retain longer the heme 
 they have received, according as they are denser and more voluminous: hence the | 
 terrestrial bodies retain, even in a great measure throughout the night, the heat com-— 
 municated to them during a fine summer’s day. The day after this they receive » 
 another accession of heat by the presence of the sun ; and so on in succession. 
 
 2d. The air being more dense in the plains and valleys, it retains a greater portion — 
 of the heat it receives in the day-time, and prevents the dissipation of the heat com- » 
 municated to the earth. For this reason, the heat continually increases in the lower 
 
 = 
 
{ . COLD. ON MOUNTAINS, 633 
 
 ‘rounds, as the sun rises above the horizon; but on the summits of the mountains 
 he case is not the same. 
 
 _ In the first place, the air in those high regions is much rarer than at the surface of 
 he earth. No sooner is the sun sunk below the horizon, than it loses the heat it 
 eceived in the course of the day ; for every person must have observed, that a dense ~ 
 iody, such as a piece of money, retains heat.longer than a body of little density, such 
 
 sa bit of cloth. If you approach a large fire, and stand some time before it, you 
 
 will find the money in your pocket burning hot; if you retire, it will be in this state 
 
 or a long time, while your clothes will retain only a common degree of heat. Hence, 
 
 he small quantity of heat which the thin air of the mountains has received in the 
 
 sourse of a summer’s day is soon dissipated ; it is not accumulated there as in the 
 
 ower regions, where the contact also of dense terrestrial bodies, violently heated, 
 
 contributes to maintain it in that state. In the second place, the exceedingly high 
 
 nsulated peaks of these mountains are only small masses, when compared with the 
 
 whole of the terrestrial bodies in the plains and the valleys. If they are heated toa 
 sertain point, the heat they have received is speedily evaporated; and this evapora- 
 
 jion is promoted by the coolness of the surrounding air, which is lowered to the tem- 
 
 yerature of ice, almost as soon as the sun has set. 
 
 _ Hence it may be easily conceived that the air, which surrounds high mountains, 
 
 equires only in a very transient mauner a certain degree of heat; that it is almost 
 
 lways below the temperature even of ice; that on this account all the aqueous 
 
 neteors there formed are converted into snow and ice, that when a certain mass is 
 
 mee formed, it will oppose the introduction of heat, either into the surrounding 
 
 ir, or into the parts which it covers, and this new obstacle will tend to increase the 
 
 old and the mass of the ice. In this manner have been formed those accumulated 
 nasses of snow and ice which cover the summits of the Cordilleras, as well as some 
 
 arts of the Alps and the Appennines ; in short, all those mountains of the universe 
 
 vhose. height exceeds a certain limit, which in the torrid zone is about 2400 toises 
 
 ‘erpendicular elevation above the level of the sea. 
 
 We must here remark that this height is less as the latitude is greater: thus, in 
 he torrid zone, you must ascend to the height of 2400 or 2500 toises to arrive at 
 hose regions of perpetual ice; but in the temperate zone, for example, these eternal 
 Jaciers will be met with at the height of 1400 or 1500 toises. The commencement 
 f those found in Switzerland, according to the measurement of Mr. Deluc, is at the 
 eight of 1500 toises above the level of the Mediterranean; and on proceeding 
 
 ‘arther north they will be found near the level of the sea. The glaciers of Norway 
 re certainly less elevated than those of Switzerland. In short, in the frigid zone 
 hat region of continual ice is at the surface of the earth. Hence it happens that in 
 hose regions the ice, as is well known, never melts. Both the arctic and antarctic 
 oles are surrounded, to the distance of several hundreds of leagues, with circular 
 ands of ice; which, according to every probability, exclude all hopes of ships being 
 iver able to traverse the frozen ocean, in order to proceed through the seas of China 
 Ind Japan to the passage known to exist between Asia:and America. 
 
 ' When the sun’s rays pass through a perfectly transparent medium, they are not 
 yund to impart to it the slightest increase of temperature, and hence if our atmo- 
 phere were perfectly transparent, it could receive no heat from the sun by direct 
 wiation. It is only when the solar rays meet with dark or opaque substances that 
 hey give out or impart sensible heat. The surface of the earth being thus warmed 
 ly the immediate influence of the sun’s rays, the temperature thus acquired is slowly 
 : -nparted to the air in contact with it; which, being rarefied, ascends upwards, gra- 
 ually giving off its heat during its ascent, tillit reaches a situation where its specific 
 -Yavity is the sameasthat of the surrounding air; when its tendency to rise ceases ; 
 
 nd, fora moment, it becomes stationary. The descending portions of air are affected 
 
634 : PHILOSOPHY. 
 
 in a way the reverse of this; giving out a portion of their latent heat as they sink 
 ' till they arrive at the surface, where, being heated by coming again in contact wit] 
 the earth, they re-ascend, to begin anew the same round of changes in their thera) 
 condition. 
 
 The heat at the surface of the earth then gives rise to a succession of ascend 
 and descending currents; and the principal cause of the difference of temperatur 
 at the top and bottom of a column of atmosphere is the exhalation and absorption 
 of heat, caused by the alternate condensation and rarefaction of the air. The im 
 perfect manner in which heat is conducted through fluids secures the lower strati’ 
 of the atmosphere from loss of heat by transmission; and hence a diminutioi 
 of temperature is observed as we ascend above the surface of the earth. | 
 
 The law which regulates the diminution of heat as we ascend in the atmosphere, 
 varies with the latitude as well as the season of the year. In ascending fron 
 Geneva to Chamouni, Saussure observed that Fahrenheit’s thermometer fell 1° fo, 
 236 feet; and on another occasion 1° for 266 feet. And from such observations hi 
 assigned, as a mean, 1° Fahrenheit for 292 feet in summer, and for 419 in winter 
 M. Ramond gives 299, M. d’Aubuisson 315, and Guy Lussac 341, for 1° Fahrenheit | 
 At the time of the experiment from which the last-named result was obtained, thi) 
 heat at the surface of the earth was very great. | 
 
 From the observations made by Humboldt among the Andes, we learn that th 
 heat does not decrease uniformly as the height increases; and also that the rati 
 of diminution is greatly affected by local circumstances. He observed, for instance 
 on one occasion, that the decrease became slower between 1000 and 3000 metre: 
 of ascent ; and was more particularly retarded from 1000 to 2500 ; but that it after: 
 wards increased between 3000 and 4000 metres. The following table shews.a few 
 
 of his results. 
 : - Height in Metres. puaiee ee herr ant | 
 From OFT0" LO00T ae 309 
 1000 DOG Gea oh eee 536 
 2000 S000" a A es 423 
 
 3000 A000 .eccccee 239 
 4000 5000 ..0s0-+- 328 
 
 His mean result for the whole ascent is 346 feet for 1° Fahrenheit. He | 
 the slow decrease of heat between 1000 and 3000 metres of height to the absorption 
 of light by the clouds, to the formation of rain, SHE to the dispersion of heat by 
 radiation from the lower strata of the clouds. 
 
 It_may however be'inferred from the above observations, that though the dimniiams| 
 tion of heat as we ascend is not perfectly uniform, yet it may be considered as nearly 
 so for all heights that we can have access to. 1 
 
 Lagrange is inclined to adopt the hypothesis of uniform decrease; but Euler 
 considers that a harmonic progression is more in accordance with appearances. 
 
 The late Professor Leslie has given a formula deduced from the capacity of air for | 
 heat under different degrees of density. His formula is this; 5 denoting the pressure’ 
 of the barometer at the lower station, and @ that at the higher; then the differ 
 ence of temperature in degrees of the centigrade thermometer is represented 4 
 
 b B 4 
 Te a } 
 
 Mr. Leslie admits that the co-efficient (25) may require alteration, and that in 
 many places it may better to take 30 for the multiplier in summer and _ in 
 winter. 
 
 It is evident that in every climate a point of elevation may be reached, where it 
 will be continually freezing ; and that the height of this point will depend both on 
 
COLD ON MOUNTAINS—DIVISIBILITY OF MATTER. 635 
 
 the local situation of the place and the season of the year. Near the equator, Bouguer 
 aoticed that it began to freeze on the sides of the lofty mountain Pinchincha, at the 
 neight of 15577 feet above the level of the sea; whereas perpetual congelation was 
 ‘ound by Saussure to take place on the Alps at the height of 13428 feet. 
 
 A curve traced on the meridian through the points at which it constantly freezes, 
 s called the line of perpetual congelation. : 
 
 The following table exhibits the vertical heights of this curve as computed by 
 Kirwin, for every 5° of latitude up to 80°; and a little way beyond that latitude 
 she curve probably coincides with the surface of the earth. 
 
 SCADA a eens Soe saemeseemennens ones oibs mobs 
 
 Mean height of Mean height of Mean height of 
 Lat. curve of Lat. curve of Lat. curve of 
 ) perp. cong. perp. cong. perp. cong. 
 } feet. feet. feet. 
 [ 0 15577 30 11592 60 3684 
 \ 5 15457 35 10664 65 2516 
 10 15067 40 9016 70 1557 
 15 14498 45 7658 75 748 
 20 13719 50 6260 80 120 
 25 13030 55 4912 
 
 _Dr. Brewster, on the supposition established by Humboldt, that the line of 
 onstant freezing is different from that of perpetual snow, has given the following 
 ormule. 
 | Let ¢ = the mean temperature in degrees of Fahrenheit, and 6 = the latitude in 
 egrees; then the height in English feet of the curve of perpetual congelation is 
 10.¢ — 32°, and of the line of perpetual snow the height is 310.¢ — 32° + 48 J, 
 For the Alps and the west of Europe in general ¢ may be taken as equal to 
 1)". cos. 2; and thence the above formule may be transformed into others, de- 
 ending on the latitude only. 
 The curve of perpetual congelation must evidently be higher in summer and lower 
 1 winter ; and though the difference is not great in tropical climates, in higher lati- 
 udes it is very considerable. 
 
 PROBLEM XxX. 
 
 Mf the attenuation of which some matters are susceptible : Calculation of the length 
 to which an ingot of silver may be wire-drawn, and of the thickness of gilding. 
 
 , We shall not here examine the question which has so much engaged the attention 
 f philosophers, whether matter be divisible or not in infinitum. To resolve this 
 uestion, it would be necessary to be acquainted with the ultimate molecule or 
 lements of bodies, which in all probability are placed beyond our reach. But nature 
 nd art present to us some instances of the attenuation of matter, which, if they do 
 ot prove its divisibility in infinitum, prove at least: that the boundaries of this diyi- 
 on are removed beyond what the imagination can conceive. 
 
 | The ductility of silver and gold supplies us with two of those examples furnished 
 yart. An ounce of gold is a cube of 54 lines on each side; so that one of its faces 
 ill consequently cover about 27 square lines. This cube a gold-beater reduces into 
 faves, which, all together, would cover 146 square feet. But 27 square lines are 
 ontained 111980 times in 146 square feet, consequently the thickness of this gold 
 af is the 111980th part of 5} lines, or the 21534th part of a line. 
 
 But we can go still farther; for this attenuation is nothing in comparison of the 
 Mowing, — 
 
 A 
 
636 . PHILOSOPHY. 
 
 A cylindric ingot of silver, weighing 45 mares, about 22 inches in length, and 13. 
 in breadth, is covered with six ounces of gold reduced to gold-leaf. The thickness | 
 of the gold in this state, called gilding, is about the 15th part of a line. But only, 
 one ounce of gold may be employed; and in this case the thickness of the gilding will 
 be only the 90th part of a line. 
 
 The ingot thus gilt is made to pass through several holes in succession, each smaller, 
 than the other, till it is reduced to a wire of the thickness of a hair. M. Reaumur 
 took a wire of gilt silver, drawn out in this manner, and having weighed half a gros. 
 of it, with the greatest nicety, measured its length, which he found to be 202 feet;| 
 whence it is easy to conclude that the gros must have been 404 feet in length; the 
 ounce 2232, the marc 23856, and the 45 marcs 1163520 or 96 leagues of 2000 toises 
 each. Here then we have an ingot of silver, 22 inches in length, drawn out in such. 
 a manner as to form a wire of 96 leagues in length. 
 
 Nay more, this gilt wire is made to pass between two rollers of polished steel, to 
 flatten it and reduce it to a thin plate. This operation, by renderi ing it } of a linein. 
 breadth, lengthens it a seventh part more at least; so that the wire by these means. 
 is converted into a thin plate 110 leagues in length, with the thickness of the 256th 
 part of aline. In regard to the gold it will be found that its thickness is only the, 
 59000th part, and even the 60000th part of a line. 
 
 Thus, if we suppose the ingot of silver to have been gilt with two ounces of gold, 
 its thickness would be the 175000th part of a line; and supposing only one ounce 
 of gold, the thickness would be the 350000th. But as there are some places of| 
 the plate unequally gilt, if we suppose that these are a half less than the rest, it| 
 will be found that the thickness of the latter will be only one 525000th of | 
 line. 
 
 Lastly, it is well known that this plate may be made to pass a second time ~ | 
 the steel rollers, bringing them nearer to each other, in such a manner as to render its 
 breadth double ; hence it follows that in the latter state there are parts of the gild-| 
 ing where its ices is only the 1000000th part of a line; which is in the same | 
 proportion as a line is to the length of 1200 toises, or half a league. 
 
 It is however certain that these gold particles have mutual adhesion and continuity 5) 
 for if this silver wire be immersed in aquafortis, the silver will be dissolved and the 
 gold will remain like a small hollow tube. Lastly, if the gilding be viewed to 
 a microscope, no trace of discontinuity will be observed. : 
 
 As the ductility of gold is far greater than that ofsilver, a much longer wire might | 
 be made with an ingot of gold of the same weight. But can we believe what is. 
 related by Muschenbroeck on this subject? This philosopher says that an artist of 
 Augsbourg made a gold wire weighing only a grain, which however was 500 feet in| 
 length. He could therefore have made a gold wire a league in length, and weighing only | 
 a dram, or the third of a gros; a wire 24 leagues in length would have weighed only. 
 one ounce; and with a pound of gold he could have made a wire 192 leagues in 
 length. A wire of this size, capable of encompassing the globe of the earth, would 
 have weighed only about 50 pounds. 
 
 But we can shew that a thread, the work of an insect, surpasses in fineness the wire 
 ascribed to the artist of Augsbourg. It has been mae that a single thread of silk) 
 360 feet in length, weighs a grain; 24 grains therefore will give 1440 toises, and 36 
 grains a league of 2160 toises: an ounce of this thread will extend 16 leagues, and, 
 a pound 128: in short, a thread of this kind capable of encompassing the globe of 
 the earth would weigh no more than 70 marcs, or 35 pounds. We shall here add,, 
 that the thread of a spider’s web, which is much finer and lighter than the thread of 
 a silk- coe of the same length as the above, would weigh only two mares, or “ 
 a poun 
 
GREAT DIVISIBILITY OF MATTER. 637 
 
 PROBLEM XXI. 
 
 Continuation of the same subject: Division of matter in the solution of bodies, and in 
 odours and light. 
 
 But new subjects of admiration present themselves to us in the prodigious small- | 
 
 yess of some parts of matter: these we shall here add, with an account of their 
 iffinity. 
 | Metallic solutions afford the first example. Dissolve a grain of copper ina sufficient 
 yuantity of volatile alkali; and you will obtain a liquor of a blue colour. If you 
 your this solution into three pints of water, the whole water will be sensibly coloured 
 blue. But three French pints make 144 cubic inches; and as each inch in length may 
 be divided into lines, then into tenths of a line visible to the eye, it will be found 
 that in these 144 cubic inches, there are 248832000 of such parts, every one of which 
 js coloured blue. A grain of copper is divided then, by these means, into at least 
 248832000 parts. But we shall go still farther; each of these parts may be seen by 
 a microscope that magnifies objects 100 times in length, consequently 10000 times 
 in surface; and every one of them will be found to be coloured: if we therefore 
 multiply the above number by 10000, or add to it four ciphers, we shall have a 
 grain of copper divided into 2488320000000 parts, visible to the eye, at least when 
 assisted by the microscope. 
 _ Let us now proceed to odours. It is said that a grain of musk is capable of per- 
 fuming, for several years, a chamber 12 feet in every direction, without sustaining 
 any sensible diminution in its volume, or its weight. Buta space such as the above 
 contains 1728 cubic feet, each of which contains 1728 cubic inches, and each of these 
 1728 cubic lines; so that the number of cubic lines is the third power of 1728. It is 
 probable, that every one of these cubic lines contains some of the odorous particles ; 
 the air of the chamber may in the course of several years be renewed 1000 times; 
 and the grain of musk, without sensible alteration, may furnish new odorous particles. 
 In calculating the tenuity of each of these, the imagination is lost. 
 
 However, notwithstanding the tenuity of these odorous particles, they do not pass 
 through glass and metals; and*there are certain effiuvia which penetrate them; such 
 as those of luminous bodies or light, magnetism, and electricity. How great then 
 must be the tenuity of the particles of which these consist! But we shall confine our 
 observations to light. 
 
 ! If those particles, the emission of which are supposed to constitute light, were not 
 
 of a smallness almost infinite, there is no body which could resist the action of the 
 
 weakest light; for their multitude, and the rapidity with which they proceed from 
 
 the luminous body, are such, that without this prodigious tenuity light would break to 
 
 pieces every body on which it might fall, instead of exciting in it that gentle vibration, 
 
 that insensible tremulous motion, in which heat consists, when it has only the density 
 
 of the light of the sun. 
 
 _ Light, indeed, in a second passes over 128880 leagues, or 257760000 toises ; con- 
 
 beauently, if a particle of light were only equal to the 257760000th part of a grain of 
 lead, a line in diameter, it would make on our organs the same impression as a 
 
 similar grain of lead impelled with the velocity of a toise per second. ‘There is no 
 
 doubt that such an impression would be very sensible to the delicate parts of our 
 
 bodies. But what would it be if millions of millions of such globules should strike 
 against it, and be followed at an interval of time infinitely small by a like quantity 
 of others, as is the case when our body is exposed to the light ! No human being could | 
 resist it. 
 
 The tenuity of a particle of light then is still far below that which we have 
 assigned to it as its first limits. el us endeavour to determine another, that may 
 
 approach nearer to the truth. / 
 
638 PHILOSOPHY. 
 
 The density of the sun’s light, such as it is when it reaches the earth, in our 
 climates, is of such a nature, that if diffused throughout a space 250000 times) 
 greater, it would have a splendour equal to that of the full moon. | 
 
 It is probable that the latter, diffused in the like manner, would be equal, at least, 
 to that of a glow-worm, which enlightens an object at the distance of 10 feet; con. 
 sequently the latter will be found by calculation to be 26500000000 times weaker, 
 It is besides very probable, that in the pupil of the eye, which at that distance 
 beholds the light of the glow-worm, there is no part which is not itself sensibly en- 
 lightened: let us suppose it to be a square line of surface, and that this square line’ 
 is divided into 10000 sensible parts ; every moment therefore there are 10000 globules 
 of light, which reach the retina, united in one imperceptible point, and’ with the 
 velocity of 257760000 toises per second, without producing however a sensible’ 
 impression, and even scarcely the perception of light. 
 
 If we suppose the same quantity of globules of light thrown by the weakest 
 light on a square line of surface, it will be found that in a line square of the sun’s 
 light, there are 625000000000000, and in an inch 90000000000000000. This quan. 
 tity of globules, moved with the velocity of 257760000 toises per second, and 
 renewed perhaps a thousand times in that interval, would produce however in the 
 palm of the hand but a slight sensation of heat; and hence there is reason to con- 
 clude, that 900 thousand millions of millions of these particles, moved with the’ 
 abaee velocity, make less impression than the shock of a leaden ball, a line in dias. 
 meter, which falls from the height of three feet. And hence arises this new cone’ 
 sequence, that if we suppose the particles of light to have the same density as lead, 
 each of them, compared with a ball of lead a line in diameter, is in a less rah 
 than a 257760000th by 90000000000000000, or a 23198400000000000000000000th 
 part to unity. | 
 
 Such, then, at least, is the tenuity of the particles of light; and by other reason-— | 
 ing perhaps we might prove that it is still much rarer; so that in all probability the 
 above ratio must be reduced to that of unity to a comparative number of 30 or 30 
 figures. But we shall confine ourselves to what has been already said, because it is | 
 sufficient for our purpose, and to shew, as we have done elsewhere, that the sun, for | 
 several successive ages, may furnish, without any sensible diminution, sufficiee 
 matter for the emission of the light which proceeds from him; and this may serve | 
 to answer an objection which has been made to the Newtonian theory of light. — 
 
 PROBLEM XXII. Ys | 
 
 What velocity ought to be given to a cannon bullet, in a horizontal direction, to prevent. 
 
 it from falling to the earth, and to make it circulate around it like a planet, suppos- | 
 ing the resistance of the air to be destroyed ? : 
 
 If a cannon bullet be fired off in a horizontal direction, from the top of a mountain, | 
 it will fall to the earth, as is well known, at a certain distance. If we now suppose | 
 that the velocity communicated to this bullet is more and more increased, it will | 
 fall at a greater and greater distance; for the parabola, or rather ellipsis, it describes, | 
 will be broader and broader. We may therefore conceive the velocity to be so great, | 
 that the bullet shall fall to the earth at the point diametrically opposite to that from 
 which it was fired. In this case, if the velocity were increased ever so little, the | 
 bullet would not touch the earth, but would return to the point from which it set © 
 out ; describing a line similar to that which it before described. It would then con- | 
 tinually move in an elliptical line, around the earth, and really be a small planet, 
 performing its revolution around it. 
 
 The question then is, to find what would be the periodical time of this revolu- 
 tion; for by knowing this time, we could easily find the velocity of the small planet, © 
 
TO MAKE SATELLITES AND PLANETS. 639 
 
 that with which the bullet set out; because nothing would be necessary but to 
 ivide the space passed over, which in this case is nearly the circumference of the 
 arth, by the time employed in passing over it. 
 
 The solution of this problem may be easily deduced from the celebrated rule of 
 “epler ; for if we suppose our small planet in motion, it must, compared with the moon, 
 erform its revolutions in sucha manner that the squares of the periodical times shall 
 eas the cubes of the distances. But the mean distance of the moon from the earth 
 _ 60 semi-diameters, and that of the small planet will be equal to the earth’s radius, 
 
 t ]semi-diameter. We shall consequently have this ratio, as the cube of 60 or 216000 
 f to 1, sois the square of the periodical time of the moon to the square of the periodi. 
 a time of the small planet. But the periodical time of the moon is oT days 8 hours, 
 
 (656 hours, the square of which is 430336; if we then say, as 216000 is to 1, so is 
 30336 toa fourth term; we shall have for this fourth term 470336 5 or in decimals 
 19923; the square root of which 1°41, will express the number of hours employed by 
 se small planet in its revolution. But 1:4]in hours and minutes is equal to lh. 24m. 
 
 3s. Thesmall planet, therefore, would perform its revolution in that time; which, 
 
 ipposing a great circle of the earth to be 24000 miles, gives nearly 282 miles per 
 inute, or 4°7 miles per second. 
 
 ‘Tf a velocity greater than the above, but less than 1491 leagues, were given to 
 tis body, it would describe an ellipsis, the perigeum of which would be in the 
 yint of departure. If the velocity of the projection were 1491 leagues per minute, 
 ‘greater, the body would not return to the earth; for in the first case it would de- 
 ribe a parabola, the summit of which would be in the point of projection, and in 
 e second it would describe an hyperbola. 
 
 | 
 t 
 
 PROBLEM XXIII. 
 _ Examination of a singular opinion respecting the Moon and the other planets. 
 ‘thas been said, and the singularity of the conjecture has given it some importance, 
 atthe moon may be nothing else than a comet, which in approaching to or receding 
 om the sun, and passing at the proper distance from the earth, may have been 
 verted from its course, and thus have become that secondary planet which accom- 
 mies our earth. For, if we suppose that such a comet, having only the projected 
 jotion necessary for describing a circle around the earth, at the distance of 60 semi- 
 meters from its centre, really passed at that distance From our globe, and in a plane 
 lined to its orbit, it must necessarily, say some po oson ae) have become our 
 pon. 
 This conjecture is supported by some remarks which seem to give it a certain 
 gree of probability. ‘The moon, say they, when viewed through a telescope, pre- 
 nts the appearance of a body which has been torrefied ; the cavities interspersed over 
 ii surface seem to be fissures, occasioned by the intense heat which caused the mois- 
 reit contained to escape in vapours; and they add, that no appearance of humidity 
 iw remains in the moon, since it has no atmosphere, All this agrees exceedingly 
 ll with a comet, which has passed very near the sun. 
 It is also to be observed, say they, that the largest planets, such as Jupiter and 
 ‘turn, have several satellites; for as their attraction extended much farther than 
 it of the earth, they had a far greater power over the comets which passed in their 
 ighbourhood, the motion of these cométs having been besides lessened in conse- 
 fence of their distance from the sun. The small planets, such as Mercury, Venus, 
 id Mars, have no satellites, on account of the smallness of their size, and the 
 locity with which comets pass them, in advancing towards, or receding from the 
 iP 
 These ideas are ingenious; but this < ssertion or conjecture, when examined accord- 
 tothe principles of geometry, cannot be maintained. 
 
640 PHILOSOPHY. _ 
 
 ify 
 
 It is found, indeed, by calculation, that whatever may be the position or magni 
 of the orbit of a comet, it cannot, when it passes near the orbit. of the earth, haw 
 the velocity necessary to make it become a satellite to it, whatever may be the proxi 
 mity at which it passes; for it can be demonstrated that every comet, when ij 
 approaches the sun within a distance equal to that of the earth, has at that moment / 
 velocity in its orbit, which is to that of the earth, as ,/ 2 is to 1, or as 1414 ti 
 1000. But this velocity is far greater than that of the moon in her orbit, and evel 
 greater than that of a planet which should circulate almost at the surface of thi 
 earth, as the following calculation willshew. | 
 The earth in about 365 days passes over an orbit of 597 millions of miles in cir 
 cumference: its velocity then in its orbit is such, that it passes over in a da: 
 1635616 miles; in an hour 68150 ; and in a minute 1136; therefore, if we multiph 
 the last number by 4414, we shall have nearly 1606 miles for the space which ever 
 
 1000? 
 
 comet, when it arrives at the distance of the earth from the sun, necessarily passes ove 
 in a minute. | 
 
 Let us now examine that of the moon in her orbit. The mean diameter of th 
 moon’s orbit is about 60 times the earth’s diameter ; consequently its circumferene) 
 will be 188 of these diameters; which, estimating the earth’s diameter at 800! 
 miles, gives for the circumference of the lunar orbit 1504000 miles. This space th’ 
 moon passes over in 27 days 8 hours, wauting a few minutes; or 274 days: th. 
 moon therefore in her orbit passes over in a day 55024 miles; in an hour 2293 
 and in a minute 88. Hence it is evident, that if a comet should pass at a distane 
 from the earth equal to that of the moon, which the comet transformed into on 
 satellite might do, it could have a velocity of no more than 38 or 40 miles per minute 
 instead of 1606, which every comet necessarily has at that distance from the sum 
 The moon then could not be a comet, which. passing too near the earth was, as W' 
 may say, subdued and carried away by it. | 
 
 Let us now see whether the comet in question, by passing much nearer the earth 
 and even close to its surface, could be attracted by it. We shall find, by a simila 
 calculation, that it could not circulate around the earth; for we have already see) 
 that a body, to circulate round our globe near its surface, would require a velocit, 
 of almost 300 miles per minute. But this is far below the velocity which a come 
 passing very near the earth would necessarily have; for if a body should be pro 
 jected from the summit of a mountain, towards the East or West, with the velocit 
 of 1600 miles per minute, it would recede from the earth without ever returning toit 
 that velocity being much greater than is necessary tomake it describe around the eart 
 any ellipsis whatever, or even a parabola. ready 4 
 
 Here then the Earth, and no doubt Mars, is excluded from the privilege of eve 
 being able to obtain a satellite in that manner, and this will hold good much mor 
 in regard to Venus and Mercury. But is this the case with Jupiter and Saturn 
 We shall examine this question also, by employing the same kind of calculations. | 
 
 The velocity of Jupiter’s revolution around the sun, is 494 miles per minute. 
 consequently the velocity of every comet advancing to, or receding from the sur 
 when at the same distance as Jupiter from that luminary, will be about 700 miles in th! 
 same time. It is found that the velocity of the first satellite of Jupiter, in its orbit, i 
 37366 miles per hour, or 622 per minute : the velocity then of every comet which shoul: 
 pass Jupiter at the distance of his first satellite, would necessarily be rather mor 
 considerable, being about a seventh part more. Hence it follows, that neither th 
 first satellite nor any of the rest was originally a comet, which this large plane 
 appropriated to itself; for the other satellites have a velocity still less than tha 
 of the first. as: 
 
 It now remains to determine whether a comet, in passing near Jupiter, could bi 
 stopped by it. This indeed does not appear to be absolutely impossible: fora sa 
 
 | 
 
a SHOCK OF A COMET. 641 
 
 tellite which should pertorm its revolution near the surface of J upiter would 
 employ a little more than three hours; which gives a velocity of 1557 miles per 
 minute. But we have already seen that the velocity of the comet would be no more 
 than 700; which therefore is not too great to make a body describe even a circle 
 around Jupiter, very near his surface. If a comet then, in advancing to or returning 
 from the sun, should fall into the system of Jupiter, between him and his first sa- 
 tellite, it might continue to circulate around that planet in an orbit, if not circular, 
 at least elliptical, and more or less elongated. 
 
 For let us suppose aB (Fig. 16.), to be the orbit 
 of Jupiter; and that this planet is at I, proceeding 
 FAR towards B, and that the comet is in c, proceeding to- 
 ea LO oS wards D, at an angle of about 45 degrees; and that 
 I B c p denotes the velocity of the comet, which we have 
 | c found to be greater than thet of Jupiter in his orbit. 
 If p= be made equal to the velocity of Jupiter, cz 
 will be the velocity of the comet, and even its route, in regard to Jupiter, supposed 
 10 be fixed and without any action on the comet. But*on account of this action it 
 would describe an inflected route, such as o F, which would make it fall almost in a 
 yerpendicular direction on the orbit of Jupiter, and with a velocity but little greater 
 han that of the first satellite. If at the moment then when Jupiter was at the 
 voint 1, so situated as to make 1F less than the distance of Jupiter from his first 
 atellite, we do not see what could prevent the comet from assuming around him 
 hat circular or elliptical motion suited to its projectile force; and if it should per- 
 orm the revolution once, it is evident that it ought to continue it. 
 | We however confess, that we have not yet examined this point so far as to be 
 ible to assert that it is fully demonstrated. To be assured of it, the following 
 juestion, which is only a branch of that of the three bodies, but which would require 
 00 much intricate analysis for this work, must be answered. Two bedies, 1 and C, 
 Fig. 17 (Fig. 17.), which attract each other in the inverse 
 ratio of the squares of the distances, and in the direct 
 
 , : 6 ratio of their masses, being projected from the points 
 z 1 and c, according to the directions 18 and ca, with 
 %y given velocities, to find the curves which they would 
 
 | describe. To simplify the problem, we might even 
 lippose one of them, 1, to be so large in regard to the second, as to be scarcely 
 urned aside from its course. 
 
 Fig. 16. 
 
 ~ 
 
 PROBLEM XXIV. 
 
 How far is there reason to apprehend the shock of a Comet; and what devastation 
 ) would be thence occasioned on the earth ? 
 
 What has been said respecting comets in the preceding problem, naturally leads 
 3 to examine a question become celebrated by the alarm it occasioned at Paris some 
 *ars ago. In the year 1774, a Memoir was written by one of the Academicians, 
 F which an incorrect account was propagated, and in which it was said the author 
 mounced the speedy approach of a comet to the earth, and that the effect of this 
 ‘proach would be at least a rising of the waters of the ocean, so as to overwhelm 
 ir continent. In consequence of these reports, the people of that capital were 
 Town into the utmost consternation and uneasiness. I knew some women who 
 ere so terrified, that they did not close their eyes for several nights successively ; 
 d I was even obliged, in order to \quiet one of them, to assure her that a very 
 eat error had been found in the Academician’s calculation, and that, on this ac- 
 unt, he had fallen into disgrace witl. the Society to which he belonged. The mo- 
 
 fe, I hope, will plead my excuse with that illustrious astronomer. I am certain 
 pag? 
 
642 PHILOSOPHY. 
 
 that in so good a cause he would have indulged in the same innocent deception ; 4 
 the object was nothing less than to restore rest, and their former lustre, to two eyes, 
 capable of deranging the observations of the most insensible astronomer. But how- 
 ever, as I have always been devoted, notwithstanding my taste for the abstract sciences, 
 to that charming portion of the human race, I shall endeavour to tranquillize them, 
 and to prove that the danger of being crushed or inundated by a comet, is not so great 
 as to disturb their repose. 
 
 Astronomers long ago conjectured that a comet might fern fatal to the earth 
 The celebrated Whiston, whose imagination was rather too powerful for his rea- 
 soning faculty, observing a comet, viz. that of 1680, accompanied with an immense 
 tail, began to conjecture that if any of the planets should happen to meet with this 
 tail, it might by its attraction condense its vapours, and be inundated by it. He 
 supposed farther, that the deluge had been produced by the same cause : and added, 
 that a comet, such as that above mentioned, if it approached near the sun in return- 
 ing, might acquire a degree of heat several thousand times greater than that of red, 
 hot iron; and consequently might consume our earth. He was of opinion also, that. 
 the general conflagration, which is one day to destroy the globe we inhabit, will be 
 occasioned in this manner. | 
 
 These ideas, in which there is more of singularity than truth, are a sufficient proof 
 ‘of what we have already observed in regard to Whiston’s disposition. It is impos- 
 sible to say what might be the case, if a comet, heated to such a violent degree, 
 should pass very near us. It is probable, considering the rapidity with which it 
 would move, when at its least distance from the earth, that we should not be much 
 incommoded by it. In regard to the danger of being inundated by the vapours of its 
 tail, it is entirely void of foundation; for it may be easily demonstrated, that these 
 vapours which float in a medium as thin as ether, must themselves be exceedingly. 
 rarefied. There is some reason to believe that all this immense tail, reduced toa 
 fluid, such as water, would scarcely furnish enough for an abundant shower. In short, 
 the comet alluded to, returns only about every 575 years; consequently it will not 
 appear again till about 448 years have elapsed. 
 
 Dr. Halley considered this danger in another manner. This philosopher observed, 
 that if the comet of 1680 had passed through the ecliptic 31 days later, its distance 
 from the earth would not have been greater than the sun’s semi-diameter, which is 
 about 441623 miles; and he adds, there can be no doubt that such a proximity be- 
 tween these two bodies would have occasioned a considerable derangement in the 
 motion of the earth: such as a change of its eccentricity and periodical time. May 
 the Author of nature, adds he, preserve us from the shock of these enormous masses, 
 or even from their contact, which is but too possible! He bowever remarks, that the 
 highly varied position of the orbits of comets, and their inclination to the ecliptic, 
 which in general is very great, seem to be arranged by the Author of nature to secure 
 us from so fatal a catastrophe. 
 
 As the astronomy of comets, since the time of Halley, has been enriched with 
 the knowledge of very many new ones, it was natural to examine whether there: 
 was any of them which, by some change in their position, and the magnitude of theit 
 orbits, might become dangerous to our earth. This labour was undertaken by De 
 la Lande, in consequence of the comet seen in 1770; and he found that there are 
 some of them which, by changing their elements a little, might approach very near 
 to the orbit described by our earth. He shewed, at the same time, that there is no 
 great cause for being alarmed at this supposed danger, as several thousands might be 
 betted to one, that if a comet should even pass through the earth’s orbit, these two 
 bodies would not fall in with each other. hi 
 
 This danger, as may be seen, was at a sufficient distance to give no great cause of 
 apprehension ; but he added, that if we should suppose such a comet:to pass at the 
 
 ." 
 
SHOCK OF A COMET. 643. 
 
 listance of 45000 miles, it would raise the waters of the ocean, and according to its 
 yosition occasion a flux capable of covering our whole continent, and of sweeping 
 iway all its inhabitants, together with their habitations. This augmented the danger 
 n a considerable degree ; for if 10000 could be betted to 1 that the earth and the 
 somet would not be at the same time in the ecliptic, at the distance of a diameter of 
 aur globe, no more than 2000 could be betted to 1 that they might not be at the dis- 
 ance of 5 diameters from each other, and consequently that we might not be drowned. 
 But the stake is so great, that even this small chance cannot be considered without 
 some uneasiness: and there are people who would not hold a chance in a lottery where 
 shere is only one blank to a hundred thousand prizes. 
 
 | But fortunately all these calculations are founded on suppositions which, though 
 they may be realized in the course of ages, cannot take place in the present state of 
 the universe ; for the orbit of no comet hitherto known falls in with the path described 
 vy the earth in the ecliptic. It is indeed true, that, as the orbits of the planets and 
 somets are subject to insensible variations, it may happen hereafter that the orbit of 
 4 comet will intersect that of the earth; but unless it should absolutely coincide 
 with the plane of the ecliptic, that position can be only momertary ; and as the revo- 
 utions of the comets are exceedingly long, there is a great probability that this posi- 
 ion will be changed when the comet passes the ecliptic. 
 
 _ But let us suppose that this position is so constant, that a comet, when it passes 
 
 ‘the ecliptic, shall be exactly in the same plane, and in the path of the earth; and let 
 ‘Isexamine, by consulting the laws of probability, what chance there is, that at the 
 ‘moment when the comet is in the ecliptic, the earth will be ina point sufficiently near 
 
 ‘0 come in contact with it. » The calculation is as follows :— 
 
 ' At the moment when the comet is in the ecliptic, there are so many positions for 
 ‘he earth in the same circle as there might be terrestrial diameters; but only three 
 of these positions are absolutely critical; for there is one which would give a central 
 shock, and the other two, at the distance of a diameter before or behind the place 
 of the comet, would give merely a superficial shock. But it is found, that the cir- 
 ‘umference of the earth’s orbit contains the diameter of the earth 72450 times; and 
 fthis number be divided by 3, the quotient will be 24150. Hence if we suppose 
 ‘comet to be in the path of the earth, 24150 might be betted to 1, that the latter 
 vould not be exposed to any shock, even of the most superficial kind. We 
 nay add also, that this dangerous position of the comet is, as we may say, the 
 \ffair of a moment ; for in crossing the earth’s orbit, it has a velocity of 4000 miles 
 
 er minute; consequently the danger would not last above 3 minutes. Some 
 
 langer cértainly might be apprehended if the earth, when a comet is in its proxi- 
 nity, should move in so irregular a manner as to fall in with it, and to block up 
 ts way. 
 
 | The danger of our globe being inundated by the rising of the waters of the ocean 
 s still more unfounded, even if the comet should pass at a very moderate distance 
 rom it, such as that of 36000 or 40000 miles, which is about a sixth part of the 
 listance of the earth from the moon. It is indeed true, that if we suppose a comet 
 ofall in exactly with the orbit of the earth, it is only 1 to about 7200, that our 
 lobe may not be at a greater distance from it than four or five times its diameter 
 ut the rapidity with which the approach would take place, and with which the two 
 lobes would afterwards recede, would not allow the waters sufficient time to rise so 
 8 to inundate our continent ; for a certain period would be required to communi- 
 ate to the enormous mass of the waters of the ocean such a movement as that 
 € the flux and reflux. A proof of this is, that the flux, even in the open seas, does 
 
 ot happen till some time after the moon’s passage of the meridian ; and that the 
 
 igh tides, at the new and full moons, do not occur on those days, but on the 
 dMlowing ones. But if a comet should arrive at the earth’s orbit, it would traverse 
 ly Pe 
 
644 ‘ PHILOSOPHY. 
 
 our lunar system nearly in the course of an nour ; consequently it could produce | 
 only a very slight motion in the open seas, such as the Pacific Ocean. Some of the. 
 small islands interspersed in it, which are almost on a level with the water, might | 
 be overwhelmed; but our continent would absolutely be sheltered from such a mis- 
 fortune. | 
 
 Tho most singular circumstance, in regard to the terror spread throughout Paris, 
 in consequence of an incorrect account being propagated of M. de la Lande’s Memoir, 
 was, that the greatest danger to which the earth had been exposed in the course 
 of several ages, was then past; for of all the comets hitherto known, that of the year’ 
 1770 approached nearest to the earth. On the first of July it was at the distance of 
 2250000 miles, which is only about nine times the moon’s distance from the earth, 
 
 We shall here observe, that a comet, nearly as large as the earth, and traversing | 
 the heavens with a velocity equal to that above mentioned, would be a grand and 
 magnificent spectacle to astronomers. What a noble phenomenon, a new star, : 
 of nearly nine degrees apparent diameter, passing over by its own motion about 180. 
 degrees of the heavens in the course of two hours! What astronomer would not 
 wish to behold such an uncommon phenomenon, were it even to occasion some ¢a- 
 tastrophe to the small uninhabited and already half inundated islands of the vast: 
 ocean ? 
 
 It has however been calculated, that this can never take place, without some great, 
 derangement in the motion of our globe. M. de Sejour has found, that if a comet, 
 as large as the earth, should pass it, at the distance of about 40000 miles, it would 
 change its periodical revolution; and this revolution, instead of 365 days 6 hours: 
 and some minutes, would become 367 days and some hours. But no physical evil 
 would thence result to the universe. Astronomers indeed would have to recaleulate: 
 their tables, which would be thus rendered useless ; chronologists would be under 
 the necessity of altering their method of computing time, and states would be. 
 obliged to change their calendars ; but this would only furnish matter for new specu-, 
 lations, and afford more occupation to the learned. in 
 
 Recent observations on comets shew that their density is in general exceedingly, 
 small. Stars invisible to the naked eye were seen through the centre of Biela’s 
 comet; and though the motions of some comets have been sensibly deranged by the 
 attraction of planets near which they have passed, when in the vicinity of the sun; 
 the motions of the planets or of their satellites have not been affected in the slightest 
 observable degree by the attraction of the comets. It is probable, therefore, that ¢) 
 direct concussion between a comet and a planet would not, to the latter, be an affair 
 of serious moment. | 
 
 THEOREM 
 
 A pound of cork weighs more than a pound of lead or of gold.—A body weighs mort 
 in summer than in winter. 
 
 These two propositions, on the first view, may appear to some of our readen 
 
 a paradox; but when they have read the following reflections, the paradox wil 
 vanish. fa | 
 When bodies are weighed in air, which is commonly the case, they are weighed ir 
 
 a fluid which, according to the laws of hydrostatics, always destroys a portion 
 of their weight equal to that of a similar volume of the fluid: hence a cubic incl 
 of lead or of gold, for example, when weighed in air, loses of its absolute weight # 
 quantity equal to the weight of a cubic inch of air; and the case is the same with 
 all other bodies. A pound of cork, under the same circumstance, loses a quantity 
 of its weight equal to that of a volume of air of the same size as the cork, Bui 
 the volume of a pound of cork is much greater than that of a pound of gold ot 
 of lead; consequently a pound of cork, when weighed in air, has a greater absolute 
 
| 
 i A POUND OF CORK HEAVIER THAN A POUND OF LEAD. 645 
 
 _ by the weight of a greater quantity of air than the latter, they still remain 
 equal. 
 
 This reasoning is confirmed by experience ; for if a pound of gold or of lead, and 
 a pound of cork, suspended from a good balance, be brought into equilibrium, and 
 ‘if the whole be then covered with the receiver of an air-pump; when the air is 
 exhausted, the cork will be immediately seen to preponderate. In this case, indeed, 
 ‘the weight of the cork is increased by the weight of an equal volume of air; and 
 that of the gold is also increased by the weight of a volume of air equal to itself. 
 
 But the former is much greater; consequently the equilibrium must be destroyed, 
 
 and the cork must preponderate. Having thus explained the first paradox, we shall 
 
 now proceed to the second. 
 
 ' In summer the air is dilated by the heat, and its density being thus lessened, the 
 necessary result is, that the same volume of air is lighter ; consequently each of the 
 bodies, brought into equilibrium, loses less of its weight than when the air is denser. 
 ‘But this effect is not produced in the same proportion: the pound of cork, for 
 example, in common air loses 4 grains of its weight, and therefore has an absolute 
 weight of 1 pound 4 grains; while the pound of gold, losing only half a grain, 
 : weighs in reality 1 pound and half a grain. In air, dilated so far as to weigh a half 
 less, a volume of air equal to the volume of cork will weigh only 2 grains; and the 
 : volume of air equal to that of the gold will weigh no more than a quarter of a grain ; 
 ‘hence the pound of cork weighed in common air will weigh in this rarefied air 1 
 ‘pound 2 grains; and the pound of gold one pound and a quarter of a grain:* the 
 cork therefore will preponderate. 
 | Corollaries.—I. From what has been said this consequence may be deduced: that 
 can weights in equilibrio at the surface of the earth, will not be so when carried 
 fo the summit of a mountain. For, on the summit of a mountain the air is more 
 Tilated, and therefore, according to the above reasoning, the equilibrium will be de- 
 tanged, and the most voluminous body will preponderate. 
 
 I. The contrary will be the case if the bodies be in equilibrio at the summit 
 of the mountain, and be then weighed at the bottom of it; or if they be weighed at 
 she surface of the earth, and be then carried to the bottom of a mine. Jn this case, 
 the most voluminous will become the lightest. 
 
 ) III. It would therefore be attended with advantage to purchase gold in summer, 
 ‘nd sell it in winter; or to purchase it in a cold place and to sellit in a stove. For 
 -yold is generally weighed with copper or brass weights, which in summer lose less 
 of their absolute weight than they do in winter: hence it follows, that in sum- 
 ner they weigh more. By these means, therefore, a larger quantity of gold will 
 be obtained in summer than in winter; consequently, by selling it in winter the 
 duyer will get less. + 
 | In purchasing diamonds, a contrary method ought to be pursued, because they 
 we weighed with copper weights, which are specifically heavier. If a weight of 
 -sopper then be in equilibrio with a weight of diamonds, in air of a mean temperature ; 
 on transporting them into cold air, the copper will preponderate ; and the contrary 
 will be the case when they are transported into warmer air. Diamonds therefore 
 yught to be purchased in cold air, or in winter, and to be sold in summer, or in 
 warm air. 
 
 _ The difference in both cases is hawever so small, that it would be a poor specula- 
 ion to purchase diamonds in winter,| with a view of selling them in summer, or to buy 
 
 ‘old insummer in order to dispose of it in winter. But the spirit of the mathematics 
 
 * The weights here given by way of example, are French. 
 
 I 
 
646 PHILOSOPHY. 
 
 is capable of shewing and. appreciating the difference; and though this pheno- 
 menon may be of little use in traffic, it is nevertheless a physical and a mathematical 
 truth. 
 
 THEOREM Ii. 
 
 Two homogeneous weights which are in equilibrium at the surface of the earth, when 
 suspended from a balance with unequal arms, will not be so when carried to the sum. 
 mit of a mountain, or to the bottom of a mine. — 
 
 Fig. ibe Let us suppose a balance (Fig. 18.) with unequal arms, aB 
 and B D, to be loaded with the two weights P and @, which are 
 
 err 7} in equilibrio, and therefore unequal: if the balance be in a ho- 
 a : Pe rizontal situation, these weights, as they tend to the centre of the 
 rok Ff i “p earth, which we suppose to be c, will form with the balance the 
 
 the larger arm, will be the least. From the point z, let fall 
 
 on the lines of direction A c and pc, the perpendiculars BE 
 
 BF: by the laws of mechanics these perpendiculars will be in 
 the reciprocal ratio of the weights, so that B x will be to B F, in the same ratio as the 
 weight Pp to the weight @: that is to say, the product of p by BF will be equal to that 
 of q by BE. 
 
 Now let the balance be removed nearer to the centre of direction; or what 
 amounts to the same thing, let the centre be brought nearer, as to c for example, 
 by which means the new directions will be ac and pce; and let Be and B f be 
 the new perpendiculars to these lines of direction: if the ratio of Bf to Be he 
 the same as that of B F toB E, or of q top, there will still be an equilibrium; 
 but it may be easily demonstrated that this ratio is no longer the same: conse- 
 quently the product of q by B é, will not be equal to that of p by Bf; and therefore 
 there willnolonger be an equilibrium. It can even be shewn that, when the centre 
 is brought nearer, the ratio of B e ton E wil] be less than that of Bf to B F; 
 hence it follows that B e will be less than is required to make these ratios equal; 
 and in this case the weight nearest the point of suspension will preponderate. 
 
 For the same reason the contrary effect will be produced, if the balance be re- 
 moved farther from the centre, by transporting it, for example, to the summit of a 
 mountain. 
 
 It may here be asked, why does the equilibrium subsist, notwithstanding this 
 demonstration? The reason is plain: the centre of the earth is always at so great 
 a distance compared with the length of such a balance, that the lines of direction are 
 sensibly parallel, at whatever height or depth above or below the surface of the earth 
 they may be placed. The difference therefore from an exact equilibrium is so small, 
 
 that it cannot be observed with the most perfect balances constructed by the art 
 of man. 
 
 \/ unequal angles c A Band c DB: consequently the angle a, at 
 / 
 
 PROBLEM XXV. 
 Of the Central Fire. 
 
 Those acquainted with the phenomena observed by different philosophers, m 
 the interior parts of the earth, cannot help acknowledging that the surface, even 
 in our climates, is subject to vicissitudes of heat and cold which we experience. At 
 a certain depth, not very great, for it is sufficient to descend only about a hundred 
 feet, the heat is constantly the same, that is to say, about 10 degrees of Reaumut’s 
 thermometer, or 544 of Fahrenheit. This is observed the same in all climates 
 and in all countries. 
 
 It is evident, therefore, that the earth, independently of the variable heah of the 
 sun, has a source of heat peculiar to itself, from whatever cause it may arise. 
 
THE CENTRAL FIRE. 647 
 ‘Nay, we shall here shew that the degree of heat which the presence of tne.sun, 
 aring several months of the year, adds to the internal heat of the earth, or that 
 hich it loses by his absence, is only a small part of the internal heat. 
 To suppose indeed that the degree of cold which freezes water 1s tne zero, 
 the 0 degree of heat, would, as before observed, be erroneous; for heat 
 id cold are merely relative terms. If the common liquors of our earth were 
 ‘the nature of spirit of wine, as the fluids of our bodies would then be 
 oof against congelation, unless they were exposed to a diminution of heat 
 xyond that at which spirit of wine freezes, it is more than probable that we 
 ‘ould experience no disagreeable sensation by living in a temperature similar to that 
 hich congeals water; on the other hand, if our liquors were of such a nature 
 | to freeze at the degree at which wax begins to become fixed, we should probably 
 sperience at this temperature the same sensation as we experience at that which 
 mmgeals water. Every degree above that term would be heat, and every degree be- 
 'w it would be cold. 
 | Besides, there is no doubt that an absolute degree of cold would congeal all liquors. 
 ut spirit of wine congeals only at 29 degrees below zero of Reaumur’s thermometer : 
 ere is still heat therefore at 28 degrees, though, on account of the disagree- 
 le sensation which we experience, we call it severe cold. We cannot how- 
 rer suppose that this is the ultimate degree of cold. Several reasons, which it 
 ould be too tedious to explain here, give reason to think that this absolute de- 
 ee of cold is a thousand degrees, at least, below the zero of Reaumur’s ther- 
 ‘ometer. 
 'But let us confine ourselves to the 240th degree, which we shall assume as that of 
 ‘e absolute privation of heat, and let us suppose a thermometer the zero of which 
 placed at that term; or let us substitute, in one of our common thermometers, the 
 gree 240 for that usually marked zero, which is only the degree of the conge- 
 tion of water: in this case we shall have 250 degrees for the term which we call 
 mperate. But taking the mean degree of heat during summer in our hemisphere, 
 will be found, that it does not exceed 26 degrees above that of the congelation 
 * water, and consequently 16 above temperate: hence we have for this degree 
 * heat the absolute degree of 266. The thermometer therefore will vary from 
 mperate to the greatest heat 16 degrees in 250, which is somewhat less than the 
 ith part. 
 \It will be found, in like manner, that the mean degree of the cold of winter, in our 
 ‘brthern hemisphere, is 6 degrees below congelation, according to the rate of Reau- 
 ur’s thermometer; that is to say, 16 degrees below temperate: hence the mean 
 minution of heat below temperate, occasioned by the absence of the sun, is about 
 ‘15th part of the heat marked by thé degree 10. Hence it follows, that from winter 
 ) summer, the variation of the heat is at most only }, or as 7 to 8. But it is highly 
 ‘obable, as M. de Mairan has shewn, in the Memoirs of the Academy, for 1765, and 
 uffon in the supplement to his Natural History, that the ratio of this variation is 
 ‘uch less. 
 The former fixes it at 4, or as 31 to 32; and the latter at 4, or as 50 to 51. 
 ut let us confine ourselves to the ratio we have formed, in order that we may set 
 it from a principle fully proved. 
 The conclusion we thence form, and it is a consequence which cannot be denied, 
 as follows: In the globe of the carth there is a degree of constant heat, which is 
 least 7 or 8 times as great as that produced by the presence of the sun while he 
 uminates it in the most advantageous manner for heating it. Here then we have 
 fire, or source of heat, which may be called central. It now remains that we should 
 amine the cause of it. 
 
643 PHILOSOPHY. 
 
 According to some philosophers, this fire is merely the effect of the continual effer 
 vescence occasioned by the mineral matters inclosed in the bowels of the earth, whe: 
 they meet and become mixed with each other. Iron, which appears to be universal] 
 diffused throughout nature, and which communicates its colour to argillaceous earths 
 produces, as is well known, a violent effervescence with the vitriolic acid, which i) 
 also very abundant. Hence, say they, is the cause which excites and maintains i, 
 the bowels of the earth that continual fire by which it is heated, and which ofte, 
 manifests itself by the eruptions of volcanoes, dispersed in considerable numbers ove) 
 its surface: voleanoes according to these philosophers, are the chimneys or | 
 of this central fire. 
 
 It would be difficult to shew the absolute falsity of this opinion: but it does no) 
 appear that a fire of this nature can be general throughout the bowels of the earth 
 The number of the volcanoes, which exist at its surface, is too small to have a caus’ 
 so general: there are even very few of them that burn without interruption. Thy 
 central fire, however, if it be real, must be constant and perpetual; and therefore iti 
 necessary that we should recur to some other cause. t 
 
 Another, which has long appeared to possess a great degree of probability, is as fol 
 lows. The central heat, say some philosophers, is nothing else than the heat whicl 
 the body of the earth, continually warmed by the sun, has acquired in consequenci 
 of the presence of that luminary. But let us render this idea more familiar by al 
 experiment. 
 
 If a globe of iron, which revolves round its axis in a determinate time, and whiel 
 has been cooled to the degree of ice, as well as the surrounding air, be exposed befor 
 a fire, the impression of the fire will first heat the surface, and the heat will gradually 
 penetrate to its interior parts; so that after a great number of revolutions the globi' 
 will acquire such a degree of internal heat, that it will be incapable of receiving more | 
 and the presence of the fire will only serve to make it retain that which has already 
 been communicated to it. 
 
 It may be readily conceived also, that the nature of the globe, or its distance irom 
 the fire, may be such, that this constant degree of heat shall not be very remote fron, 
 that of the congelation of water. - 
 
 In this case, what will be the result? as it is the surface of bodies that alan 
 begins to lose the heat they have acquired, because it loses more by its contact with; 
 the air than is furnished to it by the interior parts, it will necessarily happen, if thc 
 surrounding air be nearly at the degree of congelation, that the part of the surfac«| 
 which is illuminated obliquely, or that which by a slower revolution is opposite-te 
 the side next to the fire, will lose a little of its heat ; and as we suppose the meat’ 
 heat, which the globe has acquired, to be not far distant from the degree of congela 
 tion, like the temperature of the earth, the surface, in those parts less favourably 
 exposed to the action of the heat, may assume a degree of cold equal to that of ice’ 
 Consequently, if there were on the surface of this globe some matter, such as wax 0! 
 water, susceptible of melting and congealing alternately, it would certainly experienct. 
 these alternations : it might even happen that it would remain constantly frozen it 
 the neighbourhood of the Poles; thus it would alternately melt and be congealed it 
 the mean parts between the Poles and the equator, and it would always remain Guie 
 in the environs of the equator. 
 
 But this is exactly what takes place at the surface of the earth: exposed fori 
 great number of ages to the benign influence of the sun, the heat has been commu: 
 nicated to its most interior parts, and this internal heat is what is called the centra’ 
 ‘ire: it is continually receiving an additional quantity, and this makes up for the 
 loss of that dissipated at its surface, by the contact of the air which is less heated | 
 In a word, as the iron globe above mentioned, would possess to the depth of several 
 
 | 
 | 
 I 
 
THE CENTRAL FIRE, 649 
 
 ‘nes below its surface, a neat nearly constant, the aegree of heat which prevails to 
 ome depth below the surface of the earth is, in like manner, almost invariable.* 
 But it is difficult to believe that the mass of the earth, if deprived of all heat, 
 nd exposed to the sun, could ever acquire that heat which it seems to possess. How 
 vany ages, or millions of ages, would be necessary, before a heat, so feeble as that 
 f the sun, could melt an ocean entirely congealed, and insinuate itself into its 
 ‘owels! In our opinion, the ice melted at the line by the presence of the sun would 
 ave been again congealed during the twelve hours of his absence; so that the 
 lobe exposed in this state to the sun, would have remained in it to eternity, had 
 ‘ot some other powerful cause suddenly communicated to it that fund of heat, which, 
 iy vivifying nature, renders the earth habitable, and susceptible of vegetation. 
 
 | A third cause of the central heat remains to be examined: it is that of Buffon. 
 | According to this celebrated philosopher, the earth and other circum-solar planets 
 vere formerly a part of the sun, and were detached from its surface by a comet, 
 yhich entering in to some depth, projected the fragments to different distances. 
 Vhile they were in a state of fusion, each of them, in consequence of the laws 
 f universal gravitation, must necessarily have assumed a globular form. ‘The more 
 onsiderable masses, such as Venus, the Earth, Mars, Jupiter, and Saturn, being 
 rojected in this manner, flew off in a tangent, which, together with the attractive 
 oree of the sun, made them describe, around that luminary, orbits more or less 
 longated. Such of these new planets as had smaller fragments accidentally in their 
 eighbourhood, overcame them in some measure; and these fragments, turning 
 round the larger ones, in consequence of the same laws, became their satellites. 
 in this manner, the Earth, Saturn, and Jupiter, acquired those moons by which they 
 re accompanied. 
 
 If this generation of the earth and circum-solar planets be admitted, it is evident 
 hat these globes were at first fluid; and this may serve to explain their formation 
 ito oblate spheroids; for the earth and other planets must have necessarily been, 
 ‘uring the course of some time, either in a state of fusion or of semi-fluid paste ; 
 therwise their diurnal motion could not have given them that form which they 
 ossess. But let us set out from their supposed state of fusion. Masses of so 
 ionsiderable a size as Venus, the Earth, &c., could not certainly cool in a day ora 
 ear, nor even in twenty centuries. They first passed from a state of fusion to that 
 if solidity ; they remained long impregnated with a quantity of fire, which rendered 
 hem uninhabitable. At length their surface has gradually cooled, till they retained 
 ‘nly that degree of heat necessary for animal life, and to render them susceptyble _ 
 f vegetation. The interior parts of the earth still possess a more considerable 
 ‘egree of heat than the surface, and this heat must go on increasing towards the 
 entre. Such is the central fire. . 
 
 | But by a necessary consequence of the cause of this fire, it must always decrease, 
 © that a small portion of it is every day lost. It appears indeed, that the fertility 
 if the earth daily decreases, and that mankind degenerate, both in size and in 
 trength. This diminution, however, cannot be proved; we have not been long 
 nough possessed of an instrument proper for measuring heat ; it is not much above 
 alf a century since comparative thermometers were invented. But if it be found, 
 00 years hence, for example, that the constant heat in the caverns below the 
 bservatory of Paris, is not more than 7 or 8 degrees, instead of 93, which it is at 
 resent, the progressive cooling of the mass of the earth will be a fact which 
 an no longer be doubted, whatever may be the cause of that heat, and of its de- 
 rease. | 
 
 _* We say almost invariable, because we are acquainted with no observations of the thermometer 
 i Subterranean places, but those made in the caverns below the observatory at Paris. 
 
 | 
 
| 
 
 : 
 
 | 
 
 650 ‘PHILOSOPHY. 
 
 We must however observe, notwithstanding our respect for the illustrious philo- 
 sopher who is the author of this idea, that there are many difficulties in regard to 
 this formation of the earth and planets, which it is not easy to resolve. 
 
 Ist. If the planets were formed in this manner, it is difficult to conceive how the 
 comets could havea different origin: and if the latter were planets circulating around 
 the sun, the Sovereign Cause, who arranged the universe, could, with equal ease, haye, 
 formed the planets in the same manner, 
 
 2nd. It seems difficult to reconcile with the laws of motion and universal gravita. 
 tion, the position and dimension of the orbits of these new planets ; for according to 
 what has been demonstrated by Newton and others, since they proceeded from the, 
 sun, ina line nearly a tangent to his surface, and from a point of his surface, they 
 ought at each revolution to pass through the same point: this however is not the 
 case ; on the contrary, the orbits of the planets are nearly circular. 
 
 It appears also, that in this projection the largest masses could not go to the 
 greatest distances, and describe the largest circles ;.it would seem that the smallest. 
 planets ought to be the most distant from the sun; for if several*bodies are thrown. 
 promiscuously by any force whatever, the smallest will be projected with the ereatae 
 velocity. 
 
 In short, the effect of such a projection is beyond calculation; and it may be an. 
 also that the comet in question, while it ploughed the surface of the sun, communi. 
 cated to it an impulse which made it change its place. This comet indeed, which 
 could carry with it at once such masses as the planets, must have been of an enormous. 
 size, and impinging against the sun with immense velocity, could not fail to cause a 
 small displacement of that luminary, which is in the centre of our system, in a sort 
 
 of inert state. 
 
 Remark.— Whatever may be the fate of these ideas, the following are some of the. 
 consequences which Buffon deduces from his system on the formation of the earth, | 
 and which are too curious to be omitted in a work of this kind. | 
 
 Setting out from his principles on the formation of the earth and the planets, 
 Buffon made a series of very curious experiments, to determine in what ratio the re- 
 frigeration of different masses of matter takes place, according to their nature and 
 size; and from these experiments he concludes: 
 
 That a globe, such as Mercury, must have required 2127 years to be consolidalill 
 to the centre ; 24813 to become so cold that it could be touched ; 54192 to be reduced , 
 toits present temperature ; and in the last place, that it would require 187775 to be-. 
 come so cold as to have only the 25th part of its present temperature: for the sake of 
 brevity we shall call these the Ist, 2d, 3d, and 4th epochs. 
 
 That Venus must have employed 3596 years in the first epoch ; 41900 in the 
 second ; 91600 in the third ; and that 228540 would be required for the fourth. : 
 
 That the earth suploved in the first epoch 2936 years; in the second 34270; in | 
 the third 74800; and that 168125 will be necessary before its temperature is reduced 
 to a 25th part of what it is at present. 
 
 The earth therefore has existed 112 thousand years; and hence it follows, that | 
 Mercury passed the degree of the present temperature of the earth 30000 years 
 ago; and that it has even lost already six of the 25 degrees which remained to it. 
 
 That the moon employed only 644 years in the first epoch ; 7515 in the second; 
 16409 in the third; and 72514 in the fourth * 
 
 Hence the moon, 15000 years ago, was reduced to such a degree of coolness as t0 
 have only a 25th part of the heat of our earth. It needs therefore excite no asto-— 
 nishment that she should appear to us as an accumulation of ice, and that she 
 exhibits no signs of living nature. If she had inhabitants, they must long ago have | 
 been congealed... a 
 
CONSTRUCTION OF BAROMETERS. 651 
 
 Mars employed 1130 years in becoming solid to the centre; 13000 in the second 
 exch ; 28538 in the third; and 60,300 in the fourth: consequently this planet has 
 ten useless for 9 or 10 thousand years. jf 
 
 In regard to Jupiter, the case is different : he must have employed 9400 years in 
 (2 first epoch, and will require 110,000 for the second. But it is only 112000 
 yars since the earth and Jupiter were formed; consequently 7 or 8 thousand years 
 \ll be necessary, before Jupiter can be cooled to such a degree as to admit placing 
 2 foot upon it, without being burnt. When it attains to this epoch, it will require 
 £0400 years before it be reduced to our present temperature ; and then 483000 to 
 lie nearly the whole of its heat. This globe then will begin to be habitable, when 
 % are rendered absolutely torpid with cold. 
 
 In the last place, Saturn employed 5140 years in becoming fixed to the centre ; 
 «d required 59900 before he was fit to be touched: the duration of his third 
 éoch will be 130800 years; and of this epoch above 47000 years have already 
 apsed ; so that it will be above 84000 years before his temperature is reduced 
 1 that of the earth. 
 
 In regard to the satellites, we shall only observe that the greater part of them 
 2 in a habitable state and fit for vegetation, the fourth of Jupiter excepted, which 
 jalready advanced in its fourth epoch: the third of Saturn is nearly at the same 
 «gree of temperature as the earth, but rather somewhat warmer ; the fourth is con- 
 erably advanced in its fourth epoch; and the fifth must have been a mass of ice 
 f nearly 50000 years 
 
 PROBLEM XXVI. 
 onstruction of the Barometer.—To measure the Variations of the Gravity of the Air. 
 
 The barometer is one of those instruments for the discovery of which we are 
 Hebted to the 17th century, a period that gave birth to a great many happy ideas. 
 ‘his instrument, which serves to determine the variations that take place in the 
 javity of the air, derives its name from two Greek words, Bapos and metpey, the 
 immer of which signifies weight, and the latter to measure. It was the invention of 
 orricelli, a disciple of the celebrated Galileo, who employed it chiefly to prove 
 “e gravity of the air in which we live and breathe. But it was Pascal who first disco- 
 red its variations, by means of the famous experiment which he caused his brother- 
 -law to make on the Puy-de-Dome, a mountain in the neighbourhood of Clermont. 
 enabled him to demonstrate, in the most evident manner, the gravity of the air, 
 ‘hich some still persisted to deny, notwithstanding the experiment of Torricelli. 
 A barometer may be easily constructed without much expense. Provide a yessel, 
 me inches in depth, filled with mercury, and a glass tube 30 or 35 inches in length, 
 srmetically sealed at one end. Invert the tube, that is to say turn the sealed end 
 »wnwards, and fill it with mercury; apply your finger to the top so as to keep 
 shut, and having.turned the sealed end uppermost, immerse the open end into 
 he mercury in the vessel, and remove your finger, so as to allow the mercury in 
 ie tube 10 have a communication with that in the vessel: the column of mercury 
 yntained in the tube will then fall, but in such a manner that its upper extremity 
 ill remain about 28 inches, more or less, above the level of the mercury in the 
 sssel, if the experiment be performed at a small height only above the level of the 
 a. Inthismanner you will have a barometer constructed; and if by any contriv- 
 ice you can fix the tube thus immersed in the mercury, the end of the column of 
 ercury will be seen to fluctuate betiveen the height of 27 and 28 inches, or 29 or 
 inches English, according to the d fferent constitutions of the atmosphere. 
 
 This is a barometer of the simplest’ kind, and such as it was when it came from the 
 
 inds of Torricelli. At present, a glass tube, from 33 to 36 inches in length, is 
 nployed: it is hermetically sealed at the one end, and bent at the other, after 
 
 ee 
 
=) 
 i 
 
 ‘ 
 
 652 © PHILOSOPHY. 
 
 having been dilated at an enameller’s lamp, so as to resemble t 
 phial, as seen in the figure (Fig. 19.) This tube is filled by inclin. 
 ing it, and pouring in the mercury at different times, in suct 
 a manner, that when placed upright the mercury in the phia. 
 rises only to about half its height, as a. The difference be. 
 tween the line c a B and the line D&E, at which the mercury 
 maintains itself, is the height of the column which counterba: 
 lances the pressure of the atmosphere, as may be easily con’ 
 ceived. This glass tube, when filled with mercury, is affixes 
 to a piece of board, more or less ornamented; the interva 
 between the 28th and 3lst inch above c 8, is divided into tenths 
 and the words settled weather, fair, changeable, rain, stormy, ari 
 inscribed at equal distances, beginning at the line of 28 inches’ 
 
 ’ Such is the construction of the barometers commonly sold i 
 the shops: but to render them good, some precautions are ne 
 cessary. | 
 Ist. The diameter of the phial, or lower receptacle of the mer 
 cury must be considerably larger than that of the tube 
 otherwise the line a B, as may be easily perceived, will sensibly vary as the mer 
 cury rises or falls. i 
 2d. The mercury must be purified from air as much as possible, or at least to: 
 certain degree ; and the tube ought to be heated and rubbed in the inside, to remoy, 
 the moisture and dust which generally adheres to it; otherwise there will be a disen 
 gagement from it of air, which occupying the upper part of the tube, will, b 
 its elasticity, form a counterpoise to the gravity of the atmosphere, and cause th 
 column to remain lower than it ought to do. This air also, being dilated by heat 
 will produce on the column of mercury a much greater effect, so that its motion 
 will depend both on the heat and gravity of the air, while it ought to depend on th 
 latter cause alone. | 
 Several expedients have been adopted, for lengthening the scale of the barometer 
 sometimes the top of the tube has been bent aside ; the diagonal instead of the perpen 
 dicular of the rectangle becoming the scale. But the most popular expedient is tha 
 adopted in what is called the wheel barometer. | 
 This instrument consists of a bent tube, aBe (Fig. 20. 
 
 Fig. 20. closed at A, the height of a above c not being less than 3] or 3! 
 
 I 
 Eq 
 Z| 
 a 
 3 
 EI 
 = 
 5 | 
 3; 
 5 
 
 sg db gree 
 
 A inches, the end c being open, and the tube filled with mercury 
 The mercury will fall in the leg a 8B till the difference of th 
 5 levels of © and F be equal to the height of a column of mercur’ 
 
 | which balances the pressure of the atmosphere, and every chang 
 in the level of = will produce a corresponding but opposite on 
 in that of F; so that the change in the height of the baromete 
 column is double the change of the level Fr. On ra small iro 
 ball floats, and a string attached to the ball passes over th 
 pulley p, and to the end of the string is attached a weight w 
 somewhat lighter than the ball at r. The axis of the pulle 
 passes through the centre of a large graduated circular plate ¢ 
 This avis carries a hand w, which revolves when the pulley i 
 
 of the height of the barometer’s column. . | 
 When s falls F rises; and the hand retrogrades to a corresponding point on th 
 
 graduated are. _ 
 
 - - 
 
WEIGHT OF AIR, AND CONSTRUCTION OF BAROMETERS, 653 
 
 PROBLEM XXVII. 
 
 Does the suspension of the mercury in the barometer depend on the gravity or the 
 
 elasticity of the air? 
 
 We introduce this question, merely because it has been discussed in some books of 
 iatural philosophy, the authors of which have determined that this phenomenon 
 ought to be ascribed to the elasticity, and not to the gravity of the air. The fol- 
 owing analysis will shew how ill founded is the reasoning of those who entertain this 
 )pinion. 
 
 _ In this question there are two cases. In one of them the barometer is supposed to 
 ye placed in the open air; and this properly is the one which we here propose to 
 yxamine. In the other, it is proposed to be shut up in a room s0 close, that no air can 
 yenetrate to it; or under the receiver of an air-pump, from which the air is excluded. 
 
 It is evident, in the second case, that the cause of the suspension of the mercury is 
 
 he elasticity of the air alone ; but to extend this to the case where the barometer is 
 ‘xposed to the open air, is reasoning, we will venture to say, in a manner unworthy of 
 tr ee nosopher. 
 
 _ To ascertain to which of the two causes the suspension of the mercury in the baro- 
 neter, exposed to the open air, ought to be ascribed, let us suppose the air to be 
 leprived of its weight or elasticity; and let us examine what would be the con- 
 _equence. 
 | If the air were deprived of its elasticity, it is evident that it would fall back 
 ipon itself, and form around the earth a kind of ocean of a peculiar fluid, the height 
 f which would be much less than that of our atmosphere; but it would still have 
 he same weight, for a ball of hair which has lost its elasticity, and is reduced to a 
 ess volume, weighs as much as it did, when in consequence of its elasticity it occu- 
 ied amuch larger space. The mercury in the barometer, if immersed to the bottom 
 
 pe this fluid, would sustain neither more nor less pressure, and consequently would 
 naintain itself at the same height. 
 Let us now suppose, on the other hand, that the air, preserving its elasticity, has 
 ost its gravity. In this case, as the parts of the air would experience no impediment 
 ° recede from each other ; that is to say, as their elasticity would not be compressed 
 oy the weight resulting git the force exercised by the superior on the inferior parts, 
 heair would be dissipated, without exercising any action on the column of mercury ; 
 nless we suppose, at the top of the atmosphere, a transparent arch to confine the 
 ' lasticity of the air; for it is necessary that a spring, to exercise an action with one 
 af its extremities, ought to rest against some fixed point with the other. But as such 
 4 supposition is ridiculous, it is evident that what confines the spring or elasticity of 
 he air is its weight. 
 
 _ Since the air then, if deprived of its weight, and endowed with all the elasticity 
 vossible, would have no action on the mercury in the barometer ; and, on the other 
 tand, as it would still maintain the mercury at the same height, though deprived of 
 ts elasticity, provided it retained its weight, it may be asked to what cause this 
 
 uspension ought to be ascribed? The answer is so easy that it is needless to men- 
 ion it. . 
 
 PROBLEM XXVIII. 
 
 Ise of the Barometer to foretel the approach of fine or of bad weather. Precautions 
 : to be observed in this respect, in order to avoid error. 
 
 : fine of the principal uses of the barometer, is to foretel the approach of fine or of 
 ad weather. Experience has indeec shown that the rise of the barometer, above its 
 -nean height, is generally followed by fine weather; and on the other hand, that 
 vhen it falls below that height, it indicates the continuation or approach of rain. 
 | 
 
 panera 
 
654 PHILOSOPHY, 
 
 These rules, however, are not absolutely general and infallible. Wind also has a) 
 great influence on the rise or fall of the mercury in the barometer; and therefore! 
 we think it necessary to give a few rules, founded on observation, which may enable| 
 those who have barometers, to form a more certain opinion respecting their i) 
 cations. 
 
 Ist. The rising of the mercury announces, in general, fine weather ; and its falli is 
 a sign of bad weather, as rain, snow, hail, or storms. 
 
 2d. During very hot weather, a sudden fall of the mercury indicates a storm 
 and thunder. 
 
 3d. In winter, the rising of the mercury presages frost; and in the time of frost, 
 if the mercury falls three or four lines, it announces a thaw ; but if it rises during a 
 continued thaw, snow will certainly follow. 
 
 4th. When bad weather takes place immediately after a fall of the mercury, it 
 will not be of long duration; and the case will be the same in regard to fine wea 
 ther, if it speedily follows a rise of the mercury. 
 
 5th. But during bad weather, if the mercury rises a great deal, and continues to’ 
 do so for two or three days, before the bad weather is past, a change may be expected 
 to fine weather, which wiil be of some duration. 
 
 6th. In fine weather, if the mercury falls very low, and continues so for two or 
 three days before rain takes place, there is reason to conclude that the rain will be 
 violent, of long duration, and accompanied with a strong wind. 
 
 7th. Irregularity in the motion of the mercury, announces uncertain and variable 
 weather. 
 
 Such are the rules given by Desaguliers, according to a series of observations made 
 by Mr. Patrick, a celebrated constructor of barometers, at London. 
 
 But there can be no doubt that they are liable to exceptions and variations. 
 
 It is known, for example, that in the countries situated between the tropies, the 
 barometer scarcely varies ; on the borders of the sea it always maintains itself within 
 a few lines more or less of 28 inches.* This is a phenomenon difficult to be ex« 
 plained ; and no reason ever yet assigned for it, appears satisfactory. Those theres] 
 would be deceived who should apply the above rules to a barometer, transported 4 
 these countries. 
 
 It frequently happens, also, that the falling of the mercury takes place without any 
 rain; but in that case a considerable degree of wind prevails, if not in the lower, at| 
 ina in the upper part of the atmosphere; for Mr. Hawksbee contrived an experi- 
 
 7 
 
 ment, by which he produced that effect on the barometer artificially. 
 
 PROBLEM XXIX. i 
 
 How comes it that the greatest height of the barometer announces fine weather, and the 
 least the approach of rain, or of bad weather ? 
 
 Those not acquainted with the progress of the barometer, and who are ignorant: 
 that the mercury generally rises when the sky is serene and the air very pure, and. 
 that its fall, on the other hand, generally takes place before rain, would no doubt’ 
 judge differently, and suppose that the mercury ought to fall when the air is serene! 
 and pure, and to rise when the air is charged and impregnated with vapours ; for it 
 is natural to believe, that pure and serene air is lighter than that which holds in’ 
 solution a great deal of vapours. The progress of the mercury in the barometer is| 
 however quite the reverse; it is therefore a phenomenon which has been the subject, 
 of much discussion among philosophers, but without success; for all their oxraa 
 tions overturn themselves, and not one of them will bear examisntionl 
 
 * The natural mean height of the barometer is 20 Knglish inches. 
 
: 
 ' WEIGHT OF AIR, AND CONSTRUCTION OF BAROMETERS. 655 
 /Some philosophers have said: the air is never more serene and more transparent 
 jan when well charged with vapours, or at least when they are perfectly dissolved 
 id combined with it; for it is the property of perfect solutions to be transparent : 
 is not therefore astonishing that the mercury, being pressed down by a greater 
 eight, should in this case rise. But when the aqueous vapours are separated from 
 jeair by any cause, they disturb its transparency, and begin to be precipitated : they 
 longer contribute to its weight, since they are not suspended in it; and asa proof 
 this, they quote the celebrated experiment of Rammazini, which is as follows. 
 _ Take a narrow vessel, several feet in height, and having filled it with water, place 
 yon ita bit of cork, with a leaden weight suspended from it by a thread, so that 
 \e whole shall float. When the vessel has been thus prepared, put it into the 
 ‘ale of a balance, and load the other scale until an equilibrium is produced. If the 
 read by which the lead is attached to the cork, be then cut, it is observed that, 
 hile the lead is falling, this side of the balance is lightened, and the other prepon- 
 rates. Hence it is evident, say the above philosophers, that while a weight is 
 ling in a fluid, it exercises no pressure on the base; consequently, while the 
 ‘pours, collected in the air, are precipitating themselves, or after they begin to 
 if precipitated, the air is lighter, and the mercury becomes charged with a less 
 eight. : 
 This reasoning, which is that of Leibnitz, is exceedingly ingenious. Unfortu- 
 fag however, the experiment of Rammazini proves only that the scale of the 
 lance is unloaded during the fall of the weight; but it does not prove that the 
 ‘ittom of the vessel is eased by the quantity of the weight which is falling ; for 
 ese are two things very different. Recourse must therefore be had to another 
 planation. 
 For our part, we agree with M. de Luc,* that the only cause of the falling of the 
 ercury in the barometer, on the approach of rain, is the diminution of the gravity 
 ‘the air, when saturated with aqueous vapours. In our opinion, therefore, the air 
 il never heavier than when it is exceedingly pure; and we are inclined to think so 
 Pr various reasons. 
 _ The vapours seen floating in the atmosphere, under the form of clouds, are nothing 
 ita solution of water in air: while this combination is imperfect, it is only semi- 
 -ansparent, as is the case in regard to all solutions. The vapours, when in that 
 ate, are observed to rise in the atmosphere, and hence there is reason to conclude 
 vat they are lighter than air. The state of the air, in regard to gravity, when thus 
 _\arged with vapours, may be deduced from the gravity of the vapours themselves ; 
 id since they are lighter than the air in which they ascend, we must infer that the 
 ‘tin which they are dissolved is lighter than pure air. 
 But it may be said, how can we conceive that air combined with a fluid heavier 
 ‘an itself should become lighter? It may be replied, that if the combination here 
 eant were only the interposition of watery particles between those of the air, as 
 ight have been believed, before the improved state of chemistry had thrown light 
 ‘) a number of questions relating to the most common phenomena, this would be © 
 ‘possible. But this is not the mechanism of solutions, or of the combination 
 ‘ bodies with each other ; each particle of the solvent combines with each particle 
 | the dissolved body, and it is not improbable that this takes place here by the 
 edium of fire, which is far lighter than either air or water. We can therefore form 
 -) conclusion respecting the weight of compound particles from that of the separated 
 iticles. Besides, in this state of combination, they may be endowed with a 
 eater repulsive force ; and this even seems to be very probable, since the expansi- 
 lity ot water, when reduced into vapour, is immense. There can be no absurdity 
 
 * « Traité des Barometres, Thermometres,” &c. Geneve, 1770, 2 vols. 4to. 
 
656 PHILOSOPHY. 
 
 then in asserting that air charged with vapours is lighter than pure air. This wi 
 perhaps be demonstrated some day a priori, by chemical processes; and should th 
 be the case, philosophers will be much surprised at the difficulty which has hitheri 
 occurred in attempting to explain the falling of the mercury in the barometer, on th. 
 approach of rain. a 
 
 PROBLEM XXX. 
 Of the Compound Barometer. | 
 
 It has already been seen that a column of mercury of about 80 inches in height, . 
 necessary to counterbalance the weight of the atmosphere; and hence it follow 
 that a simple barometer can never be at a less height, unless some fluid heavier tha 
 mercury be employed. As this length has been found inconvenient, attempts hay 
 been made to shorten it; with a view, as it would seem, of confining it within th. 
 same extent as the thermometer, which may be reduced to a much less size. Th 
 method in which this has been accomplished, is as follows: | 
 
 The principle of the construction of these barometers, consists in opposing sever: 
 columns of mercury to one of air; so that these columns taken together shall hay. 
 the same length, viz. 30 inches, which one ought to have in order to be in equilibriu- 
 with the weight of the atmosphere. Consequently, 30 inches, or the common heigh 
 of the mercury, must be divided by the length intended to be given to the barometer 
 the quotient will give the number of columns of mercury which must be opposed t 
 the weight of the air. 
 
 Thus, if a barometer only 15 or 16 inches in length be required, it must be forme 
 of three glass tubes, joined together by four cylindric parts of a larger size, as ap 
 
 pears Fig.21. Two of these tubes must be filled with mer 
 Fig. 21. cury, and have a communication with each other, by mean 
 of the third, which ought to be filled with a lighter fluid) 
 Thus the first branch, from p to £, is filled with mereury; th 
 second, from & to F, is half filled with coloured oil of tartar, an 
 half with carob-bean oil; and the third, from F to 6, is fille 
 with mercury. This arrangement therefore is the same thing| 
 as if these two columns of mercury were placed one above th 
 other; for it may be easily perceived that the column of mer 
 cury F G, presses on the first by means of the intermediat: 
 column FE, exactly in the same manner as if it were above it, 
 In this kind of barometer, it is the separation of the two li) 
 quors, contained in the branch rF, that serves to indicate thi 
 variations of the weight of the atmosphere; and for this reasoi 
 these two liquors must be of different colours, and of different specific gravities, t| 
 prevent them from mixing. 
 
 To fill this barometer, stop the aperture a, and pour mercury into the two latera 
 branches through the aperture B; then pour the two liquors into the middle braneh 
 through the same aperture ; after which it must be hermetically sealed. 
 
 If a barometer only 9 or 10 inches in height were required 
 30 must be divided by 9 or 10, which will give 3; conse: 
 quently, three branches containing 9 or 10 inches of mercury. 
 and two communicating branches filled with oil of tartar an¢ 
 carob oil, will be necessary. This barometer, consisting 0! 
 five branches, is represented Fig. 22. It may be proper te 
 observe, that the height of each branch ought to be estimate¢ 
 by the difference of the level of the liquor in the upper reset= 
 voir, and that of the liquor in the lower. | 
 
 [| 
 
i THE MARINE BAROMETER.—THE MOUNTAIN BAROMETER. 657 
 
 | This construction, invented by M. Amontons, has the advantage of lessening the 
 height of the barometer, which is sometimes inconvenient ; and of rendering it fitter for 
 veing employed under certain circumstances as an ornament. Butit is to be observed, 
 shat this advantage is gained at the expense of exactness. M. de Luc says that he never 
 was able to obtain a tolerable instrument of thiskind. The intermediate column acts 
 is a thermometer; and those who have attempted to prove that this does not injure the 
 sccuracy of the instrument, did not reflect that their reasoning is true only when 
 the line of the separation of the two colours is in the middle of the Beata of the 
 cube 
 | The Marine Barometer. - 
 
 The barometer constructed in the ordinary way would evidently be useless on 
 board a ship at sea, as the pitching and rolling of the vessel would keep the mercury 
 in the tube in a state of perpetual oscillation. This inconvenience has been over- 
 come by making the greater portion of the tube of very narrow bore, but terminated 
 by a cylindrical portion of about three-tenths of an inch in diameter. The instru- 
 ment is suspended by a spring and gimbals, at a point near the top, which is found in 
 aach case by trial. Thus improved, these instruments are in very great use both in 
 the royal navy and in merchant ships; and they have rendered most important ser- 
 vice to the maritime interests of the community, by giving notice of approaching 
 storms, when frequently there have been no other indications which could have 
 anduced the mariner to make the necessary prepartions for encountering them. 
 
 The Mountain Barometer. 
 
 The barometer in this form has been extensively used by travellers to make obe« 
 servations for the purpose of determining the height of mountains. 
 _ We extract the following description from ‘* A Treatise on the principal Mathema- 
 tical Instruments used in Surveying, Levelling, and Atronomy,” by F. W. Simms, 
 civil engineer; a work which we cordially recommend to all persons desirous to 
 acquaint themselves with the use of such instruments. 
 
 In the brass box a (Fig. 23.) which covers the cistern of mercury, near the bot- 
 tom of the tube, are two slits made horizontally, precisely similar and opposite to 
 each other, the plane of the upper edges of which represents the 
 beginning of the scale of inches, or the zero of the barometer. 
 The screw b, at the bottom, performs a double office; first, itis 
 the means of adjusting the surface of the mercury in the glass 
 cistern to zero, by just shutting out the light from passing be- 
 tween it and the upper edge of the above-named slits; and 
 secondly, by screwing it up, it forces the quicksilver upwards, 
 and by filling every part of the tube, renders the instrument 
 portable. By the help of a vernier the divided scale on the 
 upper part is divided to the five-hundreth part of an inch; aud 
 even the thousandth part may be estimated. The screw c, at 
 the top, moves a sliding piece on which the vernier scale is 
 divided, the zero of which is at the lower end of the piece. 
 In taking the height of the mercury, the sliding piece is brought 
 down and set nearly by the hand, and the contact of the zero of 
 the vernier with the top of the mercurial column is pen perfected by the screw c, 
 which moves the vernier the small quantity that may be required just to exclude the 
 light from passing between the lower edges of the sliding piece, and the spherical sur- 
 face of the mercury. 
 
 The barometer is attached to the stand by a ring, in which it turns round with 
 a smooth and steady motion for the purpose of placing it in the best light for reading 
 2U 
 
 Fig. 23. 
 
658 PHILOSOPHY. 
 
 off, &c. ; and the tripod stand, when closed, forms a safe and convenient packing.cage 
 for Pein canents 7 
 
 A substitute for the Peowere was invented a few a years ago by Mr. Adie, 
 optician, of Edinburgh; it is thus described in the patent. 
 
 The principle of the instrument, which is called a Sympiesometer, consists in meas 
 ing the weight of the air by the compression of acolumn of gas. It consists of a tube ofl 
 glass A Bc D (Fig. 24.) about 18 inches long, and 0°7 ofan inch diameter inside, termi-. 
 nated above by a bulb p, and having the lower extremity bent upward, and expanding 
 
 into an oval cistern a, open at the top. The bulb’ 
 Fig. 24. D being filled with hydrogen gas, and a part of the| 
 cistern a and the tube B with almond oil, coloured 
 with anchusa root, the enclosed gas, by changing its| 
 bulk according to the pressure of the atmosphere on: 
 the oil in the cistern, produces a corresponding eleva.’ 
 tion or depression of the oil in the tube, thereby 
 indicating the variations in the weight of the atmo.| 
 sphere. The scale for measuring these changes is 
 determined by placing the instrument, with an accurate. 
 barometer and thermometer, in an apparatus where, 
 the air can be rarefied or condensed, so as to make! 
 the barometer stand at 27, 30, or any other number 
 of inches. The different heights of the oil in the 
 tube of the sympiesometer, corresponding to those’ 
 points, being marked on its scale E Fr, and the spaces. 
 between them being divided into 100 equal parts, 
 these divisions represent hundredths of an inch in the, 
 mercurial column. 
 
 To correct the error that would arise from the! 
 change produced in the gas by variations of tempera-) 
 ture, the principal scale E F is made to slide on another | 
 GH, so graduated, as to represent the amount of that 
 change eirreepnd te to the degrees of a thermometer, | 
 I K, attached to the instrument. f 
 
 To use the instrument, note the height of the ther-| 
 mometer, and set the index a, which is upon the: 
 sliding scale, to the degree of temperature on the fixed 
 scale; then the height of the oil, as indicated by the 
 scale, is the pressure of the air, or the height of the | 
 mercury in the barometer at the time. The - 
 scale is moved by means of the knob L. 
 
 This instrument has been thought to afford more delicate indications than the ba-| 
 rometer, of changes in the atmospheric pressure; and the pressure determined by an: 
 instrument of this kind has been found to agree very well with .ose given by the: 
 barometer, not differing more than from the eighth to the thirtieth part of an inch. | 
 
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 PROBLEM XXXI. 
 
 What space would be occupied by a cubic inch of air, if carried to the height of the | 
 
 earth’s semi-diameter ? 
 
 We have already mentioned that air, in consequence of its elasticity, when charged. 
 with a double weight, is reduced to one half of its volume, and so on in proportion ; 
 at least as far as has hitherto been found by the experiments made on that subject. 
 For the same reason, when freed from the half of the weight which it supports, it | 
 ‘occupies a double space; and a quadruple space, when it has only a fourth part al 
 
EXPANSION OF THE AIR, bet 
 
 the weight to support. Thus, for example, on ascending a mountain, when it is 
 found that the mercury has fallen half the height at which it stood at the bottom of 
 the mountain, it is concluded that, being freed from half the weight which it sup- 
 ported when in the plain, it has been dilated to double the volume, or that the stratum 
 of the surrounding air has only half the density of that at the bottom of the mountain ; 
 for the density is in the inverse ratio of the space occupied by the same quantity of 
 matter, 
 
 This law of the dilatation of the air, in the inverse ratio of the weight with 
 which it is loaded, has enabled geometricians to demonstrate, that as one rises in 
 the atmosphere, the density decreases, or rarefaction increases, in a geometrical pro- 
 gression; while the heights to which one rises, increase in arithmetical progression, 
 Hence, if it be known to what height we must rise to have the air rarefied one 
 fourth, for example, or reduced to three fourths of the density which it has on the 
 sorders of the sea, we can tell that at a double height its density will be the square 
 of 3, or %; at a triple height it will be the cube of 3, or 27; in short, at a hundred 
 simes: the height, it will be the 100th power of #, &e. Or, if the ratio of the den- 
 sity of the air, at the height of 1760 yards, or 1 mile, to the density of the air on 
 she borders of the sea, has been determined, and if we call this ratio p, we shall 
 aave D? for the expression of that ratio at the height of 2 miles; at three miles it 
 will p?, &c: and at n miles, it will be p”. ; 
 | But, it is known by experiment that at the perpendicular height of a mile above 
 she level of the sea. the mercury, which on the borders of the sea was at the height 
 of 28 inches, or 336 lines, falls to 22 inches 4 lines, or 268 lines, or the height of the 
 mercury at that elevation is expressed by the fraction 268, unity being the whole 
 relight. Hence it follows, that the ratio of the density of the air at that height, to’ 
 he density of the air on the borders of the sea, is expressed by that fraction: conse- 
 juently to find what this ratio would be at the height of the earth’s semi-diameter, 
 ve must first know how many miles are contained in that semi-diameter. Let us 
 uppose that there are 3000. We must therefore raise the above fraction. 268, or Sf, 
 (0 the 3000th power, which may be easily done by means of logarithms; for taking 
 he logarithm of §, which is —0-0982045, and multiplying it by 3000, we shall have 
 or the logarithm of the required number — 294-6135000; which indicates that this 
 iumber is composed at least of 295 figures. We may therefore say, that the density 
 #€ the air which we breathe at the surface of the earth, is to that which we should 
 ind at the height of the earth’s semi-diameter, as a number, consisting of 295 figures, 
 Stounity. It is needless to make a calculation to prove that the sphere even of 
 jaturn does not contain as many cubic inches as are expressed by that number; 
 nd consequently that a cubic inch of air, carried to the height of the earth’s semi- 
 jameter above its surface, would be extended in sucha manner as to occupy aspace 
 reater than the sphere of Saturn. 
 | We shall here just observe, that this rarity would be still greater, for the following 
 eason. We havesupposed the gravity uniform, which is not the case; for as gra- 
 ‘ity decreases in the inverse ratio of the distance from the centre of the earth, it 
 hence follows, that in proportion as one rises above the surface, this gravity is di- 
 xinished ; so that at the distance of a semi-diameter from the earth, it is only a fourth 
 art of what it was at the surface; every stratum of air then will be less loaded by 
 be superior strata, since they will weigh less at the same height, than on the pre- 
 eding supposition: consequently the air will be more dilated. Newton has 
 hewn the method of making the calculation; but for the sake of brevity we shall 
 nit it. 
 
 Remark.—The extreme rarity of the air, at a distance so moderate, may serve as 
 
 proof of the great tenuity of the matter with which the celestial space is filled. 
 
 or if its density were every where the same as it is at the distance of the earth’s 
 
 Ps Tih 24 
 i 
 
635 PHILOSOPHY. 
 
 semi-diameter, it may be easily perceived how little the planetary bodies can lose of 
 their motion by traversing it. The moon, during the many thousand years she has 
 been revolving round the earth, cannot yet have displaced a quantity equal to a cubic 
 foot of our air. 
 
 PROBLEM XXXII. 
 
 Ifa pit were dug to the centre of the earth, what would be the density of the air at 
 the different depths, and at the bottom of it ? 
 
 We shall begin our answer to this question by observing, that one could not pro. 
 ceed to a very great depth, without coming to air so highly condensed, that a person 
 would fioat on it, in the same manner as cork does on mercury. 
 
 This is evident, if we suppose the gravity at the different depths of the pit to be 
 uniform; for at the distance of a semi-diameter below the surface, the density must 
 be to that of the air at the surface in the inverse ratio of the density of the latter to 
 that of the air at the distance of a semi-diameter above it. But we have seen by. 
 what a number the rarity of the latter is expressed; and the same number willex. 
 press the condensation at the centre. 
 
 Quicksilver is not quite 14000 times as heavy as the air which we breathe; and. 
 therefore the air at the centre would be thousands of millions of millions, &c. of times 
 denser than mercury. But, for the sake of amusement, since we are on the subject 
 of philosophical recreations, let us examine the most probable hypothesis of the gra- 
 vity which prevails in the case stated in this problem. The gravity would not be. 
 uniform ; it would decrease on approaching the centre, being exactly as the distance 
 from the centre. But Newton has shewn that as the squares of the distances, from 
 the centre, in this case decrease arithmetically, the densities would increase geo- 
 metrically. 
 
 We must then first find what would be the density of the air at a determinate 
 depth, such as 1000 toises, for example. But this is easy, on account of the proxi- 
 “mity of that depth to the surface; for if the density at the surface be expressed, 
 by unity, that at the depth of 1000 toises, or a mile below it, will be the inverse o! 
 1000 toises aboveit. But Wi: latter was expressed by $7, consequently the expres-| 
 sion for the former will be 8, or 1 + 4%; hence the density being 1, at the distance 
 of 3000 miles from the centre, the density at the distance of 2999 will be #. Let. 
 us then square 3000, which gives 9000000, and also 2999, which gives 8994001 ; the 
 difference between these squares is 5999, by which if 9000000 be divided, we ‘shal 
 have the quotient 1500, for the number of squares decreasing gies at the 
 same rate that are contained in that square. Ifthe logarithm of &, which is 
 0-0982045, be multiplied by 1500, the product will be 147:3067500, af the loga- 
 rithm of the density at the centre, that at the surface being 1. But the number 
 corresponding to this logarithm would contain 148 figures at least; whence it follows, 
 that the density of the air, at the centre of the earth, would bel to that at the sur- 
 face, asa number consisting of 148 figures, or at least unity foliowed by 147 ipa 
 to unity. 
 
 Were it required to determine at what depth the air would have the same density, 
 as water, it will be seen by a calculation founded on the same principles, that it 
 would be at the distance of 30 miles below the surface. 
 
 It will be found, in like manner, that at the depth of 42 miles below the surfaes| 
 the air would have the same density as quicksilver. . 
 
 ‘ 
 
 PROBLEM XXXIII. 
 Of the Air Gun. 
 
 This instrument, for the invention of which we are indebted to Otto Guerike. 
 -burgomaster of Magdebourg, so celebrated about the middle of the 17th century by: 
 
 ~ 
 
—e 
 
 AIR GUN. 661 
 
 his pneumatic experiments, is a machine in which the elasticity of air, violently 
 compressed, is employed to project a ball of lead, in the same manner as gunpowder. 
 It consists of an air chamber, formed by the vacuity between two cylindric and 
 concentric tubes, placed the one within the other; the bottom of this vacuity com- 
 municates with a pump, concealed in the butt end of the gun, and furnished with a 
 piston which serves to introduce and condense the air, by means of valves properly 
 adapted for the purpose. The ball is placed at the bottom of the inner tube, where 
 itis retained by a little wadding, and at the bottom there is an aperture, closed by 
 2 valve, which cannot open until a trigger is pulled. 
 
 It may now be easily conceived, that when the air in the reservoir or chamber is 
 compressed as much as possible, if the ball be placed at the bottom of the interior 
 tube, and if the trigger, adapted to open the valve which is behind the ball, be pulled, 
 the air violently compressed in the chamber will act upon it, and impel it with 
 a greater or less velocity, according to the time it may have had to exert its 
 action. 
 
 To make an air gun then produce the proper effect, it is necessary, lst, that the 
 opening of the valve should exactly occupy the same time that the ball does to pass 
 through the length of the tube; for during that time the air will accelerate its mo- 
 tion, the expansion of the air being much more rapid than the motion of the ball. 
 [f the chamber should remain longer open, it would bea mere loss. 2d. The ball 
 must be per fectly round, and exactly fitted to the calibre of’ the piece, in order that 
 the air may not escape at its sides. As leaden balls are not always very regular, 
 chis defect may be remedied by wrapping a little tow around it. 
 
 When these requisites have been attended to, an air gun 
 will discharge a ball with sufficient force to pierce a board two 
 inches in thickness, at the distance of 50, and even of 100 
 paces. When the air chamber is once filled, it may be eni\- 
 ployed eight or ten times in succession. An English artist 
 even invented a method of placing these balls in reserve in a 
 small crooked channel, from which on discharging one ball, 
 another issued to occupy its place; so that a person could dis- 
 charge the air gun ten times running, much sooner than the 
 most expert Prussian soldier could fire half the number of times, 
 It must however be observed, that the force of the air gun 
 decreases in proportion as the air chamber is emptied. 
 
 It may be easily conceived, that if this instrument, instead 
 of being preserved in the cabinets of philosophers, should fall 
 into the hands of certain persons, it would be a most formidable 
 weapon, and the more dangerous as it makes no noise when 
 discharged. But as gunpowder, after being a long time amere 
 ingredient in artificial fireworks, became the soul of a most de- 
 structive instrument, it is not improbable that the air gun, 
 when brought to perfection, may in like manner be employed 
 by armies to destroy each other, gloriously and without re- 
 morse. * 
 
 The air gun is represented Fig. 25, where the interval be- 
 tween the two cylinders, which serves to contain the air, may 
 be easily distinguished; MN is the piston, by which the 
 
 | 
 
 * Within these few yea's the above anticipation of Montucla has been verified ; guns, not indeed 
 air guns, but steam guns on the principle of the air gun, have been made for the purposes of war- 
 fare: though, thanks to the peaceful temper of the times, they have nut yet been brought into ac- 
 tion. 
 
 At the Gallery of Practical Science, in the Lowther Arcade, London, the firing (7) of a stream of 
 
 balls by the steam gun is daily exhibited. 
 
662 PHILOSOPHY. 
 
 air is introduced into that chamber; TL the valve, by which a communication is 
 
 formed between the chamber and the cylinder; and o is the trigger. This mecha- 
 nism may be so easily understood, that no further illustration is necessary. 
 
 PROBLEM XXXIy. 
 Of the Eolipyle. 
 
 The Eolipyle is a hollow vessel made of strong metal, and generally in the form | 
 of a pear, terminating in a long tail, somewhat bent. It is filled with water or some. 
 
 other liquor, by first exposing it to a strong heat, and then immersing it in the liquor | 
 to be introduced into it. While the interior air contracts itself to resume its former | 
 
 volume, the liquor, in consequence of the pressure of the external air, must necese 
 sarily enter to supply its place. 
 
 i: 
 
 If the eolipyle, when filled in this manner, be placed on burning coals, the water | 
 
 it contains is reduced into vapour, which escapes by the narrow orifice in the tail: 
 
 or if the fluid, by the position of the eolipyle, presents itself at the entrance, being 1 
 pressed upon by the vapour, it issues through the orifice with force, and forms a_ 
 
 pretty high jet. 
 If brandy has been employed instead of water, you may set fire to it with a 
 
 taper; and, instead of a jet of water, you will have the agreeable spectacle of a 
 
 jet of fire. 
 
 This experiment serves to shew, in a sensible manner, the strength of the vapour . 
 
 produced by a fluid exposed to a strong heat. For, in the first case, this vapour 
 
 issues with impetuosity through the orifice of the eolipyle; and in the second the | 
 elastic force of the vapour, pressing on the fluid, makes it issue through the same 
 
 orifice. 
 This experiment may be rendered still more amusing in the following manner. 
 
 Provide a sort of small chariot, bearing a spirit of wine lamp, and place the belly. 
 of the eolipyle on the latter; close the orifice of the eolipyle with a stopper which | 
 
 does not adhere too firmly, and then kindle the lamp. Some time after, the stopper 
 
 will fly out, and the fluid or vapour will issue through the orifice with great violence. 
 The chariot being repelled, at the same time, by the resistance which the fluid or 
 vapour experiences from the external air, receives a motion backwards; and if the 
 exle-tree of the wheels be fixed to a vertical axis, the chariot will assume a cireular 
 
 motion, which will continue as long as the eolipyle contains any portion of the . 
 
 fluid. 
 
 It may be easily conceived, that this vessel must be made of very strong metal, 
 
 otherwise it might burst, and either kill or wound the spectators. 
 
 PROBLEM XXXV. 
 
 To construct small figures, which remain suspended in water, and which may be made | 
 to dance, and to rise up or sink down, merely by pressing the finger against the orifice 
 
 of the bottle or jar which contains: them. 
 
 Fig. 26. First construct two small hollow figures of enamel; but | 
 
 through which a drop of water can be introduced, or apply 
 
 tail (Fig. 26.), pierced at the end, so that a greater or less 
 
 right, and remain suspended in the fluid. Fill ‘the bottle 
 with water to the orifice, and cover it with parchment, which 
 must be closely tied around the neck. 
 
 in the lower part, representing the feet, leave a small hole, 
 
 to the back part of each a sort of appendage in the form of a 
 
 wt 
 } 
 
 quantity of water may be made to enter into this tube. | 
 Then bring the figure into equilibrium in such a manner, | 
 that with this small drop of water it shall keep itself up- / 
 
 I 
 
FLOATING FIGURES—BATAVIAN TEARS, 663 
 
 _ When you are desirous of putting the small figures in motion, press the parch- 
 nent over the orifice with your finger, and the figures will descend; if you remove 
 vour finger they will rise; and if you apply and remove your finger alternately, the 
 igures will be agitated in the middle of the liquor, in such a manner, as to excite 
 he astonishment of those unacquainted with the cause. 
 
 ‘The explanation of this phenomenon is as follows. When you press the water 
 hbrough the parchment which covers the orifice of the bottle, the water, being in- 
 ‘ompressible, condenses the air in the small figure, by causing a little more water, 
 han what it already contains, to enter it. The figure having thus become heavier, 
 aust sink to the bottom ; but when the finger is removed, the compresed air resumes 
 ts former volume, and aivels the water introduced bythe compression: the small 
 igure, having by these means become lighter, must re-ascend. 
 
 PROBLEM XXXVI, 
 
 . 
 To construct a barometer, which shall indicate the variations of the atmosphere, by 
 | means of a small figure that rises or sinks in water. 
 
 The principles on which this small, curious barometer. is constructed, have been 
 xplained in the foregoing problem. For since the pressure of the finger on the 
 yater, which contains the small figure in question, makes it descend, and as it rises 
 gain when the pressure is removed, it may be easily conceived that the weight 
 f the atmosphere, according as it is greater or less, must produce the same effect. 
 Tence, if the small figuré be equipoised in such a manner as to remain suspended 
 luring variable weather, it will sink to the bottom when the weather is fine; be- 
 ause the weight of the atmosphere is then more considerable. The contrary will be 
 ne case when it threatens rain, and when the mercury in the barometer falls; for 
 he weight of the atmosphere, which rests on the orifice of the bottle, being lessened, 
 he small figure must of course rise. 
 
 PROBLEM XXXVII. 
 
 Yo suspend two figures in water, in such a manner that, on pouring in more water, the 
 one shall rise up and the other sink down. 
 
 For this purpose, provide salt water, and suspend in it a small figure, or small 
 lass bottle, of such a weight, that if the water contained a little less salt, it would 
 ul to the bottom. Dispose, in the same manner, another small figure or bottle, 
 ‘pen at the lower part; so that in the same water it shall keep at the bottom, by the 
 iechanism described in the 85th problem. 
 
 When every thing is thus arranged, if fresh water, pretty warm, be poured into 
 1e salt water, which contains the figures, the first one will sink to the bottom, in con- ¢ 
 2quence of a cause which may be easily conceived; and at the same time the other will 
 se to the surface: for the air in the second figure being dilated by the heat of the 
 pater, will expel, either in whole cr in part, the drop of water which formed a por- 
 on of its weight: the figure, having thus become lighter, must consequently rise. 
 “hese two small figures therefore will change places, merely by the effusion of more 
 ater; but the second, when the water cools, will re-descend. 
 
 PROBLEM XXXVIII. 
 Of Prince Rupert's Drops, or Batavian Tears. 
 
 | This appe-lation is given to a sort of glass drops, terminating in a long tail, which 
 yssess a very singular property; for if you give one of them a pretty smart blow on 
 1¢ belly, it opposes a considerable resistance; but if the smallest bit be broken 
 
 T from the tail, it immediately bursts into a thousand pieces, and is reduced almost 
 » dust. 
 
664 PHILOSOPHY. 
 
 These drops are made by letting glass, in a state of fusion, fall drop by drop into: 
 a vessel filled with water. They are then found at the bottom completely formed. 
 A great number of them however generally burst in the water, or immediately after: 
 they have been taken from it. As these drops were first made in Holland, they are: 
 by called the French Larmes Bataviques. | 
 
 Various experiments have been made with these glass drops, to discover the cause 
 of their bursting. 'These experiments are as follow: | 
 
 Ist. If the tail of one of these drops be broken under the receiver of an air-pump, 
 by a process which may be easily conceived, it.bursts in the same manner as it would 
 do in the open air ; and if the experiment be performed in the dark, a flash of light 
 is observed at the moment of rupture. 
 
 2d. If the body of one of these drops be ground down gently on a cutler’s wheel, 
 or whet-stone, it sometimes bursts; but for the most part it does not. 
 
 3d. If a notch be made in the tail, by means of the same stone, the drop will 
 burst. 
 
 4th. The tail of one of these drops may be hie ver cut off in the following man-' 
 ner: Present the place, at which you are desirous it should be cut, to an enameller’s! 
 lamp; by these means it will be fused, and you may then separate the one part from| 
 the other, without fear of its bursting. 
 
 5th. If one of these drops be carefully heated on burning coals, and if it be ther 
 suffered to cool slowly, it will not burst, even when the tail of it is broken. 
 
 Philosophers have always been much embarrassed respecting the cause of this 
 extraordinary phenomenon; and it must indeed be confessed that it is still very 
 obscure. We can only say, that it is not produced by air, as is proved by the first 
 experiment. We think ourselves authorised to say also, from the fifth experiment. 
 that it depends on the same cause which makes all articles of glass break, if care hay 
 not been taken to anneal them, that is to say if they are not subjected to a lon 
 heat that they may cool gradually, before they are exposed to the contact of thi 
 air. This appears to result from the last experiment; but it does not seem clear it 
 what manner it is effected. It arises, in all probability, from the eruption of som 
 fluid in the inside of the drop, which rushes through the broken part of the tail, 
 It is perhaps an electrical phenomenon, and the drop may burst by the same mechanism 
 that often cracks a glass jar, when it is discharged; that is, when the equilibrium 
 is restored between its interior and exterior surface. Having explained the principa 
 phenomena of these drops, we shall leave the rest to the sagacity and researches o 
 our readers. 4 
 
 PROBLEM XXXIX. 
 To measure the quantity of rain which falls in the course of a year. 
 
 One of the meteorological objects which engage the attention of the modern 
 philosophers, is, to observe the quantity of rain that falls on the earth in the cours 
 of a year. This observation may be easily made by means of an instrument whicl 
 M. Cotte, in his Treatise on Meteorology, calls the Udometer,* but which, in ou 
 opinion, ought rather to be called the Uometer.t 
 
 This instrument consists of a box of tin plate, or lead or tin, two feet square 
 which makes four feet of surface. Its sides are six inches in depth at least, and th 
 bottom is a little inclined towards one of the angles, where there is a small pip: 
 furnished with a cock, The water which flows through this pipe, falls into anothe 
 square vessel, the dimensions of which are much less, and so proportioned, that th: 
 height of a line in the large vessel corresponds to three inches in the smaller. In thi. 
 
 t 
 
 * From v$wp water, and eTpoy a measure. 
 i 
 + From yo; rain, and feTpoy @ measure. 
 
hd é 
 
 | 
 { 
 ; 
 
 QUANTITY OF RAIN. 065 
 
 ‘resent case, therefore, the base of this vessel ought to be only two inches six lines 
 quare. From this description it may be easily conceived that very small portions 
 f a line of water, which has fallen into the large vessel, may be measured ; since @ 
 ine of height in the small one, will correspond to the thirty-sixth part ofa line in 
 he large one. 
 
 If the large vessel be properly placed, with the small one below the cock; and 
 > the small one be covered in such a manner as to prevent the air from having 
 ecess to the surface of the water it contains; it will not be necessary to examine the 
 uantity of water which has fallen after each shower, or series of rain. It may be 
 ‘xamined and measured every three, or four, or five days. It will, however, be better 
 o do it after each fall of rain. 
 
 If a register be then kept of the quantity of water which falls every time that 
 t rains, these quantities, added together, will give the quantity that falls in the course 
 \f the whole year. 
 
 _Ithas been found in this manner, by a series of observations made at Paris, for 77 
 vears, that the quantity of rain which falls there, one year with another, is 16 inches 
 } lines. 
 
 . But this quantity of water is not every where the same. In other places it is greater 
 r less, according as they are situated near to the sea or to mountains. The follow- 
 ng is a table of the principal places where observations of this kind have been 
 nade, and of the quantity of water which falls there annually. 
 
 Inch. Line. Inch. Line. 
 BE SELS Pasal tice <9) -A5's 16 8 Lonion: ..... «3. 18 9 
 Bayeux ....-... 20 0 The Hague .... 26 6 
 Beziers ........ 16 °° 3 Rome ......+. 28 O 
 i Aix in Provence 18 38 PRadda ses sores OU ra OU 
 Toulouse ...... 17 2 Petersburgh ...» 16 1 
 Lyons ........ 25°. 0 Bett ocssc cs) (iy wo 
 lS Relea 23 0 
 
 _ From an extensive collection of observations, Dr. Dalton concludes that, for 30 
 places in England and Wales, the average annual quantity of rain amounts to 35:2 
 inches ; the greatest being 67-5 inches at Keswick, among the Cumberland hills ; and 
 the least at Uxminster, in the comparatively flat county of Essex. It is probable, 
 however, that the annual quantity for the whole of England and Wales is below 30 
 inches. 
 
 It is certain that the quantity of rain which falls at the top of a hillis much 
 greater than what falls at the bottom. Near Kinfauns, in Scotland, a rain gauge, 
 on the summit of a hill 600 feet high, showed the mean annual fall of rain for five 
 successive years to be about 41°5 inches; while a gauge at the bottom of the hill 
 showed that for the same period the mean annual fall was not more than 25-7 inches. 
 Remark.—We think it necessary here to offer a remark which seems to have 
 ‘escaped all the philosophers who have made observations on the quantity of rain 
 that falls. Every time it again rains, a small quantity of water is lost; namely, that 
 which has served to moisten the bottom of the reservoir; for the water does 
 not begin to run down till the bottom is moistened to a certain degree, and covered 
 as we may say to a certain thickness with water, the quantity of which must be 
 determined and taken into the account after every fall of rain. This quantity 
 of water may be measured by the following process. Take a small sponge, 
 moistened in such a manner that no water can be squeezed from it, even when 
 pressed very hard; then fill the vessel, and having suffered the water to run from 
 it, collect with the sponge what remains on the bottom, and squeeze it into a vessel, 
 
666 PHILOSOPHY. 
 
 the base of which is an inch square, and already moistened with water. It is evident 
 that if a vessel, the base of which contains 4 square feet, gives in this manner 
 water sufficient to rise to the-height of an inch in the small vessel, there is reason to. 
 conclude that the pellicle of water which adhered to the metal was at least =}, of 
 an inch, or the 48th part of a line in thickness. At any rate, it may be safely esti 
 mated at the 30th or 36th of a line. If it has rained, therefore, two or three hundred | 
 times in the course of the year, 8 lines must be added to the quantity found. 
 
 PROBLEM XL. 
 
 Of the origin of Fountains. Calculation of the quantity of rain water sufficient to — 
 
 produce and to maintain them. ; 
 
 It would appear that the origin of fountains and springs ought not to have occa. 
 sioned such a diversity of opinions, as has, for some time, prevailed among philoso. - 
 phers. An attentive consideration of these phenomena is sufficient to shew that the | 
 origin of them is entirely owing to the rains which continually moisten the surface | 
 of the globe, and which running over beds of earth, capable of preventing them | 
 from penetrating deeper, at length force a passage to places which are lower. Every 
 person indeed must have observed, that the greater part of springs decrease in a 
 considerable degree, when a long drought has prevailed; that some of them abso- | 
 lutely dry up when this drought continues too long; that when the surface of the 
 earth has been moistened with snow or rain, they are renewed; and that they in- 
 crease almost in the same progression as the waters become more abundant. 
 
 Some philosophers however have ascribed the origin of fountains to a sublimation | 
 of the waters of the sea, which, flowing into the bowels of the earth, rise up in | 
 vapour, in the fissures of the rocks, and thence trickle down. into cavities and reser- 
 voirs prepared by nature, from which they make their way to the surface. Some | 
 have even gone so far as to imagine a sort of subterranean alembics, 
 
 But these conjectures are entirely void of foundation. If the water of the sea | 
 produced fountains in this manner, it would long ago have choaked up, with its salf, | 
 the subterranean conduits through which. it is supposed to pass. Besides, the con- 
 nection which is observed between the abundance of rain, and that of the water 
 of the greater part of fountains, would not subsist; as the internal distillation would 
 take place whether it rained or not. It is to be observed also, that the water 
 of springs always distils from above beds of clay, and not from below them; but as 
 these beds intercept the passage of vapours and water, the latter must necessarily | 
 come from above them. A sure method of destroying a spring, is to pierce this | 
 bed; but if the water came from below, a contrary effect would be produced. i 
 
 What induced philosophers to have recourse to a cause so remote, and so false, no — 
 doubt was their imagining that rain water was not sufficient to feed all the springs | 
 and rivers. But they were certainly in an error; for instead of rain water being too — 
 small in quantity to answer that purpose, it seems rather difficult to conceive in — 
 what manner it is:expended. This will be proved by the following calculation 
 of Mariotte. : 
 
 This author observes that, according to experiments which have been made, there 
 falls annually on the surface of the earth about 19 inches of water. But to render 
 his calculation still more convincing, he supposes only 15, which makes per square 
 toise 45 cubie feet, and per square league of 2300 toises in each direction, 238050000 — 
 cubic feet. 
 
 But the rivers and springs which feed the Seine, before it arrives at the Pout- — 
 Royal at Paris, comprehend an extent of territory, about 60 leagues in length and | 
 50 in breadth, which makes 3000 leagues of superficial content; by which if | 
 238050000 be multiplied, we shall have for product 714150000000, for the cubie | 
 feet of water, which falls, at the lowest estimation, on the above extent of territory, 
 
 j 
 ' 
 q 
 
 te eee 
 
land 
 
 ORIGIN OF SPRINGS—WATER MALLET, 667 
 
 ‘Let us now examine the quantity of water annually furnished by the Seine. This 
 ver, above the Pont-Royal, when at its mean height, is 400 feet in breadth, and 5 
 depth. The velocity of the water, when the river is in this state, may be esti- 
 ated at 100 feet per minute, taking a mean between the velocity at the surface and 
 jat at the bottom. If the product of 400 feet in breadth, by 5 in depth, or 2000 
 quare feet, be multiplied by 100 feet, we shall have 200000 cubic feet, for the 
 aantity of water which passes in a minute through that section of the Seine, above 
 ie Pont-Royal.. The quantity then in an hour will be 12000000; in 24 hours, 
 38000000 ; and ina year, 105120000000 cubic feet. But this is not the seventh 
 art of the water which, as above seen, falls on the extent of country that supplies 
 he Seine. 
 
 ; But how shall we dispose of the remainder of this water? The answer is easy : 
 ne rivers, rivulets, and ponds lose a considerable quantity of water by evaporation ; 
 ad a prodigious quantity is employed for the nutrition of plants. 
 
 \Mariotte makes a calculation also of the water which ought to be furnished 
 aturally by a spring that issues a little below the summit of Montmartre, and 
 which is fed by an extent of ground 300 toises in length and 100 in breadth ; 
 aking a surface of 30000 square tcises. At the rate of 18 inches for the annual 
 uantity of rain, the quantity which falls on that extent will amount to 1620000 
 abie feet. But a considerable part of this water, perhaps three-fourths, immedi- 
 tely runs off: so that no more than 405000 forces its way through the earth and 
 indy soil, till it meets with a bed of clay at the depth of two or three feet, from 
 thich it flows to the mouth of the fountain, and feeds it. If 405000 therefore be 
 ivided by 365, the quotient will be 1100 cubic feet of water, which it ought to 
 irnish daily, or about 38500 French pints; which makes about 1600 pints per hour, 
 r 27 pints per minute. Such is nearly the produce of this spring. 
 
 An objection, founded on an experiment of M. de la Hire, described in the me- 
 10irs of the Academy of Sciences, for the year 1703, is commonly made to this idea 
 especting the origin of springs. This philosopher having caused a pit to be dug in 
 
 field, to the depth of 2 feet, found no traces of moisture: from which some con- 
 lude that the rain water flows only over the surface, and does not in any manner 
 ontribute to the origin of springs. 
 
 But this experiment is of no weight, as it is contradicted by a thousand contrary 
 istances. Every one knows that water, in various places, oozes from the roofs of 
 averns and subterranean passages: it is this water which, after penetrating the 
 arth, and flowing between the joints of stones, produces stalactites, and other 
 tony concretions. It is therefore false that rain water never penetrates beyond the 
 lepth of afew feet. The fact, observed by M. de la Hire, was a particular case, 
 ‘om which it was wrong to deduce a general consequence. | 
 | It is objected also, that water is sometimes collected at heights at which it is 
 npossible that rain water could give birth to a spring. , To this it may be replied, 
 hat if the ground, where these collections of water exist, be examined, it will al- 
 vays be found that they are produced by rain or melted snow ; that these places on 
 he summits of mountains are onlya kind of funnels, which collect the waters of 
 ome neighbouring plain, continually maintained by the rain or the snow, assisted 
 y the small evaporation which takes place, in consequence of the rarity of the 
 ir. It is therefore evident to every rational mind, that the origin of springs and 
 ountains can be ascribed tono other cause, than the rain water and snow which 
 jave been collected. 
 
 PROBLEM XLI. 
 The Water or Mercurial Mallet. 
 
 _ The water mallet, as it is called, is nothing else than a long glass -flask, contain- 
 | ; 
 
668 PHILOSOPRY. 
 
 ing water, which when shaken in the flask, strikes it with a noise almost like that 
 occasioned by a small blow with a mallet. q 
 The cause of this phenomenon is the absence of the air, for as that fluid no 
 longer divides the water in its fall, it proceeds to the bottom of the flask likea 
 solid body, and produces the sound above mentioned. | 
 To construct the water mallet, provide a long glass flask, pretty strong, and ter. 
 minating in a neck that can be hermetically sealed; fill one fourth or one fifth of 
 it with water; exhaust the air from it by means of an air pump, and then close the, 
 mouth of the flask hermetically: When the flask is taken out, if you fuse the neck 
 of it gently at an enameller’s lamp, in order to shut it more securely, the instru. 
 ment will be completed. } 
 If mercury be enclosed in the flask, instead of water, it will make a much greater, 
 noise or smarter blow; and you will even be astonished that it does not break the 
 flask, If the mercury be well purified, it will be luminous; so that when made to 
 run from the one end to the other, a beautiful stream of light will be seen in the dark. 
 Remark.—In our opinion, this property of mercury may be employed to construct. 
 what might be called a philosophical lantern. For this purpose, it would be neces. 
 sary to dispose in a sort of drum a great number of small flasks, like the preceding, 
 or spiral tubes, in which purified mercury should be kept in continual movement: 
 which might be easily done if the drum were made to revolve by means of machi- 
 nery; the result would be a continued light, which would have no need of aliment, | 
 or of being fed. Who knows, whether this idea may not enable us, at some future 
 period, to dispense with the candles and lamps which we now employ to light 
 our apartments? We are however afraid, that whatever be the number of flasks. 
 arranged in this manner, they will still afford too weak a light to supply the want 
 ofa single taper. But, perhaps, there are other useful purposes to which this in 
 vention may be applied. | 
 
 PROBLEM XLII. 
 To make a Luminous Shower with mercury. 
 
 Place on the top of the air pump a small circular plate, pierced with holes, and | 
 supporting a small cylindric receiver, terminating in a hemisphere, and cover the. 
 whole with a larger receiver, having a hole in its summit, capable of admitting a 
 glass funnel filled with mercury. This funnel must be sc arranged, that it can be 
 shut with a stopper, so as to be opened when necessary. Then exhaust the air, or | 
 nearly so, from the receiver, and open the funnel which contains the mercury ; the | 
 mercury,in consequence of its weight, and of the pressure of the atmosphere, will _ 
 run down, and, falling on the convex summit of the interior receiver, will be 
 thrown up in small luminous drops, so as to resemble a shower of fire. : 
 
 This experiment may be performed also in the following manner ; provide a piece 
 of pretty compaet wood, and cut in it a small reservoir in the shape of a hemisphere, — 
 or of an inverted cone; apply it to the upper aperture of a receiver, and fill it with | 
 mercury. If you then exhaust the receiver, the pressure of the external air will | 
 force the mercury through the pores of the wood, so that it will fall down in small 
 luminous drops. 
 
 _ PROBLEM XLII. 
 
 W hat is the reason that in mines, which have sptracies, or air-holes, on the declivity of | 
 a mountain, at various heights, a current of air is established, which in winter has | 
 
 a direction different from what it has in summer ? Explanation of a similar pheno- | 
 menon, observed daily in chimneys.—Use to which a chimney may be applied in 
 summer. a 
 
 It is customary, in order to introduce air into a mine, at certain distances to sink 
 
CIRCULATION OF AIR. 669 
 
 erpendicular wells, which terminate at the horizontal or somewhat inclined gallery, 
 *here the ore is dag up; and, in general, the mouths of these wells are at different 
 eights, in consequence of the inclination of the side of the mountain, But in this 
 ise, a very singular phenomenon is observed: during the winter the air rushes 
 ito the mine through the mouth of the lowest well, and issues by that of the 
 ighest; the contrary is the case in summer. 
 
 To explain this phenomenon, it must be observed, that in the mine the temperature 
 *the air is constantly the same, or nearly so, while without it is alternately colder 
 id hotter ; that is, colder in winter, and warmer in summer. Itis to be remarked, 
 athe other hand, that the well which has the mouth highest, the gallery, and the 
 ‘her well, form all together a bent siphon with unequal branches; the effect pro- 
 aced is as follows: 
 
 ‘When the exterior air is colder than that in the mine, the column of air which 
 resses on the lower orifice p (Fig. 27.), exerts a greater pressure on the whole air 
 Fig 27 contained in the siphon Dc Ba, than that which 
 
 : presses on the upper orifice a; this air then must be 
 expelled by circulating in the direction Dc Ba. But 
 the cold air which enters by pb, being immediately 
 heated to the same degree as that in the mine, is 
 impelled, like the former, by the column which rests 
 
 on the orifice p. 
 
 The contrary takes place in summer; for the ex- 
 terior air, during that period, is warmer than the 
 air in themine. The latter then being heavier, that 
 contained in the branch A B of the siphon, prepon- 
 
 derates over the air in Bc; so that the difference 
 stween the columns which press upon a and D, is not able to produce a counterpoise. 
 he air contained in the siphon A BCD then must receive an impulse in that direction ; 
 
 id consequently must move in a direction contrary to the former. Such is the ex- 
 anation of this phenomenon. 
 
 _A similar one is observed daily in chimneys; and it is the more sensible as the flues 
 
 ‘the chimneys are higher ; for a chimney, with the chamber where it terminates, 
 id the door or a window, forma siphon similar to the preceding. Besides, the ex- 
 wior air from nine in the morning till eight or nine at night in summer, is warmer 
 ian the interior, and vice versd. In the morning then, the air must descend the 
 iimney, and issue through the door or the window; onthe other hand, as the exte- 
 or air is colder in the night than in the day, it must, during the former, enter at the 
 dor or window, and ascend the chimney. About eight or nine in the morning, and 
 eight or nine in the evening, the air is, as it were, stationary : an effect which must 
 
 2eessarily take place at the time of the passage from one direction to another. 
 Dr. Franklin, who seems to have first observed this phenomenon, says that it 
 ight be applied to some economical uses during summer ; and in that case the pro- 
 erb, “as useless as a chimney in summer,” would not be correct. One of these 
 ses is, that the chimney might be applied as a safe; for if each of its mouths were 
 osed by a piece of canyas, stretched on a frame, the alternate and almost continual 
 irent of air which would we established in it, could not fail to preserve meat from 
 drruption. 
 
 This current might perhaps be employed also for some work that requires not 
 > much a force as a continuance of it. For this purpose, it would be necessary to 
 x in the flue of the chimney a vertical axis with a helix, like the fly of a smoke- 
 ck ; the current of air would keep it in continual motion, sometimes in one direc- 
 on, sometimes in the other; and in all probability with sufficient force to raise a 
 nall quantity of water per hour. And, as it would remain inactive only three or 
 
 I, 
 
670 PHILOSOPHY. 
 
 four hours a day, it could not fail to produce a considerable effect daily. Besides, # 
 moving power would cost nothing. It would however be necessary to have the 
 wheels adjusted in such a manner, that to whatever side the axis furnished with the 
 helix turned, the machine should always move in the same direction ; which is not 
 only possible, but was executed by M. Loriot at Paris, 
 
 Remark.—The same effect is easily experienced on a small scale, ina close room or 
 chamber, which is very warm with several persons and candles in it, especially if there 
 is no fire or no fire-place. For, by unlatching the door, and setting it a very little open, 
 as an inch or half an inch, it will then be found that the air rushes strongly in near 
 the bottom, but sets as strongly out near the top, and is quiescent near the middle 
 parts. This is very easily tried by holding a candle in your hand, first near the bot. 
 tom of the small opening, where the flame is violently blown inwards; then at the 
 top, where it is carried strongly outwards; but held near the middle, the flame 
 of the candle is quite still. 
 
 PROBLEM XLIV. _ | 
 To measure the height of mountains by the barometer. | 
 
 It is very difficult, and even sometimes impracticable, to measure the height of! 
 mountains by geometrical operations. A traveller, for example, who traverses a’ 
 chain of mountains, and who is desirous of ascertaining the altitude of the principal 
 points he has ascended, cannot have recourse to that method. The barometer, how. 
 ever, supplies a convenient and pretty exact one, provided it be employed with the’ 
 ‘ necessary attention. | 
 The principle on which this method is founded will be readily conceived, when it’ 
 is recollected that ifa barometer be carried to the top of a mountain, the quicksilver’ 
 stands at a less height than at the bottom. Ist. Because it has a less column of air. 
 above it. 2d. Because this air has less density, as it is freed from the weight of a’ 
 part of the air which it supported at the bottom of the mountain. Such is the foun. | 
 dation of the rules which have been invented for applying the height at which the’ 
 mercury in the barometer stands to the purpose of measuring the height of mountains, 
 But to.givea very exact rule in regard to this operation, ig attended with no small dif 
 ficulty ; for the height of the mercury in the barometer depends on a complication of so 
 many physical causes, that it is exceedingly troublesome to subject them to calculation. | 
 M. de Luc of Geneva, who has considered this subject with the greatest care, by’ 
 combining all these causes and circumstances, seems to have discovered a method | 
 which, if not absolutely perfect, is certainly more correct than any before given. — 
 To proceed with exactness in this operation, it is necessary to have a good portable’ 
 barometer, well freed from air; and a good thermometer, which we shall suppose to | 
 be that of Reaumur, though M. de Luc, to facilitate the calculation, proposes a 
 particular kind of division. If great correctness be required, it will be necessary | 
 also that an observer should examine the progress of the barometer at the bottom — 
 of the mountain, or in one of the nearest towns, the height of which above the level | 
 of the sea is known. | 
 When you have reached the summit of the mountain, or the place the altitude | 
 of which you are desirous of ascertaining, hold the barometer in a direction perfectly 
 vertical, and examine the height of the mercury ; suspend also the thermometer in | 
 some insulated place in the neighbourhood, and observe the degree to which the 
 
 } 
 
 mercury rises. ‘ 
 
 Having then compared the height of the barometer observed on the mountam, | 
 
 with that of the corresponding barometer, observed at the same time at the bottom, 
 
 take the logarithms of these two heights, expressed in lines, and cut off from them > 
 the four last figures: the remainder will be the difference of the heights expressed — 
 
 in French toises, the logs. being to seven places of decimals. 
 
 | 
 | 
 
 F4 
 
_ : 
 
 | . 
 = 
 
 ie = 
 
 MEASUREMENT OF MOUNTAINS. 671 
 
 But this altitude requires a correction ; for it is only exact when the simultaneous 
 mperature of the two places is 163, according to the scale of Reaumur’s thermo- 
 eter. For each degree then that the thermometer has remained below 163, at the 
 yper station, one toise must be added for every 215, and the same ce be de- 
 icted for every degree above that temperature. 
 
 The same correction,* but in the contrary sense, must be made by means of the 
 ermometer left at the fixed station; that is to say, for each degree it remained 
 vove 163, one toise in 215 must be deducted, and vice versd. The height, when 
 vice corrected in this manner, will be the difference nearly between the height 
 the two places above the surface of the sea, or the height of the one above the 
 her. 
 
 Let us suppose, for exampie, that at the lower station the barometer stood at the 
 light of 27 inches 2 lines, or 326 lines; and that at the upper station it fell to 23 
 ches 5 lines, or 281 lines. 
 
 The logarithm of 326 is 2°5132176, and that of 281 is 2'4487063; their difference 
 0:0645113 ; from which if the three last figures be cut off, to answer for division 
 ' 1000, the remainder will be 645 toises. ¥ 
 We shall suppose also, that at the top of the mountain Reaumur’s thermometer 
 jood at 6 degrees above freezing, and in the lower station at 12; that is, for the 
 mer 103 below 163, consequently 103 toises are to be added to the above number 
 r every 215; and hence, by the rule of three, the number to be added will be 
 und to be 32 toises. 
 
 It will be found, by the converse correction, that the height to be deducted is 20; 
 
 sequently there will remain 12 toises to be added, and therefore the height twice 
 ywrrected will be 657. 
 
 Mr. Needham, on Mount Tourné, one of the Alps, observed the height of the 
 vometer to be 18 inches 9 lines, or 225 lines. Now if we suppose that it was 
 yserved at the same moment at the level of the sea to stand exactly at 28 inches or 
 46 lines, which is its mean height on the borders of the sea, the difference between 
 je logarithm of 336 and that of 225, cutting off the last three figures, will be found 
 | be 1742, It may therefore be concluded, that the height of Mount Tourné is 
 ‘42 toises. But as in this case we have no corresponding observation at the level 
 \ the sea, nor any observation of the thermometer made at the same time, we have 
 ployed this observation of Needham only as an example of calculation. It is 
 ywever probable that the height of Mount Tourné is between 1700 and 1800 
 ‘ises. 
 
 |The late Professor Leslie of Edinburgh has given a very simple rule for computing 
 ‘proximately the heights of mountains from barometrical observations. It is this: 
 S the sum of the heights of the mercurial columns, is to their difference, so is 52000 to 
 @ approximate height in English feet. 
 
 To find the height of the upper column at the temperature of the lower, multiply 
 ‘e height of the upper column by twice the difference of degrees in the attached 
 jntigrade thermometer ; and the product, divided by 1000, will be the correction to 
 added to the upper column, for change of temperature; and this height must be 
 ‘ed in the preceding rule. 
 
 To correct the approximate height for the expansion of the air: multiply the height 
 ‘twice the sum of the degrees on the detached thermometer, and dividing the 
 joduct by 1000, the quotient is the correction to be added to the approximate 
 light. 
 
 | 
 
 * This second correction, though not mentioned by M.de Luc, appears to us necessary, for several 
 isons, which it would be tov tedious to explain here. 
 
 —— 
 
672 PHILOSOPHY. 
 
 A concise rule, with tables for computing the heights of mountains, is givenin “4 
 Collection of Tables and Formule, by Francis Baily, Esq.” 
 
 Remarks.—As a portable barometer is an instrument difficult to be procured an¢ 
 preserved, it is almost necessary that a traveller, when he is desirous of making 
 observations, should construct a barometer for himself; but in this case, as thy 
 mercury will not be freed from air, it will always stand a few lines lower than ;¢ 
 barometer which has been constructed with every possible care. This difference maj 
 amount to two or three lines. 
 
 In regard to Reaumur’s thermometer, it is easily carried; but in what manne) 
 must a traveller proceed to have corresponding observations, either on the border 
 of the sea, or in any other determinate place, which are necessary before he car 
 employ his own with sufficient exactness? ‘This difficulty, in our opinion, seem 
 to limit, in a considerable degree, this method of determining the heights of moun. 
 tains, 
 
 Besides, it appears that, even if a traveller had on the borders of the sea, or ir| 
 any village situated for example in the centre of France, the height of which above 
 the sea is known, a diligent observer, he would not be much farther advanced, for 
 the temperature of the atmosphere may be different on the borders of the sea at Genoa’ 
 that is to say, it may rain, for example, while the weather is fine and serene on the 
 Alps and the Appenines ; or the contrary may be the case; and hence there is a new 
 difficulty to be surmounted. 
 
 This difficulty however might be obviated in part, by knowing the greatest, the 
 mean, and the least elevation, of the barometer on the borders of the nearest sea, 
 and thence determining, by meteorological conjectures, the nature of the tempera. 
 ture on the mountain to be measured, though one only passed over it; thus, if an 
 hygrometer on the mountain indicated, for example, great moisture, there would be 
 reason to conclude that the weather was inclined to rain, and that the fixed barome- 
 ter stood atits least height. On the other hand, if the air was very dry, it might 
 with probability be concluded that the weather was serene, and that the fixed baro- 
 meter stood at its greatest height: but it must be confessed that this is not sufficient, 
 to give a satisfactory exactness. 
 
 However, a great many barometrical observations have been made on the summits! 
 of mountains, and their heights have been thence deduced. Several of them have 
 also been measured geometrically ; and we here adda tabular view of the: heights 
 above the level of the sea, of the most remarkable mountains on the surface of the 
 earth, compiled from the most recent authorities. 
 
 Height. | Height. © 
 EUROPE. Feet. ‘Feet. ( 
 Mont Blanc, Alps...... eee» 15688 Monte Velino, Appenines.... 8397 
 
 | 
 Monte Rosa, Alps....--e+«.. 15084 | Simplon, Passof..... cceeee, «GOREN 
 Gros Klockner, Tyrol .... 12796 The Dole, Mount Jura...... 5523 | 
 | 
 { 
 
 | 
 Mont Perdu, Pyrenees .... 11283 Hecla, Iceland*.........s08 4887 | 
 Pic d’Ossana, ditto........ 11700 Snuhetta, Norway ........ 6122 | 
 Etna, Sicily...... Beas see 1OSTO Mont Mezin, France........ 6567 | 
 Monte Carno, Appenines.... 9523 Puy de Sanca, France...... 6200 
 Glac. de Buct, Alps........ 10] 24 Vesuvius, Italy ...vees scene 3932 . 
 Mulhbaren, Spain .......... 117893 i 
 Pic du Midi, Pyrenees.... . 9300 ASIA. 
 Olympus, Greece 2.2.00... 9754 Dhawalagiri, Himalaya...... 28007 | 
 Canigou, Pyrenees.......... 9247 | Jewahir, ditto......----+. Q5747 | 
 
 | 
 
 Mount Cenis, Alps ...,..++ 9212 Mowna Rowa, Sandwich I... 15988 
 
 “, 
 
 | 
 m.. | 
 
MEASUREMENT OF MOUNTAINS. 673° 
 Aas Feet. 
 )phir, Sumatra............ 13849 | El Pinal, City of ....ceec0. 8362 
 jgmont, New Zealand---++- 11430 Las Vigias, ditto ..... wees mErCOeD 
 ;rarat, Armenia...... essa 17200 Perotes diito..- os. Cra eer tine 
 vebanon, Palestine ........ 9600 Mexico, Giitoves «see ctelcc es 7468 
 \watska, Kamschatka...... 12000 La Puebla, ditto .......... 7200 
 St. Juan del Rio, ditto...... 6484 
 AFRICA. MountWashimgton, Appalachian 6650 
 *eak of Teyde, Teneriff.... 12180 
 itlas, highest peak of, more GREAT BRITAIN: 
 than 12000 Benmacdouie, Scotland...... 4390 
 Yable Mountain, Cape of Ben Nevis; ditto. 9. ose. <+5 4370 
 MG00d Hope..ses.seee0ee- 3520 Cairnporm, ditid...+..<.e. 4080 
 Whernside, Yorkshire...... 4050 
 hgh at Ingleborough, ditto .-... -» 3987 
 Tevado dr Sonato..cssss oe ob 2O200. Ben Lawers, Scotland...... 3858 
 llimani, Gold Mount. Peru 24450 Ben More, ditto........... > 3723 
 thimborago ++++++........ 21600 Bens Gloe; ditto’. 022. a. ee ete SHOU 
 BUDD: <0 5 es eeeeeescene - 19360 | Snowdon,Wales............ 3555 
 ntisuna, Rocky Mountains. 19136 Shehalion, Scotland ........ 3461 
 MEAN, GittO ...ea.eseess 18867 Helvellyn, England........ 3324. 
 ‘opocatepetl, ditto PN oe | ee 17903 Skiddaw, ditto..eseee ..-. 3270 
 Tount St. Elie........0... 17883 Ben Ledi, Scotland,....... 3009 
 rizaba Sresuigt elevate’ sucxaie. 5 ais ole ea FODO Ben Lomond, ditto........ 3240 
 ‘inchincha, Rocky Mountains 15931 Maegillicuddy’s Reeks, Ireland 3404 
 line of Chata, Peru........ 118380 Mourne Mountains, ditto.... 2500 
 lean height of Andes..... » 11820 Rippin Tor, Devonshire.... 1549 
 
 huito, BUY Ole. tis ea ses 0s 2) OLS 
 
 reneral Observation.—We shall here remark, that the very considerable differ- 
 tes often found between the barometric and geometrical measurement, must not 
 ‘entirely imputed to the method. The latter is certain; but the observers of 
 barometrical heights have often employed imperfect instruments; in general, 
 y have had no corresponding observations ; and they have scarcely ever taken into 
 sunt the difference of temperature at the posts of comparison ; these differences 
 
 (d therefore excite no astonishment. 
 
 temark.— We must observe that the French, in general, consider 28 Paris inches 
 she mean height of the barometer at the level of the sea; andas the following 
 arks on this subject by Mr. Kirwan, may be of use to the reader, we here sub- 
 ) them :—“ Sir George Schuckburg has shewn, from 132 observations, made in 
 ‘y and in England, that the mean height ofthe barometer at the level of the 
 the temperature of the mercury being 55°, and of the air 62°, is 30:04 inches ;* 
 may then assume the height of 30 inches as the natural mean height of the baro- 
 er at the level of the sea, in most temperatures between 322 and 82°; for if 
 mercury were cooled down to 32°, that is 23° below 55°, it would be lowered 
 hat condensation only 0:07 of an inch; and if it were heated up to 80°, that is 
 sabove 55°, it would be raised only -078 of an inch; quantities which, except in 
 lling, may be safely disregarded. 
 
 The French have heretofore considered 28 Paris inches as the mean height of 
 barometer at the level of the sea, that is 29:84 English inches, But from 1400 
 
 * Phil. Transact. 1777 p. 586. 
 Xx 
 
674 PHILOSOPHY. 
 
 observations, made at Rochelle by Fleurieu de Bellevue, and from five years’ obser. 
 vations made at Port Louis, inthe Isle of France, he concludes the mean height of 
 the barometer at the level of the sea to be 28 inches and two lines and B ofa line, 
 in the temperatures of from 52° to 55° Fahrenheit, or 30°08 English inches.* Hence 
 we may consider, in round numbers, 30 inches as the standard height of barometen, 
 at the level of the sea. And knowing the true height of any part of the earth, we 
 may, by subtracting that height, expressed in fathoms, from the log. of 30, viz, 
 747) 213, find the logarithm which indicates the number of inches at which, as its 
 natural mean, the mercury should stand at that height about the level of the sea, | 
 
 “Thus, supposing the height to be 87 feet, equal to 14-500 fathoms : ther 
 4771213 — 14:500 = 4755-7138, which is the logarithm of 29:9; this therefore is the 
 natural mean height of the barometer at the elevation of 87 feet above the level o. 
 the sea. 
 
 ‘Consequently, to all heights heretofore calculated by the French, above the 
 level of the sea, 139°32 feet must be added English measure, when the mereuria| 
 height at the level of the sea was barely supposed to be 28 French inches.” (Or 
 the Variation of the Atmosphere, by Richard Kirwan, Esq. LUL.D., F.R.S. sant, 
 P.R.I.A. Dublin, 1801.) | 
 Rule to compute heights by the Barometerin English measures, by Dr. Charles Hathon| 
 
 To complete the foregoing account of the measurement of altitudes by the baro 
 meter, I shall here annex the method of performing that operation in Englis); 
 measures, either feet or fathoms, as extracted from my Philosophical Dictionary| 
 article Barometer, or from my Course of Mathematics, vol. 2, p. 255, edit. 6th ; whicl 
 is as follows. 
 
 1. Observe the degree or height of the barometer, both at the bottom and to}, 
 of the hill, or other place, the altitude of which is required, as also the degree of thi! 
 thermometer, for the temperature of the air, in both the same places. | 
 
 2. Take out, from a table of logarithms, the logs. of these two heights of thi) 
 barometer in inches and parts, and subtract the less log. from the greater. If fron, 
 the remainder there be cut off three figures on the right hand, where the logs. con) 
 sist of seven places, the other figures on the left hand will give the altitude re | 
 quired in fathoms of 6 feet each. 
 
 3. The above result requires a small correction, when the medium tempera 
 of the air is different from 3] degrees of Fahrenheit’s thermometer, which may bi| 
 thus found, when much accuracy is desired. Add the two observed heights of thi) 
 thermometer together, and take half that sum for the mean temper of the air/ 
 Take the difference between this mean and the number or temper 31; then, as man}, 
 units as this difference amounts to, take so many times the 435th part of the fathom) 
 above found; to which add them when the mean temperature exceeds 31, but sub:| 
 tract them when it is less; and the result will be the more correct altitude of the, 
 hill, &c., as required. : 
 
 A al correction for the temperature of the barometer is sometimes employed | 
 as may be seen in the books above quoted ; but it is so small as to be seldom neces: ! 
 sary to be observed. 
 
 For an example, suppose at the foot of a mountain, the barometer be observes 
 29°68 inches, and the thermometer 57; at the same time at the top of the mountair, 
 the barometer was 25:28, and thermometer 42. Then the calculation will be a 
 below. 
 
 \ 
 
 « La Chappe thought it 28 inches 1-5 lines, See Beguelin’s Memoir. Mem. Berl. 1769. 12 Coll’ 
 
 Acad. 424. 
 ef 
 o) 
 
eo: 
 
 ARTIFICIAL SPRINGS. 675 
 29°68 log. 4724639 57 
 25°28 log. 4027771 42 
 696-868 2)99 
 or 697 nearly. 491 mean. 
 eo) subtr. 
 
 183 
 Then, as 435 ; 184::3 697 : 29 the correction. 
 29 
 
 Fe 726 fathoms = 4356 f. 
 So that the required altitude is equal to 726 fathoms, or 4356 feet. 
 
 PROBLEM XLY. 
 To make an artificial Fountain, which shall imitate a natural spring. 
 
 We here suppose that those who intend to try this experiment, have at their 
 mand a piece of ground, somewhat inclined, the bottom of which isa bed of clay, 
 t far distant below the surface of the earth. In this case, a spring absolutely 
 ailar to a natural one, and capable of answering every domestic purpose, may be 
 structed by the following process. 
 
 Uncover this bed of clay for the extent of an acre, or about 70 yards in length, 
 the same in breadth. A border of clay must be formed at the lower end, 
 ving one aperture at the lowest point, through which the water may issue. To 
 $ aperture adapt a stone with a hole in it, about an inch in diameter. Then 
 lect pebbles of various sizes, and cover this area with the largest, leaving an 
 erval of a few inches only between them. Place others, somewhat smaller, above 
 > interstices left by the former; arranging several strata in this manner above each 
 ter, always diminishing the size, till the last are only very large gravel. Cover 
 whole, to the thickness of some inches, with coarse sand, and then with some 
 ut is finer ; but if moss can be procured it will be proper to cover the very large 
 wel with it to the thickness of half an inch, to prevent the sand from falling into 
 interstices between the pebbles. 
 
 [tis evident that the rain water, which falls on the surface of this area, will pene- 
 te through the sand, flow into the interstices between the pebbles, which cover 
 : bed of clay, and at last, in consequence of the inclination of the ground, will 
 meeed towards the aperture at the bottom, through which it will issue in a stream 
 greater or less size, according to the abundance of the rain. 
 
 Now, if we suppose that the water which falls annually on this piece of ground 
 18 inches in height, it will be found that the quantity of water collected will be 
 150 cubic feet ; and if we suppose one fourth wasted by evaporation, or remain- 
 * between the joints and interstices of the stones, sand, and moss, we shall still 
 ve about 49600 cubic feet of water in the year, or about 303800 gallons; that is 
 say, almost 1000 gallons per day, a quantity much more than is necessary for the 
 gest family. 
 
 (t will perhaps be said that such a spring of water would cost exceedingly dear. 
 lis we shall admit. But we much doubt whether the construction of it would 
 tso much as that of a large cistern, which, to confine the water, requires to be 
 ed with clay or cement: besides, the water collected by a cistern is only the 
 in water which falls from the roofs of a few houses, and which is consequently 
 pure, 
 
 Besides, it might be rendered much less expensive by preparing, in the above man- 
 *, a small portion of ground, such as twenty yards square; and, to increase the 
 
 antity of rain water thus obtained, which would not exceed 5400 cubic feet, that 
 | ye ae 
 
676 PHILOSOPHY. 
 
 a 
 which fell on the neighbouring ground might be conveyed thither by small drains, 
 from the distance of some hundred yards. By these means, a very abundant reser. 
 voir of filtred water might be formed at a very small expense; and the proprietor 
 would enjoy the pleasure of having a spring exactly similar to those furnished by 
 nature. 
 We are only apprehensive that the water would flow off with too much rapidity; 
 but this inconvenience might be prevented by making the aperture through which it 
 escaped of such a size as to render it perpetual; or by adapting to it a cock, and 
 keeping it shut till it might be necessary to draw water, 
 
 PROBLEM XLVI. 
 What is the weight of the air with which the body of a man ts continually loaded? 
 
 Who would imagine that the human body is continually loaded with the weight, 
 of twenty or thirty thousand pounds, which compresses it in every direction? This, 
 however, isa truth which has been placed beyoud all doubt by the discovery of the 
 gravity of the air. 
 
 Every fluid presses on its base, in the ratio of the extent of that base, and of its 
 height. But it has been proved that the weight of the air is equivalent to the! 
 weight of a column of water 32 feet in height; consequently every square foot, al| 
 the surface of the earth, is charged with a column of air equal to one of water of 3) 
 cubic feet; that is to say 2000 pounds, as a cubic foot of water weighs 62 pound 
 and a half. The surface of the body in a man of moderate size is estimated at 14 
 square feet; and therefore, if 2000 be multiplied by 14, we shall have 28000 pou 
 for the weight applied to the surface of the body of a moderate sized man. 
 
 But how is it possible to withstand such a load ? The answer is easy: the e | 
 human frame is filled with air, which is in equilibrium with the exterior air. Of this| 
 there can be no doubt ;. for an animal placed under the receiver of an air-pump, swelli, 
 up as soon as the machine begins to be evacuated of air ; and if the operation be con) 
 tinued, it will distend so much that it will at length perish and even burst. | 
 
 It is the difference of this gravity that renders us more active or oppressed, accord. 
 ing as the body is more or less loaded. In the first case, the body being more con, 
 tracted by the weight of the air, the blood circulates with greater rapidity ; and al! 
 the animal functions are performed with more ease. In the second, the weight beinj, 
 diminished, the whole machine is relaxed, and the orifices of the vessels becom: 
 relaxed also; the motion of the blood is more sluggish, and no longer communicate)! 
 the same activity to the nervous fluid; we are dejected, and incapable of labour) 
 as well as of reflection, and this isthe case in particular when the air is at the sami 
 time damp: for nothing relaxes the fibres of our frail machine so much as moisture, 
 
 4 
 
 ) 
 
 PROBLEM XLVII. 
 
 Method of constructing a small machine, which, like the statue of Memnon, shall emi 
 sounds at sun-rise. 
 
 The story respecting the statue of Memnon, exhibited in one of the temples 0, 
 Egypt, is well known. If we can credit the ancient historians, it saluted the rising 
 sun by sounds, which seemed to proceed from its mouth. But however this ma} 
 be, a similar effect can be produced in the following manner. 
 
 Provide a pedestal, in the form of a hollow parallelopipedon a Bc (Fig. 98.) 
 and let the cavity be divided into two parts by a partition p =. The lower par) 
 must be very close, and filled with water to a third of its height : the remainder mus) 
 be filled with air. The partition p = must have a hole in it to receive a pipe, Som’ 
 lines in diameter, well soldered into it, and which reaches nearly to the bottom 9} 
 the lower cavity. This tube must contain such a quantity of water, that when thi, 
 air is cooled to the temperature of night, the water shall be nearly at the level of FG 
 
MEMNON’S STATUE. 677 
 
 Fig. 28. One of the faces of the pedestal must be so thin as 
 ER to become easily heated by the rays of the sun. Of 
 all metals, lead is the soonest heated in this manner ; 
 
 and theretore a thin plate of lead will be very proper 
 
 for the required purpose. 
 
 + aa K L is an axis which revolves freely on its pivots 
 
 F hy f | We i at k and L; round this axis is rolled a very flexible 
 
 ‘ i E | cord, which supports on the one hand the weight n, 
 
 and on the other the weight m, which moves freely in 
 
 the pipe HI. The ratio of these weights must be 
 
 | such, that m shall preponderate over nN, when the 
 
 ‘mer is left to itself; but N must preponderate when the former loses a part of 
 weight by floating in the water; this combination will not be attended with 
 
 ich difficulty. 
 
 In the last place, the axis KL supports a barrel, some inches in diameter, and a 
 
 v inches in length, implanted with spikes, which, touching keys like those of a 
 
 psichord, raise up quills and make them strike against strings properly attuned. 
 
 je air must be finished in one or two revolutions of the barrel, and it must be 
 
 seedingly simple, and consist of a few notes only. All this mechanism may be 
 
 ‘ily inclosed in the upper cavity of the pedestal. On the top of it must be placed 
 
 figure or bust, representing that of Memnon, with its mouth open, and in the’ 
 ‘itude of speaking. It would not be difficult to connect its eyes with the axis K 1, 
 
 such a manner as to render them moveable. ; 
 From this description it may be readily conceived, that the side of the pedestal, 
 
 sosed to the east, cannot receive the rays of the sun w:thout becoming hot; and 
 
 it, when heated, it will heat the air contained in the lower cavity ; this air will 
 
 ‘ke the water rise in the pipe #1, by which means the weight n will preponderate, 
 
 1 cause the axis KL to revolve, and consequently the cylinder furnished with 
 
 ‘kes, which will raise the keys; and in this manner the air that has been noted 
 
 ll be performed. But for this purpose the diameter of the axis KL must be so 
 
 portioned, that the weight n by descending, two lines for example, shail cause the 
 
 ‘inder to revolve once or twice with sufficient rapidity to make the sounds succeed 
 
 *h other quick enough to form an air. 
 
 Father Kircher, it is said, had in his museum a machine nearly of the same kind; 
 
 lescription of which has been given by Father Schott; but we think ourselves 
 
 thorised to assert, that it could not produce the desired effect ; for Schott says that 
 
 : air was impelled through a small pipe against a kind of vanes, implanted ina 
 
 all wheel; but as the air, in this manner, could issue only very slowly, it is 
 
 dent that no motion could be communicated to the wheel. If Kircher’s machine 
 
 m produced any effect, as said, the mechanism of it has not been properly de- 
 
 ibed by Schott. We will not venture however to assert, that the one in question 
 
 l answer the intended purpose, as we much doubt whether the rising sun would 
 
 efy the air, contained in its lower cavity, in a manner sufficiently sensible in all 
 
 nates. 
 
 Remark.—We shall say nothing farther in regard to the machines which may be 
 ; in motion by the compression, or the rarefaction, or condensation, &c. of the air ; 
 (if we should imitate Father Schott, we might find sufficient matter to fill a 
 arto volume. We shall therefore refer those who are fond of such machines to 
 » “ Mecanica Hydraulico-Pneumatica”’ of that Jesuit, printed at Frankfort, in 
 7, 4to; and to his “ Technica Curiosa, or Mirabilia Artis.’’ Herbip. 1664, two 
 8s. 4to, 
 
 The reader will find in these books abundance of such frivolous inventions, 
 | : 
 
678 PHILOSOPHY. | 
 
 extracted for the most part from the work of Father Kircher, who paid a good d : 
 of attention to them; and from the ‘ Spiritalia” of Hero; and from Alleoti, his 
 translator and commentator; as also the “ Philosophia Fontium,” oi Dobreremaal 
 &c. &c. 
 
 PROBLEM XLVIII. 
 
 The phenomena of Capillary Tubes. 
 
 Capillary tubes are tubes of glass, the interior aperture of which is very narrow, 
 being only half a line, or less, in diameter. The reason of this denomination | 
 be readily perceived. 
 
 These tubes are attended with some singular phenomena, in the explanation 
 of which philosophers do not seem to have agreed. Hitherto it has been easier, in 
 this respect, to destroy, than to build up. The principal of these phenomena are as 
 follow: | 
 
 I. It is well known that water, or any other fluid, rises to the same height in two. 
 tubes, which have a communication with each other; but if one of the branches be. 
 capillary, this rule does not hold good: the water in the capillary tube rises above. 
 the level of that in the other branch; and the more so, the narrower the capillary 
 tube is. i" 
 
 It seemed very easy to the first aRilomanard who beheld this phenomenon, to give 
 an explanation of it. They supposed that the air, which presses on the water in the! 
 capillary tube, experiences some difficulty in exercising its action, on account of the! 
 narrowness of the tube; and that the result: must be an elevation of the fluid on. 
 that side. if 
 
 This however was not very satisfactory ; for what reason is there to think that 
 the air, the particles of which are so minute, will not be at perfect freedom in a tube 
 half a line, or a quarter of a line, in diameter ? | 
 
 But whether this explanation be satisfactory or not, it is entirely overturned by 
 the second and third phenomena of the capillary tubes. f 
 
 II. When mercury is employed, instead of water, this fluid, instead of rising in 
 the capillary branch, to the level which it reaches in the other, remains below that! 
 level. 
 
 Ill. If the experiment be performed in vacuo, every thing takes place the same! 
 as in the open air. The cause of this phenomenon then is not to be sought for in’ 
 the air. 
 
 IV. If the inside of the tube be rubbed with any greasy matter, such as tallow, 
 the water, instead of rising above the level, remains belowit. The case is the ~ J 
 if the experiment be made with a tube of wax,or the quills of a bird, the inside 
 of which is always greasy. 
 
 V. If the end of a capillary tube be immersed in water, this fluid immediatly rises 
 above the level of that in the vessel, and to the same height to which it would rist. 
 in a siphon, if one of its branches were a capillary tube, and the other of the com: 
 mon size; so that if the surface of the water only be touched, it is immediately, 
 attracted, as it were, to the height above mentioned, and it remains suspended at 
 that height when the tube is removed from the water. 
 
 VI. Ifa capillary tube be held in a perpendicular direction, or nearly so, an i! 
 a drop of water be made to run along-its exterior surface, when the drop reaches 
 its lower aperture, it enters the tube, if it be of sufficient size, and rises to the 
 height at which it would stand, above the level, in the branch of a siphon, 0: 
 that calibre. 
 
 VII. The heights at which water maintains itself in capillary tubes, are in fh 
 inverse ratio of the diameters. Thus, if water rises to the height of 10 lines in # 
 
PERPETUAL MOTION. 679 
 
 ‘be one 3rd of a line in diameter, it ought to rise to the height of 10 lines in a 
 be one 6th of a line in diameter, and ‘to the height of 100 in a tube one 30th of 
 ‘ine in diameter. hak | 
 The falling of mercury below the level in such tubes, follows also the inverse 
 tio of the diameters of the tubes. 
 VIL. Ifa capillary tube be formed of two parts of unequal calibres, as seen Fig. 
 29, where the diameter of a B is much less than that of B c, and 
 Fig. 29. if a b be the height at which the water would maintain itself in 
 a tube such as a B, andcd that at which it would remain in 
 the larger one B c, when this tube is immersed in such a man- 
 ner that the aperture of the smaller end ps, shall be raised 
 above the level, by a height greater than ¢ d, the water will 
 maintain itself as in pr, at that height ¢ d above the level ; but 
 if the tube be immersed in such a manner that the water 
 shall reach to B, it will immediately rise to the same height 
 as if the tube were of the same calibre as that before men- 
 tioned, 
 The case is the same, if the capillary tube be immersed, beginning with the nar- 
 ver branch. 
 IX. Those persons would be deceived who should imagine, that the lightest 
 uors rise to the greatest height in these tubes: of aqueous liquors, spirit of wine 
 that which rises to the least height. Ina tube in which water rises 26 lines, spi- 
 of wine rises only 9 or 10. The elevation of spirit of wine, in general, is only 
 »half or a third of that of water. 
 This elevation depends also on the nature of the glass: in certain tubes, 
 ter rises higher than in others, though their calibres be the same. 
 To be convinced that these effects are not produced by anything without the tube 
 the liquor, it is necessary to see these phenomena, which are indeed the same in 
 vacuum, or in air highly rarefied, as in the air which we breathe. They vary also 
 vording to the nature of the glass of which the tube is formed; and they are 
 ferent according to the nature of the fluid. The causes therefore must be sought 
 in something inherent in the nature of the tube, and in that of the fluid. 
 This cause is generally ascribed to the attraction mutually exercised between 
 ss and water. This explanation has been controverted by Father Gerdil,. a Bar- 
 ite and an able philosopher, who has done everything in his power to overturn it. 
 1 the other hand, M. de la Lande has stood forth in its defence, aud is one of 
 ase modern writers who have placed this explanation in the clearest light. The 
 sder may consult also, on this subject, a very learned and profound memoir by M. 
 ‘eitbrecht, in the Memoirs of the Imperial Academy of Sciences at Petersburg. 
 
 i 
 
 PROBLEM XLIX. 
 
 _ Of some attempts to produce a Perpetual Motion, by means of capillary siphons. 
 ‘When philosophers saw water rise ina capillary tube, above the level of that in 
 sich it was immersed, or above that at which it stood in a wider tube, with which 
 formed an inverted siphon, they were induced to conjecture the vossibility of a 
 rpetual motion: for if the water, said they, rises to the height of an inch above 
 at level, let us interrupt its ascent, by making the tube only three quarters of an 
 ‘h in height: the water will then rise above the orifice, and falling down the 
 es into the vessel, the same quantity will again rise, and so on in succession. Or, 
 ‘be water that rises in the capillary branch of a siphon be conveyed, by an inclined 
 de, into.the other branch, a continual circulation of the fluid will take place ; 
 
 J hence a perpetual motion given by nature. 
 But, unfortunately, this idea was not confirmed by experiment. If the ascent 
 
580 PHILOSOPHY. — 
 
 of water, in a capillary tube, be intercepted, by cutting the tube at half the height, 
 for example, to which the water ought to rise, the latter wil) not rise above the 
 orifice to trickle down the sides. And the case will be the same in the other’ 
 attenipt. 
 
 The following, however, is a very ingenious one; and it is difficult to discover 
 the cause of its not succeeding. 
 
 Let ase (Fig. 30.) be a capillary tube, the diameter of the) 
 long branch of which is much smaller than that of the other ; itis. 
 supposed, thatif the orifice a be immersed in the water contained 
 in the vessel p u, it willrise top, the summit of the bending of 
 the tube; and that in the other branch B ¢, it will rise only to 
 the height c u above the level. 
 
 If the siphon be filled with water, and if it be immersed | 
 such a depth that the water can rise, as above said, to the bend. 
 ing B, it appears evident, and incontestable, that the water in 
 the part BH, will force down that contained in cn. But this 
 cannot take place without the water contained ina B following 
 it; hence the water will continually ascend from a to B, and fall down into the 
 vessel, through the branch Bc. Here then we havea perpetual motion. 
 
 Nothing is more specious; but unfortunately this illusion is destroyed also a 
 experience: the water does not fall through the branch B c; on the contrary, it 
 rises till the branch a B alone is full. 
 
 We think it our duty to subjoin here another idea of a perpetual motion, by 
 means of two siphons, though the siphons employed for this purpose are not alto. 
 gether capillary. It deserves the more attention, as it was not proposed by an. 
 obscure person, but by one whe is justly classed among the greatest mathematicians;] 
 we mean the celebrated John Bernoulli. 
 
 Let there be two liquors, said Bernoulli, susceptible of 
 being mixed together, the specific gravities of which are as’ 
 the lines a Band cp (Fig. 81.); it is well known that if 
 two tubes, which communicate with each other, have their 
 heights above the branch of communication in the same 
 ratio, the shortest branch may be filled with the heaviest! 
 fluid, and the longest with the lightest, and these two fluids 
 will remain in equilibrio: hence it follows, that if the longer 
 branch be cut somewhere below the length it ought to have, 
 the fluid contained in this branch will run into the lower 
 one. 
 
 Let us now suppose that the shorter branch E Fr, is filled with a fluid composed 
 of two liquors of different specific gravities, and that a filtre be placed in the point. 
 F, so as to afford a passage only to the lighter ; let the tube F G be filled with the 
 latter, and let its height be somewhat less, in order to establish an qua be- 
 tween the liquor in the branch E F, and that in F 4g. 
 
 Things being in this state, as the filtre suffers only the lighter liquor to pass, the 
 latter, in consequence of the equilibrium being destroyed, will be impelled outwards, 
 through the orifice cg; and consequently may be conveyed by a small pipe into the 
 orifice E, where it will again mix with the liquor contained in EF: and this effect 
 will always continue, because the column of liquor c ¥ will be too light to counter- 
 balance the compound column E F. Here then we have a perpetual motion; and ais, 
 says Bernoulli, is that which maintains the rivers, by means of the water of the sea; 
 for, adhering to the old ideas, in regard to the origin of fountains, he imagined it was 
 by a similar mechanism that the sea water; deprived of its salt, was conveyed to the 
 
 summits of the mountains. He only rejected the idea of those who pretend that it 
 “a 
 
 | 
 
 \ 
 
 | 
 { 
 | 
 | 
 
 Fig. 31. 
 
 |) oan 
 
FORCE OF MOISTURE. 681 
 
 ses above its level, in consequence of the property of capillary tubes; for he res 
 arked that in that case it would not flow down. 
 
 We will not venture to assert what might he the case, if it were possible to realise 
 e suppositions of Bernouilli: we are however strongly inclined to believe that it 
 ould not succeed ; and as the above reasoning, in regard to capillary tubes, though 
 appearance convincing, is belied by experience, we are of opinion that the case 
 ould be the same with this of Bernoulli. 
 
 PROBLEM L. 
 The prodigious force of moisture to raise burthens. 
 One of the most singular phenomena in physics, is the force with which the vapour 
 *water, or moisture, penetrates into those bodies which are ‘susceptible of receiving 
 . If avery considerabie burthen be affixed to a dry and well stretched rope, and 
 the rope be only of such a length as to suffer the burthen to rest on the ground, 
 1moistening the rope you will see the burthen raised up. 
 ‘The anecdote respecting the famous obelisk erected by Pope Sixtus V., before St. 
 eter’s, at Rome, is well known. The chevalier Fontana, who had undertaken to raise \ 
 is monument, was, it is said, on the point of failing in his operation, just when the 
 jlumn was about to be placed on its pedestal. It was suspended in the open air; 
 ad as the ropes had stretched a little, so that the base of the obelisk could not 
 rach the summit of the pedestal, a Frenchman cried out ‘‘ Wet the ropes.” This 
 dvice was followed; and the column, as if of itself, rose to the necessary height, 
 » be placed upright on the pedestal prepared for it. 
 This story, however, though often repeated, is a mere fable. Those who read the 
 escription of the manoeuvres which Fontana employed to raise his obelisk, will see 
 aat he had no need of such assistance. It was much easier to cause his capstans to 
 sake a few turns more than to go in quest of sponges and water to moisten his — 
 opes. But the story is established, and will long be repeated in France, because 
 ; relates to a Frenchman. 
 _ However, the following is another instance of the power of moisture, in over- 
 oming the greatest resistances: it isthe method by which millstones are made. When 
 mass of this stone has been found sufficiently large, it is cut into the form of a cylin- 
 er, several feet in height; and the question then is, how to cut it into horizontal 
 jieces, to make as many millstones. For this purpose, tenes and horizontal indenta- 
 ions are cut out quite around it, and at proper distances, according to the thickness 
 o be given to the millstones. Wedges of willow, dried in an oven, are then driven 
 nto the indentations by means of a mallet. When the wedges have sunk toa proper 
 lepth, they are moistened, or exposed to the humidity of the night, and next morning 
 he different pieces are found separated from each other. Such is the process — 
 vhich, according to M. de Mairan, is: employed in diferent places for constructing 
 nillstones. 
 
 _ By what mechanism is this effect produced? This question has been proposed by 
 ML de Mairan ; but in our opinion, the answer which he gives to it is very unsatis- 
 actory. It Sinan to us to be the effect of the attraction by which the water is 
 nade to rise in the exceedingly narrow capillary tubes with which the wood is filled. 
 4et us suppose the diameter of one of these tubes to be only the hundreth part of a 
 ine; let us suppose also that the inclination of the sides is one second, and that the 
 ree with which the water tends to introduce itself into the tube, is the fourth part 
 of a grain: this force, so very small, will tend to separate the flexible sides of the tube, 
 with a force of about 50000 grains; which make about 83 pounds. In the length 
 if an inch let there be only 50 of these tubes, which gives 2500 in a square inch, and 
 he result will be an effort of 21875 pounds. As the head of a wedge, of the kind 
 bove mentioned, may contain four or five square inches, the foree it exerts will be 
 | 
 
 Se eee 
 
682 PHILOSOPHY. 
 
 equal to about 90 or 100 thousand pounds; and if we suppose 10 of these wed a in | 
 the whole circumference of the cylinder, intended to form millstones, they | 
 exercise together an effort of 900 thousand ora million of pounds. It needs, there. 
 fore, excite no surprise that they should separate those blocks into the intervals” 
 between which they are introduced. yf 
 PROBLEM LI, | 
 Of Papin’s Digester. 
 Papin’s digester is a vessel, by means of which a degree of heat is communicated , 
 to water, superior to that which it acquires when it boils. Water indeed exposed 
 to common air, or the mere pressure of the atmosphere, however strongly it boil, 
 can acquire only a certain degree of heat, which never varies; but that inclosed ll 
 Papin’s digester acquires such a degree, that it is capable of performing operations, 
 for which common boiling water is absolutely insufficient. A proof of this will be 
 seen in the description of the effects produced by this machine. | 
 This vessel may be of any form, though the cylindric or spherical is best; but it 
 must be made of copper or brass. A cover must be adapted to it, in such a manner 
 as to leave no aperture through which the water can escape: To prevent the vessel 
 from bursting, a hole is made in the side of it, or in the cover, some lines in dias, 
 meter, with an ascending tube fitted into it, on which is placed the arm of a lever 
 kept down by a weight. This lever serves as a moderator to the heat; for if there 
 were no weight on the aperture of this regulator, the water, when it attains to a 
 certain degree of common ebullition, would escape almost entirely through the 
 aperture, either in water or in steam: if the weight be light, the water, in order to” 
 raise it, must assume a greater degree of heat. If there were no regulator of. this” 
 kind, the machine might burst into pieces, by the expansive force of the steam, | 
 For this reason, it is proper that the machine should be of ductile copper, and not 
 of cast iron; because the former of these metals does not burst like the latter, 
 but tears as it were; so that it is not attended with the same dangerous conse- 
 quences. S 
 When the machine is thus constructed, fill it with water, and having fitted on the 
 cover, let it be fastened strongly down by a piece of iron placed over it, which can | 
 be well secured by screws: then complete the filling it through the small tube whieh . 
 serves as a moderator or register, and set it over a strong fire. The water it con: 
 tains cannot boil; but it acquires such a degree of heat that it is able, ina short 
 time, to soften and decompose the hardest bodies; while the same effect could not | 
 be produced by ebullition continued for several weeks: it is even said that the heat, 
 may be carried so far, as to bring the machine to a state of ignition ; in which case | 
 it is evident that the water must be in the same ; but in our opinion this experiment 
 is exceedingly dangerous. | 
 However, the following are some of the effects of this heat, when carried only 3 
 three, four, or five times that of boiling water. | 
 Horn, ivory, and tortoise-shell, are softened in a short time, and at length reduced | 
 to a sort of jelly. 
 The hardest bones, such as the thigh bone of an ox, are sales softened, and at — 
 length entirely decomposed ; so that the gelatinous part is separated from them, and | 
 the residuum is nothing but earthy matter. When no more than the proper degree | 
 of heat has been employed in this decomposition, the jelly may be collected: it | 
 coagulates as it cools, and may be made into nourishing soup, which would be equal | 
 to that commonly used, if it had not a taste somewhat empyreumatic. This jelly | 
 may be absolutely formed into dried cakes, which will keep exceedingly well, ; 
 provided they be preserved from moisture. They may serve as a substitute for 
 meat soup, &c. 5 
 
CHANGES OF TEMPERATURE, 683 
 
 Hence it may be conceived, how useful this machine might be rendered in the arts, 
 r economy, and in navigation. 
 
 From these bones, thrown away as useless, food might be obtained for the poor in 
 mes of scarcity, or some ounces of bread, with soup made from the above cakes, 
 ‘ould form wholesome and nourishing aliment. 
 
 ‘Sailors might carry with them, during their long voyages, some of these cakes, 
 reserved in jars hermetically sealed ; they would cost much less than preparations 
 f the same kind from meat, as the matter of which the former are made is of no 
 lue. The sailors, who are accustomed to live on salt provisions, would be less 
 xposed to the scurvy. At any rate, these cakes might be reserved till a scarcity 
 f fresh meat or of any other kind of provisions, which so often takes place at sea. 
 ; would be a great advantage to have collected into a small volume the nourishing 
 art of several oxen; for since a pound of meat contains, at least, an ounce Of gela- 
 nous matter reduced to dryness, it thence follows, that 1500 pounds of the same 
 eat, which is the whole weight of a bullock, would give only 94 pounds, which 
 ight be easily contained in an earthen jar. 
 
 In the last place, it would be of great use to the arts, to be able, with a machine 
 this kind, to soften ivory, horn, bone, and wood, so as to render them susceptible 
 being moulded into any form at pleasure. 
 
 PROBLEM LII. 
 
 hat is the reason that in winter, when the weather suddenly becomes mild, the air 
 in houses continues, even for several days, to be colder than the exterior air ? 
 
 This question will present no difficulty to those who are acquainted with the 
 enomena of the communication of heat. It is well known, indeed, that the rarer 
 body is, the less time it requires to become hot, or to cool; and, on the other 
 nd, that the denser it is, the more abstinately it retains the heat it has 
 2quired. 
 
 Hence it may be easily conceived, that when cold has prevailed for some time, all 
 
 4e bodies of which our houses are composed, are cooled to the same degree as the 
 xterior air. But when the exterior air, by any particular cause, becomes suddenly 
 varmer, these bodies do not immediately assume the same temperature: they lose 
 nly gradually that which they had acquired; and during this time the interior air, 
 vhich is surrounded by them, retains the same degree of cold. 
 Houses, strongly built—that is to say, constructed of good squared stone, which 
 ave thick walls—must, for this reason, retain much longer the cold they have re- 
 ived from the exterior air; and, for the same reason, the air within these houses 
 rill remain longer at a temperature below that of the atmosphere, than in houses 
 uilt in a slighter manner: for the same reason, also, it will be less cold in such 
 ‘ouses, at the commencement of winter, than in slighter houses. 
 
 PROBLEM LIIl. 
 
 IY some natural signs, by which a change of the present temperature of the air can 
 : be predicted. 
 
 This part of philosophy, we must confess, is still in its infancy. No person has 
 ver yet been able to make a series of observations, sufficient to shew the connec- 
 on which subsists between the temperature of the air, and different physical signs 
 rhich are commonly supposed to precede them. We shall here confine ourselves 
 > a few of those signs, which ate commonly considered as indications of good or 
 ad weather, but we will not warrant them as infallible. 
 
 Ist. When a strong hoar frost is seen on the ground in the morning, during win- 
 
 er, it will not fail to rain the second or third day after, at farthest. 
 | ‘ 
 
694 PHILOSOPHY. 
 
 2d. It has also been remarked, that it commonly rains on the day when the sun 
 appears red or pale ; ; or the next day when the sun at the time of setting is involved 
 in a large cloud; in this case, if it rains, the next day is exceedingly windy, 
 The same thing almost always takes place also, when the sun at setting appears: 
 pale. 7 
 
 3d. When the sun is red at the time of his setting, it is commonly a sign of fine” 
 weather the next day; on the other hand, if he rises red, rain or a strong wind 
 commonly takes place the day after. | 
 
 4th. When a white mist or vapour is seen to rise from the water or mars 
 places, after the sun has set, or a little before he rises, one may conjecture, with 
 some degree of probability, that next day will be fine. 
 
 5th. When the moon is pale, it indicates rain; when red, wind may be expectell a 
 and when of a pure and silver colour, it is a sign of fair weather, according to this” 
 
 verse ¢ 
 
 Pallida luna pluit, rubicunda flat, alba serenat. 
 
 6th. When small black clouds, detached from the rest, and lower, are seen wan= 
 dering here and there, or when several clouds are seen collected inthe west, at sun 
 rise, it isa sign of a future tempest. If these clouds, on the other hand, disperse, 
 it is asign of fine weather. When the sun appears double or treble through 
 clouds, it prognosticates a storm of a long duration, It is the sign of a great storm 
 also, whentwo or three broken and spotted circles are seen around the moon. 
 
 7th. When an iris, or rather halo, is seen around the moon, it isa sign of rain; 
 and if a halo is seen around the sun, during bright and serene weather, it is also 
 a sign of rain: but if the halo appear in the time of rain, it is a sign of fine 
 weather. 
 
 8th. Ifanimals shew signs of fear and uneasiness, while the weather is exceedingly 
 calm and close, it is almost certain that a storm will ensue. The barometer, in 
 this case, falls exceedingly low all of a sudden. 
 
 9th. Indications of rain not being far distant, may be gathered from the actions 
 
 of various animals, as follows : 
 When birds are seen more employed than usual in searching among their feathers, 
 
 for the small insects which torment them ; ; 
 When those which are accustomed to remain on the branches of trees, retire a 
 
 their nests ; 
 
 When the sea-gulls, and other aquatic fowls, and particularly geese, make a ereter 
 noise than usual; 
 
 When the swallows fly very low, and seem to skim over the surface of the. earth; 
 
 When pigeons return to the pigeon-house before their accustomed time ; 
 
 When certain fish, such as the porpoise, sport at the surface of the water ; 
 
 When the bees do not quit their hives, or fly only to a very short distance; 
 
 When sheep bound in an extraordinary manner, and push each other with their 
 heads ; 
 
 When asses shake their ears, or are very much stung by the flies ; 
 
 When flies and gnats sting more severely, and are more srounlcedie than usual ; 
 
 When agreat number of worms issue from the earth ; 
 
 When frogs croak more than usual ; | 
 
 When cats rub their heads with their fore-paws, and lick the rest of their bodill 
 with their tongue ; 
 
 When foxes and wolves howl violently ; 
 
 When the ants quit their labour, and conceal themselves in the earth; 
 
 When the oxen, lying together, frequently raise their heads, and lick each others 
 muzzles ; 
 
 When the cecks crow before their usual hour; 
 
TO SEPARATE MIXED LIQUORS. 685 
 
 When domestic fowls flock together, and squeeze themselves into the dust ; 
 When toads are heard crying in elevated places. 
 
 10th. During the time of rain, if any small blue space of the heaven be observed, 
 
 ne may almost be assured that the rain will not be of long duration; this sign is 
 ‘ell known to huntsmen. 
 -lith. Very violent storms, when accompanied with earthquakes, are almost always 
 receded by an extraordinary calm in the air, and of that alarming kind which 
 »ems the silence of nature about to be convulsed. Animals, more sensible of these 
 atural indications than man, are frightened by it, and hasten to their retreats. 
 ‘ometimes a hollow subterranean noise is heard, When all these signs are 
 nited, the inhabitants of the unfortunate countries, subject to these destructive 
 courges, ought to fly from their houses, that they may avoid the danger of being 
 uried under their ruins. . 
 
 We shall not entertain our readers with the prolix description which Ozanam, or 
 is continuator, here gives of one of these storms which spread devastation 
 hroughout the kingdom of Naples, in the time of the famous queen Jean. 
 12th. An English navigator says he observed that an aurora-borealis was always 
 ollowed, in the course of a few days, with a violent gale from the south-east; and 
 e gives this notice to navigators, about to enter the Channel, that they may be upon 
 heir guard.* 
 
 PROBLEM LIV. 
 To separate two liquors, which have been mixed together. 
 
 _ This operation is merely an application of the property of capillary tubes, and of 
 hat law of nature by which homogeneous fluids, when near each other, unite toge- 
 her. This is observed to be the case with two drops of mercury, or water, or 
 il, when they come into contact. It is even probable that, before they are in contact, 
 hey lengthen themselves, and mutually approach each other. 
 
 | However, if you are desirous of separating water, for example, from the oil with 
 which it has been mixed, take a bit of cloth or sponge, well moistened in water, 
 ind place it, immersing it. by one end, in the vessel containing the liquors to be 
 eparated ; the other end must be made to pass over the edge of the vessel, and to 
 tang down much lower than the surface of the liquor: this end will soon begin to 
 lrop, and in this manner will attract and separate all the water mixed with the 
 ail 
 
 _ If it be required to draw off the oil, the rag or sponge must be first immersed in 
 hat liquid. , | 
 
 - But those who should imagine that wine or alcohol can be separated in this 
 nanner from water, would be deceived: in order that the operation may succeed, 
 he two liquors must be nearly immiscible together, otherwise they will both pass 
 over at the same time. This process, therefore, cannot be employed for separating 
 water from. wine, though it has been given in the preceding editions of the Mathema- 
 Heal and Philosophical Recreations, with many others equally childish. 
 
 _ The colouring part of the wine appears indeed to remain behind, because it is less _ 
 ittenuated than the phlegm and spirit; but in reality these two liquors, of which 
 wine essentially consists, are not separated from each other. 
 
 PROBLEM LV. 
 | What is the cause of the ebullition of Water! ? 
 
 | Though this question, on the first view, may appear as of little importance, it 
 leserves to be examined ; for those who might imagine that the bubbling observed 
 
 * See Philos. Transact. vol. LXV. p. 1. 
 
 | - 
 
686 PHILOSOPHY. 
 
 in water which boils, is the necessary consequence of the heat it receives, wou d : 
 be deceived. That the contrary is the case may be proved by the following expe. 
 
 2 i| 
 
 riment. 
 Immerse, with the necessary care, any vessel, such as a bottle filled with water, for 
 
 example, into kettle containing water in a strong state of ebullition; the water in 
 the bottle will not fail to assume, in a short time, a degree of heat absolutely equal . 
 to that of the water which boils; this will be proved by means of a thermometer, 
 yet the smallest sign of ebullition will not be observed in it. 
 
 What then is the cause of that observed in the water, which is immediately | 
 exposed to the action of the fire? 
 In our opinion, the boiling up is the effect of portions of the water, which toueh 
 the sides of the vessel, suddenly converted into vapour by coming into contact with 
 these sides; for when a vessel rests on burning coals, its bottom tends to receive a 
 degree of heat much greater than that uecessary to convert immediately into vapour a 
 drop that falls upon it. The pellicle of water which touches the bottom must, theres — 
 fore, be continually converted into vapour; and this indeed is the case; for bubbles — 
 of an elastic fluid are continually seen rising from the bottom, and these bubbles, 
 carried by an accelerated motion to the surface, in consequence of their lightness, 
 
 produce there that bubbling which constitutes ebullition. 
 
 But the water contained in the bottle, immersed in the boiling liquid, cannot 
 assume a degree of heat greater than that of boiling water; because, however 
 strong the ebullition may be, the water does not acquire a greater degree of heat, 
 On the other hand, a piece of metal, heated only to the degree of boiling water, 
 does not convert the water it touches into vapour; the water therefore contained in 
 the interior vessel, though become equally warm, cannot boil. Such is the explana-_ 
 tion of the two phenomena; and their necessary connection with each other, as well 
 as with the assigned cause, proves the truth of that cause. 
 
 PROBLEM LVI. 
 
 What is the reason that the bottom of a vessel, which contains water in a high state 
 of ebullition, is scarcely warm ? 
 
 Before we attempted to enquire into the cause of this phenomenon, we thought 
 it proper first to assure ourselves of the fact, for fear of exposing ourselves to ridi+ 
 cule, like those who explain in so ingenious a manner the phenomenon of the child 
 in Silesia with the golden tooth; a phenomenon however which was only a decep- 
 tion, as well as that which occurred to the marquis of Vardes, explained with so 
 much sagacity by Regis, and which however was the trick of a servant. And the 
 case is the same with many others, which ought first to be confirmed, before we 
 attempt to explain them. We brought water therefore to a strong state of ebulli- 
 tion, in an iron vessel, and having touched the bottom of it, while the water was 
 boiling, we indeed found that it had but a very moderate heat; ft did not begin to 
 be burning hot, till the moment when the ebullition had almost ceased. 
 
 In our opinion, this effect is produced in the following manner: we have already 
 shewn, that the ebullition is occasioned by the pellicle of water, which touches the 
 bottom of the vessel, being continually converted into vapour. This conversion 
 into vapour cannot take place, without the bottom always losing some of that heat, — 
 which it acquires by the contact of the coals or fire. But during the interval be- — 
 tween the moment when the vessel is taken from the fire, and that when it is 
 touched, as no new igneous fluid reaches it, though it still continues to boil, it is 
 probable that the remainder of this fluid is absorbed by the water which touches the 
 bottom, and which is converted into vapour. 
 
 Without giving this explanation as absolutely demonstrative, we are strongly 
 
HYGROMETER. 687 
 
 clined to think that such is the real case; and what seems to give it more pro- 
 bility is, that while the bottom of the vessel, from which the boiling proceeds, is 
 it little hot, the sides have the heat of boiling water; so that the finger would be 
 mnt, were it kept as long on them as it can be kept on the bottom. But no sooner 
 the boiling ceased, than the bottom itself receives part of the heat of the water, 
 the finger cannot then touch it without being burnt. 
 
 Remark.—The solution of the following little problem depends, in all probability, 
 | a similar cause. 
 To melt lead in a piece of paper. 
 
 Wrap up a very smooth ball of lead in a piece of paper, taking care that there be 
 » wrinkles in it, and that it be every where in contact with the ball; if it be held, 
 this state, over the flame of a taper, the lead will be melted without the paper 
 ing burnt. ‘The lead, indeed, when once fused, will not fail in a short time to 
 erce the paper, and to run through. 
 
 PROBLEM LVII. 
 
 o measure the moisture and dryness of the air. Account of the principal Hygrometers 
 invented for that purpose ; their faults, and how to construct a comparative Hygro- 
 meter. 
 
 The air is not only susceptible of acquiring more or less heat, but also of becoming 
 ore or less humid. It belongs therefore to philosophy, to measure this degree 
 * moisture; especially as this quality of the air has a great influence on the human 
 dy, on vegetation, and many other effects of nature. 
 
 This gave rise to the invention of the hygrometer, an instrument proper for mea- 
 ring the humidity of the air. a 
 
 But it must be allowed, that the instruments hitherto invented for this purpose 
 ) not give that result which might have been expected. We have hygrometers 
 deed, which indicate that the air has acquired more or less moisture than it had 
 fore; but they are not comparative, that is to say, they do not enable us to com- 
 ire the moisture of one day or place with that of another.* It is, however, proper 
 ‘at we should make known the different kinds of hygrometers, were it only that 
 e may be able to appreciate their utility. 
 
 I. As fir-wood is highly susceptible of participating in the dryness or humidity 
 ‘the air, an idea has been conceived of applying this property to the construction 
 ‘ahygrometer. For this purpose, a very thin small fir board is placed across be- 
 veen two vertical immoveable pillars, so that the fibres stand in a horizontal direc- 
 on; for it is in the lateral direction, or that across its fibres, that fir and other 
 nds of wood are distended by moisture. The upper edge of the board ought to 
 : furnished with a small rack, fitted into a pinion, connected with a wheel, and the 
 tter with another wheel having on its axis an index. It may be easily perceived, 
 vat the least motion communicated by the upper edge of the board to the rack, by 
 3 rising or falling, will be indicated in a very sensible manner by the index ; conse- 
 ently, if the motion of the index be regulated in such a manner, that from extreme 
 yness to extreme moisture it may make a complete revolution, the divisions of this 
 rele will indicate how much the present state of the atmosphere is distant from 
 ther of these extremes. 
 
 This invention is ingenious ; but it is not sufficient. The wood retains its moisture 
 long time after the air has lost that with which it is charged ; besides, the board 
 
 * This is not altogether correct. M.de Luc has described, in the Philosophical Transactions, the 
 ethod of constructing a hygrometer, which approaches very near to what might be desired in this 
 spect. We have added it to this article. 
 
 ! 
 
688 PHILOSOPHY. | 
 
 gradually becomes less sensible to the i impressions of the air, and therefore produces: 
 little or no effect. 
 
 IJ. An hygrometer may be made APS with the beard of a wild oat, fixed on a’ 
 small column, placed in the centre of a round box: the other extremity of the beard | 
 passes through the centre of the cover of the box, the circumference of which is. 
 divided into equal parts; in the last place, a smallindex, made of paper, is adapted to 
 the extremity of the beard. In order to afford access to the air, it is necessary 
 that the sides of the box should be open, or cut into holes. 
 
 When this instrument is exposed to drier or moister air, the small index, by turn-. 
 ing round, either in the one direction or the other, indicates the state of the 
 atmosphere, 
 
 But this. hygrometer, which is exceedingly sensible at first, gradually this 
 loses this property: consequently, it isa very imperfect instrument, as well as the 
 following. 
 
 III. Suspend a small circular plate by a string, or piece of catgut, fastened to its 
 centre of gravity ; ; and let the other end of the string be attached to a hook. Ac- 
 cording as the air is more or less moist, you will see the small plate turn round, in 
 one direction or in another. This small machine may be covered by a bell-glass, to 
 prevent its being deranged by the agitation of the air; but the bell must be 
 elevated above the base on which it is placed, that the air may have access to the’ 
 string. 
 
 The hygrometers commonly sold, are constructed on this principle. They consist. 
 of a kind of box, the fore part of which represents an edifice with two doors. On 
 one side of the metal plate which turns round, stands the figure of a man with an 
 umbrella to defend him from the rain; and on the other, a woman with a fan. The 
 appearance of the former of these figures, indicates damp, and that of the other, 
 dry weather. This pretended hygrometer can serve for no other purpose than to 
 amuse children; the philosopher must observe that, as thevariations of humidity 
 are transmitted to this instrument only by degrees, it will indicate moisture or 
 drought, when the state of the atmosphere is quite contrary. 
 
 If a piece of cat-gut, made fast at one extremity, be 
 conveyed over different pulleys, as a, B, C, D, £, F, G, | 
 &c., (Fig. 32.), soas to make several turns back wards, 
 and forwards; and if a weight p, be suspended from 
 the other extremity, it may be easily seen that it must| 
 rise or fall in a more sensible manner, in consequence, 
 of the moisture or dryness of the air, according as 
 the number of the turns backwards and forwards is’ 
 greater. But this will be indicated better if an index 
 H K, turning on a pivot I, and placed in such a manner. 
 that the part 1K shall be much longer than 1 u, be 
 made fast to the extremity of the cord n ; the slightest 
 change in the moisture of the air, will be manifested : 
 by the point xk of the index, 
 
 V. An hygrometer may be constructed also in 
 the following manner, Extend a cord, five or six 
 feet in length, between the pegs a and B (Fig. 33), 
 and suspend from the middle of it c, a weight p, by. 
 a thread pc. If an index p F, turning on the pivot, 
 
 . E, and having the part = F much longer than p £, be 
 adapted to the thread p c, as seen at p; as the cord 
 
 a CB will be shortened by moisture and lengthened 
 
HYGROMETER. : 689 - 
 
 y drought, the weight Pp, as well as the point p, will rise or fall, and consequently 
 yake the index pass over a certain portion of the area 4, the divisions on which will 
 ‘adicate the degree of moisture or dryness. 
 VI. Put into the scale of a balance any salt that attracts the moisture of the air, 
 nd into the other a weight, in exact equilibrium with it. During damp weather, 
 be scale containing the salt will sink down, and thereby indicate that the state of 
 fhe atmosphere is moist. An index, to point out the different degrees of drought 
 r moisture, may be easily adapted to it. 
 _ This instrument, however, is worse than any of the rest ; for salt, immersed in moist 
 ir, becomes charged with a great deal of humidity ; but loses it very slowly when the- 
 ir becomes dry: fixed alkali of tartar even imbibes moisture, till it is reduced toa 
 iquid or fluid state. : 
 
 VII. Music also may be employed to indicate the dryness or moisture of the 
 ir, The sound of a flute is higher during dry than during moist weather. Ifa 
 yiece of cat-gut then, extended between two bridges, be put in a state of vibration, 
 + will emit a tone with which a tonometre must be brought into unison. When 
 che weather becomes moister, the string will emit a lower sound; and the contrary 
 will be the case when the air becomes drier. 
 
 VIII. M. de Luc of Geneva, to whom we are indebted for an excellent work on 
 
 thermometers and barometers, attempted to construct a compa- 
 
 Fig. 34. rative hygrometer, and published a paper on that subject in the 
 
 Philosophical Transactions. The description of this hygrome- 
 ter is as follows. . 
 
 It has a great resemblance to a thermometer. The first and 
 principal part is a cylindric reservoir of ivory, about 25 inches in 
 length, the cylindric cavity of which is 24 lines in diameter, and 
 the thickness 1 or j, of aline. This piece of ivory must be 
 cut from about the middle of an elephant’s tooth, both in regard 
 to its thickness and length; and it is necessary that the cavity 
 should be pierced in a direction parallel to that of the fibres. 
 A representation of this piece is seen Fig. 34, where it is de- 
 noted by the letters a Bc. 
 
 The second piece is a tube of turned copper, one end of 
 which fits exactly into the ivory cylinder, while the other re- 
 ceives into its cylindric cavity a glass tube of about a quarter 
 of aline internal diameter. A representation of it is seen Fig. 
 34, No. 2. 
 
 These three pieces are strongly fixed toeach other, Fig. 34. 
 by introducing into the ivory cylinder the end of the No. 3. 
 copper tube destined to fill it, having first puta little qi 
 fish glue between them. To unite these parts better | 
 together, the neck of the ivory cylinder ought to be 
 surrounded by a virol of copper. 
 
 A glass tube of about 30 inches in length, and of such a size as to fit 
 into the same cavity, is also introduced into it,-as seen Fig. 34, No. 3,. 
 which represents the instrument completely constructed. 
 
 - Itis then filled with mercury, in such a manner that it shall rise to 
 about the middle of the glass tube, and the ivory reservoir is immersed in 
 water ready to freeze, taking care to maintain it at that temperature for 
 several hours; for the ivory will require ten or twelve before it absorbs 
 all the moisture it is capable of receiving. As soon as this reservoir is 
 immersed in the water, the mercury is seen to rise, at first very quick, 
 and then more slowly, until it at length remains stationary towards the 
 
 ay, 
 
s . be’ b 
 . , ye 
 iy, 
 . Se yy: 
 = be 
 N 5 , 
 - ' a 
 a 
 - =} 
 
 690 PHILOSOPHY. ~ ; 3 | 
 bottom of the tube. This place, which ought to be some inches above the inser: 
 tion of the glass tube into the copper one, must be marked o, which signifies the 
 zero of dryness or the greatest humidity. This point, as we have said, must bi 
 some inches higher than the copper tube; for it has been remarked, that if thr 
 instrument be immersed in hot water the mercury falls still lower, and this inter’ 
 val below zero is left for the purpose of marking these divisions. 
 
 We must here acknowledge, that we do not properly understand how M. de Lu 
 proceeds in order to render his instrument comparative : something, in our opinion 
 still remains to be done to give it that property. We must therefore refer th’ 
 reader to the original memoir, in the ‘‘ Journalde Physique” of the abbé Rozier 
 for the year 1775. It will be sufficient to observe, that this hygrometer is ver, 
 sensible; scarcely has it been placed in air more or less humid, than it gives indi 
 cations of that sensibility, by the rise or fall of the mercury ; ‘but it requires, an 
 always will require, to be accompanied with a thermometer; for the same degre 
 of humidity has a greater effect on it during warm weather, than during cold; be 
 sides, the mercury rises or falls independently of moisture, merely by the effect ¢ 
 heat. This instrument, therefore, requires a double correction; the first to kee 
 an account of the dilation which the mercury experiences by heat, a correctio 
 which will be minus whenever the heat exceeds the term of freezing: the seconé 
 to reduce the effect of the moisture observed, to what it would have been ha 
 the temperature been at freezing. . 
 
 It may be readily conceived, of how great advantage it would be, in regard t 
 the improvement of this hygrometer, to find a degree of dryness, or of less humidity 
 fixed and determinable in every country, to serve as a second fixed term, like the 
 of water reduced to the temperature of melting ice, namely that of the greates 
 humidity ; this would tend greatly to simplify the graduation of the instrumen 
 which, according to the method of M. de Luc, appears to us to be complex an} 
 uncertain. But this is enough on the present subject, respecting which the natu 
 of our work will not permit us to enter into farther details. | 
 
 All the preceding contrivances have been superseded by the hygrometer invelite 
 by Professor Daniel of King’s College, London. 
 
 We shall here give a concise abridgment of the account of this instrument, fr 
 the article on the subject in the Library of Useful Knowledge. 
 
 M. Le Roi having suggested the temperature at which dew begins to be depc 
 sited as a measure of the moisture of the air, De Luc proved that the quantity an 
 force of vapour in vacuo, are the same as in an equal volume of air of th 
 same temperature, or that these two elements of vapour depend on the ten 
 perature. i 
 
 Dr. Dalton investigated the force of vapour at every degree of temperature fot 
 0° to 212° Fahrenheit, and expressed this force by the height of the mercurial barc 
 metric column which it could support; and he has given the results ina tabula 
 form, which are thus easily applied to hygrometric purposes. The dew point / 
 found by pouring cold water into a glass, and noting the temperature at which, i 
 the open air, dew ceases tobe formed on the sides of the glass. This is the poin 
 at which, in air of that temperature, dew would just begin to be formed. -Hene 
 may be inferred not only the force exerted by the vapour, but its quantity in a per 
 pendicular column of the whole atmosphere, and the force of evaporation. 
 
 , Thus if the dew point be at 45° Fahrenheit, the force of vapour, by Dalton’s table 
 is ‘316 of an inch, or about the 95th part of the pressure of the air as measured b’ 
 30 inches of the mercurial column; or if the specific gravity of steam be ‘70, th 
 weight of the steam or vapour ina given volume of air will be 136th part of th 
 whole. - Now as the force of a whole atmosphere of steam, at the surface of thi 
 earth, would be the weight of a perpendicular column of it, and in a mixture of stear’ 
 
! 
 ARTIFICIAL COLDS. 3 . 691 
 ad air the force exerted by each is as their relative weights: when the dew point 
 45° the superincumbent column of vapour in the atmosphere, being one 95th part 
 f the whole atmospheric pressure, is equivalent to the pressure of 4-3 of water; or 
 ae vapour, if condensed, would give that depth of water. Dalton has hence shewn how 
 ) find the force of evaporation at a given time; for the quantity of water evapo- 
 ated from a given surface is proportioned to the maximum force of vapour at the 
 emperature of the surface; the vapour continuing in contact with the surface of 
 he water. Hence, asan example, if the dew point is 45° when the temperature of 
 he air is 50°, we have by Dalton’s table ‘375 — -316 = -059 the force of evapo- 
 ation. . 
 
 On this principle is constructed Daniel’s hygrometer, which in its most improved 
 ‘orm is represented at Fig. 35. 
 
 The ball @ is of black glass about 1°25 inch in 
 diameter, and connected with a transparent glass ball ds 
 of the same size, by a bent tube one eighth of an inch in 
 diameter. A portion of sulphuric ether sufficient to fill 
 about three fourths of the ball a is introduced; a small 
 mercurial thermometer, with an elongated bulb, is fixed 
 inside the limb a b, and the atmospheric air being ex- 
 pelled, the whole is hermetically sealed. The ball d is 
 covered with muslin, and the whole is supported on a 
 brass stem fg, on which is another delicate thermometer. 
 The tube can be removed from the spring tube h; and the 
 whole instrument, with a phial of ether, packed neatly in 
 _ a box, which may be carried in the pocket. 
 pees The dew point is ascertained thus. The ether being 
 i brought into the ball a by inclining the tube, the balls are placed perpendicularly, 
 he temperature of the air is noted by the thermometer attached to the stand, and 
 ther is gradually dropped on the muslin cover of d; and the cold produced by the 
 waporation of this ether, condenses the elastic etherial vapour within the ball, 
 vhich produces a rapid evaporator from the ether in a, and lowers in consequence 
 he temperature within the instrument. When the black ball is thus cooled to 
 he dew point, a film of condensed vapour like a ring surrounds the ball; and if 
 jhe thermometer inside the limb c be noted at that instant, we obtain the true 
 lew point of air at the temperature shewn by the other thermometer. 
 
 Having thus found the dew point, and the temperature of the external air, the 
 noisture contained in a cubic foot of air may be found from the following formula, 
 
 Weight in grains = ae x p, where ¢ is the temperature of the external air, 
 nd p the elasticity of the aqueous vapour at the temperature shewn by the interior 
 hermometer, which for every degree of the thermometer is given in Dr. Dalton’s 
 lables of the elastic force of steam. See the 5th vol. of the Manchester Memoirs. © 
 
 PROBLEM LVIII. “ 
 
 In the supposition of what we have before shewn, in regard to the tenuity of th 
 particles of light, and their great velocity ; what loss of its substance may the sun 
 | Sustain, in a determinate number of years ? . 
 
 One of the most specious objections made to the Newtonian theory of light, is, 
 hat if light consisted of a continual emanation of particles, thrown off from lumi- 
 ous bodies, the sun would have sustained such a loss of his substance, that he must 
 ave have been extinguished or annihilated, since the time at which he is commonly 
 upposed to have been created. This objection we have always considered as 
 f little weight ; and we have long been convinced that, assuming as basis what can 
 2yv2 
 
692 PHILOSOPHY, 
 
 be easily proved in regard to the tenuity of the particles of light, and their arts 
 velocity, a very probable hypothesis could be formed, from which it might be shewn, 
 that no sensible diminution could have taken place in the sun, during the course 
 of the 6000 years which he is commonly supposed to have existed. ' 
 
 We have since seen, in the Philosophical Transactions, a similar calculation by 
 Dr. Horsley, to shew the frivolity of such an objection. But as there are differen) 
 methods of considering the same question, our reasoning on the subject is as follows | 
 it has nothing in common with that of the learned Englishman, but the prodig@ 
 tenuity of the particles of light. 
 
 To form this calculation, we suppose and require it may be granted, that at eal 
 instantaneous emanation of light from the sun, this luminary projects in every possi 
 ble direction all the particles of light at its surface. / 
 
 We require, it may be granted also, that this emanation is not absolutely continued 
 but composed of a multitude of instantaneous emanations or jets, which succeed eacl 
 other with prodigious rapidity : we shall suppose that there are 10000 in a second 
 As the retina of the eye preserves for about 3 of a second the impression it receives 
 it is evident that the impression made by the sun will be absolutely continued i it 
 regard to us. 
 
 We shall suppose also, what is almost proved, that the diameter of a parti 
 of light is scarcely the 1000000000000th part of an inch. 
 
 According to these suppositions, it is evident that the sun, at each emanation, a 
 prives himself of a luminous pellicle, the thickness of which is as before stated: 
 consequently, in the course of a second, it will be the 100000000th part of an inek 
 and in 100000000 seconds this luminary therefore will have lost an inch in thick 
 ness. But 100000000 are nearly three years: in three years, then, the sun will her; 
 Jost only an inch in thickness. 
 
 Hence it appears, that in the course of 3000 years, this loss will amount to 100, 
 inches, or 83: feet in depth; and during the 6000 years, which we suppose the su 
 to have Ganvedl it will be 1662. Hence it follows, that before the sun can lose on 
 second only of his apparent duuneien forty millions of years must elapse ; for th 
 diminution of a second in the apparent diameter of the sun corresponds to 36000 
 yards: if in the course of 6000 years, the diminution is only about 54 yards i, 
 depth, it will be found, by the rule of proportion, that it will require 40 million, 
 of years to make it extend to the depth of 360000 yards in thickness, or one secon 
 of apparent diameter. | 
 
 We need therefore entertain no fear of the sun becoming extinct. Our childre. 
 and grand children, at least, are secured from being witnesses of that fatal even’ 
 
 We shall here add, that we have not taken the benefit of all the advantages w” 
 might have employed; for we might have extended this period much farther; an: 
 Dr. Horsley indeed finds a much greater interval between the present moment an 
 the final consumption of the sun. But we have confined ourselves to those suppt 
 sitions which are most admissible. 
 
 PROBLEM LIX. | 
 To produce, amidst the greatest heat, a considerable degree of cold, and even to free: 
 water. On artificial congelations, &c. 
 
 It is a very singular phenomenon, and highly worthy of admiration, that a col 
 far exceeding that of winter can be produced even in the middle of summer; an, 
 what adds to the singularity is, that this production of cold does not take plac 
 unless the ingredients employed become liquid. Sometimes even by re-acting 0 
 each other they produce a strong effervescence. We shall first take a cursory vie\ 
 of the different means of producing cold; and then endeavour to give some explané 
 tion of the phenomenon. 
 
 : | 
 
ARTIFICIAL COLDS. 693 
 
 I. Take water cooled only to the temperature of our wells, that is to say, to 10 
 sgrees of Reaumur’s thermometer, and for every pint throw into it about 12 ounces 
 ’ pulverised sal ammoniac; this water will immediately acquire a considerable 
 gree of cold, and even equal to that of congelation. Ifa smaller vessel then 
 mtaining water be put into the one containing this mixture, the water in the former 
 ill freeze, either entirely or in part. If it freezes only in part, make a mixture in 
 jother vessel, similar to the first, and immerse in it the half-frozen water: by these 
 eans it will be entirely congealed. 
 
 If you employ this water half frozen, or at least greatly cooled in the interior 
 sssel, and throw into it sal ammoniac, the cold produced will be much more 
 msiderable: a cold indeed several degrees below that of ice will speedily be the 
 sult. 
 ‘Tf this mixture be made ina flat vessel on a table, with a little water placed be- 
 veen them, the ice formed below will make the vessel adhere to the table. 
 ‘The solution of the salt must be accelerated as much as possible, by stirring the 
 jixture with a stick; for the speedier the solution, the greater will be the cold. 
 Il. Pulverise ice, and for one part of it mix two parts of marine salt; stir well 
 1e mixture, and a cold equal to that of the severest winter will be produced in the 
 liddle of the mass. By these means Reaumur was able to produce a cold 13 de- 
 ‘ees below congelation. 
 'Saltpetre, employed in the same quantity, will produce a cold only 8 or 4 degrees 
 alow freezing. It is a mistake therefore, as Reaumur observes, to imagine that 
 iltpetre produces a greater effect than marine salt. Saltpetre is employed only 
 ecause it is cheaper; and besides, when artificial cold is applied to domestic pur-_ 
 oses, it is not necessary that it should be considerable. 
 
 Instead of saltpetre, Alicant soda, or the ashes of green wood, which contain an 
 quivalent salt, might be employed: the same effect would be obtained, and at a 
 yuch less expense. 
 
 ‘Ill. A cold much greater, however, than any of the preceding, may be produced 
 1 the following manner. Take snow and well concentrated spirit of nitre, both 
 poled to the degree of ice; pour the spirit of nitre on the snow, and a cold 17 de- 
 rees below that of congelation will be immediately excited. 
 
 If you are desirous of producing a cold still more considerable, surround the snow 
 nd spirit of nitre with ice and marine sal3; Which will produce a cold 12 or 13 de- 
 rees below zero; if you then employ the snow and spirit of nitre cooled in this 
 hanner, a cold equal to 24 degrees below zero will be the result. This cold is much 
 reater than that produced by Fahrenheit; for ‘it did not exceed 8 degrees of his 
 hermometer below zero, which arnounts to 173 degrees of Reaumur, below the same 
 erm. 
 
 But this is nothing in comparison of what the philosophers of Petersburgh per- 
 ormed, towards the end of the year 1759. Assisted by a cold of 30 degrees and 
 nore, they cooled snow and spirit of nitre below that temperature, and by these 
 aeans obtained a degree of cold which, reduced to the scale of Reaumur’s ther- 
 aometer, was more than 170* degrees below zero. It is well known that at this 
 erm mercury freezes, and of the consequences of this experiment we have spoken 
 lsewhere. 
 
 IV. There is still another method of producing a cold superior to that even which 
 necessary to freeze water. It is founded on a very singular property of evaporable 
 luids. Immerse the bulb of a thermometer in one of these fluids, such as well 
 lephlegmated spirit of wine, and then swing it backwards and forwards in the air, to 
 
 _* This number, when corrected, ought to be only 3h below water-freezing on Reaumur ; or 39, that 
 4371 below water-freezing, on Fahrenheit. See the remark at the end of Problem 18. 
 
P| 
 3 
 
 + 
 
 “eng 
 
 ee 
 
 ? 
 
 694 PHILOSOPHY. 
 
 excite a current like that of the wind, which promotes the evaporation of the fluic 
 you will soon see the thermometer fall: by employing ether, the most evapora 
 of all liquors, you may even make the thermometer fall to 8 or 10 degrees belo 
 zero. 
 
 Very curious things might be said in regard to this property of evaporation ; b 
 to enlarge farther on the subject would. lead us too far. We shall therefore on 
 observe, that this method of cooling liquors is not unknown in the east. Travelley) 
 who are desirous of drinking cool liquor, put their water into jars made of poro, 
 earthern ware, which suffers the moisture to ooze through it. These vessels a 
 suspended on the sides of a camel, in such a manner as to be in continual motio 
 which answers the same purpose as if they were exposed to a gentle wind, and whi 
 causes the moisture to evaporate. By these means the remaining liquor is so mu 
 cooled, as to approach the degree of congelation. 
 
 We shall now offer a few observations on the cause of these singular effects, b 
 ginning with the means explained in the first three articles. | 
 
 When ice and marine salt, or spirit of nitre and snow very much cooled, are mix, 
 together, it is observed that cold is not produced unless these substances be dissolve 
 From this circumstance there is reason to conjecture, that the mixture absorbs t. 
 igneous fluid diffused throughout the surrounding bodies, or those surrounded by t. 
 mixture, which amounts to the same thing. ‘The melting mixture produces, i in th 
 case, the same effect as a dry sponge applied to a moist body: as long as it is mere 
 confined around it, no change will take place in it; but as soon as the sponge is. 
 liberty to extend itself to its full volume, it will abies a considerable part of t' 
 moisture contained in that body. It must be confessed, that we do not see the m 
 chanism by which the frigorific mixture produces the same effect; but we may co 
 sider the above comparison as capable of giving some idea of it. 
 
 In regard to the reason why an evaporable liquor cools the bodies from vid 
 evaporates, it appears that the most probable reason is the affinity which that liqn 
 has to fire; so that each of its molecule, in flying off, carries with it some of tho 
 of the fire contained in that.body. But how comes it that these molecule of tl 
 evaporable liquor do not combine rather with the fire which the air can furnish to: i 
 and with which that element seems to have less adhesion than to solid bodies, since | 
 cools more readily? This question we cannot answer; but we give the above el 
 planation rea as conjecture. 
 
 Remark.—In addition to what has been given on this subject by Montucla, ‘| 
 shall here observe, that the best experiments yet made known on frigorific mixture 
 without the aid of snow, are a of Mr. Walker, of Oxford: some of these a 
 as follows: 
 
 Take strong fuming nitrous acid, diluted with water (rain or distilled water is bes 
 in the proportion of 2 parts in weight of the former to one of the latter, well mixe 
 and cooled to the temperature of the air, 3 parts; of Glauber’s salts 4 parts; of 1 
 trous ammonia 35 parts,* each by weight, aud reduced separately to fine powde 
 The Glauber’s salt is to be first added to the diluted acid; the mixture must th. 
 be well stirred, and the powdered nitrous ammonia is immediately to be introduce 
 stirring the mixture again. The salts should be procured as dry and transparent | 
 possible, and are to be used newly powdered. | 
 
 These are the best proportions, when the common temperature is 50°. Accor | 
 as the temperature, at setting out, is higher or lower, the quantity of diluted ac 
 must be proportionably diminished or increased. This mixture is little inferi. 
 
 * A powder composed of sal ammoniac 5 parts, and nitre 4 parts, mixed together 1 
 stituted for the nitrous ammonia, 8 ott ee 
 
ARTIFICIAL COLDS. 695 
 
 to one made by dissolving snow in nitrous acid; for it sunk the thermome- 
 ter from 32° to 20°; that is in all 52°. In this experiment 4 parts diluted acid 
 ‘were used. 
 
 - Crystallized nitrous ammonia, reduced to very fine powder, sunk the ther- 
 mometer, during its solution in rain water, from 56° to 8°; when evaporated gently 
 to dryness, and finely powdered, it sunk the thermometer to 49°. Mr, Walker 
 has frequently produced ice by a solution in water of this salt alone, when the 
 thermometer stood at 70°, If an equal weight of mineral alkali, finely powdered, be 
 added to the mixture, the temperature will be lowered 10° or 11° more. 
 
 As it is evident that artificial frigorific mixtures may be applied to domestic pur- 
 { poses, in hot climates, especially where the inhabitants can scarcely distinguish summer 
 from winter by the sense of feeling, it may not be amiss to give a few hints respect- 
 bis the easiest method of using them. 
 
 In most cases, the following cheap one may be sufficient: Take any quantity of strong 
 ite acid, diluted with an equal weight of water, and cooled to the temperature 
 
 of the air, and add to it an equal weight of Glauber’s salt, in powder. This is the 
 ‘proportion when the temperature, set out with, is 50°; and will sink the thermome- 
 ter to 5°; if the temperature be higher than 50°, the quantity of salt must be pro- 
 portionally increased. 
 
 The obvious and best method of ascertaining the quantity of any salt necessary to 
 produce the greatest effect by solution, in any liquid, at any given temperature, is to 
 add the salt gradually, till the thermometer ceases to sink, stirring the mixture all 
 the time. If a more intense cold be required, double aqua-fortis, as it is called, 
 ‘may be used. Glauber’s salt, in powder, added, will produce very nearly as much 
 cold as when added to diluted nitrous acid. A somewhat greater quantity of the salt 
 is required. At the temperature of 50°, about three parts of the salt, to 2 of the 
 acid, will sink the thermometer from that temperature to nearly 0° ; and the conse- 
 quence of more salt being added is, that it retains the cold rather longer. This 
 mixture has one great advantage in its favour: it saves time and trouble. A little 
 water ina phial immersed in a tea-cup full of this mixture, will be soon frozen, 
 eyenin summer ; and if the salt be added in crystals, not pounded, to double aqua- 
 fortis, though in a warm temperature, the cold produced will be sufficient to freeze 
 water or cream; but if diluted with one fifth of its weight of water, and cooled, it 
 will be nearly equal to the diluted nitrous acid before mentioned, and will require 
 ‘the same proportion of the salt. 
 
 A mixture of Glauber’s salt and diluted nitrous acid, sunk the thermometer from 
 70°, the temperature of the air and ingredients, to 10°. , 
 
 ‘The cold in any of these mixtures may be kept up a long time, by occasionally 
 adding the ingredients in the proportions indicated. 
 
 Take equal parts of sal ammoniac and nitre, in powder; and cool them by im- 
 mersing the vessel which contains them in pump water newly drawn, its tempera- 
 ture being. generally 50°. On three ounces of this powder pour four ounces, wine 
 measure, of pump water, at the above temperature, and stir the mixture ; its tempera- 
 ture will be reduced to 149, and consequently it will soon freeze the contents of 
 any small vessel immersed init. The cold may be continually kept up and regu- 
 lated for any period of time, by occasionally pouring off the clear saturated liquor, and 
 adding more water; taking care to supply it constantly with as much of the powder 
 asitcan dissolve. This is a convenient mixture; for if the solution be afterwards 
 evaporated to dryness in an earthen vessel, and reduced to powder, it will answer 
 ‘the purpose as well as at first ; as its power does not seem to be lessened by being 
 repeatedly treated in this manner. 
 
 All the ingredients employed by Mr, Walker being taken at the temperature of 50°, 
 
 the following table will exhibit the resilt of a great many experiments: 
 | 
 
 | 
 
 I 
 
 i 
 
696 PHILOSOPHY. 
 Temperature. | 
 Sal ammoniac 5, nitre 5, water 16 parts ...---ee+-e+> Ae. Ssh 10° | 
 Do. 5, do. 5, Glauber’s salt 8, water 16 ........ 4 | 
 *Nitrous ammoniac 1, water 1 sececsereceeceeccee eeeteees 4 
 Do. —————_— 1, soda ], water ] “ssccereerescsereeeeree ‘4 
 tGlauber’s salt 3, dilute nit. acid 2 ..--.+>- o Sanloemiate duties 3 
 Glauber’s salt 6, sal ammoniac 4, nitre 2, dees nit. acid 4 10 
 Do. 6, nitrous ammoniac 5, dilute nit. acid 4 ...... 14 
 Phosphorated soda 9, dilute nit. acid. 4 seseer---eeee eH 12 } 
 Do. 9, nitrous ammon. 6, dilute nit. acid 4 .... aT } 
 +Glauber’s salt 8, marine acid 5 .....+-+.+--.- Peis hae tuts 0 
 tDo. 5, dilute vitriolic acid 4 ........... pirat cet bon 3 
 
 | 
 
 The salts marked thus (*) may be recovered by evaporating the mixture, ani 
 may be used again repeatedly; those marked thus (}) may be recovered for usi 
 by distillation and crystallization: the dilute nit. acid was red fuming nitrous aci( 
 2 parts, rain water 1 part: the dilute vit. acid was strong vitriolic acid and rail) 
 water, equal parts. | 
 By a judicious management, frigorific mixtures, with the aid of snow or poundei 
 ice, mercury even may be frozen into a solid mass. Mr. Walker immersed a hal| 
 pint glass tumbler containing equal parts of vitriolic acid, the specific gravity 0) 
 which was 1°5596, and strong fuming nitrous acid, in mixtures of nitrous acid ant 
 snow, until the mixed acids in the tumbler were reduced to — 30°: he then gra, 
 dually added snow, which had been also previously cooled in a frigorific mixtur. 
 to — 15°, to the mixture in the tumbler, stirring the whole, and found, after some mi 
 nutes, that the mercury in a thermometer immersed in the fluid had become con 
 
 | 
 
 gealed or frozen. 
 Quicksilver may be congealed by adding newly fallen snow to strong fuming ni) 
 
 trous acid, previously cooled to between — 25° and — 30°, which may be easil. 
 and speedily effected by immersing the vessel containing the acid in a mixture ¢ 
 snow and nitrous acid. 
 
 But the most powerful frigorific mixture yet discovered, is produced by eque 
 parts of muriate of lime and snow. An account of a very remarkable experimen! 
 of this kind is givenin Tilloch’s Philosophical Magazine, Vol. III. It was performe 
 by Messrs. Pepys and Allen, Into a mixture of equal parts of muriate of lime a) 
 33°, and snow at 32°,a bladder containing no less than 56 pounds of mercury wa} 
 immersed, after the mixture had liquefied by stirring, and when its temperatur 
 was found to be — 42°; as soon as the cold mixture had deprived the mercury ¢ 
 so much of its heat that its own temperature was raised from — 42° to “i yy th 
 mercury was taken from it, and put into another fresh mixture, the same in ever 
 respect as the first. 
 
 In the mean time, the muriate of lime was kept cooling, by immersing the ver 
 sel which contained it into a mixture of the same ingredients: 5 pounds of th 
 muriate were, by these means, reduced to — 15°; a mixture being made of thi 
 muriate and snow, at the temperature of 32°, in the course of three minutes / 
 gave a temperature of — 62°, or 94° below the freezing point of water. . 
 
 The mercury reduced to — 30° by immersion in the second mixture, and sus, 
 pended in a net, was put into the new made mixture, and the whole was covere. 
 with a cloth to impede the passage of heat from the surrounding atmospheré) 
 After an hour and forty minutes, the 56 pounds of mercury were found solid an, 
 fixed. The temperature of the mixture, at this time, was — 46°; that is 16° highe 
 than when the mercury was put into it. 
 
 Several of those who were present at this experiment having, without attending ¢ 
 
 j 
 i 
 
ARTIFICIAL COLDS. 697 
 
 : 
 
 the consequences, taken pieces of the frozen mercury into their hands, experienced 
 ‘a painful sensation, which they could compare to nothing but that produced by a 
 purn or a scald, or by a wound inflicted with a rough edged instrument. The parts 
 of the hand which were in contact with the metal lost all sensation, and became 
 white, and to appearance dead: a phenomenon which alarmed the sufferers not a 
 little: however, soon throwing away the pieces from them, as they would have 
 done hot coals, the injury scarcely penetrated the skin; and in a little time the 
 parts, by friction, resumed their usual sensation and colour. 
 | The late Professor Leslie devised an elegant method of reducing the tempera- 
 ture sufficiently low to freeze water, in any climate, and at any season of the year. 
 His method is shortly this: under the receiver of an air pump, place one vessel 
 containing sulphuric acid, and another containing a small quantity of water. The 
 ‘air being partly withdrawn from the receiver by the air pump, vapour is raised 
 ‘abundantly from the water, and absorbed by the acid. Thus a degree of cold is 
 ' produced which freezes the water in a very short time. 
 | A saucer of porous earthenware is best adapted for holding the water, and instead 
 of sulphuric-acid, other absorbents may be used, such as parched oatmeal, the 
 powder of mouldering whinstone, or the dry powder of pipeclay. 
 ' Mr. Leslie placed a hemispherical vessel of porous earthenware, containing a pound 
 ‘and a quarter of water, over a body of parched oatmeal, one foot in diameter and 
 one inch deep; and by working the pump for some time, the whole of the water 
 “was frozen. 
 
 PROBLEM LX. 
 To cause water to freeze, by only shaking the vessel which contains it. 
 
 During very cold weather, put water into a close vessel, and deposit it in a 
 ‘place where it will experience no commotion; in this manner it will often acquire 
 a degree of cold, superior to that of ice, but without freezing. If the vessel how- 
 ever be agitated ever so little, or if you give ita slight blow, the water will im- 
 mediately freeze with singular rapidity. This will be the case, in particular, when 
 the water is in vacuo. i 
 
 This phenomenon is exceedingly curious; but in our opinion, it is susceptible of 
 an explanation which must appear highly probable to those acquainted with the 
 phenomena of congelation. Water does not congeal unless its molecule assume a 
 | new arrangement among themselves. When water cools, at perfect rest, its mo- 
 ‘leculee approach each other, and the fluid which keeps it in fusion gradually escapes ; 
 
 but something more is necessary to determine the molecule to arrange themselves 
 ‘ina different manner, under angles of 60 or 120 degrees. ‘This determination they 
 ‘receive by the shock given to the vessel; they were in equilibrio ; the shock de- 
 - stroys that equilibrium, and they fall one upon another, forming themselves into 
 | groups, in such a manner as their approach to each other requires. 
 
 Another phenomenon of congelation is as follows. If you boil water, and then 
 expose it to the frost, close to an equal quantity of unboiled water, the former will 
 freeze sooner than the latter. 
 
 This is a fact proved by experiments, made at Edinburgh, by Dr. Black, and, in 
 
 our opinion may be easily explained; as congelation is occasioned by the molecule 
 of the water approaching each other: it must congeal the sooner, if these molecule, 
 before being exposed to the frost, are already closer. But water which has boiled 
 possesses, in this respect, an advantage over that which has not boiled; for the 
 "effect of boiling is to deprive it of a great deal of its combined air; these molecule 
 then, ceteris paribus, must arrive sooner at the term of proximity, at which they 
 adhere to each other, and form a solid body. We are convinced, that for the same 
 
 mee 
 
698 PHILOSOPHY. 
 
 reason, water impregnated by artificial means with air, would be longer in freezin, 
 than common water. 
 
 PROBLEM LXI. 
 Of the figure observed sometimes in Snow : Explanation of that phenomenon. 
 
 It often happens, and it has long been remarked with admiration, that the smal 
 flakes of snow have a regular figure. Such is the case: in particular, when the snov 
 falls gently, and in very small flakes. This figure is hexagonal or stellated ; some 
 times it is a plain star with six radii; at other times the star is more complex 
 and resembles a cross of Malta, having six salient and six re-entering angles. I 
 sometimes happens that each branch presents ramifications, like the barbs of a fea 
 ther; but it would be too tedious to describe them all. We shall therefore confim 
 ourselves to a representation of the most remarkable, as seen Fig. 36. 
 
 This phenomenon has always occasioned great em 
 barrassment to philosophers, since the time of Des. 
 cartes and Kepler, who seem to have been the firs: 
 who remarked it. Bartholin wrote a dissertation 
 
 - “De Figura Nivis Hexangula,” in which he reason) 
 very badly on the subject. It was indeed difficul: 
 to reason justly on it,* until M. de Mairan observed 
 as he did with great sagacity, the phenomena of con 
 gelation, and until chemistry had discovered those 0} 
 
 . the crystallization of bodies, when they pass from a fluid to a solid state. | 
 
 Chemistry indeed has taught us that all bodies, the elements of which, floating ir 
 a fluid, calmly approach each other, assume regular and characteristic figures. Thus 
 sulphur, when it becomes fixed, forms long needles ; regulus of antimony has on its 
 surface the figure ofa star. Salts, when they crystallize slowly, assume regular 
 figures also. Marine salt forms cubes, alum octaedra, gypsum a kind of wedges, 
 regularly irregular, the lamine of which break into triangles of determinate angles; 
 calcareous spar, called Icelandic Crystal, forms oblique parallelopipeda under inva-| 
 riable angles, &c. 
 
 On the other hand, M. de Mairan, while observing the progress of congelation, 
 saw that the small needles of ice, which are formed, are implanted one into the 
 other at regular and determinate angles, which are always 60 or 120 degrees. 
 
 Whoever is acquainted with these phenomena, will see nothing in ice and snow, 
 but a crystallization of water, condensed in cold air: one particle of frozen water 
 meets another, and unites with it, at an angle of 60 degrees; a third is added, and is . 
 determined by the action of the point of the first angle, to unite itself in the same’ 
 manner, &c, This is the simplest of the stars of snow, as represented by No. 1. | 
 
 If new needles of ice are added, which will for the most part be the case, they 
 must place themselves on the first radii, either by making an obtuse or an acute angle - 
 towards the centre. In the first case, the result will bea star, the radii of which 
 have a kind of barbs like a feather, as in No. 2, or like a star, as No.3. The last 
 arrangement however is rare, and that of No. 2 is more common. Some are also. 
 seen, though in less number, much more complex; but whatever may be their com-) 
 position, their elements are always angles of 60 or 120 degrees, it 
 
 M. Lulolf of Berlin conjectured that these figures were occasioned by the sal 
 ammoniac, or rather volatile alkali, with which snow is impregnated: and in support 
 of this idea, he mentions a very pretty experiment. Having exposed some water to — 
 
 [ 
 
 * We find however that Gassendi referred the regular figure of snow to crystallization. Seead — 
 Diog. Laert. Not. opp. vol. I, p. 577. \| 
 
 i 
 } 
 
. 
 
 ee Ee 
 
 = _e 
 ——— 
 
 RECIPROCATING FOUNTAINS. 699 
 
 freeze near the common sewer, he found the surface of it entirely covered with small 
 stars of ice, while the frozen water which was at a greater distance, exhibited 
 nothing of the kind. He acknowledges however, that he was never able, by any 
 
 - process, to detect this principle in snow water, or snow melted in close vessels. No 
 
 philosopher at present will indeed believe, that either sal ammoniac or volatile alkali 
 exists in snow, unless accidentally ; and there is no necessity of supposing it, in 
 
 order to explain its crystallization in stars. 
 
 PROBLEM LXII. 
 Lo construct a fountain, which shall alternately flow and intermit. 
 
 We have already described the mechanism of a fountain which produces this effect, 
 and which is well known to those acquainted with hydraulics; but as it cannot be 
 adapted to the purposes we have here in view, we shall give another method of solv- 
 ing the problem. 
 
 Let a pcp (Fig. 37.) be a vessel of any form, 
 which receives by the pipe = a continual influx 
 of water, capable of filling it to the height cH, in 
 the interval, for example, of two hours, Let FGI 
 be a siphon, the upper orifice of which, immersed in 
 the liquor, is F; let Fa be the shorter, and cx the 
 longer, branch, the orifice of which, 1, must be con- 
 siderably below the level of F; lastly, let the bore 
 of this siphon be such, that it can draw off the li- 
 quor contained in the part a1, in the course of half 
 an hour. These suppositions being made; if the 
 vessel be empty, and if the water be suffered to run 
 in by the pipe £, it will fill the vessel to the height 
 G, in two hours, for example; but when it reaches the bending «, the siphon Fat 
 will be filled, and the water flowing into it, in the course of somewhat more than 
 half an hour,* it will empty, not only the water collected as far as GH, but that also 
 which the pipe = may have furnished during that time; because this discharging pipe 
 
 Fig. 37. 
 
 “rar will exhaust much more rapidly, than that which furnishes, viz. px. The 
 
 surface of the water, always descending, will at length fall to the level of the orifice 
 F, and the air introducing itself, the play of the siphon will be interrupted: the 
 water will then begin to rise again to the bending of the siphon at c, so that the 
 play of the siphon will recommence, and this will be the case as long as the pipe E 
 
 ‘can furnish water. 
 
 Remark.—It is necessary to observe, that the siphon will not perform its effect, 
 unless it be a capillary tube as far as the bending ; for if its diameter, at this place, 
 be 5 or 6 lines, the water, when it reaches to a little above the bending, will flow 
 off without filling the whole pipe; and the pipe would run off a quantity of water 
 equal only to that furnished by the pipe x. This observation was made, and with 
 great justice, by the Abbé Para du Phanjas, who had recourse therefore, in this case, 
 to several capillary tubes, uniting in one. 
 
 Another remedy ¢onsists in making the calibre of the discharging pipe capillary, 
 throughout its whole length, and proportionally wide in a horizontal direction, in 
 order that it may have the same surface, and that the same quantity of water may 
 flow through it. By these means the discharging pipe, though single, will perform 
 its office. 
 
 * This time will be exactly 40 minutes ; for it is the sum of a sub-quadruple progressicn, the first 
 
 term of which is 30 minutes, the second 7}, &c. 
 
700 PHILOSOPHY. 
 
 PROBLEM LXIIi. 
 
 To construct a fountain which shall flow and stop a certain number of times succes- 
 sively ; and which shall then stop, for a longer or shorter period, and afterwards 
 resume its intermitting course ; and so on. 
 
 The solution of this problem depends on a very ingenious combination of two 
 intermittent fountains, similar to the preceding. Let us suppose a similar fountain, 
 the periodical flowing of which is exceedingly quick, that is to-say, 2 or 3 minutes, 
 and its intermission the same, making altogether an interval of 4 or 5 minutes: let 
 this fountain be fed by another intermittent fountain, placed above it, the duration 
 of the flowing of which is an hour, and the intermittence 2, or 3, or 4: it will 
 thence follow, that the lower one will furnish water only while the upper one sup- 
 plies it; that is to say, during an hour; and in the course of this hour the lower 
 fountain will have 12 or 15 periods of flowing, interrupted by as many periods 
 of cessation; after which time, as the fountain or pipe & of Fig. 37 will not furnish 
 more water for two or three hours, the lower fountain will absolutely cease for one, 
 or two, or three hours. Here then we have a fountain which will be doubly inter- 
 mittent, as it will remain a considerable time without flowing, and when it flows it 
 will ‘be intermittent. 
 
 Remarks.—I. With three fountains of this kind, combined together, periods of 
 flowing and intermission, so singular as to appear almost inexplicable, might be pro= 
 duced. But it may be readily conceived, that they would all depend on the same 
 principle. . 
 
 II. By means of these principles, a fountain to flow continually, but which should 
 become larger and decrease alternately, might be easily constructed. Nothing would 
 be necessary for this purpose, but to combine with the fountain of the preceding 
 problem, a continued fountain: it is evident that it would become larger, when the 
 water flowed through the siphon F @1; and that when it stopped, it would assume 
 its usual state. 
 
 If this continued fountain were combined with the double intermittent one of this 
 problem, the result would be a fountain uniform and continued for several hours of the 
 day, and which would afterwards become larger and decrease alternately for an 
 hour. : 
 
 PROBLEM LXIV- 
 
 Construction of a fountain which shall cease to flow when water is poured into it ; and 
 shall not begin to flow again till some time after. 
 
 For this purpose, we must suppose a very close reservoir, half filled with water,’ 
 as ABCD (Fig. 38.), having a discharging pipe x, some lines only indiameter. This 
 reservoir forms part of another vessel H B F D, in which it is placed; and a portion 
 of the vessel u G ¥F remains empty: I K is a pipe 
 which proceeds from the top of the interior reservoir, 
 nearly to the bottom of the vessel, rp; the upper 
 part of the vessel is funished with a rim, so as to re- 
 semble a cup, and the part H G, is pierced with a 
 number of small holes ; some moss, with coarse sand, 
 or even grass, must be put into this cup, but in sucha 
 manner that the air may have access through the 
 bottom of H G, into the cavity H c. 
 
 These things being supposed, let the small reservoir 
 be half filled with water, which will flow out through 
 the discharging pipe £; if water be then poured into the cup at the top, it will fall into — 
 
RECIPROCATING FOUNTAINS. 7OL 
 
 the lateral reservoir H c, and close the aperture kK of the pipe Kr. This aperture 
 being closed, the air contained in that part above the interior reservoir, can no longer 
 expand itself: the water flowing through & will fall at first slowly, and at length 
 stop. But if a small pipe be inserted in the corner F, to afford a passage to the 
 water which has fallen into the reservoir # c, when this water is discharged, that at 
 E will again begin to flow. 
 
 If water be poured incessantly into the cup H G, and if its escape at F be concealed, 
 this machine will excite great astonishment, as it will seem to flow only when no more 
 water is poured into it. 
 
 This machine might be constructed in the figure of a rock, with a fountain issuing 
 from the bottom of it; and the upper part might represent a meadow, or forest, &c. 
 On pouring water over it from a watering pot, to represent rain, the small fountain 
 would be seen to stop, and to continue in that state as long as water was poured over 
 
 it. The use to which this idea might be applied will be seen hereafter. 
 
 PROBLEM LXV. 
 
 To construct a fountain which, after flowing some time, shall then sink down to a certain 
 point ; then rise again ; and so onalternately. 
 
 Though we have not found anything satisfactory on this subject, it is nevertheless 
 possible; for we shall mention hereafter some ‘instances of fountains, the basons 
 of which exhibit this phenomenon. We shall therefore content ourselves for the 
 present with having proposed the problem to our readers. 
 
 RemarKks.—Containing the history and phenomena of the principal intermittent foun- 
 tains known, as well as of some lakes and wells which have similar properties. 
 
 History of the famous lake of Tschirnitz. 
 
 In the preceding problems we have explained the principles of the phenomena 
 exhibited by a great number. of fountains, or collections of water, the properties of _ 
 which have at all times furnished matter of reflection to philosophers, and been a 
 
 “subject of admiration to the vulgar. But much is to be deducted from what the 
 vulgar relate, or imagine they see, in regard to this subject. Many of these springs, 
 when examined by philosophers, or accurate observers, lose the greater part of what 
 they had of the marvellous. In several of them, however, there still remains enough 
 
 to exercise the sagacity of the searchers into nature. The object of this work ~ 
 obliges us, in some measure, to make known the most remarkable of these fountains. 
 But we shall confine ourselves to those, the facts respecting which are confirmed by 
 good descriptions ; for it is of no utility torepeat what is uncertain or incorrect. 
 
 I. The greater part of those springs which originate from accumulations of ice, 
 are observed to be intermittent. Such are some of those seen in Dauphiné, on the 
 road from Grenoble to Briangon. They flow, as we have been assured, more abun- © 
 
 -dantly in the night than in the day time, which on the first view seems difficult to-he 
 
 ‘reconciled with sound philosophy; but we shall shew that this may be explained 
 without much difficulty. 
 
 The author of the Description of the Glacieres of Switzerland, speaks of a similar 
 spring, at Engstler, in the canton of Berne: it is subject to a double intermittence, 
 
 that is to say, an annual and a daily: it does not begin to flow till towards the month 
 
 of May; and the simple peasants, in the neighbourhood, firmly believe that the 
 
 Deity sends them this spring every year for the use of their cattle, which about that - 
 
 period they drive to the mountains. Besides, like those of which we have already 
 spoken, it is during the night that it flows in the greatest abundance. 
 
 The annual reappearance of this fountain, on the approach of spring, may be 
 easily explained: for it is only towards this period that the mass of the earth, being 
 
 “afficiently heated, begins to melt the ice from below. It is at this period, therefore, 
 
i es 
 
 702 PHILOSOPHY, | 
 
 that the fountain in question can flow. We make use of the expression from below; 
 for it is in this manner that these enormous masses of ice are melted. No doubt 
 indeed can be entertained of it, when it is observed that they continually give birth) 
 to large currents of water, even while their upper surface exhibits the strata of the) 
 preceding year scarcely altered. But how comes it that the greater part of these. 
 fountains furnish the largest quantity of water in the night time? This phenomenon’ 
 deserves to be explained. 
 
 It arises, in our opinion, from the alternation of heat and cold, occasioned by the’ 
 presence and absence of the sun, in the mass of the earth covered by this accumula- | 
 tion of ice. But as a certain time is necessary before the heat of the sun can produce | 
 its effect, and be communicated to the distant parts, it happens that the moment, 
 of their greatest heat is posterior, by several hours, to that of the greatest heat | 
 of the air, which takes place about three in the afternoon: it is only some hours | 
 then after sun-set, that the greatest liquefaction of the ice, which is in contact with. 
 the earth, can be produced; and if we take into consideration the space which the 
 water thence arising must pass through, in confined channels between the valleys and. 
 under the ice, it will not seem astonishing that it should not make its appearance till | 
 towards night, It will therefore be about eleven o'clock, or midnight, that these 
 streams, produced by the melting of the ice, will furnish the greatest quantity of water. | 
 
 II. The intermittence in this case depends upon causes which may be easily dis- | 
 covered: it is not even areal intermittence: but the fountains we are about to 
 describe are really intermittent. ° 
 
 A spring of this kind is seen at Fontainebleau, in one of the groves of the Park, | 
 It would probably be better known, and would not be inferior in celebrity to that | 
 of Laywell, if courts were more frequented by philosophers. 
 
 This fountain flows from a sandy bottom, into a bason six or eight feet square: 
 there is a descent to it by several steps, in the last of which, or close to the water, — 
 is dug a small channel, which suffers it torun off. The following are the phenomena | 
 observed in this fountain. | 
 
 The bason being supposed to be half full, as is the case when a large quantity 
 of water has been drawn from it, the water rises to the edge of the last step, and 
 runs off by the channel for some minutes. This discharge is followed by a bubbling, | 
 sometimes so strong as to be heard at a considerable distance. This is a sign of the | 
 speedy falling of the water. It immediately begins, indeed, to fall a few inches | 
 below the level of the channel; but this height is variable. It is then stationary | 
 for some time; but afterwards rises; and continues in this manner alternately. | 
 Fach fiux of this kind employs about seven or eight minutes. Sometimes however 
 it seems to sport with the curious, and remains half an hour, or even a whole hour, | 
 without repeating the same play. 
 
 The description of a fountain, nearly similar to the preceding, may be seen in the 
 Philosophical Transactions, Nos. 202 and 424; and in Desaguliers’s Course, vol. 2: 
 it is situated at one of the extremities of the small town of Brixham, near Torbay, 
 in Devonshire: the people in the neighbourbood call it Lay-Well. It is on the | 
 declivity of a small hill, and distant from the shore a full mile; so that it can have no 
 communication with the sea. The bason, according to the latest description, is eight © 
 feet in length, and four feet and a half in breadth. A current continually flows into 
 the bason, and the water escapes at the other extremity, through an aperture, three 
 feet broad, and of a proportionable depth. 
 
 Sometimes the water flows uniformly for several hours, without rising or falling ; 
 and hence some credulous people believe, that the presence of certain persons has an _ 
 influence on this fountain, which interrupts its play. But, for the most part, it has 
 a very sensible and very speedy flux and reflux. For about two minutes the water 
 rises some inches, after which it falls for about the same period, and thena short — 
 rest ensues; so that the total duration is about five minutes. This takes place 
 
 { 
 
 , 
 
TT: 
 
 RECIPROCATING FOUNTAINS, 703 
 
 twenty times in succession, after which the fountain seems to rest for about two 
 hours, and during that time the water flows in a uniform manner. This, according 
 to the author of the description, is a peculiarity by which this fountain is distin- 
 guished from all others that have come within his knowledge. But we have seen 
 that the one of Fontainebleau experiences something of the same kind: a very strong 
 analogy even is remarked between them, and it appears almost evident from the 
 descriptions, that their periodism is not in the spring, but only in the discharge. 
 This is certain, at least in regard to that of Fontainebleau ; as the nature of the 
 ground does not permit us to suppose any thing similar to that which requires a pe- 
 riodical flowing in the spring itself. 
 
 _ However, we shall here describe a third fountain, much more considerable than 
 either of the preceding two, and which presents a very striking intermittence ; it is 
 situated in Franche-Comté, and a very good description of it was published in the 
 “Journal des Scavans,” for October 1688. ; 
 _ This fountain is, or at least was at that period, near the high road leading from 
 Pontarlier to Touillon, at the extremity of a small meadow, and at the bottom 
 of some mountains which hang over it; it flows from two different places, into two 
 basons, on account of the roundness of which it has acquired the name of La Fon- 
 taine ronde. The upper bason, which is larger than the other, is about seven paces 
 in length, and six in breadth; and in the middle of it there is a stone cut ina sloping 
 form, which serves to render the motion of its reciprocation sensible. 
 
 When the flux is about to commence, a bubbling is heard within the fountain, 
 and the water is immediately seen to issue on all sides, producing a great many 
 air bubbles: it rises a full foot. 
 
 During the reflux, the water falls nearly the same time, and by the same gradations. 
 The total duration of the flux and reflux, is about half a quarter of an hour, includ- 
 ing about two minutes of rest. 
 
 The fountain becomes almost dry at each reflux, and at the end of it is heard a 
 sort of murmuring noise, which announces its cessation. ; 
 
 The small town of Colmars, in Provence, presents also a fountain of the same 
 kind. It is situated in the neighbourhood of the town, and is remarkable for the 
 
 frequency of its flux. When it is ready to flow, a slight murmur is heard; it after- 
 
 wards increases for half a minute, and then throws up a jet of water as thick as the 
 arm; it then decreases for five or six minutes, and stops a short time, after which 
 itagain begins to flow. In this manner the duration of its flowing and intermit- 
 tence together is about seven or eight minutes: so that it flows and stops about 
 eight times in an hour. Gassendi and Astruc have given a more detailed account 
 of this fountain; the former in his works, and the latter in his ‘ Histoire Naturelle 
 du Languedoc et dela Provence.” 
 
 The fountain of Fonzanches, in the diocese of Nismes, deserves also to be men- 
 tioned. Fonsanches is situated between Sauve and Quissac, not far from, and on 
 ‘the right of the Vidourle. It issues from the earth, at the extremity of a pretty 
 ‘steep declivity, looking towards the east. Its intermittence is very striking; it 
 flows and stops regularly twice in the course of the day, or of twenty-four hours ; 
 the duration of its flux is 7 hours 25 minutes, and that of its intermission 5 hours 
 or nearly ; so that its flowing is retarded every day about 50 minutes. But it 
 would be erroneous thence to conclude, that it has any connection either with 
 the motion of the moon, or with the sea, though it has been called Za Fontaine au 
 flux et reflux. It would be absurd to suppose channels proceeding thence to the 
 sea of Gascony, which is 130 leagues distant. Besides, as the retardation of 50 
 minutes is not exactly that of the tides, or of the moon’s passage over the meri- 
 dian, the analogy of the one movement with the other can no more be maintained, 
 than if this retardation were much greater or less. 
 
Paes 
 
 704 PHILOSOPHY. 
 
 We shall terminate this paragraph with a description of the famous fountain calle 
 Fontestorbe, situated in the diocese of Mirepoix. The account we shall give of iti 
 extracted from Astruc’s description, published in the work before mentioned. 
 
 Fontestorbe is situated at the extremity of a chain of rocks, which advance almos 
 to the banks of the river Lers, between Fougas and Bellestat, in the diocese of Mi 
 repoix. At a considerable height above the bed of the river is a cavern, 20 or 3 
 feet in length, 40 in breadth, and 30 in height. On the right side of this cavern i 
 the fountain in question, in a triangular aperture of the rock, the base of which i 
 about 8 feet in breadth. It is through this aperture that the water issues, whe’ 
 the flux takes place. What characterizes its intermission, in a very singular man’ 
 ner, is, that it is intermittent only during the time of drought; that is to say, ii 
 the months of June, July, August, and September: it then flows for 36 or 37 mi) 
 nutes, rising 4 or 5 inches above the base of the triangular aperture, after which i 
 ceases to flow for 32 or 33 minutes. Ifit happensto rain, the time of intermissio) 
 is shortened, and when it has rained three or four days in succession, it becomes an. 
 nihilated; so that the fountain then continues, though with a periodical increase. 
 but at length, when the rain has lasted a considerable time, the flux is continuel 
 and uniform, and remains in this state throughout the winter, until the return o 
 dry weather, when the fountain again becomes periodical and intermittent, by th 
 same gradations inverted. | 
 
 The reason of the greater part of the phenomena here described, may be deduce(’ 
 from the principles explained in the preceding problems. For this purpose, nothing 
 is necessary but to conceive a cavity of greater or less extent, formed by the sink’ 
 ing down of a bank of clay, and which serves as a reservoir to a collection of water’ 
 furnished by a spring. Let this cavity have a communication outwards by a kind o' 
 crooked chanuel, the interior aperture of which is near the bottom of the cavity 
 and the exterior one much lower; this channel will evidently perform the part 0 
 the siphon of Prob. LXIJ. Fig. 37, and will produce the same phenomena, sup 
 posing however that the exterior air has access to the cavity. #| 
 If the spring then which fills the cavity, here described, always furnishes less wa: 
 ter than the supposed siphon can evacuate, the water «will flow only periodically 
 for before it can issue, it must rise to the summit or angle of the two branches Oo 
 the siphon; it will then flow and evacuate the water contained in the cavity, ant 
 it will again stop till more water rises. | 
 
 But, if the concealed spring, which feeds the reservoir, be variable: that is to say’ 
 if it be much more abundant in winter, and during rainy weather, than in summer’ 
 or during dry weather, the apparent spring will be intermittent only during the lat 
 ter; the duration of its intermissions or rest will decrease, according as the cou 
 cealed spring becomes more abundant; and when the concealed spring gives as muel' 
 water as the siphon can evacuate, the apparent spring will become continued: it wil 
 at length gradually resume its intermittence, according as the interior spring decreases 
 in volume. | 
 
 Here then the phenomena of the spring of Fontestorbe are explained, by the same 
 mechanism as that of the other springs purely intermittent. It appears, that in the 
 latter the.concealed spring derives its origin from subterraneous water, which re- 
 ceives little or no augmentation from exterior water; and that, on the contrary, the 
 spring of Fontestorbe is fed by water arising from rain and melted snow. 4 
 
 We shall add only a few words more, respecting some fountains of this kind, men- 
 tioned in various authors. Such is that in the environs of Paderborn, called Bullerborn, 
 which flows, it is said, for twelve hours, and rests during the same period: that of 
 Haute-Combe, in Savoy, near the lake of Bourget, which flows and stops twice in an 
 hour ; that of Buxton, in the county of Derby, mentioned by Childrey in his 
 “ Curiosités d’ Angleterre,” which flows only every quarter of an hour; one near the 
 
; RECIPROCATING FOUNTAINS, 705 
 
 ake Como, celebrated in the time of Pliny the younger, which rises and falls periodi- 
 sally, three times a day, &c. 
 
 iI, We shall now describe phenomena of another kind, namely, those exhibited 
 
 vy certain wells or springs, which rise and fall at certain periods, while no place is 
 snown by which the water is discharged. There is a well near Brest subject to this 
 yeriodical falling and rising, the explanation of which has afforded considerable occu- 
 yation to philosophers. The description we shall give of it isextracted from the Jour- 
 jal de Trevoux, October 1728; it was written by Father Aubert, a jesuit, who 
 ippears to have been a very correct and well informed philosopher. 
 . This well is situated at the distance of two leagues from Brest, on the border of an 
 am of the sea, which advances as far as Landernau. It is 75 feet from the edge of 
 the sea at high water, and nearly double that distance at low water. It is 20 feet 
 ndepth, and its bottom is lower than the surface of the sea at high water, but 
 aigher than the same surface at low water. 
 
 , It would not he astonishing, or rather would be altogether in the natural order of 
 ings, if the well should sink down at low water, and rise at high water; but the 
 tase is quite contrary, as will be seen by the following detailed account of the phe- 
 10mena observed. 
 
 _ The water of the well is lowest, that is to say is only 11 or 12 inches above its 
 yottom, when the sea is at its highest. It remains in that state about an hour, 
 ‘eckoning from the time of high water ; it then increases for about two hours and a 
 aalf, during the time the sea is ebbing; after which it remains stationary for about 
 wo hours. It then- begins to decrease for about half an hour before the time of low 
 water, and this continues for the first four hours of the sea’s fowing. In the last 
 jlace, it remains in the same state of falling for about three hours, that is during 
 he last two hours of the sea’s rising, and the first hour of its ebbing; after which it 
 gain begins to rise, as before mentioned. It was observed during the great drought, 
 n the year 1724, that this well was for some hours dry, while the sea flowed, and that 
 t became full as the sea ebbed. We do not know whether this well be still in 
 xistence. What adds to the singularity of the phenomenon is that the neighbouring 
 vells, which might be supposed to experience the same vicissitudes, are subject to 
 othing of the kind. 
 _ According to Desaguliers, a small lake at Greenhithe, between London and 
 aravesend, exhibits the same phenomena; and this author adds, that he heard at 
 aambourn, in Berkshire, of a spring which is full in dry weather, and dry during 
 ainy weather. It is much to be wished that he had ascertained the truth of these 
 ircumstances. 
 IV. But every thing hitherto said, though very remarkable, is nothing when com- 
 ared with the singularity of the famous lake of Tschirnitz. This lake, which is of 
 onsiderable extent, is situated near a small town of the same name, in Carniola. 
 tis about three French leagues in length, and one anda half in breadth, having a 
 ery irregular form. 
 _ The singularity of this lake consists in its being full of water during the greater 
 art of the year; but towards the end of June, or the first of July, the water runs 
 ff by eighteen holes or subterranean conduits, so that what was the abode of fish and 
 bundance of aquatic fowls, becomes the haunt of cattle, which repair thither to pas- 
 ure on the grass which is found there in great plenty. Things remain in this 
 tate for three or four months, according to the constitution of the year; but after 
 tat period, the water returns through the holes'by which it had been absorbed, and 
 7ith so considerable a force that it spouts up to the height of several feet, so that 
 1 less than twenty-four hours the lake has resumed its former state. 
 It is however to be observed, that there are some irregularities in the time ana 
 uration of this evacuation, It par ehes ee that the lake is filled and emptie? 
 
706 PHILOSOPHY. 
 
 two or three times in the year. One year it experienced no evacuation, but j 
 never remains empty above four months. Notwithstanding these irregularities, th 
 phenomenon deserves a place among the most extraordinary singularities of natur 
 See on this subject a work by M. Weichard Valvasor, a learned man of that country 
 entitled ‘‘ Gloria ducatus Corniole,” &c., 1688, 4to. This author enters into detail 
 which entitle him to credit ; and besides this, it is a fact well known, and mentione 
 by various intelligent travellers. 
 
 M. Valvasor deduces, with great probability, the phenomena of this lake from sul; 
 terranean cavities, which communicate with it, by the apertures already mentione 
 and which are full of water supplied by the rain. When the rain ceases for a cons 
 derable time, so that the water is evacuated toa certain point, a play of siphot, 
 takes place, by which means the whole lake is emptied. But for the details « 
 this explanation we must refer to the work before mentioned, or to the Acts i 
 
 Leipsic for the year 1688. 
 
 PROBLEM LXVI. | 
 
 Of the Speaking Trumpet, and ear trumpet. Explanation of them. Construction 9 | 
 the enchanted Head. | 
 
 As the sight is assisted by telescopes and microscopes, so similar instruments hay| 
 been contrived for assisting the faculty of hearing. One of these, called the speal 
 ing trumpet, is employed for conveying sound to a great distance: the other, calle 
 the ear-trumpet, serves to magnify to the ear the least whisper. j 
 
 Among the moderns, Sir Thomas Moreland bestowed the most labour in endej, 
 vouring to improve this method of enlarging and conveying sound, and on this sil 
 ject he published a treatise, entitled ‘‘De Tuba Stentorophonica,” a name whic 
 alludes to the voice of Stentor, celebrated among the Greeks for its extraordinal| 
 str ength. The following observations on this subject are in part borrowed from thi! 
 curlous work. 
 
 The ancients, it would seem, were acquainted with the speaking trumpet: for v) 
 are told that Alexander had a horn, by means of which he could give orders to b 
 whole army, however numerous. Kircher, on the authority of some passages in 
 manuscript, preserved in the Vatican, makes the diameter of its greatest aperture |) 
 have been seven feet and a half. Of its length he says nothing; and only adi 
 that it could be heard at the distance of 500 stadia, or about 25 miles. 
 
 This account is no doubt exaggerated; but however this may be, the speakir| 
 trumpet is nothing else than a long tube, which -at one end is only large enough i 
 receive the mouth, and which goes on increasing in width to the other extremit,, 
 bending somewhat outwards. The aperture at the small end must be a little fla 
 tened to fit the mouth; and it ought to have two lateral projections, to cover pa 
 of the cheeks. 
 
 Sir Thomas Moreland says, that he caused several instruments of this kind tol 
 constructed of different sizes, viz. one of four feet and a half in length, by whic 
 the voice could be heard at the distance of 500 geometrical paces; another 16 for 
 8 inches, which conveyed sound 1800 paces; and a third of 24 feet, which rendere 
 the voice audible at the distance of 2500 paces. 
 
 To explain this effect, we shall not say, with Ozanam, that tubes serve, in g) 
 neral, to strengthen the activity of natural causes; that the longer they are th 
 more this energy is increased, &c.; for this is not speaking like a philosopher 5 it, 
 taking the effect for the cause: we must reason with more precision. 
 
 The cause of this phenomenon i is as follows. As the air is an elastic fluid, so thi. 
 every sound produced im it is transmitted spherically around the sonorous body, 
 when a person speaks at the mouth of the trumpet, all the motion which would t 
 communicated to a spherical mass of air, of four feet radius for example, is eqn 
 
2 
 
 . DUCKS AND DRAKES. 107 
 
 municated only to a cylinder, or rather cone of air, the base of which is the wider 
 ‘end of the trumpet. Consequently, if this cone is only the hundredth part of the 
 whole sphere of the same radius, the effect will be as great as if the person should 
 speak a hundred times as loud in the open air; the voice must therefore be heard 
 ata distance a hundred times as great. 
 
 The ear trumpet, an instrument exceedingly useful to those almost deaf, is nearly 
 the reverse of the speaking trumpet; it collects, in the auditory passage, all the 
 sound contained within it; or it increases the sound produced at its extremity, in 
 ja ratio which may be said to be as that of the wide end to the narrow one. Thus, 
 for example, if the wide end be 6 inches in diameter, and the aperture applied to 
 'the ear 6 lines, which in surfaces gives the ratio of 1 to 144, the sound will be 
 imereased 144 times, or nearly so; for, we: do not believe that this increase is 
 exactly in the inverse ratio of the surfaces; and it must be allowed that, in this 
 respect, acoustics are not so far advanced as optics. 
 
 _ The tube of the ear trumpet is now often made of india-rubber covered. with an 
 ornamental net work. It is made of considerable length, and being flexible, the 
 wearer can converse with a person across a table by passing over the end to which 
 the mouth-piece is attached, and applying the other end to his ear. 
 
 ' Remark.—It is a certain fact, proved by experience, whatever may be the cause, 
 that sound confined in a tube, is conveyed to a much greater distance than in the 
 open air. Father Kircher relates, in some of his works, that the labourers 
 employed in the subterranean aqueducts of Rome, heard each other at the distance 
 of several miles, A Wie? 
 
 ' Ifa person speaks, even with a very low voice, at the extremity of a tube, some 
 ‘nches in diameter, another who has his ear at the extremity will hear distinctly 
 what is said, whatever be the number of the circumvolutions of the tube. 
 
 This observation is the principle of a machine, which excites great surprise in 
 those unacquainted with the phenomena of sound. A bust is placed upon a table; 
 ‘rom one or each of its ears a tube is conveyed through the table and one of its feet, 
 30 as to pass through the floor, and to end in a lower or lateral-apartment. Ano- 
 cher tube, proceeding from the mouth, is conveyed in a similar manner, into the 
 same apartment. A person in company is desired to ask the figure any question, by 
 whispering into its ear. A confederate of the one who exhibits the machine, by 
 ‘\pplying his ear to the extremity of the first tube, hears very plainly what has been 
 aid: and placing his mouth at the aperture of the other tube, returns an answer, 
 which is heard by the person who proposed it. If motion be communicated at the 
 ‘ame time, tothe lips of the machine, by any mechanical means, the ignorant will be 
 nuch surprised, and inclined to believe that this phenomenon is the effect of magic. 
 {6 may be easily seen, however, that the cause is very simple. 
 
 PROBLEM LXVII. 
 
 hen boys play at Ricochet, or duck and drake, what is the cause which makes the stone 
 | rise above the surface of the water, after it has been immersed in it? 
 _ This play is well known, as most boys amuse themselves with it, when near a 
 niece of water of any extent. But the cause why the stone rebounds, after it has 
 ouched the surface of the water, seems to be involved in a certain degree of ob- 
 curity ; and we will even say that some philosophers have mistaken it, by ascribing 
 tto the elasticity of the water. As water has no elasticity, it is evident that this 
 ‘xplanation is not well founded. 
 _ This rebounding however depends on a cause which approaches very near to elas- 
 icity. Itis the effort made by every column of water, depressed by a shock, to 
 ‘ise up and resume its former situation, in consequence of a sort of equilibrium which 
 2242 
 
2 } 
 708 PHILOSOPHY, 
 
 must prevail between it and its neighbours. But let us enter into a more detaiie¢ 
 analysis of what takes place on this occasion. 
 
 When the stone, which must be flat, is thrown obliquely at the surface 
 of the water, and in the direction of its edge, it is evident that it is carried Dj 
 two kinds of motion compounded together, one horizontal, which is quicker, anc 
 the other vertical, which is much slower. The stone when it reaches the surface o 
 the water, impels it by the effect of the latter only, and depresses a little the colum1 
 of water which it meets; this produces a resistance which weakens the vertica 
 movement, but without destroying it; so that it continues to dip, depressing othe) 
 columns; hence there result new resistances, which at length annihilate this motion 
 so far as it is vertical. The stone has then reached the greatest depth to which i 
 can attain, and must necessarily describe a small curve, the convexity of which i 
 opposite to the bottom of the water; but, at the same time, its motion so far a 
 it is horizontal, has lost little or nothing. On the other hand, the column, depressec 
 by the shock of the stone, reacts against it, being pushed by the neighbouring 
 columns: and hence there arises a vertical motion communicated to the stone, whicl 
 is combined with the remaining part of its horizontal motion. The result then mus 
 be an oblique motion, tending upwards ; which causes the stone to rebound aboy 
 the water, making it describe a very much flattened small parabola; it then agair 
 strikes the water obliquely, which produces a second rebounding; then a third, ¢ 
 fourth, and so on, always decreasing in extent and height, till the motion is entirel} 
 annihilated. 
 
 PROBLEM LXVIII. . 
 
 Mechanism of paper Kites. Various questions in regard to this amusement. 
 
 Every one is acquainted with the amusement of the paper kite, a very curiow 
 small machine, which in its mechanism displays great ingenuity. ‘To some howeve 
 it may appear astonishing that an object of this nature should form the subject of a1 
 academic memoir ; for there is one on paper kites in the Transactions of the Academ: 
 of Berlin for the year 1756. But this surprise will cease when it is known that M 
 Euler was a profound geometrician, at an age when most young per 
 sons see nothing in the paper kite but an object of amusement: t¢ 
 him therefore it could hardly fail of being a subject of meditation 
 It presents indeed several curious questions, and which for the mos: 
 part cannot be treated without the higher analysis. This memoi 
 therefore may be ranked among the juvenilia of a great mathemati 
 cian. We shall not follow him in his profound calculations; wi 
 shall content ourselves with treating the subject in a less igo 
 manner, but much easier to be understood. 
 
 The kite, as is well known, isa plane surface aBcD (Fig. 39.) 
 as light as possible, shaped like an irregular rhombus; that is to say 
 formed of two triangles, B ac and BDC, in which the angle a of thi 
 former is much greater than the angle p of the latter. The head i 
 towards 4, and p is the tail, to which is generally affixed a long cord 
 having pieces of paper attached to it at certain lengths: some mucl 
 shorter are placed at the corners B and c, which cause the smal’ 
 machine, when elevated, to appear at a distance like a monstrous 
 bird balancing itself in the air by the help of its tail and its wings. | 
 
 At a point of the axis aD, and towards the point , is affixed a small cord, som 
 hundreds of feet in length, rolled upon a stick, to be let out or taken in as occasiot 
 may require. But it is necessary that this cord should be made fast to the kite in‘ 
 certain manner ; for, in the first place, two other small cords proceeding from a poin’ 
 near the place where it is attached must be extended to the points B and ©, to pre 
 
 Fig. 39. 
 
| 
 
 \ 
 
 ) 
 : | 
 
 | 
 
 | 
 
 { 
 
 | 
 
 j 
 
 PAPER KITES. ' 709 
 
 & 
 
 « 
 vent the machine from turning on the axis ap; and secondly, from the same point 
 
 sf the cord, another small cord must proceed to a point near the head a; so that 
 
 the angle formed by the cord with the axis aD, shall be acute towards a, and inva- 
 
 dable: a fourth even is made to proceed from this point of the cord to a point 
 
 near D. 
 
 These arrangements being made; when the kite is to be committed to the wind, 
 an assistant holds the cord at the distance of some yards, and the inferior surface 
 of the kite being exposed to the wind, it is thrown up into the air. The person 
 who holds the cord then begins to run againt the wind in order to increase the action 
 
 of the air on its surface. If a considerable resistance is experienced, a little of the 
 
 ‘cord is successively unrolled, and the kite rises: it is necessary to know how to 
 
 govern it by unrolling or winding up the cord properly ; that is to say, letting it go 
 when it is found by the effort experienced that the kite can still rise, and winding it 
 
 up when it becomes slack. <A kite properly constructed, when the time and place 
 
 are favourable, can rise to the height of three or four hundred feet, and even more. 
 
 To analyse this amusement, and explain 
 Fig. 40. what takes place, let us suppose that aD 
 (Fig. 40.) represents the axis of the kite, 
 to which is attached the cord Eo, held at 
 c by the person who directs it. Let FE 
 be the direction of the wind, all the cur. 
 rents of which we suppose united in one, 
 acting on the centre of gravity of the sur- 
 u face of the kite; and which, for the sake 
 —___=£ — of simplifying, we shall suppose not to 
 differ from that of the body itself, or to be 
 
 very near it. 
 
 Let FE Pavel the force with which the wind, to which the kite is exposed, 
 impels its surface ina perpendicular direction; draw © G perpendicular to that sur- 
 face, and make ELa third proportional to EF and EG, and draw LM, parallel to GF; 
 EL will represent the force with which the wind impels the lower surface of the 
 Kite, i in the perpendicular direction, and Lm will be the effort exercised by this impulse 
 
 aA F 
 Mee 
 
 in the direction ML or AED. 
 
 We shall first observe, that by the latter the kite would tend to be precipitated 
 downwards; but the angle arc being acute, there thence results an effort in the 
 Miection E a, which counterbalances the former; otherwise the kite could not 
 support itself, and this is the reason why this angle must necessarily be acute. 
 
 If we now make EH equal to EL, and draw E1 perpendicular to the horizon, 
 cand H1 perpendicular to E H, we shall have two new forces; one of which, 1H, will 
 act in the direction x p, and tend to throw down the kite: but this force is anni- 
 hilated, as well as the former m L, by the power in c, which draws according to the 
 
 acute anglea Ec. The other, £1, is that which tends to make the kite rise ina yer- 
 ' tical direction. 
 
 Hence, if the force EI be greater than the weight of the kite, it will be raised 
 into the air, and if we suppose that the extremity of the cord is fixed in c, it will 
 
 | turn around the point cas it rises; but by turningin this manner it must necessarily 
 
 happen that the wind will fall with more obliquity on its surface a D; so that there 
 will at length be an equilibrium, ‘The kite then will rise no farther, cee the cord 
 is let out ; in which case it will rise parallel to itself, and as in ascending it will meet 
 with freer air and stronger wind, it will still turn alittle around the angle c; or the 
 
 + angle c will become greater, and approach more and more to a right angle. 
 
 Such is the mechanism by which the paper kite rises into the air. It may be rea- 
 
710 PHILOSOPHY. a 
 
 dily seen, that if the velocity of the wind, with the surface and weight of the kite, be 
 known, as well as the constant value of the angle a Ec, the height to which it will 
 rise may be determined. 
 
 ’ A question, which here naturally presents itself, is, what ought to be the value of 
 the angle a £ F, in order that the small machine may rise with the greatest facility? 
 We shall not give the analysis of this question, but shall only say, that if the wind 
 be horizontal, this angle must be 54° 44’, or the same which the rudder of a ship 
 ought to make with the keel, that the vessel may be turned with the greatest 
 facility, supposing the currents of water which impel it to have a direction parallel to 
 the keel. 
 
 We shall here observe, that it is not absolutely necessary that the angle a Eo 
 should be invariable, and determined to be such, by asmall cord proceeding froma 
 point of cx to another point near the head; but in this case the point x, where this 
 cord is attached to the kite, must not be the same as the centre of gravity of the 
 surface of the kite, and the centre of gravity must be as far as possible towards the 
 centre of the tail p. It is for this reason, that a cord with bits of paper fixed in it 
 is added to the point D; by which means the centre of gravity is thrown towards 
 that point. Those who amuse themselves with kites, were certainly not conducted 
 to this mode of construction a priori: the origin of this appendage must have been 
 a desire to give to the small machine the appearance of a bird with a long tail, ba- 
 lancing itself in the air. But accident on this occasion has been of great utility; 
 for M. Euler found by a calculation, of which no idea can be here given, that this 
 small tail contributes a great deal to the elevation of the kite 
 
 In short, this amusement, however frivolous it may appear, presents some other 
 mechanical considerations which require a great deal of address, and a very intri- 
 cate calculation; but for*farther particulars, we must refer to the Memoir of M. 
 Euler before mentioned. | 
 
 Remark.—By observing the before-mentioned rules, various figures may be given : 
 to this small machine; such as that of an eagle, or a vulture, &c. We remember. 
 to have once seena kite which resembled a man. -It was made of linen-cloth cut, ) 
 and painted for the purpose, and stretched on a light frame, so constructed as to ) 
 represent the outline of the human figure. It stood upright, and was dressed in a 
 sort of jacket. Its arms were disposed like handles on each side of its body, and. 
 its head being covered with a cap, terminating in an angle, favoured the ascent of 
 the machine, which was twelve feet in height; but to render it easier to be trans- 
 ported, it could be folded double by means of hinges adapted to the frame. The 
 person who directed this kind of kite was able to raise it, though the weather was | 
 very calm, to the height of nearly 500 feet ; and when once raised he maintained it | 
 in theair by giving only aslight motion tothe string. The figure, by these means, , 
 acquired a kind of libration like that of a man skaiting on the ice. The illusion | 
 occasioned by this spectacle, which might seem fit only for amusing school-boys, did 
 not fail to attract a great number of curious spectators. | 
 
 PROBLEM LXIX. 
 
 Of the Divining Rod ; and opinion which we ought to form of it. 
 
 In the article which Montucla has given under this head, he expresses strongly his | 
 opinion that the performances which are said to have been effected by the aid of the | 
 divining rod, are either illusions or philosophical quackeries. | 
 The rod itself is merely a forked hazel twig, and it is said that, if held by the tips | 
 between the fingers and thumb there are certain persons in whose hands it spon- | 
 taneously cevolves with considerable force, when near or over either a spring of . 
 water or treasure concealed in the earth; and it is said to have been available in dis- 
 
 * 
 
 covering criminals, 
 
 ‘a 
 
 & 
 
DIVINING ROD. Tit 
 
 | 
 , | 
 _ Montucla mentions a person named d’Aymar, who, by the aid of the divining rod 
 (or la baguette devinatoire, as the French call it), pointed out a murderer, of whom 
 he had been sent in pursuit, among many other persons whom he found in prison 
 with him. But the belief among the better informed people of the time was, that 
 d’Aymar had accidentally been a witness of the murder; and gave credit to his rod 
 for the knowledge which he had derived from a more feasible source. 
 
 _ He was a treasure-finder too; but having failed most egregiously in an experiment 
 where he had to experiment only, he was thenceforth accounted a cheat, and died in 
 poverty. 
 
 | Peranque, another of these impostors, exercised his vocation in the southern pro- 
 vinces of France. The wonder in him was his powers of vision. He could see 
 springs in the bowels of the earth, even at great depths; and he could trace out 
 their course and give a good estimate of their depth. Another worthy, of the same 
 district, discovered secret springs, by a faculty which manifested itself in a less 
 agreeable manner ; he was seized with violent illness when he passed over the place 
 where they were. 
 
 _ A female at Lisbon had the sharp-sighted faculty of divination. When but a child 
 she discovered (and it is said truly) that the family cook was enceinte, by seeing the 
 child in its mother’s belly. The human body was, to her sight, transparent. She 
 was even able to point out to the physicians the viscera affected with disease; but it 
 was essential, to the exercise of her faculty, that the body should be exposed to her 
 view—naked. 
 
 It is pretended that many wells at Lisbon were dug in consequence of her indica- 
 tions; and that she discovered by her sight an obelisk which had been long buried 
 in the earth, and which was dug out and erected as an ornament to the city. 
 
 It is not to be wondered at, that in these times of general credulity, when the 
 little scientific knowledge possessed by the learned was often employed for the 
 purposes of delusion, such follies obtained a certain degree of credence ; but it is 
 lamentable to reflect that, even among the most enlightened classes, many were 
 believers in the powers of the divining rod. 
 
 In the present edition of Montucla’s Recreations, however, the article Divining Rod 
 s retained chiefly from a circumstance of which we now proceed to give a short 
 account. 
 
 Soon after the publication of the first edition of the English translation of Montucla, 
 Dr. Charles Hutton, the translator and editor, received a clever anonymous letter from 
 a lady, declaring that, incredible as it might seem to the Doctor, and unaccountable as 
 + was to herself, she did actually possess the power of discovering hidden springs by 
 the aid of the baguette; that she had known several others who had the property ; 
 md detailing very fully the circumstances connected with her becoming aware 
 of possessing such a faculty. She pointed out to the Doctor how he might address 
 a letter to her, and a correspondence ensued, from which it eventually appeared that 
 shere was no hoax whatever on the part of the lady, who was a person of exalted 
 ‘ank and superior talents. 
 
 On her coming to town with her family, the Doctor waited upon her to pay his 
 sespects, when it was arranged that she should visit him at his residence, and give 
 deular proof of the power of the divining rod in her hands. 
 
 We give the result in Dr. Hutton’s own words. 
 
 « Accordingly, at the time appointed, the lady, with all her family, arrived at my 
 house at Woolwich Common; when, after preparing the rods, &c., they walked out 
 to the grounds, accompanied by the individuals of my own family, and some friends ; 
 when Lady shewed the experiment several times, in different places, holding the 
 tods, &c. in the manner she had described in her letter. In the places where I had good 
 reason to know that no water was to be found, the rod was always quiescent ; but 
 
712 PHILOSOPHY. 
 
 in the other places, where I knew there was water below the surface, the rods 
 turned round slowly and regularly till the twigs twisted themselves off below her 
 fingers, which were considerably indented by forcibly holding the rods between 
 them. 
 
 “¢ All the company present stood close round the lady, with all eyes intently fixed 
 on her hands and the rods, to watch if any particular motion might be made by the 
 fingers ; but in vain; nothing of the kind was perceived ; and all the company could 
 observe no cause or reason why the rods should move in the manner they were seen 
 to do. -After the experiments were ended, every one of the company tried the rods 
 in the same manner as they saw the lady had done, but without the least motion 
 from any of them.” The Doctor adds: ‘In my family, among ourselves, we have 
 since then several times tried if we could possibly cause the rod to turn, by means 
 of any trick in twisting of the fingers, held in the manner the lady did; ae in vain; 
 we had no power to accomplish it; and he expresses his conviction ‘‘ that theif 
 appears to exist such evidences of the reality of the motion, as it-seems next to im-' 
 possible to be questioned. me | 
 
 In conclusion, Dr. Hutton having requested permission to use the name of the’ 
 lady, in connection with an account of the experiment which she had made in his, 
 presence, she declined, from a dislike which she had to appearing in print; but 
 added that, ‘‘ They (the circumstances) are known to so many, that I am of opinion | | 
 they will obtain credit in a great degree, without a name being formally attached to 
 them.” | 
 
 Both parties have long been removed beyond the reach of the press; there 
 can therefore be no impropriety in stating now, that the lady in question was the‘ 
 Hon. Lady Milbanke, wife of Sir Ralph Milbanke, Bart. (afterwards Noel), a: 
 mother of the present dowager Lady Byron. | 
 
 aie = al pi i cin 
 
“MAGNETISM. 713° 
 
 PART TWELFTH. 
 
 OF THE MAGNET, AND ITS VARIOUS PHENOMENA, 
 
 OF all the phenomena exhibited to us by nature, magnetism, or the properties 
 of the loadstone, and electricity, may with justice be considered as the most extra- 
 ordinary, as the causes of the effects produced by them have occasioned the greatest 
 lifficulties to philosophers ; for it must be confessed that, not withstanding all their 
 ittempts to explain them, we are as yet acquainted only with facts. They have 
 been able indeed to apply certain hypotheses to some of these phenomena; but if we 
 examine these hypotheses with an unprejudiced eye, and without suffering ourselves 
 to be the dupes of illusion, we must perceive that they have little solidity, and 
 that they are subject to difficulties which cannot be removed, as long as we make it a 
 tule to reason only from the known properties of matter, and the laws of motion. 
 Posterity perhaps will be more successful; and, assisted by time and accumulated 
 experiments, will see more clearly into these matters; or perhaps they may for ever 
 remain an impenetrable secret to the human mind. 
 
 In this part of our work we shall confine ourselves, in speaking of the magnet, to 
 its properties, and the philosophical amusements which may be performed by their 
 means. Electricity will furnish matter for the succeeding part. 
 
 ARTICLE I, 
 Of the Nature of the Magnet. 
 
 The magnet is a metallic stone, commonly of a greyish or blackish colour, com- 
 pact and very heavy, and is usually found in iron mines. It affects no particular 
 form ; exhibits no external marks, to distinguish it from the meanest productions 
 of the bowels of the earth. But its property of attracting or repelling iron, and 
 of directing itself to the north, when at full liberty to move, gives it a title to be 
 classed among the most singular objects of nature. 
 
 This stone, properly speaking, is merely an iron ore, but of that kind which is 
 palled poor; because it contains only a small quantity of metal. Modern metallur- 
 gists indeed have been able to extract iron from it; but, besides that it is difficult 
 to be fused, it is so unproductive, that it would not pay for the expense of work- 
 ng it. | 
 But it may here be asked, why is not every kind of iron ore magnetic? This isa 
 question to which, in our opinion, no answer has ever been given. Its magnetic 
 virtue arises, no doubt, from a peculiar combination of iron with the heterogeneous 
 articles to which it is united; and perhaps it contains some principle which does not 
 mter into the other ores of that metal: but it must be allowed that this does not 
 solve the difficulty. Possibly, chemical analysis may some day discover in what 
 this combination consists; and our profound ignorance respecting the physical causes 
 of the action of the magnet, may arise from chemists having hitherto neglected to 
 make this production of nature a subject of their researches. 
 
714 - MAGNETISM. - 
 
 Formerly, the loadstone was exceedingly rare. The name magnes, by which it 
 was known both among the Greeks and the Romans, seems to have originated in 
 Magnesia, a province of Macedonia, where it was found in great abundance, or which 
 furnished the first magnets known. But the loadstone has been since found in 
 almost every region of the earth, and particularly in iron mines. The island of Elba, 
 so celebrated for its mines of that metal, worked from the earliest ages, is said to 
 furnish the largest and best magnets. 
 
 ARTICLE Il. 
 Of the principal Properties of the Magnet. 
 
 The ancients were acquainted with no other property of the magnet, than that 
 which it has of. attracting iron; but the moderns have discovered several others; 
 such as its communication, its direction, declination and inclination, to which we 
 may add its annual and daily. variation. 
 
 SECTION I. 
 
 Of the attraction which prevails between the Magnet and Iron, or between one Magnet 
 and another. 
 
 EXPERIMENT I. 
 Which proves the atiractive power of the Magnet over Iron. 
 
 Every person is acquainted with the attractive force which the magnet has upon 
 iron. If filings of that metal be presented to a magnet, and even at some distance, 
 you will see the filings dart themselves towards the stone, and adhere toit. The: 
 case will be the same with any small and light bit of iron, such as a needle; this will, 
 approach the magnet, as soon as it is within a certain distance of it, according to the 
 force of the stone. f 
 
 This experiment may be performed also in the following manner. Suspend a long 
 iron needle, in a state of equilibrium, by means of a silk thread, or rather on a point 
 or pivot, so as to leave it at full liberty to move: present to it a magnet, at the | 
 distance of several inches, or even of several feet; then, if the magnet be endowed 
 with proper force, you will see one end of the needle turned towards it, until it be 
 as near to it as possible, and then stop in that position; so that if the situation of the | 
 magnet be changed, the needle will continually follow it. If the needle float on 
 water, which it may be easily made to do by placing it on a small bit of cork, it will 
 not only turn one of its ends towards the magnet, but it will approach till it come | 
 in contact with it. 
 
 All these phenomena will take place, even if there be between the magnet and the 
 needle a plate of copper or glass, or a board, or any other body whatever, iron ex- | 
 cepted; which proves that the magyetig virtue is not intercepted by any of these 
 bodies but the last. 
 
 If the magnetic virtue then is produced by molecule agitated, or put in motion 
 in any manner whatever, the tenuity of these molecule must be very great, or at least. 
 superior to that of any other emanations with which we are acquainted, such as 
 odours; since they freely traverse all metals and even glass. If they produce no 
 effect through iron, it is, in all probability, because they find such a facility of moying _ 
 
 in it, or have such affinity with it, that they do not pass beyond it, but are thus | 
 
 intercepted. | 
 
 We may here remark, that if a light and well-polished sewing needle be dropped | 
 gently on water, it will swim on the surface, and thus remain in a state of more | 
 delicate suspension than can be attained in any other way, for exhibiting the pheno- 
 mena of magnetic attraction. | 
 
 a. 
 4 
 ie 
 Aw 
 
PROPERTIES OF THE MAGNET. 715 
 
 EXPERIMENT II. 
 
 To find the Poles of a Magnet. 
 
 ‘If a magnet be immersed in iron filings, and then drawn out, it will be found covered 
 {lover with them; but it will be observed that there are two places, diametri- 
 ally opposite to each other, which are the poles, where the filings are closer, 
 nd where the small oblong fragments stand as it were upright, while in other parts 
 hey lie flat. 
 
 By this experiment we are enabled to find the poles of the magnet. very mag- 
 let indeed has two poles, or two opposite points, which, as will be seen hereafter, 
 ossess different and peculiar properties. One of these points is called the north 
 ole, and the other the south ; because, if the magnet be freely suspended, the for- 
 jer will turn of itself to the north, and consequently the other will be directed to 
 
 ie south. When it is intended to perform experiments with a magnet, these two 
 ‘oints must be first determined. 
 
 EXPERIMENT IIr. 
 Properties of the Poles of the Magnet, in regard to each other. 
 
 Provide a magnet; and having determined its two poles, make it float on the 
 rater by placing it on a piece of cork of a proper size ; if you then present to the 
 orth pole of this stone the same pole of another, the former will be repelled, instead 
 f attracted ; -but if you present to its north pole the south pole of another, it will 
 e attracted. 
 
 In like manner, if to the south pole of the former you present the south ae of 
 he latter, the first will recede; but if you present to this south pole the north 
 ole of the second, it will approach. 
 
 The poles then of the same name, repel; and those of a different name, attract 
 ach other. 
 
 EXPERIMENT Iv. 
 To produce New Poles in a Magnet. 
 
 If a magnet be cut in a direction perpendicular to the 
 axis passing through its two poles a and s, Fig. 41, there 
 will be formed by the section two new poles, such as Fand 
 E: so that if a be the south pole of the whole stone, £ 
 will be a north pole, and Fa south pole. By this bisection, 
 ecfore, the north side of the stone will acquire a south pole, and the south sidea 
 orth pole. 
 
 This production of magnetic poles may be simply illustrated thus: Suspend from 
 he end of a strong magnetic bar a piece of iron, as a key; and from the lower end 
 f that key, a smaller one may be made to hang in consequence of its induced mag- 
 etism. To this may be appended a smaller piece of iron, such as a nail, and we 
 jay go on thus adding piece after piece, till the lower one will sustain only such a 
 Fay small weight as a light needle. 
 
 The polarities of the lower end of each piece, if examined before another is added, 
 ill be found to be the same as that of the lower end of the magnet, each piece 
 ecoming for the time an actual magnet ; as are, in fact, the individual ay of 
 ‘on filings taken up by a magnet placed among them. 
 
 This may be further illustrated by the following experiment. Take a piece of 
 ‘on shaped like the letter y, and suspend it by one of the branches of the fork 
 » the north pole of a magnet; its lower end will instantly acquire a northern po- 
 wity, and will attract and support a small key, or any other small piece of iron. 
 Vhile the key is thus suspended, apply to the other branch of the fork the south 
 
716 MAGNETISM. 
 
 pole of another magnet, and the key will immediately drop off, the latter magnet 
 tending to induce southern polarity at the lower end of the suspended iron, while 
 the other tended to induce northern polarity, and the result is that the polarity at 
 that point is effectually destroyed. | 
 
 If, on the contrary, the north pole of the latter magnet be applied instead of the 
 southern one, a stronger polarity will be induced at the lower end of the suspended. 
 iron; which will therefore, in its turn, support a heavier piece of iron than when * 
 y shaped piece is suspended by one magnet only. 
 
 If the north pole of a magnet be placed in the middle of an iron bar, both extn 
 mities of the bar are rendered north poles, while the middle isa south pole. And 
 if the north pole of a magnet be placed perpendicularly in the centre of a round 
 iron plate, the plate will have a south pole at the centre, and every part of its cir. 
 cumference will have the property of a weak north pole. Ifthe plate be cut i:to 
 the form of a star pointed towards the outer part, each point will bea stronger 
 north pole than the rim of the uncut circular plate. 
 
 f 
 
 Remarks.—A magnet, however good it may be, unless it be very large, will 
 scarcely support a few pounds of iron; and, in general, the weight which a magnet! 
 can carry, is always very much below its own weight. But means have been 
 found out, by employing what is called arming, to make it produce a much more’ 
 considerable effect. We shall therefore describe the method of arming a magnet. 
 
 First give the magnet a figure nearly regular, and square its sides where the two 
 poles are situated, so that these two sides may form two parallel planes. Then 
 make, of soft ‘pceMor steel is not so good—two pieces, such as that seen Fig. 42, 
 the long and flat side of which may be of the same length and breadth as’ 
 the faces of the magnet where the poles are situated. The proper thick- 
 ness however of this side, as well as the projection of the foot, and. its. 
 thickness, can be found only by repeated trials. These two pieces must! 
 embrace the magnet on the two faces where the poles are situated, the 
 feet passing below as if to support it; and they must be fastened in that| 
 situation by transverse bands of copper, surrounding the magnet, and com-| 
 pressing the long branches of these pieces against the faces of the poles.’ 
 Then provide a piece of soft iron, of the form seen Fig. 43, a little longer than) 
 
 the distance between the bands of iron applied to the poles of 
 
 Fig. 43. the magnet, and in thickness somewhat more than the flat’ 
 
 ) faces of the lower part of the feet of the arming. In regard’ 
 
 Pz_snaiililhin gas to the height, it must be regulated by what may appear most’ 
 
 . convenient. Pierce a bole in it towards the middle, to re-| 
 
 Fig. 44. ceive a hook for the purpose of suspending from it the weight 
 
 to be supported by the magnet. Fig. 44 represents an armed 
 
 magnet, which will be sufficient to give an idea of the whee 
 arrangement, without any farther explanation. _ 
 
 A magnet armed in this manner will support a much greater. 
 weight, than one not armed. A stone, for example, of 2) 
 or 3 ounces, will by these means support 50 or 60 ounces’ 
 of iron; that is, 20 or 30 times its own weight. 
 
 Lemery says he saw a magnet, of the size of a moderate 
 apple, which supported 22 pounds. Some have been seen, 
 of 11 ounces, which could support 28 pounds. The sum of 5000 livres (above | 
 £200 sterling) was asked for it. MM. de Condamine, of the Royal Academy | 
 of Sciences, possessed one, given to him by Maupertuis, which was capable, | 
 we believe, of supporting a weight much greater than any other magnet 
 known, We do not remember its dimensions, or its weight, neither of which was | 
 
PROPERTIES OF THE MAGNET. 717 
 
 ery considerable ; but, if we recollect right? he used to say that it could support 
 sixty pounds.* 
 _ Il. Researches have been made to discover whether there be any other bodies, 
 yesides iron, susceptible of being attracted by the magnet; from which it appears 
 ndeed that there are not. Yet Muschenbroeck says, that the magnet seemed to exer- 
 tise an action on a stone which he calls Lough-neagh. We are not acquainted with 
 this stone; but it is probably some kind of iron ore in which that metal is, very little 
 wnineralised. 
 _ In his ‘‘ Cours de Physique Experimentale,” chap. VII., he gives an account of 
 some trials, made ona great many different kinds of matter, to ascertain whether they 
 were susceptible of being attracted by the magnet. He found that this stone, with- 
 out any preparation, attracted the whole or a great many of the particles in different 
 xinds of sand and earth, which he enumerates, Several others presented no par- 
 ticles susceptible of attraction by the magnet, until they had been exposed to the 
 action of fire, by bringing them to ared heat, and burning them with soap, or char- 
 coal, or grease: after which, says he, they were attracted by the magnet, with almost 
 ‘as much force as iron filings; such, adds he, are the earth of which bricks are made, 
 and which becomes red when burnt; also different kinds of bole and coloured sand. 
 Others, when burnt in this manner, presented only a few particles susceptible of 
 being attracted by the magnet: of these he gives a long enumeration, which we shall 
 not here repeat. 
 This will not appear surprising, if we compare the two following facts: the first is, 
 that the magnet never attracts iron, but when in its metallic state; and that it 
 has no action on this metal when calcined, or reduced to the state of an oxide: 
 the second is, that iron is universally diffused throughout nature, and in almost all 
 bodies more or less distant from its metallic state, as will be seen hereafter. Bodies 
 which contain it in its metallic state, are all, or in part, susceptible of being attracted 
 bythe magnet without any preparation ; but in others, it is not susceptible of attraction 
 till it has been burnt with fat matters, which restore it to its metallic state. Such is 
 the only cause of the phenomenon which seems to have embarrassed Muschenbroeck. 
 ‘It would have occasioned no difficulty, had he been as well acquainted with 
 chemistry, as with the other branches of philosophy. 
 _ An English navigator says, he observed that grease, which happened to fall on the 
 (mariner’s compass, disturbed the magnetic needle, and that brass produced the 
 “game effect. If this observation be correct, we must conclude that the tallow and 
 brass accidentally contained some ferruginous particles; for in our opinion it may be 
 considered as certain, that iron alone, in its metallic state, has the power of acting 
 on the magnet, and is susceptible of being attracted by it. 
 
 EXPERIMENT ve 
 i The direction of the Magnetic Current. 
 ' 
 
 ’ Place an unarmed magnet on a piece of pasteboard, and throw iron filings around 
 it: if you then tap gently on the pasteboard, you will see all the filings arrange © 
 themselves around the magnet, in curved lines, which, approaching each other like the 
 meridians in a map of the world, meet at its two poles. 
 
 This experiment favours the opinion of those who think that the magnetic pheno- 
 mena depend on a fluid, which issues from one of the poles of the stone, and enters 
 at the other, after having circulated around it. 
 
 7 
 
 * The attractive force of magnetism, temporarily induced by galvanism, is much greater than that 
 of any combination of simple magnets. We believe that at the Royal Gallery of Practical Science 
 
 in London, 3 or 4 cwt. is often suspended by magnetism induced in this way. 
 
 ' 
 
 7) 
 
718 MAGNETISM. 
 
 EXPERIMENT VI. 
 Which proves that the Magnet and Iron have a mutual action on each other. 
 
 Place two magnets, or a magnet and a piece of iron, on two bits of cork, madet 
 float in a bason of water. Having then turned the north pole of the one toward 
 the south pole of the other, provided two magnets are employed, if the two piekel 
 of cork bg left to themselves, you will see them proceed towards each other; th. 
 weaker moving faster than the other. The case will be the same, if a bit of iron by 
 - presented to the north pole of the magnet. This attraction then is reciprocal ; 7 
 it may be said that the iron attracts the magnet, as much as the magnet does thi 
 iron. This indeed must necessarily be the case, since there is no action without re! 
 action, and as the latter is always equal to the former. a 
 
 Remark.—Muschenbroeck endeavoured to determine in what ratio the action of thy 
 magnet decreases, according to the distance; and he thought he observed that iti, 
 attractive force decreased in the quadruple ratio, or as the 4th powers of the distances, 
 Thus, if at a line distance, a particle of iron is attracted with a force equal to 1; al, 
 2 lines distance, that force will be 16 times less; at 3 lines, 81 times less; at 4 
 lines, 256 times less; and so on of the rest. This action perhaps decreases stil], 
 more rapidly ; for ina ship of war laden with large iron cannon, it is not observed 
 that they have a sensible action on the compass. In our Opinion, however, it would) 
 be prudent to remove them to as great a distance as possible. | 
 
 Vote.—Recent observations shew that the iron in ships hasa very perceptible action| 
 on the compass ; so much even, in the British Channel, as a point of deviation in the 
 direction of the needle having been produced by a change in the direction of the’ 
 ship’s head. 
 
 SECTION I. 1 
 Of the Communication of the Magnetic Property. 
 
 Magnetism, or the property of attracting iron, and of turning towards a certain’ 
 point of the heavens, ‘is not so peculiar to the magnet, as to be incapable of being’ 
 communicated ; but no bodies have yet been found susceptible of this communication, | 
 except iron and steel. About a century ago it was believed that contact alone, or 
 the continued presence of a magnet, could produce this effect ; but a method has since 
 been discovered to render a piece of iron magnetic, without the magnet ; and these’ 
 artificial magnets are even susceptible of acquiring a strength rarely found in natural 
 magnets. We shall give an account, in the following experiments, of the different 
 methods of communicating the magnetic virtue. 1 
 
 y 
 | 
 
 EXPERIMENT VII. 
 
 Method of Magnetising. . a 
 
 Provide a magnet, either armed or unarmed, and make one of the feet of the’ 
 armour, or one of the poles, to pass over a plate of tempered steel, such as the blade | 
 of aknife, but proceeding always in the same direction, from the middle, for example, | 
 towards the point. After performing this operation a certain number of times, the’ 
 plate of iron will be found to be magnetised, and, like the magnet itself, it will’ 
 attract iron, if placed within the sphere of its attraction. ; 
 The case will be the same, if a long slender bit of steel be left a long time attached | 
 to a magnet: the steel, by remaining in that situation, will acquire the magnetic 
 property ; it will have poles like the magnet, so that the north pole will be at the © 
 end which was near the south pole of the stone, and the end which touched the — 
 north pole will become the south pole. 7 | 
 
 yi 
 i 
 
PROPERTIES OF THE MAGNET. 719 
 
 EXPERIMENT VIII. 
 Method of making an Artificial Magnet with bars of steel. 
 
 We shall here shew the method of making, with bars of steel, an artificial magnet much 
 stronger than anatural one. For this purpose, provide twelve bars of tempered steel 
 about six inches in length, six lines in breadth, and two inthickness. Care must be 
 taken before they are tempered, to make a mark witha punch, or in any other manner, 
 at one of their extremities. Arrange six of these bars in a straight line, but so as to 
 be in contact, and that the marked ends shall be directed towards the north; take 
 an armed magnet, and place it on one of these bars, with its north pole towards the 
 marked end, and the south pole towards the other end; then move the stone over the 
 whole line, beginning at the unmarked end of the first, and repeat this operation 
 three or four times. 
 
 _ When this is done, remove the two bars in the middle, and substitute them for 
 those at the two extremities, which must be placed in the middle; then move the 
 stone in the same direction over the four bars in the middle only ; for it is needless 
 so comprehend those at the extremities; and invert the whole line, that is to say 
 surn up the face which was turned downwards, and magnetise the bars again in the 
 same manner, taking care to transpose the bars at the extremities into the place 
 of the middle ones. 
 | By these means you will have six magnetised bars, which must be formed into two 
 harcels, each containing three. In these parcels the northern extremities must be 
 iowards the same side; but when the one parcel is placed upon the other, care must 
 de taken that the northern extremities of the bars of the one may rest upon the south 
 sxtremities of those of the other. These two parcels must touch at their upper 
 sart, and be separated on the other side; this separation may be effected by means of 
 1 bit of wood placed between them. 
 
 Then place the six bars which were not touched, in the same manner as the pre- 
 ‘eding six, and magnetise them as above described, by means of the double parcel of 
 the former ; that is to say, by drawing the two extremities north and south of this 
 louble parcel over the new series of bars: you will thus have six bars much more 
 strongly magnetised than the former. ‘Then make a line of the six former, and 
 nagnetise them in the same manner with the double parcel formed of the second, 
 according to the same method, and you will obtain bars of steel capable of support- 
 ing 16 times their weight and more. 
 | This is the process of Mr. Mitchell, fellow of the Royal Society of Liéadoa Mr. 
 Canton, a celebrated observer of the phenomena of the magnet, has given a method 
 wf effecting the same thing ; and M. Duhamel, of the Academy of Sciences, another, 
 which may be seen ina small treatise on Artificial Magnets, printed in 1775. We 
 shall say nothing farther on this subject, but only remark, that by these processes 
 the weakest commencement of magnetism is sufficient to produce magnetic bars of the 
 fees force. It is not even necessary to have a magnet ; for, in the following 
 2xperiment, we shall describe various methods of communicating magnetism without 
 one. 
 
 EXPERIMENT IX. 
 
 To produce the magnetic virtue in a bar of iron, without the use of a magnet. 
 
 To propose communicating the magnetic virtue without the use of a magnet is, 
 no doubt, a sort of paradox. This however has been effected in consequence of some 
 theoretic considerations in regard to the nature of the magnet, and the manner in 
 which the magnetic fluid acts on iron. A magnet therefore is not necessary to pro- 
 duce the commencement of magnetism, which may be afterwards increased to a consi- 
 derable degree by the process before explained. 
 
720 MAGNETISM. 
 
 Canton, Mitchell, and Antheaume have given different methods for magnetising: 
 without a magnet. According to Mr. Canton, take a poker, and having placed it) 
 between your knees, in a vertical direction, with the point downwards, affix length-| 
 wise to its upper part, by means of a silk thread, a small plate of soft ternpered, 
 steel; then holding this apparatus in the left hand by the silk thread, take a pair 
 of tongs, and, holding them almost vertically, rub the small bar fast the bottom: 
 upwards, about a dozen of times, with the lower end of them: by these means) 
 you will communicate to ita magnetic force, capable of making it support a small key, 
 
 Mr. Mitchell employed another method. Place, says he, a small bar of steel ina’ 
 straight line between two iron bars, in the direction of the magnetic meridian, and in’ 
 such a manner that they shall be somewhat inclined towards the north: then take a’ 
 third bar, and holding it almost vertically, but with the upper extremity a little. 
 inclined towards the south, glide the lower extremity of this bar along the three 
 other bars situated ina straight line, taking care to make 1t move from north to 
 south: the result will be a commencement of the magnetic virtue in the bar of 
 steel. 
 
 M. Antheaume’s method is as follows: First fix a board in the direction of flee! 
 magnetic current; that is, inclined at an angle of about 70 degrees towards the ho- 
 rizon, so that its horizontal projection shall make one of about 20 degrees. Then: 
 place in a line, on that board, two square bars of iron, four or five feet in length, or. 
 even more, and 15 lines in thickness: they must be filed square at the extremities, 
 which are opposite to each other. Each of these extremities must be furnished’ 
 with a small square of iron plate, two lines in thickness, so as to project beyond the 
 upper face of the bar the height of a line, and filed square on that side, to form) 
 above the bar a kind of knee. The three other sides of this square of iron plate: 
 must slightly touch the corresponding faces, and be cut into a bevel. In the last 
 place, a small bit of wood must be placed between the arming of the extremities 
 of these two bars. 
 
 When every thing is thus arranged, glide the bar of steel to be magnetised, over 
 the two knees above described, by making it move gently from one of its ends to: 
 the other, as an iron bar is magnetised, on the two knees of its arming. M. An-) 
 theaume says, he was surprised to find that he could magnetise, by this method, not 
 only small bars of steel, as Messrs. Canton and Mitchell did, but bars of a foot in 
 length, and several lines in thickness. 
 
 The same philosopher says, he observed that steel de caume, or 4 la rose, BA 
 English steel, are the fittest for this purpose; that the operation succeeds best with 
 the first kind, when tempered hard in the usual manner, and that English steel re-: 
 quires to be tempered in bundles: lastly, that if steel tempered and annealed be em- 
 ployed, the temper is a matter of indifference. 
 
 Remark.—Even the rubbing of one piece of iron over another is not necessary to 
 produce the magnetic virtue. It has been observed that a bar of iron, kept fora 
 long time in the direction of the meridian, or in a situation nearly approaching to it, 
 acquires the magnetic virtue. - The steeple of Notre Dame de Chartres having been’ 
 considerably damaged by a great storm in 1690, some bars of iron taken from it were 
 found to be magnetic. But what is still more remarkable, pieces of these bars, which 
 were almost destroyed by rust, formed excellent magnets. The Abbé de Vallemont 
 wrote at that time an account of them, which was published in 1692, ; 
 
 Gilbert, an English physician and philosopher, who wrote a treatise on the magnet, 
 
 - in 1640, had then observed that the small bars of an iron window frame, placed north’ 
 and south, which had remained many years in the same position, were become mag- 
 netic. He relates also,* that the wind having bent an iron bar which supported an‘ 
 
 * Book iii, chap. 13. 
 
 1 
 y 
 
oe 
 7 
 
 PROPERTIES OF THE MAGNET. 721 
 
 ornament on the church of St. Augustine, at Rimini, when the monks belonging to 
 t were desirous, ten years after, to straighten this bar, they were much surprised to 
 ind that it possessed all the properties of a good magnet. Muschenbroek speaks of a 
 jimilar circumstance, in regard to some pieces of iron taken from the tower of Delft. 
 We read also in the Memoirs of the Academy of Sciences, for the year 1731, that 
 shere was at Marseilles a bell which moved on an iron axis, standing in an east and 
 west direction, and resting with its two ends on stone; that the rust of these ends 
 nixing with the dust rubbed from the stone, and with the oil used to facilitate its 
 notion, formed together a hard and heavy mass, which, when detached, was found to 
 yossess all the properties of the magnet. It is believed that this bell had existed in 
 ‘hat situation 400 years. 
 
 _ Tt will be found, on trial, that iron palisades, which have been some time up, 
 ire strongly magnetical; the upper end being a south pole, and the lower a north 
 role. 
 
 | Gilbert observes also, that if a bar of iron, placed north and south, be brought to 
 red heat in a forge, and be then beat on the anvil in the same position, it will ac- 
 juire the magnetic virtue; and that if this virtue be not immediately sensible, it 
 will become so by repeating the operation. But it is to be observed, that for this pur- 
 yose the length of the iron must be 100 or 150 times its diameter. The case is the 
 jame with a bar of iron, if it be heated, and then cooled in the direction of the 
 ‘neridian. 
 
 | The following conjecture of this philosopher, however, has not been verified. He 
 ‘ays that if a spherical form be given to a magnet, and if its two poles be at the 
 »xtremities of a diameter, this spherical magnet, when placed in complete equili- 
 orium, and suspended on its poles, will turn round its axis in twenty-four hours; for 
 is the earth, adds he, is but a large magnet, it must have a similar motion. This 
 vould have been a pretty strong proof of the motion of the earth, at least round its 
 ixis; but M. Petit, an industrious philosopher of the last century, having taken tlie 
 ‘rouble to make the experiment proposed by Gilbert, the small magnetic globe re- 
 nained perfectly motionless. This does not however prevent the motion of the 
 varth from being certain, and it may even be considered as a large magnet, though 
 father Grandamy concluded, from the failure of Gilbert’s experiment, that the earth 
 vas motionless. 
 
 _ We add another method of making artificial magnets. Tix the needle to be mag- 
 jetised horizontally in the magnetic meridian; and apply to its middle a long iron 
 yar, as a poker, held vertically. Immediately opposite, at the lower side of the 
 teedle, apply the upper end of another similar poker, or iron bar. Then, keeping 
 he bars vertical, draw them towards the opposite ends of the needle, drawing the 
 ‘\pper bar towards the end intended for the south pole. Separate the bars. from the 
 teedle; and first removing them to a distance, bring them again, as before, to the 
 niddle of the needle, and, as before, slide them in Opposite directions towards the 
 ‘nds; and repeat the operation several times. By this simple process a small needle 
 “aay be magnetised to saturation. 
 
 | To preserve bar magnets, keep them in pairs, laid parallel to each other, with their 
 oles turned in opposite directions; and connect each adjoining pole with a piece 
 'f soft iron. 
 
 _ When the magnet is made of the horse-shoe form, the poles ought also to be joined 
 y @ piece of soft iron. 
 
 But magnetism may be imparted by simple percussion. Mr. Scoresby found that 
 Tectangular steel bar acquired a feeble magnetism, by being hammered vertically, 
 1¢ lower end resting on stone or pewter; but that it received a considerable acces- 
 on of power, when it was similarly hammered while it was placed on a parlour 
 oker, kept also in a vertical position. “a i poker itself became strongly magnetic ; 
 
722 MAGNETISM, 
 
 and in this state exerted upon the bara much more powerful influence than th 
 earth alone could have done. 
 
 ARTICLE III. 
 Of the Direction of the Magnet; and of its Declination and Variation. 
 
 EXPERIMENT xX. 
 To find the Direction of the Magnet. 
 
 Having found the poles of a magnet, if you place it on a small bit of cork, ant 
 make the cork to float on water, it will always place itself in one direction. 
 
 The case will be the same with a magnetic needle made to float on water by thy, 
 same means, or suspended on a fine pivot, so as to be at full liberty to move: i 
 will always assume the same direction. [ 
 
 To make a needle place itself north and south, it is not even necessary that a 
 should be magnetised. When exceedingly light, and perfectly free to move, it spon. 
 taneously assumes that direction. 
 
 Ifa very slender common needle be made to float on the surface of water in ¢ 
 state of perfect rest; at the end of some hours it will be found in the same direc. 
 tion as that suddenly assumed by the magnetic needle. 
 
 The direction, according to which a needle, whether magnetised or not, arranges’ 
 itself, is called the magnetic meridian, and must be carefully distinguished from the 
 terrestrial or true meridian ; for we shall soon shew that, in general, they form an angle 
 with each other. Philosophers agree, almost unanimously, in thinking that this 
 property of the magnet is produced by a current of a particular fluid, surrounding 
 the earth, and which, passing through the magnet lengthwise, or from one pole tc 
 the other, makes it assume its proper direction. 
 
 What is very singular is, that not only the magnetic meridian, in almost all places, 
 of the earth, is different from the terrestrial meridian ; declining sometimes to the 
 east and sometimes to the west; but that this declination varies annually, as is proms 
 by the following experiments. 
 
 EXPERIMENT XI. 
 
 i 
 
 If a magnetic needle be suspended on a pivot, in the direction of a meridian line, 
 traced out with great care, and at a distance from any iron, you will generally find, 
 that the direction it takes will form an angle with the meridian. In 1770, for ex. 
 ample, it wasat Paris 19° 55’ west. 
 
 If the experiment be repeated some years after, it will be found that this RA 
 is not the same; but that it is increased or decreased. In 1750, for example, it 
 was at Paris 17° 15’ west; in 1760 it was observed to be 18° 45’; in 1770, 19° 55’ 
 or even 20 degrees and some minutes. And, at London, in 1800, it was about 998 
 30’; in 1818, about 24° 30’; and in 1836, about 24°. 
 
 It appears also, that the annual change in the variation has been gradually dimi 
 nishing ; from 1622 to 1692, the annual change at London was about 10’; from 1723 
 to 1753, about 8’; from 1787 to 1795, about 5’; in 1818 it was reduced to Zero, ay 
 at London at present the variation is decidedly decreasing. 
 
 t 
 
 Remark.—In the greater part of our continent, as well as in all North America, 
 except that part of it which is nearest the Gulf of Mexico, the declination is at 
 present west, and goes on continually increasing. In all North America and all. 
 the Gulf of Mexico, as well as part of the Pacific Ocean, between the tropics. 
 and on the southern coast, the declination is east, and goes on continually decreas- 
 ing. | 
 The celebrated Dr. Halley, having taken the trouole to collect a prodigious 
 
 | 
 
 - 
 
a . “7 - 
 hs 
 5 4 “ 
 
 5 
 
 = 
 
 i 
 | 
 ' PROPERTIES OF THE MAGNET. 120 
 
 number of observations, made by different navigators, published in 1700 a very 
 curious chart, in which he connected, by lines, all those places of the earth where 
 the declination of the magnetic needle was the same. It is there seen, for example, 
 that the line on which the magnetic needle in 1700 had no declination, divided nearly 
 the southern part of the Atlantic Ocean, and cut the equator towards the first 
 dezree of longitude, or at its intersection with the first meridian; it thence pro- 
 ceeded in a curved line to New England, and, traversing New Mexico and California, 
 stretched to the north of the Pacific Ocean. In all probability it reached Asia, 
 ‘then passed to the north of Tartary, and proceeding through China, traversed New 
 ‘Holland. On the south and west of this line, the declination was east; on the” 
 ‘north and east it was west. 
 
 _ By other observations, made at a later period, it appeared that this.line was dis- 
 placed; and that it had in some manner a motion towards the south-west, changing 
 ‘a little its form. According to those observations collected by Messrs. Mountain 
 ‘and Dodson of the Royal Society, it traversed, in 1744, the middle of the Atlantic 
 Ocean ; nearly intersected the equator towards the twelfth degree of longitude, to 
 the east of the first meridian; proceeded thence to the middle of Florida; and 
 |passing nearly along the coast of Louisiana, it traversed Old Mexico, from which it 
 extended to the point of California, then to the north of the Pacific Ocean, and in- 
 \tersected the first meridian towards the 44th degree of north latitude; from which 
 ‘it turned southwards, and traversed Japan, the largest of the Philippines, the king- 
 ‘doms of Pegu and Arracan, and formed a point on the east near the island of Cey- 
 ‘lon. It then returned, and traversing the Moluccas, proceeded in a curved line 
 ‘towards the south pole, leaving New Holland on the west. Such was the position 
 ‘of this line in 1744, and thence we may determine nearly its present position. 
 
 Dr. Halley’s chart exhibited also the line which joined all the points where the 
 ideclination was 5° to the east or the west; those where it was 10°, 15°, &c. It is 
 ‘observed, at present, that they have all had a motion nearly similar to that of the | 
 ‘line without declination. . 
 
 Dr. Halley’s object, in this painful labour, was not mere curiosity: he intended 
 these charts to be employed in determining the longitude at sea. If an accurate 
 chart of these lines of declination were indeed constructed, it is evident that, by 
 ‘observing the latitude and the real declination of the compass, the precise point of 
 ‘the earth, where the observation was made, would be determined. Let us suppose, 
 /for example, that the declination has been observed in the Atlantic Ocean, to be 
 7} west, the latitude being 32 north. It is evident, in this case, that the ship’s 
 ‘place will be the point where the parallel of 32° north, intersects the line of 73° 
 declination. Nothing then would remain, but to improve the means of determining 
 ‘the declination with great exactness, which is a thing not impossible. 
 
 It isto be regretted that we have no old observations of the declination of the 
 magnetic needle. The reason of this no doubt is, that the declination was not 
 properly ascertained by philosophers, till towards the end of the sixteenth century. 
 ‘It is seen however by the observations which have been made, that at Paris, at 
 London, and in great part of Germany, the declination formerly was east; for in 
 1580 it was found at Paris to be 11° 30’ east. After that time it decreased till 
 1666, when -it vanished entirely ; it then became west, increasing continually in that: 
 direction; for in 1670 it was observed to be 1* 30’, in 1680 to be 2° 40’, in 1701 to 
 ‘be 8° 25’; in 1770 it was observed to be within a few minutes of 20°. The Royal 
 Society of London recorded their annual observations of the magnetic needle for 
 many years; and it is a great pity that they have of late discontinued such useful 
 observations. 
 
 But what is the cause of the magnetic declination? On this subject we shall offer 
 ‘the following conjectures. Messrs. de la’ Hire, senior and junior, made a curious 
 
 3A 2 
 
7 24 MAGNETISM. . 
 
 a ae, 
 
 experiment, which may serve to throw some light on the cause of this phenomenon, | 
 They took a very large magnet, and having given it a globular form as nearly as possi. 
 ble, they sought for its poles, which were found exactly at the extremities of a dia-| 
 meter; and then they traced out on it its equator, and twelve meridians. On thig| 
 magnetic globe, which was about a foot in diameter, and which weighed nearly a 
 
 hundred pounds, they applied a magnetic needle, and observed that there were 
 places where it declined towards the west, and others where it had no declination, | 
 and which formed one or two continued lines on the surface, as Dr. Halley had 
 determined on the surface of the earth, though of a form absolutely different. 
 
 It is more probable, says the historian of the Academy, that the cause of the 
 declination observed on the magnetic globe, was merely the inequality of its con. 
 texture, and of the magnetic force of its different parts. There is reason also to 
 conjecture that the earth, being a large magnet, or at least a globe containing in its 
 bosom large magnetic masses, it is the unequal distribution of these masses that_ 
 occasions on its surface the variety of the direction of the magnetic needle. But 
 there is this difference, that in the bowels of the earth new masses are continually — 
 generated ; whereas the magnet of Messrs. de la Hire experienced nothing of the 
 kind. Hence it happens that, on the surface of the earth, the direction of the magnet | 
 is variable; while on the surface of the magnetic globe, it was necessary that it 
 should be constant. | 
 
 It must however be allowed, that in this explanation it is difficult to assign & 
 reason why, for two centuries at least, the line without declination has been seen 
 to move constantly from east to west. Effects arising from causes so variable as the 
 destruction and generation of masses in the bosom of the earth, ought to experience | 
 greater irregularities, and the progress of se magnetic needle ought to be sometimes | 
 east and sometimes west. 
 
 Dr. Halley proposed a physical hypothesis, to account for the variety in the mag- | 
 netic declination. He supposed two fixed magnetic poles, and two moveable, in | 
 certain positions. But this hypothesis has been simplified by Albert Euler, in a 
 curious memoir, which may be seen in the Transactions of the Academy of Berlin | 
 for the year 1757. Euler supposes only two magnetic poles, one at 14° 53’ from the | 
 north pole of the earth, and the other 29° 23’ from the south pole. The meridian 
 in which the former is situated passes through the 258th degree of longitude, and 
 that of the second through the 303rd. He then assumes, as a principle, that the 
 magnetic needle always ranges itself in the plane passing through the two magnetie | 
 poles and the place of observation ; and he determines, by calculation, the inclina- 
 tion of that plane to the meridian, in the different places of the earth. By means 
 ‘of these data, calculation gives, with great exactness, the quantity of the declination — 
 observed of late years, and the position of the lines of declination, as they were 
 found by Messrs. Mountain and Dodson, for 1774, at least in the Atlantic Ocean; 
 for Albert Euler is obliged to consider as false the position given to the line of de- 
 clination, in the north part of the Pacific Ocean, by those members of the Royal 
 Society, and what he says on this subject is highly probable. 
 
 It may be easily conceived, that by making these poles to vary, the lines of deeli- 
 nation will vary also, and that according as they approach or recede from each other, — 
 they may change their form, as has indeed been observed. j 
 
 While engaged in investigations connected with the attration of iron on shipboard, 
 Mr. Barlow, of the Royal Military Academy at Woolwich, found that the attractive — 
 force of iron on the magnetised needle, depended not upon the mass, but upon the 
 surface ; a shell and a solid ball of the same diameter giving precisely the same — 
 results. He found however that a certain thickness of the shell is requisite to a full 
 development of the attractive force ; and that when the thickness was less than the 
 thirtieth of an inch, the effect was sensibly weakened. 
 
 “i 
 +e 
 
- Oe ee 
 
 THE MAGNETIC NEEDLE, 725 
 
 He found also that a ball, in the neighbourhood of a compass, produces no disturb- 
 ance in the needle, when the centre of the needle is in any part of a plane passing 
 through the centre of the ball, and perpendicular to the direction of the dipping needle 
 at the place where the experiment is made. This plane forms what may be called, 
 or rather it is parallel to what may be called, the magnetic equator. 
 Another plane of neutrality is a vertical plane in the line of the dip, or the mag- 
 ‘ netic meridian. 
 It is not however the whole of the attractive force of the ball that vanishes in 
 these planes, but the part only which occasions deviations in the natural position 
 ; 
 
 of the needle. 
 
 Magnetism of non-ferruginous bodies. 
 
 _ Nickel and cobalt have occasionally been found magnetic, and sometimes to exhibit 
 
 polarity. Brass also, under certain circumstances, has been observed to be magnetic, 
 
 _ especially after it has been hammered. 
 
 Copper and zinc, the constituents of brass, have never been observed to be mag- 
 
 netic in their separate state. 
 
 _- The magnetism of brass has been found sometimes to interfere with the needle in 
 
 _ compasses. 
 
 __ Carbon phosphorous, or sulphur combined with iron, increases its susceptibility for . 
 
 magnetism ; but there is a limit beyond which an excess of these substances renders 
 
 the compound totally insusceptible of magnetism. On the other hand a small 
 ‘quantity of arsenic has been found to deprive a strongly magnetic mass of nickel 
 
 of the whole of its magnetism. 
 
 Minerals of various kinds, and even some of the precious stones, exert a feeble 
 influence on the magnetic needle; and late inquiries appear to have established the 
 fact that all bodies whatever are, in a greater or less degree, susceptible of mag- 
 netism. 
 
 Whether this susceptibility arises from iron being a constituent of all bodies, or 
 from the magnetic property being inherent in all, but more decidedly developed in 
 one class of bodies than in others, are questions which have as yet received no satis- 
 factory solution. — 
 
 _ In the year 1824, M. Arago shewed that if a plate of copper, or of any other 
 substance, be placed immediately under a magnetic needle, it diminishes sensibly the 
 ‘extent of the needle’s oscillations, but does not affect their individual duration ; 
 but the needle is brought to rest sooner than when no such substance is placed 
 under it. 
 
 _ When, instead of the needle being made to oscillate, the plate below it is made to 
 Tevolve with a certain velocity; the needle is found to deviate from its natural 
 position in the magnetic meridian; the deviation increasing as the rotation of the 
 ‘plate becomes more rapid; and if the plate revolve with sufficient rapidity, 
 ‘the needle will at length revolve also, and always in. the same direction with the 
 plate. 
 
 Conversely, when a circular plate of copper, balanced on a point at its centre, was 
 placed under a strong bar magnet, to which a rapid rotatory motion was given; the 
 
 copper soon began to turn in the same direction, and acquired by degrees a very rapid 
 plate velocity. 
 
 | This tract of investigation has been pursued by many scientific persons, both in 
 _this country and on the continent; and the results are in a high degree interesting. 
 
 _ Mr. Canton, a member of the Royal Society of London, discovered some years 
 ago a new motion of the magnetic needle, which is founded on the following ex-. 
 periment. 
 
726 . MAGNETISM. 
 
 EXPERIMENT XII. 
 Diurnal Variation of the Magnet. 
 
 Provide a pretty large magnetic needle, 12 or 15 inches in length, and nicely 
 suspended. It must be surrounded by a circle, the centre of which is the point 
 of suspension, divided into degrees, and haif degrees, or quarters, at least in that 
 part of its circumference which is opposite to the point of the needle. The whole 
 apparatus must be covered in such a manner, as to prevent it from being subject to 
 any impression from the air. 
 
 If this needle be observed at different hours of the day, it will be found that it is 
 scarcely ever at rest. According to Mr. Canton, the declination will be greatest 
 in the morning, and least in the evening: about noon it will be a mean between these 
 two extremes. He assigns also a very probable reason, which is as follows: 
 
 It is a fact proved by experience, that a magnet, when heated, loses a little of its 
 force. But as the eastern parts of the earth have noon when the sun rises to us, it 
 is at that time, or nearly so, that they are most heated. The magnetic needle, the 
 direction of which is, in all probability, an effect compounded of the attraction of all 
 the magnetic parts of the earth, will at sun rise be a little less impelled towards the 
 east, than if the sun were not on that side; consequently it will yield to the action 
 of the western parts, and will turn a little more towards that side. Mr, Canton 
 even renders this explanation sensible, by means of two magnets,.each of which is 
 heated alternately. 
 
 But, whatever truth there may be in this explanation, the phenomenon is now 
 
 well known: and meteorologists do not fail to observe, at different times, the 
 
 declination of the magnetic needle, which often varies between morning and night, 
 20’ and more.* . 
 
 Mr. Barlow, of Woolwich, has devised a mode of rendering the diurnal oscilla-- 
 
 tions much more perceptible, by diminishing the ordinary directive power of the 
 needle, through the influence of two magnets, so placed with respect to the 
 needle as to counteract or neutralize the terrestrial action. The effort of the 
 ordinary or overmastering action of the earth upon the needle being thus 
 
 removed, Mr. Barlow anticipated that the cause, whatever it might be, which pro- 
 
 duced the daily variation, would exhibit effects more perceptibly; and these antici- 
 pations have been amply realised by the success of his own experiments, and those of 
 
 his colleague, Mr. Christie, as detailed in several papers in the Philosophical ‘Trans- 
 
 actions- 
 
 The general result of the experiments of the latter is, that the deviation of the — 
 horizontal needle from its mean position is easterly during the forenoon; greatest — 
 
 about eight o’clock; thence returning quickly to its mean position between nine and 
 
 ten o’clock; after which it became westerly; at first increasing rapidly, so as to — 
 
 reach its minimum about one o’clock in the afternoon, and then slowly receding 
 
 during the rest of the day it arrived at its mean place about ten o’clock at : 
 
 night. 
 
 Similar changes in diurnal intensity have been observed both by Hansteen at Chris- | 
 tiana, and Mr. Christie at Woolwich; and the results, on the whole, accord very © 
 well, though deduced by totally different methods. Mr. Christie says that he found the | 
 terrestrial magnetic intensity the least between ten and eleven o’clock in the morn- 
 
 ing ; between which hours, in this country, the sun passes the magnetic meridian. It 
 increases from this time until nine or ten o’clock in the evening, when it again 
 ~ decreases, till it obtains its minimum, as already stated. It appears, too, that these 
 
 * See “ Traité de Météorologie du P. Cotte.” 
 
DIP OF THE MAGNETIC NEEDLE. Cee 
 
 variations of magnetic position are in some degree connected with the diurnal changes 
 of temperature. 
 
 The mean diurnal changes of variation have been observed to be greatest in June, 
 and least in December. s 
 
 SECTION III. 
 Of the inclination or dip of the Magnetie Needle. 
 
 EXPERIMENT XIII. 
 
 | To observe the inclination of the Magnet. 
 
 _ If the needle of a compass, not yet magnetised, be placed in perfect equilibrium 
 on its pivot, so as to remain parallel to the horizon; and then be touched with a mag 
 net, it will lose this equilibrium, and will dip its northern extremity below the horizon, 
 . This experiment is well known to those who construct compasses; for after the 
 needle is magnetised, they are obliged to file the heavier end till it be in equilibrium 
 with the other. ‘The same effect might be produced by loading the other end with 
 1 small weight, and it would even be of advantage if this weight were moveable ; 
 for as the inclination is variable, different forces are required to form an equilibrium 
 to the effort made by the needle to dip. It is therefore necessary to add a small 
 weight to one of the ends of the needle, according to the different positions of the 
 ship, in order that it may remain perfectly horizontal. 
 
 EXPERIMENT XIV. 
 To observe the inclination of the Magnetic Needle. 
 
 Provide a magnetic needle, made of very straight steel wire, terminating in a point 
 it each extremity. The middle of it must be flattened, and formed into a small 
 tircle, having its centre ina line with the two points of the needle. A piece of very 
 ine steel wire passing through this circle, in a perpendicular direction, must serve it 
 is a pivot; so that when suspended horizontally in two holes, made in two vertical 
 »lates of brass, it may be indifferent to every position, and remain in equilibrium in 
 my situation whatever. These two plates must be affixed to the edge of a brass band, 
 Jent into a circular form, and of a diameter somewhat greater than the length of the 
 ieedle, the pivots of which will be in the centre. This brass circle must be sus- 
 vended in a ring, and one of its diameters must be placed ina vertical direction. 
 Divide the inside of it into degrees, and quarters of a degree, if possible ; but in 
 juch a manner that the division beginning with zero, at the extremities of the hori- 
 ‘ontal diameter, may end with 90 degrees at the extremities of the vertical diameter. 
 Phe position of this diameter may be ascertained by means of a wire and plummet, 
 uspended from its upper extremity, and which must pass through the lower 
 ‘xtremity, that it may be in its true position. 
 , Provide also a wooden stand in the form of an oblong parallelopipedon, in the. 
 ipper part of which let there be a circular hollow proper for lodging the instrument 
 n the direction of its length. Inthe last place, there must be a small wedge, for the 
 vurpose of being placed under this stand, till the plane of the instrument, or that 
 assed over by the needle in its motion, shall be exactly vertical. 
 _ When the needle has been magnetised, apply to the two sides of the instrument, 
 n grooves made for that purpose, two circular pieces of glass, to preserve the 
 \eedle and its pivots from the contact of the exterior air, and from moisture which 
 3 hurtful to magnetism. : 
 _ By the description of this instrument, it may be readily seen, that it’ must be dis- 
 osed in a vertical situation, either by suspending it or placing it on its supporter, 
 vhich may be easily done by means of the wire and plummet. . 
 
rs 
 Fi 
 
 728 - MAGNETISM. 
 
 The plane of the instrument also, or that passed over by the needle, must be ir 
 the plane of the magnetic meridian. For this purpose, lay the instrument flat ona, 
 horizontal table ; the needle, when it stops, will indicate the magnetic meridian ; then 
 draw on the table a line in that direction, and make the long side of the supporte: 
 coincide with it. By means of the small wedge and the plumb line, it may then bi 
 adjusted in the proper position. 'The needle, after very long vibrations, will at/ 
 length stop, and indicate by its point the number of degrees it is distant from the 
 plane of the horizon, which will give the inclination or dip desired. 
 By these meansitis found that the inclination at Paris is at present 72 degrees. 
 
 Remarks.—I. Though the construction of such an instrument does not seem diffi- 
 cult, it is shewn by experience that it requires a peculiar kind of skill and dexterity 
 which few possess. Unless the instrument indeed be perfect, the magnetic needle 
 does not recover its position when displaced, or when the instrument is turned in ¢ 
 contrary direction ; that is, in such a manner that the plane which looked towards 
 the east may look towards the west. 
 
 _ [1.—The inclination of the magnetic needle is no less variable than its declination, 
 It is observed that-it is different in different parts of the earth ; but itis erroneous 
 to suppose, as some philosophers in the last century did, that it has any relation 
 to the latitude. It is observed, for example, that it is at Paris at present 72° 25! 
 North; at Lima, about 18° South ; at Quito, about 15°S. ; at Buenos- Ayres, about 
 601° S.; at the Isle of France, 523° North. | 
 
 This is sufficient to destroy the idea that it has the least relation to the latitude. 
 
 The dip, like the variation of the magnetic needle, has undergone changes, sitce it 
 was first observed. In London, the dip was 73° 20’ in 1680; in 1723, it was 749 42’; 
 since which time it appears to have been progressively, though by no means reg wlan 
 diminishing. In 1773, it was 72° 19’; in 1786, 72° 8’; in 1805, 70° 21’; in 1818, 
 70° 34’; in 1821, 70° 3’; in 1828, 69° 47’; and in 1830, 69° 38’. At Paris, in 1814, 
 M. Bouvard found the dip at the Observatory, 68° 30’; and in June 1829, M. Ars 
 found it 67° 41’.. Captain Sabine, by comparing the present dip with that observed for, 
 the last fifty years, concludes that the mean annual diminution may be taken at about 
 3’; and Mr. Barlow finds that both the dip and variation accord nearly with what 
 would result from a uniform revolution of the magnetic pole round the pole of 
 the earth—the whole period of revolution being about eight hundred and fifty 
 years. 
 
 It would appear, then, both from theory and recent observations, that at present 
 the dip is changing more rapidly than the variation; and Mr. Barlow’s suggestion 
 would lead to the expectation that they will continue to decrease together till about. 
 the year 2085, when the longitude of the magnetic pole will then be 180°, the. 
 variation nothing, and the dip only 56°, which will be its minimum. They will "¢hen’ 
 increase together for the next 260 years, when the needle will have attained its, 
 greatest easterly variation, and will then return northerly, the dip increasing and the | 
 variation decreasing till about 2510, when the magnetic pole will be on the meridian 
 of London, the variation nothing, and the dip 77° 43’. But the’ whole of these 
 anticipations depend on whether the hypothesis on which they are based has any 
 foundation in nature ; and this can be determined only by future experiments. 
 
 As observations of the inclination are considered to be of no utility in navigation, | 
 it needs excite no astonishment that we have so few. Besides, it is much more, 
 difficult ‘at sea to observe the inclination than the declination, on account of the. 
 rolling of the ship. Father Feuillée however made a considerable number of them, | 
 during his voyages in Europe and America; but, according to all appearance, they | 
 are only within a few degrees of the truth. It is nevertheless to be wished that | 
 
; DIP OF THE NEEDLE, 729 
 
 these observations were more numerous; for though, on the first view, they do not 
 ‘seem of great utility, they may become so hereafter. Let us not cease to accumu- 
 Jate facts, though in appearance useless. Some unexpected light often arises from 
 ‘an observation long considered to be frivolous and unimportant. 
 
 II. We may remark also, that the motion of the magnetic needle experiences 
 -yery singular variations on the approach, or by the effect, of igneous meteors. A 
 needle has been deprived of its magnetic property by thunder, or even magnetised in 
 acontrary direction, The Aurora Borealis seems also to have a very sensible action 
 onthe magnetic needle; but for farther information cn this subject, we must refer 
 to Father Cotte’s “ Traité de Météorologie.” 
 
 ARTICLE IV. 
 
 Of certain means proposed for freeing the magnetic needle from its declination, or for 
 making compasses without declination. 
 It would be so great an advantage to have compasses which should point exactly 
 
 _to the north, that the attempts made to devise combinations to destroy the declina- 
 
 -tion of the needle need excite no surprise; but unfortunately these attempts have 
 
 hitherto been fruitless, and in our own opinion will always be so. They however 
 
 ideserve to be known, were it only to guard our readers against the illusions of those 
 
 who imagine that they have solved this problem. 
 
 One of these inventions is described by Muschenbroek. It consists in combining, 
 for a determinate place, two needles of equal force, in such a manner, that the one 
 may decline on the one side, and the other on the other side, from the magnetic 
 ‘meridian, by a quantity equal to the declination. Thus, one of them will decline 
 double, and the other will be exactly in the meridian. Let us suppose, for example, 
 that the declination is 20 degrees to the west, as it was at Paris in 1770. If two 
 magnetic needles of the same force be suspended on the same pivot, forming together 
 an angle of 40 degrees, it is evident that neither of them being able to place itself 
 ‘in the magnetic meridian, they will equally decline from it: thus the one will decline 
 20 degrees to the west of that meridian, or 40 degrees from that of the earth; con- 
 sequently, the other needle will necessarily be in the meridian, and will have no 
 declination. 
 
 It is astonishing that any one should imagine that a combination of magnetic 
 ‘needles, capable of making one of them coincide with the terrestrial meridian, could 
 be obtained in this manner. It may be readily seen, that these two needles, if of 
 equal force, will always arrange themselves in such a manner, that the magnetic 
 ‘meridian will divide into two equal parts the angle comprehended between them. 
 ‘Thus, if we suppose that the magnetic meridian, instead of declining 20 degrees 
 from the terrestrial meridian, declines only 10 degrees to the west, one of the needles 
 will be carried 20 degrees more to the west; consequently will have 30 degrees 
 of declination. At the same time therefore the other needle will be carried 10 de- 
 
 grees from the meridian towards the east. 
 
 The last translator of Pliny has given a method, nearly similar, for annihilating 
 the declination, and which differs in nothing from that of Muschenbroek, except that 
 ‘one of the needles must be larger than the other. But Muschenbroek had before 
 proposed and analysed this combination of two unequal needles, and it appeared to 
 him as unlikely to succeed as the other. It is opposed indeed by the same, or by 
 similar reasons, and nothing can rest on slighter grounds, than the physical theory 
 by which the author alluded to appears to have been guided ; for he seems to think, 
 that the cause of the declination, is a sort of weakness in the needle, which does not 
 permit it to reach the north. ‘This is an idea not only void of foundation, but even 
 incompatible with the most probable theory of the magnetic motion ;' for as the 
 
 —_ 
 
730 MAGNETISM. 
 
 mibuetid needle, which during the first half of the 17th century declined to the oan 
 afterwards approached the north, and passed it, to proceed to the west, it would be’ 
 necessary to say that it was sick; since it was cured about 1660, and that it after. | 
 wards got diseased in a contrary direction. We cannot therefore sufficiently wonder | 
 
 at the precipitation of some journalists and some authors, who hastened to announce, - 
 with the greatest eulogiums, this pretended discovery, as likely to change the face . 
 of navigation. Unfortunately, nothing could be more chimerical; and a better: 
 acquaintance with the magnetic phenomena would have preserved both from this ‘ 
 
 error. 
 
 We have seen formerly at Paris a Genoese pilot, named Mandillo, who pretended | 
 to have found another combination of magnetic needles, proper for correcting the | 
 
 declination. He placed two needles of equal force above each other, but in sucha 
 
 manner that each of them had full liberty to move; he then brought them together, 
 
 for Paris we shall suppose, so that their deviation was double the inclination 
 observed ; for in this position they would each diverge by the effect of the repulsion ‘ 
 of their Piles or points of the same denomination, and so much the more as they - 
 were brought nearer to each other. By these means, one of the needles, as in the 
 preceding process, was carried to the meridian. But the sieur Mandillo pretended | 
 that this would every where be the case, which is evidently false ; for the deviation 
 
 of the two needles being the effect of the repulsion of the two poles of the same | 
 
 name, this repulsion, and consequently the deviation, will be the same, whatever be | 
 the angle of the magnetic meridian with the terrestrial meridian; otherwise we | 
 must suppose that this repulsion decreases at the same time as the declination, which © 
 is impossible. This objection we mentioned to Mandillo; but to no purpose. A . 
 
 man who imagines he has found the means of correcting the declination of the © 
 magnetic needle, or has discovered the solution of the problem of the longitude,* is « 
 
 as obstinate in his opinions, as he who thinks he has found the quadrature of the 
 
 circle. 
 
 We shall here mention an idea of M. de la Hire on this subject. It was founded — 
 on a belief of his having discovered that the poles of a natural magnet had changed | 
 their place, as the magnetic poles of the earth had done at the same time. He 
 thence conceived the idea of magnetising steel rings, presuming that their poles 
 
 \ 
 j 
 
 : 
 ! 
 I 
 
 would change in the same manner. But it may be readily seen that in this case, the ‘ 
 line marked originally north and south on the ring would remain motionless, and |. 
 would always indicate the real north. This principle however has been found to be : 
 false, and even if true, the consequence deduced from it by M. de la Hire did not ° 
 
 necessarily follow. 
 
 ARTICLE V. 
 Of certain Tricks which may be performed by means of the Magnet. 
 For some years past, the properties of the magnet have been employed to perform 
 
 several tricks, which excited a considerable degree of astonishment in those who first _ 
 
 beheld them. No means indeed more secret, and at the same time more proper for | 
 
 action, could be employed than magnetism ; ; since its influence is stopped by no body | 
 
 with which we are acquainted, except iron. 'This idea was first conceived by the - 
 
 celebrated Comus, who varied, in a singular manner, the different tricks performed 
 by this agent; so that all Paris flocked, with the utmost eagerness, to the places 
 where they were exhibited. He was admired by the ignorant, who considered him 
 
 as a sorcerer, while the learned endeavoured to discover the artifice, which however — 
 
 was a profound secret, as long as no one suspected magnetism to be the principal 
 cause of it. 
 
 * In the time of Montucla, deemed the extreme of absurdity. 
 
 { 
 
TRICKS WITH THE MAGNET. 731 
 
 We shall here endeavour to give an.idea of some of these tricks, as they will 
 ‘orm a fund of rational amusement to those who know how to perform them. 
 
 SECTION I. 
 | Construction of a Magic Telescope. 
 _ Those who exhibit these tricks often employ a pretended magic telescope, by 
 neans of which one can see, it is said, through opake bodies. It is nothing else 
 ban an instrument in the form of a telescope, at the bottom of which, that is 
 cowards the object glass, there is a magnetic needle, which assumes its proper 
 lirection, when the telescope is placed upon the side which that object glass 
 jorms. 
 To construct this telescope, provide a turned tube of ivory, wider towards the 
 md where the object glass is placed; but the ivory must be of sufficient thinness 
 admit the light through to the inside, The narrow end is furnished with an eye- 
 ‘ass, which serves to shew more distinctly the inside of it. The other end also is 
 urnished with a glass, which has the appearance only of an object glass, the pos- } 
 erior surface of it being opake, so as to serve for the base or bottom of a sort 
 f compass or magnetic needle, which turns on a pivot fixed in its centre. When 
 he telescope rests on the end containing the object glass, this needle assumes a 
 iorizontal position, and points towards the north, or towards a magnetic needle in 
 he neighbourhood. It is necessary also to have a real telescope, similar in appear- 
 mee to the other, in order that it may be shewn instead of it, which may be done by 
 lexterously substituting the one for the other. 
 , When you wish to employ the pretended magic telescope, place it with the 
 ybject glass downwards upon any thing you intend. to examine, and if there be a 
 nagnet, or piece of magnetised iron below it, the needle will turn to that side. 
 
 SECTION II. 
 
 Several figures being given, which a person has arranged close to each other in a boz, 
 | to tell through the lid or cover what number they form. 
 
 If you are desirous of employing the ten ciphers, take ten small squares, of an 
 ach and a half on each side, and on the upper face of each make a groove; but let 
 hese grooves be in different directions; that is to say, the first intended for the 
 jmber 1 must proceed directly from the top to the bottom; the second must 
 -eviate to the right, so as to form an angle equal to a tenth part of the circum- 
 erence; the third an angle of two-tenths; and so of the rest; which will give ten 
 ifferent positions. Then introduce into these grooves small bars of steel, well 
 1agnetised, taking care to turn their north poles to the proper direction; cover these 
 rooves and the face of the squares with strong paper, in order to conceal the bars. 
 fou must also provide a narrow box, capable of containing in its breadth one 
 £ these squares, and of such a length that they can all be arranged in it. 
 | Then desire a person in your absence to take several of these squares, and arrange 
 hem in the box in any manner, at pleasure, so as to form any number whatever, 
 nd to shut the box; after which you are to tell the number which has been 
 -ormed. 
 
 Deposit your pretended telescope on the place of the first square, that is on the 
 oft, if the figure below it be unity ; the needle will turn in such a manner that the 
 orth point or pole will be before you. If the figure be 4, it will turn to the fourth 
 ivision of the circle, which is equally divided into ten parts; and so of the rest. 
 t will thence be easy to discover the figure in each place, and consequently to 
 ell it. 
 
 A word written in secret, with given characters, may be discovered in the same 
 hanner; also an anagram, formed of a proposed word, as Roma, which gives amor, 
 
+8 
 P| 
 t 
 
 | 
 
 mora, orma, maro, &c.; or a question which has been selected from several persons! 
 and put into the box. In short, this trick may be varied in a great many ways) 
 exceedingly agreeable, but all depending on the same principle. 
 
 The box of metals, for example, is only a similar variation of the same trick 
 You put six plates of different metals in a box, and bid a person take any one 01 
 them, and put it into another box, and shut it. You may then easily tell which one 
 he has taken. These plates are of such a form, that they can occupy in the smal’ 
 wox only one position. Each of them, that of iron excepted, contains in its thick. 
 ness a magnetic bar, arranged in situations which are known, and these situations are 
 discovered by means of the pretended magic telescope; consequently the nature o} 
 the metal must be known. No magnetic bar is placed in the plate of iron, because 
 this would be useless; but one side of the plate may be magnetised, or if it be 
 not magnetised, the indeterminate direction of the needle will announce that it is 
 irou. 
 
 (32 MAGNETISM. 
 
 SECTION IIL , 
 
 The Learned Fly, or the Syren. 
 
 This trick is somewhat more complex than the preceding, and depends partly on 
 philosophical principles, and partly on a little deception. You must provide a table, 
 with a box sunk in its thickness, and the box must be furnished with a broad brim, 
 inscribed with numbers, the hours of the day, or answers to certain questions. You 
 then desire a person to point out a number, or to name any hour in the day, or to 
 ask what o’clock it is, or to select any one of certain questions written upon cards 
 which you present to him. A fly, a syren, or a swan, floating in water, indicates, 
 in their order, the figures of this number, or answers the question proposed. 
 
 All this is performed by means of a strongly magnetised bar, supported by a brass 
 circle, concealed in the rim of the bason, which contains the water. It is evident 
 that if the motion, necessary to point out the letters, or numbers, required for the 
 answer, can be given to this bar, the fly or syren, placed on a small boat containing 
 another magnetic bar, will proceed towards it, and appear to answer the question. 
 Such are the philosophical principles of this trick. ‘The deception is as follows. 
 
 The table, which is some inches in thickness, is hollow, and the cavity contains 
 a certain mechanism put in motion by a string, which, passing through the feet of 
 the table, traverses the floor, and is conveyed into a neighbouring apartment, sepa- 
 rated from that where the trick is exhibited, only by a very slight partition, This 
 string terminates at a sort of table, on which are marked the divisions of the bason ; 
 and the whole is combined in such a mahner, that when the end of the string is 
 brought opposite to a certain figure, such as 4 for example, the magnetic bar will 
 be under the 4, inscribed on the edge of the vessel. 
 
 When the syren then is desired to tell what o’clock it is, the person behind the 
 partition, and who hears the question, has nothing to do but to pull the string, and 
 to bring the end of it opposite to the required hour on the table, which is before 
 him. The magnetic bar will arrange itself below, and the tractable syren, beginning 
 to move, will go and point out the hour. 
 
 Ifa question has been selected, the person who exhibits the trick repeats it under 
 a pretence of interrogating the syren. The confederate, who hears it, causes the 
 magnetic bar to move to the answer. 
 
 It would not be difficult to establish between both a secret communication of such 
 a nature, that, without speaking, the syren should appear to guess the question, and 
 to give an answer to it. 
 
 The principle works on the magnet are, ‘A Treatise de Magnete,” by Gilbert, 
 an English philosopher, printed in 1633: it contains traces of that spirit of observa- 
 tion which has since caused philosophy to make so great a progress. The “ Ars 
 
 a 
 
| TRICKS WITH THE MAGNET. 733 
 
 ‘agnetica” of Kircher: thisis a kind of encyclopedia of everything written before 
 2 author's time on that subject, enlarged by a great many of his own ideas, the 
 ater part of which however display more imagination than judgment. The ‘*Mag- 
 jtologia” of Father Leothaud, 1668, in 4to: a work of very little importance. 
 ither Scarella’s treatise, entitled ‘‘ De Magnete,” in four volumes, quarto, printed 
 : Brescia, in 1759, may supply the place of all the preceding, as it contains a com- 
 ehensive account of everything useful or solid, said or written on the magnet, 
 ‘that period ; to which the author, a very enlightened philosopher, has added his 
 yn ideas. The small treatise on Artificial Magnets, translated from the English, 
 ith an historical preface by the translator, will make the reader fully acquainted 
 ith that part of the theory of the magnet; or, in want of it, recourse may be had 
 » the “‘ Memoire sur les Aimants Artificiels,” by M. Antheaume, which gained 
 je prize proposed by the academy of Petersburgh in 1758. Several papers also by 
 |. Dutour, inserted in the “ Mémoires presentés a l’Académie, par des Scavans 
 apgers, ” deserve to be known, and to be studied, by those who may be desirous 
 cultivating and enlarging this theory. 
 
 To the above may be added an excellent treatise on the subject by Dr. Roget, in 
 ie Library of Useful Knowledge, and a treatise on the subject by Sir David 
 rewster, in the Encyclopedia Britannica. 
 
brea i 
 es 
 
 t 
 734 ELECTRICITY. | 
 | 
 
 PART THIRTEENTH. 
 
 OF ELECTRICITY. | 
 
 ELEcTRICcITY is an almost inexhaustible source of singular and surprising phenomena, 
 which must excite the curiosity of the most indifferent observer of nature. What) 
 indeed can be more extraordinary, and at the same time more difficult to be reconciled) 
 with the known laws of natural philosophy, than to see mere friction excite, in cer-. 
 tain substances, the power of attracting and repelling such light bodies as are near 
 them ; to see this power communicated, by contact, to other bodies, and even to. 
 very great distances ; to see fire issue from a body in that state; and a thousand | 
 other phenomena, the enumeration of which would be too tedious? We shall men- } 
 tion only the famous experiment of Leyden, where a rank of persons, holding each | 
 other by the hands, or having a communication by means of an iron wire, or rod, i 
 suddenly receive from an invisible agent an internal commotion, which might even. 
 be so violent as to kill those who experience it. j 
 
 It must however be allowed, that the case has not yet been the same with’ 
 electricity as with magnetism. The latter, by the invention of the magnetic needle, | | 
 has served to render navigation more secure, and to discover the new world, a source | 
 of new riches, new wants, and of new evils to the old one. But electricity has | 
 not yet produced any thing of so much importance to mankind, and to the arts, if} 
 we except the analogy now fully proved between the electric fire and lightning: an } 
 analogy which has given rise to a pretty sure preservative from the effects of that 
 dreadful meteor; for in regard to the cures effected by electricity, it must be acknow- | 
 ledged that they are either rare, or not well ascertained. 
 
 But we must not treat all researches on this subject as useless; for when we con- | 
 sider the phenomena exhibited by electricity, we cannot help allowing that it is one. 
 of the most general and most powerful agents in nature. Is it possible to deny that 
 ‘the identity of the electric fire and lightning is a noble and grand discovery ? What 
 can we say of a multitude of other analogies observed between electricity and mag- 
 netism, the nervous fluid, the principle of vegetation, &c. They seem to promise a | 
 copious harvest to those who shall continue to cultivate this fertile field. 
 
 SECTION I. 
 
 Of the Nature of Electricity ; and the distinction between bodies Electric by friction | 
 or by communication. 
 
 a 
 
 Electricity is a property which certain bodies acquire by friction, that is to say 
 the power of attracting or repelling light bodies which may be near cue If you 
 rub, for example, a stick of Spanish wax, with your hand, or rather with a piece of 
 eloth, and then make the wax pass within a few lines of small bits of paper or straw, 
 you wil see them rush towards the wax and adhere to it, as if cemented, until the 
 virtue acquired by the friction is dissipated. The ancients bad observed that yellow | 
 amber, when rubbed in this manner, attracted light bodies: hence the name of | 
 
 q 
 
. NATURE OF ELECTRICITY. 736 
 
 electricity; for they called that substance electrum. But their observations went no 
 farther. 
 The moderns have found the same property in a great many other bodies; such 
 as grey amber,-and in generalin all resins, which ean bear a certain degree of friction 
 without becoming soft; as sulphur, wax, jet, glass, the diamond, crystal, the 
 greater part of the precious stones, silk, woollen, the hair of animals, and very 
 dry wood. 
 
 In regard to bodies which do not acquire electricity by friction, it has been ob- 
 _ served that they can acquire it by communication ; that is to say by contact, or by 
 _ being brought very near to those of the first species; and that they can transmit it, 
 _ by the same means, to other bodies of the same nature. Those bodies, which cannot 
 
 be rendered electric by friction, are metals, and water, either liquid or congealed ;* 
 also earthly and animal bodies. But we must observe that, properly speaking, metals 
 - and the aqueous fluidare the only true conductors of electricity ; and that the rest are 
 
 not so, unless they participate in the metallic nature, or contain more or less moisture. 
 _ Electricity seems even to prefer the metallic bodies, for transmitting itself from 
 one body to another. If you place a body then of the latter kind, such as a metal 
 tod, or a piece of moist wood, in the neighbourhood of a body, or in contact with a 
 _ body of the first kind, electrified by friction, and with precautions to be mentioned 
 hereafter, it will itself become electric; which may be readily seen by the motion 
 it will communicate to light bodies in its neighbourhood. ' 
 
 All bodies then are susceptible of being electric, but in two different ways : one 
 
 _kind are in some measure electric of themselves, as that virtue can be excited in 
 them by friction; for this reason they are called electric per se: the other kind are 
 electric only by communication, and for this reason, they are commonly called 
 electric by communication, or non-electric ; but it would be better to call them con- 
 _ ductors of electricity: and this is the appellation which we shall most frequently 
 employ. 
 _ It may be here proper to observe, that those of the first class are not susceptible 
 _ of receiving electricity by communication, or they receive itin that manner with diffi- 
 culty. Hence it happens that, in the experiments we are about to describe, the bodies 
 _to be electrified by communication, must be placed either on cakes of resin, or be 
 suspended by silk strings; otherwise the electricity produced in them would be im- 
 mediately dissipated, by the contact of bodies susceptible of being electrified by 
 communication, with which they might be in contact. 
 
 SECTION II. 
 Description of the Electric Machine, and of the apparatus necessary for performing 
 electrical experiments. 
 
 When philosophers began to cultivate the theory of electricity, they employed 
 nothing for the purpose of exciting it but a glass tube, about 3 inches in diameter, 
 and from 25 to 30 inches in length. It was rubbed lengthwise and in the same 
 direction, with the bare hand provided it was very dry, or wrapped up in a bit 
 of flannel or cloth; and this tube was afterwards presented to the body intended to 
 be electrified. It was in this manner that Gray and Dufay made their first experi- 
 ‘ments. 
 
 __A globe suspended between two wooden pillars, which was made to revolve 
 ‘rapidly by means of a handle or wheel, was afterwards substituted instead of this 
 ‘tube: the dry harid was applied to the globe thus arranged, or it was made to rub 
 against a cushion: this operation excited the electricity, which was collected as we 
 
 * It bas since heen observed that glass heated till it becomes red or more, and flame, are conductors 
 ‘of electricity. On the other hand water, which in its state of fluidity is a conductor of electricity, 
 |ceases to be so when strongly frozen. 
 
pean 
 
 736 ELECTRICITY. 
 
 may say by means of a metallic fringe suspended from the globe, or disposed in some | 
 other manner. 
 
 These machines were succeeded by one much simpler. It consists of a foot or 
 stand a, Fig, 45, upon which are raised two : 
 uprights, B and c, secured and united at the | 
 top by means of the circular piece p. These 
 uprights must be of a greater or less height, 
 according to the diameter of the circular 
 glass plate r, placed between them; for | 
 the edge of the plate must not approach too | 
 near the wooden frame, either at the top or 
 the bottom. 
 
 This circular plate of glass ©, is the moa 
 
 essential part of the machine. It has a hole 
 in the centre, of sufficient size to admit a steel | 
 axis, which turns in the two supporters, and 
 _ towards the side c, is continued outwards, where it terminates in a square extremity, : 
 fitted into a handle, which serves to turn the plate. | 
 
 Leather cushions, stuffed with hair, are applied to the supporters on each side, 
 both at the top and bottom; so that the glass plate, as it revolves, is rubbed by | 
 the cushions to the distance of some inches from its edge. | 
 
 On the longitudinal part of the stand is placed a conductor, supported by a glass” 
 foot in the form of a column. This conductor is a copper cylinder, terminating at 
 one end ina ball a, of the same metal, and having at the other end two arms bent : 
 almost in a semicircular form. At the extremities of these arms are two hemi- 
 spherical figures, H and 1, which present to the glass plate their circular bases, fnr- ; 
 nished with four sharp steel spikes, all of the same length. The foot of the con- 
 ductor ean be made to advance or recede on the bottom part of the machine, which 
 supports it, in such a manner that the before mentioned spikes can be brought | 
 nearer to, or removed from, the surface of the glass plate; for it is these spikes, as” 
 will be seen hereafter, that attract the electric fluid, excited or put in motion by the | 
 friction of the small cushions on the circular glass plate. 
 
 When you are desirous of’ producing electricity, place the machine on a firm table, 
 and make it fast by means of screws; then fix the conductor in such a manner, that 
 the spikes may approach very near to the glass plate, and put the latter in motion | 
 by turning the handle. The conductor will almost immediately exhibit signs of elec- 
 tricity, either by emitting sparks on the finger being applied to it, or by attracting 
 and repelling light bodies placed near it. 
 
 The common electrifying machine, consisting of aglass cylinder, was first intro- 
 duced by the Germans, and for ordinary purposes it may still be considered as the 
 most perfect of any. _ 
 
 This machine is represented at Fig. 46, 
 where A B is the cylinder of glass support- 
 ed upon two pillars = F, which may be 
 either of glass, or baked wood, though it is 
 not essential that the pillars should be non- 
 conductors. They are firmly fixed into a 
 stand or horizontal board G a, which may 
 either rest on the floor, or be set upon 
 and clamped toa table. Rotatory motion 
 is given to the cylinder either by a simple 
 winch, as w, or by a multiplying wheel 
 fixed to the lower part of the pillar E, 
 
 Fig. 46. 
 
as Pa 
 
 ELECTRICAL EXPERIMENTS. = a 
 
 and acting by means of a band ona smaller wheel affixed to’the axis of the 
 cylinder. 
 
 A rubber represented at m is supported on a pillar r, fastencd to the board « u. 
 By means of a screw s, this pillar can be brought so as to make the cushion apply 
 with sufficient tightness to the cylinder. A silk flap” o, attached to the rubber, 
 passes over the cylinder. The conductor c D is placed upon an insulating stand, 
 and receives the electricity from the cylinder by means of a series of metallic points, 
 
 as at p. 
 
 If the cushionis insulated, a conductor is sometimes Perica to it, and while the 
 
 other conductor is electrified positively by the electricity collected from the revolving 
 cylinder, that applied to the cushion is at the same time exhausted of its electricity, 
 or electrified negatively. 
 
 When only the positive conductor is used, and the cushion is insulated, it is 
 _ necessary that a communication should be established between the cushion and the 
 earth, to afford asupply of electric matter to be put in action by the cylinder. 
 
 Remark.—Some other instruments are necessary for electric experiments ; but we 
 shall here mention those only. which are commonly used, reserving the description 
 of the rest till we come to speak of the different experiments in which they are 
 necessary. 
 
 I. You must be provided with a few stools, either square or circular, covered with 
 resin, about 15 or 18 inches in breadth: and, the better to ensure the effect, they 
 may be made to rest on four glass bottles or feet., These stools serve to insulate 
 the persons or bodies intended to be electrified. 
 
 II, As it is sometimes dangerous to draw out the electricity by the finger, it will 
 be proper to have an instrument, called the discharger, which js a circular.piece of 
 
 metal, Fig. 47, affixed by the middle to a handle, made of glass 
 
 Fig. 47. or Spanish wax: but the first is preferable and stronger. By 
 
 touching bodies, electrified in the highest degree, with one of 
 
 the balls of this instrument, sparks may be extracted from them 
 
 without danger ; because the glass handle intercepts the pas- 
 
 sage of the electricity, from the discharger to the person who 
 holds, it. 
 
 III. You must have also a chain of metal, or of serena pieces of wire connected 
 together. This chain serves for transmitting the electricity to a distance from the 
 first conductor 1; which is done by suspending it. from silk cords, attached to the 
 ceiling, or ecnive it between two supporters. 
 
 IV. It will be proper to provide likewise a long tube of metal, or of gilt paste-. 
 board, three or four inches in diameter. ‘This tube, having a communication with 
 the conductor by means ofa chain, forms a second conductor, which becomes 
 charged with a great deal of electricity, and may be employed in a variety of experi- 
 ments, The longer and larger this tube is, the greater will be the quantity.of elec- 
 tricity with which it will be charged. For reasons which will be mentioned here- 
 after, it is necessary that it should have no points or sharp eminences. 
 
 V. A few glass plates are also necessary, to insulate those bodies which may be 
 required to retain their electricity. 
 
 VI. You must provide likewise a few pieces of metal, some pointed and others 
 terminating in spherical or round ends; some affixed to glass handles, and others 
 furnished with handles of some substance that transmits aE as before men- 
 tioned. 
 
 VII. The cushions must be occasionally rubbed over with a kind of amalgam 
 which serves to increase the friction. That which appears to answer best, is an 
 ‘nalgam of tin aud mercury, snch as that placed at the back of mirrors, mixed 
 
 3B 
 
Sis ELECTRICITY. 
 
 with one half of chalk or Spanish whitening, the whole reduced to an impalpable 
 powder. 
 
 Such are the principal parts of the apparatus necessary for the most common 
 electric experiments; to which we shall now proceed, beginning with those that 
 are simplest. 
 
 EXPERIMENT I. 
 The Electric Spark. 
 
 Every thing being arranged, as above described, and the air in the apartment 
 being dry, put the machine in motion. If a person then present his finger to the 
 conductor, at the distance of two or three lfnes, or more, according to the strength 
 of the electricity, two sparks will proceed at the same time, from the conductor and 
 the finger, accompanied with a ‘snapping noise, which will even occasion some 
 
 pain. When the person, whom we suppose to be on the floor, touches the con- — 
 
 ductor, it will give no more signs of electricity, because he will then be in com- 
 ‘munication with the whole mass of non-electric bodies with which he is in contact. 
 
 If a piece of chalk be placed on the conductor it will be found that considerably 
 longer sparks may be drawn from the chalk than the conductor itself; and that the 
 sparks are of a zig-zag form; and are indeed beautiful miniature representations 
 of forked lightning. 
 
 EXPERIMENT II. 
 ~ Communication of Electricity to Several Persons. 
 
 Cause the person, above mentioned, to stand on a cake of resin, by which means 
 he will be insulated from the floor, and put the machine in motion. ‘This person, as 
 well as the conductor, will thus be electrified ; so that all those, not in the same 
 state, who happen to touch each other, will elicit electric sparks. 
 
 Twenty persons, and more, may be electrified in this manner, provided they are 
 insulated. 
 
 EXPERIMENT III. 
 Attraction and Repulsion. 
 
 Present to the person electrified, or to the conductor, some gold leaf or tin-foil, 
 or bits of straw, or paper, or other light non-electric bodies. When at a proper 
 
 distance, you will see them dart towards the electrified body ; but as soon as they 
 
 have touched it they will be repelled. If you then receive them on a non-electric 
 
 body, as soon as they touch it, they will return towards the electrified body, and 
 
 will be again repelled ; and so on alternately. 
 
 EXPERIMENT IV. 
 
 Some Electric Amusements, founded on the preceding property. The Gold Fish; 
 
 the Electric Dance; the Luminous Rain. 
 
 The property which electric bodies have of repelling each other, when in that 
 state, and of attracting each other when one of them only is electrified, has given rise 
 to several very agreeable amusements, which we shall here explain. ee. 
 
 I. Cut a piece of strong gold leaf into the form of a rhombus, two opposite angles 
 of which may be very obtuse, while the other two are very acute. Present this 
 metallic leaf to the conductor, in such a manner that one of the acute angles shall 
 be first raised, and immediately put below it a metallic plate. You will then see the 
 leaf place itself between the conductor and the plate, and remain in that state al- 
 most motionless. 
 
 Cut leaves of metal of this kind into the shape of the human figure, having an 
 
 acute angle at the top, like a pointed cap, and lay them flat on the plate: if you then ~ 
 
{ = 
 \ 
 
 ELECTRICAL EXPERIMENTS. 739 
 
 place them below the conductor, on another plate, you will see them start up, leap 
 towards the conductor, then fall down, turning round with more or less rapidity, so 
 as to represent a kind of dance; and if the experiment be performed in the dark, 
 you will observe luminous aigrettes dart alternately from the head and the feet, 
 which will form a very agreeable spectacle. 
 
 II. Cut a piece of the same gold leaf into a figure very much lengthened on one 
 side, but on the other much less acute; to this part if you choose you may give the 
 form of the head ofa fish. If you lay hold of it by the acute angle, and present the 
 obtuse one to the conductor, at the distance of a foot, if the electricity be strong, it . 
 will escape from your fingers, and will fly with an undulating motion towards the 
 the conductor, above which it will place itself at the distance of the eighth. part 
 of an inch, turning towards it the obtuse angle. Sometimes it will approach so near 
 as to come into contact with it, and will be immediately repelled, forming the repre- 
 sentation of a small fish going to attack or bite the conductors. This amusement 
 therefore has been called the Gold Fish. 
 
 III. The luminous rain may be produced in the following manner. Suspend from 
 the conductor a circular plate, 5 or 6 inches in diameter; then provide a metallic 
 plate in the form of a saucer, and surround the edge of it with a glass cylinder 5 or 
 6 inches in height. Cover this plate with very fine light shavings of metal, and 
 place it under the plate suspended from the conductor. When the latter is strongly 
 electrified, you will see all these small leaves of metal ascend from the lower to the 
 upper plate, and sparkle; being then repelled to the lower one, they will again 
 sparkle, or will sparkle between the plates, when one which is electrified meets with 
 one not electrified: by these means the glass cylinder will be filled with a great deal 
 of light, which will exhibit the appearance of a shower of fire. 
 
 / 
 EXPERIMENT V. 
 
 Repulsion between bodies equally electrified. 
 
 Suspend from the extremity of the conductor two threads of any non-electric 
 matter; such as flax, hemp, or cotton, which will hang down perpendicularly, and 
 touch each other, if their upper extremities are in contact. If you then work the 
 machine, and produce electricity in the conductor and these threads, you will see them 
 repel each other, and form an angle of greater extent, as the electricity is stronger. 
 When the electricity decreases, they will approach each other. 
 
 This experiment proves a very important fact in the theory of electricity; which 
 is, that two bodies, similarly electrified, repel each other: and hence we are enabled 
 
 to explain several electric phenomena and amusements. 
 
 EXPERIMENT VI. 
 Construction of an Electrometer. 
 
 By the preceding experiment we are furnished with the means of determining the 
 strength of electricity ; and the two threads, above mentioned, may be considered 
 as a sort of electrometer. However, as two threads of this kind may be subject to 
 various movements, independent of electricity, electricians have almost universally 
 adopted the following instrument, which is equally simple. 
 
 The whole of this machine consists of two small balls, two lines in diameter, made 
 of cork or the pith of the elder tree, and fixed to the two extremities of a thread 
 capable of conducting electricity. This thread is made to pass over the conductor 
 in such a manner, that the two balls hang at the same height. As soon as electricity 
 is produced in the conductor, and consequently in the small balls, they diverge from 
 each other, and the magnitude of the angle formed by the threads will convey some 
 idea of the intensity of the electricity. We say conveys an idea of this intensity; for 
 
 it is not possible, either by this or any other method with which we are acquainted 
 3B 2 
 
4 
 
 to determine when the electricity is double, triple, or quadruple, &c. ; but we are at 
 least enabled to conclude, that one degree of electricity is greater or less than ano- 
 ther; or that two degrees of electricity are equal, according as the divergency of the 
 balls is greater or less, or the same, which in general is all that is necessary to be known. 
 
 740 ELECTRICITY. 
 
 EXPERIMENT VII. 
 To kindle Spirit of Wine by means of the Electric Spark. 
 
 When a person is electrified, if another standing on the floor approaches him, 
 having in his hand a spoon filled with spirit of wine, well dephlegmated, and some- 
 what heated ; and if the electrified person presents his finger to the spirit of wine, 
 or, what is still better, the point of some blunt instrument, or the point of a sword, 
 an electric spark will proceed from the liquor, and set it on fire. 
 
 If the painful sensation produced by the electric spark, could leave any doubt of its — 
 being real fire, this experiment must be a convincing proof of it. 
 
 EXPERIMENT VIII. 
 Properties of sharp Points or Spikes. 
 
 Instead of a conductor, such as that above described, if you employ an n angular bar 
 of metal, or a bar terminating in one or more points ; on approaching your finger to 
 one of the angles or points, when the machine is put in motion, but not in such a man- 
 ner as to produce an electric spark, you will feel something exhaled like a gentle 
 breath of wind, and even with a sort of crackling noise. 
 
 But, if the experiment be performed in the dark, you will enjoy a very beautiful 
 spectacle; for when the electricity is strong, you will see luminous gerbes issue from 
 the angles of the conductor, and these gerbes will be considerably increased on pre- 
 _ senting your finger to them. 
 
 You will discover, at the same time, that the cause of this gentle breath, accom- 
 panied with noise, is nothing else than the eruption of the electric fluid, whatever it 
 may be, from the electrified body, which rushes towards your finger. Hence it fol- 
 lows that it is a body, since it re-acts against another body. It will be found also that 
 this dispersing electricity has a peculiar smell. 
 
 It is to be observed, that when the electrified body is angular, it loses much sooner 
 the electricity which has been communicated to it. These angles and points seem 
 to be so many spontaneous discharges of the electric matter; they ought therefore to: 
 be avoided in bodies intended to be electrified, and in which you are desirous to main- 
 tain the electricity as long as possible. 
 
 EXPERIMENT 1X. 
 Difference between Pointed and Blunt Bodies. 
 
 Electrify strongly in the dark a common conductor, or anyother body whatever, 
 not angular, and when it is strongly charged, present to it a blunt body, such as the 
 finger or a spike rounded at the end, holding it so near it as to elicit the electric spark. - 
 But if you present to it a very sharp instrument or spike, you will see a luminous 
 star formed at the extremity of it, even before it is brought so near; and if the elec- 
 trified body does not every moment receive a fresh supply of electricity, it will-thus 
 soon be deprived of it. | 
 
 If this spike be supported by a one of resin, it will itself become electrified; but 
 the electricity of the conductor will not be entirely destroyed. 
 
 It appears from this experiment, that if the luminous gerbes, in the preceding one, 
 are formed by a matter which fiows off from the electrified body, the case in the pre- 
 sent instance is contrary: they are formed by a matter which throws itself towards 
 the point presented to the electrified body. What indeed can be said when we observe 
 a non-electrified body become so by this method; but that the electric matter, fire, — 
 
ELECTRICAL EXPERIMENTS. 741 
 
 or fluid, whatever it be, proceeds from the electrified body to another, especially as 
 it is certain that the former thereby loses either the whole ora part of its electricity, 
 according to circumstances ; that is to say, according as the other stands on the floor, or 
 is insulated ? 
 
 But however this may be, the following is a singular and remarkable property fi 
 pointed bodies. The extraordinary use which Dr. Franklin made of it will be 
 shewn hereafter. 
 
 EXPERIMENT X. 
 Method of knowing whether a body be in a state of Electricity. 
 
 When two bodies are equally electrified, if they be brought into contact with 
 each other, no sign of electricity will be manifested between them, by ae or any 
 electric emanation. 
 
 This may be easily proved; for if a person electrified by touching ihe conductor, 
 gives his hand to another electrified in the same manner, there will be no spark. 
 
 These two persons, however, may know that they are electrified by the following 
 sign. Let each of them take in his hand a thread, made of any non-electric substance, 
 ora cork ball suspended from such a thread; if these two balls or two threads repel 
 each other, it may be concluded that the persons are in an electric state. 
 
 EXPERIMENT XI. 
 Distinction between the two kinds of Electricity. 
 
 Provide two electric machines, one of them constructed as they were formerly, 
 that is, with a glass globe, and the other with a globe of sulphur instead of glass: if 
 a conductor be then electrified by ‘each of them at one of its ends, you will see with 
 astonishment, if the machines are moved with equal velocity, that scarcely any sparks 
 can be extracted from the conductor. The case certainly would not be the same, if 
 the conductor were electrified by means of two glass globes at the same time, or 
 
 ' with two globes of sulphur; the sparks would be much stronger than if one globe 
 had been put in motion. 
 
 Remark.—This experiment, which Dr. Franklin says he made at the request of his 
 friend Mr. Kinnersley, seems to me to leave no doubt in regard to the difference be- 
 tween electricity communicated by glass and that communicated by sulphur; and 
 consequently establishes the distinction of vitreous and sulphureous or resinous 
 electricity; a distinction before asserted by Dufay. 
 
 Dufay indeed had observed, that, though two bodies electrified by glass or sulphur 
 mutually repelled each other, yet when one of them was electrified by the one of these 
 substances, and the other by the other, instead of repelling they attracted each other. 
 We do not think that any stronger proof of the two states can be desired. 
 
 If to this be added the above experiment of Dr. Franklin, how can we elude the 
 consequence which he and Dufay deduce from it? For, it is well known that two 
 bodies equally electrified by a glass globe, may touch each other without producing a 
 spark, and without the electrical virtue being diminished in either of them. Since 
 these bodies then electrified, one by the glass and the other by the sulphur, mutually 
 destroy each other’s electricity, the one must be of a nature contrary to the other, 
 and entirely different. 
 
 Some able philosophers, however, notwithstanding these reasons, persist in reject- 
 ing this distinction; but in our opinion they labour under the influence of prejudice, 
 or, being seduced by peculiar ideas, keep their eyes shut against the light. We are 
 inclined to think, that if the Abbé Nollet had not previously formed his system on 
 electricity, he would have adopted the distinction of the two kinds. 
 
742 ELECTRICITY. 
 
 However, as this is the proper place, we shall here give an idea of Dr. Franklin's 
 system in regard to electricity. According to this celebrated philosopher, all bodies in 
 their natural state contain in their substance, or on their surface, a certain quantity 
 ef a fluid, which is the electric fluid. The air, which when dry is not a conductor of 
 electricity, prevents its dispersion. But the friction of certain bodies, glass for 
 example, collects on the surface of thema greater quantity of the fluid; so that ifthe 
 glass be in contact, or very near to a body electric by communication, such as a mass 
 of iron, the fluid accumulated on the surface of the glass tends to pass into the mass 
 of iron, in order to preserve an equilibrium. By these means this mass acquires a 
 greater quantity of the electric fluid, and is then electrified positively. But if the 
 electric body, instead of acquiring by friction a greater quantity of the electric fluid, 
 ioses some of what it had, as is the case with sulphur, the body in contact with it will 
 
 lose apart of its own natural electricity, and will then be electrified negatively. The 
 one will have more electric fluid than it has in its natural state, and that of all the 
 bodies which have a communication with the earth; the other will have less. Such 
 is the nature of positive and negative electricity. 
 
 It must, however, be allowed that it does not clearly appear how friction should 
 accumulate, on the surface of the rubbed body, a greater quantity of the electric 
 fluid. It is not even known whether the effect of friction is to accumulate the fluid 
 on the glass, or to diminish the quantity of it ; whether it lessens it on sulphur and 
 resins, or increases it. Hence it is uncertain which is the positive and which the 
 negative electricity ; but we know beyond a doubt that their effects are contrary, and 
 this is sufficient. Several reasons, however, make it probable that the electricity pro- 
 duced by the friction of glass, is the positive or accumulated electricity. 
 
 Notwithstanding this uncertainty, Franklin’s theory has a great advantage over that 
 of the Abbé Nollet. The latter supposes a matter diffused throughout all bodies, 
 and even in the atmosphere, which with all other electricians he calls the electric 
 fiuid. In this he agrees with Dr. Franklin; but he thinks the effect of friction is to 
 make this fluid sometimes issue from the pores of the body rubbed, and sometimes to 
 attract it. Electricity therefore, or the electric fluid, is sometimes effluent and some- 
 times affluent ; and it is by means of this effluence or affluence that this philosopher 
 explains all the phenomena of electricity. But the great defect of this system is, 
 that every thing in it is, as we may say, arbitrary. What cannot be explained by the 
 affluent fluid, may be explained by the effluent. These are the different matters of 
 Descartes, or his: subtle matter, which may be applied to everything. On the other ~ 
 hand, in the system of Franklin, the effects are much better connected with the causes, 
 even supposing them hypothetical. Why does aspark issue when a body, positively 
 or negatively electrified, is brought near to another which is in its natural state ? 
 The answer is easy. The electric fluid accumulated on the one side, and extended 
 in the form of an atmosphere, as it were, on the surface of a body, puts itself in 
 equilibrium, when it comes in contact with another electric atmosphere less condensed: _ 
 the fluid divides itself equally between the two bodies ; which cannot be done with- 
 out an exceedingly rapid movement that produces light. But what is most remark- 
 able in the hypothesis of Franklin, and is almost the touchstone of truth, is, that 
 even the bare description of the simplest experiment, to those who have properly 
 comprehended this hypothesis, is sufficient to enable them immediately to guess 
 tne result. The case is not the same with the system of the Abbé Nollet: 
 none of the effects about to be produced are foreseen, and if every thing be 
 explained, it is because no effect is connected with its cause. Had the phenomenon 
 been quite contrary, it might have been explained with the same ease: effluence 
 could be employed instead of affluence; one is the remedy or supplement of the — 
 other. 
 
ELECTRICAL EXPERIMENTS, 743 
 
 We must however acknowledge, that there are some facts difficult to be recon- 
 ciled with the motion of the electric fluid, which is a necessary consequence of 
 Franklin’s system. 
 
 For example, when the finger is brought near to a body, electrified either positively 
 or negatively, why do we see a double spark proceed from each of these bodies? 
 It would appear that it ought to proceed only from that which is endowed witb 
 positive electricity. 
 
 In a certain experiment, in which a quire of paper is pierced by the electric spark, 
 why isthe rough edge of the hole turned in a direction contrary to that in which 
 it ought to be, if the fluid accumulated on the surface of the electrified body were 
 the only one that proceeded to the body negatively electrified? We shall omit 
 several others, which have been remarked by the partisans of the Abbé Nollet, and 
 only observe that there is still reason to suspend our opinion on the mechanism of 
 this phenomenon. 
 
 EXPERIMENT XII. 
 The Leyden Flask, and Shock. 
 
 There is not perhaps in natural philosophy a phenomenon more astonishing than 
 that which we are about to describe. Provide a flask of very thin white glass, with 
 a long neck, and fill about two thirds of it with water, or metallic filings, or raspings 
 oflead. Close it witha cork stopper, and introduce into it, through the cork, an 
 iron wire, so as to be immersed with one end inthe water or filings, while the other 
 projects some inches beyond the cork, and terminates in a blunt or crooked extre- 
 mity. 
 
 When the flask is thus prepared, lay hold of it by the belly, and present the iron 
 wire to the conductor of the electrifying machine while in action. By these means 
 the flask will be charged. While the wire is in contact with the conductor, if you 
 then endeavour to touch the latter, or the iron wire, with the other hand, you will 
 experience throughout your whole body a violent shock, which will seem to affect 
 more particularly, sometimes your breast, sometimes your shoulders, and sometimes 
 your arm or wrist. 
 
 The same effect will be experienced if you retire, holding the bottle by the 
 belly with one hand, and touch the iron wire with the other. 
 
 Nay, a chain may be formed of as many persons as you choose, holding each other 
 by the hand, and without being insulated. The first person holds the bottle in his 
 hand, or only touches it, while the iron wire is in contact with the conductor ; 
 
 ‘and the last touches the conductor; all those who form the chain will experience 
 the before-mentioned shock at the same instant. When the flask is of considerable 
 size, and has been well charged, the shock is sometimes so violent, that those sub- 
 jected to it suffer a momentary loss of respiration. The celebrated Muschenbroek, 
 to whom M. Cuneus exhibited this phenomenon, which he had discovered by acci- 
 dent, received, according to every appearance, a violent shock ; since in announcing 
 
 _ it to the French philosophers, he protested that he would not expose himself to it 
 a second time for the whole kingdom of France. It is however probable that he 
 afterwards became bolder. As this singular experiment was first performed at Ley- 
 den, it is generally called the Leyden experiment, and the bottle, so prepared, is dis- 
 tinguished by the name of the Leyden flask or phial. 
 
 The French philosophers once formed a chain 900 toises in length, by means of 
 200 persons, all connected by iron wires; and all these persons experienced the 
 shock at the same instant. Another time they tried to transmit the shock along 
 
 * an iron wire 2000 toises in length, and the experiment succeeded, though the wire 
 passed over the wet grass, and newly ploughed land. In short, they comprehended 
 
so 
 
 744 ELECTRICITY. - ~ | 
 
 in the chain the water in the grand bason of the Tuilleries, which is nearly an acre 
 in extent, and the shock was transmitted very well across it. 
 
 Remarks.—I. As some inconvenience resulted from the weight of the water or 
 filings, put at first into the flask, the idea was afterwards conceived of covering the 
 inside of it with a metallic coating. This may be done in several ways. The most 
 simple is to pour into the bottle strong gum water, and to moisten with it the part 
 intended to be coated. The superfluous gum water is then poured out, and very 
 fine copper filings are put into the bottle; these filings adhere to the gum water, 
 and form an internal coating, which must be in contact with the irou wire, that the 
 bottle may be charged. 
 
 The effect of the Leyden flask may be increased also by covering a great part of 
 the outside of it with tin-foil. 
 
 If. The fiask may be charged in another manner, that is to say, externally. For 
 this purpose, hold it suspended in the one hand by the hook or iron wire on the 
 outside, and bring the outside into contact with the electrified conductor. If it be 
 then touched on the outside withthe other hand, you will experience a shock. You 
 may form, in like manner, a chain of several persons, the last of whom, or the one 
 farthest distant from the person who holds the iron wire, by touching the outside of 
 it, will produce the same phenomenon throughout the whole chain. 
 
 If. Dr. Franklin observed the following very singular circumstance, which takes 
 place in performing the Leyden experiment: if you are desirous of charging the in- 
 side of the jar or flask, the outside must communicate with some body which is a 
 conductor of electricity; for if the flask be placed on a cake of resin, it will be in 
 vain to electrify, by.the conductor of the machine, the wire which touches the water 
 or metallic coating in the inside: the flask will not become charged. Is it necessary, 
 before it can be charged, that in proportion as the electricity is accumulated on one 
 side, it should be diminished on the other? This is the conclusion which Dr. Frank- 
 lin forms, and which appears indeed to be agreeable to reason. «But how is it that 
 the electric fluid is expelled from one side, while the other becomes more highly 
 charged with it? This appears to us to be a considerable difficulty. 
 
 IV. The jar seems to be impermeable to electricity, at least when cold, or whell | 
 it has only the temperature of the atmesphere. Dr. Franklin once tried to grind 
 away the belly of a charged flask, which was of the usual thickness. He ground 
 down § of its thickness, without its being discharged, which would have been the’ 
 case if the fluid in the inside had communicated with that on the outside. It is to 
 be wished that this philosopher had continued to diminish the thickness, until a dis- 
 charge had taken place. 
 
 But when the glass is dilated and softened Bae a heat which brings it nearly to a 
 state of fusion, it then not only becomes a conductor of electricity, but the charged 
 jar discharges itself spontaneously. 
 
 V. If a chain, suspended from the conductor, be introduced into the flask while 
 held in the hand, it becomes charged in the like manner ; but if the flask be lowered 
 in such a manner, as no longer to support any portion of the chain, it then gives no 
 more signs of electricity. 
 
 There is reason thence to conclude that the electricity with which the inside 
 of the bottle is charged, ought to have for its support some non-electric matter, or a 
 conductor of electricity. It would be in vain to attempt to charge an empty bottle, 
 or a bottle not covered in the inside with a metallic coating. 
 
 VI. If the inside of the bottle be charged, making it communicate by the hooked 
 wire with a conductor positively electrified, the outside will then be negatively” 
 electrified ; for the outside will attract the small ball of cork suspended from the 
 
ELECTRICAL EXPERIMENTS. 745 
 
 conductor, while the hook of the flask repels it. But it is well known that an elec- 
 trified body repels another electrified like itself; it attracts only bodies not electri- 
 
 fied, or electrified in a contrary sense. Since the outside of the bottle then, elec- 
 trified by the hook, attracts the small ball of cork, the electricity of which is of the 
 same nature as that of the conductor, or of the inside of the jar, the exterior elec- 
 tricity must be of a nature entirely different. 
 
 VII. If you have two equal flasks, equally charged, and in the same manner, and 
 
 _if you then bring them near to each other, so as that the hooks or the-sides of them 
 
 be in contact, they will not be discharged; but if you apply the hook of the one 
 to the side of the other, a discharge will immediately take place. 
 
 If one of the flasks be charged by a globe of sulphur, and the other by a globe 
 of glass, if the hook of the one be then made to approach the hook of the other, or 
 the side of the one the side of the other, they will be discharged. 
 
 VIIL. If several persons, instead of holding each other by the hands, present to 
 each other the tips of their fingers, ‘at the distance of one or two linés, at the mo- 
 ment when the last touches the conductor, you will observe between all the fingers 
 an electric spark, and each will experience a shock. 
 
 IX. If the persons, instead of holding each other by the hands, form a communi- 
 cation with each other by holding glass tubes filled with water, and stopped with a 
 cork, through which passes an iron wire immersed in the fluid, and which is in con- 
 tact with each person, at the moment when the last person touches the conductor, 
 or the wire immersed in the flask, you will perceive a train of light in the water 
 in each tube, which will be wholly illuminated by it. 
 
 X. The chain being formed, if one or two or more persons form a new one, con- 
 nected on one side with those who form the first chain, and on the other side with 
 another person of the same chain, those of the latter will experience nothing; the 
 electric fluid seems to proceed from one end to the other of the first chain, by the 
 
 - shortest route. 
 
 me 
 
 XI. Some beautiful effects of colour may be obtained by passing the shock 
 
 through different sorts of wood. And if the shock be passed through a lump 
 
 of sugar, or over a piece of chalk, they will, in the dark, exhibit a sort of phos- 
 phoric illumination for some time. 
 
 EXPERIMENT XIII. 
 
 Another method of giving the shock, namely, by an electric pane of glass.—To pierce 
 a quire of paper by the electric spark. 
 
 Does the singular effect observed in the preceding experiment depend on the figure 
 of the Leyden jar, or merely on the nature of the glass? ‘This question, which 
 naturally occurs, is answered by the following experiment: it proves that the effect 
 alluded to depends entirely on the nature of the glass. 
 
 Take.a pane of glass of any dimensions, and cover both sides of it with tin-foil, 
 leaving on each side a border of the glass uncovered; place the glass horizontally on 
 a non-electric supporter, and make the chain of the conductor to fall on its surface. 
 If you then put the machine in motion, the glass will become charged like the Ley- 
 den flask ; that is, if resting one side of the discharger on the upper surface, the other 
 be applied to the lower one, you will extract a strong and powerful spark. If the 
 glass be large, it will be dangerous to touch one of the surfaces with the one hand, 
 and the other with the other. 
 
 If you are desirous to pierce a quire of paper with the electric spark, you must 
 proceed as follows. Extend a piece of iron wire on a table, and place over it the 
 pane of glass, so that the end of the wire shall touch the coating of the lower 
 side: on the upper coating place a quire of paper ; then electrify the upper surface by 
 means of the chain of the conductor, which must be made to fall upon it. When 
 
 ae 
 
746 ELECTRICITY. 
 
 you think the electricity is very strong, bring one of the ends of the discharger into 
 contact with the wire, and apply the other end to the paper. A very strong spark 
 willissue from it, with a noise like the report of a pistol; and the quire of paper will 
 be pierced through and through. 
 
 If the experiment be made with a piece of glass of about 35 inches square, 150 
 sheets of paper, and even more, may thus be pierced. 
 
 A very remarkable circumstance will be noticed in making this experiment. 
 There will not be an indentation on one side of the paper, and a protrusion on the 
 other, but a protrusion on both sides; as if the electricity had passed out of the 
 paper on both sides. | 
 
 This method of performing the Leyden experiment is attended with the advantage 
 of greatly increasing its effect; for the surface of the largest flask can contain no 
 more than two or three square feet. But a plate of glass, 36 inches in every direc- 
 tion, contains 9 square feet, and the effect is thereby increased in the same propor- 
 tion. 
 
 It may be easily seen, that, in performing this experiment, you must take care 
 not to stand in the circle between the upper and lower surface, otherwise you might 
 run the danger of being killed. 
 
 EXPERIMENT XIV. 
 Means of increasing, as it were indefinitely, the force of electricity—The Electric 
 Battery. 
 
 A single flask, charged with electricity, does not produce a very great effect: but 
 this effect is increased according as the volume of the jar is augmented. It would 
 however be inconvenient, and perhaps impossible, to obtain jars beyond a certain 
 size ; for this reason several jars have been substituted in the room of one; but 
 the united effect of them is exceedingly dangerous, unless great precautions are 
 employed. 
 
 '~ For this purpose, instead of long-necked flasks, several large cylindric jars, of a 
 much greater length than breadth, must be employed. It is not necessary how- | 
 ever that their diamzter should be very great, because cylinders of a small diame- 
 ter have, in proportion to their solidity, a greater surface, and surface is what is 
 required to be increased. They are lined on the inside with a coating of tin-foil, 
 which covers the bottom and sides to within two inches of the brim. And they are 
 coated, in the same manner, on the outside: after which, they are arranged close 
 to each other in a box, lined also in the inside with tin foil and copper filings. The 
 tin-foil communicates with a wire ring projecting from each jar, and to these rings 
 is attached the chain, by means of which a communication is established between 
 any body and the exterior part of the battery. 
 
 To establish a communication with the inside of the jars, a piece of iron wire, 
 twisted at the lower end, and terminating at the upper end in a ring, must be 
 made to pass through the cork stopper of each jar, so as to descend to the bottom 
 of it. A metal rod, having a ball at each end, passes through all the rings of the 
 same row: and, to establish a communication between the rods, the chain of the 
 conductor is made to rest on them; in consequence of this arrangement, you may 
 
 charge, if necessary, only one or two rows of the jars, by 
 Fig. 48. making the chain to rest on one rod only, or on two, &c. 
 
 Such is the construction of an electric battery, the repre- 
 sentation of which, supposing it to consist of only nine jars, 
 each 15 inches in height, 3 inches in diameter, and containing 
 12 inches of coated surface, which gives altogether 63 square 
 feet, is seen Fig. 48. A similar battery of 64 jars would give — 
 48 square feet, and yet form only a box of two feet some inches 
 
 AME 
 
ELECTRICAL EXPERIMENTS, 147 
 
 square, and from 15 to 18 inches in height. The effect of such a battery would be 
 prodigious. 
 
 The method of using this apparatus is as follows. To charge the jars, make the 
 chain, which proceeds from the conductor of the machine, to rest on the rods, and 
 turn the glass globe or plate for some time. Experience will shew how many turns 
 of the machine will be necessary for this purpose; if the jars are over-charged, 
 they will discharge themselves with a loud report. When they are in the proper 
 state, if you wish to discharge them, nothing is necessary but to lay hold of the 
 chain, which communicates with the outside by means of the discharging rod, and 
 to bring the end of it into contact with the conductor ; a strong spark will be elicited, 
 and the jars will be discharged. 
 
 lf a person, holding the end of the chain, should touch with his finger, either 
 the conductor of the machine, or one of the rods which touch the inside of the 
 jars, he might be killed in consequence of the terrible shock he would experience. 
 If a flask indeed, 5 or 6 inches in diameter, strongly charged, give by its discharge 
 a violent shock in the arm and breast, we may form some notion of the effect 
 which would be produced by the discharge of 12, 15, 20, 30 or 50 square feet. 
 Electricians therefore ought to be very attentive to themselves, and to the specta- 
 tors, for fear of some fatal accident. 
 
 All philosophers, who perform electrical experiments on a large scale, have at pre- 
 sent a similar apparatus, of greater or less size. It is by these means that they fuse 
 metals, which can be reduced even to a calx; that they communicate the magnetic 
 virtue to a needle, or change its poles, or imitate the effects of thunder, and so on, 
 as will be seen hereafter. 
 
 EXPERIMENT XV. 
 To kill an animal by means of electricity. 
 
 Affix the chain, which communicates with the outside of the battery, to one of 
 the animal’s feet ; and then with the discharging rod form a communication between 
 the animal’s skull and one of the rods that communicate with the inside: the animal, 
 if it be even a sheep or perhaps an ox, will be struck dead. 
 
 Remark.—It has been observed that the flesh of animals, killed in this manner, is 
 immediately fit for being eaten: for the shock which kills them is similar to that of 
 lightning, and it is a well known fact that animals killed by lightning pass very 
 speedily to astate of putrefaction. This artificial lightning might therefore be em- 
 ployed to kill those animals which are intended to be immediately used as food : 
 they will be what is called mortified ina minute. But as the operation is danger- 
 ous, Dr. Franklin humorously advises philosophers to be on their guard, lest in at- 
 tempting to mortify a pullet they should mortify their own flesh. 
 
 EXPERIMENT XVI, 
 Production of Magnetism by Electricity. 
 Fig. 49. Provide a steel needle, like that of a compass, some inches 
 A in length, and place it between two plates of glass, so that its 
 Fa aay j two ends, a and z, shall project a little beyond them. Then 
 make one of its ends a communicate with the conductor of the 
 electric machine, or any one of the transverse rods of the electric battery ; charge 
 the battery strongly, and discharge it through the needle by means of the discharging 
 rod, bringing the end of the chain, which communicates with the outside of the jars, 
 to the end B of the needle. All the electric fluid will pass through the needle, en- 
 tering by the end a, and issuing at B ; and the needle will then be magnetized in such 
 
 a manner that the end a will turn to the north. 
 When a needle has been magnetised, if the end a be turned towards the north, 
 
 e 
 
748 : - ELECTRICITY. 
 
 performing a contrary operation, that is to say, making the electric fluid pass from 
 B to a, the needle will be un-magnetised ; and by repeating the same operation it 
 will be magnetised in a contrary sense; that is, in such a manner that the end B will 
 turn to the north. : 
 
 It may be readily conceived, that this will depend on the quantity of the electric 
 fluid. If it be less in the second operation than in the first, some small portion 
 of magnetism may remain; if it be much more considerable, the poles may be changed 
 by the first shock. : | 
 
 EXPERIMENT XVII. 
 To Fuse Metals by means of Electricity. 
 
 This experiment is one of the most curious that are performed by electricity, 
 Take an iron wire half a line in diameter, and suspend from it a weight of about 6 
 pounds; then by means of a battery, consisting of from 16 to 25 jars, make the 
 electric fluid pass along it; the wire will immediately stretch itself, and sometimes 
 will break. But this could not be the case, were it not softened or mollified in some 
 part of its extent. 
 
 Another Method.—Take a piece of very thin gold leaf, and having cut it intoa 
 slip of two inches in length, and a line in breadth, put it between two plates of glass, 
 very close to each other; then place these plates in such a manner as to form part 
 of the electric circle of a strong battery, consisting of 50 or 60 jars. The gold leaf 
 will pass through the state of fusion; and what proves it is, that several of its parts 
 will be incorporated with the glass itself.* 
 
 But if you place between the glasses and the gold leaf two bits of card, and 
 squeeze the plates of glass closely together, the electric spark made to pass, as 
 above, through the gold leaf in the direction of its length, will reduce it, in a great 
 measure, into that kind of purple powder, known in chemistry, under the name 
 of the precipitate of Cassius ; because this preparation was first invented or simpli- 
 fied by that chemist. The two cards will be dyed of the same colour, which ma 
 be heightened by repeating the operation with new bits of gold leaf. e 
 
 Silver leaf treated in the same manner gives a powder of a beautiful yellow 
 colour. 
 
 Copper leaf gives a green powder. 
 
 Tin-foil gives a white powder. < 
 
 Platina, treated in this manner, is reduced, after repeated shocks, to a blackish 
 powder, which when applied to porcelain produces a dark olive colour. . 
 
 In short, we are assured, by different chemical proofs, that these calces are exactly 
 the same as those produced by longer processes. 
 
 For these experiments we are indebted to M. Comus, so celebrated for his industry 
 and address, and who united to the most extraordinary talents in this way the most 
 profound knowledge in different parts of philosophy. A circumstantial and truly 
 interesting detail of them may be seen in the “‘ Journal de Physique” for the year 
 
 1773. i 
 EXPERIMENT XVIII. 
 Which proves the identity of Lightning and the Electrical Spark. 
 
 On a high insulated place, such as the summit of a tower, fix in a vertical direc. 
 tion an iron rod, terminating in a very sharp point. The higher this point is in the 
 atmosphere, the better will the experiment succeed. This bar must be supported 
 by some base to insulate it from every body capable of conducting electricity. 
 
 * We have seen this effected by a battery formed of four phial glasses, each about two inches 
 in diameter, and eight inches high. 
 
 a 
 
 +. 
 i 
 
ELECTRICAL EXPERIMENTS. 749 
 
 ’ 
 
 , Then wait tilla storm takes place, and when a thunder cloud passes over the rod, 
 or near it, touch it with an iron rod attached to a glass handle. You will not fail 
 so extract sparks from it, sometimes very large, and accompanied with a loud noise. 
 [t will be dangerous however to approach too near it; for the rod is sometimes so 
 ighly charged with electricity, that the sparks proceed to the distance of some feet, 
 d make a noise like the report of a pistol. Mr. Richman, professor of mathematics 
 at Petersburgh, and member of the Imperial Academy of that city, fella victim, as is 
 well known, to an experiment of this kind ; for ina moment of forgetfulness, having 
 approached too near the machine, he was struck dead, and all those effects observed 
 in persons killed by lightning were seen on his body. . 
 
 _ This accident has induced some philosophers, who study electricity, to arrange their 
 machine in such a manner, that it can never become too much charged with electricity. 
 For this purpose, they place at some distance from the rod a piece of sharp pointed 
 metal, which communicates with the floor or the mass of non-electric bodies. This 
 point, when the electricity is moderate, will attract none of it ; but when very strong, 
 it will draw it off, as we may say, and discharge it insensibly ; so that it will accumu- 
 late only in such a-moderate quantity as to be incapable of doing michief. The nearer 
 ithe point is to the rod, the more it will absorb of the electricity. 
 
 _ By its means it may be known in obscurity whether the cloud be electrified posi- 
 tively or negatively ; for in the first case you will observe at the point simple lumin- 
 ‘ous star, or very short gerbe; inthe second, you will observe a large and beautiful 
 gerbe. 
 
 It is customary also to place near the bar a metal ball suspended by a silk thread ; 
 and, at alittle farther distance, a bell communicating with the body of the building. 
 The use of this apparatus, is to inform the observer that the bar is electric; for at 
 ‘the moment when it is charged with electricity, it attracts the ball which possesses 
 none, electrifies it, and propels it against the bell; the sound of which announces 
 that the electric cloud has produced its effect. The degree of the electricity alsois 
 ‘indicated by the same means ; for if it be very strong, the vivacity of the ringing is 
 
 proportioned to it, and the observer is warned to be on his guard. 
 
 This experiment, without either a tower or a terrace, may be performed in 
 chamber. Nothing will.be necessary but to place in the chimney a bar of iron, in- 
 sulated by means of silk strings, which keep it firm on every side. The point of ihis 
 bar must rise some feet above the top of the chimney: 12 or 15 feet, and even less 
 will be sufficient. Every time then that an electric cloud passes over the chimney, 
 the bar will emit signs of electricity, if touched with caution, or by means of a few 
 electric bells arranged near it. 
 
 Instead of this apparatus, Father Cotte, an assiduous observer of all meteorological 
 phenomena, places in a transverse direction, between two elevated places, an iron 
 chain, the links of which are furnished with sharp points. The two ends of the 
 chain are supported by silk strings, covered with pitch. From the middle of the 
 chain proceeds another, of the same form and size as those used for electric experi- 
 ments, which is conveyed into the apartment, either through the chimney or the 
 window, by means of silk strings which support it. At the end it ought to be fur- 
 nished with a metal ball, which will produce sparks much more considerable than the 
 chain itself would do. The multitude of points, with which the chain is covered, 
 furnish such a quantity of electric matter, that the ball must not be touched without 
 great circumspection. 
 
 Remark.—This curious experiment, highly interesting to philosophy, was. proposed 
 and announced by the celebrated Dr. Franklin, in letters addressed to Mr. Collinson, 
 fellow of the Royal Society ; but it was performed, for the first time, at Marly, by 
 M. Dalibart and M. Raulet the Curé of that place, on the 10th of May 1752. It was 
 
750 ELECTRICITY. 
 
 afterwards exhibited before the king and the whole court. Since that time it has 
 been repeated by various philosophers, and at present nothing is more common than| 
 this electrical apparatus, which shews the identity of the electric fluid and lightning, 
 But it is to America, and to Dr. Franklin in particular, that we are originally ine 
 debted for it. 
 
 From this discovery we can deduce the explanation of several phenomena; for. 
 which no proper cause had been before assigned. Of this kind are those fires often, 
 observed during storms, on the top of steeples, at the extremity of the masts and yards’ 
 of ships, which the ancients distinguished by the names of Castor and Pollux, and. 
 which are known to the moderns under that of the fire of St. Elmo. It is nothing: 
 else than the electric fluid of the clouds attracted by the points of these steeples, or 
 the iron at the summits of the masts. Cesar relates, that a great storm having come. 
 on while his army was arranged in the order of battle, flames were seen to issue from. 
 the points of the soldiers’ pikes. This phenomenon has nothing wonderful in it to: 
 those acquainted with electricity. The flames observed were the electric fluid, which: 
 escaped from these points; the clouds, in all probability, being electrified negatively 
 which, according to Dr. Franklin, is often the case. 
 
 EXPERIMENT XIX. 
 Which proves the same fact in another manner.— The Electric Kite. 
 
 It is sometimes difficult, if not impracticable, to raise aniron rod to a great heights. 
 and therefore another artifice has been devised to deprive the clouds, in some measure, 
 of their electric fluid or lightning. It is by means of the paper ee a small machine 
 more employed before that time by young persons and school-boys than by philoso- | 
 phers; but the use made of it by some of the latter has in some measure en- ' 
 nobled it. 
 
 Provide a kite, covered with silk, of a pretty large size, such as 5 or 6 feet in- 
 length at least; for the larger it is the higher it will rise, on account of the weight | 
 of the cord being less, in comparison of the force with which the wind tends to 
 elevate it. Adapt to the head of it, a very delicate rod of iron, extending on the one. 
 side, along the lower axis of the kite, to the point where the cord is affixed to it, and, 
 on the other, terminating in a sharp point projecting beyond the kite; so that when | 
 the machine is at its greatest height, it may be nearly vertical, and rise above it about 
 afoot. The cord may be of common pack-thread, with a very flexible copper-wire | 
 twisted round it, nearly in the same manner as on the lower strings of some musical in- | 
 struments, butmuchcloser. Thisis done, because hemp, unless moistened, is a bad | 
 conductor of electricity. To the extremity of this rod is attached another of silk, 
 some feet in length, to insulate the kite, when it has reached its greatest height; 
 and near this silk string is connected with the cord of the kite a small tube of tin-. 
 plate, about a foot in length and an inch in diameter, for the purpose of drawing sparks 
 from it. 
 
 When these arrangements are made, expose the kite to the wind, when you 
 observe a storm approaching, and suffer it to rise to its greatest height. If the silk 
 string be then made fast to some fixed object, and in such a manner that the string 
 shall not be moistened by the rain, you will not fail to observe very often exceedingly 
 strong signs of electricity, and sometimes so powerful that it would be dangerous to 
 touch the string or tube without the utmost caution. 
 
 For this purpose, affix to the end ofa glass tube, or a large stick of Spanish wax 
 of about a foot long, a piece of iron some inches in length, having a small metal 
 chain hanging down from it to the earth. Without this precaution, weak sparks only 
 would be elicited, because the piece of iron, being itself insulated, would on the first 
 touch be electrified like the cord of the kite. 
 
 M. de Romas, the first person in Europe who employed this method of drawing 
 electricity from the clouds, caused a kite 7} feet in length, and 3 in breadth, at its 
 
 x. 
 
eh mw 
 
 ELECTRICAL EXPERIMENTS. Vol 
 
 _widest diameter, to rise to the perpendicular height of 550 feet, and produced by it 
 the most extraordinary effects. Having imprudently touched the tube of tin-plate 
 _with his finger, he received a violent shock ; but happily the electricity had not nearly 
 acquired its utmost strength; for the storm increasing, some time after he felt, at the 
 distance of more than three feet from the cord, an impression similar to that made by 
 a spider’s web; he then touched the tube of tin-plate with the discharging-rod, and 
 extracted a spark of an inch in length and three lines in diameter. The electricity 
 ‘then increasing in a very great degree, at the distance of more than a foot, he ex- 
 tracted a spark three inches in length and three lines in diameter, the snapping of 
 »which was heard at the distance of 200 paces. 
 + But what was most remarkable in this experiment is, that while the electricity 
 ‘was nearly at its highest degree, three straws, one of them a foot in length, stood 
 ‘upright in consequence of the attraction of the tin-plate tube, and balanced them-_ 
 “selves for some time between it and the earth, always turning round, till one of them 
 cat length rose to the tube, and produced an explosion in three claps which were heard 
 in the middle of the town of Nerac, the experiment having been performed in the 
 ‘suburbs. The spark which accompanied this explosion, was seen by the spectators 
 like a spindle of fire, 8 inches in length, and 4 or 5 lines in diameter. The straw 
 which had occasioned this spark at last moved along the string of the kite, some- 
 times receding from it, and sometimes approaching it, and producing a very loud 
 ‘snapping when it came near it. Some of the spectators followed it with their eyes, 
 .to the distance of more than 50 toises. 
 
 Farther details respecting this experiment, no less interesting than curious, may be 
 ‘seen in the ‘‘ Mémoires des Scavans Etrangers,” published by the Royal Academy of 
 ‘Sciences, vol. ii. It was followed by a great many others of the same philosopher, 
 
 which prove that, even during calm weather, a kite of this kind is sometimes so 
 highly electrified, as to make the cord to sparkle, and to give violent shocks to those 
 ‘who inadvertently touch it. 
 
 We have already said that M. de Romas was the first person in Europe who made 
 this curious experiment ; but it had been made some months before in Pennsylvania 
 by Dr. Franklin; for he sent an account of it to Mr. Collinson, his correspondent at 
 London, in the month of October, 1752. This discovery however was not known in 
 France till a long time after; and M. de Romas even announced it enigmatically to 
 ‘the Academy of Sciences, about the middle of the same year. Thus, while we 
 -adjudge the first merit of the invention to Dr. Franklin, we cannot help acknowledg- 
 ing that M. de Romas concurred in this respect with the celebrated philosopher of 
 Philadelphia. 
 
 : EXPERIMENT XX. 
 The House struck by Lightning. 
 
 Dr. Lind is the author of this experiment, which serves to prove the difference 
 | between the effects of the explosion of thunder,when received on a blunt end or ball, 
 ‘or on the sharp point of an uninterrupted conductor. It displays in the fullest light 
 
 the advantage of good conductors for preserving houses from lightning. ; 
 aB (Fig. 50.) is the model of a small house, of which 
 
 cis the summit of the roof; A Disa wall in which is 
 formed the square hole, GF HE, destined to receive a 
 square board ; in this board is placed diagonally an iron 
 rod, which according to the position of the board can be 
 disposed in the direction F E or GH, as seen in the figure. 
 Lc is an iron rod terminating in a ball 1, which ends at 
 the point c. From u to 1 is another rod of the same 
 kind, the end of which 1 terminates in a chain of a 
 length proper for the purpose intended. 
 
 When this arrangement has been made, place the board 
 
752 ELECTRICITY. 
 
 * 
 in such a manner that the rod, sunk in it, may be in the direction FE; leaving an 
 interruption from G to H. Then make the chain pass round the body of a jar, like 
 those used for an electric battery, and charge the jar as highly as possible. Then affix 
 to one of the legs of the discharging rod, furnished with a glass handle, the chain of 
 - the conductor, and with the other leg of the rod touch the ball 1, which rises above’ 
 the roof of the house, and the rod cc. An electric circle being thus formed, a strong 
 explosion will ensue, and the board F G # &£ will be thrown from its place on account 
 of the jump which the electric matter must make from G to H, to reach the conduc. 
 tor, which is interrupted in that place. Ba 
 
 But, instead of a rod terminating in a ball L, substitute a rod ending ina sharp 
 point, and place the board Faux in such a manner, that the small rod rE ¥ ma 
 be in the direction au; if you then repeat the same operation as before, 
 the electricity will silently pass along the rod L@u1, without displacing any 
 thing. : fe 
 This is an exact representation of what takes place when a building is struck by 
 lightning. _The top of the building receives the shock, and the lightning follows 
 the first metallic conductor it finds, without doing it any damage, provided it be of 
 a sufficient size; but if this conductor be any where interrupted, it makes an exe 
 plosion, and blows to pieces the walls, the wainscoting, &c., till it finds a new con-— 
 ductor. At every interruption a new explosion takes place, to the great danger of 
 those who are in the neighbourhood ; for as the body of a man is a pretty good con- 
 ductor of electricity, on account of the fluids with which it abounds, it attacks him 
 in preference, and infallibly destroys him. 
 
 But if the rod, elevated above the house, terminates in a sharp point, and if the 
 conductor is not interrupted, nothing of this kind will take place. ‘There may be 
 some slight explosion at the point of the bar; but the electric fluid or lightning 
 thence follows the conductor to its extremity, which is sunk in the earth to a depth 
 sufficient to reach moisture. 
 
 M. Sigaud de la Fond, professor of natural philosophy, rendered this experiment 
 still more sensible, by the disposition he gave to the small house. It was such that 
 the electric explosion blew up the roof, and separated the walls. t 
 
 EXPERIMENT XXtf, 
 
 The Ship struck by or preserved from Lightning. 
 
 This experiment, in some respects, is merely a variation of the preceding. We 
 have introduced it, however, because it is no less amusing, and is equally calculated 
 to prove the utility of uninterrupted metallic conductors, for preventing damage by — 
 lightning. a 
 In the middle ofa small boat, representing the hull of a vessel, raise a tube about. 
 eight inches in height, and half an inch in diameter, to represent the main-mast. Fill _ 
 this tube with water; and, having closed both its extremities with two pieces of 
 cork, introduce through them two pieces of iron wire, so that the ends of them 
 shall be at the distance of half an inch from each other, in the inside. The lower 
 
 piece of wire must be immersed in the water in which the vessel floats, and the 
 upper one must terminate, without the tube, in a small knob. | 
 Now, if a communication be established between the exterior surface of an electrie x 
 
 battery and the lower wire, and if the end of the iron chain, which is connected 
 with the inside of the battery, be applied to the end of the upper wire, the explosion 
 of the electric matter, passing from the end of one wire to that of the other in the 
 tube, will be such, even if a small part only of the chain be employed, that it 
 wili shatter the tube to pieces ; and the bottom of the vessel being pierced, it will _ 
 tnk. Such is the manner nearly in which the main-mast of a ship is shivered by — 
 ightning, so that the vessel is in danger of being lost. * 
 
ELECTRICAL EXPERIMENTS, | 753 
 
 But if instead of two wires, one only be made to pass through the two pieces of 
 cork and the water with which the tube is filled, and if the same communication be 
 established with the electric battery, the charge of sixty-four jars may be trans- 
 mitted through the tube, without doing it any injury. Sometimes however the 
 force of the electric matter, or of this small flash of artificial lightning, will be so 
 great, as to destroy the metallic wire. 
 
 This experiment was invented by Mr. Edward Nairne, and might easily be adapted 
 to represent, in a manner more agreeable to reality, the phenomena of a vessel struck 
 by lightning ; but we have chosen rather to describe it as given in the Philosophi- 
 cal Transactions for the year 1773. It clearly shews how dangerous the interrup- 
 tion of metallic conductors is, and how the smallest conductor, if properly con- 
 tinued, will carry off the electric fluid. 
 
 GENERAL REMARK. 
 
 On the analogy between lightning and the electric fluid.— The means of securing houses 
 Srom the effects of lightning. 
 
 Though the identity of lightning and electricity is sufficiently proved by the pre- 
 ceding experiments; to establish it more completely, we shall mention some of 
 the phenomena most commonly observed in the progress of lightning, when it 
 falls-on a house or any other object. . 
 
 The first of these phenomena, or what takes place most frequently, is, that the 
 lightning runs along metallic bodies, wherever it meets with them in its Wa}. 
 For want of metallic bodies, it explodes, or attaches: itself to moist bodies, or to 
 animals, which are composed almost entirely of fluids. Hence it is often observed, 
 when the lightning falls on steeples, that from the weather-cock or cross on the 
 summit, which receives the first shock, it runs along the iron work, proceeding thence 
 to the roof or to the inside of the building, and there explodes; for as it no longer 
 meets with any thing besides wood or stone, which are bad conductors, it cannot 
 conveniently pursue its course; it therefore often strikes men who are in the 
 steeple, in consequence of a bad custom which prevails of ringing the bells on such 
 occasions. Sometimes it falls on the bell, and follows the rope to its extremity ; 
 and if the rope at that moment is held by a man, he seldom escapes destruction ; 
 for being a better conductor than hemp, the lightning seems to give him a fatal pre- 
 ference. 
 
 It very often happens that the lightning melts the lead of the cross, which it 
 strikes rather than other bodies, which are worse conductors. 
 
 - We may thence explain also, why it has happened that a man with a sword by his 
 side has been struck by lightning, without sustaining any hurt, and that the point of 
 the sword had been found melted in the scabbard : it is because the electric fluid pre- 
 ferred passing through the sword, entering at the hilt, and issuing at the extremity; 
 and as this extremity terminates in a sharp point, it found it more compact, and 
 reduced it to a state of fusion. This effect may be imitated; by causing a large 
 quantity of the electric matter to pass through a sharp-pointed wire. 
 
 When lightning falls on a tree, if there be any animals beneath, they rarely escape, 
 especially if the tree be of a resinous or oily nature. The reason of this is, that 
 wood is a bad conductor; the lightning therefore abandons it if there be a better 
 conductor, such as an animal, in the neighbourhood. ‘Hence it happens that the wal- 
 nut tree is reckoned to be particularly dangerous: its oily sap renders it a worse con- 
 ductor of electricity than any other. 
 
 But it is when lightning falls on a house that its predilection for metallic bodies 
 principally appears. Almost all the accounts of the effects of lightning agree, in 
 representing the electric matter as preferring to run along the wires fixed to the 
 bells, or the metallic edges of cornices, or looking-classes, or pictures, &c-y exploding 
 
 & 
 
= + 7 
 * : ‘ { - aa 
 cf 
 er 
 
 754 ELECTRICITY. 
 
 every time that this route, which it finds most commodious, is interrupted. It has 
 been seen to pass in this manner, through several apartments, and even through | 
 several stories. This route indeed is so well established by the observations which | 
 
 have been made, that there is every reason to believe, that if these metallic conduc- | 
 ’ tors had been wanting, or had been insufficient, it would have: occasioned great 
 damage. . 
 
 One of the best related, and most remarkable accounts of such events, is that of 
 the lightning which struck the hotel occupied by Lord Tilney at Naples, on the 20th 
 of March 1773. We are indebted for it to Sir William Hamilton, who was present | 
 in the apartment, through which the lightning passed, together with M. de Saussure, — 
 professor of Natural history of Geneva, and they both soon after examined the whole 
 hotel with the utmost care, in order to trace out the progress of the meteor. The 
 circumstances of this event were as follow. | 
 
 The apartments of Lord Tilney, which consisted of nine rooms on a floor, were | 
 decorated with great elegance, like most of those belonging to persons of rank in that 
 country. <A very large cornice went round all the rooms, and this cornice was gilt 
 in the Italian manner; that is to say coated with tin-foil, covered: by a yellow var-_ 
 nish, in imitation of gold. From this cornice proceeded a great number of plat-— 
 bands, which served as frames to the tapestry, and which were gilt in the same man- 
 ner, as well as the borders of the panels of the wainscoting, the frames of the pic- 
 tures, and the mirrors, the door-posts, &c. The apartments above were ornamented 
 inthe same style. This hotel had a profusion of such ornaments, and it is to be 
 observed that all the rooms communicated with each other by means of the bell- wires 
 which, for the sake of convenience, were very numerous. 
 
 Lord Tilney had a party to dinner, and Sir William Hamilton says, that on this 
 occasion there were in the hotel upwards of 500 persons, including the domestics. A 
 loud clap of thunder was heard, and in an instant the whole apartment, where the 
 company were assembled, seemed to be on fire. Every one thought himself struck 
 by the lightning, and the terror and confusion which this circumstance produced 
 may be easily conceived. No person however was either killed or wounded; and 
 this. no doubt was owing to the prodigious quantity of metal conductors, which 
 enabled the lightning to pass through them. 
 
 Sir William Hamilton and M. de Saussure, having examined, soon after, and néxt 
 day, the different apartments, observed that the greater part of the extensive cor- 
 nices were damaged, and black in a number of places, particularly at the corners, and ‘ 
 where the bell-wires passed through them ; the gilt varnish was detached in many 
 parts, and thrown down in the form of powder; in some other places the cords of 
 the bells were burnt.’ In one room where two paintings were suspended, one above 
 the other, between the cornice and the door, the lightning had passed from the cor- 
 nice to the gilding of the frame of the picture immediately below; then to that of 
 the second picture, and thence to the frame of the door, and its course was marked 
 on the wall, which was whitened according to the custom of the country, by the 
 impressions of the smoke. In another room, the lightning had also passed from the 
 cornice to the frame of a picture, which was in contact with it, and thence to the 
 interior border of the frame of the door, making an explosion each time; it had then 
 descended along the frame of the door, and had split part of a small socle, at which 
 the mouldings terminated. The same phenomena nearly had taken place in the 
 upper story. 
 
 It may be seen, by this description, that the lightning had preferred passing through 
 all these metallic materials ; and there can be no doubt that it was owing to the 
 great profusion of gilding, and to the wires of the bells, that some of the persia 
 present escaped being killed. fe 
 
 The predilection which the electric matter, or lightning, shews fos metallic con-— 
 
 Yi 
 
ELECTRICAL EXPERIMENTS, 700 
 
 ductors, induced Dr. Franklin, about the year 1752, to propose, at Philadelphia, a 
 new method of preserving edifices from this destructive meteor. It consists in plac- 
 ing on the tops of the houses an iron rod, terminating in a point, and continued 
 downwards by several more rods joined together. The lowest rod ought to be sunk 
 in the earth to a sufficient depth to meet with moisture, which, being a good conduc- 
 tor, will convey off the electric matter, by transmitting it to the whole mass in the 
 earth. In regard to the size of the rod, Dr. Franklin observes that three or four 
 lines in diameter will be sufficient. 
 
 In the year 1755, a great many houses in North America, and particularly in Penn- 
 sylvania, Maryland, and Virginia, where thunder is very common, and often falls on 
 buildings, were furnished with conductors of this kind. It is allowed that several of 
 these houses were struck by lightning; but the circumstances always observed in 
 America, were as follow: Ist, that the damage done to these houses was less. 2d, 
 
 - that when they were struck, the lightning, instead of occasioning the same havoc as 
 in others, passed off by the conductors, leaving only a slight impression in the neigh- 
 bouring parts. In these cases the point of the conductor was, for the most part, 
 found to have been fused. 
 
 The object of these rods is not indeed, as was at first supposed in Europe, to 
 deprive an immense cloud of its electricity, but to furnish a conductor to that electri¢ 
 matter, when by an accident, which cannot always be avoided, a cloud highly charged 
 with electricity has fallen on an edifice. 
 
 This expedient, however, found powerful opponents, especially in France. One 
 of the most conspicuous was the celebrated abbé Nollet, the rival of Dr. Franklin in 
 the theory of electricity; but it must be allowed that nothing could be weaker than 
 the arms with which the French philosopher combated the American. They were 
 mere assertions, unsupported by any proofs, or rather contrary to the result of expe- 
 riments. According to Nollet, these pointed rods of iron are more calculated to 
 attract the lightning than to preserve from it ; and it is not a rational project for a 
 philosopher, says he, to exhaust a stormy cloud of the electric matter it contains. 
 To answer these assertions, it is sufficient to be acquainted with the effects of light- 
 ning. They prove, in the most evident manner, that if the places where it fell had 
 been furnished with good conductors, no explosion would have taken place. Besides, 
 it is not true that a sharp-pointed rod attracts the electricity of a storm-cloud; on 
 the contrary, if a sharp-point be presented to a flake of cotton, suspended from the 
 
 ~ conductor of the electric machine, it immediately repels it. Is it therefore better to 
 wait till a storm-cloud, charged with electricity, and driven by the wind. against a 
 
 - building, shall explode and pour into it a torrent of the electric fluid, than to draw it 
 
 off by degrees, so that when it approaches the edifice, it shall be entirely deprived of 
 it? In regard to the impossibility of freeing a cloud entirely from its electric matter, 
 it is needless to make many observations; as all that is meant is merely to supply the 
 
 electric matter, poured forth from a storm cloud, with the easy means of escaping. 
 
 But, when it is considered, that every time almost, that lightning has fallen in any 
 
 _ place, without doing damage, it has followed conductors as small as a bell-wire, or 
 gilding., &c, and that it has never exploded but when its course was interrupted, 
 
 there can be no doubt that a rod, half an inch or an inch in diameter, would afford a 
 passage to all the electric fluid that could be produced by the largest cloud. 
 Sharp-pointed conductors, considered as preservatives against the effects of light- 
 
 ning, were opposed in England by the noted electrician Wilson, on the following 
 occasion. The method proposed by Dr. Franklin, for preventing the effects of light- 
 ning, having excited the attention of the government in 1772, the Royal Society of 
 
 London were consulted on the means of securing from this destructive agent the 
 new powder magazines at Purfleet. The Society having appointed Mr. Cavendish, 
 
 Dr. Watson, Dr. Franklin, Mr. Wilson, and Mr. Robertson to examine this subject ; 
 
 a2 
 
: 
 4° 3 
 ; 
 ~ 
 " 
 
 four of these gentlemen were of opinion, that the magazines should be furnished with _ 
 sharp-pointed conductors. Mr. Wilson alone maintained that the points of the con- | 
 ductors ought to be blunt, and he refused to sign the report. It may be easily seen 
 that Mr. Wilson’s motive was an apprehension that sharp-pointed conductors might 
 attract the electric fluid at too great a distance. Dr. Franklin, in a paper written on 
 purpose, which contains an account of new and ingenious experiments, endeavoured 
 to make him change his opinion; but did not succeed. The magazines of Purfleet 
 however were furnished with conductors according to the idea of Dr. Franklin and 
 the other three commissioners. 
 
 The sequel of this business gave rise to the most extraordinary transactions that 
 ever occurred in the Royal Society. 
 
 706 ELECTRICITY. 
 
 EXPERIMENT XXII. 
 
 Of some amusements founded on electric Repulsion and Attraction.—The Electric 
 Spider, §c. 
 
 Cut a small bit of cork, or of the pith of the elder tree, into the form of a spider, 
 and fix to it six or eight cotton or linen threads, a few lines in length. Suspend this 
 small figure from a hook by a silk thread, and place on one side of it, and at the same 
 height, the knob of a small jar positively charged, and on the other that of a jar 
 negatively charged, or merely a similar knob not electrified, and communicating with 
 the general mass of non-electric bodies. You will then see this figure first attracted 
 towards the electrified knob, and then repelled by it: and as the filaments of the 
 threads will mutually repel each other, the spider will appear as if at work, and ex-~ 
 tending its legs to lay hold of the second knob. But it will have no sooner touched 
 it, than it will seem to fly from it; for when deprived of its electricity by the touch, | 
 it will be attracted by the first knob, from which it will be afterwards repelled; and 
 this play will continue as long as there is any electricity in the jar. ! 
 
 A common conductor charged with electricity will supply the place of the elec- 
 trified jar; and, instead of the electrified knob, the finger may be employed. The 
 spider, after having touched the conductor, will appear to throw itself on the finger, 
 as if to seize it, and will embrace it with its legs. 
 
 EXPERIMENT XXIII. 
 The Electric Wheel and Turnspit. 
 
 Construct a wheel consisting of eight or ten glass radii, implanted in a common’ | 
 centre, about six or eight inches in length, and each furnished at the extremity with 
 a ball of lead. El 
 
 This wheel must.be placed, in perfect equilibrium, on a small vertical axis, which 
 turns in a piece of glass, so that the slightest impulse can put it in motion. The 
 stand, by which it is supported, must be susceptible of being insulated. 
 
 Then provide two jars, one charged positively and the other negatively; and, 
 having insulated the above wheel, place the two jars one on each side of it, so 
 that the balls shall pass at the distance of a quarter of an inch from the knob of 
 each jar. ' 
 
 It may be easily conceived, that if this small machine be in perfect equilibrium 4 
 when one of the balls approaches one of the knobs, that for example which belongs to _ 
 the flask charged positively, it will be attracted by it, and the machine will tend to _ 
 turn round; but the ball, by passing near to that knob, will be electrified positively, 
 and consequently will be immediately repelled. 2 
 
 The same thing will take place in regard to the flask which is charged negatively: 
 the non-electrified ball will be attracted by it, and in ‘passing near it will be elece | 
 trified negatively ; it will therefore be repelled, as soon as it has passed it. eon 
 
 _ As the same thing takes place in regard to all the other balls, the result will be 4 
 
ere i 
 \ 
 \ 
 
 fod 
 ELECTRICAL EXPERIMENTS. 157 
 
 a circular motion, which will be accelerated more and more, and will continue as long 
 as the two jars are in a state of electricity. But they may be easily kept in motion, 
 by making the knob of a jar strongly charged touch that of one of them, and the 
 knob of the other the side of the same jar: by these means the one will be charged 
 positively, and the other negatively. 
 
 When the electricity is very strong, and the machine is well constructed and in 
 equilibrio, it acquires a motion capable of turning a weight of several pounds placed 
 on its vertical axis. 
 
 The electricians of Philadelphia employed this apparatus to turn a spit, when a 
 party of them met to amuse themselves with philosophical experiments. Being per- 
 suaded, no doubt, that Reason must sometimes throw itself into the arms of FollP, 
 they assembled on the banks of the Skuylkill, a river which runs past Philadelphia, 
 and having killed a turkey, by the electric shock, they placed it on a spit adapted to 
 an electrical jack, and roasted it at a fire kindled by the electric spark ; they then drank 
 to the health of the European and American philosophers who cultivated electricity, not 
 amidst the noise of musketry, but of electric batteries discharged at each toast. Dr. 
 Franklin, the first of the philosophical electricians, calls this an electrical feast. 
 
 EXPERIMENT XXIV. 
 The Electric Alarum and Electric Harpsichord. 
 
 Suspend from the conductor of an electric machine, three bells, at the distance of 
 about an inch from each other ; but the outer ones must be suspended by threads which 
 transmit the electric fluid, and that in the middle by a silk thread, or other electric 
 subtance. The bell in the middle must communicate, at the same time, with the 
 floor, by means of a small chain or metallic wire. 
 
 At equal distances, between each of these three bells, suspend ‘vy silk threads two 
 small balls of metal, in such a manner, that when pulled a little to the right or left, 
 they shall strike against the bells. 
 
 If the conductor be now electrified, you will immediately see these small clappers 
 put themselves in motion, and strike the bells alternately, which will form a small 
 alarum ; and if the electric apparatus be concealed it will be difficult for those pre- 
 sent to discover the cause of it. 
 
 _ The cause however of this continued play may be easily discovered ; for, by the 
 construction of the apparatus, the two lateral or outer bells are electrified, as soon 
 as the electric machine is putin motion. The small balls suspended between them 
 and that in the middle will therefore be attracted by these bells; but as soon as they 
 touch them they will be repelled, being electrified in the same manner as they are; 
 they will then be carried towards the middle bell, which having a communication with 
 the floor, will immediately deprive them of their electricity. They must therefore 
 fall back towards the electrified bells, which will attract them again; and this play 
 will continue as long as the electric machine is kept in action. 
 
 _ Remark.—On this principal, an instrument called the electrical harpsichord has 
 deeuinvented. The following is a short account of this ingenious machine, for which 
 we are indebted to Father de la Borde, a Jesuit, who gave a description of it in a 
 mall work published in 1759. 
 
 Suppose an iron rod, supported by silk strings, and furnished with two rows of 
 yells, each two of which are capable of emitting the same sound, because two will 
 erequired for each tone. One of these bells must be suspended from the rod by a 
 vire, so that when it is electrified the bell may be electrified also. The other must 
 ye suspended by a silk string, and between each pair of bells a small ball of steel must 
 e suspended by the same means. 
 The bell suspended from the bar at the top, by the silk string, is furnished with a 
 
Y. . ¥ y : : ee 
 é : i aS 
 3 r. : 
 
 ; 
 758 ELECTRICITY, | 
 wire which proceeds downwards, and is fixed by another silk string. To its lower 
 extremity is fastened a small lever, which in its usual position rests on another 
 insulated bar, communicating as well as the other with the conductor of an electrical 
 machine. 
 
 In the last place, below this second bar, 1s a harpsichord so disposed, that when 
 one of its keys is touched, the other extremity of it raises up the corresponding lever; 
 this intercepts the communication of the bell with the electrified conductor, and 
 establishes a communication with the general mass of the earth. 
 
 After this description, it may be easily conceived, that if one of the keys be 
 touched, while the electric machine is in motion, one of the bells being electrified, the 
 steel ball willimmediately advance towards the other, and being electrified by it, will 
 he repelled towards the first, which will deprive it of its electricity, and therefore 
 it will return to the other. This motion will take place indeed with great rapidity, 
 and the result will be an undulating sound resembling the vibrations of an organ. 
 When the lever falls down, the two bells are equally electrified, and in a moment’ the 
 steel ball stops. 
 
 Father de la Borde, naving constructed this machine, could py practice perform 
 on it, with a considerable degree of correctness, simple airs. But was it of sufficient 
 importance to be made the subject of a particular treatise, since neither the science of 
 music nor the theory of electricity could be much benefited bv it? 
 
 EXPERIMENT XXV. 
 The Electrical Horses which pursue each other ; -or the Electrical Horse-race. 
 
 With two small iron plates, or two small wires, construct a sort of cross, having @ 
 piece of copper in the centre, so as to represent two magnetic needles crossing each 
 other at right angles. The ends of these four branches must terminate in a point, and — 
 about an inch of the extremities of them, more or less according to the size of the 
 machine, must be bent back so as to form somewhat less than a right angle. On 
 these bent ends fix a small bit of light card, and place on each the figures of horses 
 having their tails turned toward the points. Then arrange the whole ona steel pivot, 
 raised in a perpendicular direction, that the cross with itsload may preserve itself in 
 a horizontal direction, and have a very easy rotary motion. i 
 
 Having then insulated the machine and its plate, if the latter, or the point of steel, 
 be made to communicate with the electrified conductor, you will soon see these four 
 branches of iron assume, as if spontaneously, a rotary motion in a direction contrary 
 to that in which their extremities are bent; so that the four horses will seem to 
 pursue each other ina circular course. And this play will continue as long as the 
 electricity lasts, and even longer, on account of the acquired velocity. 
 
 If the experiment be made in the dark, and without the horses, that is to say with 
 the four points alone, you will see pencils of light or electric fire issue from them, — 
 which will form a very agreeable spectacle, as the result will be a ring of fire; and 
 this ring may be rendered larger by giving unequal lengths to the branches of the cross. 
 
 Several stages of wires, placed in the form of a cross, each stage always decreas- 
 ing in size, might be constructed, and by these means a luminous pyramid would be. : 
 formed. 
 
 The cause of this apparently spontaneous motion may be easily conceived. It is 
 the impulse of the electric matter, issuing from the points, which meeting with - 
 air, experiences a reaction, and is impelled backwards. e 
 
 Remark.—Some have pretended to deduce from this experiment a pretty strong 
 objection against the hypothesis of Dr. Franklin; for whether this small machine be q 
 electrified either positively or negatively, the motion takes place in the same direc- 
 tion, which has astonished those even who are decided Franklinians. To us this 
 objection appears to have little weight; for in our opinion it might be said, that in 
 
ELECTRICAL EXPERIMENTS. 759. 
 
 the case or negative electricity, the electric fluid which is thrown into the points, can- 
 not be absorbed, without communicating to them an impulse, which acts exactly in 
 the same direction as the repulsion experienced by the electric fluid, on issuing 
 from the points when they are electrified positively. 
 
 EXPERIMENT XXVI. 
 
 _ To cause writing in luminous characters to appear suddenly, by the means of 
 electricity. 
 
 This electric amusement is founded on a well known observation, that if several 
 metallic wires be disposed together in such a manner that their ends, without being 
 in contact, shall approach very near to each other, so as to be at the distance of a line 
 or halfa line; if the first be electrified, while the last has a communication with the 
 mass of non-electric bodies. sparks will continually be emitted between the ends of 
 these wires. 
 
 The same thing will take place if the last of these wires terminate in a point ; 
 for as it will by these means lose its electricity, there must be a continual afflux 
 of new matter; and this cannot be the case without causing a spark in each of the 
 small intervals, that separate the ends of the wires. 
 
 This being understood, it may be easily conceived that a series of sparks, forming 
 any representation whatever, with some limitations, which will be seen hereafter, 
 might be produced, by arranging the ends of the wires along the outlines of any 
 figure. If the last of the wires be then touched by the finger, or, what will be 
 still better, with the exterior furniture of the Leyden flask, there will be instantly 
 formed, in the intervals between these wires, sparks representing the contour of the 
 figure. 
 
 But, as this would be attended with difficulties, it may be easily executed in the 
 following manner. Cut a leaf of tin-foil into small pieces, of a line or half a line 
 square, or into the form of a rhombus somewhat elongated. Then delineate on 
 paper the letters you intend to represent, and having puta plate of glass, about a 
 line in thickness, on the drawing, cement to the glass the small squares of tin-foil, 
 or the rhombs above described, according to the outlines; taking care that the 
 angles correspond to angles, and that they be at the distance 
 from each other of about half a line, as seen in the delinea- 
 tion of the letter S, (Fig. 51.) Then connect the extremity 
 of one letter with the commencement of the following one, 
 by a small bent metallic plate, terminating on both sides in a 
 sharp point, as seen in the same figure; a small plate of the 
 like kind at the commencement of the first letter, and 
 another from the end of the last, must proceed to the edge of the glass, and be. 
 yond it. 
 
 _ Let us now suppose that the first of these small plates has a communication with 
 
 the electric conductor, and that you touch the second, or vice versd; each angle 
 of the small squares will convey the electric fluid by a spark to its neighbour; and 
 if the experiment be performed in the dark, these two letters will be perceived as 
 if delineated by a series of luminous sparks. 
 
 If the last plate communicate with a mass of non-electric bodies, and if the elec- 
 tricity be strong, an explosion will take place between each square, which will render 
 this writing luminous. A very brilliant effect is produced by passing an electric 
 shock through the circuit proposed as above. 
 
 Fig. 51. 
 
 Remark.—But it is to be observed, that all the letters of the alphabet cannot be 
 represented in a manner so simple as the two here given by way of example. Thus 
 the O cannot be represented by this method, as the electric fluid, instead of going 
 
7609 ELECTRICITY. 
 
 round in a circle, would proceed from the first to the last square. The A also would 
 remain truncated at its upper part, as the electric matter would pass through it. A 
 particular artifice therefore is necessary to obviate this inconvenience, which occurs 
 in regard to a great many other letters, such as E, F, H, &c. . 
 This artifice consists in delineating one half of the letter on one side of the glass, 
 and the other half on the other, and in forming a communication between them by a 
 small metallic band, which, proceeding from the upper to the lower side of the glass, 
 may convey the electric matter from the 
 last square of the first half of the o (Fig. 
 52.), for example, to the first square of the 
 second half of the same letter, and then 
 uniting, by a similar band, the last square 
 of that second half with the first square 
 of the following letter. If Fig. 52 be carefully examined, the mechanism of this 
 amusement will be easily comprehended. The letters, or parts of letters, represented _ 
 on the upper side of the glass, must be strongly shaded, and those on the lower 
 lightly. As the propagation of the electric fluid is instantaneous, no inconvenience 
 can arise from this method of transmission. 
 It may be readily seen that such an artifice, in the times of superstition, migh 
 have been employed to terrify the ignorant. If a number of people, assembled in a 
 dark place, should see, after a clap of thunder, a luminous writing on the wall, con- 
 taining a pretended decision of the Deity, what would they not be capable of doing ? 
 and with what terror would a man be struck, who on waking should see written on 
 the glass, This day thou shalt die? 
 
 EXPERIMENT XXVII. 
 Electric Fire Works. 
 
 We shall here describe a new kind of spectacle. We will not absolutely warrant 
 its success, but we are inclined to think that our idea is susceptible of being carried 
 into execution. 
 
 An exhibition of fire-works is generally composed of a fixed decoration, repre- 
 senting an edifice, suited to the subject, and of various moveable pieces of fire ; such 
 as rockets, gerbes, cascades, fixed or revolving suns, &c.; and all these pieces, in 
 -our opinion, may be represented by merely electric fire. 
 
 Let us first assume, by way of example, a decoration of architecture, which is ~ 
 illuminated by a series of lamps, disposed in such a manner as to trace out its prin- | 
 cipal parts. Now, might not a series of points, rendered luminous by electricity, be 
 substituted instead of these lamps? The preceding experiment will furnish us with 
 the means of accomplishing this end; for since letters, the figures of which are much 
 more complex, may be rendered apparent by a series of similar points, lines, the 
 greater part of which are straight and parallel or perpendicular to each other, may be 
 represented with much more ease, by attending to the directions given in that expe- 
 riment. The following however is another method. 
 
 On a piece of very dry and well planed resinous wood, trace out the design 
 of your decoration, and mark by points the places where lamps would be suspended, 
 were it to be illuminated; then place at each of these points a piece of iron wire, 
 one or two lines in length, and terminating on the outside in a very delicate sharp 
 point; and form a communication between all these wires, by a long piece of wire 
 connected with them. If the electric matter be then excited in a powerful manner, 
 there is no doubt that each of these points will emit in the dark a small luminous 
 gerbe, which will trace out the design of your architectural decoration ; for it is 
 well known that a bar of iron, when strongly electrified, throws out in the dark 
 
ps . 
 
 ' 
 
 ELECTRICAL EXPERIMENTS. 761 
 
 from all its angles large luminous gerbes, which are sometimes several inches in 
 
 length. 
 
 Nothing is easier than to represent a gerbe of fire: a group of iron wires, termi- 
 nating in a point, will produce an assemblage of small gerbes, which together will _ 
 form one of a considerable size. 
 
 If a fixed sun is to be represented, it may be done by means of ten or twelve 
 
 points disposed in the form of radii, at the extremity of a wire which terminates ina 
 _ button, and if twelve points be arranged in a proper manner, they may form a star by the 
 
 emanation of the electric fluid: nothing will be necessary but to dispose them in the 
 same manner as the rockets are in common fire works, to represent the same thing. 
 
 If several pieces of iron wire, terminating in a point, and having a communication 
 with a common handle, be arranged in a semicircular form, in a direction inclined to 
 the horizon, they will represent a cascade, by the electric gerbes which issue from 
 
 - their points. 
 
 If the figure of a revolving sun be required, you must construct a cross similar to 
 that described in the 25th experiment; but instead of making it turn round ona 
 vertical axis, it must be brought into perfect equilibrium on a horizontal one. The 
 luminous gerbes which issue from the bent points will form a circle of fire, if the 
 motion be rapid, or something that has a near resemblance to a sun. 
 
 What may give to this exhibition an air of reality, is that it is possible to accom- 
 pany it with the noise of an electric battery, which will excite the idea of marroons 
 and -saucissons, a discharge of which accompanies in general other fire-works, if not 
 continually, at least at certain intervals. This might be done by means of small elec- 
 tric batteries, discharged partially and successively. 
 
 This, as already said, is merely an idea, which has need of being subjected to 
 experiment; but in our opinion an ingenious artist might turn it to advantage. It 
 may easily be conceived, that the electricity in this case ought to be strong ; but 
 what could not be done by one electric machine, might be performed, in all proba- 
 bility, by two, or three, or four. 
 
 EXPERIMENT XXVIII. 
 On the Electricity of Silk. 
 
 We shall here present the reader with a few more singular experiments made by 
 Mr. Symmer, who published them in the Philosophical Transactions for the year. 
 1759. 
 
 lst. During exceedingly dry, cold weather, when a north or north-east wind pre- 
 ‘yails, take two new silk stockings, the one black and the other white, and after 
 having heated them well, put them on the same leg: the action of putting them on 
 will itself electrify them. If you then pull them off, one within the other, making 
 them both glide at the same time on the leg, they will be found so much elec- 
 trified, that they will adhere to each other with a greater or less force. Mr. Symmer - 
 once saw them support, in this manner, a weight which was equal at least to sixty » 
 
 times that of one of them. 
 
 — ——$ ee oe 
 
 2d. If you draw the one from within the other, pulling one by the heel and the 
 other by the upper end, they will still remain electrified, and you will be asto- 
 
 nished to see each of them swell in such a manner as to represent the volume 
 
 of the leg. 
 3d. If one of these stockings be presented to the other at some distance, you will 
 
 see them rush towards each other, become flat, and adhere with a force of several 
 
 ounces. 
 4th. But, if the experiment be performed with two pairs of stockings, combined 
 
 ‘in the same manner, the one white and the other black, on presenting the white 
 stocking to the white, and the black to the black, they will mutually repel each 
 
762 ELECTRICITY, 
 
 other. If the black be then presented to the white, they will attract each other, 
 and become united, or will tend to unite, as in the third experiment. 
 
 Sth. The Leyden flask may be charged by these stockings. 
 
 It appears thence to result, that silk rubbed against silk is capable of electrifying; 
 but for this purpose a preparation must be given to one of the pieces: for two white 
 or two black stockings, placed one within the other, cannot electrify. But it is not the 
 black as black, opposed to the white as white, which produces this effect. The Abbé 
 Nollet has shewn, that the preparation here alluded to, is the operation of galling, 
 which precedes that of dying black; for two white ribbons, one of which only is 
 galled, if rubbed against each other properly, will produce the same phenomena 
 
 of adhesion, attraction, and repulsion. There can be no doubt that the case is the 
 
 same in regard to stockings. 
 
 The partisans of the Franklinian doctrine respecting electricity, will not find it 
 difficult to explain the other phenomena which have been mentioned. Each stocking 
 is electrified in a different manner, one positively and the other negatively; it 
 appears that it is the white which is electrified positively, or in the manner of 
 glass. The swelling up, observed in each of the insulated stockings, is only the 
 effect of repulsion between the bodies similarly electrified ; for all the parts of the 
 same stocking have received the same electricity. For the like reason, two stockings 
 of the same colour necessarily repel each other. 
 
 But, if a black stocking be presented to a white one, as their electricities are 
 different, these two bodies will attract each other; this phenomenon is well kuown, 
 and if not general does not fail to take place between two bodies electrified one 
 
 positively the other negatively, or the one in the manner of glass, and the other in 
 
 that of sulphur. 
 
 A. very remarkable phenomenon here is, that two bodies, the one electrified po- 
 sitively, and the other negatively, according to the language of the Franklinians, 
 may be applied to each other, without their electricity being destroyed.- Mr. Sym- 
 mer remarks this with some astonishment; and hence he was induced to deviate 
 from the Franklinian doctrine in assigning reasons for it, which, as the Abbé Nollet 
 observes, approach near to the explanation of the latter. 
 
 ~ 
 
 It has since been remarked, that the surfaces of two bodies, one of which is — 
 
 electrified positively, and the other negatively, may be applied to each other, with- 
 
 out their electricity being destroyed. This is the principle of the electrophorus, an 
 
 electric instrument, invented a few years ago. Nay more, these two surfaces 
 applied in this manner, retain their electricity muchlonger; but it does not mani- 
 
 fest itself when they are separated. Electricity is a mine, which, the more it is 
 
 searched, presents new phenomena difficult to be explained. How this is to be ex- 
 
 plained, according to the Franklinian theory, we donot know; and, though attached 
 to the science we shall not attempt it. 
 
 EXPERIMENT XXIX. 
 Which proves that electricity accelerates the course of fluids. 
 Provide a capillary tube; or a tube terminating in an aperture so narrow, that 
 the water which runs ‘throtgh it can issue only drop by drop. If this water be 
 electrified, you will immediately see it run out in a continued stream. 
 
 RemarKks.—On the consequences of this experiment, and the cures performed, or — 
 
 said to be performed, by electricity. 
 
 It is probable that it was this experiment which gave rise to the application of 
 electricity to medicine; for it was natural to reason in this manner; as electricity 
 accelerates the course of fluids, it is probable that it will accelerate that of the blood, 
 and of the nervous fluid in animals. But there are certain diseases which appear 
 
ELECTRICAL EXPERIMENTS. 763 
 
 to be merely the consequence of an accumulation of the nervous fluid, such as the 
 palsy, and different maladies depending on the same cause; as deafness, blindness, 
 &c; consequently, electricity, by accelerating either the course of the blood, or 
 that of the nervous fluid, may remove that accumulation, and so will produce a 
 cure of the disease. 
 
 Some physicians therefore began to electrify patients attacked by the palsy; and 
 it must be allowed, as we have the testimony of persons free from every kind of 
 suspicion, such as M. Jallabert of Geneva, and others, that the attempt was at- 
 tended with some success. It is certain that this celebrated professor, and citizen 
 ‘of Geneva, if he did not cure radically, at least greatly relieved a person of the 
 name of Noguez, afflicted with a paralytic disorder. This man, who was incapable 
 of raising up his arm, after being electrified three months, was able to raise up a 
 large hammer. 
 
 This cure, which was published in some of the journals, made a considerable 
 noise, as may be well supposed ; and a multitude of electricians, in various parts 
 of Europe, undertook the cure of paralytics, the dumb, the blind, &c. We have a . 
 collection, in three volumes, by M. Sauvages, not of these cures, for there are few 
 to which that appellation belongs; but of the progress of this application of elec- 
 tricity. There were some however very well established, such in particular is that 
 of a young man of Colchester, to whom Mr. Wilson restored the use of his eye- 
 sight, which he had lost after a violent fever. In regard to the greater part of the 
 rest, the application was of no effect. ; 
 
 It cannot however be denied, that on the first application of the electricity, the 
 patients in general experienced some relief. Paralytics felt, inthe palsied part, a kind 
 of heat and pricking, which seemed to announce the return of sensation; the blind 
 sometimes saw sparks of light; but in general nothing farther took place, and these 
 beginnings, which seemed to promise the greatest success, were not followed by 
 happy consequences. 
 
 Some Italian philosophers have asserted something more marvellous. About the 
 year 1750 they announced, at Padua, that electricity exalted and attenuated odours 
 to such a degree, that they passed through glass; that purgative drugs put into a 
 vessel carefully, and hermetically sealed, produced their effect on the person who 
 held the vessel in his hand, while it was electrified. This no doubt would have 
 been a noble discovery in medicine; but unfortunately this pretended discovery, 
 announced with great solemnity to all Europe, vanished entirely before the enlight 
 ened eyes of the Abbé jNollet, who undertook a journey to Italy to examine it. 
 He found, at least, that there was precipitation and misconception in all these fine 
 assertions, which could not be realized in his presence. Having repeated the expe- 
 riment himself several times, in his closet, he never found that the most penetrat- 
 ing odour passed through the pores of a glass vessel, when properly closed, whether 
 electrified or not electrified ; and the case was the same with the purgative emanae, 
 tions of cassia and rhubarb. 
 
 M. le Roy, one of the French philosophers, who cultivated this branch of philo- 
 sophy with the greatest care, was induced to try the effects of electricity on some 
 of his patients; the first of whom had been afflicted with a hemiplegia for three 
 years; another with a gutta serena; anda third with deafness. The electric mat- 
 ter conveyed several times through the palsied parts of the first, seemed in the 
 commencement to revive sensation ; the patient perspired a great deal, an effect 
 which could not be produced by all the medicines before administered to him. 
 After being electrified four or five months, sensation and the power of motion re- 
 turned to the palsied fingers, and the patient could lay hold of a glass, and convey it 
 to his mouth; he could even raise a weight of 40 or 50 pounds; but this com- 
 ~ mencement of a cure was all that could be effected; and as the patient received no 
 
764 ELECTRICITY. 
 
 additional benefit from a continuation of the same treatment, for four months more, — 
 it was laid aside as entirely useless. 
 
 M. le Roy received as little satisfaction in regard to his blind patient, though in 
 order to free the optic nerve from its obstruction, he had invented an apparatus, 
 by means of which he gave him gentle shocks through the head. The patient, at 
 the moment of the electric explosion through his head, perceived a flash, but after 
 the electricity had been applied some months, M. le Roy became tired, as before, of 
 administering a useless remedy. 
 
 The patients labouring under deafness were not more fortunate. M. le Roy 
 directed the electric fluid from one ear to the other. At each shock a kind of noise 
 was heard in the head, which one of them compared to all the petards of La Gréve. 
 But the auditory nerves were not cleared, nor was the deafness removed. An 
 account of the treatment of these patients may be seen in the Memoirs a the 
 Academy of Sciences, for the year 1755. 
 
 We have read, somewhere in the Philosophical Transactions, of an intermittent 
 fever being cured by electricity. This is not impossible; as the electricity, by acce- 
 lerating the motion of the fluids, might act as a tonic. 
 
 Some years ago, the Abbé Sans, Canon of Perpignan, announced at Paris several] 
 cures which had been performed in his country by electricity. He published them in 
 a particular work, with various certificates annexed; but his operations on M. de la 
 Condamine, attacked with total insensibility in one half of his body, and total deaf- 
 ness, were attended with no effect. These infirmities indeed had taken root for 
 several years, and therefore it would have been unjust to require success in any 
 attempt made to remove them. But we never heard that this electrician had ever 
 much suceess at Paris. 
 
 To conclude, it appears to us that too great hopes were at first conceived in regard 
 to the application of electricity to such diseases ; but that it was not entirely 
 without effect, and that in recent cases it might be tried with some hope of success. 
 The rheumatism, according to M. le Roy, seemed to oppose the least resistance to 
 this remedy, which perhaps acted by re-establishing perspiration. He obtained pro- 
 fuse sweats to the greater part of his patients. Jn short, there can be no doubt 
 that it occasions in the human body an universal orgasm, which under certain circum- 
 stances might be critical and advantageous. 
 
 EXPERIMENT XXX. 
 Natural and Animal Electricity. 
 During cold weather, lay hold of a cat, and draw your hand over her back several 
 times, in a direction contrary to that of the hair; by these means you will often excite 
 very strong sparks, which will emit a snapping noise. 
 
 Remark.—This however is not the only animal which exhibits the electric pheno- 
 mena by friction: even men, under certain circumstances, emit sparks which are— 
 absolutely of the same nature. There are few people to whom this circumstance is 
 not known by experience. It is during cold winters, and after being well heated, 
 that they exhibit this phenomenon. ~ On these occasions, if they throw off their shirt 
 in the dark, sparks more or less vivid, and accompanied with a sensible snapping, 
 will issue from it. Some, in consequence of a particular temperament, are more sub- 
 ject to this phenomenon than others. In all probability, these persons are very 
 hairy; for hair, as it approaches the nature of silk, is electric by friction; and 
 according to every appearance, it is the friction of the dry, warm linen, against the 
 hair, which is also dry and warm, that produces this electricity, and the sparks which 
 accompany it. In combing a person’s hair of a morning, in dry frosty weather, it 
 becomes very elastic, and manifests strong signs of electricity. x 
 
ELECTRICAL EXPERIMENTS. 765 
 
 These luminous appearances were formerly classed among the phosphoric pheno- 
 mena; but since the new discoveries in electricity, there can be no doubt that they 
 belong to the latter. 
 
 It would have been easy for us to enlarge this part of the Philosophical Recrea- 
 tions much more, by introducing into it a great number of other curious and surprising 
 experiments relating to the theory of electricity ; but as we must confine ourselves 
 within narrow limits, we shall conclude this part with a list of the principal works to 
 which those who are desirous of obtaining a thorough knowledge of electricity may 
 recur. Of this kind are ‘‘ Essai sur Electricité de M.1’ Abbé Nollet,” and in particu- 
 lar a work entitled ‘‘ Recherches sur les Causes particuliéres des Phénoménes élec- 
 triques,et sur les Effets nuisibles ou avantageux qu’on peut en attendre,” Paris 1754, 
 12mo; and to these may be added ‘‘ Lettres sur VElectricité,” 3 vols. 12mo. Though 
 the Franklinian theory seems, in general, to have many more partisans than that of 
 
 ‘the Abbé Nollet, it must be allowed that the latter applied to the study of this branch 
 
 of science with the greatest success. Besides the above, we may refer also to various 
 memoirs of the same author, in which he discusses the theory of Dr. Franklin, pub- 
 lished in the ‘* Mémoires de l’Academie, for, 1755 and 1760,” &c. ; and “* Recherches 
 sur les Mouvements de la Matiére électrique, par M. Doutour,’”’ 1760, 12mo. These 
 are the best of the works in which the theory of the French philosopher is illustrated 
 and defended. 
 
 The Franklinian theory was made known for the first time, in France, by a work 
 entitled ‘‘Expériences et Observations sur l’Ectricité, faites a Philadelphie en 
 Amerique par M. Benjamin Franklin,” Paris 1756, 12mo; translated from the 
 English. We have since seen an edition of Dr. Franklin’s works, in two vols. 4to. ; 
 the first of which contains all his experiments, and a great many interesting facts in 
 regard to electricity. From these works the reader may soon acquire a sufficient 
 knowledge of the theory of electricity. We ought to add also different memoirs of M. 
 le Roy, one of the principal partisans of Dr. Franklin, published in the ‘* Memoires 
 de l’Académie, for 1754,” &c. A very interesting work also on this subject, is a 
 Treatise by Father Beccaria, entitled “ Dell’ Electricismo naturale € artificiale,’ 
 which appeared at Turin in 1759, 4to. It contains experiments which are strongly 
 in favour of the Franklinian theory, and a great many new observations concerning 
 the electricity of the clouds. We must not here omit to mention, that Father 
 Beccaria is one of those philosophers who were most successful in cultivating the 
 science of electricity, and that he discovered a number of new and very extraordinary 
 phenomena. In some of the volumes of the Philosophical Transactions, likewise, are 
 a great many curious papers on different electric phenomena, by Mr. Nairne and 
 Mr. Wilson, Drs. Lind, Watson, &c.; but it would be too tedious to particularise 
 
 them. 
 
 ; 
 
 In the year 1752, a work entitled “Historie de l’Electricité,” was published in 
 
 three small vols. duodecimo. The author has collected pretty well every thing that 
 
 had been done or said in regard to electricity before that period ; but intermixed with 
 
 insipid witticisms and illiberal sarcasms, Since that period, however, Dr. Priestly, one 
 
 of the ablest of the English philosophers, has given a new History of Electricity, 
 which is much better and more instructive. A French translation of it appeared at 
 Paris in 1771, 3 vols. 12mo. To these we may add the following works :—Adams’s 
 Essay on Electricity, 8vo. 1787; a complete Treatise on Electricity by Cavallo, 
 3 vols. 8vo; and a learned Work on the subject, in 4to. 1779, by Lord Mahon, now 
 Earl Stanhope. 
 
 A most excellent Treatise on Electricity, by Dr. Roget, Secretary to the Royal 
 Society, has recently been published in the Library of Useful Knowledge. 
 
766 ELECTRICITY. 
 
 To the preceding articles on Magnetism and Electricity, we shall here add a short 
 
 account of the modern discoveries in kindred branches of science. 
 If we take a plate of zinc z, (Fig. 53.) and one c of 
 Fig. 53. copper, and partially immerse them in diluted sulphuric acid, 
 keeping their upper edges in contact, but their lower edges 
 apart, we shall find that a continued current of electricity 
 passes from the zinc to the acid, from the acid to the copper, 
 and from the copper to the zinc, and so on through the 
 acid, copper, and zinc in continued succession, in the same 
 direction. 
 If, instead of joining the metal plates, they be connected 
 bya wire, asin Fig. 54, the same effect will be produced, 
 The course of the electric current will be, in the 
 fluid from the zinc towards the copper, from the cop- 
 per through the wire to the zinc, &c. By separating 
 the wire as at y, the current may be passed through 
 any body which it may be wished to subject to its 
 action. The direction of the current is indicated by 
 the arrows. When united, the wire w, proceeding 
 from the copper, is imparting electricity to x, which” 
 is in connection with the zinc plate. The wire w is” 
 therefore considered as being in a positive and x in 
 a negative state of electricity. Either of these 
 arrangements is called a galvanic circle. . 
 
 Though the electrical effects of this simple apparatus are in general very feeble, j 
 it has been found possible, by a small apparatus of this kind to exhibit some of the 
 more energetic effects of galvanism. In Dr. Thompson’s Annals of Philosophy, Dr. 
 Wollaston has described what he calls an elementary battery, by which he raised the 
 temperature ofa slender piece of wire to a red heat. 
 
 The following is Dr. W.’s description of the miniature instrument : ) 
 
 “The smallest battery I have formed of this construction consisted of a thimble ‘ 
 without its top, flattened till its opposite sides were about ¥, of an inch asunder. The 
 bottom part was then nearly one inch wide, and the top about y%, and as its length 
 did not exceed ¥, of an inch, the plate of zinc to be inserted was less than 3 of a square — 
 inch in dimensions. . 
 
 ‘« Previously to insertion, a little apparatus of wire, through which the communica-_ 
 tion was to be made, was soldered to the zinc plate,and its edges were then coated 
 with sealing wax, which not only prevented metallic contact at those parts, but 
 also served to fix the zinc in its place, by heating the thimble so as to melt the wax. — 
 
 “A piece of strong wire, bended so that its two extremities could be soldered to the — 
 upper corners of the flattened thimble, served both as a handle to the battery, and as — 
 a medium to which the wires of communication from the zine could be soldered. 
 
 ‘* The conducting apparatus consisted, in the first place, of two wires of platinaabout — 
 ay Of an inch in diameter, and one inch long, cemented together by glass in two parts; — 
 so that one end of each wire was united to the middle of the other. These wires 
 were then turned, not only at their extremities for the purpose of being soldered to 
 the zine and to the handle, but also in the middle of the two adjacent parts for re- — 
 ceiving the fine wire of communication. ad 
 
 ** One inch of silver wire 44, of an inch in diameter, containing platinum at its cen- — 
 tre gd, part of the silver in diameter, was then bended so that the middle of the platina ~ 
 would be freed from its coating of silver by immersion in dilute nitrous acid. The 
 portion of silver remaining on each extremity served to stretch the fine filament of — 
 platina across the conductors during the operation of soldering; a little sal-ammo- 
 
eI 
 oi 
 
 ELECTRICAL EXPERIMENTS. 767 
 
 niac being then placed on the points of contact, the soldering was effected without 
 difficulty, and the two loose ends were readily removed by the silver attached to 
 them. 
 
 ‘It should here be observed, that the two parallel conductors cannot be too near 
 each other, provided they do not touch, and that, on this account, it is expedient to 
 pass a thin file between them, (previously to soldering on the wire) in order to 
 remove the tin from the adjacent surface. The fine wire may thus be made as short 
 as from 4, to 4, of an inch in length; but it is impossible to measure this with preci- 
 sion, since it cannot be known at what points the soldering is in perfect contact. 
 
 ‘* The acid which I have employed with this battery consists of one measure of sul- 
 
 _pauric acid, diluted with about 50 equal measures of water; for though the ignition 
 effected by this acid is not permanent, its duration for several seconds is sufficient for 
 exhibiting the phenomena, and for shewing that it does not depend upon mere contact, 
 
 - by which only an instantaneous spark should be expected. 
 
 ‘«* Although in this description I have mentioned a wire 45, of an inch in diameter, I 
 am doubtful whether this thickness is the best. Iam, however, persuaded that no_ 
 thing is gained by using a finer wire; for though the quantity of matter to be heated 
 is thus lessened, the surface by which it is cooled does not diminish in the same ratio, 
 so that where the cooling power of the surrounding atmosphere is the principal obsta- 
 cle to ignition, a thicker wire, which conveys more electricity in proportion to its 
 cooling surface, will be more heated than a thin one; a fact which I not only ascer- 
 tained by trials on these minute wires, but afterwards took occasion to confirm an 
 
 the largest scale, by means of the magnificent battery of Mr. Children, in the sum- 
 
 mer of 1813.” 
 
 By increasing the size of the plates, batteries of this kind have been constructed, 
 capable of producing very powerful effects. 
 
 Many effects of galvanism require for their production the combined influence of 
 several pairs of plates, arranged so as to form what has been called a compound galva- 
 nic circle. 
 
 Let there be taken an equal number of silver coins, round pieces of zinc, and cir- 
 cular discs of paper, somewhat smaller than the metallic ones, but soaked in salt 
 water. Of these forma pile, beginning with silver s, then zinc z, then card c, and so © 
 on ; Ss, Z, C, successively, as often as you think proper, finishing however with z. If 
 then the uppermost disc be touched with the finger of one hand, previously wetted, 
 while a finger of the other hand, also wetted, touches the lowest disc, a distinct 
 shock will be felt similar to that from a Leyden phial, or still more resembling that 
 from a weakly charged electric battery. By retouching, as often and as quickly as 
 you please, a succession of shocks is received; the strength being greater, the greater 
 the number of plates ; and the direction of the fluid is from the zinc at the top of the 
 pile, to the silver at the bottom. 
 
 Another arrangement was adopted by Volta. 
 
 Fig. 55. Taking a number of glasses filled with an acid or 
 
 poe saline solution, he placed in each a plate of zinc 
 and a plate of copper (or silver) connecting them 
 as in Fig.55. The course of the electric current 
 is from the last plate of copper on the right 
 through the zine plate outside, and the wires, to 
 the copper c on the left; the end of the battery 
 terminated by the zine plate—being that from 
 which the electricity is given out to the wire—is, 
 consequently, the positive end or pole of the battery ; and the opposite end, terminat- 
 
 - ed by a copper plate, is the negative pole. 
 
768 ELECTRICITY. 
 
 In the pile described above the zinc also is the positive, and the silver or copper % 
 the negative pole. . 
 The apparent discordance. between the direction of the electric current in the , 
 simple and the compound galvanic circles is explained by the consideration that in the ¥ 
 former the conducting wire communicated directly with the plate which is immersed — 
 in the fluid; but in the latter it proceeds, not from the immersed plate, but from one — 
 outside associated with it, and therefore of a different kind. 
 
 The form of the galvanic battery has been variously modified, according to the 
 views of experimenters; but in the space to which we must confine ourselves we _ 
 have not room for further details. f 
 
 A striking difference observable between electricity, as excited by galvanism, oras _ 
 produced by friction of electric or non-electric bodies, is; that though the action 
 of the latter, while it lasts, may have any degree of energy, that energy is exerted 
 only for an instant; it is accumulated by degrees, and is discharged at once; and the 
 effect of the discharge is beyond the control of the operator; while in galvanic (or 
 Voltaic) electricity, the battery continues to supply while in action, uninterruptedly — 
 and indefinitely, vast quantities of electricity, which is not lost by returning to its — 
 source (the earth), but circulates with undiminished force in a perpetual stream. — 
 The effect of this continuous current in bodies subject to its action is therefore more 
 definite ; and as it increases with the time, it may at length acquire a force incom- — 
 parably greater than even that produced by electric explosion. 
 
 But the intensity of electricity developed by galvanic combinations is increased 
 according to the number of alternations in the elements which form those combina- 
 tions ; and is totally independent of the extent of surface exposed to the action — 
 of the fluid in which the plates are immersed; though the quantity developed in a 
 given time is greater, the more extensive that surface is. % 
 
 If the Voltaic battery is of sufficient size, its electricity may be transferred to © 
 a common electric battery, by connecting the inner and outer coatings of the electric 
 battery respectively with the poles of the Voltaic one, when the charge will be — 
 instantly communicated to the former. On removing the Voltaic battery, this com- — 
 municated electricity may be discharged; and on renewing the communication, a 
 similar charge will be received ; and the same process may be repeated indefinitely. 
 Instead of removing the Voltaic battery, we may allow it to remain connected with — 
 the electric battery, when a rapid succession of sparks may be obtained from it by 
 connecting a wire with the outer coating, and presenting the other end of the wire to — 
 the knob of the phial. If the Voltaic battery is very powerful, these rapid explo- 
 sions are so powerful as to throw the iron wire into a state of intense combustion. 
 With a series of a thousand plates, each spark, or discharge, is attended with a sharp 
 sound, and will burn thin metallic leaves. It is remarkable that this often happens 
 when the same Voltaic battery has not power to produce such effects by itself, 
 unconneeted with the electrical battery, The Voltaical battery imparts in an instant 
 to the electrical one the whole of the charge it is capable of communicating. ¥. 
 
 When a Voltaic battery is composed of a considerable number of alternations ; 
 of plates, on bringing together the wires from the opposite poles, the transfer 
 of electricity begins while they are at a sensible distance from each other, and is # 
 accompanied, as in ordinary electricity, by vivid light; the sparks occurring every — 
 time the contact is broken, as well as when it is renewed. ‘ 
 
 We extract the following interesting paragraph from the article Galvanism, in the — 
 Library of Useful.Knowledge ; recommending, at the same time, the article itself to a 
 the attention of our readers. 
 
 “The most splendid exhibition of ‘electric light is obtained by placing pieces 
 of charcoal, shaped like a pencil, at the end of two Wires, in the interrupted Voltaic — 
 
 “4 
 
, = 
 ayy \ 
 
 ELECTRICAL EXPERIMENTS. 769 
 
 circuit. When the experiment was tried with the powerful battery of the Royal 
 Institution, a bright spark passed between the two points of charcoal, when they 
 came within the distance of about the thirtieth of an inch; and immediately after- 
 wards more than half of each pencil of charcoal! (the length of which was an inch, 
 and the diameter one sixth of an inch), became ignited to whiteness. By withdrawing 
 the points from each other, a constant discharge took place through the heated air, 
 in a space of at least four inches, forming an arch of light in the form of a double 
 cone, of considerable breadth, and the most dazzling briliiancy. Any substance in- 
 troduced into this arch instantly became ignited; platina melted in it, as wax in the 
 flame of a candle; quartz, the sapphire, magnesia, lime were fused; fragments 
 of diamond, charcoal, and plumbago, seemed converted into vapour, apparently with- 
 out having undergone previous fusion,” 
 
 , A battery of a hundred pairs of plates, of six inches square, will exhibit these 
 phenomena on a smaller scale. Charcoal prepared from the harder woods is to be 
 preferred. : 
 
 Light thus obtained from voltaic electricity is the most intense that art has yet 
 produced. It often assumes in succession different prismatic colours, and exhibits 
 some of the rays which are deficient in the sclar beams. Its brightness distresses 
 the eye, even bya momentary impression, and effaces the light of lamps in an apart- 
 ment otherwise brilliantly illuminated, so as to leave an impression of darkness on 
 the sudden cessation of the galvanic light. 
 
 The light is given out with equal splendour whether the experiment. is made in 
 air or in gases which contain no oxygen; such as azote or chlorine; it is therefore 
 ‘independent of combustion, and it is found too, that during the ignition neither the 
 gas nor the charcoal undergoes any chemical change. 
 
 In common electricity, heat is not evolved when the fluid passes freely through 
 a conducting substance, but only when some resistance is opposed to its passage, 
 and then its equilibrium is suddenly restored by an effort accompanied with light, 
 heat, and sound. But in using the voltaic battery, heat is evolved when the con- 
 nection is perfect, and the stream of electricity is conducted from one pole to the 
 other in the most silent manner. Ifthe connecting wire pass through water con- 
 tained in a vessel, the water becomes heated even to boiling ; and the ebullition 
 continues as long as the connecting wire passes through the vessel. Even the bat- 
 tery itself is heated when the apparatus is in an active state. 
 
 As was remarked of the light produced by voltaic agency, so it may be said of heat 
 produced by it, that it is more intense than any other which art has yet produced. ° 
 
 We shall now advert to a few of what may be called the magnetic effects of vol- 
 taic electricity. ; 
 
 Let the poles of a powerful voltaic battery be connected by a metallic wire, a 
 part of which is made straight. Take a magnetic needle nicely balanced on its 
 
 pivot, and allowing it to settle in the magnetic meridian, bring over and parallel to 
 ‘it the aforesaid straight part of the rod in the galvanic circuit, and it will be found 
 that the needle will instantly change its position, its ends deviating from their 
 natural position, towards the east or west, according to the direction of the elec- 
 tric currentin the wire; so that by reversing the direction of the current, the devia- 
 tion of the needle is also reversed, the end of the needle next the negative pole of 
 the battery being always deflected towards the west. 
 
 _ The deviation of the needle is the same while the uniting wire is parallel to the 
 needle and above it; even when it is not directly over it, but the direction of the 
 ‘deviation is reversed when the uniting wire is parallel to and below the needle. 
 
 If the wire is not parallel to the needle, various phenomena both with respect 
 to deviation and dip are presented; but for an account of these phenomena we 
 
 3 D 
 
770 ELECTRICITY, 
 
 must refer to works on this comparatively new branch of science, and conclude with 
 a short account of Electro Magnetic Induction. 
 
 During the passage of an electric current through a conducting wire, it tends” 
 to induce magnetism in such bodies in its vicinity, as are capable of being excited. 
 The connecting wire in action has a sensible attraction for iron filings, which it 
 holds in suspension like an artificial magnet, while the electric current circulates 
 through the wire; but the moment the galvanic circuit is interrupted, the filings 
 fall off. — 
 
 Mr. Watkins of Charing Cross, a most successful cultivator of this branch of 
 science, first noticed that if a thick copper wire be extended between the poles of 
 a voltaic battery, and some fine iron filings be gently sifted upon it, they adhere to 
 the wire all round in distinct transverse bands, the particles of which cohere as long 
 as the current is maintained. On a broad and thin copper ribbon, substituted for 
 the wire, the filings arrange themselves in parallel bands across the ribbon. 
 
 To try whether steel might not receive permanent magnetism from voltaic action, 
 Sir Humphrey Davy fastened several steel needles in different directions by fine 
 silver wire, to a wire of the same metal about the thirtieth of an inch in thick- 
 ness, and eleven inches long, some parallel, others transverse, above and below, 
 in different directions, and placed them in the electrical circuit of a battery of 
 thirty pairs of plates, each of 45 square inches, and tried their magnetism by iron” 
 -filings. They were all magnetic. Those that were parallel to the wires attracted 
 filings in the same way as the wire itself; but those in transverse directions had each 
 two poles. All the needles that were placed under the wire, when the positive end 
 of the battery was east, had their north poles on the south side of the wire, while 
 those that were placed over the wire had their south poles to the south, whatever 
 was the inclination of the needle to the horizon. On breaking the connection all 
 the steel needles that were on the wire in a transverse direction retained their 
 magnetism, which was as powerful as ever; while those that were parallel to the 
 silver wire appeared to lose it at the same time as the wire itself. 4 
 
 Contact with the wires was not necessary for magnetising the needle, the effect 
 being produced instantaneously by mere juxta-position of the needle in a transverse 
 direction, even through a thick pane of glass. 3 
 
 The efficacy of electric magnetic induction is greatly increased by placing the ‘ 
 needle or bar to be magnetised within a helix of wire, as represented in Fig 56. It 
 _is not necessary that the bar should be placed in the: 
 axis of the helix, as it may lie in any situation within” 
 it, or be enclosed in a tube of glass, which will also be” 
 convenient as a support for the coils of wire, and for 
 the introduction of different needles in succession. 
 The needle should not be allowed to remain more 
 than a moment in the tube; for the magnotisiil 
 effects of the helix are produced almost instan- 
 taneously ; andit has been observed, that a needle leftin the helix for a few minutes 
 sometimes has its first acquired polarity impaired, or even destroyed. _ 
 
 If the needle to be magnetised is not very hard, its whole length need not be in- 
 serted in the glass tube ; for if held in the hand, so that only one half of it is within the 
 helix, it wili become equally magnetic with one that has been wholly acted upon. It 
 will be understood that the.ends of the helix Pp and p are connected with the two poles 
 of the galvanic battery. The two cups at Pp and p have a little mercury put in them — 
 to render the contact with the wires from the battery and the helix more complete. — 
 
 A very powerful temporary magnet may be obtained by bending a thick cylinder 
 of soft iron into the form of a horse shoe, and surrounding it with a coil or helix o 
 thick copper wire; the coils being prevented from touching each other by a covering 
 
 Fig. 56. 
 
BLECTRICAL EXPERIMENTS.-—ELECTROTYPRE. iil 
 
 of silk or some other non-conducting material. ‘When this wire is made part of the 
 
 galvanic current of a battery, even of moderate power, the iron is rendered so highly 
 magnetic that it will lift up a very heavy weight by means of a piece of iron applied 
 to its poles, which acts precisely for the time like those of a horse-shoe magnet. 
 
 Fig. 57. exhibits an arrangement of this 
 kind, w w being the twoends of the wire 
 coiled round the iron to be magnetised, 
 and bent so as to dip into the cups, P and 
 N, for forming connexions with the battery. 
 
 More recent discoveries in this branch 
 of science have gone far to show that 
 light, electricity, and magnetism are essen- 
 tially the same,-but developed under dif- 
 ferent circumstances. We have seen that 
 galvanism produces electricity, and that 
 both are capable of imparting to iron the 
 magnetic properties of attraction and 
 polarity ; and perhaps the most singular of all the discoveries that have been made 
 with reference to these subjects is, that magnetism, under certain circumstances, is 
 capable of exciting electric action, to produce electric sparks, and even a continuous 
 stream of electric fluid. 
 
 But we must here close our brief notice of this interesting subject; and refer our 
 readers to works which are expressly devoted to its discussion. 
 
 ELECTROTYPE. 
 
 A most unlooked-for and wonderful application of voltaic electricity has recently 
 been made. We copy the following account of it from the ‘“ Atheneum Journal” 
 of October 26th, 1839; merely remarking that the applications of the principle which 
 have since been made fully justify the anticipations of the journalist. 
 
 ‘We lately published M. Jacobi’s letter to Mr. Faraday, in which he described his 
 
 attempts to copy in relief engraved copperplates, by means of voltaic electricity. We 
 have since received a communication from Mr. Thomas Spencer of Liverpool, from 
 which it appears that that gentleman has for some time been independently engaged 
 on the same subject ; and that he has not only succeeded in doing all that M. Jacobi 
 has done, but he has successfully overcome those difficulties which arrested the pro- 
 gress of the latter. It is unnecessary here to enter on the question of priority between 
 these gentlemen. To Mr. Spencer much credit is certainly due for having investi- 
 gated, and successfully carried out, an application of voltaic electricity, the value 
 of which can hardly be questioned. The objects which Mr. Spencer says he pro- 
 posed to effect were the following: ‘To engrave in relief upon a plate of copper—to 
 deposit a voltaic copper-plate, having the lines in relief—to obtain a fac-simile of 
 a medal, reverse or obverse, or of a bronze cast—to obtain a voltaic impression from 
 plaster or clay—and to multiply the number of already engraved copper-plates.’ The 
 results which he has obtained are very beautiful; and some copies of medals which 
 he has forwarded to us are remarkably sharp and distinct, particularly the letters, 
 which have all the appearance of being struck by a die. 
 
 *“‘ Without entering into a detail of the steps by which Mr. Spencer brought his 
 process to perfection, many of which are interesting, as shewing how slight a cause 
 ‘May modify the resnlt, we shall at once give adescription of his process. 
 
 ** Take a plate of copper, such as is used by an engraver ; solder a piece of copper 
 wire to the back part of it, and then give it a coat of wax ; this is best done by heat- 
 ing the plate as well as the wax; then write or draw the design on the wax with'a 
 
 black lead pencil or point. The wax must now be cut through with a graver or steel 
 op 2 
 
Mae ELECTRICITY. 
 
 point, taking special care that the copper is thoroughly exposed in every line. The 
 shape of the tool or graver employed must be such that the lines made are not V _ 
 shaped, but as nearly as possible with parallel sides. The plate should next be im- 
 mersed in diluted nitric acid; say three parts water to one of acid: it will be at once 
 seen whether it is strong enough, by the green colour of the solution and the bubbles 
 of nitrous gas evolved from the copper. Let the plate remain in it long enough for 
 the exposed lines to get slightly corroded, so that any minute portions of wax which 
 might remain may be removed. The plate thus proposed is then placed in a trough 
 separated into two divisions by a porous partition of plaster of Paris or earthenware ; 
 the one diyision being filled with a saturated solution of sulphate of copper, and the 
 other with a saline or acid solution. The plate to be engraved is placed in the divi- — 
 sion containing the solution of the sulphate of copper, and a plate of zinc of equal 
 size is placed on the other division. A metallic connexion is then made between the 
 copper and zinc plates, by means of the copper wire soldered to the former; and the 
 voltaic circle is thus completed. The apparatus is then left for some days. As the 
 zinc dissolves, metallic copper is precipitated from the solution of the sulphate on the 
 copper -plate, wherever the varnish has been removed by the engraving tool. After 
 the voltaic copper has been deposited in the line engraved in the wax, the surface of 
 it will be found to be more or less rough, according to the quickness of the action. 
 To remedy this, rub the surface with a piece of smooth flag or pumice stone with 
 water. Then heat the plates, and wash off the wax ground with spirits of turpentine 
 anda brush. The plate is now ready to be printed from at an ordinary press. 
 
 ‘‘ In this process, care must be taken that the surface of the copper in the lines be 
 perfectly clean, as otherwise the deposited copper will not adhere with any force, but 
 is easily detached when the wax is removed. It is in order to ensure the perfect — 
 cleanness of the copper, that it is immersed in diluted nitric acid. Another cause of | 
 imperfect adhesion of the deposited copper Mr. Spencer has pointed out, is the pre-_ 
 sence of a minute portion of some other metal, such as lead, which, by being preci-~ 
 pitated before the copper, forms a thin film, which prevents the adhesion of the sub-_ 
 sequently deposited copper. This circumstance may, however, be turned to advan- — 
 tage in some of the other applications of Mr. Spencer’s process, where it is desirable 
 to prevent the adhesion of the deposited copper. : 
 
 “In copying a coin or a medal, Mr. Spencer describes two methods: the one is by _ 
 depositing voltaic copper on the surface of the metal, and thus forming a mould, from : 
 which fac-similes of the original medal may readily be obtained by precipitating cop- 
 perintoit. The other is even more expeditious. Two pieces of clean milled sheet- 
 lead are taken, and the medal being placed between them, the whole is subject to | 
 
 pressure ina screw-press, and a complete mould of both sides is formed in the lead, 
 shewing the most delicate lines perfect (in reverse.) Twenty, or even a hundred of 
 these, may be so formed on a sheet of lead, and the copper deposited by the voltaic ; 
 process with the greatest facility. Those portions of the surface of the lead which © 3 
 are between the moulds may be varnished, to prevent the deposition of the lead; or 
 a whole sheet of voltaic copper having been deposited, the medals may be afterwardal i 
 cut out. When copper is to be deposited on a copper mould or medal, care must be — 
 taken to prevent the metal deposited adhering. This Mr. Spencer effects by heating 
 the medal, and rubbing a small portion of wax overit. This wax is then wiped off, a 
 sufficient portion always remaining to prevent adhesion. 4 
 ‘“‘Enough has been said to enable any one to repeat and follow up Mr. Spencer’s — 
 interesting experiments: The variations, modifications, and adaptations of them are 
 endless, and many new ones will naturally suggest themselves to every scientific — 
 reader; and for their gratification, the medals produced by this process, and for- 
 warded to us by Mr. Spencer, will be left with our publisher for some days, and open 
 to inspection.’’ 
 
a 
 
 CHEMISTRY. ? 773 
 
 ‘ PART FOURTEENTH. 
 
 CHEMISTRY. 
 
 -TxoseE who are only initiated-in Chemistry must conceive an idea of this science very 
 
 different from that which is entertained by the vulgar. According to the common class 
 of mankind, Chemistry is the chimerical art of transmuting metals, or at most of pro- 
 ducing some extraordinary phenomena, rather curious than useful. But in the eyes of 
 the philosopher, to whom it is known, it is the most extensive and most interesting 
 of all the branches of natural philosophy. We may even venture to assert, that it is 
 doubtful whether the appellation ofa great philosopher can be given to any person 
 unacquainted with chemistry. It is at least certain, that though without its assistance 
 we can account for some of the phenomena of nature, such as the motion of the 
 celestial bodies, the effects of the gravity of the air, &c. ; yet there are a far greater 
 number which can be explained by chemistry alone. Indeed chemistry is not less 
 extensive than nature itself. Animals, vegetables, and minerals, all fall within its 
 province. It is it that analyses them, combines their principles, examines the pheno- 
 mena resulting from them; and thus renders us more intimately acquainted with 
 their nature. Hence are deduced a multitude of useful processes; so that it may 
 with truth be said, that many of the arts are nothing else than a continual application 
 of chemistry. Of this kind are the arts of glass-making, dyeing, metallurgy, &c. 
 We may even add, that the most common of the arts, those that are most necessary, 
 are often chemical processes ; such, for example, is that of washing; respecting the 
 truth of which the person who practises it entertains no doubt; though still it is’an 
 operation which chemistry alone can explain. This explanation consists in the pro- 
 perty which fixed alkalies have of rendering fat substances soluble in water, and of 
 forming a soap with them. Those who know that one of the first operations of this 
 art is to soak the linen in a sort of ley, made of wood ashes or vegetable fixed alkali 
 (pot-ash), will readily be convinced of the truth of what we have advanced. In this 
 part of our work we need not adduce any other examples. 
 
 Chemistry also, of all the branches of philosophy, exhibits the most singular and ~ 
 most curious phenomena. Who will not be astonished to see iron filings, immersed 
 
 ina liquid as cold as itself, immediately produce a violent ebullition, and vapours 
 
 susceptible of inflammation? Can any one, without admiring the operations of 
 nature, see this metal, so solid, afterwards destroyed in some measure by the above 
 fluid, and united with it in sucha manner asto pass with it through the closest 
 
 -filtre? Who will not be filled with wonder, on seeing another limpid liquor sud- 
 
 denly dissolve this union, and cause the iron to fall to the bottom of the vessel, in 
 the form of an impalpable powder? It is needless, and would require too much time, 
 to enumerate here any more of these phenomena, as we shall hereafter give a parti- 
 cular account of those that are the most remarkable and most interesting. But it 
 will first be necessary to bring the reader acquainted with the principal substances 
 employed as agents in these operations. - 
 
7/4 CHEMISTRY. 
 
 ARTICLE I, 
 Of Salts. 
 
 ~The name of Salts, or saline matters, is given to all those bodies which, when q 
 
 immersed in water or exposed to damp air, resolve themselves into liquids: of this 
 we have an example in the well known marine salt, in nitre or saltpetre, in alum, 
 
 vitriol, tartar, sal ammoniac, &c. When immersed in water, these disappear, and — 
 
 become intimately mixed with the whole liquor. This is what is called solution or 
 dissolution. 
 
 Salts, to be dissolved entirely, require a greater or less quantity of water, accord-~ 
 
 ing to the nature of them. | Marine salt, to be dissolved entirely, requires twice its — 
 weight of water; alum, twelve times its weight; selenite, six or seven hundred — 
 
 times, &e. 
 
 Though acids, alkalies, and their mixtures, generally speaking, are called saline _ 
 
 substances, it is necessary in chemistry to consider them separately, in order to” 
 
 obtain a knowledge of their principal properties. 
 
 SECTION I. 
 Of Acids. 
 
 We shall not here say, with the author of the ‘ Dictionnaire de Physique por= _ 
 
 tatif,” * and some of the old chemists, that an acid is formed of long, sharp, and 
 cutting particles; for nothing is more destitute of foundation; and by the help 
 
 ‘ of such a definition it would be impossible, in an apothecary’s shop, to distin- Zz 
 
 guish an acid from an alkali, or from a neutral salt. The following is a more 
 correct description. 
 
 Acids in general have a liquid form, and leave on the tongue an impression ‘of 
 sourness and coldness. They possess also the property of changing the colour of 
 blue vegetable juices to red; and they combine readily with alkalies, earths, and 
 metals, with which they form neutral salts. 
 
 To try an acid by vegetable colours, take sirup of violets, or blue paper, and 
 
 pour the liquid acid over it; the violet or blue colour will soon assume a red _ 
 
 hue. When any liquid therefore, poured over sirup of violets or blue paper, ° 
 
 changes. the colour to red, we may be assured that it is an acid, or that an acid 
 predominates in it. 
 
 Acids are by modern chemists divided into mineral, vegetable, and animal, ac- 
 cording as they are derived from one or the other of these sources. 
 
 The old chemists were acquainted with no more than three mineral acids, namely, | 
 
 the vitriolic, the nitric, and the marine, so called because obtained from vitriol, nitre, 
 and marine salt. The mineral acids now known are much more numerous. The 
 principal of these acids are the sulphuric or vitriolic, the nitric, the muriatie or 
 marine, the carbonic, the boracic, the fluoric, the succinic, the arsenic, the molybdic 
 and the tungstic. ‘ 
 
 The vegetable acids are also now pretty numerous. The following are known: 
 the acetic acid, or pure vinegar; the tartareous, extracted from tartar ; the gallic, 
 
 froxa gall-nuts; the citric, from lemons; the malic from apples; the benzoic, from — 
 
 flowers of benjamin ; the camphoric, from camphor oxygenated to acidity; the pyro- 
 tartareous, the pyroligneous, and the pyromucous. 
 
 The three last mentioned, however, have been proved to be merely acetic acid» 
 
 * According to this author, iron is composed of vitriol, sulphur, : ion i 
 I i L ( and earth ; ferm - 
 ment occasioned by the introduction of acids into alkalies ; ered alkalies coa ate they tian 
 tals; sulphur is an inflammable mixture composed of fire, oil, water, 
 
 pound of sulphur, vitriol, &c, !! and earth; copper is a com- 
 
 gulate they form crys-_ 
 
OF ACIDS. 715 
 
 rendered impure by empyreumatic oil; and ought therefore to be no longer considered 
 as distinct acids. 
 
 The following animal acids are known: the phosphoric, or phosphorus and oxygen 
 in union to saturation; the lactic, obtained from milk; the saccholactic, extracted 
 from sugar of milk; the formic, from ants; the sebacic, from fat ; the bombic, from 
 silk-worms; the uric, from the human calculus; the prussic, which is the colouring 
 matter of Prussian blue. 
 
 Acids formerly were considered either as simple bodies, or various hypotheses were 
 
 admitted in regard to their origin and composition. It is now however proved by ex- 
 experiment, that every acid consists of a peculiar body combined with oxygen, which 
 is the basis of oxygen gas, and thence has been called oxygen, that is the generator 
 of acidity. The bodies which enter into union with oxygen, and form acids, are 
 called the radicals, or bases of the respective acids, which they constitute. 
 _ When the bases are perfectly saturated with oxygen, the acids are called perfect 
 acids, and are denoted by the termination ic, to distinguish them from the imperfect 
 acids, in which the bases predominate, and to which the termination ous is given. 
 Hence we have nitric and nitrous, sulphuric and sulphurous acids, &c. 
 
 Some bases can combine with an excess of oxygen; the acids thence resulting are 
 called oxygenated or oxygenised. 
 
 Of the Vitriolic Acid. 
 
 The vitriolic or sulphuric acid, the most powerful of all, is furnished by vitriol 
 
 either green or blue, or by alum; for, vitriol is a salt formed merely by the combina- 
 tion or union of an acid with iron and copper; alum, in like manner, is the combi- 
 nation of an acid with argillaceous earth; and the acid in these three substances is 
 absolutely the same. 
 - It will be sufficient here to observe, that the acid is extracted by means of fire. 
 These matters are inclosed, with certain precautions, in a retort, and being ex posed 
 to a strong fire, the violence of the heat obliges the acid, which is susceptible of being 
 reduced to vapour, to abandon the metal or earth, to which it was united, and to 
 pass over into the receiver in the form of a liquid. 
 
 Another method, still more simple, of obtaining the vitriolic acid, is by the com- 
 bustion of sulphur; for it has this substance for its base, and on that account it is 
 now known by the name of the sulphuric acid. 
 
 If sulphur be burnt slowly under a bell glass, the fumes arising from it will be 
 condensed on the interior surface of the vessel, and run down in the form of a liquid, 
 which will be an acid with an excess of base, that is sulphurous acid. In this pro- 
 cess, the sulphur, by its combustion, receives oxygen from the atmospheric air, which 
 is a mixture of oxygen gas and azotic gas, and at the same time moisture, which 
 evables it to assume the liquid form. 
 
 In the large way of manufacture, the sulphuric acid is obtained by burning sul- 
 phur, with the addition of a little nitre, in an apartment lined all round with lead, a 
 metal on which that acid exercises little or no action. The floor is covered to some depth | 
 with water, and the aperture or door is shut as soon as the materials are inflamed. 
 The nitre is employed to facilitate the combustion. The nitric acid, one of the 
 principles in the nitre, is in a great measure decomposed, and gives up its oxygen to 
 the sulphur, which by this supply, and the oxygen it derives from the air in the 
 room, becomes acidified. When the combustion is over, and the fumes have had 
 sufficient time to be absorbed by the water in the apartment, the door is opened to 
 admit a fresh supply of air, and to introduce a new charge of sulphur and nitre, 
 which is deflagrated as before. This operation is successively repeated for several 
 
 weeks, till the water, now mixed with acid, is found by examination to be of a certain 
 specific gravity. The fluid is then distilled or simply boiled, which drives off a por- 
 
(16 CHEMISTRY. » 
 
 tion of the redundant water, and leaves the acid in a concentrated state. When this 
 is properly done; its weight is equal to about double the weight of an equal volume 
 of water. a 
 
 The sulphuric acid of commerce is never quite pure ; always containing a portion — 
 of sulphate of lead and sulphate of potash. The former comes from a partial disso- 
 lution of the lead in the chamber in which it is made, and the latter from the nitre 
 used in the process. 
 
 The vitriolic acid is the most powerful of all, and is able to separate all the 
 others from their combinations with alkalies, earths, and metals. In a subsequent 
 article we shall give some experiments to illustrate this chemical play, which is ex- 
 ceedingly curious, and the cause of a thousand singular effects in nature. . 
 
 Sulphurous acid, like sulphuric, is a combination of sulphur and oxygen; having 
 less of oxygen, or more sulphur, than sulphuric acid. In the gaseous state it is” 
 invisible, but of a strong suffocating smell. It is readily absorbed by water, and 
 then forms liquid sulphurous acid. It unites with various bases, forming the salts 
 called Sulphites. 
 
 Of Carbonie Acid. 
 
 Carbonic acid is a combination of carbon and oxygen, there being in it 27-5 parts 
 of carbon to 72°5 of oxygen. Ina state of vapour, it is invisible, and unfit for com-— 
 bustion or respiration. By pressure, water may be made to absorb three times its 
 bulk of this gas; and it imparts to the water an acidulous and not unpleasant taste. — 
 In the proportion in which it exists in atmospheric air, it is favourable to the growth — 
 of vegetables; but in a large proportion it is highly injurious. In combination 
 with earth, alkalies, and metallic oxides, it forms with them salts called Carbonates. — 
 
 Carbonic acid gas is found in abundance in many natural waters, as those of Pyr- 
 mont, Spa, and Selzer. Various imitations of these pleasant waters, sold under the 
 names of single and double soda water, are manufactured by houses in London, 
 equal in every respect to the natural waters imported from the continent. ; 
 
 Of the Nitrous Acid. 
 
 Nitre or saltpetre, a substance well known, furnishes the nitric acid. Saltpetre 
 indeed is the result of the union of this acid with a matter to which chemists give 
 the name of fixed vegetable alkali, because the ashes of vegetables yield the ‘same 
 substance. They are separated from each other by processes ~which it is not our 
 object here to describe ; and in this manner is obtained the liquor called nitrous acid; 
 from which the nztrie aetd is procured merely by separating, by means of heat applied 
 to it in a proper apparatus, that portion which has not been fully saturated with oxygen. 
 It is lighter, and in general less active than the vitriolic acid. It is commonly of a 
 dark yellow colour ; and when well concentrated, continually emits reddish vapours, 
 which seem to circulate in the vessel that contains it. The ratio of its weight to 
 that of water is then as 3 to 2. . 
 
 The nitrous acid, when in a moderate state of concentration, is called also aqua-— 
 fortis. It is the proper solvent of silver and copper. When fully saturated with 
 oxygen, or in the state of nitric acid, it is colourless. | 
 
 Nitric oxide gas is formed by depriving nitric acid of some portion of its oxygen. 
 It is obtained by putting some diluted nitric acid into a retort, and then adding to it 
 some bits of copper, iron, silver, or some other metal. The metals are dissolved by 
 the acid, and a gaseous product is obtained, which is nitrie oxide gas. It does not 
 possess acid properties, and if inhaled it is destructive to animal life. If suffered to es-_ 
 cape into the air, it attracts oxygen, and a red vapour is formed, which is nitrous acid gas. 
 
 When nitrous oxide gas is received into the lungs, it does not at once prove fatal, — 
 because it meets there, and is diluted with the atmospheric air which is present in — 
 
 red 
 
ACIDS. | 777 
 
 - that organ. The sensations produced by it are highly pleasurable. Great éxhilare- 
 
 tion, an irresistible propensity to laughter, a rapid flow of vivid ideas, and a propen- 
 sity for jumping and other muscular exertions, are its ordinary effects. It has there- 
 fore been called laughing gas. We would however advise all who desire to eschew 
 apoplexy, to resort to other means of exciting pleasurable sensations. 
 
 Nitric acid may be used to determine whether an instrument is made of steel or 
 wrought iron. For a drop of the acid let fall on steel will, when it has dried, leave 
 a black mark ; but not if dropped upon iron. 
 
 Of the Marine or Muriatic Acid. 
 
 Marine salt, so generally employed, and so well known, is the substance from 
 
 which the marine acid is extracted ; for common salt is a combination of this acid 
 
 with a substance called by chemists the fixed mineral alkali, or soda. They are 
 
 separated from each other by certain processes, and the liquor thence resulting is 
 
 marine or muriatic acid. 
 
 The marine acid has characters and properties, which render it very distinct from 
 the preceding ones. In its highest state of concentration, it is only a little 
 heavier than water, being in the ratio of about 19 to 17. Its colour is a lemon 
 yellow, and its smell approaches to that of saffron. 
 
 Of Fluorie Acid. 
 
 Fluoric acid is of a very peculiar nature ; it is found in fiuor spar. It is obtained 
 by pouring sulphuric acid upon the powdered spar in a leaden retort, and applying a 
 gentle heat; the sulphuric repels the fluoric, and unites with the lime in its stead. 
 The gas evolved is condensed in water to form liquid fluoric acid. To preserve the 
 liquid acid, it must be kept in bottles lined on the inside with wax dissolved in oil ; 
 or in vessels of lead or platina. It dissolves silica, so that it cannot be kept ina 
 
 glass vessel. 
 
 If the surface of a pane of glass be covered with wax, and a drawing of any kind 
 be made uponit by cutting out the wax with proper instruments,—and the piece being 
 put into a leaden receiver, and the gas disengaged from fluor spar by an Argand 
 
 | lamp thrown upon it, the drawing made on the glass will be etched in, and the 
 
 drawing will be as permanent as the glass itself. 
 This acid is also used for destroying the polish on glass for lamp shades, &c. 
 
 Remark.—To enter into a particular account of all the mineral acids, would re- 
 quire more room than the nature of this work will admit; and it is the less ne- 
 cessary, as information respecting them may be found in almost every work on 
 chemistry. 
 
 Of the Vegetable Acids. 
 
 The preceding remark must be applied also to this class of acids, as we mean to 
 confine ourselves to the acetous, which is well known. Vinegar may be deprived 
 of its superfluous water in a great many different ways; and in this state its strength 
 is little inferior to that of many of the mineral acids. 
 
 The following is a simple method of concentrating it. Expose vinegar to the 
 severe cold during winter : it will then in part freeze, and if the ice be removed 
 the remainder will be vinegar of a much superior quality. By repeating this opera- 
 tion several times, you will obtain vinegar more highly concentrated, as the cold has 
 been more intense. Artificial cold may be afterwards employed, and in a much 
 
 | greater degree than the greatest cold ever experienced in our climates. 
 
 If vinegar.be mixed with charcoal dust, and exposed to a strong heat in a distill- 
 ing apparatus, the water will be first thrown off when a boiling heat is attained; 
 
| 
 
 778 CHEMISTRY. 
 
 and then by a stronger heat the pure concentrated acetous acid will be drawn 4 
 over. This and the Prussic acid are the only vegetable acids that rise in distilla- © 
 tion combined with water. 
 
 SECTION Il. 
 Of Alkalies. 
 
 Alkalies are divided into fized and volatile. The former have no smell: the 
 latter have a pungent and penetrating odour. The general properties of pure alka- 
 lies are: 
 
 Ist. That they have a peculiar caustic or burning taste, termed alkaline. 
 
 2d. That they change the blue juices of vegetables green; the yellow infusion of 
 turmeric, brown; and the red infusion of Brazil wood, violet. 
 
 3d. That combining with acidsthey form neutral salts. 
 
 When a liquor therefore poured over sirups or infusions of the above kind, or 
 upon paper that has been tinged with them, changes the colour in the manner de- 
 scribed, it is alkaline, or an alkali predominates in it. 
 
 ' Alkalies however are seldom pure in the strict sense of the word, being generally 
 combined with carbonic acid, or fixed air; and in this state they are termed mild, 
 to distinguish them from caustic or pure alkalies, that is alkalies which have been 
 freed from the fixed air. 
 
 The common method of depriving them of their fixed air is by throwing into an 
 alkaline solution, or ley, a quantity of quick lime. The lime, by its greater affinity 
 for the fixed air, seizes on it, and leaves thealkali in a caustic state; for limestone — 
 or chalk is pure lime saturated with fixed air, which it gives up when exposed toa 
 strong heat; and it is then called quicklime. When saturated with the carbonic 
 acid or ped air, from the alkali it again becomes mild lime or chalk, and falls to 
 the bottom of the vessel. 
 
 Of fixed Alkalies. 
 
 There are two kinds of fixed alkali, namely, the mineral or soda, and the vege- 
 table or potash. They are called.fixed, because though exposed toa strong heat 
 they are not dissipated, but fuse like metals, and like them can endure a red heat. 
 They facilitate the fusion of stones, earth, and sand; and for this reason are much 
 used, especially the first, in glass houses. They are both of extensive use in the 
 arts. 
 
 Fixed mineral alkali is obtained from marine salt ; for when this salt is deprived 
 of the acid which enters into its composition, the remainder is fixed mineral alkali; 
 but to extract it from this substance requires a tedious and expensive process. The 
 most common method of obtaining it, is to burn certain plants which grow on the 
 sea coast, or are washed on shore by the tide. Of this kind is the plant kali or 
 glass wort, from which the denomination of alkaliis derived, and various other ma- 
 rine plants, such as varec or sea-wrack, fuci, &c. The ashes of these plants con- 
 tain abundance of fixed alkali, which may be extracted and purified by lixiviating 
 them, and then evaporating the ley. This is what is known in commerce under the 
 name of soda. ; 
 
 Fixed vegetable alkali is commonly obtained by the combustion of the greater 
 part of other plants, and of wood, such as common fire wood. A great deal is 
 made in this manner, in different forests, where immense quantities of wood are 
 burnt for that purpose in pits; the ashes which remain contain abundance of this 
 fixed alkali, known in the shops under the name of potashes. By lixiviating them, 
 and then evaporating the ley, an alkali much more active may be obtained. 
 
 Another method of obtaining fixed vegetable alkali, much purer, is to employ 
 wine lees, and the tartar which adheres to the sides of wine casks. These substances 
 
ALKALIES, 779 
 
 are formed into packets or masses of the size of the fist, and then burnt until they have 
 ‘assumed a white colour. By these means very pure fixed alkali may be produced. It 
 is known in the shops under the name of salt of tartar, or alkali of tartar. Itis ab- 
 solutely the same as common potash. In this state however they are not entirely 
 free from a combination of carbonic acid. 
 
 Common potash is composed of the metal called potassium, and oxygen, in the 
 proportion of 100 of potassium to 20 of oxygen. Potassium is procured in large 
 “quantities by keeping potash in fusion with intensely hot iron. Potassium is solid at 
 the usual temperature of the atmosphere, but quickly becomes soft and ductile when 
 held in the hand. Its specific gravity is to that of water as 865 to 1000. It is per- 
 fectly white, and has the lustre of polished silver. 
 
 The two fixed alkalies, the vegetable and mineral, differ from each other 
 principally by one peculiar property. The fixed vegetable alkali attracts the mois- 
 ture of theair so strongly, that to preserve it ina solid state it must be put into 
 well closed vessels. If left exposed to the air, it deliquesces of itself, and in this 
 state is called oil of tartar per deliquium; an appellation very improper, for it is not 
 an oil. 
 
 On the other hand, fixed mineral alkali, instead of attracting moisture, loses its’ 
 own, and effloresces ; that is, falls into dust: and for this reason it can be preserved 
 with much more convenience than the other. 
 
 Volatile Alkali. 
 
 This alkali, or ammonia, is produced by the combustion of most animal matters, 
 or by the putrefaction of animal or vegetable substances. The smell of putretied 
 bodies is owing to the alkali disengaged from them during that operation, by which 
 nature reduces them in some measure to their first principles, in order that they 
 may serve for new compositions. The strong odour which proceeds from privies, 
 is a highly volatile alkali. It is called volatile, because a heat, inferior even to that 
 of boiling water, is sufficient to disperse it in vapours, which always discover 
 themselves by their penetrating smell, 
 
 Ammonia is a compound of hydrogen, and nitrogen in the proportion of one 
 part, by weight, of the former, to four parts of the latter. It may be in a short 
 time formed by the following process, so as to become evident to the sense of smell- 
 ing. 
 
 - some filings of tin or zinc pour some moderately diluted nitrous acid, and 
 after a short time stir into the mixture some quick lime, and a very strong pun- 
 gent smell of ammonia will be produced. 
 
 All animal and vegetable substances, in a state of putrefraction, furnish ammonia : 
 but in England it is generally procured by a dry distillation of bones, horns, and 
 other animal substances. It is found in mineral waters, and whenever iron rusts 
 in water, having a free communication with the air. Carbonate of ammonia is pro- 
 cured in large quantities from the waste liquor which is collected in the manufac- 
 tories of coal gas. 
 
 Ammoniacal gas for chemical experiments may be procured by mixing one part 
 of powdered sal ammoniac with two parts of quick lime ina retort, and applying 
 the heat of a lamp which will disengage the gas in abundance. 
 
 When liquid ammonia is combined with carbonic acid, it takes a concrete form, 
 and a beautiful white colour being then what is called volatile salts. 
 
 Muriate of ammonia is formed by combining ammonia with muriatic acid. It is 
 known in commerce by the name of sal ammoniac. 
 
- 780 a CHEMISTRY. 
 
 2 
 
 SECTION III. 
 Of Neutral Salts. 
 When a salt is neither acid nor alkaline, if it neither turns sirup of violets nor blue 
 
 paper red or green,* it is called neutral. The reason of this appellation is evident. — 
 
 ‘ 
 
 PO zo 
 
 Of this kind are marine salt, nitre, the different species of vitriol found in a natural 
 
 state, and various other salts, both natural and artificial. 
 A neutral salt consists of an acid combined with an alkali, an earth, ora metal. 
 
 We shall here give a few examples, enumerating the most common combinations of — 
 
 the three principal mineral acids, with different substances. 
 Thus, vitriolic acid, combined with zinc, forms white vitriol (sulphate of zinc) ; 
 With copper, blue vitriol ; 
 With iron, green vitriol ; . 
 With argillaceous earth, alum ; . ie a 
 With calcareous earth, selenite ; ; iit 
 With volatile alkali, ammoniacal vitriol ; 
 With fixed mineral alkali, Glauber’s, Epsom, or Seidlitz salt ; 
 With fixed vegetable alkali, vitriolated tartar. 
 
 Salts so formed are called sulphates ; as, sulphate of iron, of copper, of zinc, of — 
 
 ammonia, &c. according as one or other of these substances is united with the acid. 
 The nitric acid, combined with fixed vegetable alkali, forms nitre ; 
 With fixed mineral alkali, quadrangular or cubic nitre; 
 With volatile alkali, a nitrous ammoniacal salt ; 
 
 With silver, a peculiar salt fusible by a moderate heat, known under the name of 
 
 lapis infernalis, on account of its causticity. 
 These combinations are called nitrates of the substances joined to the nitric acid, 
 
 The marine acid, combined with fixed mineral alkali, forms common marine salt, 
 
 or muriate of soda; 
 
 With fixed verelnble alkali, febrifuge salt of Sylvius, or muriate of potash ; 
 
 With volatile alkali, sal ammoniac ; 
 
 With mercury, if digested on it long enough to oxidate the metal completely, it 
 forms corrosive sublimate. 
 
 When this muriate is not perfectly saturated with oxygen, it forms a salt called 
 mercurius dulcis. 
 
 All neutral salts, composed of marine or muriatic acid, combined with alkalies, ; 
 
 earths, or metals, are called muriates. 
 
 The vegetable acid, called the tartareous, combined with fixed vegetable alkali, ‘ 
 
 forms tartar : 
 
 With fixed mineral alkali, the salt called vegetable salt, Seignettes salt, or Sal : 
 
 polychrest : 
 
 Acetic acid, when combined with vegetable alkali, forms a salt called foliated earth 
 of tartar (acetate of potash) : 
 
 With copper it forms verdigris Recetas of copper); a salt well known in com-— 
 
 merce, and a violent poison : 
 
 With lead it forms a salt called saccharum saturni (acetate of lead), which is 
 also a poison, and employed in the arts: 
 
 With mercury, acetate of mercury, a salt of great use in cases of syphilis. 
 
 We shall confine ourselves to this brief account of the best known compositions of ' 
 the different acids with different substances. The number however might have been ty 
 considerably increased ; for every acid may be combined with almost all the alkalies, — 
 
 earths, and metals. > 
 
  * This rule is liable to some exceptions. It may however be followed without any danger of — 
 
 much mistake. 
 
OF OXYGEN. 781 
 
 ARTICLE II. 
 Of Oxygen. 
 We have already had occasion to mention this substance, for a knowledge of which 
 we are indebted to modern chemistry. It has never been obtained alone; but its 
 presence and effects are now so well known, that its existence may be considered as 
 demonstrated. We shall here observe that almost all the phencmena explained for- 
 merly, by admitting the agency of an ideal principle, to which the old chemists gave 
 the name of phlogiston, are now known to be produced by a play of chemical affinities, 
 in which oxygen performs a conspicuous part. Thus, for example, in what used to 
 be called the calcination, but now the oxydation, of metals, it was believed that the 
 metals parted with phlogiston. It is however shewn by experiment, that instead of 
 parting with anything, they receive an increase of weight ; and it is known also, by 
 _ taking proper care to perform the experiments in close vessels, that the air not only 
 loses in its volume, but suffers also a diminution of its weight equal to what the 
 metal has gained. But besides, the oxygen may be again separated from the metal, 
 which yields a quantity of gas equal in weight and volume to what had been taken 
 from the air by the metal. & 
 
 The oxygen gas, thus separated from an oxide, is found to be that part of the atmo- 
 
 sphere which serves for respiration; it is therefore called also vital air. When a 
 metal, such as mercury for example, has been converted into an oxide, by exposure with 
 heat, to atmospheric air, in a close vessel, the air which remains after all the oxygenous 
 part has been taken up by the metal, is found to bea suffocating gas, distinguished by 
 the name of azotic gas. Itis not only unfit for-respiration, but incapable of maintaining 
 combustion ; whereas oxygen gas, which may be separated from the oxide of mercury, 
 or of manganese, &c., is better adapted for promoting combustion than atmospheric 
 
 -air itself; so that many substances, which will not inflame in common air, burn with 
 great violence in oxygen gas. We shall mention one for the sake of illustration. If 
 a piece of small iron wire, with a bit of lighted tinder to begin the combustion, be 
 introduced into a vessel containing oxygen gas, it will burn with much greater in- 
 tensity than flax or cotton, even though dipt in oil, will do in atmospheric air. 
 
 From this, and other considerations, which it would be too tedious here to enume- 
 rate, it is inferred that the oxygen in calces or oxides, in acids, and in all other com- 
 binations, the gaseous ones excepted, is in a concrete state; but that in the gaseous 
 state it is united to something else, namely the matter of heat, or of light, or of 
 both: and from this theory the process of combustion is thus explained: during the 
 process, the oxygen of the oxygenous part of the atmosphere, unites itself to the 
 burning body ; and the heat and light which were united with the oxygen, and which 
 held it in chemical solution, are liberated, producing that phenomenon commonly 
 
 called burning. The heat and light therefore are not furnished by the burning body, 
 but by the air. 
 
 When carbonaceous matters, such as coals, wood, &c., are employed in the process, 
 
 a portion of the liberated heat is again taken up by a new gas which has been formed, 
 
 namely carbonic acid gas, or fixed air, which we have already had occasion to mention, * 
 
 It is the same, in every respect, as the gas that may be separated from limestone or 
 
 chalk, by exposing it to heat. The matter of heat or caloric givesit the gaseous form ; 
 and oxygen communicates to it its acid properties. All acids indeed, as already men- 
 tioned, are indebted for their acidity to oxygen. Acidifiable bases, fully saturated 
 
 with oxygen, produce acids; others, such as the metals, become oxides; but 
 if means could be devised to saturate them compietely it is probable they would all 
 produce acids. Some of them indeed have been brought to this state. 
 
 * There are a variety of gases which all differ in many of their properties from common air; but 
 to enter upon them would extend the present article beyond the limits which must be allotted to it. 
 
722 CHEMISTRY. 
 
 Having had occasion, in speaking of the vitriolic acid, to mention the part which 
 oxygen has in the formation of acids, it is needless to say any thing farther on that 
 subject. 
 
 * 
 
 ARTICLE ITIf. 
 Of Affinities. 
 
 It is also necessary that we should here say a few words respecting Affinities ; 
 as they are a key which serves to explain a great number of chemical compositions — 
 and decompositions. 
 
 Affinity is the force with which two substances tend to combine and maintain — 
 themselves in a state of union. Thus, for example, if vitriolic acid be poured over — 
 chalk, it drives out the carbonic acid, or fixed air, before united to it; lays hold 
 of the calcareous earth; combines with it, and forms a mixture which is neither 
 earth nor acid; but if fixed alkali, whether vegetable or minera), be added to the 
 solution, the calcareous earth will be expelled from its place; the vitriolic acid will 
 seize on the fixed alkali, abandoning the former, and will thus form a new salt—an 
 alkaline sulphate. 
 
 The molecule of the calcareous earth, and those of the vitriolic acid, have there- 
 fore a stronger tendency to unite, than those of the same earth and carbonic acid. 
 The union is so perfect, that the fluid, though one of the ingredients in the compound — 
 be an earth, passes through a filtre. And hence it appears, that this result is not a 
 mere division and interposition of the particles of the stone, between those of the 
 solvent as was supposed, and is still believed by some who are not acquainted with 
 the principles of chemistry. But, we might ask such persons, why do the particles 
 of iron, dissolved in an acid, maintain themselves in the liquor, notwithstanding their 
 excess of specific gravity? for, according to their philosophy, this is inexplicable. 
 Butif each particle of the iron be united to each particle of the solvent, the difficulty 
 will vanish ; and if we admit this principle, and also an inequality of force in the © 
 above tendency, all the phenomena of chemistry may be so easily explained, that the - 
 existence of such a force in the particles of bodies caunot be denied. 
 
 Besides, we have positive proofs of the force with which polished surfaces adhere, 
 independently of any surrounding fluid. Nothing then is more natural, than to 
 conceive a similar force between the minute particles of bodies: it would be suffi-— 
 cient to suppose them to have small facets of different forms and sizes, by which - 
 they adhere with a force that may be subject to very complex laws; since it may 
 depend on the extent of the facet, and the density and form of the particle ; for 
 these may produce a great many variations. 
 
 These affinities or tendencies are indeed very unequal, and by way of example we 
 shall observe, that the force with which calcareous earth combines with vitriolic acid, 
 is less than that with which it combines with any alkali. On this account an alkali 
 may be substituted instead of calcareous earth. All acids, in general, have more 
 affinity for alkalies than for calcareous earths; for the latter than for metals; and 
 for some metals than for others; which furnishes an easy method of decomposing 
 certain mixtures. In the course of this Part we shall give a few curious and in- 
 structive examples. 
 
 Chemical affinity is sometimes rendered evident by the heat which is produced on — 
 mixing two cold bodies. A mixture of equal parts of water and oil of vitriol in- — 
 stantly acquires the temperature of boiling water. If potassium be dropped upon 2 
 ice, the ice will be partially melted, the water decomposed, and a brilliant flame en- 
 gendered by the action of the two substances upon each other. : 
 
 Sometimes the effect of the admixture of different bodies is to produce another in — 
 which the properties of the constituents are totally lost. For example,— Equal ~ 
 parts of tin and iron, malleable and ductile metals, form an alloy which is neitheay 
 
 <n re 
 
SOLUTION AND PRECIPITATION, . 783 
 
 malleable nor ductile, being a very brittle metal. Liquid ammonia and muriatic acid, 
 both strong smelling fluids, if mixed in equal proportions, produce muriate of am- 
 monia, a fluid devoid of smell. 
 
 It often happens, if one body be added to a mixture of two others, that it will 
 combine with one in preference to the other. This property which bedies possess 
 of singling out those substances with which they form the strongest affinity, is called 
 
 elective attraction. 
 
 ARTICLE Iv, 
 Solution and Precipitation. 
 
 Solution is an operation, by which a fluid combines with the molecule of a solid, 
 or of another fluid; so that each particle of the one contracts an adhesion with each 
 particle of the other. This union or adhesion is produced by the affinity of these 
 particles for each other; because, if a greater or less affinity does not exist, there 
 can be no solution. 
 
 Solution does not consist in a mere attenuation of the body dissolved, and an in- 
 terposition of its molecule between that of the fluid. When there is only an inter- 
 position of this kind, a separation soon takes place. 
 
 Precipitation is effected when the molecule of the dissolved body, being aban- 
 doned by the solvent, fall to the bottom of the liquor. This happens sometimes in 
 
 consequence of a mere diminution in the force of the solvent, produced by diluting 
 
 it with a great deal of water; but it is produced, for the most part, by presenting to 
 the solvent some body for which it has a greater affinity than for the body already 
 dissolved. For example, if a fixed alkali be poured into nitrous acid, holding in so- 
 lution calcareous earth, the acid will seize on the alkali, on account of its greater 
 affinity, and will abandon the earth, which will fall to the bottom of the vessel. 
 
 At other times, precipitation takes place, in consequence of presenting to the so- 
 lution a body which, by combining with the dissolved body, forms a new mixture, in- 
 soluble in the solvent. Of this we have an example in the following operation: 
 If calcareous earth be dissolved in nitrous or marine acid, on pouring into the so- 
 lution vitriolic acid, the latter will seize on the earth, and form with it sulphate 
 of lime, well known under the name of selenite. But as selenite is not soluble in 
 
 ‘these acids, nor even in water, unless it be in very large quantity, it falls to the 
 
 bottom. The same thing takes place when vitriolic acid is poured into a solution 
 of mercury in nitrous acid: the particles precipitated form what is called white 
 precipitate, - 
 ARTICLE V. ’ 
 Of Effervescence and Fermentation.—Difference between them. 
 
 Nothing is more common than for those who have but little knowledge of che- 
 mistry, to confound these two phenomena; which however are essentially different : 
 and it must be allowed that, till within a few years, the French chemists confounded 
 these terms, though they did not confound the operations which they denote. 
 
 Effervescence is the motion, accompanied with heat, which often takes place during 
 asolution. Thus, for example, when a little nitrous acid is poured over copper filings ; 
 or when vitriolic acid is poured over those of iron; or when a small quantity of the 
 same acid is poured over calcareous earth, a violent ebullition is excited, till the 
 combination is formed, when the commotion subsides, and the liquor becomes trans- 
 
 parent. Such, in a few words, is effervescence.* Hence it is said that acids, in 
 
 general, effervesce with alkalies, with metals, and with calcareous earths. 
 
 * In this case, one of the gases, alluded to in a former note, is produced, namely, hydrogen 
 
 gas, or inflammable air. It has the property of burning when cnce inflamed: if mixed with at- 
 
 ‘mospheric air, and then set fire to, it explodes like gunpowder. 
 
: , | | 
 
 ; ¥ 
 784 CHEMISTRY. - 
 - But fermentation is entirely different: it is the intestine and spontaneous motion, 
 produced in certain liquors, extracted from vegetable matters ; and which, from being 
 sweet and insipid, renders them spirituous and vinous. Must, for example, or the 
 expressed juice of grapes, is not wine; there is not a single drop of spirit in it; bat 
 when exposed to a moderate heat, a play of affinities takes place, by which the liquor 
 becomes turbid of itself, is internally agitated, throws up bubbles, which are found 
 to be fixed air, and when the disengagement of these bubbles ceases, it is entirely a- 
 new liquor, spirituous, intoxicating, &c. The case is the same with beer, produced 
 by the fermentation of malt, or the strong decoction of barley, prepared in a certain 
 manner. This, as may be seen, is an operation very different from effervescence, as 
 above described. When a person, therefore, speaking on chemical subjects, con-— 
 founds these two words, the ears of the enlightened chemist are as much shocked as 
 those of a philosopher would be, were he to hear abhorreuce of a vacuum employed to 
 explain any of the phenomena of nature. 
 
 ARTICLE VI. 
 Of Crystallization. 
 
 This appellation is given to that peculiar arrangement, which the greater part of 
 salts, and even other bodies, affect to assume, after their solution in a liquid, when 
 their parts, brought sufficiently near to each other by evaporation, dispose themselves 
 in groups. As rock crystal was the first.body in which this regular arrangement 
 was observed, its name has been given to that discovered in many other bodies, and 
 particularly salts, by the subsequent researches of chemists and naturalists. 4 
 
 Dissolve common salt in water, and evaporate the solution to a certain degree ; if 
 it be then left at rest in a cool place, the saline particles, brought near to each other, 
 and falling together to the bottom of the vessel, or attaching themselves to the sides’ 
 of the vessel, will form masses, in which the cubic figure will be easily distinguished ; 
 as prisms of six sides terminating in pyramids, implanted in each other, are distin- 
 guished in rock crystal. If crystallization be promoted at the surface, by evaporat- 
 ing the liquid, it takes place in the form of truncated square pyramids, composed of 
 small cubes, heaped together in a certain order, one upon the other; as has been 
 shewn by M. Rouelle, who has explained the phenomenon with great ingenuity. 
 
 If the salt held in solution be saltpetre, the crystals formed will be hexagonal 
 prisms, terminated by hexagonal pyramids. 
 
 In short, each salt affects a peculiar form. 
 
 Alum crystallizes in exact octaedra, that is, figures formed of two quadrangular 
 pyramids, having a common square base. 
 
 Vitriol of iron forms crystals which are oblique- pale cubes, ¢ or cubes the six faces 
 of which are rhombuses with unequal sides, 
 
 The crystals of blue vitriol are compressed Toieeedent the form of which cannot be 
 described in a few words. Verdigrise, or the salt produced by a combination of vine- 
 gar and copper, forms crystals which are oblique-angled parallelopipedons. 
 
 Crystallized or candied sugar forms quadrangular prisms, cut obliquely bya an 
 inclined plane. 
 
 But, as before observed, there are a multitude of other bodies, besides salts, whieh 
 possess the same property of forming themselves into masses, and affecting the same 
 regular figures. Most ores and pyrites are distinguished by their particular form: 
 mineralized lead, for example, has a great tendency to the right-angled or oblique- 
 angled cubical form. Even stones, in such cases, observe a certain regularity. The 
 crystals of gypsum, or plaster of Paris, are shaped like the point of a lance; and 8YP- 
 sum therefore is properly a salt. Calcareous spar, known under the name of Icelan- 
 dic crystal, is always an oblique-angled parallelopipedon, inclined in the direction of 
 its diagonal, and at determinate angles. In short, when metals cool slowly, their 
 
CHEMICAL EXPERIMENTS. 785 
 
 particles are at liberty to arrange themselves in a regular form ; and it was long ago 
 remarked that this was the case with ee ha and has since been observed in regard _ 
 to iron, copper, zinc, &c. 
 
 As this phenomenon is one of the most curious in chemistry, it would afford mat- 
 ter fora very long article ; but as we have given a short view of the subject, we must 
 refer the reader, for further information, to the “ Essai de Cristallographie ” of Romé 
 Delisle, which appeared in 1772, in 8vo, and to the Abbé Haiiy’s late work on the 
 same subject. 
 
 We shall now give a series of chemical experiments, which will be partly an appli- 
 cation of the before-mentioned principles, or which will exhibit curious phenomena. 
 
 ARTICLE VII. 
 Various Chemical Experiments 
 
 EXPERIMENT I. 
 
 How a body of a combustible nature may be continually penetrated by fire without being 
 consumed. 
 
 Put into an iron box a piece of charcoal, sufficient to fill it entirely, and solder 
 on the lid. Ifthe box be then thrown into the fire it will become red, and it may 
 even be left in it for several hours or days. When opened after it has cooled the 
 charcoal will be found entire, though there can be no doubt of its having been pene- 
 trated by the matter of the fire, as well as the whole metal of the box which con- 
 tains it. 
 
 The cause of this effect is as follows. Before charcoal, or any other combustible 
 body can be consumed, oxygen must have access to it; but the contact of the atmo- 
 sphere is prevented by the iron case being made quite close. 
 
 Hence too it happens, that coals covered with ashes require much longer time to be 
 consumed, than if they were exposed to the open air. This phenomenon, though 
 well known, could not be explained by any philosopher unacquainted with the nature 
 and properties of oxygen. 
 
 EXPERIMENT II. 
 Apparent Transmutation of tron into copper or silver. 
 
 Dissolve blue vitriol in water, till the latter is nearly saturated ; and immerse into 
 the solution small plates of iron, or coarse filings of that metal. These small plates 
 of iron, or filings, will be attacked and dissolved by the acid of the vitriol ; while its 
 copper will be precipitated, and deposited in the place of the iron dissolved. 
 
 If the bit of iron be too large to be entirely dissolved, it will be so completely: 
 covered with the cupreous particles, that it will seem to be converted into copper. 
 This isan experiment commonly exhibited to those who visit copper mines. At 
 least we -have seen it performed at Saint-Bel in the Lyonnese. A key immersed 
 some minutes in water, collected at the bottom of a copper-mine, was entirely of a 
 copper colour when drawn out. 
 
 If you dissolve mereury in marine acid, and immerse in it a bit of iron; or if this 
 solution be rubbed over iron, it will assume a silver colour. Jugglers sometimes 
 exhibit this chemical deception, at the expence of the credulous and ignorant. 
 
 Remark.—In.this case, there is no real transmutation, but merely the appearance 
 of one. The iron is not converted into copper; the latter, held in solution by the 
 liquor impregnated with the vitriolic acid, is only deposited in the place of the iron 
 
 with which the acid becomes charged. Every time indeed that a menstruum, hold- 
 
 ing any substance in solution, is presented to another substance, which it can dissolve 
 
 with more facility, it abandons the former, and becomes charged with the latter. This 
 3. E 
 
786 * CHEMISTRY. he Tas 
 
 is so certain, that when the liquor which has deposited the copper is evaporated, it 
 ' produces crystals of green vitriol, which, as is well known, are formed by the combi- 
 nation of the vitriolic acid with iron. And this also is practised in the mine before 
 mentioned. The liquor in question, which is nothing but a pretty strong solution of 
 blue vitriol, is put into casks, or large square reservoirs ; and pieces of old iron being 
 then immersed in it, are at the end of some time converted into a sort of sediment, 
 from which copper is extracted. The liquor thus charged with iron is evaporated to 
 a certain degree, and wooden rods are immersed in it, which become covered with 
 crystals of green vitriol. ; 
 
 This experiment may be made also by dissolving copper in the vitriolic acid, and 
 then diluting the solution with a little water. This is a new proof that the liquor 
 only deposits the copper with which it is charged. 
 
 EXPERIMENT IIt. 
 Different Substances successively precipitated, by adding another to the Solution. 
 
 In the former experiment we have seen copper precipitated by iron; we shall now 
 shew iron itself precipitated. For this purpose, throw into a solution of iron a small 
 
 bit of zinc ; and in proportion as the latter dissolves, the iron will fall to the bottom — 
 
 of the vessel: it may easily be known to be iron, because it will be susceptible of 
 being attracted by the magnet. 
 
 If you are desirous to precipitate the zine, nothing will be necessary but to 
 throw into the solution a bit of calcareous stone, such as white marble, or any 
 
 other stone capable of making lime ; the vitriolic acid will attack this new substance, 
 
 and suffer to be deposited at the bottom of the vessel a white powder, which is 
 zine. 
 
 To precipitate the lime or calcareous earth, pour into the solution liquid volatile 
 . alkali (spirit of hartshorn); the earth, being abandoned by the acid, will deposit 
 itself at the bottom of the vessel. 
 
 The calcareous earth may be precipitated also, and much better, by pouring into 
 the liquor a solution of fixed alkali, such as fixed vegetable alkali; or by throwing 
 into it fixed mineral alkali. 
 
 Remark.—It is by a similar effect that hard water decomposes soap, instead of — 
 dissolving it, and suffers to be deposited a greater or less quantity of calcareous earth. 
 
 The manner in which this takes place is as follows. 
 
 Water in general is hard only because it holds in solution selenite or gypsum (a ~ 
 
 combination of vitriolic acid with calcareous earth), which it has dissolved in its 
 passage through the bowels of the earth, or which has been formed by the water first 
 becoming impregnated with vitriolic salts, and afterwards in its course meeting with 
 and dissolving a portion of calcareous earth. 
 
 On the other hand, soap is an artificial combination of fixed alkali with oil, or with 
 some other greasy substance, and which have no great affinity. 
 
 When soap therefore is dissolved in water impregnated with selenite, the vitriolic 
 acid of the latter, having a greater tendency to unite with the fixed alkali than with 
 the calcareous earth, which enters into the composition of the selenite, abandons 
 that earth, and combines with the fixed alkali, in such a manner that the soap is 
 
 decomposed ; and as the oil is immiscible with water it is diffused through it in 
 
 the form of white flakes, while the calcareous earth of the selenite falls to the 
 bottom. 
 
 - 
 
 Here we have a new example of the use of chemistry, to account for certain com- _ 
 mon effects which are inexplicable to the philosopher unacquainted with that — 
 
 science. 
 
& 
 
 CHEMICAL EXPERIMENTS. 787 
 
 EXPERIMENT IV. 
 
 By the mixture of two transparent Liquors to produce a blackish Liquor. —Method of 
 making good Ink. 
 
 Provide a solution of green or ferruginous vitriol, and an infusion of gall-nuts, or 
 of any other astringent vegetable substance, such as oak-leaves, well clarified 
 and filtred; if you then pour the one liquor into the other, the compound will 
 immediately become obscure, and at last black. 
 
 If the liquor be suffered to remain at rest, the black matter suspended in it will fall 
 to the bottom, and leave it transparent. 
 
 Remark.—This experiment may serve to explain the formation of common ink;: 
 for the. ink we use is nothing else than a solution of green vitriol mixed with an 
 infusion of gall-nuts, and a little gum. The blackness arises from the property 
 which the gall-nuts have of precipitating, of a black or blue colour, the iron held in 
 solution by the water impregnated with vitriolic acid; but as the iron would soon 
 fall to the bottom, it is retained by the addition of gum, which gives to the water 
 sufficient viscidity to prevent the iron from being precipitated. 
 
 The reader perhaps will not be displeased to find here the following recipe for 
 making good ink. 
 
 Take one pound of gall-nuts, six ounces of gum arabic, six ounces of green cop- 
 peras, and one gallon of common water or beer: pound the gall-nuts, and infuse them 
 in a gentle heat for twenty-four hours, without bringing the mixture to ebullition ; 
 then add the gum in powder. When the gum is dissolved, put in the green vitriol: 
 if you then strain the mixture, you will obtain very fine ink. 
 
 EXPERIMENT V. 
 To produce inflammable and fulminating Vapours. 
 
 Put into a moderately sized bottle, with a short and wide neck, three ounces of 
 oil or spirit of vitriol, and twelve ounces of common water- If you then throw into 
 this mixture at different times, an ounce or two of iron filings, a violent effervescence 
 will take place, and white vapours will arise from it. On a taper being presented to 
 the mouth of the bottle, these vapours will inflame, and produce a violent detona- 
 tion, which may be repeated several times, as long as the liquor continues to furnish 
 similar vapours. 
 
 EXPERIMENT VI. 
 The Philosophical Candle. 
 
 Provide a bladder, into the orifice of which is inserted a metal tube, some inches 
 in length, and so constructed, that it can be fitted into the neck of a bottle contain- 
 
 ‘ing the. same mixture as that used in the preceding experiment. 
 
 Having then suffered the atmospheric air to be expelled from the bottle by the 
 elastic vapour of the solution, apply to the mouth of it the orifice of the bladder, 
 after carefully expressing from it the common air.* The bladder, by these means 
 will become filled with inflammable air; which if you force out against the flame 
 of a taper, by pressing the sides of the bladder, will form a jet of a beautiful green 
 flame. This is what chemists Call a philosophical candle. ' 
 
 EXPERIMENT VII. 
 To make an Artificial Volcano. 
 For this curious experiment, which enables us to assign a very probable cause 
 for volcanoes, we are indebted to Lemery. 
 
 ® Great care must be taken not to omit this precaution ; for a-mixture o. inflammable and atmo- 
 spheric air will explode with violence, instead of burning. 
 
 3E 2 
 
 - 
 
788 CHEMISTRY, 
 
 Mix equal parts of pounded sulphur and iron filings, and having formed the whole 
 into a paste with water, bury a certain quantity of it, forty or fifty pounds for 
 example, at about the depth of a foot below the surface of the earth. In ten or 
 twelve hours after, if the weather be warm, the earth will swell up and burst, and 
 ames will issue out, which will enlarge the aperture, scattering around a yellow 
 and blackish dust. : 
 
 It is probable that whatis here seen in miniature, takes place on a grand scale 
 in volcanoes; as it is well known that they always furnish abundance of sulphur, 
 and that the matters they throw-up abound in metallic and perhaps ferruginous par- 
 ticles; for iron is the only metal which has the property of producing an efferves- 
 cence with sulphur, when they are mixed together. , at 
 
 But it may be easily conceived, from the effect of a small quantity of the above 
 mixture, what thousands or millions of pounds of it would produce. There is no 
 doubt that the result would be phenomena as terrible as those of earthquakes, 
 and of those volcanic eruptions with which they are generally accompanied. 
 
 EXPERIMENT VIII. 
 To make Fulminating Gold. 
 
 First to make Aqua Regia, by mixing four parts of spirit of nitre with one of sal 
 ammoniac. Put some fragments of pure gold into this liquor, and when they are 
 dissolved, add to the liquor a solution of fixed alkali, called also oil of tartar per 
 deliquium. The gold will be precipitated to the bottom in the form ofa yellow 
 powder, which must be collected by pouring off the supernatant liquid. If this 
 
 powder be then washed in warm water and dried, you will- have fulminating gold. 
 
 To make it detonate, put a very small quantity of it on the point of a knife, and 
 hold it over the flame ofa taper. As soon has it has acquired a certain degree of heat, 
 the powder will inflame with a terrible explosion, far greater than that which would 
 be produced by a like quantity of gunpowder. 
 
 Gold prepared in this manner does not detonate by the application of heat alone; 
 mere touching is sufficient to produce the same effect. We have seen some particles 
 of fulminating gold, which had got between the neck of a bottle and the stopper, 
 during the act of shutting it, make a sudden explosion, so as to break the bottle 
 to pieces, and wound the person who held it. The same thing would infallibly take 
 
 7 
 
 place if this gold were triturated in a mortar; or if one should attempt to fuse it, in ~ 
 
 order to reduce it toa metallic mass, without the necessary preparations. 
 
 Remarks.—The gold would not be fulminating, if the aquaregia were made by a 
 mixture of spirit of nitre and spirit of marine salt, or spirit of nitre to which marine 
 salt had been added, (for it is prepared in all these ways); and if the gold were 
 precipitated by fixed alkali, in order to render the gold fulminating, volatile alkali 
 must be united either with the aqua regia or the precipitant. 
 
 If aquaregia, therefore, made with spirit of nitre and spirit of marine salt, be em- 
 ployed to dissolve gold, it must be precipitated by volatile alkali. By these means 
 you will still obtain fulminating gold. 
 
 To deprive it of its fulminating property, pour over it vitriolic acid, or asolution 
 of fixed alkali. By this process a combination is formed which destroys in the 
 gold its detonating quality; if it be then washed, it will be found in a powder, 
 which may be reduced without any danger by the usual means. 
 
 EXPERIMENT ix. 
 To make Fulminating Powder. 
 
 Mix together three parts of nitre, two of well dried alkali, and one of sulphur ; 
 if a little of this mixture be put into an iron spoon over a gentle fire capable how- 
 
 * 
 
CHEMICAL EXPERIMENTS, 789 
 
 ever of melting the sulphur, when it acquires a certain degree of beat, it will de- 
 tonate with a loud noise, like the report of a small cannon. 
 
 This would not be the case, if the mixture were exposed to a heat too violent ; 
 the parts only most exposed to the fire would detonate, and by these means the 
 effect would be greatly lessened. 
 
 If thrown on the fire it would not detonate, and would produce no other effect 
 than pure nitre, which indeed detonates, but without any explosion. 
 
 Mix eight grains of nitrate of potash with four grains of phosphorus, place the 
 mixture on a warm anvil, and striking it smartly with a warm hammer, a violent 
 detonation will be produced. 
 
 EXPERIMENT X. 
 
 A liquor which becomes coloured and transparent alternately, when exposed to or 
 
 removed from the contact of the external air. 
 
 Digest copper, that is to say dissolve it slowly by means of a gentle heat, ina strong 
 solution of volatile alkali. As the solvent attacks the copper, it will acquire a beau- 
 tiful blue colour. If you pour some of this liquor into a small bottle till it is nearly 
 full, and then close it well with a stopper, the colour will gradually become fainter, 
 and at last disappear. On opening the bottle, the colour willreturn; and this alter- 
 nation may be produced as often as you choose. 
 
 EXPERIMENT XI. 
 Pretended Production of Iron. 
 
 Take clay, or the ashes of burnt vegetables or animals, and draw over them an 
 artificial magnet. By these means you will often attract some particles of iron, which 
 will adhere to the magnet. You may then rest assured that no iron, in a metallic 
 state, remains in the earth or ashes. 
 
 Then mix the remaining earth or ashes with pounded charcoal, and having formed 
 the mixture into a paste with linseed oil, put the whole into a crucible, and expose it 
 to a red heat for some time, but not intense enough to prodnce vitrification. When 
 the mass is cold, and reduced to dust, if the artificial magnet be again drawn over it, 
 a great many iron particles will be attracted by it, and will adhere to it. 
 
 Remark.—Some have pretended to give this experiment as a proof that iron may 
 be produced by clay and linseed oi]. A celebrated chemist, member of the Academy 
 of Sciences, even entertained this idea, notwithstanding the opposition he experienced 
 from one of his brother members. But, in our opinion, no chemist at present will 
 see in it a production of iron. 
 
 It would indeed be wrong to suppose, after the iron found at first by the magnet 
 has been extracted, that no more remains in it. The magnet attracts iron only in its 
 metallic state, or when it approaches near to it; but some still remains in the state 
 of an oxide, and in this state it is not susceptible of being attracted by the magnet, as 
 may be proved by an experiment made on oxide formed artificially by the torrefaction 
 of iron, or on the rust of that metal. 
 
 Besides, it is well-known that of all the metals iron is that most universally dif- 
 fused throughout the earth ; it is the colouring principle of common or red clay, and 
 when clay is so coloured it contains iron. 
 
 What then is the effect of the torrefaction of clay with charcoal dust and linseed 
 oil, or any other fat oily body which has a strong affinity for oxygen? Such bodies 
 having a greater affinity for that principle than iron has, by the aid of a proper tem- 
 perature, they decompose the oxide, and seizing on its oxygen leave the iron in a 
 metallic state, and consequently susceptible of being attracted by the magnet. This 
 
 “is the whole secret of the operation, 
 
790 CHEMISTRY. 
 
 But it may be said, what reason is there to think that these wood-ashes contain iron ? 
 In answer to this question, we shall observe, that iron being diffused in great abun- 
 dance throughout all nature, it is a constituent part in a great many vegetable produc- 
 tions : in some of them it is found even in a metallic state ; that it is susceptible of great 
 attenuation ; and that when dissolved in any liquid it passes with it through the filtre, 
 at least in part. Hence it may easily ascend with the sap of plants; it circulates in the 
 human body with the blood: in short, some even assert that plants are coloured by 
 this metal with the concurrence of light; so that without iron and light all plants 
 would be entirely white. 
 
 EXPERIMENT XII. 
 
 With two liquids mixed together, to form a solid body, or at least a body which has 
 
 consistence. 
 
 Make a highly concentrated solution of fixed alkali, and another of nitrate of lime: 
 if the two solutions be mixed together, there will be an abundant precipitation of a 
 matter which will assume a sort of solidity, 
 
 This phenomenon appeared so wonderful to the old chemists, that the operation by 
 which it is produced was called the chemical miracle. There is however nothing won- 
 derful in it; for what takes place is as follows: : 
 
 The two solutions being mixed, the nitrous acid abandons the earth, to seize on the 
 fixed alkali, and with it remains in solution and transparent in the supernatant liquor. 
 
 The earth is then precipitated, and forms the solid body which results from the 
 
 mixture. 
 
 We shall here give another operation, which might with more justice be called a 
 chemical miracle. We are indebted for it to a remark of M. de Lassonne, first 
 physician to the queen. 
 
 EXPERIMENT XII. 
 To form a combination, which when cold is liquid and transparent ; but which when 
 warm becomes thick and opake. 
 
 Put equal quantities of fixed alkali, either mineral or vegetable, and of well pul- 
 verised quick-lime, into a sufficient quantity of water, and expose it toa strong and 
 speedy ebullition. Then filtre the product, which at first will pass through with diffi- 
 culty, but afterwards with more ease, and preserve it in a bottle well stopped. This 
 liquor, when made to boil either in the bottle or in any other vessel, will become 
 turbid, and assume the consistence of very thick glue; but when cold it will recover 
 its fluidity and transparency; and this alternation may be repeated as often as you 
 choose. 
 
 M. de Lassonne made many experiments to discover the cause of this singular 
 phenomenon ; and he assigns one which may be seenin the ‘‘ Mémoires de l’Académie 
 des Sciences ” for 1773. 
 
 EXPERIMENT XIV. 
 
 To make a flash, like that of lightning, appear in a room, when any one enters with a 
 lighted candle. ) 
 Dissolve camphor in spirit of wine, and deposit the vessel containing the solution 
 in a very close room, where the spirit of wine must be made to evaporate by speedy 
 and strong ebullition. If any one then enters the room with a lighted. candle, the 
 air will inflame; but the combustion will be so sudden, and of so short duration, as 
 not to occasion any danger. 
 It is not improbable that the same effect might be produced, by filling the air of an 
 apartment with the dust of the seed of a certain kind of lycoperdon, which is in- 
 flammable ; for this seed, which is exceedingly minute, and like fine dust, inflames in 
 
 Rates 
 
 By 
 
SYMPATHETIC INKS, 791 
 
 the same manner, as the pulverised resin used for the torches of the Furies, and for 
 representing lightning at the opera. And perhaps it would be better to substitute it 
 for resin, as it does not produce that strong smell which results from the latter when 
 burnt, and which is so disagreeable to the spectators. 
 
 EXPERIMENT XV. 
 Of Sympathetic Inks ; and some tricks which may be performed by means of them. 
 
 Sympathetic inks are certain liquors which afne, and in their natural state, are 
 colourless; but which, by being mixed with each other, or by some particular cir- 
 cumstance, assume a certain colour. 
 
 Chemistry presents us with a great many liquors of this kind, the most curious 
 of which we shall here describe. 
 
 Ist. If you write with a solution of green vitriol, to which a little acid has been 
 added, the writing will be perfectly colourless and invisible. To render it visible, 
 nothing will be necessary but to immerse the paper in an infusion of gall nuts in 
 water, or to draw a sponge moistened with the water over it. Those who have observed 
 the fourth experiment may readily see, that in this case the ink has been formed on 
 the paper. In the making of ink the two ingredients are combined before they are 
 used for writing ; here they are not combined till the writing is finished: this is the 
 whole difference. 
 
 2d. If you are desirous of having an ink that ‘sha!l become blue, you must write 
 with a solution of green vitriol, and moisten the writing with a liquor prepared in 
 the following manner: 
 
 Make four ounces of tartar, mixed with the same quantity of nitre, to detonate 
 on charcoal; then put this alkali into a crucible with four ounces of dried ox blood, 
 and cover the crucible with a lid, having in it only one small aperture ; calcine the 
 mixture over a moderate fire, till no more smoke issues from it; and then bring the 
 whole to a moderate red heat. Take the matter from the crucible, and immerse it, 
 while still red, in two quarts of water, where it will dissolve by ebullition; and 
 when the liquor has been reduced to one half, it will be ready for use. If you then 
 moisten with it the writing above mentioned, it will immediately assume a beautiful 
 blue colour. In this operation, instead of black ink, there is formed Prussian 
 blue. 
 
 3d. If you dissolve bismuth in nitric acid, and write with the solution, the letters 
 will be invisible. ‘To make them appear, you must’ employ the following liquor. 
 
 Boil a strong solution of fixed alkali with sulphur, reduced to a very fine powder, 
 until it dissolves as much of it as it can: the result will be a liquor which exhales 
 vapours of a very disagreeable odour, and to which if the above writing be exposed, 
 it will become black. 
 
 4th, But of all the kinds of sympathetic ink, the most curious is that made with 
 cobalt. It isa very singular phenomenon, that the characters or figures traced out 
 with this ink, may be made to disappear and to re-appear at pleasure: this property is 
 peculiar to ink obtained from cobalt; for all the other kinds are at first invisible, 
 until some substance has been applied to make them appear; but when once they 
 have appeared they remain. That made with cobalt may be made to appear and to 
 disappear any number of times at pleasure. 
 
 To prepare this ink, take zaffre, and dissolve in it aqua regta (nitro-muriatic acid) 
 till the acid extracts from it every thing it can; that is the metallic part or the 
 cobalt, which communicates to the zaffre its blue colour ; then dilute the solution, 
 which is very acrid, with common water. If you write with this liquor on paper, 
 the characters will be invisible; but when exposed toa sufficient degree of heat, they 
 will become green. When the paper has cooled they will disappear. 
 
792 CHEMISTRY. 
 
 It must however be observed, that if the paper be heated too much, they will not 
 disappear at all. : 
 
 Remark.—With this kind of ink, some very ingenious and amusing tricks, such as 
 the following, may be performed. 
 
 Ist. Zo make a drawing which shall alternately represent summer and winter. 
 
 Draw a landscape, anddelineate the ground and the trunks of the trees with the usual _ 
 colours employed for that purpose; but the grass and leaves of the trees with the 
 liquor above mentioned. By these means you will have a drawing, which at the 
 common temperature of the atmosphere will represent a winter piece; but if it be 
 exposed to a proper degree of heat, not too strong, you will see the ground become 
 covered with verdure and the trees with leaves, so as to present a view in summer. 
 
 . Screens painted in this manner were formerly made at Paris. Those to whom 
 they were presented, if unacquainted with the artifice, were astonished to find, when 
 they made use of them, that the views they exhibited were totally changed. * 
 
 2d. The Magic Ovacle. 
 
 Write on several leaves of paper, with common ink, a certain number of questions, 
 and below each question write the answer with the above kind of sympathetic ink. 
 The same question must be written on several pieces of paper, but with different 
 answers, that the artifice may be better concealed. 
 
 Then provide a box, to which you may give the name of the Sybil’s cave, or any. 
 other at pleasure, and containing in the lid a plate of iron made very hot, in order 
 that the inside of it may be heated to a certain degree. 
 
 Having selected some of the questions, take the bits of paper containing them, 
 and tell the company that you are going to send them to the sybil or oracle, to ob- 
 tain an answer ; introduce them into the heated box, and when they have remained 
 in if some minutes take them out, and shew the answers which have been written. 
 
 You must however soon lay aside the bits of paper; for if they remain long in 
 the hands of those to whom the trick is exhibited, they would see the answers 
 gradually disappear, as the paper becomes cold. 
 
 EXPERIMENT XVI. 
 Of Metallic Vegetations. 
 
 To see a shrub rise up in a bottle, and even throw out branches, and sometimes 
 
 a kind of fruit, is one of the most curious spectacles exhibited by chemistry. The 
 operation by which this delusive image is produced, has been called chemical or 
 metallic vegetation, because performed by means of metallic substances: and it is 
 not improbable that some respectable persons, who thought they saw a real palin- 
 genesy, have been deceived by a similar artifice. However this may be, the fol- 
 lowing are the ‘most curious of these vegetations, which in fact 
 
 are only a sort of crystallizations. 
 
 Arbor Martis, Tree of Mars. 
 
 Dissolve iron filings in spirit of nitre (aquafortis) moderately 
 concentrated, till the acid is saturated; then pour gradually 
 into the solution a solution of fixed alkali, commonly called 
 oil of tartar per deliquium. A strong effervescence will take 
 place; and the iron, instead of falling to the bottom of the 
 
 * These drawings are now very commonly exhibited for sale in the shop windows in London. 
 
: CHEMICAL EXPERIMENTS. 793 
 
 vessel, will afterwards rise, so as to cover its sides, forming a 
 multitude of ramifications heaped one upon the other, which 
 will sometimes pass over the edge of the vessel, and extend 
 themselves on the outside, with all the appearance of a plant. 
 If any of the liquor be spilt, it must be carefully collected, and 
 be again put into the vessel, where it will form new ramifica- 
 tions, which willcontribute to increase the mass of the vege- 
 tation. 
 
 Two of these vegetations, copied from a memoir of M. 
 Lemery, Junior, and inserted among those of the Academy of 
 Sciences for 1706, are represented Fig. 58 and 59. A very Brea Ote explanation of 
 the phenomenon may be found among those of 1707. 
 
 Arbor Diane, Tree of Diana. 
 
 This kind of vegetation is called the Tree of Diana, because it is formed by means 
 of silver; as the former is called the Tree of Mars, because produced by iron., We 
 shall here give two processes for this purpose, one of them by Lemery, and the 
 other by Homberg. 
 
 Dissolve an ounce of pure silver in a sufficient quantity of aquafortis, exceedingly 
 pure, and of a moderate strength, and having put the solution into a jar, dilute it 
 with about twenty ounces of distilled water. Then add two ounces of mercury, 
 and leave the whole at rest. In the course of forty days, there will rise from the 
 mercury a kind of tree, which throwing out branches will represent a natural vege- 
 tation. 
 
 Should this process, which is in other respects very simple, be thought too tedi- 
 ous as to time, the following one of Homberg may be employed. 
 
 Form an amalgam of a quarter of an ounce of very pure mercury, and half an 
 ounce of fine silver reduced into filings or leaves; that is to say, mix them together 
 by trituration in a porphyry mortar, by means of an iron pestle. Dissolve this amal- 
 
 gam in four ounces of very pure nitric acid, moderately strong, and dilute the solu- 
 
 tion in abouta pound and a half of distilled water, which must be stirred, and then 
 preserved in a bottle well stopped. Pour an ounce of this liquor into a glass, and 
 throw into it a small bit of an amalgam of mercury and silver, similar to the former, 
 and of the consistence of butter. Soon after you will see rising from the ball of 
 amalgam a multitude of small filaments, which will visibly increase in size, and 
 
 throwing out branches will form a kind of shrubs. 
 
 Homberg, in the Memoirs of the Academy for 1710, gives a method of making a 
 a similar vegetation, either with gold or silver, in the dry way; that jis to say by 
 distillation without any solution. 
 
 There is still another kind of vegetation, mentioned by M. de Morveau, which he 
 
 calls Jupiter’s beard, because tin forms a part of its composition: the process for 
 
 making it may be seen in his ‘* Essais Chimiques.”’ 
 
 Non-metallic Vegetation. 
 
 Cause to decrepitate, on burning charcoal, eight ounces of saltpetre, and place it 
 in a cellar, in order that it may produce oil of tartar per deliquium: then gradually 
 pour over it, to complete saturation, good spirit of vitriol, and evaporate all the 
 moisture. The result will be a white, compact, and very acrid saline matter. Put 
 this matter into an earthen dish, and having poured over it a gallon of cold water, 
 
 leave it exposed to the open air. At the end of some days the water will evapo- 
 
 rate, and there will be found all around the vessel ramifications in the form of needles, 
 
 variously interwoven with each other, and about 15 lines in length. When the 
 _ water is entirely evaporated, if more be added the vegetation will continue. 
 
 XY 
 
794 CHEMISTRY. 
 
 It may be readily seen that this is nothing but the mere crystallisation of a neu- 
 
 tral salt, formed by the vitriolic acid and the alkali of the nitre employed; that is to” 
 
 say vitriolated tartar. 
 EXPERIMENT XVII. 
 To produce heat, and even flame, by means of two cold liquors. 
 
 Put oil of guiacum into a bason, and provide some spirit of nitre, so mu¢h con- 
 centrated, that a small bottle, capable of holding an ounce of water, may contain 
 
 nearly an ounce and a half of this acid. Make fast the bottle containing the acid, 
 
 to the end of a long stick; and after taking this precaution, pour about two thirds 
 
 of the acid into the oil in the bason; the result will be a strong effervescence, fol- 
 
 lowed by a very large flame. If an inflammation does not take place in the course 
 
 of a few seconds, you have nothing to do but to pour the remainder of the nitrous 
 acid over the blackest part of the oil; a flame will then certainly be produced, and — 
 
 there will remain, after the combustion, a very large spongy kind of charcoal. 
 
 Oil of turpentine, oil of sassafras, and every other kind of essential oil, may be 
 made to inflame in the like manner. ‘The same phenomenon may be produced with fat 
 oils, such as olive oil, nut oil, and others extracted by expression, if an acid, formed 
 
 by equal parts of the vitriolic and nitrous acids well concentrated, be poured into— 
 
 them. 
 EXPERIMENT XVIII. 
 
 To fuse tron in a moment, and make it run into drops. 
 
 Bring a rod of iron to a white heat, and then apply to it a roll of sulphur; the iron | 
 
 will be immediately fused, and will run down in drops. It will be most convenient 
 to perform this experiment over a bason of water, in which the drops that fall down 
 will be quenched. On examination they will be found reduced to a kind of cast 
 iron. 
 
 This process is employed for making shot used in hunting, as the drops, by falling 
 
 into the water, naturally assume a round form. 
 
 We shall here add two little experiments, merely because they are usually given in 
 
 books of Philosophical Recreations. 
 
 EXPERIMENT XIX. 
 To fuse a piece of money in a walnut-shell, without injuring the shell. 
 
 Bend any very thin coin, and having put it into the half of a walnut shell, place the 
 shell on a little sand, in order that it may remain steady. Then fill the shell witha 
 mixture made of three parts of very dry pounded nitre, one part of the flowers — 
 
 of sulphur, and a little saw-dust well sifted. 
 
 If you then inflame the mixture, as soon as it has melted you will see the metal — 
 completely fused in the bottom of the shell in the form of a button, which. will 
 
 become hard when the burning matter around it is consumed. The shell employed 
 for the operation will have sustained very little injury. The cause of this, no doubt, — 
 is, that the activity of the fire, assisted by the vitriolic acid contained in the sulphie 
 acts with such rapidity, that it has not time to burn the shell. 
 
 * 
 
 A small ball of lead, closely wrapped up in a bit of paper,* may be fused in the } 
 
 same manner, by exposing it to the flame of a candle. The paper will not be hurt, 
 except in the bottom, where it will have a small hole through which the metal has 
 ruil. 
 
 4 
 
 ; 
 
 he 
 +a 
 
 * If the paper be not in perfect contact with the lead, the experiment will not succeed. — 
 
 2 
 
 rt 
 .. 
 
CHEMICAL EXPERIMENTS, 795 
 
 EXPERIMENT XX. 
 Action of quicksilver upon nitrous acid. 
 
 Put about half an ounce of quicksilver into an oil flask, and on it pour about an 
 ounce of diluted nitrous acid. The nitrous acid will be decomposed by the quick- 
 silver with great rapidity; the bulk of the acid will be changed into a beautiful 
 green, while its surface will appear of a dark crimson colour; and a vivid and pleasing 
 effervescence will go on while the acid acts on the quicksilver. When a part of the 
 metal is dissolved, the acid will by degrees become paler, till it is as pellucid as water. 
 
 EXPERIMENT XXI. 
 
 Throw a little pulverized muriate of lime into a spoonful of burning charcoal, and 
 a beautiful red flame will be produced, the colour of which may be heightened by 
 agitating the mixture during inflammation. 
 
 EXPERIMENT XXII. 
 To split a piece of money into two parts. 
 
 Fix three pins in the table, and lay the piece of money upon them ; then place a 
 heap of the flowers of sulphur below the piece of money, and another above it, and 
 set fire to them. When the flame is extinct, you will find, on the upper part of the 
 piece, a thin plate of metal, which has been detached from it. 
 
 It is to be observed that the value of about three-pence might be detached in this 
 manner from a piece of gold such asa guinea, by employing sulphur to the value 
 of fifteen or twenty pence; so that this experiment can never become dangerous to 
 the public. Besides, the piece of money loses, in a great measure, the boldness of its 
 impression: those therefore who might attempt to debase the current coin in this 
 manner, would become victims to their dishonesty. 
 
 EXPERIMENT XXIII. 
 Illustration of continued identity under apparent change of form. 
 
 Dissolve separately equal weights of sulphate of copper and crystal of carbonate 
 of soda, in sufficient quantities of boiling water ; pour them together, while hot, into 
 a flat pan, and when the water has evaporated a little, and the whole is suifered to 
 cool, the salts will shoot ; the sulphate of copper in blue, the soda in white crystals, 
 similar to what they were before they were dissolved. 
 
 EXPERIMENT XXIV. 
 Examples of different Fluids forming Solids, when mized. 
 
 Dissolve muriate of lime and carbonate of potash, separately, in water, so as to 
 form a saturated solution of each. On pouring the two transparent fluids together, 
 muriate of potash and carbonate of lime will be formed; and if the mixture be well 
 stirred, a solid mass will be produced. 
 
 Again: Take a saturated solution of sulphate of magnesia (Epsom salts), and pour 
 into it a like solution of caustic potash, or soda, and the mixture will immediately 
 become solid. 
 
 These instances of sudden conversion of two fluids into a solid, have been called 
 chemical miracles. 
 
 What we have said is sufficient to-inspire our readers with a desire to become 
 better acquainted with this useful science. We shall therefore point out to them a 
 few works which will assist them in prosecuting that design. Though chemistry has 
 _ experienced a complete revolution since. the discoveries of Lavoisier, Priestley, 
 Black, Higgins, Cavendish, and others, some useful information may still be ga- 
 
796 CHEMISTRY. — 
 
 thered from the works of the old chemists: we therefore hope we may be allowed 
 
 to introduce here “‘ Les Elemens de Chimie Theoretique et Practique’’ of Macquer, ; 
 
 3 vols. 12mo; the first of which contains the theoretical, and the other two the 
 practical part. To this work we may add the ‘Manuel de Chimie” of Baumé, 
 
 which contains a great many curious processes useful in the arts. Boerhaave’s Ele. 
 
 ments of Chemistry were formerly held in great estimation; but at present this work 
 is of less value: it may however serve as a good introduction to the modern che- 
 
 mistry. Wiegleb’s Chemistry, transl ate dby Hopson, though founded on the old — 
 
 theory, may also be perused with advantage, by those who are desirous of acquiring 
 a thorough knowledge of this science. Also the chemical works of Bishop Watson. 
 
 The best systems of chemistry, and elementary works according to the new theory, — 
 
 are those of Lavoisier, Fourcroy, Chaptal, Gren, Jacquin, Buillon-Lagrange, Brisson, 
 and Thomson. As books of reference we can recommend Henry’s Manual and 
 Parkinson’s Chemical Pocket Book. A great many curious and interesting papers 
 
 on chemistry in general, and its application to the arts, may be found also in the - 
 
 Philosophical Magazine. 
 
 DISSERTATION ON THE PHILOSOPHER’S STONE ; ON AURUM POTABILE ; 
 AND ON PALINGENESY. 
 
 We have here mentioned the two most celebrated chimeras of the human mind: for — 
 
 though the quadrature of the circle in geometry, and the perpetual motion in me- 
 
 chanics, have been much celebrated also on account of the fruitless attempts made to _ 
 
 solve these two problems, their celebrity is inferior to that of the first two above 
 
 mentioned, nearly in the same ratio as the importance of squaring the circle is — 
 
 inferior to that of acquiring immense riches, or of rendering ourselves almost immor- 
 
 tal. A great many persons therefore, seduced by these chimeras, have at all times — 
 
 bestowed incredible labour on researches respecting them. 
 Such is the character of the human mind, 
 
 —— Quid non mortalia pectora cogunt 
 Auri sacra fames, viteeque immensa cupido ! 
 
 We shall therefore here offer a few observations on these chemical problems, either ; 
 
 because they afford matter which comes within our province, or because what we 
 shall say may serve as a preventive of that illusion to which so many persons have 
 been dupes. 
 
 SECTION I. 
 Of the Philosopher’s Stone. 
 
 The philosopher’s stone (formerly called the work, by way of excellence, or chry- 
 sopea,* the transmutation of the base and imperfect metals into gold or silver) has 
 since time immemorial been an object to which the attention of multitudes of people, 
 either versed in chemistry or scarcely initiated in the science, has been directed. 
 The vulgar even think that it is the sole object of chemistry; and it must indeed be 
 allowed that it was in some measure the fault of those who first cultivated that noble 
 branch of philosophy: there were few of them who did not suffer themselves to be 
 blinded by the illusion of attempting to make gold. 
 
 earee Liab « 
 
 But at present fewer people are infatuated with the philosopher’s stone; at any rate 2 
 
 * Xpucomem, auri fabricatio, or gold-making. 
 
~* 
 
 PHILOSOPHER'S STONE, 197 
 
 few of the enlightened chemists employ themselves with the means of making gold ; 
 but there are still many other persons, who though they have scarcely an idea of the 
 simplest operations of chemistry, waste their time in vain attempts to regenerate that 
 precious metal. They are often seen proceeding at hazard, and still imagining them- 
 selves on the point of succeeding ; in the midst of poverty consoling themselves with 
 the agreeable idea, that this indigence will be succeeded by the possession of immense 
 treasures. They call themselves adepts, because they pretend to have reached to 
 the summit of philosophy, quasi summam sapientiam adepti; they speak enigmati- 
 cally and in an unintelligible manner, because mankind in general do not deserve to 
 possess such a secret. Filled with empty pride, they cast a sardonic smile of con- 
 tempt on the rational chemists, and on those who endeavour to deduce phenomena 
 from clear and established principles. 
 
 p We might say to the searchers after the philosopher’s stone, Before attempting to 
 make gold, first decompose and re-compose it; for if there be any method of ascer- 
 taining and demonstrating the constituent principles of any substance, it is that of 
 decomposition and re-composition. We might say also to those alchemists, Before you 
 make for us the precious metals, such as gold and silver, make for us only lead ;* for 
 before you proceed to the most difficult, method requires that you should execute the 
 easiest. But we are acquainted with no chemical operation which resolves either of 
 these problems. Gold, as stubborn in regard to decomposition as to composition, 
 always remains the same, in whatever manner it be treated: it is only more or less 
 attenuated, but is never in the state of calx. It has been kept for several years in 
 fusion, without losing the least part of its weight. 
 
 But let us hear the alchemists, and learn what are their pretensions in regard to the 
 
 formation of metals. 
 
 According to them, metals are all formed of an earth, which they call mercurial, 
 
 : but more or less mature, more or less mixed with heterogeneous matters; so that, to 
 convert the imperfect into perfect metals, nothing is necessary but to free them from 
 these heterogeneous matters, and to mature them. 
 
 All this is very fine: but who has proved the existence of this mercurial earth ? 
 who has proved that the difference among metals consists in this greater or less ma- 
 turity? by what meansisit to be produced? To these questions no solid answer can 
 be given. The partisans of this idea, seduced by words, have no just and precise 
 conception of what they say. 
 
 According to other alchemists, mercury contains in principle all the perfect metals ; 
 it has the splendour of them, and nearly the weight; it is even heavier than silver. 
 If it is fluid and exceedingly volatile, it is because it is alloyed with impurities which 
 degrade it. The question thenis to fix the mercury, by freeing it from these impuri- 
 ties. We should then have the mercury of the philosophers, which would require only 
 a certain degree of baking to be brought to a red heat, and the result would be gold; 
 
 - brought to a white heat, it would furnish silver; nay, this matter would have such 
 an activity on the impure parts of other metals, that by throwing a pinch of it into a 
 crucible filled with melted lead, it would transmute it into silver or gold, according as 
 it had been carried to a white or a red heat. But the great matter is, how to destroy 
 the impurities by which quicksilver is debased. Aristeus, a celebrated adept, teaches 
 us the process in the clearest manner, in his ‘‘ Code de Verité.” ‘‘ Take,” says he, 
 ‘‘king Gabertin, and the princess Beya his sister, a young lady, beautiful, fair, and 
 exceedingly delicate : marry them together, and Gabertin will die almost immediately. 
 Be not however alarmed; after eighty days, Gabertin will revive from his ashes, and 
 become more beautiful and more perfect than he was before his death ; will beget with 
 
 * Some have pretended to make iron; but it is now proved that, in the operation employed for 
 this purpose, the iron is only restored to its metallic form, 
 
798 CHEMISTRY. 
 
 Beya ared child, more beautiful and perfect than themselves.” After this, will any 
 one pretend to say that the alchemists explain themselves obscurely? What true 
 adept, for there are true and false, and every one thinks himself among the former, 
 will not evidently see in this allegory the whole process of the fixation of na 
 and of the powder of projection ? 
 
 This language, and this affectation of obscure allegories, are no doubt very propel 
 for making these pretended adepts be considered as finished and contemptible quacks, 
 or perhaps as people whose brains have been deranged by the heat of their furnaces, 
 But the partisans of their researches and follies allege pretended facts; and it is our 
 business to make them known. 
 
 It is related that Helvetius, a physician and celebrated professor in Holland, having 
 declaimed one day with great violence, in one of his lectures, against the vanity and 
 absurdity of pretending to make gold, was visited by an adept, who gave him a cer- 
 tain powder, a pinch of which, thrown into a crucible filled with melted lead, would 
 transform it into gold: that the learned Dutchman did so, and obtained from his 
 lead a considerable quantity of that metal. Helvetius then hastened to find the 
 adept; but the latter had given him a false address, and was not to be found; for the 
 chemists of this order never fail to disappear at the moment when they have givena 
 proof of their profound knowledge. 
 
 The same thing occurred, it is said, to the emperor Ferdinand. An adept came to 
 him, and offered to transform mercury into gold. Mercury was put into a crucible in 
 the presence of the prince, and the adept having performed certain operations, a 
 button of gold was found in the bottom of the vessel. But while those present were 
 employed in examining and assaying the gold, the adept disappeared, to the great 
 regret of the emperor, who already beheld in idea the immense treasures which he 
 hoped to obtain by the acquisition of this grand secret. 
 
 At the sale of the effects left by M. Geoffroy, in 1777, there were three nails, 
 which, as it was said, were a proof of the possibility of at least transmuting silver 
 into a common metal, such as iron. They were the work, as asserted, of a celebrated 
 adept, who wished to prove to Geoffroy the possibility of the transmutation of metals, 
 One of these nails was converted into silver, by being dipped in an appropriate 
 liquor; the head of the other only having been dipped, the remainder of it was iron; 
 and the point of the third having been dipped, that part was silver, and thes 
 head iron. 
 
 Notwithstanding these authorities we have no belief in the philosopher’s stone. It 
 is very probable, that in all these pretended transmutations, there was some decep- 
 tion, even if the above accounts were true. In short, we shall believe in the philo- 
 sopher’s stone when we have seen any adept perform before us the same operations $ 
 but he must permit us to furnish him with the crucibles, rods, and ingredients ; for’ 
 it is more than probable, that if gold has been made in this manner, it either 
 existed in the matters employed, or some of it was introduced into them by 
 sleight of hand. 
 
 However, the alchemists pretend that all the fables of auitiquite are nothing else 
 than the process of the grand work explained symbolically. The conquest of the 
 golden fleece, the Trojan war, the events which followed it, and the whole mytho- 
 logy, are only emblems of the chrysopea, prudently veiled by the ancient philosophers, 
 who did not wish that their secret, become common, should be employed to produce | 
 an immense increase of the precious metals, which must then have lost their value, 
 and have ceased to be the medium of commerce among mankind. The reader may 
 see, in a curious work by Dom Pernetty, entitled ‘‘ Les Fables Egyptiennes et Grec- 
 ques,” 3 vols. 8vo., including the ‘‘ Dictionnaire Mytho-hermetique,” how far human 
 sagacity may be ed) to find an explanation of such fables. But every thing” 
 may be explained in the same manner. We have heard of an adept, in the Faubourg 
 
POTABLE GOLD. 799 
 
 Saint Marceau, who, being persuaded that the whole Roman histury was a fiction, 
 intended to give a chemical explanation of it, which would serve as a supplement to 
 the Fables Egyptiennes et Grecques. We have even heard that the history of the 
 combat of the Horatii and the Curiatii was explained in it, with an appearance of 
 truth, capable of making us doubt whether that famous circumstance in the Roman 
 history ever really took place. 
 
 SECTION II. 
 Of Potable Gold. 
 
 If there be no reason to think that gold’ will ever be made, is it not possible to 
 employ this precious metal for prolonging life? Gold is a metal unalterable, and as 
 difficult to be destroyed as to be made; it is the soyereign of the metallic world, as 
 the sun, to which it is assimilated, is inthe system of the universe. Nature therefore 
 must have concealed in this valuable body the most useful remedies; but to make it 
 useful, in this respect, it is necessary that it should be introduced into the body ina 
 liquid form; it must in short be rendered potable: let us endeavour then to make 
 potable gold. A life indefinitely prolonged is certainly worth all the treasures in the 
 world. Such is in substance the reasoning of the alchemists, and therefore they have 
 subjected gold to a multitude of operations, by means of which they have pretended 
 to render it soluble, like a salt in water. The substance they produce has indeed the 
 appearance of it; but to speak the truth, it is only gold very much attenuated, and 
 by these means suspended in the liquid: in short, it is in no manner combined 
 with the fluid, and it even gradually deposits itself at the bottom in the metallic 
 form. 
 
 However, the following is a process for making a kind of potable gold. We 
 shall examine afterwards, supposing it to be a real solution of @old, whether it 
 would possess properties so marvellous, and so salutary to the human body, as is 
 pretended. 
 
 First dissolve gold in aqua regia: mix this solution with fifteeen or sixteen times 
 the quantity of any essential oil, such as that of rosemary, stirring it round, and 
 separate the aquaregia, which occupies the bottom, from tke essential oil. If you 
 then dissolve this essential oil in four or five times its weight of well rectified spirit 
 of wine, you will havea yellowish liquor, known under the name of the potable gold 
 of Mademoiselle Grimaldi. 
 
 Vitriolic ether, and ethereal liquids of different kinds, possess the same property 
 as essential oils; namely, that of setzing on the gold dissolved in the aquaregia, A 
 kind of potable gold therefore may be made with ether. This gold may then be 
 taken in drops on sugar, in the same manner as when ether is taken; for this liquor 
 is not miscible with water. 
 
 The celebrated drops of General Lamotte are not different from the potable gold of 
 Mademoiselle Gramaldi. It has been remarked that one gross of gold was diluted in 
 216 gross of spirituous liquor, and as the bottles must have weighed two gross, and 
 as General Lamotte sold his for 24 livres, it results that with one gross of gold he 
 made at least 108 bottles, from the sale of which he received at least 2592 livres. In 
 reality he made 136, which were worth to him 3264 livres. 
 
 It hence appears, that if General Lamotte’s drops were not useful to the health, 
 they were exceedingly useful to his purse. But what will not quackery effect among 
 mankind, when supported by ignorance and the love of life ? 
 
 But let us examine whether there be any foundation for the wonderful properties 
 ascribed to potable gold. A very little reasoning will shew that nothing can rest on 
 a slighter foundation. What proofs indeed can the alchemists produce, that potable 
 gold is salutary to the human body ? Because gold is the most fixed of all metals, 
 ‘because it has the beautiful colour of the sun’s rays, because it is represented in 
 
809 CHEMISTRY. 
 
 chemical characters by the characteristic sign of that luminary, are we thence to con- 
 clude that, when reduced to a liquid form, and conveyed into the blood, it regenerates 
 that fluid, renovates youth, and restores health? What person, accustomed to : 
 deduce just consequences from any principle, will ever form such a conclusion ? All” : 
 the virtues of potable gold are founded merely on analogies, invented without any ~ 
 physical foundation, by fervid imaginations, and by heads deranged by the heat of 
 theirfurnaces. This is the most favourable opinion that can be entertained; for it 
 is probable that such ideas are as much connected with imposture, as with credulity 
 and want of reasoning. 
 
 SECTION III. 
 Of Palingenesy. 
 
 Palingenesy is a chemical operation, by means of which, a plant, or an animal, as 
 some pretend, can be revived from its ashes. This, if true, would no doubt be one 
 of the noblest secrets of chemistry and philosophy. If some authors are to be cre-— 
 dited, several learned men of the 17th century were in possession of it ; but at present, 
 as this pretended secret, in consequence of the great progress made in chemistry, is 
 considered as a mere chimera, we shall here confine ourselves to examining the foun-_ 
 dation of those principles which have induced some respectable authors, such as the 
 Abbé Vallemont* and others, to believe in the possibility of this process. & 
 
 According to the honest Abbé, nothing is simpler and easier to be explained. We 
 are indeed told, says he, by Father Kircher, that the seminal virtue of each mixture is 
 contained in its salts, and these salts unalterable by their nature, when put in motion 
 by heat rise in the vessel through the liquor in which they are diffused. Being then ; 
 at liberty to arrange themselves at pleasure, they place themselves in that order in~ 
 which they woulfl be placed by the effect of vegetation, or the same as they occupied — 
 before the body, to which they belonged, had been decomposed by the fire: in 
 short, they form a plant, or the phantom of a plant, which has a perfect resem- 
 blance to the one destroyed. 
 
 This reasoning is worthy of an author who could believe that he who robs another 
 of his money can exhale corpuscles different from those exhaled by a man who carries — 
 his own, and thereby make the divining rod turn towards the places where he has : 
 passed, or remained for some time. Does it not shew great weakness to believe - 
 that the mere immorality of an action can produce physical effects ? It would indeed 
 be offering an insult to our readers, to attempt to shew the folly and absurdity of the i 
 above reasoning of the good Abbé, and of Father Kircher. Let us therefore only ex-_ 
 amine the facts which he relates. r 
 
 An English chemist, named Coxe, asserts that having extracted and dissolved the , 
 essential salts of fern, and then filtered the liquor, he observed, after leaving it at rest 
 for five or six weeks, a vegetation of small ferns adhering to the bottom of the vessel. — 
 The same chemist, having mixed northern potash with an equal quantity of sal am-— 
 moniac, saw some time after a small forest of pines, and other trees, with which he 
 was not acquainted, rising from the bottom of the vessel. . 
 
 The following fact is considered by the author as more conclusive. The celebrated _ 
 Boyle, though not very favourable to palingenesy, relates, that having dissolved ine 
 water some verdigris, which, as is well known, is produced by combining copper with ; 
 the acid of vinegar, and having caused this water to congeal by means of artificial 
 cold, he observed at the surface of the ice small figures, which had an exact resem=— 4 
 blance to vines. 
 
 Notwithstanding these facts, and others quoted by the Abbé from Daniel Majoul : 
 Hanneman, and various authors, if the partisans of palingenesy can produce none 
 
 + 
 
 q 
 ‘ 
 
 * See “* Les Curiosités de Ja Végétation,” &c. 
 
PALINGENESY. : S01 
 
 more conclusive, it must be confessed that they support their assertions by very weak 
 proofs. Every true chemist sees in these phenomena nothing but a simple ramified 
 crystallization, which may-be produced by different well known compositions: 
 the most beautiful of these crystallizations, called improperly vegetations, are pro- 
 duced by the combination of bodies from the animal kingdom. 
 
 The last experiment, related by Boyle, might occasion more embarrassment ; but 
 as it is the only one, of a great many, made with the essential salts of a variety of 
 plants, that succeeded, there can be no doubt that the figures he saw were the mere 
 effect of chance ; for how many other philosophers, who made the same attempt, 
 saw nothing but what is exhibited by the surface of frozen water, which sometimes 
 forms ramifications exceedingly complex ? 
 
 The partisans of palingenesy however quote other authorities, to which they attach 
 great importance. We are told by Sir Kenelm Digby, on the authority of Quercetan, 
 
 _ physician to Henry 1V. of France, that a Pole shewed twelve glass vessels herme- 
 
 tically sealed, each containing the salts of different plants: that at first thase salts 
 had the appearance of ashes, but that when exposed to a gentle and moderate heat, 
 the figure of the plant, as a rose for example, if the vessel contained the ashes of a rose, 
 
 was observed gradually to rise up, and that as the vessel cooled the ‘whole disappear- 
 ed. Sir Kenelm adds, that Father Kircher had assured him, that he performed the 
 
 ‘same experiment, and that he communicated to him the secret, but it never had 
 
 succeeded. The story of this Pole is related by various other authors, such as Bary, 
 
 “in his Physique, and Guy de la Brosse, in his book on the Nature of Plants. 
 
 Lastly, we are told by Kircher himself, in his Ars Magnetica, that he had a long- 
 necked phial, hermetically sealed, containing the ashes of a plant which he could 
 
 .Tevive at pleasure, by means of heat; and that he shewed this wonderful phenome- 
 
 non to Christina, queen of Sweden, who was highly delighted with it; but that hav- 
 ing left this valuable curiosity one cold day in his window, it was entirely destroyed 
 by the frost. Father Schott also asserts, that he saw this chemical wonder, which, 
 according to his account, was a rose revived from its ashes; and he adds, that a cer- 
 tain prince having requested Kircher to make him one of the same kind, he chose 
 rather to give up his own than to repeat the operation. 
 
 The process, indeed, as taught by Kircher, is so complex and tedious, that it would 
 require no small patience to follow it. Father Schott relates it at full length, in his 
 work entitled “‘ Jocoseria Nature et Artis,” and he calls it the Imperial secret, 
 because the emperor Ferdinand purchased it from a chemist, who gave it to Kircher. 
 
 This emperor was exceedingly fortunate ; for it was to him that the philosopher, 
 who had the secret of the philosopher’s stone, addressed himself, and gaye a proof of 
 his art by transmuting, as is said, in his presence, three pounds of mercury into two 
 pounds and a half of goid. 
 
 We must however content ourselves with pointing out the places where the curious 
 may find this singular process; for besides the length of the description, nothing 
 seems less calculated to succeed. Digby, therefore, and many others who followed 
 
 _ this method, did not obtain a favourable result; and there is reason to believe that 
 _ their zeal for palingenesy would induce them to omit nothing that was likely to insure 
 _ them success, 
 
 Dobrezensky, of Negropont, has also given a process for the resurrection of 
 plants, which seems to have been attended with no better success. We are at least 
 told by Father Schott, that the attempts of Father Conrad proved ineffectual, and 
 he therefore supposes that Dobrezensky did not reveal all the circumstances of the 
 process, but kept the most important to himself. 
 
 What then can be said in opposition to all these authorities? In our opinion, the 
 
 “the Polish physician was a quack, and we shall describe hereafter a method of pro- 
 
 _ ducing a false palingenesy, which, if performed with art and in a proper place, may 
 
802 CHEMISTRY. 
 
 impose on credulous persons. . To be convinced that Dobrezensky of Negropont was 
 a mere impostor, we need only read the ‘‘ Technica Curiosa,”’ or the ‘ Jocoseria” 
 Nature et Artis,” of Father Schott; for he had the impudence to pretend that he 
 could pull out the eye of an animal, and in the course of a few hours restore it, by ~ 
 means of a liquor, which he no doubt sold as a remedy for sore eyes. He even tried © 
 it on acock. A person who could assert such an impudent falsehood, in regard to ~ 
 one fact, would do the same in regard to another. . 
 
 The authority of Father Schott will certainly be of little weight with those who | 
 have read his works. ; 
 
 In regard to the testimony of Kircher, we confess that we find some embarrass-— 
 ment: a Jesuit certainly would not wilfully have told a falsehood. But Kircher ~ 
 was a man of a warm imagination, passionately fond of every thing singular and ~ 
 extraordinary, and who had a strong propensity to believe in the marvellous. What 
 can be expected from a man of that character? He often thinks he sees what — 
 he does not see, and if he deceives others he is first deceived himself. 
 
 Some persons go still farther, and assert that an animal may be revived from its — 
 ashes. Father Schott, in his ‘‘ Physica Curiosa,” even gives the figure of a sparrow 
 thus revived in a bottle. Gaffarel, in his ‘‘ Unheard of Curiosities,’ believes in this” 
 fact, and considers it as a proof of the possibility of the general resurrection of bo-— 
 dies. This pretended revival, however, is a chimera, still more absurd than the 
 former; and which, at present, it would be ridiculous to attempt seriously to~ 
 refute. 
 
 In short, what reasonable man can with Kircher believe, that if the ashes of a~ 
 plant be scattered on the ground, plants of the like kind will spring up from them, 
 as he says he frequently experienced? Who can admit as a truth, that if crabs be 
 burnt, and then distilled, according to a process given by Digby, there will be 
 produced in the liquor small crabs of the size of a grain of millet, which must be 
 nourished with ox’s blood, and then left to themselves in some stream? yet we are 
 told by Sir Kenelm that this he himself experienced. It is therefore impossible to— 
 clear him from the charge of imposture, unless we suppose that by some means or 
 other he was led into error. However, it is certain that Digby, with great zeal and 
 a considerable share of knowledge, had a strong propensity to all the visions of the - 
 occult and cabalistic sciences. In our opinion, he was one of those visionaries - 
 known under the name of Rosterusians. . 
 
 An illusory hind of Palingenesy. 
 
 We have already mentioned a kind of sleight of hand, by means of which, cree 
 dulous people might easily be imposed on, and induced to believe in the reality 
 of palingenesy. -We shall now discharge our promise by describing it. 
 
 Provide a double glass jar of a moderate size, that is, a vessel formed of two jars. 
 placed one within the other, in such a manner, that an interval of only a line in- 
 diameter may be left between them. The vessel may be covered by an opake top or 
 lid, so disposed, that by turning it in different directions, the inner jar may be raised — 
 from or brought nearer to the bottom of the exterior one. In the interior jar, on a 
 base representing a heap of ashes, place the stem of an artificial rose. Into the 
 Jower part of the interval between the two jars introduce a certain quantity of ashes, 
 or some solid substance of a similar appearance ; and let the remainder be filled with 
 a composition made of one part of white wax, twelve parts of hog’s lard, and one or 
 two of clarified linseed oil. This oily compound, when cold, will entirely conceal 
 the inside of the jar; but when brought near the fire, if done with dexterity, it will” 
 dissolve, and by shaking the lid, under a pretence of hastening the operation, the 
 ‘compound may be made to fall down into the bottom of the exterior jar. The rose” 
 in | the interior one will then be seen; and the credulous spectators, who must not be. 
 
 QE 
 
PHOSPHORUS. 803 
 
 suffered to approach too near, will be surprised and astonished. When you wish to 
 make the rose disappear, remove the jar from the fire, and by a new sleight of hand 
 make the dissolved semi-transparent wax flow back into the interval between the 
 jars. By accompanying this maneeuvre with propey words, the gaping spectators will 
 be more easily deceived, and will retire firmly persuaded that they have seen one 
 of the most curious phenomena that can be exhibited by the united efforts of che- 
 mistry and philosophy. 
 
 SUPPLEMENT I. 
 
 OF THE DIFFERENT KINDS OF PHOSPHORUS, BOTH NATURAL AND 
 ARTIFICIAL. 
 
 One of the mostinteresting objects in chemistry is Phosphorus; for it is a very 
 singular and curious spectacle, to see a body, absolutely cold, emit a light of greater 
 or less vivacity, and others kindle of themselves, without the application of fire. 
 What person, who has any taste for the study of nature, will not be struck with 
 astonishment, on viewing such phenomena ? 
 
 These phenomena are the more remarkable, as most philosophers have hitherto 
 failed in their attempts to explain them. We, however, except the artificial kinds 
 of phosphorus, respecting which they have advanced things consistent with proba- 
 bility, and founded on chemical causes fully established. But in regard to the 
 natural kinds nothing satisfactory has yet been offered. The explanation of them 
 depends, no doubt, on a more profound knowledge of the nature of fire and light. 
 
 Some kinds of phosphorus are natural ; others are the production of art, and par- 
 ticularly of chemistry. Hence we are furnished witha natural division of this Sup- 
 plement. We shall begin with the Natural. 
 
 ARTICUE I. 
 Of the Natural Kinds of Phosphorus. 
 
 SECTION. I. 
 Of the Luminous appearance of the Sea. 
 
 Though navigators must have observed this phenomenon for many centuries past, 
 as it is common to every sea, and as there is scarcely any climate where it does 
 not, under certain circumstances, present itself, it appears that very little attention 
 has been paid to it till within a late period. Most sea-faring people believed that 
 this light was merely. the reflection of that of the stars, or of that of the vessel it- 
 self; others, considering it as a real light, imputed it to the collision of sulphur 
 and salts; and, satisfied with this vague explanation, they scarcely condescended to 
 
 _ pay attention to the phenomenon. 
 
 As it is highly worthy of profound research, and is attended with very remarkable 
 circumstances, we shall here give a description of it, as it appeared to us on cur 
 passage from Europe to Guyana, in the year 1764. 
 
 Ido not recollect that we beheld the sea luminous till our arrival between the tro- 
 pics ; but at that period, and some weeks before we reached land, I almost constantly 
 observed that the ship’s wake was interspersed with a multitude of luminous sparks, 
 
 - and so much the brighter as the darkness was more perfect. The water around the rud« 
 
 | 
 
 _ der was, at length, entirely brilliant ; and this light extended, gradually diminishing, 
 3 ¥F2 
 
804 CHEMISTRY. : 
 
 along the whole wake. I remarked also, that if any of the ropes were immersed 
 in the water they produced the same effect. 
 
 But it was near land that this spectacle appeared in allits beauty. It blew a 
 fresh gale, and the whole sea was covered with small waves, which broke after 
 having rolled for some time. When a wave broke, a flash of light was produced; 
 so that the whole sea, as far as the eye could reach, seemed to be covered 
 with fire, alternately kindled and extinguished. This fire, in the open sea, that is — 
 at the distance of 50 or 60 degrees from the coasts of America, had a reddish cast. — 
 I have made this remark, because I do not know that any person ever examined 
 the phenomenon which I am about to describe. ; 
 
 When we were in green water,* the spectacle changed, The same fresh gale 
 continued ; but in the night time, when steering an easy course, between the 3d 
 and 4th degree of latitude, the fire above described assumed a tone entirely white, 
 and similar to the light of the moon, which at that time was not above the horizon. 
 The upper part of the small waves, with which the whole surface of the sea was 
 curled, seemed like a sheet of silver; while on the preceding evening it had resem- 
 bled a sheet of reddish gold. I cannot express how much I was amused and inte- 
 rested by this spectacle. | 
 
 The following night it was still more beautiful; but at the same time more alarm- — 
 ing, in consequence of the circumstances under which I then found myself. The 
 ship had cast anchor at a considerable distance from the land, waiting for the new 
 moon, in order to enter the harbour of Cayenne. Being anxious to get on shore, I~ 
 stepped into the boat with several other passengers; but scarcely had we got a © 
 league from the ship, wheu we entered a part of the sea where,there was a prodi- 
 gious swell, as a pretty smart gale then prevailed at south-east. We soon beheld ~ 
 tremendous waves rolling in our wake, and breaking over us. But what a noble — 
 spectacle, had we not been exposed to danger! Let the reader imagine to himself a — 
 sheet of silver, a quarter of a league in breadth, expanded in an instant, and shin- © 
 ‘ing with a vivid light. Such was the effect of these billows, two or three of which 
 only reached us, before they broke. This was a fortunate circumstance, for they 
 left the boat half filled with water, and one more, by rendering me a prey to the 
 sharks, would certainly have saved me from the trouble of new modelling the work 
 of the good M. Ozanam. 
 
 There is scarcely.a sea in which the phenomenon of this light is not sometimes 
 observed; but there are certain parts where it is much more luminous than in others. | 
 In general, it is more so in warm countries, and between the tropics, than any 
 where else ; it is remarkably luminous on the coasts of Guyana, in the environs of ~ 
 the Cape Verd islands, near the Maldives and the coast of Malabar, where, accord- 
 ing to M. Godeheu de Riville, it exhibits a spectacle very much like that above 
 described 4 
 
 A phenomenon so surprising could not fail to excite the attention of philosophers ;— ‘ 
 but till lately they confined themselves to vague explanations ; ; they ascribed it to ™ 
 sulphur, to nitre, and other things, of which there is not a single atom in the sea, : 
 and they then imagined that they had reasoned well. : 
 
 M. Vianelli, an Italian philosopher, is the first persen, it seems, who endeavoureds © 
 by the help of observation, to explain the cause of this light; and he was thence ~ 
 conducted to a very singular discovery. Observing that the sea water, near Chi-— 
 
 * The water of the sea, at least of the Atlantic Ocean, at a distance from the coasts, is of a dark ~ 
 blue colour; but near land, that is to say, 20 or 25 leagues from the coast of Guyana, the water Z 
 suddenly changes its colour, and becomes a beautiful green. This is a sign of being near land. This — 
 change, in all probability, is produced by the muddy, yellowish water of the river of the Amazons; — 
 for it is well known that blue and yellow form green. But a remarkable circumstance is, that this — 
 change is absolutely abrupt; it does not take place by degrees, but suddenly and in an ine ‘ 
 which appeared to me, from the deck of the vessel, to be scarcely a fuot in extent. 
 
NATURAL KINDS OF PHOSPHORUS. 805. 
 
 -oggia, had a very luminous appearance, and that the light was concentrated into 
 small brilliant points, he conceived the idea of examining it with a microscope. _ By 
 these means he found that those luminous points were small insects, resembling 
 worms, or rather caterpillars, composed of twelve articulations ; that they differed from 
 our luminous worms in this respect, that the light proceeded from every part of their 
 bodies, and that when at perfect rest the light ceased, but that it re-appeared when 
 they were put into a state of agitation. This explains why certain parts are made 
 to sparkle by the strokes of an oar, the dashing of water against therudder, and 
 the breaking of the waves, while the rest of the water remains dark. These ob- 
 servations were confirmed by the Abbé Nollet, who soon after undertook a tour 
 to Italy. 
 
 It appears however that the insect which gives a luminous appearance to the water 
 of the sea, is not every where the same. M. Godeheu de Riville, having observed 
 - some of these luminous points in the Indian seas, between the Maldives and the coast 
 
 of Malabar, saw an insect quite different from Vianelli’s worm with twelve rings. This 
 insect has a near resemblance to that called the water flea, and is enclosed between 
 two transparent shells, somewhat like a kidney half open. The luminous matter 
 seems to be contained in a vessel, which may be compared to a bunch of grapes; it 
 consists of small round grains, and when the insect is pressed it emits a luminous 
 liquor. It then mixes with the water, and, as it is of an oily nature, it collects itself 
 on its surface in the form of small round drops. According to every appearance the 
 insect suffers this phosphoric liquor to escape only in consequence of some shock or 
 agitation, or of other circumstances; and hence the reason why the seais not lumin- 
 ous except when agitated, and at certain times more than others.* 
 
 M. Rigault observed, in the seas between Europe and America, another insect, 
 different from the worm of Vianelli, or the water flea of M. Godeheu, being rather a 
 kind of polype, almost spherical, and with only one arm, 
 
 In the last place, M. Leroy, a physician of Montpellier, observed, in sea water, 
 globules of a phosphoric matter, on which he made different experiments to ascertain 
 what circumstance rendered them luminous, and by what means they were deprived 
 of that quality. From these experiments he was induced to conclude, that though 
 Vianelli and others have, on good grounds, ascribed the luminous appearance of the 
 sea to insects, or to a liquor which they contain, and which they emit on certain 
 occasions, this is not the only cause; but that it may arise also from a phosphoric 
 matter in the water of the sea, and which is produced there by a peculiar combination 
 of the principles dispersed throughout it; that this matter is not always luminous, 
 
 ‘but becomes so from different causes ; such as the shock of the particles of the water 
 against each other, the contact of the air, and its mixture with certain liquors. 
 
 SECTION II, 
 Of some Luminous Insects. 
 
 If those beings which we tread under foot hold a very low, and, we might even 
 say, contemptible rank in the animal kingdom, Nature, which seems to observe a 
 general system of compensation, has given to several of them properties very extra- 
 ordinary, and for which the largest animals might envy them: such is that of emitting 
 light, with which many of them are endowed. We are acquainted indeed with no 
 large animal which enjoys it while living; but there are several insects which are 
 luminous, and it appears that they can become soat pleasure. Of what utility is this 
 light to them, and how is it produced? These are problems which we shall not 
 attempt tosolve; we shall only confine ourselves to facts. 
 
 © See “ Mémoires des Sgavaus étrangers,’* vol. iii. 4Idem. “a8 
 
806 ; CHEMISTRY. 
 
 1. Of the Glow-worm. 
 
 Every person is acquainted with this small insect, the light of which is often 
 
 observed under the hedges in the fine evenings of summer. 
 
 This insect, called by the Greeks lampyris, and by the Latins cicindela, exhibits 
 nothing remarkable in its external appearance. It has a pretty near resemblance to 
 the sea louse, only that it is much smaller, and proportionally thinner. It is its last 
 ring, where the anus is situated, that emits the light, by which it is distinguished 
 from other animals of the same class. Thislight is of a.pale greenish colour, and the 
 animal can shew or conceal it at pleasure. It is supposed that it is by this light, 
 which is peculiar to the female, that it attracts the male: the latter has wings, and 
 is destitute of this luminous quality. This however is rather a matter of conjecture, 
 and is contested by Baron de Geer, a celebrated Swedish naturalist, in consequence 
 of some observations made by him. 
 
 An insect so singular was, no doubt, worthy of being celebrated by the poets; 
 and indeed M. Huet, bishop of Avranches, has made it the subject of a small poem, 
 entitled Lampyris, which is much esteemed by those who are fond of Latin poetry. 
 It begins in the following manner. 
 
 Que nova per cecas splendescit stellula noctes 
 Sepibus in nostris? An ab ethere lapsa sereno 
 Astra cadunt, tacitis an captant frigora silvis, 
 Si quando ardentis coeperunt teedia coeli? 
 
 Non ita, sed duris frustra exercita matris 
 
 Imperiis, sentes lustrat Lampyris opacos, 
 Siforté amissum possit reperire monile. 
 
 The poet then feigns that the nymph Lamppyris, having lost her necklace, is ex- 
 * pelled by her mother, and that she wanders about through the woods, searching for it 
 by the help of a lantern. ‘These ideas were much applauded a century ago; but we 
 do not know whether the case be the same in the present one. 
 
 2. Of the Fire Fly of warm climates. 
 
 Such is the luminous insect of our climates; but the warmer countries have been 
 more favoured by nature. Their luminous insects have wings. They are found in 
 Italy after the Alps are crossed ; and they become more frequent, according as the 
 traveller approaches the southern parts of Italy. They exhibit a very curious spec- 
 tacle, during the fine nights of summer, when they are seen flying about in every 
 direction, and one cannot move a step in a meadow without observing some of these 
 small animals, whose route is marked by a train of light. I never enjoyed this 
 spectacle in Italy ; but I have seen it in South America. 
 
 It appears, however, that the fire flies of Italy, and those of America, are entirely 
 different from the luminous insect of our countries. I confess that during my resi- 
 dence in America I did not pay much attention to them. I was employed with 
 occupations of greater importance ; but I know with certainty that this insect emits 
 light as it flies about. The part of its body which is luminous seems to be concealed 
 by its wings, or by the covering of them when closely applied to the body. I have 
 
 . ‘ : 
 er cee ee ee ee A, 
 
 SS ee a ee ee 
 
 never seen a good description of this insect, which has a great resemblance 10 a com- : 
 
 mon fly. 
 It may be readily conceived, that these luminous insects must have inspired some 
 
 persons with the hope of obtaining from them a perpetual phosphorus. Many at- 
 
 tempts have been made for this purpose; but though the posterior part of the animal, 
 when it is cut in two, retains its light for some time, it gradually becomes extinct : 
 and every effort hitherto made to retain it has proved fruitless. Some authors 
 
NATURAL KINDS OF PHOSPHORUS. 807 
 
 indeed have proposed means for accomplishing this object; but these people were 
 
 either impostors, or labouring under a deception: it is certain that their pretended 
 means will not succeed. 
 
 3. Of the Cucuyo of America. 
 
 A valuable acquisition of this kind, possessed by America, is the Cucuyo. The 
 Caribs have given this name toa large beetle, found in the islands of the Gulf of 
 Mexico, and even in Mexico itself. Its luminous quality is seated in the eyes, and in 
 two parts of its body covered by the sheaths of its wings. It is asserted that five 
 or six of these beetles will afford a sufficient light to enable a person to walk in the 
 darkest night; that the natives of the country tie them together alive, and by these 
 means form them into a sort of necklaces, to guide them through the woods, and 
 
 that they employ them in their huts to give them light to perform their nocturnal 
 labours, But this we can hardly believe. 
 
 4, Of the Beetle of Guyana. 
 
 A luminous insect, which had a great resemblance to the Cucuyo, and which perhaps 
 
 was the same, was some years ago brought to France by a very singular accident. 
 A great deal of wood for cabinet-makers having been imported from Cayenne, in 
 1764, and the following years, a cabinet maker purchased apiece of it, and kept it 
 by him till he should find use for it. His wife hearing some noise one night, like the 
 buzzing of an insect when flying, observed soon after a strong light adhering to the 
 window. Recovering from the terror which this spectacle at first inspired, she ran 
 up to it, and found an insect of the coleoptera kind, (that is, insects whose wings are 
 covered by a sheath,) which emitted from the posterior parts of its body a bright 
 light that illuminated-the whole of the apartment. The insect was put into the _ 
 hands of M. Fougeroux, who wrote a description of it, which was inserted in the 
 Memoirs of the Academy of Sciences for 1766. 
 _ There is great reason to think, or rather it is certain, that the animal had been 
 brought over in the piece of wood in the state of nymph, concealed in some hole ; 
 the time of its development being arrived, it issued from its retreat, and appeared 
 under the form of a beetle. 
 
 If this insect was not the Cucuyo of the American isles, or of New Spain, it must 
 be considered as a fourth kind endowed with the property of emitting light. 
 
 SECTION III. 
 Of some other Phosphoric Bodies. 
 We hall here take a cursory view of a great number of other phosphoric bodies. 
 
 1. The Eyes of different animals. 
 
 As several animals, such as the tiger, and the cat, which is only a tiger in minia- 
 ture, the wolf, the fox, &c., among quadrupeds; and the owl, and others, among 
 birds, have been destined by nature to search for their food in the night time, it was 
 necessary they should have a lamp to guide them. This lamp is contained in their 
 own eyes; for they are luminous, and it is no doubt by this light that they are guided 
 in the dark. As their retina is exceedingly sensible, the light of their eyes renders 
 objects to them sufficiently luminous. Besides, nature has favoured them with a 
 very large aperture in the pupil, so that the quantity of light which reaches the 
 retinais increased. Such, inall probability, is the mechanism by which these animals 
 see in the night time: the extreme sensibility of their retina renders the light of the 
 day incommodious to them, and even blinds some of them. ~ 
 
 It appears that these animals have it in their power to render their eyes luminous 
 
808 CHEMISTRY. 
 
 at pleasure. I have often seen those of a cat, which I kept, entirely destitute 
 of light, while at other times they were like a burning coal. 
 
 The dog also is endowed in some measure with the same property. I have several 
 times seen the eyes of that animal sparkle. 
 
 In short, it is asserted that some men also are endowed with this property. Ti- 
 berius, it is said, could see in the night time; and the same thing is related of many 
 others. The most singular instance of this faculty is that of a hermit, who, ac- 
 cording to Moschus, in his ‘‘ Pré Spirituel,’’ had never occasion for a lamp while 
 reading at night, or employed with any other occupation. ‘Those who could believe 
 such ridiculous tales, would almost deserve to be sent to feed cum asinis et 
 jumentis. , 
 
 2. Clayton’s Diamond. 
 
 This diamond was much celebrated; and if it was not one of the finest of its 
 kind, this defect was more than compensated by the singular property it possessed. 
 When rubbed in the dark againt any dry stuff, or against the fingers, it shone with a 
 faint whitish light. The celebrated Boyle made a great many observations on this 
 diamond, an account of which he communicated to the Royal Society in 1668; and 
 he does not hesitate to call it a precious stone, unique of its kind,—gemma sui generis 
 unica; for at that time no other was known which possessed the same property. 
 We have however heard that, since that time, other diamonds have been found, which 
 could be rendered brilliant in the dark by friction. This singular diamond was pur- 
 chased by Charles II. 
 
 We shall here say a few words respecting the carbuncle, which, as some pretend, 
 shines also in the dark; but we must observe, that this property ascribed to it is 
 absolutely fabulous. The carbuncle is a ruby; but no ruby, nor any other kind 
 of precious stone, shines in the dark; and this supposed phenomenon is merely a 
 popular tale. 
 
 We may make remark also that this light is not properly phosphoric, but is of the 
 
 same nature as electric light. The diamond indeed is susceptible of becoming elec- 
 tric by friction ; its light is of the same nature as that emitted by sugar when grated, 
 and by various other bodies when rubbed. 
 
 3. Rotten Wood. 
 
 It is not uncommon to find, in the forests, pieces of rotten wood, which emit a 
 very vivid light of a white colour, inclining to blue; it has even sometimes happened 
 that this light has been the occasion of great terror. 
 
 Unfortunately, every kind of rotten wood is not phosphoric; and the cause which 
 renders it so is not known. | 
 
 We must class also, among the number of puerile tales, what is related by Josephus 
 of a plant called Baaras, said to be luminous in the dark. This plant, it is said, ean- 
 not be plucked up without the most ‘imminent danger; but when the root of 
 the plant has been loosened, a dog is tied to it, and the animal, by making efforts 
 to join its master, at length tears it up. Is it possible that authors can thus sport 
 with the credulity of mankind ! 
 
 We must place in the same rank what is related by Pliny of another plant, called 
 WNyctegretum, which grows, it is said, in Gedrosia, and which, when torn up by the 
 root and dried in the sun’s rays for a month, becomes luminous in the night time. 
 This it not absolutely impossible ; but if so, the plant would be known to our natu- 
 ralists, as well as the Aglao-phytis, and the Lunaris, to which the same property was 
 
 ascribed, according to the testimony of Aclian. When a circumstance is related by 
 
 lian, one may bet a hundred to one that it is a fable. 
 
NATURAL KINDS OF PHOSPHORUS. 809 
 
 4. The Worms in Oysters. 
 
 This natural phosphorus was first remarked by M. dela Voye, who communicated 
 his discovery to M. Auzout, in 1666. 
 
 _ $mall oblong worms, which shine in the dark, are often engendered in oysters. 
 According to the description given of them, some are as large as a small hair pin, and 
 about five or six lines in length: others are much smaller. He found also three 
 kinds; the first with legs, to the number of about twenty-five on each side; the 
 second kind were red, and similar, except in size, to our common glow worms; the 
 third were of a singular form, having a head like that of the sole. They readily re- 
 solve, on the least touch, intoa viscid matter, which retains its luminous property for 
 about twenty seconds. 
 
 Such are the observations of M. de la Voye, which do not entirely agree with those 
 _ of M. Auzout, who observed only a viscid matter, extended in length. But it is to 
 _be remarked, that the latter made his experiments only on old oysters ; whereas the 
 former made his on oysters quite fresh. 
 
 5. Putrid Flesh. 
 
 Putrid flesh is also susceptible of becoming sometimes luminous in the dark. 
 Lemery says that a great quantity of such luminous flesh was seen at Orleans, in 
 1696; some of it was entirely so; other pieces were luminous only in some points, 
 which had the appearance of small stars. People were at first afraid to eat of it; but 
 they soon learnt by experience that there was no danger, and that it was as good as 
 - any other. It was remarked that in some butchers’ shops the meat was almost all 
 luminous; in others only part of it was so. 
 
 Fabricius ab Aquapendente relates the same thing of a lamb, purchased by some 
 young men at Rome. One half of it, which was left, being put by, they observed in 
 the evening that several parts of it were luminous. They immediately sent for the 
 above physician, who having examined the phenomenon with attention, observed that 
 the flesh and the fat shone witha silver coloured light; and that a piece of goat's 
 flesh, which had touched the lamb, shone in the same manner. The fingers of those 
 who touched it became luminous also. He observed likewise that the luminous 
 places were softer. This phenomenon would, no doubt, be observed more frequently 
 if the butchers’ shops, and places where meat are kept, were oftener visited 
 in the dark. 
 
 6. Different kinds of Fish, or the Parts of Fish. 
 
 But this phenomenon is most frequently exhibited by fish, and their different 
 parts. 
 
 It is generally when fish, or any of their parts, approach the state of putrefaction, 
 that they acquire this phosphoric property. Leo Allatius, in a letter to Fortunio 
 Liceti, says that he was once much frightened by fresh water crabs thrown into a 
 corner by a careless servant. He describes the whole of this adventure at great length ; 
 but want of room will not permit us to enter into any farther details respecting it. 
 
 According to Pliny, and other authors,the sea-worm is susceptible of shining in this 
 manner. Those who reside near the sea coast may have an opportunity of ascertaining 
 the truth of this fact. 
 
 The celebrated Thomas Bartholin observed the same thing, in regard to some 
 polypes, which he was dissecting: he gives this name to the fish called at present 
 the cuttle fish, since he says that it contains a black liquor, which may be employed 
 ~asink. This light, adds he, flowed from beneath the skin, and was the more abun- 
 dant the nearer the animal approached to a state of putrefaction. 
 
 We shall conclude this subject by mentioning some experiments of Dr. Beale, in- 
 
810 CHEMISTRY. 
 
 serted in the Philosophical Transactions, for the year 1666, Fresh mackarels having 
 been boiled in water with salt and herbs, the cook stirring the water a few days after, 
 to take out some of the fish, observed that on the first movement of the water it 
 became luminous, as well as the fish, which emitted.a strong light through it: the 
 water also appeared to be transparent: whereas in the day time it was opake. 
 
 Drops of this water were exceedingly luminous ; and wherever they fell, they 
 left a luminous spot, as large as a six-pence. On rubbing the hands with it, they 
 became entirely luminous. 
 
 We have here confined ourselves to facts; for nothing farther can be said on the 
 subject, as it is difficult to assign any probable or well-founded cause for this light. 
 Theglobulous matter of Descartes was exceedingly convenient for explaining all 
 these phenomena ; for it was sufficient to say, that putrid fermentation, being a kind 
 of intestine motion, this motion, according to every appearance, put in action the 
 globulous matter in which light consists. But unfortunately this matter, at present, 
 is considered as a chimera. ; 
 
 ADDITION B¥Y THE FRENCH CENSOR. 
 
 There are some inaccuracies in what has been before said in regard to the luminous 
 insects, in the Ist, 2d, 3d, and 4th paragraphs of the second section. The author 
 seems to have trusted too much to his memory, which has led him into error, and not 
 to have been acquainted with every thing written on the subject. We shall there- 
 fore supply this deficiency. 
 
 Ast. The male of the glow worm is a winged insect of the class of the coleoptera, 
 or insects which have sheaths to their wings. It is not entirely destitute of the pro- 
 perty of being luminous. M. Fougeroux says, that he often caught in the dark some 
 of these males, which were attracted by the light of the female, and he observed that 
 they emitted light themselves after copulation. 
 
 2d. The fire-fly of Italy, commonly known under the name of the Jucciola, is nota 
 fly; it is also an insect with sheaths to its wings, and in form approaches near to the 
 male of the glow worm. At first one might be induced to believe that it is the same 
 insect, to which the climate has given the property of shining in the dark, as is the 
 case with that of our country under certain circumstances. But there are some 
 differences, which wil] not allow of their being confounded; and what seems abso- 
 
 lutely to exclude this identity is, that in places were the lucctolais found, the common _ 
 
 glow worm is never seen, though it exists also in Italy. 
 
 In regard to the luminous insect of the warm countries of America, I must remark, 
 with the author, that I am unacquainted with any correct description of it. 
 
 3d. What has been said, in regard to the luminous insect of Cayenne, is not entirely 
 correct ; and the account given of its being discovered requires to be rectified. In 
 
 the month of September, 1766, two women, in the Faubourg Saint-Antoine, saw this ~ 
 
 insect in the evening flying through the air, like a stream of light, and at length settle 
 ona window. They at first thought it was one of those falling stars so common 
 during the summer nights ; but as the light continued, they went to inform the occu- 
 piers of the house against which the animal rested. It was caught, and given to M. 
 Fougeroux, in order that he might examine it. That it came from Cayenne was 
 ouly conjecture ; but by comparing it with the insects of that country, it appeared to 
 be an inhabitant of the same ora neighbouring climate. It was acoleoptera, known 
 under the name of mareschal, and of the class of those which, when placed on their 
 back, dart into the air like a spring which unbends itself: on this account it has been 
 distinguished by the name of edater. This insect is an inch and a half in length; its 
 light is contained in two elongated protuberances, placed on the posterior and lateral 
 part of its corselet. It emits light also in certain positions, by the separation of its 
 body from the corselet; and in all probability by that of the rings of its body from 
 
ARTIFICIAL KINDS OF PHOSPHORUS. 811 
 
 each other. This light is of a beautiful green colour, and of such strength, that if the 
 insect be put into a paper cornet, one can see to read the smallest characters by it, at the 
 distance of some inches. The insect is found also in Jamaica, and has been described 
 by Brown, under the denomination of elater major fuscus phosphoricus. Another 
 smaller kind of phosphoric fly is found in Jamaica, and also in Saint Domingo. 
 
 4th. What the author says of the cucuyo of America, that it emits light from its 
 eyes, and two parts above the sheaths of its wings, is not correct. It is possible that 
 travellers, unacquainted with natural history, who have spoken of it, may not have 
 examined it with attention. 
 
 5th. There are some other luminous insects, which the author has not ‘mentioned. 
 The lantern bearer, or aeudia, which Reaumer places. in the class of the pro-cigales, 
 (cicada spumaria), or a class which approaches near to that of the cigales, (grass- 
 hopper), the vielleur beetle of Surinam; like the author we are unacquainted with 
 any description, sufficiently correct, to enable us to determine in what they differ from 
 the cucuyo, and from each other. Such also is the lantern-bearer of China, described 
 by Linnzus, in the Transactions of the Academy of Stockholm; but as the animal 
 was dead, that learned naturalist had no opportunity of ascertaining what part of it is 
 luminous ; he suspects it is the proboscis, which does not appear improbable. There 
 is also an insect of the same kind in Madagascar, known under the name of hese- 
 cherche, which shines in the night-time. But we have never seen a description of it. 
 
 6th. Clayton’s diamond was long considered to be the only one which had the pro- 
 perty of shining in the dark. But M. Dufay found, bya great number of experi- 
 ments made on different diamonds, that several of them possessed the same property ; 
 though he was not able to discover the cause, why some of them possessed it, while 
 others were destitute of it. Beccari, a celebrated philosopher of Bologna, made at 
 the same time similar experiments, which confirm the discovery of Dufay. This phi- 
 losopher found that the class of phosphoric bodies is much more considerable than — 
 is commonly supposed, and it follows from his experiments, that the phosphoric bodies 
 which have chiefly attracted the attention of philosophers, did so, not on account of 
 that property maintaining itself for a longer time, but because a very great number of 
 bodies appear luminous to an eye immersed in profound obscurity, when they are 
 speedily removed from the light to a place of darkness. 
 
 7th. The sea worms or borers possess this property in an eminent degree; not 
 when they approach to a state of putrefaction, as has been before said, but when liv- 
 ing and fresh, so as to be fit for eating. The observations of Beccari, Monti, and 
 Calcati of Bologna, on these marine fish, are very old; and they confirm and illus- 
 
 trate what has been said by Pliny on the same subject.* 
 
 ARTICLE II. 
 Artificial kinds of Phosphorus. 
 
 What nature produces under certain circumstances, art, assisted by observation, has 
 found means to imitate, in the artificial kinds of phosphorus. But before we explain 
 these curious operations, we must make a distinction which the modern chemists have 
 introduced, and which is necessary. , 
 
 The appellation Phosphorus is still given to those bodies which emit light without 
 any sensible heat ; but when a body emits light, and at the same time inflames of 
 itself, when exposed to the air itis called pyrophorus. Hence we say the pyrophorus 
 of Homberg, to denote that composition of alum and animal or vegetable matter 
 which takes fire when exposed to the air. The English phosphorus is both phospho- 
 rus and pyrophorus ; fdr when exposed to the air in a mass, it burns, and consumes 
 
 * See a Memoir of M. Reaumur on the same subject, ‘‘ Mem. de Acad.” 1723. 
 
812 CHEMISTRY. 
 
 like sulphur, of which it is a singular species, but very much attenuated; and when 
 mixed with a liquor it becomes luminous, without heat. 
 
 SECTION I, 
 Phosphoric experiment : or how to burn gunpowder without an explosion. 
 
 Expose a towel or cloth to a strong heat, till it becomes very hot, and carry it into 
 a dark place. While it is cooling, throw upon it, from time to time, some grains of 
 gunpowder, which at first will inflame. Leave it to cool a little, till the powder no 
 longer detonates. If you then cover it with powder, the latter, when it acquires the 
 same heat as the cloth, will emit in the dark a faint light or weak flame, which will 
 consume all the sulphur, without causing the nitre to detonate. 
 
 It is hence seen, that common sulphur is susceptible of two combustions, one 
 gentle and calm, which is not capable even of kindling the charcoal, otherwise the 
 nitre would detonate ; and the other violent, which burns and kindles such combus- 
 tible bodies as afe in contact with it. 
 
 SECTION II, 
 Of the Bologna Stone. 
 
 This kind of phosphorus is called the Bologna stone, because first made from a 
 stone found only at the bottom of mount Paterno, near that city. A shoemaker, 
 named Vincenzo Casciarolo, was the first who observed the property which these 
 stones have, of shining in the dark, after they have been calcined. He was employ- 
 ed on the grand work, as it was called; and from the brilliant appearance of these 
 stones, he conceived an idea that they contained either metals, or some principle by 
 which he should obtain what he was in quest of. He therefore brought them to a 
 red heatina crucible, and having afterwards carried them into a dark place, he was 
 struck by their luminous appearance, and published an account of his discovery. 
 This phosphorus is made by the modern chemists in the following manner. 
 
 They take one of these stones, and having freed it from all its heterogeneous parts, 
 file it all round with a large file, in order to obtain a certain quantity of dust. They 
 
 _ then dip the stone in the white of an egg, and roll it in the dust, until it be entirely 
 
 covered with it to a certain depth. When the stone is dry, it is placed in a furnace 
 
 1 
 ! 
 4 
 : 
 
 filled with charcoal, in such a manner as to be completely surrounded by it. The © | 
 
 charcoal is then kindled, and when the whole is consumed, the stone is found cal- 
 cined according to the required degree. If it be carried into a dark place, it will be 
 seen to shine with a singular brilliancy, which however becomes gradually weaker, 
 and after some minutes entirely ceases. But this brightness may be renewed by 
 exposing the stone for some time to the day-light. These stones are preserved ina 
 dry place wrapped up in dry cotton. They however gradually lose their property of 
 imbibing the light; but it may be restored to them by a second calcination. 
 
 The Bologna stone, according to the observations made by naturalists, is one of 
 those known under the name of fusible spar. Vitriolic acid enters into their compo- 
 sition, and this inspired Margraf, a celebrated chemist, with the idea of trying whether 
 all the other spars were not endowed with the same property. He found that when 
 treated in a proper manner they all become luminous. The process for calcining 
 and preparing them, according to his method, is as follows. 
 
 When properly freed from their heterogeneous parts, they are brought to a red 
 heat. in a crucible, and then reduced to very fine powder in a glass or porphyry 
 mortar. This powder is formed into small cakes, a line or more in thickness, and 
 of any size at pleasure, by mixing it with gum tragacanth and the white of an egg; 
 and these cakes are then calcined in the following mauner, after they have been 
 dried in a strong heat. 
 
 : 
 
ARTIFICIAL KINDS OF PHOSPHORUS. 813 
 
 _ A common reverberating furnace is filled to three fourths of its height with char- 
 coal; the cakes are laid flat above it, and are covered with more charcoal, The 
 furnace is then kindled, and-when the whole charcoal is consumed, and the furnace has 
 cooled, the cakes are found calcined. After being cleaned from the ashes, by means 
 of a pair of bellows, they are preserved for use, as before described. When an 
 experiment is to be made, they are exposed for some time to the light, after which 
 they are carried into a dark place, where they exhibit the appearance of burning coals 
 to those who have kept their eyes shut for a few minutes. 
 
 The most probable cause of this phenomenon, according to the ablest chemists, is 
 as follows : 
 
 When it is considered that phosphorus of the same kind is made only by burning 
 with charcoal stones which contain vitriolic acid,* there is reason to think that in this 
 operation there is formed a kind of sulphur, exceedingly combustible, and in which the 
 action of the light alone is capable of producing that slow combustion, almost with 
 out heat, of which common sulphur, as already seen, is itself susceptible. This 
 combustion manifests itself only by the faint light it emits. It ceases with the 
 absence of the cause which produced it; and the stone no longer is luminous. 
 
 Among several reasons which confirm this explanation, there is one which seems 
 to be of great weight: after the stone has ceased to shine, if placed in a dark 
 place on a plate of iron, which has been heated, but not to such a degree as to emit 
 light, it immediately becomes luminous, without having been exposed to the action of 
 the sun’s light. Tothis reason we may also add, the odour exhaled by the Bologna 
 stone; for it is exactly the same as that of sulphur. But in regard to this subject 
 we must refer the reader to Macquer’s * Dictionnaire de Chimie,’’ under the head 
 Stony Phosphorus, where explanations will be found, which, on account of their 
 length, cannot be admitted into this work. 
 
 SECTION III. 
 Baldwin’s Phosphorus. 
 
 This phosphorus, as well as the following, has a great affinity to the Bologna stone. 
 The method of making it is as follows: 
 
 Dissolve very pure white chalk in good spirit of nitre; filter the solution, and 
 evaporate the liquor till the residuum is very dry. Then put the residuum into a 
 good crucible of a proper size, and place it for an hour in a reverberating furnace. 
 If the matter calcined in this manner be put into a bottle, with a glass stopper, you 
 will have Baldwin’s. phosphorus. 
 
 This phosphorus has the property of shining in the dark, like the Bologna stone, 
 if the bottle containing it be exposed open to the light. But as it has the fault of 
 attracting moisture, it soon loses this property. 
 
 SECTION IV. 
 Homberg’s Phosphorus. 
 
 Take one part of sal ammoniac in powder, and two parts of quick-lime slaked in 
 the open air: mix them well together; and having filled a crucible with the mix- 
 ture, place it over a slow fire. As soon as the crucible is red, the mixture will 
 begin to fuse; but as it rises up and swells in the crucible, it must be stirred with 
 an iron rod, to prevent it from running over. When the matter is fused, pour it into 
 a copper bason, and when cool it will appear of a grey colour, and as if vitrified. If 
 it be struck with any thing hard, such as a piece of iron, copper, or other substance 
 of the like kind, the whole part which has been struck will for a moment seem on 
 
 ® Margraf at least asserts so, though Dufay says he made Bologna phosphorus with stones purely 
 calcareous. 
 
814 CHEMISTRY. 
 
 fire. Butas this matter is very brittle, the experiment cannot be often repeated. To | 
 
 remedy this defect, M. Homberg immersed in the crucible, containing the fused mat- 
 ter, small rods of iron or copper, which by these means became covered with it as 
 with enamel. ‘The experiment may be performed on rods incrusted in this manner, 
 as they will bear to be struck several times, without the matter being deranged. 
 
 It is to be observed that the phosphoric enamel, which adheres to these rods, 
 readily attracts the moisture of the air: for this reason they must be deposited in 
 a dry warm place, where they will retain their property for along time. 
 
 SECTION V. 
 
 Canton’s Phosphorus, or Phosphorus in Powder. 
 
 Another kind of phosphorus, analogous to that of Baldwin, and to the Bologna 
 stone, may be made in the following manner: 
 
 Calcine oyster shells, by keeping: them in a common fire for half an hour, and then 
 pulverize them; mix the finest part of this powder with one third of its weight of fine 
 flour of Btiahur, and put the mixture in a crucible, filling it to the brim, and keep 
 it for half an hour at least in the midst of burning coals, till it is perfectly red. Then 
 suffer it to cool, and having once more pulverized the matter it contains, if neces- 
 sary, you will obtain a phosphorus, which, if exposed for a few minutes to the day- 
 light, will appear luminous in the dark. 
 
 Those who have comprehended the nature of the Bologna phosphorus, will readily 
 see that Canton’s phosporus is properly the same thing; for the Bologna stone and 
 all the fusible spars, which have been found to possess a phosphoric property, are 
 nothing but combinations of vitriolic acid with calcareous earths. 
 
 SECTION VI. 
 Homberg’s Pyrophorus. 
 The following chemical discovery was entirely owing to chance. The celebrated 
 
 : 
 : 
 : 
 
 a 
 
 Homberg had been assured that a white oil, in no manner fetid, which had the pro- — 
 
 perty of fixing mercury, could be extracted from human excrement. He therefore 
 subjected this matter to experiment ; and extracted from it a white oil without any 
 odour. It did not fix mercury; but having exposed the residuum of his distillation 
 to the air, he was surprised to see it take fire. Such is the origin of his pyro- 
 phorus. . 
 
 It has however been since found, that it is not necessary to employ matters so 
 filthy as those from which Homberg first extracted this pyrophorus. The common 
 process for this purpose is simple, and is as follows: 
 
 Mix in an iron pan, placed over the fire, by means of an iron spatula, three parts 
 of alum and one of sugar, until the matter becomes perfectly dry, and reduced to a 
 blackish carbonaceous substance: if there be any lumps in it of considerable size 
 they must be broken. Put this matter into a matrass with a narrow neck, about 
 eight inches in length, and place the matrass. in a crucible, capable of containing 
 the belly of it when surrounded by half an inch of sand. Then immerse the 
 crucible in burning coals, and bring both it and the matrass to a state of ignition, 
 heating it at first gradually, and then strongly urging the fire, till a sulphurous flame 
 is seen to issue from the neck of the matrass. The fire must be maintained in this 
 state for about a quarter of an hour; then suffer the fire to become gradually ex- 
 tinct, and when the neck of the matrass is no. longer red, close it with a cork 
 stopper, otherwise the pyrophorus would inflame, 
 
 When the whole is perfectly cold, pour the pyrophorus speedily into several phials, © 
 
 which can be well shut, and instantly close them. Sometimes it inflames in passing 
 from the matrass to the bottle; but this is of no consequence, as it is extinguished 
 as soon as the bottle is closed. 
 
_ ARTIFICIAL KINDS OF PHOSPHORUS. 815 
 
 To make an experiment with pyrophorus, put a small quantity of it upon a piece 
 of paper. Soon after, it inflames, becomes red like a burning coal, and sets fire 
 to any combustible bodies it may be in contact with. The inflammation may be 
 accelerated by putting it on paper somewhat moist, and breathing upon it. 
 
 SECTION VII. 
 Of the Phosphorus, or Pyrophorus of Kunckel, called also the English Phosphorus. 
 
 This is the most curious composition of modern chemistry. Who would have 
 believed that a luminous body could be extracted from putrid urine? Nay more, 
 a body susceptible of inflammation, and capable of inflaming, by its contact, other 
 combustible bodies? Such, however, is the origin of this phosphorus, which in some 
 measure may be considered as abject; but to the philosopher nothing is abject in 
 nature; and the most disgusting object sometimes contains principles capable of pro- 
 ducing the most singular and uncommon effects. 
 
 The discovery of the phosphorus of urine, like many others, was the effect 
 of chance. A citizen of Hamburgh, an enthusiast in regard to the philosopher’s 
 stone, was making some experiments with urine. He was not the first nor the only 
 person who imagined that the substance proper for fixing mercury ought to be found 
 in human excrement; and by repeated trials on this matter he found phosphorus. 
 This discovery made a great noise in the chemical world. But Brandt, the author 
 of it, was not disposed to part with his secret for nothing. Kunckel, an able che- 
 mist, united with one Krafft to endeavour to draw from him the process; but Krafft 
 deceived Kunckel, purchased from Brandt the secret of making this phosphorus, and 
 being desirous to carry on a lucrative traffic, refused to impart it to his associate. 
 The latter, incensed on account of this treachery of Krafft, and knowing that he had 
 made great use of human urine, endeavoured by researches to discover the secret, and 
 at length found it. The honour of it therefore has remained with him ; for this 
 phosphorus is commonly called Kunckel’s Phosphorus.* 
 
 On the other hand, Krafft went to England, and having shewn this phosphorus 
 to the king and queen, the celebrated Boyle, whose curiosity was highly excited by so” 
 singular a phenomenon, endeavoured also to discover the secret. He knew only, like 
 Kunckel, that Krafft laboured on urine. He began therefore to make experiments 
 on that matter, and found out likewise the method of extracting phosphorus from it. 
 He communicated the process to the public, in the Philosophical Transactions for 
 1680, and according to every appearance taught the different operations more pare 
 ticularly to a German chemist, settled at London, named Godfreyd Hanckwitz; for 
 he was a long time the only person who made phosphorus. 
 
 Though Boyle published the process for making this phosphorus in 1680; though 
 Homberg taught it in 1692; and though described in various books, phosphorus was ~ 
 made only in England, and by Hanckwitz alone. A foreigner, who came to France 
 in 1737, offered however to disclose the whole process, and the ministry promised 
 him a reward for it. Several chemists and members of the Royal Academy of Sci- 
 ences were requested to be present at the operation, which was performed at the 
 Jardin Royal des Plantes, and attended with perfect success. M. Hellot wrote an 
 account of the process, and published it in 1738, in the Memoirs of the Royal 
 Academy of Sciences. Since that time only the method of making phosphorus has 
 peen known; but it is one of the nicest operations of chemistry, and does not suc- 
 ceed but in very expert hands. 
 
 But none of the modern chemists has paid so much attention to the composition 
 of phosphorus as Margraf, who has rendered the process more certain, more exact 
 
 * Leibnitz asserts that this account, generally given in regard to Brandt, is entirely yoid of foun. 
 dation. He gives a history of phosphorus, which may be seen in his works, vol. ii. 
 
816 CHEMISTRY. 
 
 and less tedious; for which reason we shall take him as our guide, in what we are _ 
 
 going to say on this subject. 
 
 Ist. Provide good urine, and let it purify itself; then put it into a glass vessel 
 placed over the fire, and evaporate the phlegm, till it be reduced to the consistence 
 of honey or of cream. 
 
 It must here be observed, that this matter contains a particular salt, called fussible 
 salt of urine; that this salt is composed of an acid different from all the rest, called 
 the phosphoric, because it is a necessary ingredient in phosphorus, by its combination 
 with another principle, and because this acid is extracted by the deflagration of phos- 
 phorus, as the vitriolic acid is by that of common sulphur. 
 
 2d. Then mix four pounds of minium with two pounds of sal ammoniac in powder, 
 and distil the mixture, which will furnish a volatile alkali highly concentrated. This 
 alkali however is useless. But marine acid will attack the minium or calx of lead, 
 
 and will form with it a compound, known to chemists under the name of corneous — 
 lead. Corneous lead, ready made, may be employed; but we have thought — 
 
 proper to describe the method of making it, because all our readers may not be 
 chemists. 
 
 3d. Mix this corneous lead, by little and little in an iron pan, with eight or nine 
 pounds of the extract of urine, mentioned in the first article, taking care to stirit con- 
 tinually ; add to it halfa pound of charcoal dust, and continue to dry it till it be re- 
 
 duced toa black powder. Then throw the matter into a retort to distil it in a 
 
 moderate heat, and extract from it all the products; which are volatile alkali, a fetid 
 oil, and a kind of sal ammoniac, which adheres to the neck of the vessel. Then bring 
 
 the retort to a moderate red heat, and when nothing more passes over, unlute the — 
 
 apparatus, and reserve the residuum, which is a kind of caput mortuum. This resi- 
 duum contains the phosphorus, and must be distilled in a much more violent heat. 
 
 ——s_ 
 
 It is a sign that it is well prepared ; as a small bit of it, when thrown on the coals, ex. — 
 
 hales an odour of garlic, and burns with-a small lambent flame. 
 
 4th. Put the residuum into a good Hessian retort. MM. Margraf recommends those q 
 
 of Waldenbourg as the best ; but none of them are brought to France. Those of — 
 
 Hesse will answer the purpose, though they suffer the phosphoric matter to transpire 
 a little; but this defect may be in part remedied, by a luting of earth mixed with 
 cow’s hair. 
 
 5th. Having filled three fourths of the retort with the above matter, place it ina 
 furnace, having over it a chimney or tube, five or six inches in diameter, and eight or 
 nine feet in height. ‘The chimney serves to increase the activity of the fire, by the 
 rapidity of the current of air, and for introducing through the door of it, at 
 different times, the quantity of charcoal uecessary to carry on the operation about six 
 hours. 
 
 6th. Lute the neck of the retort with that of a middle sized balloon, half filled — 
 
 with water, and having in it a small hole, by means of fat luting; and secure the 
 joint by a bandage of linen, covered with a luting of lime and the white of an egg. 
 The hole in the balloon affords a passage to the vapours, which otherwise would 
 occasion it to burst. It may be slightly closed with a pellet of paper, which must 
 be occasionally taken out during the distillation. Care must be taken also to close 
 with a luting of clay the notch of the furnace, through which the neck of the retort 
 passes; and to raise between the furnace and the balloon a brick wall, to prevent the 
 heat from being communicated to the former. 
 
 7th. These arrangements having been made twenty-four hours before, set fire to 
 the furnace, and expose the retort to a gradual heat for an hour and a half ; after 
 
 which the fire must be excited, to bring it to a white heat. This operation will — 
 cause to pass into theballoon,first luminous vapours, and then drops of pure phos- | 
 
 phorus, which falling into the water in the balloon will become fixed. This operation | 
 
 eS OO ee ee 
 
 ee ee a 
 
ARTIFICIAL KINDS OF PHOSPHORUS, 817 
 
 will be completed in four or five hours; whereas by Hellot’s process it would require © 
 ‘twenty-four. 
 
 8th. As the phosphorus obtained by this violent distillation is black, on account of 
 the fuliginous vapour it carries along with it, you must distil ita second time ina 
 smaller retort, and in a moderate heat. This heat will be sufficient to render it pure, 
 for when once formed it is exceedingly volatile.¢ 
 
 9th, The phosphorus must then be formed into small sticks, by putting it into glass 
 tubes somewhat conical, and immersed in tepid water ; for in such a heat it runs like 
 tallow. These operations must be performed in water, to prevent inflammation of 
 the phosphorus, and when the water is cold the phosphorus will be fixed in 
 sticks, which must be taken out and put into phials filled with water, and. care- 
 fully shut. 
 
 It has not yet been discovered to what useful purpose the English phosphorus can 
 be applied, except that its nature and composition have thrown light on certain points 
 of chemistry. Butit may be readily conceived, that it may be employed for perform- 
 ing various philosophical tricks, exceedingly curious, such as the following. 
 
 To write tn characters which will appear luminous in the dark. 
 
 __ For this purpose, it will be first necessary to make liquid phosphorus. Place a 
 
 grain of phosphorus at the bottom of a small bottle, and having bruised it, pour imme- 
 diately over it about half an ounce of very clear oil of cloves. Ifthe whole be then 
 put to digest ina gentle heat, such as that of a dunghill, the phosphorus will be 
 almost entirely dissolved. When the bottle is taken out, the matter it contains will 
 shine in the dark, on the bottle being unstopped or agitated a little. ; 
 
 If you dip a small brush in this oil, and describe it with any characters, on a wall, 
 they will appear luminous in the dark. - 
 
 You may also, if you choose, render your face and hands luminous. Nothing will 
 be necessary but to rub over them a little of this oil, which has no sensible heat, 
 because the phosphoric fire is very much rarified. 
 
 This phosphorus amalgamates very well with mercury, and forms a luminous com- 
 pound, Put ten grains of phosphorus into a pretty large long phial, with two ounces 
 of oil of lavender. The phosphorus will dissolve in it, provided it be exposed toa 
 gentle heat. If you then add half a dram of quicksilver, you will obtain an amalgam, 
 which will be entirely luminous in the dark. 
 
 For the same purpose, phosphorus may be mixed with pomatum; it will become 
 luminous, and may’be rubbed over the face and hands without any danger. 
 
 SECTION VIII. 
 
 Composition of a hind of a pyrophorus, which emits flames when brought into contact 
 oer with a drop of water. 
 
 For this composition we are indebted to the celebrated chemist Glauber. Mix 
 together iron filings, cadmia, tartar, and nitre ; then form them into a paste, and dry 
 it well in a strong heat, such as that of a potter’s furnace. Ifa few drops of water 
 be thrown over this mass, it will emit flames and sparks. Such is the description 
 given of this process by Beccher. The following is another, extracted from the 
 Natural Magic of Martius. Pulverize quick-lime, tutty; and storax calamite, each 
 an ounce; live sulphur and camphor, each two ounces; and having mixed them well 
 together and sifted them, wrap them upin a piece of very thick linen cloth. Put 
 this cloth into a crucible, cover it with another crucible, which must be tied closely 
 to the former, and lute the joining with potter’s earth. When the luting is perfectly 
 dry, put this double crucible into a potter’s furnace, and leave it there till thegnatter 
 is entirely calcined. This may be known by the colour of the crucibles, which ought 
 
 3G 
 
* 818 CHEMISTRY. 
 
 to be of a pale red: when the whole is cool, if you throw a drop of water or spirit on™ 
 this matter, it will emit sparks. 
 It was no doubt by means of a similar composition, that a German Jew drew 
 
 sparks from the head of his cane, by spitting on it. This invention indeed is very — 
 
 proper for being employed by jugglers, to excite the wonder of the populace, and 
 extort money from them. The Jew, here alluded to, it seems, turned to great 
 advantage this chemical secret. 
 
 Remark.—There are some other pretended kinds of phosphorus; but properly 
 speaking they are not so: they are merely electric phenomena. 
 
 Of this kind is the light seen in the inside of certain barometers, called for this rea- 
 son luminous. In the old editions of the Mathematical Recreations, it was called Dutal’s 
 Phosphorus, because that physician, but after Bernoulli, was able to make luminous 
 barometers ; it is however now known that this does not arise from phosphorus, but 
 is merely an electric light. M. Ludolff, a German philosopher, has clearly proved 
 that this phenomenon is the effect of electricity, produced in the tube of the baro- 
 meter, by the friction of the mercury. 
 
 The case is nearly the same with mercury, which becomes luminous when inclosed 
 in a very clear glass vessel, exhausted of air. We have already described this phe- 
 nomenon : it is also electric. 
 
 The light emitted by a diamond, rubbed in the dark, or a bit of sugar when grated, 
 is of the same kind. 
 
 SUPPLEMENT I. 
 
 PERPETUAL LAMPS. 
 
 The subject of perpetual lamps has too intimate a connection with that of phos- 
 phorus to be here omitted; for if we were urged to explain the accounts, given of fire 
 found in the tombs of the ancients, and from which some pretend to conclude they 
 were acquainted with the art of maintaining a lamp lighted for ages, it would be 
 necessary to have recourse to phosphorus. But these facts rest on so slight a foun- 
 dation, some of them even bear such evident marks of fiction, and the greater part of 
 those which the honest Fortunio Liceti,.a strong partisan of perpetual lamps, has 
 collected as proof of this discovery, are so evident proofs of the contrary, that a 
 moderate degree of acuteness is sufficient to shew that nothing is less entitled to cre- 
 dit, If to this be added the physical reasons which contradict the possibility of an 
 inflammable liquor, burning continually without being consumed, the perpetual lamps 
 must be considered as a chimera, unworthy the attention of a philosopher, and fit 
 ouly to be banished to the country of potable gold and palingenesy. If we introduce 
 them therefore into this work, it is merely on account of the celebrity of the subject, 
 and because we know that some persons are fond of these singular and extraordinary 
 tales. 
 
 ARTICLE I. 
 
 Examination of the facts alleged as a proof of the existence of perpetual lamps. 
 
 Before the improved state of philosophy had shewn the impossibility of real un- 
 extinguishable fire, the learned were much divided in their opinions on this subject ; 
 
 but of Ml the champions in favour of perpetual lamps, none has made greater efforts _ 
 
 to obtain credit to their existence than Fortunio Liceti, in his book entitled ‘‘ De 
 Reconditis Antiquorum Lucernis.” 
 
 a ee ee 
 
 . ee 
 
 eo 
 
PERPETUAL LAMPS, ; 819 
 
  -If credit may be given to this author, “nothing was more common among the ancients 
 shan perpetual lamps. The lamp of Demosthenes, that which burnt in the temple of 
 Minerva at Athens, the Vestal fire at Rome, all furnish him with so many proofs of 
 the possibility of unextinguishable fire. We cannot help smiling to see so much 
 learning so idly employed: for who does not know that these fires were called per- 
 petual, merely because it was a point of religion to preserve them from being extin- 
 guished, and to supply them with continual aliment ? . 
 The other partisans of perpetual lamps, while they smile at the simplicity of Liceti, 
 support their reasoning on facts, which seem to carry with them alittle more weight ; 
 they are as follow. 
 
 1. The Lamp of Tulliola. 
 
 The tomb of Tulliola, the beloved daughter of Cicero, and whose death cost him 
 
 - so many tears, was discovered, it is said, under the pontificate of Pius III. It is pre- 
 
 tended that in this tomb there was a lamp actually burning, but which became extin- 
 guished on the admission of air. 
 
 2. The Lamp of Olybius. 
 
 But it is the lamp of Olybius, which, above all others, supplies the partisans of 
 perpetual lamps with one of their strongest arguments. 
 
 In the year 1500, as we are told, some peasants, digging the earth to a considerable 
 depth at Atesta, in the neighbourhood of Padua, came to a tomb, in which they 
 found two earthen urns, one within the other. The inner, it is said, contained a 
 burning lamp, placed between two phials, one filled with liquid gold, and the other 
 with liquid silver. On the large urn was the following inscription : 
 
 Plutoni sacrum munus ne attingite, fures ; 
 Ignotum est vobis hoc quod in orbe latet ; 
 Namque elementa gravi clausit digesta labore, 
 
 Vase sub hoc modico, maximus Olybius. 
 Adsit fecundo custos sibi copia cornu, 
 Ne tanti pretium depereat laticis. 
 
 The second is said to have been inscribed also with these lines: 
 
 Abite hinc, pessimi fures ; 
 Vos quid vultis vestris cum oculis emissitiis? 
 Abite hinc vestro cum Mercurio 
 Petasato caduceatoque. 
 Maximus maximum donum Plutoni hoc sacrum fecit. 
 
 Such is the manner in which this curious discovery is related by Gesner. But 
 what follows is still stronger. Liceti gives a letter of one Maturantius, who tells 
 his friend Alphenus, that he had got possession of this valuable treasure. ‘‘ Both 
 the vases,” says he, ‘‘ with the inscriptions, the lamp, and the phials, have fallen into 
 my hands, and are now in my possession. If you saw them you would be astonished. 
 I would not part with them for a thousand crowns of gold.” This is no doubt the 
 language of a man who believes he possesses an inestimable rarity. We do not 
 
 - however know that it exists in any collection. 
 
 It appears that in this case, as in regard to the tomb of Tulliola, an accident prevented 
 enlightened people from being witnesses to the phenomenon; for we read in the 
 credulous Porta, thatas the peasants who found this treasure handled it too roughly | 
 the lamp broke in their hands, and was extinguished. 
 
 3. The Lamp of Pallas, the son of Evander. 
 
 We are told also that, about the year 800 of the Christian era, the tomb of the 
 3 G2 , 
 
f d 
 _ 820 CHEMISTRY, 
 
 famous Pallas the son of Evander, killed as is well known by Turnus, was found at 
 Rome. It was known to be that of Pallas by these verses : 
 
 Filius Evandri Pallas, quem lancea Turni 
 Militis occidit, more suo jacet hic. 
 
 It contained a burning lamp, which consequently must have burnt nearly 2000 
 years, since it was shut up in the year 1170 before the Christian era. 
 
 4, The Lamp in the temple of Venus. 
 
 This lamp, and the temple of Venus, in which it was suspended, are mentioned 
 by St. Augustine. He says it burnt perpetually, and that the flame adhered so 
 strongly to the combustible matter, that neither wind, rain, nor tempests could ex- 
 tinguish it, though continually exposed to the air, and to the inclemency of the sea- 
 sons. This author endeavours to explain the mechanism of it, and after offering a 
 conjecture, which in part is pretty correct, namely, that a wick of asbestos was per- 
 haps employed, he concludes by saying that it might have been the work of demons, 
 in order to blind the pagans more and more, and to attach them to the infamous 
 deity worshipped in this temple. 
 
 Here then, according to the partizans of perpetual lamps, we have unextinguish- 
 able fire, the existence of which is fully confirmed by the testimony of the most 
 enlightened man of his age; and who, notwithstanding his knowledge, is obliged to 
 have recourse to the artifice of demons to explain this phenomenon. 
 
 5. Lamps of Cassiodorus. 
 
 The celebrated Cassiodorus, who, it is well known, was as much respected on 
 account of his employments as of his talents, tells us himself that he made perpetual 
 lamps for his monastery at Viviers. Each monk, itis probable, had one of them for 
 his own use. His words are, “ Paravimus etiam nocturnis vigiliis mecanicas lucer- 
 nas conservatrices illuminantium flammarum, ipsas sibi nutrientes incendium, que 
 humano ministerio cessante prolixé custodiant uberrimi luminis abundantissimam cla- 
 ritatem, ubi olei pinguedo non deficit, quamvis jugiter flammis ardentibus torreatur.” 
 Some partizans of perpetual lamps may here say: “Is it possible to refuse credit to 
 testimony so authentic, so clear, and so respectable ?” 
 
 Such are the principal facts adduced in favour of perpetual lamps: but we may 
 venture to say that they will not stand the test of critical examination. In regard 
 to the first three, what dependance can be placed on facts related in so vague a 
 manner, and accompanied with such incoherent and romantic circumstances? None 
 of these facts are supported by any other testimony than that of men who lived a 
 long time after ; no person whose testimony is of any weight asserts that he actually 
 saw them. But in disputes which are contrary to the common laws of nature, they 
 must at least be certified by enlightened men, above all suspicion of credulity or 
 ignorance. 
 
 The tale respecting the tomb of Tulliola is as old as the year 1345, a period 
 when all Europe was sunk in the grossest ignorance. A body is said to have been 
 found init ; and in that case it could not be the body of Tulliola; for the Romans, — 
 in the time of Cicero, always burnt their dead. In consequence of this and similar 
 circumstances, some authors have conjectured tbat the tomb alluded to was that of 
 the wife of Stilico; but the Christians never placed lamps in their tombs. The 
 account therefore of a lamp found in this tomb has every appearance of a fic- | 
 tion. 
 
 But what shall we say of the tomb of Olybius, and the lamp with two phials, 
 
PERPETUAL LAMPS. 821 
 
 one filled with fluid gold, and the other with fluid silver? This double urn was 
 found by peasants, who according to some authors handled the lamp, contained in 
 the second urn, so clumsily as to break it; and yet Maturantius pretends that he had 
 it in his possession. Who saw the lamp iplirninen? what evidence have we that the 
 peasants beheld it in that state? and whose testimony in this case would be admis- 
 sible? Some vapour, exhaled from a place shut up for so many ages, might easily 
 impose on rude and ignorant people. 
 
 What is the meaning of the inscription? where is there any allusion in it to perpe- 
 tual fire? Is it necessary that a gift sacred to Pluto should be a burning lamp? 
 If there be any truth in the discovery of this tomb, it ought only to be concluded that 
 it belonged to some alchemist, of an age not very remote; for it is well known that 
 the Romans had no idea of chemistry, and none of them ever attempted the transmu- 
 tation of metals. If this folly had then been in existence, some traces of it would 
 
 certainly be found in their writings; but on this subject they all observe the most 
 profound silence. This chimera was communicated to us by the Arabs, with some 
 real knowledge in regard to chemistry. 
 
 But if the Romans were unacquainted with chemistry, how could they construct 
 perpetual lamps, which would be one of the greatest productions of that science? | 
 
 The story of the tomb of Pallas, the son of Evander, is scarcely worth refutation. 
 Who can be so weak as to believe that the verses, eread y quoted, were written in 
 the time of Aineas. One needs only to have seen the language of the twelve tables, to 
 be able to judge how little resemblance the ancient language of the Romans, and con- 
 sequently that of the period of the kings of Alba, bore to these Latin verses. 
 
 In regard to the lamp of Venus, which occasioned so much difficulty to St. Augus- 
 tine, we shall observe that this author does not say that it was never supplied with 
 new aliment. What seems to be most singular is, that it could not be extinguished, 
 either by wind or by rain; but in this there is nothing wonderful, since our oilmen 
 sell flambeaux which have the same property. A method of making similar fire may 
 be found in various books of chemistry. Besides, even admitting that this lamp was 
 perpetual and inextinguishable, who is so ignorant as not to know that the Pagan 
 priests were the greatest impostors, and that they might employ mao artifices to 
 supply the lamp with new aliment? 
 
 The lamps of Cassiodorus may be explained with equal ease; they were lamps — 
 which, like those of Cardan, supplied themselves with oil by means of a reservoir ; 
 and Cassiodorus only meant to say, that these lamps lasted a long time, in comparison 
 of the common lamps of that period, which stood frequently in need of having oil 
 poured into them. 
 
 These reflections did not escape several ingenious writers, such as Aresi, a bishop, 
 
 and author of “Symbola seu Emblemata sacra; Buonamici, a philosopher contem- 
 porary with Liceti; and particularly Octavio Ferrari, to whom we are indebted for a 
 curious and learned work ‘‘ De Veterum Lucernis sepulchralibus.” All these 
 authors, and especially the last, overturn the arguments of Liceti, and fully shew that 
 _ the facts he has adduced, in favour of perpetual lamps; rest on a weak foundation, and 
 that they abound with absurdities and contradictions. They even ridicule the weak - 
 ness of this learned man, who, by an excess of credulity almost beyond belief, finds, 
 in the lamp of the tomb of the necromancer Merlin, described by the poet Ariosto, a 
 proof of perpetual lamps. 
 : We shall conclude this article with the following very just reflections of Octavio 
 ‘Ferrari, before mentioned, which naturally suggest themselves to the mind. If the 
 secret of constructing perpetual and inextinguishable fire had been known to the 
 ancients, would an art so useful have remained buried in oblivion ? but, even admit- 
 ting that it might be lost, for want of philosophical and chemical knowledge, is it 
 possible that Pliny, who enumerates the common inventions, as well as those most 
 
Fa 
 
 i aa 
 
 899 : CHEMISTRY. 
 
 celebrated, should say nothing of this perpetual fire—a thing so wonderful? When 
 Plutarch makes mention of the lamp of Jupiter Ammon, because it burnt a whole — 
 year, is it to be supposed that he would observe silence respecting aneies in compa- — 
 rison of which the former was a contemptible trifle? 
 
 We must therefore say, that both history and sound criticism oppose every idea of 
 such an invention having ever existed. Weshall now examine how far it is consistent 
 with the principles of philosophy. 
 
 — 
 
 ARTICLE Il. 
 On the Physical Possibility of making a Perpetual Lamp. 
 
 Having proved the weakness of all the facts brought as proofs in favour of per- 
 petual lamps, it remains that we should examine how far they are possible, according 
 ~ to the principles of sound philosophy. 
 
 To obtain a perpetual lamp, it would be necessary to have as follows : 
 > st. A wick which could not be consumed. 
 
 2d. Some aliment which could not be consumed, or a substance which, after havi 
 served as aliment to the fire, should return into the vessel, without losing its inflam- — 
 mable quality. q 
 
 3d. It would be necessary also that the flame should be able to exist a long time in 
 a place absolutely close, and of small dimensions; for such were the tombs in which 
 these perpetual lamps are said to have been found. 
 
 But all these things are impossible, as will be seen by what follows. 
 
 Oe eee a 
 
 SECTION I. 
 Impossibility of procuring a perpetual Wick.—History of the Amianthus ; manner of 
 spinning it, and forming it into a wick; examination of its supposed incom- — 
 bustibility. 
 The curious properties ascribed to the amianthus, which are in part real, are well ~ 
 known. We shall here give the history of it; but we shall not be so prolix as the © 
 inexhaustible Abbé Vallemont. The amianthus, called also incombustible flax, and 
 asbestos, is a mineral substance found in several parts of the earth. Jt consists of 
 fibres of a white colour, more or less greyish, which adhere strongly to each other. 
 Means however are found to separate them, and when well washed they have the — 
 appearance of the whitest flax. The amianthus is found in the Pyrenées, the Alps, — 
 &c. The most beautiful, in our opinion, is that found in or near the mine of Pesey in 
 Savoy. We have seen some, the filaments of which were above a foot in length, and — 
 exceedingly white. 
 But the singular property which characterizes this substance is, that it remains — 
 unhurt in the fire, and when taken out is purer and whiter than it was before. This 
 property therefore has been made the basis of a thousand moral and pious compari-— 
 sons, which we shall not here repeat. ; 
 It is proper to observe, that the druggists, a sort of men who throw all natural 
 history into confusion by their corrupted nomenclature, are acquainted with asbestos — 
 under no other name than that of feathered alum. But this denomination arises — 
 from profound ignorance. Alum isa salt, and is soluble in water, whereas the ami- t 
 anthus is insoluble in that liquid. The amianthus therefore is not alum. "What has 
 given occasion to this false denomination is, that there is indeed a feathered alum, or 
 alum crystallized in fibres, which has some resemblance to the amianthus ; but this” 
 alum is exceedingly rare, and the druggists, when it is called for, substitute the ami- — 
 anthus in its stead. 
 . However, it appears that the property of the amianthus has been long known; for 
 we are told by Pliny, in his Natural History, B. xix. chap. 1, that when certain 
 indian princes died, their bodies were wrapped up in a winding sheet made of live © 
 
 : 
 . 
 : 
 
PERPETUAL LAMPS, Oo aoe 
 
 flax, and that they were thus burnt, in order that: their ashes might not be mixed 
 with those of the funeral pile. It is certain that this is not impossible, and we 
 cannot doubt the testimony of Pliny, who says in another place, that he saw cloth 
 and nets, which when dirty required only to be thrown into the fire, and that when 
 taken out they were clean, and not in the least injured. Plutarch asserts the same 
 thing. But Pliny is evidently mistaken, when he says that this live flax was pro- 
 duced from a plant, found only in the hottest districts of India, torrified by the rays 
 of the sun, as if it were fond of growing in the midst of flames. He~knew the 
 production, but was deceived in regard to its origin. This custom however seems to 
 have become extinct in India: we know no traveller who speaks of bodies being 
 now burnt in that manner. 
 
 It is needless to quote more authorities to prove tne possibility of making incom- 
 bustible cloth of this kind. We have seen purses, brought from the Pyrenées, which 
 possessed this property. They were indeed exceedingly coarse, but it is certain that 
 they might be made much finer. 
 
 It however requires great industry to be able to spin the amianthus, and to form it 
 into cloth. The following is the method, as given by Ciampini, in his treatise ‘‘ De 
 Incombustibili Lino, deque illius filandi modo ;” Rome, 1691. 
 
 To spin this stone, says Ciampini, it must be first soaked in warm water; when it 
 has remained in the water some time, it is taken out, rubbed between the hands, 
 opened and spread out, frequently dipping it in the water, in order to clean it from 
 the earthy particles. This operation is repeated five or six times, until the filaments 
 are well detached from each other, after which they are collected. 
 
 They must then be dried on some apparatus, through which the water can easily 
 drain off. The next thing is to provide two small cards, finer than those used to 
 card the wool employed for making hats and stuffs, and the incombustible flax must 
 be placed between these two cards, that a few of the filaments may be drawn out at 
 a time, in order to be spun with a small spindle. 
 
 But it is to be observed, that as the filaments of this flax are in general very short, 
 it is necessary to spin along with it some fine cotton or wool, which may embrace 
 and unite them. ~ 
 
 Care however must be taken to use always a little more amianthus than of the 
 substance you have chosen to spin along with it. The reason of this is, that when 
 the thread is made into cloth or into purses, the work is thrown into the fire; the 
 cotton or wool then burns, and being consumed, nothing remains but pure amianthus. 
 It is almost in the same manner that gold and silver are spun with silk; and that old 
 gold and silver lace is burnt to obtain the pure metal. 
 
 Ciampini says, that those who spin this substance must moisten their fingers, and 
 particularly the thumb and the fore finger, to render the operation easier, and to 
 prevent the fingers from being excoriated, because the amianthus is corrosive. He 
 says also that the use of cards may be dispensed with; and that it is sufficient to 
 put the filaments of the amianthus in regular order, in such a manner that they may 
 easily separate to insinuate themselves into the cotton or wool added, in order that 
 they may be spun together. When the cloth or purses are dirty, they are thrown 
 into the fire, and on being taken out are whiter and more brilliant than they were 
 before. He recommends moistening them with a little oil or,essence, when they 
 come from the fire; because oil nourishes the amianthus, and causes the thread to 
 remain smoother. 
 
 We shall here observe, that to form the amianthus into wicks, it is not necessary 
 that it should be purified or spun. It will be sufficient to take the longest filaments, 
 and to tie them together with a white silk thread, in a quantity proportioned to the 
 size of the wick. It is astonishing to see with what avidity the amianthus attracts 
 
« 
 
 y 
 
 824 ; CHEMISTRY. 
 
 and imbibes the vil. It may be employed as it is found in filaments, in the druggists’ 
 shops, and the lamp will not fail to burn and to emit a strong light. 
 
 Ciampini however is mistaken when he ascribes to the amianthus a corrosive 
 quality ; its stony and no ways saline nature will not admit of such a property 
 
 Having given this short history of the amianthus, it remains that we should 
 examine the consequences that may be hence deduced. 
 
 If we can believe the partisans of perpetual lamps, since the first step towards the 
 execution of such a work as a perpetual and incombustible wick, we have the object 
 accomplished ; for the amianthus supplies us with such a wick, since it is incombus- 
 tible, and since the trials made of it have been attended with success. Father Kir- 
 cher assures us, that he had a lamp with a wick of this kind, which answered ex- 
 ceedingly well. 
 
 We will not deny that it is possible, by means of the amianthus, to make a wick, 
 which will last for a very long time; but we will assert that it would not be per- 
 petual ; for though the amianthus is boasted of as being incombustible, this property 
 is not absolute. We will even venture to:say, that the amianthus is at length anni- 
 hilated by fire, like every other body. It is true that cloth of the amianthus, when 
 thrown into the fire, is taken out sound and entire, but not absolutely so. It is ob- 
 served that it loses a little of its weight every time it is exposed to the fire. - It 
 would therefore be at length destroyed, and perhaps in the course of a very short 
 time, such as a few days, if it were only made red hot and suffered to coo}, or if it 
 were left all that time in a very strong fire. Consequently a wick of amianthus 
 would, at the end of a certain period, he entirely destroyed. 
 
 Some have tried to make wicks of bundles of gold threads, exceedingly fine. 
 This perhaps would be the means of obtaining a wick almost perpetual in its duration ; 
 but it has never been possible to light them; and even if this could be done, another 
 inconvenience would prevent their being ofany use: the gold filaments would be fused 
 by the flame, and consequently would be rendered unfit for answering the intended pur- 
 pose ; as it is well known, that if a piece of silver wire be presented to the flame of a 
 taper, it is instantly fused; and the case would be the same with a gold wire, for it is 
 more fusible than silver. 
 
 SECTION II. 
 Impossibility of procuring indestructible aliment for the perpetual lamps. —Pretended 
 recipes for making indestructible oil. 
 
 But we shall even suppose a wick absolutely unalterable to have been found, and 
 that it does not become choaked up with fuliginous matter, from the combustible sub- 
 stance by which it is fed. This however would be only a small part of what is 
 necessary for obtaining a perpetual lamp. Some kind of aliment, which shall expe- 
 rience no diminution, or which having served to maintain the flame without expe- 
 riencing any alteration, may return by a perpetual circulation into the vessel 
 from which it issued will also be requisite. Is all this possible ? 
 
 But let us now hear the alchemists, or the partisans of perpetual lamps. We shall 
 be entertained with their ideas respecting the manner in which an oil, such as these 
 lamps would require, might be obtained. 
 
 Some, considering that the amianthus resists fire, have tried, but without success, 
 to extract an oil from it. 
 
 Others, observing that gold and silver, but particularly the former of these metals, 
 are indestructible by fire, conceived an idea of searching in them for that valuable 
 oil which would put us in possession of perpetual lamps. This is the noble secret 
 with which Liceti pretends that Olybius was acquainted; but the metals are as inca- 
 pable of producing oil as the amianthus, . 
 
 It may however be said, that if it were possible to reduce gold to a liquid state, 
 
 \ 
 
 _ —— ee eS ee ee 
 
PERPETUAL LAMPS. 825. 
 
 we might perhaps obtain an mcombustible oil, as gold is unalterable in the fire. But 
 besides the impossibility of converting gold into a liquid, how do we know that it 
 would be inflammable like oil. . 
 
 The Abbot Trithemius, or the person who in his name has written a great many 
 falsehoods, pretends to give two recipes for making incombustible oil. We shall 
 here lay one of them before our readers, with the whole process for making a perpe- 
 tual lamp. , 
 
 Mix, says that celebrated visionary, or the person who speaks in his name, 
 four ounces of sulphur and four ounces of alum; sublime them, and convert 
 them into flowers. Take two ounces and a half of these flowers with half an 
 ounce of borax and Venetian crystal, and pulverise the whole in a glass mor- 
 tar. Put the powder into a phial, and having poured into it spirit of wine, four 
 times rectified, cause it to digest. Pour off the spirit of wine, and having 
 added some new, repeat the same thing three or four times, until the sulphur runs 
 without smoke, like wax, on hot plates of brass. You must then. prepare a proper 
 wick, which may be done in the following manner: Take filaments of the asbestos 
 stone, of the length of the finger; form them into a packet half as thick as the finger, 
 and tie them with a white silk thread. The wick being thus made, cover it with 
 the sulphur prepared as before described, and immerse it in the sulphur, in a vessel of. 
 Venetian glass. Place the whole upon a sand bath for twenty-four hours, so that 
 you may always see the sulphur boil; and the wick being by these means well pene- 
 trated and impregnated with that aliment, put it into a small glass vessel with a 
 wide mouth. The match must rise a little above it. Then fill this glass vessel 
 with the prepared sulphur, and place the vessel in warm sand, that the sulphur may 
 melt and surround the wick. If it be then kindled, it will burn with a perpetual 
 flame. 
 
 Such is the first kind of fire of the Abbot Trithemius. A very small knowledge 
 of chemistry is sufficient to shew that it would be ridiculous to expect from this» 
 process inextinguishable and perpetual fire. None therefore of the partisans of per- 
 petual lamps, not even Liceti himself have any confidence in it, nor yet in the second ; 
 from which Liceti concludes that none of the moderns possess, or ever possessed, this 
 valuable secret. 
 
 Some alchemists assert that an incombustible oil may be obtained by another pro- 
 cess. They pretend that oil of vitriol edulcorated on gold, which they call oleum 
 vitrioli aurificatum, will give this valuable liquor. But it is well known that the appel- 
 lation of oil is here improperly applied ; for what is called oil of vitriol has in reality 
 nothing of an oily or inflammable nature; and we shall believe in perpetual lamps 
 when an alchemist shall shew us a common lamp, filled with oil of vitriol, and furnished 
 with any wick whatever, in which the flame shall exist only one second. . 
 
 SECTION III. 
 Impossibility of continually maintaining fire in a. place absolutely close. 
 
 It is a fact, well known to all those who cultivate natural philosophy, that flame 
 cannot be maintained in a close place. If alighted taper be placed under a glass re- 
 ceiver, so as to prevent its having communication with the external air, the flame 
 will gradually decrease in size, become faint, extend itself in length, and at last be 
 extinguished. Dr. Hales and others have even calculated what quantity of air is 
 consumed in acertain time by the combustion of a taper of certain dimensions ; so that 
 we may easily predict in what time the flame will infallibly be extinguished. 
 
 A flame however might perhaps be maintained in a large place, though hermetically 
 sealed ; but it is well known that the cavities of the ancient tombs were exceedingly 
 small; and to increase the difficulty it is said that the perpetual lamps burnt in the 
 vessels in which they were inclosed. Such at least was the case with that of Oly- 
 
 ° 
 
’ 
 
 826 . CHEMISTRY. ~ a 
 bius; but if the vessel of Olybius had been three feet in Sianeter: which by no means 
 appears, it is certain that a lamp could not have existed-im it two atc without » 
 being extinguished ‘. . 
 
 -We shall not enlarge farther on this mabicae as it would be wasting time to accu- 
 mulate arguments to combat the chimera of perpetual lamps; agid we have reason 
 to think that every enlightened philosopher, at present, will pe of the same opinion. 
 
 Remark.—NotwithstatWding ‘the reasons which are conte deduced from the 
 principles of sound philosophy, we have seen, in some journal, that. a Neapolitan 
 prince was in possession of the secret of perpetual lamps.. But as several years have 
 elapsed since this circumstance was announced, and as the secret has not yet been 
 divulged there is reason to think that the information was premature. It isno new 
 thing to see chemists, employed in researches respecting the philosopher’s stone, 
 announce their discovery before the operation is finished. Some even in consequence 
 of the goéd colour -of their matter, like that described by Philalethes and the learned 
 Morien,* have gone so far as to purchase estates for a large sum of money. 
 
 But, unfortunately, every thing is still deficient, and the good alchemist dies in the 
 hospital, protesting that nothing was wanting to his matter, but an “imperceptible 
 degree of coction, to render him the richest man on the earth. 
 
 In regard to the perpetual lamp of Naples, we shall change our opinion when we 
 learn with certainty that it has been tried, and that it has burnt only one year. 
 
 * Two celebrated adepts. 
 
 FINIS. 
 
 Printed by Nuttall and Hodgson, Gough Square, London, 
 
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